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[Jean-Baptiste Morée](https://orcid.org/0000-0002-0710-9880), [Youhei Yamaji](https://orcid.org/0000-0002-4055-8792), [Masatoshi Imada](https://orcid.org/0000-0002-5511-2056)

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[Dome structure in pressure dependence of superconducting transition temperature for <math>  <mrow>    <msub>      <mi>HgBa</mi>      <mn>2</mn>    </msub>    <msub>      <mi>Ca</mi>      <mn>2</mn>    </msub>    <msub>      <mi>Cu</mi>      <mn>3</mn>    </msub>    <msub>      <mi>O</mi>      <mn>8</mn>    </msub>  </mrow></math>: Studies by <i>ab initio</i> low-energy effective Hamiltonian](https://mdr.nims.go.jp/datasets/0a883283-5c20-45cf-b288-9b9ff308ea2c)

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Dome structure in pressure dependence of superconducting transition temperature for ${\rm HgBa}_2{\rm Ca}_2{\rm Cu}_3{\rm O}_8$: Studies by ab initio low-energy effective HamiltonianPHYSICAL REVIEW RESEARCH 6, 023163 (2024)Dome structure in pressure dependence of superconducting transition temperaturefor HgBa2Ca2Cu3O8: Studies by ab initio low-energy effective HamiltonianJean-Baptiste Morée ,1,2 Youhei Yamaji ,3 and Masatoshi Imada 1,41Research Institute for Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555, Japan2RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan3Research Center for Materials Nanoarchitectonics (MANA) and Center for Green Research on Energy and Environmental Materials(GREEN), National Institute for Materials Science (NIMS), Namiki, Tsukuba-shi, Ibaraki 305-0044, Japan4Physics Division, Sophia University, Kioi-cho, Chiyoda-ku, Tokyo 102-8554, Japan(Received 26 December 2023; revised 12 April 2024; accepted 23 April 2024; published 13 May 2024)The superconducting (SC) cuprate HgBa2Ca2Cu3O8 (Hg1223) has the highest SC transition temperature Tcamong cuprates at ambient pressure Pamb, namely, T optc � 138 K experimentally at the optimal hole dopingconcentration. T optc further increases under pressure P and reaches 164 K at optimal pressure Popt � 30 GPa,then T optc decreases with increasing P > Popt generating a dome structure [Gao et al., Phys. Rev. B 50,4260(R) (1994)]. This nontrivial and nonmonotonic P dependence of T optc calls for a theoretical understandingand mechanism. To answer this open question, we consider the ab initio low-energy effective Hamiltonian(LEH) for the antibonding (AB) Cu3dx2−y2/O2pσ band derived generally for the cuprates. In the AB LEH forcuprates with N� � 2 laminated CuO2 planes between block layers, it was proposed that T optc is determinedby a universal scaling T optc � 0.16|t1|FSC [Schmid et al., Phys. Rev. X 13, 041036 (2023)], where t1 is thenearest-neighbor hopping, and the SC order parameter at optimal hole doping FSC mainly depends on theratio u = U/|t1| where U is the onsite effective Coulomb repulsion: The u dependence of FSC has a peak atuopt � 8.5 and a steep decrease with decreasing u in the region u < uopt irrespective of materials dependenton other ab initio parameters. In this paper, we show that |t1| increases with P, whereas u decreases with Pin the ab initio Hamiltonian of Hg1223. Based on these facts, we show that the domelike P dependence ofT optc can emerge at least qualitatively if we assume Hg1223 with N� = 3 follows the same universal scalingfor T optc , and Hg1223 is located at the slightly strong coupling region u � uopt at Pamb and u � uopt at Popt bytaking account of expected corrections to our ab initio calculation. The consequence of these assumptionsis the following: With increasing P within the range P < Popt, the increase in T optc is accounted for bythe increase in |t1|, whereas FSC is insensitive to the decrease in u around � uopt and hence to P as well. AtP > Popt, the decrease in T optc is accounted for by the decrease in u below uopt, which causes a rapid decrease inFSC dominating over the increase in |t1|. We further argue the appropriateness of these assumptions based on theinsight from studies on other cuprate compounds in the literature. In addition, we discuss the dependencies of uand |t1| on each crystal parameter (CP), which provides hints for design of even higher T optc materials.DOI: 10.1103/PhysRevResearch.6.023163I. INTRODUCTIONUnconventional SC occurs in cuprates [1] with a diversedistribution of T optc . At Pamb, known values of T optc range fromT optc � 6 K in Bi2Sr2CuO6 (Bi2201) [2] to T optc � 138 K inHgBa2Ca2Cu3O8 (Hg1223) [3,4]. T optc further increases underpressure. In the case of Hg1223 and other Hg-based cuprates,T optc has a domelike structure as a function of P [3,5]. Anexample is shown in Fig. 1(b) for Hg1223: T optc increases withpressure and shows the maximum 164 K at Popt � 30 GPa[3,6], which is the highest known value of T optc in the cuprates.A wide range of T optc � 6–164 K in the cuprates hasinspired studies on chemical substitution and pressurePublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.application to gain insights into the microscopic mechanismof the diversity in T optc . For example, for Y-based [8–16] andHg-based [17,18] high-Tc cuprates, the uniaxial pressures Paand Pc were applied. (In this paper, Pa refers to the simul-taneous compression along axes a and b in Fig. 2, whilekeeping |a| = |b|, and Pc refers to the compression along axisc. The axes are represented in Fig. 2 for the tetragonal cellin Hg1223.) This decomposition of pressure revealed, in thecase of HgBa2CuO4 (Hg1201, T optc � 94 K [19]), that T optcdecreases with out-of-CuO2 plane contraction caused by Pc(∂T optc /∂Pc � −3 K/GPa) but increases with in-plane con-traction caused by Pa (∂T optc /∂Pa � 5 K/GPa) [17].However, microscopic mechanisms leading to the P depen-dence of T optc are not well understood, while understanding themechanism of them certainly helps future materials design.Since it is difficult to isolate these hidden mechanisms byexperiments only, further theoretical studies of cuprates underP are desirable.2643-1564/2024/6(2)/023163(22) 023163-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-0710-9880https://orcid.org/0000-0002-4055-8792https://orcid.org/0000-0002-5511-2056https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.6.023163&domain=pdf&date_stamp=2024-05-13https://doi.org/10.1103/PhysRevB.50.4260https://doi.org/10.1103/PhysRevX.13.041036https://doi.org/10.1103/PhysRevResearch.6.023163https://creativecommons.org/licenses/by/4.0/MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)FIG. 1. Summary of the main results and theoretical predictionof the dome structure in the P dependence of Tc. (a) P dependenceof |t1| and u deduced at the most sophisticated cGW -SIC+LRFBlevel. These values are estimated from the calculated results at thecGW -SIC level by supplementing the correction from the cGW -SICto the cGW -SIC+LRFB levels defined in Eqs. (2) and (3). Thepressure P is measured from Pamb. P dependence of FSC estimatedat the cGW -SIC+LRFB level denoted as F estSC is also plotted, whereF estSC is deduced from the universal u dependence found in Ref. [7] andusing u for Hg1223 shown here. See below Eqs. (2) and (3) for the de-tailed corrections of cGW -SIC and cGW -SIC+LRFB levels, wherethe notations for the quantities improved in such ways are denotedas ucGW −SIC+“LRFB” (ucGW −SIC) and |t1|cGW −SIC+“LRFB” (|t1|cGW −SIC) in-stead of u and |t1|, respectively, to indicate the cGW -SIC+LRFB(cGW -SIC) levels explicitly. Diamond symbols show results at P =Pamb, 30 GPa and 60 GPa, and dashed lines show linear interpolationsbetween diamonds. (b) Experimental T optc [3] (black curve) and thepresent theoretical optimum Tc denoted as T estc deduced from Eq. (1)proposed in Ref. [7] by replacing FSC with F estSC . Shaded areas in(a) and (b) indicate the uncertainty described below Eqs. (2) and(3). Qualitative dome structure of T optc is reproduced in the presentprediction, T estc .In this paper, we propose a microscopic mechanism for theP dependence of T optc for the carrier-doped Hg1223 based onan ab initio study. For ab initio studies, the density functionaltheory (DFT) has been widely applied [20,21]. However,its insufficiency in strongly correlated electron systems isalso well known. Instead, we apply the multiscale ab initioscheme for correlated electrons (MACE) [22–28], which hassucceeded in correctly reproducing the SC properties of thecuprates [7,28–30] at ambient pressure and has motivatedfurther studies on hypothetical Ag-based compounds [31].MACE consists of a three-step procedure that determinesthe LEH parameters for the single-band AB Hamiltonian; thisprocedure has several different accuracy levels, which aredefined below and whose details are given in Appendix A.At the earliest stage of the MACE, the simplest level denotedas LDA+cRPA [22,23] or GGA+cRPA was employed; atthis level, we start from the electronic structure at the localdensity approximation (LDA) or generalized gradient approx-imation (GGA) level, and the effective interaction parametersare calculated on the level of the constrained random phaseapproximation (cRPA) [22]. The next level is denoted ascGW -SIC [27], in which the starting electronic structure ispreprocessed from the LDA or GGA level to the one-shot GWlevel, and the one-particle part is improved using the con-strained GW (cGW ) [25] and the self-interaction correction(SIC) [26]. The most recent and accurate level is denoted ascGW -SIC+LRFB [28], which is essentially the same as thecGW -SIC, except that the GW electronic structure is furtherimproved: The level renormalization feedback (LRFB) [28] isused to correct the onsite Cu3dx2−y2 and O2pσ energy levels.Although the cGW -SIC+LRFB level is the most accurateand was used to reproduce the SC properties of the cuprates[7,28–30], we mainly employ the simplest GGA+cRPAversion for the purpose of the present paper, because thequalitative trend of the parameters can be captured by thissimplest framework. (See Appendix A for a more detaileddiscussion.) We also reinforce the analysis by deducing morerefined cGW -SIC+LRFB level in a limited case from theexplicit cGW-SIC level calculations to remove the knowndrawback of GGA+cRPA as we detail later.We derive and analyze the pressure dependence of ABLEH parameters including various intersite hoppings and in-teractions; however, we restrict the main discussion to |t1|and u, since they are the principal parameters that controlTc in the proposal [7]. Other LEH parameters are given inthe Supplemental Material [32]. In the following, we mainlydiscuss |t avg1 | and uavg, which are the ab initio values of |t1|and u at GGA+cRPA level, averaged over the inner and outerCuO2 planes. (See Fig. 2 for a representation of the CuO2planes.)This paper is organized as follows. In Sec. II, the centralresults of the present paper are outlined to capture the essenceof the results before detailed presentation. In Sec. III, we givethe crystal structure of Hg1223, the hole concentration, anda reminder of the GGA+cRPA scheme. In Sec. IV, we givethe DFT electronic structure at the GGA level as a function ofP. In Sec. V, we show the pressure dependence of AB LEHparameters at the GGA+cRPA level. In Sec. VI, we discussthe adequacy of the assumptions made in Sec. II. We alsodiscuss the consistency of our results with the experimental Pdependence of T optc in Fig. 1. Our summary and conclusion aregiven in Sec. VII. In Appendix A, methodological details ofthe MACE scheme are summarized. In Appendix B, computa-tional details used in this paper are described. In AppendixesC and D, we detail the corrections used in Secs. II and VI.In Appendix E, we discuss in detail the P dependence at theintermediate stage of the present procedure. In Appendix F,we detail the dependence of AB LEH parameters on crystalparameters (CPs) around optimal pressure.II. OVERVIEWThe main results obtained in this paper are summarized as(I) and (II) below.(I) |t avg1 | increases with P. This increase in |t1| is causedspecifically by the uniaxial pressure Pa, in agreement withprevious experimental studies on, e.g., Hg1201 [17].(II) uavg decreases with P. The decrease in u is causedmainly by (I), namely, by the increase in |t1|, but is slowedby the increase in U at P < Popt. The increase in U is alsocaused by Pa.The nontrivial pressure dependence of T optc can be under-stood from (I) and (II), which is derived from our ab initio023163-2DOME STRUCTURE IN PRESSURE DEPENDENCE OF … PHYSICAL REVIEW RESEARCH 6, 023163 (2024)FIG. 2. Left panel: Crystal structure of Hg1223. We show the block layer, the inner CuO2 plane (IP), the outer CuO2 plane (OP), and theinterstitial Ca atoms. The thick gray lines represent the primitive lattice vectors a, b, c. The cell parameters are a = |a| = |b| and c = |c|; otherCPs are defined in Table I. Middle and right panels: Pressure dependence of the CP values in Å. We show the optimized CP values (squares)and the extrapolated CP values from Zhang et al. [33] (solid lines); for details, see Appendix B 1. The open squares show the modifications ofthe optimized CP values at Popt = 30 GPa that are considered in Appendix F. For comparison, we also show the experimental CP values fromArmstrong et al. [34] (open crosses) and Hunter et al. [35] (open circles), and the values of a and c from Eggert et al. [36] (dots).Hamiltonian even at the preliminary level GGA+cRPA, if weassume the following (A) and (B). [The reality of (A) and (B)will be discussed later in Sec. VI.](A) The universal scaling for T optc given theoretically asT estc � 0.16|t1|FSC (1)recently proposed for the cuprates with N� = 1, 2, and ∞[7,30] is also valid for Hg1223 with N� = 3.(B) FSC follows a universal u dependence revealed inRef. [7], where FSC has a peak at u = uopt � 8.0–8.5. In ad-dition, at Pamb, Hg1223 is located at slightly strong couplingside u � uopt, while the highest pressure P = 60 GPa appliedso far is in the weak coupling side u < uopt. In fact, we justifylater u � uopt at optimal pressure Popt � 30 GPa for Hg1223.To understand the consequences of the assumptions (A)and (B) appropriately and to complement the consequencesquantitatively, we correct the errors anticipated in our ab initioGGA+cRPA calculation using the following (C) and (D).[Details of (C) and (D) are given in Appendixes C and D,respectively.](C) We correct the values of uavg and |t avg1 | obtained atthe GGA+cRPA level by deducing the most sophisticatedcGW -SIC+LRFB level. Since GGA+cRPA is known to un-derestimate u in Bi2201 and Bi2Sr2CaCu2O8 (Bi2212), itis desirable to improve the AB LEH to the more accuratecGW -SIC+LRFB level. However, the explicit calculation atthe cGW -SIC+LRFB level is computationally demanding,while the corrections from the explicitly calculated cGW -SICto the cGW -SIC+LRFB levels are known to be small and arerelatively materials insensitive. Thus we represent the correc-tion by a universal constant with admitted uncertainty. Theestimates of u and |t1| improved in such ways are denoted asucGW −SIC+“LRFB” and |t1|cGW −SIC+“LRFB” The procedure con-sists in the two steps (C1) and (C2):TABLE I. Irreducible Cartesian atomic coordinates (x, y, z) within the unit cell given by the (a, b, c) frame in Fig. 2. The atom index l iseither Cu(i) (Cu atom in the IP), O(i) (O atom in the IP), Cu(o) (Cu atom in the OP), O(o) (O atom in the OP), O(ap) (apical O atom), Ca, Ba,or Hg. The coordinates of other atoms in the unit cell may be deduced by applying the transformations (y, x, z) to O(i) and O(o) and (x, y, −z)to Cu(o), O(o), Ca, Ba, and O(ap). The atomic coordinates are entirely determined by the seven CPs a, c, dzCa, dzCu, dzbuck, dzBa, and dzO(ap). Notethat dzbuck [the displacement of O(o) due to the Cu(o)-O(o)-Cu(o) bond buckling] may be either positive or negative. The CP values are listedin Fig. 2 as a function of P.Atom index l Cu(i) O(i) Ca Cu(o) O(o) Ba O(ap) Hgx 0 a/2 a/2 0 a/2 a/2 0 0y 0 0 a/2 0 0 a/2 0 0z 0 0 dzCa dzCu dzCu − dzbuck dzCu + dzBa dzCu + dzO(ap) c/2023163-3MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)(C1) cGW -SIC calculation: Starting from the whole anddetailed pressure dependence of uavg and |t avg1 | for Hg1223calculated at the GGA+cRPA level, we calculate explicitlythe level of the cGW -SIC denoted as ucGW −SIC and |t1|cGW −SICin limited cases of pressure choices of Hg1223 to reduce thecomputational cost.(C2) Estimate at the cGW -SIC+LRFB level: We useucGW −SIC+“LRFB” = xLRFBucGW −SIC (2)and|t1|cGW −SIC+“LRFB” = yLRFB|t1|cGW −SIC (3)and estimate constants xLRFB and yLRFB in Hg1223 fromthe already explicitly calculated results for other compoundsHg1201, CaCuO2, Bi2201, and Bi2212. The estimated valuesare xLRFB = 0.95 (with the range of uncertainty 0.91–0.97)and yLRFB = 1.0. See Appendix C for detailed procedure toestimate xLRFB and yLRFB for the case of Hg1223. The concreteeffect of (C) for Hg1223 is to increase u from the cRPAlevel by the ratio ucGW −SIC+“LRFB”/uavg � 1.29 at Pamb, � 1.15at 30 GPa, and � 1.08 at 60 GPa; also, the � 13%–14%increase in |t avg1 | from Pamb to 30 GPa becomes � 17% by thiscorrection.(D) After applying (C), we further correct the value of|t1|cGW −SIC+“LRFB” by considering the plausible error in crystalparameters at high pressure. Structural optimization by abinitio calculation is known to show quantitative error and itis preferable to correct it if experimental value is known. Wecompare our structural optimization and the experimental cellparameter a if it is available (this is the case at P < 8.5 GPa)and assume that this trend of the deviation continues forP > 8.5 GPa, where experimental data are missing. Namely,at P > 8.5 GPa, we assume that our calculation overestimatesthe experimental a by � 0.05 Å, and we correct a by �a =−0.05 Å accordingly. The concrete effect of (D) is that theincrease in |t1|cGW −SIC+“LRFB” from Pamb to 30 GPa is now� 22%.The final estimates of ucGW −SIC+“LRFB” and|t1|cGW −SIC+“LRFB” are shown in Fig. 1(a). Since (C1) iscomputationally demanding, we perform (C) and (D) only atPamb, 30 GPa and 60 GPa, and infer the correction at otherpressures by linear interpolation for the pressure dependence.Even by considering only (A) and (B) above, the presentmechanism qualitatively accounts for the microscopic trendof the dome structure: At P < Popt, (I), namely, the increasein |t1|, plays the role to increase Tc, whereas the decrease inu does not appreciably affect FSC and thus Tc, because FSCpasses through the broad peak region in the u dependence. AtP > Popt, (II), namely the decrease in u, drives the decrease inFSC and thus Tc surpassing the increase in |t1|, which generatesa dome structure. If we take into account (C) and (D) in ad-dition to (A) and (B), the dome structure in the P dependenceof experimental T optc is more quantitatively reproduced (seeFig. 1). In addition to the above results, we discuss the depen-dence of AB LEH parameters on each CP, which provides uswith hints for future designing of even higher T optc materials.TABLE II. Definitions of the uniform pressure P and uniaxialpressures Pa, Pc, Pbucka , and Pbuckc considered in this paper. Each CP ismarked with a check mark if its value is modified by the applicationof the pressure, and with a cross if not. If the CP value is modified,the value is that in the P dependence in Fig. 2. If not, the value is thatat Pamb in Fig. 2.P Pa Pbucka Pc Pbuckca � � � × ×dzbuck � × � � ×c, dzCa, dzCu, dzBa, dzO(ap) � × × � �III. FRAMEWORK OF METHODWe start from the crystal structure of Hg1223 and thepressure dependence of the CP values in Fig. 2. We abbreviatethe inner and outer CuO2 planes shown in Fig. 2 as IP andOP, respectively. The crystal structure is entirely determinedby the seven CPs defined in Table I, which consist of the twocell parameters a and c and the five characteristic distances dzl .The CP values considered in this paper are listed in Fig. 2, as afunction of P. In the main analyses of this paper, we consider(i) CP values obtained by a structural optimization, which aredenoted as optimized CP values. For comparison, we alsoconsider (ii) the theoretical calculation of the CP values inZhang et al. [33] for the region between Pamb and 20 GPa,and extrapolate the pressure dependence up to 60 GPa. Detailsabout (i) and (ii) are given in Appendix B 1. We also consider(iii) the experimental CP values from Armstrong et al. [34]between Pamb and 8.5 GPa. (The values at Pamb correspond tothe SC phase with the experimental SC transition temperatureT expc � 135 K close to T optc � 138 K.) It is known that theoptimized CP values slightly deviate from the experimentalvalues, and it is indeed seen in Fig. 2. From the comparison ofthe optimized and experimental CPs, we take into account thecorrection (D) addressed in Sec. I.We simulate at the experimental optimal hole concentrationp, which allows a reliable comparison with the P dependenceof T optc [3]. We use the same procedure as that in Ref. [30]employed for Hg1201: We partially substitute Hg by Au. Weconsider the chemical formula Hg1−xsAuxs Ba2Ca2Cu3O8 withxs = 0.6 in order to realize the average hole concentration perCuO2 plane pav = 0.2 [5,37,38]. This choice is discussed andjustified in Appendix B 2.In addition, we examine the distinct effects of the uniax-ial pressures along axis a and axis c, whose definitions aregiven in Table II and discussed below. The nontrivial pointis: Experimentally, what are the variations in CP values whenthe crystal structure is compressed along a (c)? First, thecompression along a obviously modifies the cell parameter aas well as the amplitude |dzbuck| of the Cu-O-Cu bond bucklingin the OP, but it should not affect the other CPs dzCu, dzCa, dzBa,and dzO(ap). Thus, we define the uniaxial pressure Pbucka alonga as follows: The compression along a modifies the valuesof a and dzbuck, and all other CP values are those at Pamb.We also consider a simplified definition, denoted as Pa: Thecompression along a modifies only the value of a, and all otherCP values are those at Pamb. As we will see, Pa is sufficient todescribe the main effect of the compression along a. Second,023163-4DOME STRUCTURE IN PRESSURE DEPENDENCE OF … PHYSICAL REVIEW RESEARCH 6, 023163 (2024)FIG. 3. (a)–(g) Uniform pressure dependence of the GGA band structure. We show the GGA bands outside (dashed black color) and inside(solid black color) the M space, the AB bands (red color), and the 29 other bands in the band window, which are disentangled from the ABband (dashed cyan color). High-symmetry points are, in coordinates of the reciprocal lattice: G = [0 0 0], D = [1/2 0 0], and X = [1/2 1/2 0].(h)–(n) Uniaxial pressure dependence of the GGA band structure. (o)–(q) Uniform pressure dependence of the Cu3dx2−y2 onsite energy εlx , thein-plane O2pσ onsite energy εlp, and the Cu3dx2−y2/O2pσ hopping in the unit cell t lxp in the IP (l = i) and OP (l = o). We also show the resultat the uniaxial pressure Pa = 60 GPa [denoted as 60(Pa)]. All quantities are obtained using the optimized CP values.the compression along c modifies the values of dzl , that is, allCP values except that of a. This uniaxial pressure is denotedas Pc. For completeness, we also consider a second definition,denoted as Pbuckc : The compression along c modifies all CPvalues except those of a and dzbuck. This allows to discuss theeffect of the relatively large value of |dzbuck| at P > Popt. In themain analyses of this paper, we consider Pa (Pc) to simulate thecompression along a (c). We also give complementary resultsby considering Pbucka and Pbuckc .We first compute the electronic structure at the DFT level.The P dependence of the GGA band structure is demonstratedin Fig. 3, from which we derive the LEH spanned by theCu3dx2−y2/O2pσ AB bands by employing the GGA+cRPAscheme sketched in Appendix A. Computational details ofDFT and GGA+cRPA scheme are described in Appendix B.Then, we define the AB LEH as follows. In the AB LEH formultilayer cuprates [30], there is only one AB orbital centeredon each Cu atom. Then the AB LEH readsH =∑l,l ′Hl,l ′ =∑l,l ′[Hl,l ′hop + Hl,l ′int], (4)where l, l ′ = {i, o, o′} with i being an IP site, and o, o′ be-longing to the two equivalent OPs. in which we distinguishthe hopping and interaction parts between planes l and l ′, as,respectively,Hl,l ′hop =∑(σR),(σ ′R′ )t l,l ′ (R′ − R)ĉ†lσRĉl ′σ ′R′ , (5)Hl,l ′int =∑(σR),(σ ′R′ )U l,l ′ (R′ − R)n̂lσRn̂l ′σ ′R′ , (6)where σ, σ ′ are the spin indices. By using these notations,(lσR) is the AB spin orbital in the plane l and in the unitcell at R, with spin σ . c†lσR, clσR, and n̂lσR are, respectively,the creation, annihilation, and number operators in (lσR), andt l,l ′ (R′ − R) and U l,l ′ (R′ − R) are, respectively, the hoppingand direct interaction parameters between (lσR) and (l ′σ ′R′).The translational symmetry allows us to restrict the calcu-lation of LEH parameters to t l,l ′σ,σ ′ (R) and U l,l ′σ,σ ′ (R) between(lσ0) and (l ′σ ′R).In this paper, we focus on the intraplane LEH Hl = Hl,lwithin the plane l and analyze only the first nearest-neighborhopping t l1 = t l,l ([100]) and the onsite effective interactionU l = U l,l ([000]), because these two parameters were pro-posed to essentially determine T optc at least for single- andtwo-layer cuprates [7]. (Other LEH parameters are given inthe Supplemental Material [32].) Then within this restricted023163-5MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)FIG. 4. Uniform pressure dependence of the AB LEH parameter values in the IP (l = i), the OP (l = o), and the average value over theIP and OP (l = avg). We show the basic energy unit |t l1|, the correlation strength ul = U l/|t l1|, the onsite effective Coulomb interaction U l ,the screening ratio Rl = U l/vl , and the onsite bare interaction vl . In addition, we show the charge transfer energy �Elxp, and the amplitude ofthe hopping t lxp between the Cu3dx2−y2 and in-plane O2pσ ALWOs at the GGA level. We show the quantities obtained using the optimized CPvalues (a)–(g), the experimental CP values from Armstrong et al. [34] at Pamb and 8.5 GPa [crosses in (a)–(g)], and the CP values from Zhanget al. (h)–(n).range, Hl is rewritten asHl = |t l1|[H̃lhop + ulH̃lint] = |t l1|H̃l , (7)in which H̃lhop = Hlhop/|t l1| and H̃lint = Hlint/U l are the di-mensionless hopping and interaction parts, expressed inunits of their respective characteristic energies |t l1| and U l .The full dimensionless intraplane LEH is H̃l = Hl/|t l1|, andthe dimensionless ratio ul = U l/|t l1| encodes the correlationstrength. As mentioned in Sec. I, we also discuss the values of|t avg1 | = (|t i1| + |t o1 |)/2 and uavg = (ui + uo)/2. Average valuesof other quantities with the superscript l are defined similarly.We compute the above LEH parameters |t l1| and U l asfollows. We use the RESPACK code [30,39]. The standardcalculation procedure is presented in detail elsewhere [30,39].First, we compute t l1 ast l1 =∫�drw∗l0(r)h(r)wlR1 (r), (8)in which wlR is the Wannier function of the AB orbital (lR),R1 = [100], � is the unit cell, and h is the one-particle part atthe GGA level. Then we compute U l as follows. We computethe cRPA effective interaction WH, whose expression is foundin Appendix B 5, Eq. (B6). We use a plane wave cutoff energyof 8 Ry. We deduce the onsite effective Coulomb interactionasU l =∫�dr∫�dr′w∗l0(r)w∗l0(r′)WH(r, r′)wl0(r)wl0(r′). (9)We also deduce the onsite bare Coulomb interaction vl byreplacing WH by the bare Coulomb interaction v in Eq. (9), andthe cRPA screening ratio Rl = U l/vl . The obtained values of|t l1|, U l , vl and Rl are plotted in Fig. 4.IV. PRESSURE DEPENDENCE OF ELECTRONICSTRUCTURE AT DFT LEVELNow, we show the result of GGA calculation as a functionof P and clarify what can be learned within the DFT levelalready. The band dispersion is shown in Figs. 3(a)–3(l). Wealso show in Figs. 3(o)–3(q) the onsite energy of the Cu3dx2−y2and in-plane O2pσ atomiclike Wannier orbital (ALWO). (Asexplained in Appendix B, we denote these ALWOs as M-ALWOs because they are in the M space.) We also show theCu3dx2−y2/O2pσ hopping amplitude |t lxp| = |tCu(l ),O(l )x2−y2,pσ| in theunit cell.We first elucidate main mechanisms of the following items[MW ], [Mε], and [Mt] when P increases:[MW ] Broadening of the M band dispersion in Figs. 3(a)–3(g)[Mε] Decrease in onsite energies of M-ALWOs relative tothe Fermi level in Figs. 3(o) and 3(p).[Mt] Increase in hoppings between M-ALWOs in Fig. 3(q).A simple interpretation of [MW ], [Mε], and [Mt] is thatP works to reduce the interatomic distances. This causes twodistinct effects: First, the electrons in the CuO2 plane feel thestronger Madelung potential from ions in the crystal. Indeed,the amplitude of the Madelung potential scales as 1/d , whered is the interatomic distance between the ion and the Cuor O atom in the CuO2 plane. The variation in Madelungpotential modifies the M-ALWO onsite energies and causes[Mε] (for details, see Appendix E 1). Second, the overlap andhybridization between M-ALWOs increases, which causes[Mt]. Both [Mε] and [Mt] increase the splitting of the B/NB(bonding/nonbonding) and AB bands, which causes [MW ]:The bandwidth W of the M bands increases from W � 9 eVat Pamb to W � 12 eV at P = 60 GPa [see Figs. 3(a)–3(g)].Simultaneously, the bandwidth WAB of the AB band increases023163-6DOME STRUCTURE IN PRESSURE DEPENDENCE OF … PHYSICAL REVIEW RESEARCH 6, 023163 (2024)TABLE III. Summary of the variations in AB LEH parameterswith P < Popt and P > Popt in Fig. 4. We use ↗, ↗↗, �, ↘, or↘↘ if the quantity increases, strongly increases, remains static,decreases, or strongly decreases, respectively. The variation in u =U/|t1| is controlled by that in |t1| and U . The variation in U = vRis controlled by that in the onsite bare interaction v and the cRPAscreening ratio R.|t1| u = U/|t1| U = vR v RP < Popt ↗↗ ↘ ↗ ↗ ↗P > Popt ↗↗ ↘↘ � / ↘ ↗ ↘ / ↘↘from WAB � 4 eV at Pamb to WAB � 5.5 eV at P = 60 GPa,which is caused by [Mt]. Indeed, the increase in |t l1| and thusWAB � 8|t1| originates from the increase in |t lxp|, as discussedlater in Sec. V A.Effects of the uniaxial pressures Pa and Pc to [MW ], [Mε],and [Mt] can also simply be accounted for when we con-sider the anisotropy of the overlap of the two M-ALWOs andthe direction of the pressure. For instance, [MW ] is causedby Pa rather than Pc [see Figs. 3(h)–3(n)], because the ABbandwidth WAB and W are mainly determined by the over-lap between Cu3dx2−y2 and O2pσ ALWOs in a CuO2 plane.This increase in the bandwidth with Pa was also mentionedin Ref. [40] in the case of Hg1201. On the other hand, theapplication of Pc shifts a few specific bands: Hg5d-like bandsare shifted from −4/ − 5 eV at Pamb to −7 eV at Pc = 30 GPa.However, Pc does not modify WAB. Effects of uniaxial pressureon [Mε] and [Mt] are also obviously and intuitively under-stood in a similar fashion: We clearly see in Figs. 3(o)–3(q)that [Mε] and [Mt] are caused by Pa rather than Pc. For moredetails of the pressure effects, see Appendix E 1.V. PRESSURE DEPENDENCE OF ABEFFECTIVE HAMILTONIANNow, we discuss the P dependence of AB LEH parametersin Figs. 4(a) and 4(b), in which the two main mechanisms(I, II) are visible: (I) |t l1| increases, whereas (II) ul decreases.In this section, we discuss the mechanisms of (I) and (II) thatare summarized in Table III, and demonstrate that (I) and(II) are indeed physical and robust. We discuss mainly |t avg1 |and uavg, and discuss briefly the difference between values inthe IP and OP. A comparison with experiments will be madeseparately in Sec. VI.A. Increase in |t1| with PThe increase (I) in the P dependence of |t avg1 | [see Fig. 4(a)]is purely caused by the reduction of cell parameter a whenthe crystal is compressed along axis a. Indeed, (I) is purelycaused by the application of Pa [see Fig. 5(a)], whose onlyeffect is to reduce a. The underlying origin is simply theincrease in overlap between AB orbitals on neighboring Cuatoms due to the decrease in cell parameter a when increas-ing Pa as already discussed in Sec. IV at the DFT level.We note that |t l1| has a similar P dependence as that of |t lxp|[see Figs. 4(a) and 4(g)]. This is obvious because the ABorbital is formed by the hybridization of Cu3dx2−y2 and O2pσM-ALWOs.Note that, at P > Popt, |t o1 | is reduced with respect to |t i1|;this is because of the buckling of Cu-O-Cu bonds in the OP.Indeed, the decrease in |t o1 | − |t i1| and also |t oxp| − |t ixp| occursin the Pbucka dependence [see Fig. 5(h)] but not in the Padependence [see Fig. 5(a)], and the value of dzbuck is modi-fied by the application of Pbucka but not by the application ofPa. Furthermore, the P dependence of |t o1 | − |t i1| is consistentwith that of |dzbuck|: The decrease in |t o1 | − |t i1| starts at Poptand is amplified at larger pressures [see Figs. 4(a) and 4(g)],which is consistent with the increase in |dzbuck| from 0.05 Å to0.20 Å between Popt and 60 GPa (see Fig. 2). The origin ofthe decrease in |t o1 | − |t i1| can be understood as follows: When|dzbuck| increases, the overlap between Cu3dx2−y2 and O2pσ M-ALWOs in the OP is reduced. Note that the buckling induceddecrease in |t1| has also been observed in the two-layer cuprateBi2212 [30].FIG. 5. Uniaxial pressure dependence of the AB LEH parameter values. Notations are the same as those in Fig. 4. All quantities areobtained using the optimized CP values. We show the Pa and Pc dependencies (a)–(g), and the Pbucka and Pbuckc dependencies (h)–(n).023163-7MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)Comparison of results obtained from different CP valuesshows that (I) is physical and robust. If we consider both (i)the optimized CP values and (ii) the CP values from Zhanget al., the P dependencies of |t l1| and |t lxp| are very similar for(i) and (ii) [see Figs. 4(a), 4(g), 4(h), and 4(n)]. This is intuitivesince the P dependence of a is similar for (i) and (ii), and the Pdependence of dzbuck at P > Popt is also similar (see Fig. 2). Ifwe consider (iii) the experimental CP values from Armstronget al. [34] at P < 8.5 GPa, the increase in |t avg1 | and |t avgxp | isfaster. This is in accordance with the faster decrease in a for(iii) with respect to (i) and (ii) (see Fig. 2), and implies theuncertainty of the estimate of |t avg1 | at Popt, as discussed laterin Sec. VI.B. Decrease in u with P < PoptAt P < Popt, the decrease (II) in uavg is largely inducedby the increase (I) in |t avg1 |; however, the increase in U avg[see Fig. 4(c)] partially cancels the decrease in uavg. Thus, wediscuss the P dependence of U avg below.The increase in U avg is caused by two cooperative factors(i, ii) whose main origin is the reduction in a. These are (i) theincrease in onsite bare interaction vavg [see Fig. 4(c)], and (ii)the reduction in cRPA screening represented by the increase inthe average value Ravg of the cRPA screening ratio Rl = U l/vl[see Fig. 4(d)]. In the following, we discuss the microscopicorigins of (i) and (ii).On (i), the increase in vl mainly originates from the in-crease in charge transfer energy �Elxp between Cu3dx2−y2and O2pσ M-ALWOs. This is because the increase in �Elxpreduces the importance of the Cu3dx2−y2/O2pσ hybridization.(The latter is roughly encoded in the ratio Olxp = |t lxp|/�Elxp.)The reduction in hybridization increases the Cu3dx2−y2 atomiccharacter and thus the localization of the AB orbital. This isdiscussed and justified in item (a) in Appendix E 2. This sim-ple view is consistent with the systematic correlation betweenvl and �Elxp in this paper [see Figs. 4(e), 4(f), 4(l), and 4(m)and Appendix F], and also in the literature [27,30]. Still, notethat the correlation between vl and �Elxp is slightly reduced atP > Popt [see Figs. 4(e), 4(f), 4(l), and 4(m) at P > Popt]. Thisis because |t oxp| is reduced with respect to |t ixp| at P > Popt dueto the nonzero dzbuck, which contributes to reduce Ooxp [see alsoitem (c) in Appendix E 2].The increase in �Elxp mainly originates from the reduc-tion in a. Indeed, the increase is mainly caused by Pa [seeFig. 5(f)]. This is because the reduction in a increases theenergy of Cu3dx2−y2 electrons with respect to that of O2pσelectrons (see Appendix E 1). Although the reduction in a isthe main origin of the increase in the P dependence of �E avgxp ,note that �Elxp depends not only on a but also on other CPs(see Appendix F).The concomitant increases in vl and |t l1| seem counterin-tuitive but can be explained as follows. The counterintuitivepoint is that the increase in vl suggests a more localized ABorbital whereas the increase in |t l1| would be more consistentwith a delocalization of the AB orbital. Although the ABorbital is more localized, the increase in |t l1| is explained bythe increase in |t lxp| with Pa in Fig. 4(g). This is discussedin detail in item (b) in Appendix E 2, which is summarizedbelow. We apply Pa and examine the a dependencies of |t avg1 |,|t avgxp |, and �E avgxp , and the average values Oavgxp and T avgxp ofOlxp and T lxp = |t lxp|2/�Elxp. The increase in �E avgxp with a isfaster than the increase in |t avgxp |, but slower than the increasein |t avgxp |2. As a result, when a decreases, |t avg1 | ∝ T avgxp ∝ 1/a3increases. On the other hand, Oavgxp ∝ a decreases, hence theincrease in vl .On (ii), the decrease in cRPA screening [the increase inRl in Fig. 4(d)] is due to the broadening [MW ] of the GGAband dispersion (whose origin is the reduction in a as dis-cussed in Sec. IV). Indeed, [MW ] causes the increase incharge transfer energies between occupied bands and emptybands, which reduces the amplitude of the cRPA polariza-tion (see Appendix E 3 for details). The increase in Rl ismonotonous, except for the small dip in the P dependence ofRo at P � 24 GPa in Fig. 4(d). The dip may originate fromthe change in the sign of dzbuck at P � 24 GPa (see the nextparagraph).Comparison of results obtained from different CP valuesshows that (i) and (ii) are essentially correct, independentlyof the uncertainty on CP values. Let us consider the resultsobtained from the CP values from Zhang et al. in Figs. 4(h)–4(n) and compare them with the results obtained from theoptimized CP values in Figs. 4(a)–4(g). The increase in vavg iswell reproduced [see Figs. 4(e) and 4(l)]. The increase in Ravgwith P is qualitatively reproduced [see Figs. 4(d) and 4(k)];however, the P dependence of Rl is not exactly the same, andwe discuss the difference below.First, there is a small dip in the P dependence of Ro at P �24 GPa in Fig. 4(d) (optimized CP values). This dip is notobserved in Fig. 4(k) (CP values from Zhang et al.). This maybe because the sign of dzbuck does not change at P � 24 GPa ifwe consider the CP values from Zhang et al., contrary to theoptimized CP values (see the P dependence of dzbuck in Fig. 2).Second, at Popt = 30 GPa, the value of Ri is similar but thevalue of Ro is larger in Fig. 4(k) with respect to Fig. 4(d). Thisis because the values of both dzCa and dzCu are larger in Zhanget al. with respect to the optimized CP value (the difference is0.1 Å as seen in Fig. 2). As shown in Appendix F, the largervalue of dzCa increases Ro. At the same time, the larger valueof dzCa (dzCu) decreases (increases) Ri. (Both effects cancel eachother.)Finally, if we consider the experimental CP values fromArmstrong et al., the increases (i) and (ii) are faster [seeFigs. 4(d) and 4(e)]. This is consistent with the faster decreasein a in Armstrong et al. with respect to the optimized CPvalues and those from Zhang et al. (see Fig. 2).C. Decrease in u with P > PoptAt P > Popt, the decrease in uavg is faster because Ravgdecreases. Let us start from the P dependence of U avg: AtP > Popt, U avg ceases to increase [see Fig. 4(c)] and may evendecrease if we consider the CP values from Zhang et al. [seeFig. 4(i)]. The origin is not the P dependence of vavg, whichincreases monotonically [see Figs. 4(e) and 4(l)], but ratherthat of Ravg, which shows a dome structure with a maximumat Pscr � 30–40 GPa and a decrease at P > Pscr [see Figs. 4(d)and 4(k)]. The decrease in Ravg dominates the increase in vavg.The decrease in Ravg looks physical, and robust with re-spect to the uncertainty on CP values. It is still observed023163-8DOME STRUCTURE IN PRESSURE DEPENDENCE OF … PHYSICAL REVIEW RESEARCH 6, 023163 (2024)if we consider the CP values from Zhang et al. instead ofthe optimized CP values [see Fig. 5(k)], even though the Pdependence of Rl is modified.The decrease in Ravg is the result of a competition betweenPa and Pc. (The effect of Pc is dominant at P > Popt.) Asseen in Fig. 5(d), applying only Pa causes (i) the nonlinearincrease in Ravg, which dominates at P < Popt but saturatesat P > Popt. On the other hand, applying only Pc causes(ii) the decrease in Ravg, which becomes dominant at P >Popt. [(i) and (ii) are interpreted in terms of the cRPA po-larization in Appendix E 3.] The microscopic origin of (ii)is the decrease in both dzCu and dzO(ap) when Pc is applied(see Appendix F).Note that, in the OP, the destructive effect of Pc on Ro andthus uo is canceled by the buckling induced decrease in |t o1 |.Indeed, the Pc dependence of uo in Fig. 5(b) shows a 6%increase from Popt to 60 GPa. This increase originates fromthe buckling of Cu-O-Cu bonds in the OP, because it does notappear in the Pbuckc dependence of uo in Fig. 5(i), and the valueof dzbuck is modified by applying Pc but not by applying Pbuckc .The buckling reduces |t o1 | as discussed in Sec. V A, which isthe main origin of the increase in uo from Popt to 60 GPa.VI. DISCUSSIONHere we discuss in detail how the experimental P depen-dence of T optc is predicted by considering (I) and (II) togetherwith the assumptions (A, B) and the corrections (C, D) inSec. I. We also discuss that (A) through (D) are all physicallysound.First, we emphasize that only by considering (A) and (B),the dome structure in the P dependence of T optc is qualitativelyunderstood. Since (B) implies that FSC stays at a plateauregion around the peak of parabolic P dependence betweenPamb and Popt as is seen in Fig. 1(a). Then the dominant Pdependence of FSC arises from t1, which causes an increase inT estc in Eq. (1). On the other hand, FSC rather rapidly decreaseswith increasing P above Popt, which dominates over the effectof increase in |t1|.The location of Hg1223 assumed in (B) is justified from(C). Without (C), we would have uavg � 7.2 at Pamb and � 6.8at Popt: Both values are below uopt � 8.0 − 8.5, so that FSCwould quickly decrease with P, and (B) would not be valid.On the other hand, if we apply (C), we have u � 9.3 � uoptat Pamb and � 7.8 � uopt at Popt [see Fig. 6(a)], so that (B)becomes valid.Let us discuss more quantitative aspects. Although FSCdoes not vary substantially with increasing P below Popt, thereis a small (� 5%) decrease in FSC from Pamb to Popt even afterapplying (C) [see Fig. 1(a)]. If we apply (C) without (D), the �13%–14% increase in |t avg1 | from Pamb to Popt becomes the �17% increase in |t1|. However, the increase in T estc estimatedfrom Eq.(1) is only � 10% due to the � 5% decrease in FSC.If we apply (D) after (C), the increase in |t1| becomes � 22%,so that the increase in T estc becomes � 17% and reproducesthat in T optc . Note that the quantitative agreement between theincreases in T estc and T optc is very good at xestLRFB = 0.95 at leastfor small P [see Fig. 1(b)]. For completeness, note that (D)has a limitation: It relies on the a dependence of |t1| at theFIG. 6. (a) P dependence of estimated u at the cGW -SIC+LRFBlevel denoted here as ucGW −SIC+“LRFB” The diamond symbols showucGW −SIC+“LRFB” with the choice of xLRFB = 0.95 correcting explicitcalculations at the cGW -SIC level employing Eq. (2). The dashedlines show linear interpolations between the diamond symbols. Thecolored shaded area corresponds to the range xLRFB = 0.91 − 0.97.(b) u dependence of FSC extracted from Schmid et al. [7], Fig. 10.We also show F estSC at 0, 30 and 60 GPa.GGA+cRPA level. [For more details, see the last paragraphof Appendix D.]Now, we argue that (A)–(D) are adequate from the physicalpoint of view. On (A), it was shown that the scaling Eq. (1)is equally satisfied for N� = 1, 2 and ∞ [7]. This is on theone hand due to the fact that the interlayer coupling is smallfor all the cases and within a CuO2 layer on the other hand,the superconductivity is mainly dependent on t1 and U only,and the dependence on other parameters is weak within therealistic range. In the present case of Hg1223 with N� = 3, theinterlayer coupling is again small. For instance, the ratio be-tween the interlayer offsite Coulomb repulsion V i,o and U avgis V i,o/U avg = 0.13 at Pamb, and the superconducting strengthis expected to be governed by the single-layer physics, whichis the same as the cases of N� = 1, 2, and ∞.On (B), the statement that ucGW −SIC+“LRFB” at Pamb isabove uopt � 8.0–8.5 is indeed satisfied in the ab initio esti-mate by considering the correction (C). As mentioned earlier,the GGA+cRPA estimate is uavg � 7.2 < uopt at Pamb. How-ever, (C) yields ucGW −SIC+“LRFB” � 9.3 � uopt at Pamb, anducGW −SIC+“LRFB” � 7.8 � uopt at Popt [see Fig. 6(a)].On (C), the calculation of ucGW −SIC+“LRFB” and|t1|cGW −SIC+“LRFB” is detailed in Appendix C.On (D), it is plausible that our calculation overestimates aby � 0.05 Å at P > 8.5 GPa, because the same overestimationis already observed at P = 8.5 GPa in Fig. 2. The derivationof improved |t1|cGW −SIC+“LRFB” is detailed in Appendix D.In Fig. 1(b), although the pressure dependence of T optc isnicely reproduced for P < Popt, the estimated T estc decreasesmore rapidly than the experimental T optc at P > Popt. Theorigin of this discrepancy is not clear at the moment. Onepossible origin is of course the uncertainty of the crystal pa-rameters at high pressure because there exist no experimentaldata. Another origin would be the limitation of the inferencefor the LRFB correction taken simply by the constants xLRFBand yLRFB. The third possibility is the possible inhomogeneityof the pressure in the experiments. The complete understand-ing of the origin of the discrepancy is an intriguing futureissue.023163-9MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)VII. SUMMARY AND CONCLUSIONWe have proposed the microscopic mechanism for thedomelike P dependence of T optc in Hg1223 as the consequenceof (I) and (II) obtained in this paper together with the as-sumptions (A, B) and the corrections (C, D) mentioned inSec. II and supported in Sec. VI. We have also elucidatedthe microscopic origins of (I) and (II), which are summarizedbelow.(I) The increase in |t1| is caused by the reduction in the cellparameter a when the crystal is compressed along axis a.(II) The decrease in u is induced by (I), but is partiallycanceled by the increase in U at P < Popt. The increase inU is caused by two cooperative factors: (i) the increase inonsite bare interaction v, whose main origin is the reductionin Cu3dx2−y2/O2pσ hybridization, and (ii) the reduction incRPA screening at P < Popt. Both (i) and (ii) originate fromthe reduction in a. At P > Popt, U ceases to increase withincreasing P, because the cRPA screening increases due tothe compression along axis c, more precisely the reductionin distance dzCu between the IP and OP [dzO(ap) between the OPand apical O], which screens AB electrons in the IP (OP).The elucidation of the above mechanisms offers a platformfor future studies on cuprates under P and design of newcompounds with even higher T optc : For instance, Tc may becontrolled by controlling |t1| via the cell parameter a. How-ever, the increase in |t1| is a double-edged sword for theincrease in Tc: On one hand, it is the direct origin of theincrease in T optc ∝ |t1| at P < Popt in Hg1223. On the otherhand, it is a prominent cause of the decrease in u and thusFSC and T optc at P > Popt. Conversely, in the OP, the bucklingof Cu-O-Cu bonds reduces |t1|: This reduces T optc ∝ |t1|, butthis may also increase FSC and thus T optc if the value of uis in the weak-coupling region [u < 7.5 in Fig. 6(b)]. Forinstance, the buckling may be identified as the main originof the higher T optc in Bi2212 (T optc � 84 K [2]) compared toBi2201 (T optc � 6 K [2]): The buckling reduces |t1| and thusincreases u in Bi2212 with respect to Bi2201 [30], so thatBi2212 is near the optimal region whereas Bi2201 is in theweak-coupling region [7]. This explains the larger |t1|FSC inBi2212 [7] despite the smaller |t1|.ACKNOWLEDGMENTSWe thank Michael Thobias Schmid for useful discus-sions. This work was supported by MEXT as a Programfor Promoting Researches on the Supercomputer Fugaku(Basic Science for Emergence and Functionality in Quan-tum Matter-Innovative Strongly-Correlated Electron Scienceby Integration of Fugaku and Frontier Experiments, JP-MXP1020200104 and JPMXP1020230411) and used com-putational resources of the supercomputer Fugaku providedby the RIKEN Center for Computational Science (ProjectIDs hp200132, hp210163, hp220166, and hp230169). Wealso acknowledge the financial support of JSPS KakenhiGrant-in-Aid for Transformative Research Areas, Grants No.JP22H05111 and No. JP22H05114 (“Foundation of MachineLearning Physics”). Part of the results were obtained underthe Special Postdoctoral Researcher Program at RIKEN. Theleft panel of Fig. 2 was drawn using software VESTA [41].APPENDIX A: METHOD OF MACEHere, as a complement to Sec. I, we summarize andcomment the MACE methodology that is used to derive theeffective Hamiltonian. We also mention other diagrammaticapproaches employed in the literature.The MACE methodology consists of three steps (i)–(iii)that are summarized below.(i) First, starting from the crystal structure, the electronicstructure of the material is calculated at the simplified den-sity functional theory (DFT) [20,21] level. This frameworkuses the LDA or GGA exchange-correlation functionals anda single-determinant wave function. The electronic struc-ture is either left at the LDA(GGA) level [in case theLDA(GGA)+cRPA is employed] or preprocessed to the GWlevel (if cGW -SIC is employed) supplemented with LRFB (ifcGW -SIC+LRFB is employed), as explained in Sec. I.(ii) The description of the L space is improved by de-riving a low-energy effective Hamiltonian (LEH) restrictedto the L space. In this LEH, the two-particle part is calcu-lated at the constrained random phase approximation (cRPA)[22,23] at the GGA+cRPA level. At the cGW -SIC andcGW -SIC+LRFB levels, the one-particle part of the LEH isalso improved by removing the exchange-correlation doublecounting term [25] and the self-interaction term [26] (see alsoSec. I). This properly describes high-energy (H) states suchas core and semicore bands from closed shells, but fails todescribe many-body effects and strong electronic correlationin the low-energy (L) subspace near the Fermi level, even withthe above preprocessing. In the case of cuprates, this L spaceis composed of the AB orbital centered on each Cu atom inthe CuO2 plane. The correlation strength is quantified withinthe ratio u whose value is typically above 7 for the high-Tccuprates [27,28,30].(iii) The LEH is solved by a many-body solver, e.g..many-variable Variational Monte Carlo (mVMC) [42]. (SeeRef. [43] for a benchmark of the mVMC solver.)This three-step MACE procedure allows to correctly de-scribe the Mott physics in the mother compound and the SCphase in the carrier doped compound [7,29]. In the mVMCsolution, FSC rapidly increases with u in the range 7 � u � 8.5[7], which suggests an increase in Tc with u [7], in agreementwith the positive correlation between u and T optc [30] in thesame range of values of u. This range corresponds to the weak-coupling and plateau regions [7 � u � 9 in the u dependenceof FSC in Fig. 6(b)]. These results led to the identification ofthe possibly universal scaling Tc � 0.16|t1|FSC in the solutionof the AB LEH at the cGW -SIC+LRFB level [7].To predict the SC character of the material with the aboveMACE procedure, insights may be obtained even prior tothe computationally expensive solution (iii), by examiningintermediate quantities within the hierarchical structure ofMACE. Notably, the scaling T optc � 0.16|t1|FSC proposed inRef. [7] and the u dependence of FSC in Fig. 6(b) suggest it ispossible to anticipate the crystal structure dependence of Tc bystudying the crystal structure dependence of LEH parameters(ii), particularly |t1| and u. Following this idea, we tacklein this paper the derivation of the AB LEH (ii) for Hg1223as a function of pressure, without performing explicitly thesolution (iii), which is left for future studies. Of course, the023163-10DOME STRUCTURE IN PRESSURE DEPENDENCE OF … PHYSICAL REVIEW RESEARCH 6, 023163 (2024)explicit many-body solution of the LEH (iii) is necessary toreach the final conclusion.Furthermore, qualitative insights into the SC may beobtained by deriving the LEH parameters at the simpleGGA+cRPA level up to the process (ii), whereas cGW -SIC+LRFB brings a mostly quantitative correction to theLEH parameters [30]. Note that this quantitative correctionby cGW -SIC+LRFB is still important to stabilize the SCstate with mVMC (iii) in practice: The improvement by cGW -SIC+LRFB increases U and thus u by 10%–15% in Bi2201and Bi2212 [30], which allows a quantitative estimate ofthe SC order in the mVMC solution. On the other hand,at the simple GGA+cRPA level, u may be underestimated.Nonetheless, GGA+cRPA still reproduces the dependenceof u in the LEH parameters on the materials, and the CPsincluding pressure effects systematically in accordance withcGW -SIC+LRFB [30], which allows us to extract qualita-tively correct trends in the LEH parameters by avoiding thelarge computational cost [30] of cGW -SIC+LRFB. For in-stance, in the comparison between Bi2201 (T optc � 6 K [2])and Bi2212 (T optc � 84 K [2]), u is larger for Bi2212 at thecGW -SIC+LRFB level, and this qualitative result is also re-produced at the GGA+cRPA level in Ref. [30], Appendix C.Following the above idea, we mainly employ theGGA+cRPA scheme to derive the AB LEH (ii) for Hg1223.We also employ the cGW -SIC+LRFB scheme in a limitedcase in Appendix C, as explained in Sec. I.Besides the MACE approach employed in the present pa-per and described above, several other approaches allow totake into account diagrams beyond GW , which is necessary todescribe the strong electronic correlations in cuprates. Theseother approaches include the dynamical vertex approximation(D�A) [44], the unification of parquet and GW � methods[45], the quasiparticle self-consistent GW (QSGW ) with lad-der diagrams [46], as well as the QSGW +DMFT approachemployed in Refs. [47,48]. (The QSGW +DMFT approachtreats local many-body effects with the quantum impuritysolver, but treats nonlocal correlations at the QSGW level.)The MACE methodology takes into account the strongcorrelation effect in an alternative fashion without relyingon the perturbative diagrammatic expansion of the vertexcontribution for the relevant fluctuations. In Ref. [7], boththe universal u dependence of FSC and the scaling T optc �0.16|t1|FSC have been obtained using the accurate quantummVMC solver for the AB band where the dominant many-body quantum fluctuations reside. This quantum many-bodysolver fully incorporates the effect of temporal and spatialquantum fluctuations on an equal footing. In this approach,the nonlocal interactions are treated at the many-body level inaddition to the local interactions.APPENDIX B: COMPUTATIONAL DETAILS1. Choice of crystal parameter valuesThe CP values obtained from neutron diffraction powderand energy-dispersive synchrotron x-ray diffraction experi-ment [34–36] are summarized in Fig. 2.There is an uncertainty on the CP values, especially afterP � 9.2 GPa. Indeed, the experimental P dependence of theCP values varies between different works. In addition, to ourknowledge, the CP values at P > 9.2 GPa have not been com-pletely determined in experiment. References [34,35] provideall CP values, but only up to P � 8.5–9.2 GPa. Reference [36]provides the values of a and c up to P � 26 GPa, but not thevalues of dzl . Thus, the CP values are not available within therange Pamb < P < 45 GPa that corresponds to the domelike Pdependence of T optc in Ref. [3].To verify the robustness of our results with respect to theuncertainty on CP values, we consider CP values obtained bytwo different theoretical calculations (i and ii), up to 60 GPa.We consider (i) CP values obtained by a structural optimiza-tion (denoted as optimized CP values) and (ii) CP valuesobtained in Zhang et al. [33]. We determine first (ii), then (i),as explained below.On (ii), the values in Zhang et al. have been obtained froma theoretical calculation, by means of interatomic potentials.At P < 9.2 GPa, these values are in reasonable agreementwith the different experimental values from Refs. [34,35].Although the values of a are overestimated with respect toRefs. [34–36], they are in good agreement with Ref. [36] at24 GPa.However, the CP values from Zhang et al. are availableonly up to 20 GPa; thus, we extrapolate their P dependenceup to 60 GPa, as follows. We fit the P dependence of a byconsidering the Murnaghan equation of statea(P)a(Pamb)=[1 + κ ′κP]−1/κ ′, (B1)as done in Ref. [36]. We deduce the values of the two param-eters κ and κ ′, which are, respectively, the bulk modulus andits pressure derivative. The same procedure is applied to c,dzO(ap), dzCu, dzBa,O(o) = dzBa + dzbuck, and dzCa,Ba = dzCu − dzCa +dzBa, whose values are extracted from Ref. [33]. In the case ofdzbuck, we fit the Cu(o)-O(o)-Cu(o) bond angle as a functionof P in Ref. [33], Fig. 5 with Eq. (B1). Then we deduce dzBafrom dzBa,O(o) and dzbuck, and dzCa from dzCa,Ba, dzCu, and dzBa.We checked that values of κ for these CPs from Ref. [33] arereproduced with a difference lower than 0.5%. These valuesof κ are 1.81 × 10−3 GPa−1 for a, 4.61 × 10−3 GPa−1 for c,7.01 × 10−3 GPa−1 for dzO(ap), 2.94 × 10−3 GPa−1 for dzCu,0.64 × 10−3 GPa−1 for dzBa−O(o), and 1.535 × 10−3 GPa−1for dzCa−Ba. We obtain the CP values in Fig. 2.On (i), the optimized CP values are obtained by startingfrom (ii), and performing a structural optimization. We im-pose the following constraint: The volume V = a2c of the unitcell remains constant. This allows us to avoid the relaxationof the volume to its value at Pamb. Other computational detailsare the same as those for the self-consistent calculation (seeAppendix B 3). Results are shown in Fig. 2.We deem (i) more reliable than (ii) because the structuraloptimization allows the rigorous minimization of the free en-ergy of the crystal; thus, we consider (i) in the main analysesof this paper and (ii) as a complement. Still, (ii) is useful tocheck the robustness of results obtained from (i): We showthat both (i) and (ii) yield the same qualitative P dependenceof AB LEH parameters (see Sec. V). Of course, it would be023163-11MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)desirable to determine accurately all CP values from Pamb to60 GPa in future experimental works.Note that, at P = 60 GPa, the negative value dzbuck �−0.2 Å obtained for both (i) and (ii) is physical, as discussedbelow. First, the P dependence of dzbuck at P > Popt looksrobust, because it is similar for (i) and (ii) (see Fig. 2). Second,the negative value of dzbuck has a physical origin: the “colli-sion” between the in-plane O in the OP and the Ca cation.Indeed, when P increases, the distance dzCu − dzCa between theOP and Ca cation is reduced (see Fig. 2). If we see the ionsas rigid spheres, the Ca cation “collides” with the in-planeO in the OP, so that the in-plane O is pushed outside of theOP. This explains why dzbuck becomes negative and |dzbuck|increases. In addition, the rigidity of Cu-O-Cu bonds mayplay a role in the increase in |dzbuck|: When a is decreased,|dzbuck| is also increased to prevent the reduction in distancedCu−O = √(a/2)2 + (dzbuck )2 between Cu and in-plane O.2. Hole concentrationNext, we take into account the experimental optimal valuepopt of the hole concentration p, which realizes T expc (theexperimental value of Tc) close to T optc � 138 K at Pamb.Experimentally, hole doping in the CuO2 planes is realizedby introduction of excess oxygen atoms and/or partial substi-tution of atoms, e.g., Hg by Au, so that the chemical formulaof Hg1223 becomes Hg1−xsAuxs Ba2Ca2Cu3O8+δ . In that case,a rough estimate of the total hole concentration is ptot =2δ + xs, which corresponds to the average hole concentrationper CuO2 plane pav = ptot/3 = (2δ + xs)/3.At Pamb, previous studies [3,5,37,38] suggest the optimalvalue of pav is popt � 0.14–0.20. In Ref. [37], the xs de-pendence of T expc is explicitly studied: For δ = 0.3, we haveT expc � 133 K at xs = 0, then T expc decreases with xs, so thatthe maximum value of T expc � 133 K is reached at pav =2δ/3 � 0.2. This value of T expc corresponds to T optc � 138 K[3]. Also, the value popt � 0.2 is consistent with Ref. [38] inwhich T expc � 115–133 K at pav � 0.19–0.24, and also withpopt � 0.19 in Ref. [5]. However, Ref. [3] reports popt � 0.14which corresponds to T optc = 138 K. Thus, the maximal valueof T expc � 133–138 K is realized for experimental popt �0.14–0.20 [3,37]. We checked that the LEH parameters areinsensitive to the variation in pav in the range � 0.14–0.20, asdiscussed below.Thus, in our calculations, we realize pav = 0.2 by realizingptot = 0.6. We do not consider excess oxygen, so that δ = 0.0;instead, we consider xs = 0.6 to compensate the absence ofexcess oxygen and realize ptot = 0.6.Also, we checked that our calculations correspond to op-timal hole doping popt � 0.14–0.20 not only at Pamb but alsounder pressure, which allows a reliable comparison with theP dependence of T optc [3]. According to Ref. [5], popt is re-duced under pressure: We have popt � 0.19 (T optc � 134 K) atPamb but popt � 0.163 (T optc � 150 K) at P = 12 GPa. Linearextrapolation of the above pressure dependence of popt yieldspopt � 0.12 at Popt = 30 GPa. However, we have checked thatthis reduction in popt does not affect substantially the AB LEHparameters. We consider xs = 0.4 to realize pav = 0.133, andcompare with results obtained at pav = 0.2. The values ofTABLE IV. Values of |t l1|, U l , and ul as a function of the averagehole concentration per CuO2 plane pav, at Pamb and Popt. We use theoptimized CP values.P pav |t i1| |t o1 | |t avg1 | U i U o U avg ui uo uavgPamb 0.133 0.526 0.519 0.522 3.97 3.98 3.98 7.55 7.68 7.61Pamb 0.2 0.528 0.523 0.525 3.85 3.79 3.82 7.31 7.26 7.28Popt 0.133 0.596 0.591 0.594 4.10 3.87 3.99 6.88 6.54 6.71Popt 0.2 0.598 0.594 0.596 4.13 3.97 4.05 6.91 6.67 6.79|t l1| and ul at Popt change by only 1%–2% (see Table IV).For completeness, we have also considered pav = 0.133 atPamb: In that case, the values of |t l1| change by only 1% andthe values of ul increase by only 3%–6% with respect topav = 0.2. Thus, the pav dependence of AB LEH parametersis weak, and considering the same value of pav = 0.2 at allpressures is acceptable.3. DFT calculationWe perform the conventional DFT calculation as fol-lows. We use Quantum ESPRESSO [49,50], and optimizednorm-conserving Vanderbilt pseudopotentials (PPs) [51] byemploying the GGA-PBE functional [52] together with thepseudopotentials X_ONCV_PBE-1.0.upf (X = Hg, Au, Ba,Ca, Cu, and O) [53]. The substitution of Hg by Au is doneusing the Virtual Crystal Approximation (VCA) [54]. TheHg1−xsAuxs fictitious atom is abbreviated as “Hg” from nowon. We consider nonmagnetic calculations, a plane wavecutoff energy of 100 Ry for wave functions, a Fermi-Diracsmearing of 0.0272 eV, a 12 × 12 × 12 k-point grid for theBrillouin zone sampling in the self-consistent calculation, anda 8 × 8 × 3 k-point grid and 430 bands for the followingnon-self-consistent calculation.We obtain the GGA band dispersion in Fig. 3. In this banddispersion, the medium-energy (M) space near the Fermi levelis spanned by the 44 Cu3d , O2p, and Hg5d-like bands from−10 eV to +3 eV by defining the origin at the Fermi level.First, we separate the M space from other bands as follows.We compute the 44 atomiclike Wannier orbitals (ALWOs)spanning the M space (denoted as M-ALWOs), as maximallylocalized Wannier orbitals [55,56], using the RESPACK code[30,39]. The initial guesses are d , p, and d atomic orbitals cen-tered, respectively at Cu(l), O(l) (with l = i, o representingthe inner and outer planes, respectively) and at Hg atoms. 44ALWOs are constructed from the GGA band number from no.41 to no. 87, which are numbered from the energy bottom ofthe GGA cutoff. We preserve the band dispersion in the GGAusing the inner energy window from the bottom of the lowestband in the M space [the band in black between −7 eV and−10 eV in Figs. 3(a)–3(l)] to the bottom of the lowest emptyband outside the M space [the dashed band in black betweenthe Fermi level and +2 eV in Figs. 3(a)–3(l)]. Then, the threebands above the 44 M bands are disentangled [57] from thelatter.We obtain the M-ALWOs. They are denoted as (l jR),where R is the coordinate of the unit cell in the space [xyz]expanded in the (a, b, c) frame in Fig. 2, j is the orbital index023163-12DOME STRUCTURE IN PRESSURE DEPENDENCE OF … PHYSICAL REVIEW RESEARCH 6, 023163 (2024)and l is the index (defined in Table I) giving the atom locatedin the cell at R, on which (l jR) is centered. We then expressthe GGA one-particle part h(r) in the M-ALWO basis, ashl,l ′j, j′ (R) =∫�drw∗l j0(r)h(r)wl ′ j′R(r), (B2)in which wl jR is the one-particle wave function of (l jR).From Eq. (B2), we deduce the onsite energy εll = hl,lj, j (0)of the M-ALWO (l j) at any R, and the hopping t l,l ′j, j′ (R) =hl,l ′j, j′ (R) between the M-ALWO (l j0) and the M-ALWO(l ′ j′R). In this paper, we discuss in particular the Cu3dx2−y2and in-plane O2pσ onsite energies and the Cu3dx2−y2/O2pσhopping in the unit cell t lxp = tCu(l ),O(l )x2−y2,pσ. These quantities aregiven in Figs. 3(o)–3(q).4. Low-energy subspaceThen we focus on the L space, which is spanned by theCu3dx2−y2/O2pσ AB band shown in red in Figs. 3(a)–3(g). Toconstruct the AB maximally localized Wannier orbitals, theinitial guesses are the dx2−y2 atomic orbitals centered on eachof the three Cu(l) atoms in the unit cell. The band window isessentially the M space, but we exclude the Nexcl = 14 lowestbands from it to avoid catching the B/NB Cu3dx2−y2/O2pσcharacter. Then, in the band window, we disentangle the 29other bands from the AB band.5. Constrained polarization and effective interactionThen we compute the cRPA polarization at zero frequency.It is expressed as [39][χH]GG′ (q) = − 4Nk∑kempty∑nuoccupied∑no(1 − TnokTnuk+q)MGno,nu(k + q, k)[MG′no,nu(k + q, k)]∗�no,nu (k, q) − iη, (B3)in which q is a wave vector in the Brillouin zone, G, G′ arereciprocal lattice vectors, nk is the Kohn-Sham one-particlestate with energy εnk and wave function ψnk , and Tnk = 1 if nkbelongs to the L space, and Tnk = 0 else. The charge transferenergy�no,nu (k, q) = εnuk+q − εnok (B4)encodes the difference in onsite energies of nuk + q and nok,and the interstate matrix elementMGno,nu(k + q, k) =∫�drψ∗nuk+q(r)ei(q+G)rψnok (r) (B5)encodes the wave functions ψnk , and also encodes the overlapbetween ALWOs since the latter are constructed from ψnk . Wededuce the cRPA effective interaction asWH = (1 − vχH)−1v, (B6)in which v is the bare Coulomb interaction. We deduce theonsite Coulomb repulsion in Eq. (9).APPENDIX C: CORRECTION OF u AND |t1|:IMPROVEMENT FROM THE GGA+CRPA LEVELTO THE cGW -SIC+LRFB LEVELHere we give details on the calculation of xLRFB and yLRFBin Eqs. (2) and (3) which allows us to deduce ucGW −SIC+“LRFB”and |t1|cGW −SIC+“LRFB” in Hg1223. [This corresponds to thecorrection (C) mentioned in Sec. II.]First, we address again the computational load of the directcGW -SIC+LRFB calculation for Hg1223. This calculationrequires the LRFB preprocessing, whose extension to thecuprates with N� = 3 is computationally demanding, becauseone needs to solve the three-orbital Hamiltonian consistingof three CuO2 planes in total by an accurate quantum many-body solver (see Ref. [30] for details) by taking into accountthe inter-CuO2 plane hopping and interaction parameters. Weleave such an extension for future studies. Instead, we employthe procedure (C1) and (C2) mentioned in Sec. II, because italready allows us to reach physically transparent understand-ing.In the procedure (C1), we improve the AB LEH from theGGA+cRPA level to the cGW -SIC level. Since the ratiosucGW −SIC/uavg and |t1|cGW −SIC/|t avg1 | may have strong mate-rials dependence and also pressure dependence, due to thediversity of the global band structure outside of the AB band,we need to perform this procedure with respect to each ma-terial and pressure separately. For instance, in Hg1223, wehave ucGW −SIC/uavg � 1.36 at Pamb and � 1.21 at 30 GPa.The calculated cGW -SIC level of the parameters is shown inTable V; computational details of the cGW -SIC calculationare given at the end of this Appendix.To perform (C2), we employ the material independentconstants xLRFB and yLRFB to correct the cGW -SIC resultsobtained in (C1), because this procedure is only to readjustmainly the onsite Coulomb interaction U and this correctionis materials insensitive. This readjustment arises from thecorrection of the relative chemical potential between the ABand B/NB bands to keep the electron fillings of the Cu3d andO2p orbitals, while the band structure of AB and B/NB bandsby readjusting their chemical potentials and this chemicalpotential shift are indeed material insensitive in the knownfour compounds [30] because of the similar AB and B/NBband structures of the cuprates in general. In fact, our explicitcalculations of xLRFB and yLRFB for several other cuprates(Hg1201, CaCuO2, Bi2201, and Bi2212) show that, near opti-mal hole doping, xLRFB � 0.91–0.97 and yLRFB � 0.99–1.06are rather universal and almost independent of the material.Thus, it may be reasonable to assume that Hg1223 near theoptimal hole doping has similar values of xLRFB of yLRFB, andthe narrow range of uncertainty allows accurate estimation ofthe Hamiltonian parameters.Still, the small uncertainty on xLRFB � 0.91–0.97 causesa possible quantitative error on the P dependence of T estc[see Fig. 1(b)], even though the qualitative dome structure isrobust. We thus narrow the estimate of xLRFB as follows. In023163-13MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)TABLE V. Values of u and |t1| calculated for Hg1201, CaCuO2, Bi2201, Bi2212, and Hg1223, at the average hole concentration perCuO2 plane pav close to the optimal hole concentration. N� is the number of adjacent CuO2 layers sandwiched between block layers. OnHg1201, CaCuO2, Bi2201, and Bi2212, we show the values of ucGW −SIC+LRFB and |t1|cGW −SIC+LRFB at Pamb taken from Ref. [30], and the valuesof ucGW −SIC and |t1|cGW −SIC at Pamb calculated in this paper. The values of xLRFB and yLRFB at Pamb are calculated from Eqs. (2) and (3). OnHg1223, we show the values of uavg and |t avg1 | at the GGA+cRPA level, and the values of ucGW −SIC and |t1|cGW −SIC calculated in this paperat Pamb, 30 GPa and 60 GPa. We also show the values of ucGW −SIC+“LRFB” and |t1|cGW −SIC+“LRFB” estimated from Eqs. (2) and (3) with thechoices of xLRFB = 0.91, 0.95, and 0.97 and yLRFB = 1.0 inferred by analyzing other compounds (see the main text). We also show the valuesof |t1|cGW −SIC+“LRFB” obtained after applying the correction (D).Hg1201 CaCuO2 Bi2201 Bi2212 Hg1223 Hg1223 Hg1223P( GPa) 0 0 0 0 0 30 60pav 0.1 0.1 0.2 0.2 0.2 0.2 0.2N� 1 ∞ 1 2 3 3 31/N� 1 0 1 0.5 0.333 0.333 0.333uavg — — — — 7.22 6.80 6.35ucGW −SIC 8.06 8.39 9.06 9.97 9.83 8.21 7.23ucGW −SIC+LRFB 7.35 8.10 8.34 9.37 — — —xLRFB 0.91 0.97 0.92 0.94 — — —Estimated xLRFB — — — — 0.91/0.95/0.97 0.91/0.95/0.97 0.91/0.95/0.97ucGW −SIC+“LRFB” — — — — 8.95/9.34/9.54 7.47/7.80/7.96 6.58/6.87/7.01|t avg1 | — — — — 0.528 0.596 0.643|t1|cGW −SIC 0.526 0.526 0.498 0.436 0.485 0.569 0.615|t1|cGW −SIC+LRFB 0.544 0.521 0.527 0.451 — — —yLRFB 1.034 0.990 1.058 1.034 — — —Estimated yLRFB — — — — 1.0 1.0 1.0|t1|cGW −SIC+“LRFB” — — — — 0.485 0.569 0.615|t1|cGW −SIC+“LRFB” [after (D)] — — — — 0.485 0.593 0.642Fig. 7 we see a small but systematic linear dependence ofxLRFB on 1/N�. Linear interpolation of the 1/N� dependenceof xLRFB yields xestLRFB = 0.951 � 0.95 at N� = 3. Thus, weassume xLRFB = 0.95 in Hg1223; for completeness, we alsoadmit the range of uncertainty xLRFB � 0.91–0.97. On yLRFB,there is no clear 1/N� dependence of yLRFB, so that we simplyassume yLRFB = 1.0. (Note that the results shown in Fig. 1and Fig. 6 do not depend on the value of yLRFB.) We deducethe values of ucGW −SIC+“LRFB” and |t1|cGW −SIC+“LRFB” that areshown in Table V.The universality of calculated xLRFB and yLRFB may beunderstood as follows. The LRFB corrects the value of �ExpFIG. 7. 1/N� dependence of calculated xLRFB (blue symbols) forHg1201, CaCuO2, Bi2201, and Bi2212 listed in Table V and their lin-ear fitting (red dashed line). Red diamond is the estimate for Hg1223(xestLRFB = 0.951) obtained from the interpolation at 1/N� = 1/3.by an amount �μ whose value is similar for all optimallydoped compounds (we obtain �μ � 1.1–1.4 eV in Ref. [30],Table IV). This universality in �μ is consistent with theuniversality in xLRFB and yLRFB.Note that ucGW −SIC+“LRFB” and |t1|cGW −SIC+“LRFB” arerough estimates of the actual cGW -SIC+LRFB result. In theactual cGW -SIC+LRFB calculation, more complex factorssuch as the self-doping of the IP and OP [38] and the Coulombinteraction between the IP and OP may affect the result ofthe LRFB calculation. (Clarification of these factors is leftfor future studies.) Nonetheless, the simple above estimatesupports the assumption (B) in Sec. I.Computational details of the cGW -SIC scheme. We applythe cGW -SIC scheme to Hg1201, Bi2201, Bi2212, CaCuO2,and Hg1223 as follows. On Hg1201, Bi2201, Bi2212, andCaCuO2, we consider the same computational conditions andhole concentration as in Ref. [30]. On Hg1223, we first pre-process the 44 bands within the M space from the GGA levelto the GW level. (The GW preprocessing is presented in detailin Ref. [30], Appendix A2.) The random phase approximation(RPA) polarization is calculated using 100 real frequenciesand 30 imaginary frequencies; the maximum modulus of thefrequency is 19.8 Ha. The exchange-correlation potential issampled in the real space using a 120 × 120 × 540 grid tosample the unit cell. In the calculation of the GW self-energy,we reduce the computational cost by employing the schemesketched in Ref. [30], Appendix E, with the cutoff energyε = 0.01 eV. Other computational details are the same asthose in Appendix B. We obtain the GW electronic structure,in which the M bands are preprocessed at the GW level and theother bands are left at the GGA level. Then, we derive the AB023163-14DOME STRUCTURE IN PRESSURE DEPENDENCE OF … PHYSICAL REVIEW RESEARCH 6, 023163 (2024)LEH. We start from the GW electronic structure and constructthe AB MLWO. The band window is the M space but weexclude the Nexcl lowest bands from it. (We use Nexcl = 9 atPamb, and Nexcl = 10 at 30 GPa and 60 GPa.) Then we use thecRPA to calculate the two-particle part and U . We also use thecGW to calculate the one-particle part and |t1|. (Details aboutthe cGW scheme can be found in Ref. [30], Appendix A5.)APPENDIX D: CORRECTION OF |t1| BY CORRECTINGTHE CELL PARAMETER aHere we give details about the correction (D) mentioned inSec. I. To correct the P dependence of |t1|, we correct (i) theP dependence of a in Fig. 2, then combine the corrected (i)with (ii) the a dependence of |t1| estimated in Appendix E 2,Eq. (E1).On (i), the P dependence of a is shown in Fig. 2. Theexperimental values of a are available at Pamb [34] and P =8.5 GPa, but not at P > 8.5 GPa. At Pamb, the experimental aand optimized a are in very good agreement (the differenceis � 0.004 Å). However, at P = 8.5 GPa, the optimized aoverestimates the experimental a by � 0.05 Å. We assumethat such an overestimation also happens at P > 8.5 GPa, andwe correct the P dependence of optimized a accordingly. Thevalues of the P-dependent correction �a(P) are �a(P) = 0 Åif P = Pamb and �a(P) = �a = −0.05 Å if P > Pamb, and theP-dependent corrected a is denoted asã(P) = a(P) + �a(P). (D1)On (ii), Eq. (E1) gives |t1|(a) ∝ 1/a3. Combination withEq. (D1) yields|t1|[ã(P)] = |t1|[a(P)]1 + 3�a(P)a(P) + 3[�a(P)a(P)]2+[�a(P)a(P)]3 , (D2)which allows us to determine |t1|[ã(P)] as a function of|t1|[a(P)]. The last two terms in the denominator of Eq. (D2)are negligible because |�a(P)/a(P)| � 0.014   1, so that wehave|t1|[ã(P)] = |t1|[a(P)]1 + 3�a(P)a(P). (D3)[Note that |t1|[ã(P)] � |t1|[a(P)] because �a(P) � 0.]We use Eq. (D3) to correct the P dependence of|t1|cGW −SIC+“LRFB”. The values of |t1|cGW −SIC+“LRFB” thatare obtained after applying (D) are shown in Table V at Pamb,30 GPa and 60 GPa.For completeness, we mention a limitation of the correc-tion (D): It relies on the dependencies (i) and (ii) mentionedabove, and (ii) is determined at the GGA+cRPA level.The only way to improve slightly the approximation in (D)and Eq. (D3) would be to take the optimized CPs at P =30 GPa and reduce the cell parameter a by 0.05 Å (theestimated difference between optimized a and experimentala), then perform explicitly the cGW -SIC calculation from theCP with the reduced a, then deduce |t1|cGW −SIC+“LRFB” anducGW −SIC+“LRFB”. However, this improvement is computation-ally expensive, and we do not expect it to change the resultssignificantly. Thus, we do not consider it here.APPENDIX E: PRESSURE DEPENDENCEOF INTERMEDIATE QUANTITIES1. Pressure dependence of the DFT band structureand Madelung potentialHere, as a complement to Sec. IV, we show that [MW ]is robust with respect to the definition of uniaxial pressureand with respect to the uncertainty on CP values. First, [MW ]is caused by Pbucka rather than Pbuckc (see Fig. 8), which isconsistent with Fig. 3 in which [MW ] is caused by Pa ratherthan Pc: The main origin of [MW ] is indeed the reductionin a, and the variation in dzbuck with Pbucka does not affectthis result. Also, if we use the CP values from Zhang et al.instead of the optimized CP values, [MW ] is well reproduced(see Fig. 9).In addition, we discuss the mechanisms of [Mε] and [MW ]in terms of Madelung potential created by ions in the crystal.As shown in Sec. IV, [Mε] and [MW ] are mainly caused bythe reduction in a. This may be understood as follows. Themain contribution of the Madelung potential felt by electronsin the CuO2 plane is from the positive Cu and negative Oion within the plane. Then the energy of an electron at theCu3d orbital gets higher when the surrounding O ions be-come closer to the Cu site, namely, if a is reduced. On thecontrary, an electron at the O2pσ orbital feels opposite for thereduced a. This makes the difference of the electronic levelsfor the Cu3dx2−y2 and O2pσ larger. More precise calculationincluding long-range Coulomb potential by DFT supports thissimple view is essentially correct.The Pa induced increase in energy of Cu3d bands is illus-trated in Figs. 10(a), 10(b), and 10(c). The application of Paincreases the absolute energy of Cu3d bands. (The absoluteenergy is defined as the energy without renormalization withrespect to the Fermi level.) Note that examining the pressuredependence of absolute energies does make sense, becausethe chemical composition of the crystal is not modified by theapplication of pressure.The application of Pc increases the energy of not only Cu3dbands but also O2p bands in Figs. 10(a), 10(d), and 10(e),so that [MW ] does not occur. This can also be understood interms of Madelung potential from in-plane O anions. When Pcis applied, the distance dzCu between the IP and OP is reduced.This reduces not only (i) the interatomic distance between theO anion in the OP (IP) and the Cu in the IP (OP), but also(ii) the interatomic distance between the O anion in the OP(IP) and the O in the IP (OP), The concomitant reduction in(i) and (ii) causes the concomitant increase in Cu3d and O2pelectronic levels.2. Pressure dependence of the onsite bare interactionand Cu3dx2−y2/O2pσ charge transfer energyHere, as a complement to Sec. V B, we discuss the follow-ing points:(a) The increase in onsite bare interaction v is caused bythe reduction in Cu3dx2−y2/O2pσ hybridization when �Expincreases.(b) The concomitant increases in |t1| and v when adecreases can be understood by further analysis of the a de-pendence of quantities.023163-15MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)FIG. 8. Pbucka and Pbuckc dependence of the GGA band structure obtained using the optimized CP values. Notations are the same as those inFig. 3.(c) The reduction in the correlation between vl and �Elxpat P > Popt originates from the nonequivalence of the IP andOP, especially the buckling of Cu-O-Cu bonds in the OP. }On (a), a first remark is that the Cu3dx2−y2/O2pσ hy-bridization reduces v by reducing the atomic Cu3dx2−y2character of the AB orbital. In the AB orbital, the onsitebare interaction is vavg � 14.5–15.5 eV, but in the Cu3dx2−y2M-ALWO, the onsite bare interaction vavgx � 25.5 eV is larger[see Fig. 11(a)]. This is because the Cu3dx2−y2 M-ALWO hasatomic character and is more localized than the AB orbital.In the limit of zero hybridization, the AB orbital is equivalentto the Cu3dx2−y2 M-ALWO if we neglect the effect of otherorbitals: In that case, vavg = vavgx . However, the hybridizationis always nonzero in the realistic cuprate, so that the atomicCu3dx2−y2 character of the AB orbital is reduced.Second, the importance of the Cu3dx2−y2/O2pσ hybridiza-tion decreases with P. The importance of the hybridizationis roughly encoded in the ratio Oxp = |txp|/�Exp betweenthe Cu3dx2−y2/O2pσ hopping amplitude and Cu3dx2−y2/O2pσcharge transfer energy. Oxp decreases when the hybridizationis reduced and becomes zero when the hybridization is negli-gible. And Oavgxp decreases with P [see Fig. 11(b)].Thus, the atomic Cu3dx2−y2 character of the AB orbitalincreases with P: We interpret this as the origin of the increasein v. To confirm this, we show explicitly that vavg � vavgx in thelimit of zero hybridization. Let us consider the Oavgxp depen-dence of vavg in Fig. 11(c), which is obtained by combiningthe P dependencies of vavg and Oavgxp in Figs. 11(a) and 11(b).In the ab initio calculation, we have Oavgxp � 0.7–0.8, so thatthe Oxp dependence of vavg can be explicitly obtained onlywithin this range. However, the value of vavg in the limit ofzero hybridization may be estimated by performing a linearextrapolation of the ab initio Oavgxp dependence of vavg. Theextrapolation yields vavg � 24 eV at Oavgxp = 0 [see Fig. 11(c)],which is similar to vavgx � 25.5 eV. This suggests the ABorbital becomes the Cu3dx2−y2 M-ALWO in the limit of zerohybridization, as mentioned above.On (b), we analyze the a dependence of |t avg1 |, �E avgxpand |t avgxp |, as well as the average values Oavgxp and T avgxp ofOlxp = |t lxp|/�Elxp and T lxp = |t lxp|2/�Elxp; the key point is that,when a decreases, Oavgxp decreases whereas |t avg1 | ∝ T avgxp in-creases. To obtain the a dependence of the above quantities,we take the values of a as a function of pressure in Fig. 2and combine them with the Pa dependence of |t l1|, �Elxp and|t lxp| in Figs. 5(a), 5(f), and 5(g). Note that we consider thePa dependence instead of the P dependence. This is becausethe application of Pa modifies only the value of a: This allowsto extract accurately the a dependence while avoiding the dzldependence of quantities.We interpolate the a dependencies of the above quantities,using the fitting function f (a) = Caβ , where β and C are thefitting parameters. We examine the values of β, which encodethe speed of variation in quantities with a. The obtained valuesof β are shown in Fig. 12.First, we have ∣∣t avg1∣∣ ∝ T avgxp ∝ 1/a3. (E1)Indeed, the value of β for |t avg1 | is β(|t avg1 |) � −2.88, whichis very close to −3. This is consistent with the 1/r3 decay ofthe density-density correlation function [58]. Also, β(T avgxp ) �−2.89 is almost identical to β(|t avg1 |).Second, we haveOavgxp ∝ a. (E2)FIG. 9. P dependence of the GGA band structure obtained using the CP values from Zhang et al. Notations are the same as those in Fig. 3.023163-16DOME STRUCTURE IN PRESSURE DEPENDENCE OF … PHYSICAL REVIEW RESEARCH 6, 023163 (2024)FIG. 10. Pa and Pc dependencies of the band structure and Fermienergy. We show the bands inside the M space (solid black color)and outside M space (dotted black color). These band structurescorrespond to those in Figs. 3(h), 3(i), 3(k), 3(l), and 3(n), exceptthat the band energies are not renormalized with respect to the Fermienergy. The latter is given by the horizontal line in red color.Indeed, the Oavgxp dependence of a is almost linear: β(Oavgxp ) =0.98 is very close to 1.The above equations (E1) and (E2) show that both |t i1|and vi increase when a decreases. Indeed, in item (a), wehave clarified that vavg increases when Oavgxp decreases, andEq. (E2) shows that Oavgxp decreases when a decreases. Notethat β(�E avgxp ) < β(|t avgxp |) < 0: This is the origin of thepositive value of β(Oavgxp ). On the other hand, 2β(|t avgxp |) <β(�E avgxp ) < 0: This is the origin of the negative value ofβ(|t avg1 |) � β(T avgxp ).On (c), the nonequivalence of the IP and OP causes aslight difference in the �Elxp dependence of vl for l = i andl = o [see Figs. 4(e), 4(f), 4(l), and 4(m) at P > Popt]. Thisis simply because |t oxp| is reduced at P > Popt due to theFIG. 11. (a) P dependence of the average value vavg over theIP and OP of the onsite bare interaction. We show vavg in the AB(black dots) and Cu3dx2−y2 (green dots) Wannier orbitals. (b) Pdependence of the average value Oavgxp of the ratio Olxp = |t lxp|/�Elxp,which encodes the Cu3dx2−y2/O2pσ hybridization. (c) Oavgxp depen-dence of vavg in the AB (black dots) and Cu3dx2−y2 (green dots)Wannier orbitals. The dashed lines show the linear extrapolationto Oavgxp = 0 (the limit in which the Cu3dx2−y2/O2pσ hybridizationbecomes negligible).FIG. 12. Cell parameter a dependence of |t avg1 |, �E avgxp , |t avgxp |, andthe average values Oavgxp and T avgxp of Olxp = |t lxp|/�Elxp and T lxp =|t lxp|2/�Elxp (dots). We start from the optimized CP values at Pamb andapply Pa to modify only the value of a. The dashed curves show theinterpolation of the a dependence by the function f (a) = Caβ , whereβ and C are the fitting parameters. The legend shows the obtainedvalues of β.increase in |dzbuck| [see Fig. 4(g)]. This contributes to reduceOoxp = |t oxp|/�Eoxp, which increases vo [see item (a)]. Thisexplains why, at P > Popt, vo � vi even though �Eoxp < �Eixp[see Figs. 4(e) and 4(f)]. On the other hand, if we apply onlyPa (which modifies only a without modifying dzbuck), |t oxp| isnot reduced with respect to |t ixp| [see Fig. 4(g)], and the �Elxpdependence of vl is very similar for l = i and l = o [seeFigs. 4(e) and 4(f)].The nonequivalence of the IP and OP also causes a slightdifference in the P dependence of �Elxp for l = i and l = o inFigs. 4(f) and 4(m). This is because �Elxp depends not only ona, but also on dzl [see Fig. 13(f) in Appendix F]. For instance,�Eixp (�Eoxp) increases (decreases) when dzCa decreases. And,dzCa � 1.48 Å in the optimized CP values is smaller than dzCa �1.59 Å in the CP values from Zhang et al. This is why �Eixp >�Eoxp for the optimized CP values, but �Eixp < �Eoxp for thevalues from Zhang et al. [see Figs. 4(f) and 4(m)].3. Pressure dependence of the screeningHere, as a complement to Sec. V C, we discuss the uniaxialpressure dependence of Ravg, from which the dome structurein the uniform pressure dependence of Ravg originates. } First,we discuss the Pa dependence of Ravg in Fig. 5(d).(i) At Pa < Popt, the increase in Ravg is explained by thebroadening [MW ] of the GGA band dispersion. More pre-cisely, the origin is the increase in charge transfer energies(B4) (schematically denoted as � in this Appendix) betweenoccupied and empty bands, due to [MW ] discussed in Sec. IV.The increase in � participates in the decrease of the amplitude023163-17MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)FIG. 13. CP dependencies of |t l1|, ul , U l , Rl = U l/vl , vl , �Elxp and |t lxp|. We show quantities in the IP (l = i) and OP (l = o) in red andblue, respectively. The quantities are obtained using the optimized CP values at Popt = 30 GPa, and modifying the values of a, c, and dzl by�a, �c, and �dzl , respectively. Note that when �dzCu is applied, �c = 2�dzCu is also applied so that all interatomic distances in the block layerremain unchanged. The horizontal dashed lines represent the values at Pamb for comparison.of the cRPA polarization (B3), schematically denoted as |χ | ∝1/�. This reduces the cRPA screening and thus increasesRavg.(ii) At Pa > Popt, the increase in Ravg ceases. This is be-cause the effect of [MW ] is progressively reduced: We have∂|χ |/∂� ∝ −1/�2, so that the larger Pa and thus �, thesmaller the decrease in |χ | when � is further increased, andthe less important the effect of [MW ]. In addition, when Paincreases, the charge transfer energy �M−empty between the Mbands and empty bands outside M space is reduced, becausethe energy of the Cu3d bands increases [see Figs. 10(a), 10(b),and 10(c)]. This may contribute to increase |χ | ∝ 1/�M−emptyand cancel the effect of [MW ] at high pressure.Second, we discuss the decrease in the Pc dependence ofRavg in Fig. 5(d). This is because [MW ] does not occur whenPc is applied, contrary to Pa. Thus, � does not increase. On theother hand, �M−empty is reduced because the energy of Cu3dbands increases [see Figs. 10(a), 10(d), and 10(e)]. As a result,|χ | ∝ 1/�M−empty increases.APPENDIX F: CRYSTAL PARAMETER DEPENDENCEOF EFFECTIVE HAMILTONIAN PARAMETERSAT OPTIMAL PRESSUREHere, as a complement to Sec. V, we analyze the CPdependencies of AB LEH parameters around Popt. We startfrom the optimized CP values at Popt and modify separatelythe values of each CP. The modified values are given in Fig. 2(open squares). The CP dependencies of AB LEH parametersare shown in Fig. 13.We summarize the main results below:(i) As for |t l1|, the a dependence is the strongest.(ii) As for ui, the dzCa and dzCu dependencies are thestrongest.023163-18DOME STRUCTURE IN PRESSURE DEPENDENCE OF … PHYSICAL REVIEW RESEARCH 6, 023163 (2024)(iii) As for uo, the dzCa and dzO(ap) dependencies are thestrongest.Also, (ii) and (iii) suggest the origin of the decrease in Rlat P > Pscr in Sec. V C, Figs. 4(d) and 4(k): The decreases inRi and Ro are caused, respectively, by the decreases in dzCu anddzO(ap).a dependence of AB LEH parameters. At Popt, the optimizedvalue of a � 3.69 Å is the same as that from Zhang et al.. Still,this value might be overestimated. Indeed, the P dependenceof experimental values [34,35] shows the faster decrease atlower pressures (see Fig. 2). Thus, we consider the modifica-tion �a of a at Popt, such that −0.05 Å � �a � 0 Å at Popt.The a dependence of |t1| is strong [see Fig. 13(a)], asdiscussed in Sec. V A. We note that the 15% increase in |t1|from Pamb to Popt becomes 18%–19% if �a = −0.05 Å. Thus,the 3% difference between the increase in |t1| and that in T optcmay be understood by admitting the above uncertainty on a atPopt (see the discussion in Sec. VI).dzCa dependence of AB LEH parameters. The optimizedvalue dzCa � 1.48 Å is lower than that from Zhang et al.(dzCa � 1.59 Å). Thus, we consider 0.0 Å � �dzCa � +0.2 Åto examine the dzCa dependence of AB LEH parameters.Increasing dzCa causes the rapid decrease in ui and in-crease in uo [see Fig. 13(b)], due to the decrease in �Eixpand increase in �Eoxp [see Fig. 13(f)]. Indeed, vl and Rlare correlated with �Elxp. The correlation between vl and�Elxp has been discussed in Appendix E 2, and the increasein �Elxp also contributes to increase Rl by reducing thecRPA screening between Cu3dx2−y2/O2pσ B/NB and ABbands.The increase (decrease) of �Elxp originates from the pos-itive Madelung potential created by the Ca cation, whichstabilizes electrons in the vicinity of the Ca cation. WhendzCa increases, the Ca cation becomes closer to (farther from)the O atoms in the OP (IP). Thus, the O2pσ orbitals in theIP (OP) are destabilized (stabilized) [see Fig. 14(b)]. TheCu3dx2−y2 orbitals are also destabilized, but less than O2pσorbitals because Cu atoms are farther from Ca compared toin-plane O. The above simple view is supported by the factthat the variation in εlpσwith dzCa and the variation in LEHparameters with dzCa are twice faster in the IP compared tothe OP [see Fig. 14(b) and Fig. 13]. This is because the IPis surrounded by twice as many Ca cations than the OP (seeFig. 2). However, note that the average values of LEH parame-ters do not vary substantially, because the �dzCa dependenciesof LEH parameters in the IP and OP compensate each other.This explains why the increase in �E avgxp from Pamb to Poptoriginates from Pa rather than Pc (see Sec. V B).dzCu dependence of AB LEH parameters. The optimizedvalue dzCu � 2.82 Å is lower than that from Zhang et al.(dzCu � 2.91 Å). Thus, we consider 0.0 Å � �dzCu � +0.2 Åto examine the dzCu dependence of AB LEH parameters.Increasing dzCu causes the rapid increase in ui [seeFig. 13(b)], due to the decrease in both vi and Ri [seeFigs. 13(d), 13(e)].The decrease in vi is caused by the decrease in �Eixp [seeFigs. 13(e), 13(f)]. �Eixp decreases because the in-plane O an-ions in the OP become farther from those in the IP. As a result,the O2pσ electrons in the IP are stabilized [see Fig. 14(c)],FIG. 14. CP dependencies of the Fermi energy εF and onsiteenergies εli of the Cu3dx2−y2 and O2pσ ALWOs. The quantities areobtained using the optimized CP values at Popt = 30 GPa, and mod-ifying the values of a, c, and dzl by �a, �c, and �dzl , respectively.Note that when �dzCu is applied, �c = 2�dzCu is also applied so thatall interatomic distances in the block layer remain unchanged. Inpanel (e), we also show the dzO(ap) dependence of the onsite energyεO(ap)pzof the apical O2pz orbital.because the Madelung potential from O anions in the OP isweaker.However, the decrease in �Eixp may not be sufficient toexplain the decrease in Rl . We see that from �dzCu = 0.0 Åto �dzCu = −0.2 Å, Ro slightly decreases and Ri sharply de-creases [see Fig. 13(d)]. The decrease in Ro is not consistentwith the increase in �Eoxp which contributes to increase Ro;also, the decrease in Ri is very sharp compared to the smoothdecrease in �Eixp.Instead, the decrease in Rl may be caused by an increasein cRPA screening between adjacent CuO2 planes. This is in-tuitive because �dzCu = −0.2 Å reduces the distance betweenthe CuO2 planes in the real space. This increases the overlapand hybridization between M-ALWOs in the IP and OP, whichmay increase the cRPA screening (see also the discussionabout the dzO(ap) dependence of the screening below). Theinterplane cRPA screening particularly affects the IP, becausethe IP is adjacent to two OPs, whereas the OP is adjacent toonly the IP; this explains the sharp decrease in Ri.dzBa dependence of AB LEH parameters. The optimizedvalue dzBa � 1.96 Å is similar to that from Zhang et al.(dzBa � 1.98 Å). Still, for completeness, we consider −0.2 Å� �dzBa � 0.0 Å in order to examine the dzBa dependence ofAB LEH parameters.Decreasing dzBa does not cause a significant variation in ul[see Fig. 13(b)]. Still, we note that vo and �Eoxp slightly in-crease [see Figs. 13(e) and 13(f)]. This is because the positiveMadelung potential from Ba cation felt by the OP is stronger(see the above discussion on the dzCa dependence of �Elxp).023163-19MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)FIG. 15. (a) �dzO(ap) dependence of the partial density of states ofthe apical O2pz M-ALWO. (b) �dzO(ap) dependence of the amplitude|tO(o),O(ap)(o)pσ ,pz| of the apical O 2pz/in-plane O2pσ hopping. The quan-tities are obtained using the optimized CP values at Popt = 30 GPa,and modifying the value of dzO(ap) by �dzO(ap).Note that the positive Madelung potential from Ba cation doesnot affect the IP, because the IP is separated from the Ba cationby the OP (see Fig. 2).dzO(ap) dependence of AB LEH parameters. The optimizedvalue dzO(ap) � 2.22 Å is slightly lower than that from Zhanget al. (dzO(ap) � 2.32 Å). Thus, we consider 0.0 Å� �dzO(ap) �+0.2 Å to examine the dzO(ap) dependence of AB LEH param-eters. In addition, we consider −0.2 Å � �dzO(ap) � 0.0 Å toprobe the effect of apical O displacement at higher pressures.In the dzO(ap) dependence of uo [see Fig. 13(b)], there is asharp decrease in uo when dzO(ap) decreases. This decrease hasalso been observed in the case of Bi2201 and Bi2212 [30]. Ithas two origins: (i) the decrease in vo due to the decrease in�Eoxp [see Figs. 13(e) and 13(f)], and more prominently (ii)the decrease in Ro [see Fig. 13(d)]. (ii) is due to the cRPAscreening of AB electrons by the apical O, and this screeningincreases when dzO(ap) decreases as in Bi2201 and Bi2212 [30].Note that, contrary to Ro, Ri does not decrease significantlywhen dzO(ap) decreases: This is because the IP is protected fromthe cRPA screening from apical O by the OP, which separatesthe IP from the apical O (see Fig. 2).A possible origin of (ii) is the increase in hybridizationbetween the apical O 2pz orbital and AB orbital in the OP. Weshow in Fig. 15(a) the partial density of states of the apicalO2pz M-ALWO. We see that bands at the Fermi level haveslight apical O2pz character, in addition to the dominant ABcharacter. This originates from the hybridization between theAB orbital and the apical O2pz orbital. The apical O2pz partialdensity of states at Fermi level increases when dzO(ap) de-creases, which suggests the increase in hybridization betweenapical O 2pz and AB orbitals. This is further supported bythe increase in amplitude |tO(o),O(ap)(o)pσ ,pz | of the apical O 2pz/in-plane O2pσ hopping when dzO(ap) decreases [see Fig. 15(b)],because the AB orbital is partly constructed from the in-planeO2pσ orbital.[1] J. G. Bednorz and K. A. Müller, Possible high Tc superconduc-tivity in the Ba-La-Cu-O system, Z. Phys. B 64, 189 (1986).[2] J. B. Torrance, Y. Tokura, S. J. LaPlaca, T. C. Huang, R. J.Savoy, and A. I. Nazzal, New class of high Tc structures: In-tergrowth of multiple copper oxide perovskite-like layers withdouble sheets of BiO, Solid State Commun. 66, 703 (1988).[3] L. Gao, Y. Y. Xue, F. Chen, Q. Xiong, R. L. Meng, D. Ramirez,C. W. Chu, J. H. Eggert, and H. K. Mao, Superconductivity upto 164 K in HgBa2Cam−1CumO2m+2+δ (m = 1, 2, and 3) underquasihydrostatic pressures, Phys. Rev. B 50, 4260 (1994).[4] P. Dai, B. C. Chakoumakos, G. F. Sun, K. W. Wong, Y. Xin, andD. F. Lu, Synthesis and neutron powder diffraction study of thesuperconductor HgBa2Ca2Cu3O8+δ by Tl substitution, PhysicaC: Supercond. 243, 201 (1995).[5] A. Yamamoto, N. Takeshita, C. Terakura, and Y. Tokura, Highpressure effects revisited for the cuprate superconductor fam-ily with highest critical temperature, Nat. Commun. 6, 8990(2015).[6] M. Nuñez-Regueiro, J. L. Tholence, E. V. Antipov, J. J.Capponi, and M. Marezio, Pressure-induced enhancement of Tcabove 150 K in Hg-1223, Science 262, 97 (1993).[7] M. T. Schmid, J.-B. Morée, R. Kaneko, Y. Yamaji, andM. Imada, Superconductivity studied by solving ab initiolow-energy effective Hamiltonians for carrier doped CaCuO2,Bi2Sr2CuO6, Bi2Sr2CaCu2O8, and HgBa2CuO4, Phys. Rev. X13, 041036 (2023).[8] M. F. Crommie, A. Y. Liu, A. Zettl, M. L. Cohen, P. Parilla,M. F. Hundley, W. N. Creager, S. Hoen, and M. S. Sherwin,c-axis stress dependence of normal and superconducting stateproperties of YBa2Cu3O7, Phys. Rev. B 39, 4231 (1989).[9] C. Meingast, O. Kraut, T. Wolf, H. Wühl, A. Erb, and G.Müller-Vogt, Large a-b anisotropy of the expansivity anomalyat Tc in untwinned YBa2Cu3O7−δ , Phys. Rev. Lett. 67, 1634(1991).[10] G. L. Belenky, S. M. Green, A. Roytburd, C. J. Lobb, S. J.Hagen, R. L. Greene, M. G. Forrester, and J. Talvacchio, Effectof stress along the ab plane on the Jc and Tc of YBa2Cu3O7 thinfilms, Phys. Rev. B 44, 10117 (1991).[11] U. Welp, M. Grimsditch, S. Fleshler, W. Nessler, J. Downey,G. W. Crabtree, and J. Guimpel, Effect of uniaxial stress on thesuperconducting transition in YBa2Cu3O7, Phys. Rev. Lett. 69,2130 (1992).[12] C. Meingast, J. Karpinski, E. Jilek, and E. Kaldis, Uniaxialpressure dependence of Tc of single crystalline YBa2Cu4O8via thermal expansion, Physica C: Supercond. 209, 591(1993).[13] M. Mito, T. Imakyurei, H. Deguchi, K. Matsumoto, T. Tajiri,H. Hara, T. Ozaki, H. Takeya, and Y. Takano, Uniaxial straineffects on cuprate superconductor YBa2Cu4O8, J. Phys. Soc.Jpn. 81, 113709 (2012).[14] M. Mito, H. Matsui, T. Imakyurei, H. Deguchi, T. Horide, K.Matsumoto, A. Ichinose, and Y. Yoshida, Effects of selectivelattice deformation on YbBa2Cu4O8 and YBa2Cu3O7 epitaxialfilms, Appl. Phys. Lett. 104, 102601 (2014).[15] M. Mito, T. Imakyurei, H. Deguchi, K. Matsumoto, H. Hara,T. Ozaki, H. Takeya, and Y. Takano, Effective disappearance ofthe Meissner signal in the cuprate superconductor YBa2Cu4O8under uniaxial strain, J. Phys. Soc. Jpn. 83, 023705 (2014).[16] M. Mito, H. Goto, H. Matsui, H. Deguchi, K. Matsumoto,H. Hara, T. Ozaki, H. Takeya, and Y. Takano, Uniaxial strain023163-20https://doi.org/10.1007/BF01303701https://doi.org/10.1016/0038-1098(88)90987-8https://doi.org/10.1103/PhysRevB.50.4260https://doi.org/10.1016/0921-4534(94)02461-8https://doi.org/10.1038/ncomms9990https://doi.org/10.1126/science.262.5130.97https://doi.org/10.1103/PhysRevX.13.041036https://doi.org/10.1103/PhysRevB.39.4231https://doi.org/10.1103/PhysRevLett.67.1634https://doi.org/10.1103/PhysRevB.44.10117https://doi.org/10.1103/PhysRevLett.69.2130https://doi.org/10.1016/0921-4534(93)90580-Jhttps://doi.org/10.1143/JPSJ.81.113709https://doi.org/10.1063/1.4867907https://doi.org/10.7566/JPSJ.83.023705DOME STRUCTURE IN PRESSURE DEPENDENCE OF … PHYSICAL REVIEW RESEARCH 6, 023163 (2024)effects on superconducting transition in Y0.98Ca0.02Ba2Cu4O8,J. Phys. Soc. Jpn. 85, 024711 (2016).[17] F. Hardy, N. J. Hillier, C. Meingast, D. Colson, Y. Li, N. Barišić,G. Yu, X. Zhao, M. Greven, and J. S. Schilling, Enhance-ment of the critical temperature of HgBa2CuO4+δ by applyinguniaxial and hydrostatic pressure: Implications for a universaltrend in cuprate superconductors, Phys. Rev. Lett. 105, 167002(2010).[18] M. Mito, K. Ogata, H. Goto, K. Tsuruta, K. Nakamura, H.Deguchi, T. Horide, K. Matsumoto, T. Tajiri, H. Hara et al.,Uniaxial strain effects on the superconducting transition inRe-doped Hg-1223 cuprate superconductors, Phys. Rev. B 95,064503 (2017).[19] S. N. Putilin, E. V. Antipov, O. Chmaissem, and M. Marezio,Superconductivity at 94 K in HgBa2CuO4+δ , Nature (London)362, 226 (1993).[20] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys.Rev. 136, B864 (1964).[21] W. Kohn and L. J. Sham, Self-consistent equations includ-ing exchange and correlation effects, Phys. Rev. 140, A1133(1965).[22] F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S.Biermann, and A. I. Lichtenstein, Frequency-dependent localinteractions and low-energy effective models from electronicstructure calculations, Phys. Rev. B 70, 195104 (2004).[23] F. Aryasetiawan, K. Karlsson, O. Jepsen, and U. Schönberger,Calculations of Hubbard U from first principles, Phys. Rev. B74, 125106 (2006).[24] M. Imada and T. Miyake, Electronic structure calculation byfirst principles for strongly correlated electron systems, J. Phys.Soc. Jpn. 79, 112001 (2010).[25] M. Hirayama, T. Miyake, and M. Imada, Derivation of staticlow-energy effective models by an ab initio downfoldingmethod without double counting of Coulomb correlations: Ap-plication to SrVO3, FeSe, and FeTe, Phys. Rev. B 87, 195144(2013).[26] M. Hirayama, T. Misawa, T. Miyake, and M. Imada, Ab initiostudies of magnetism in the iron chalcogenides FeTe and FeSe,J. Phys. Soc. Jpn. 84, 093703 (2015).[27] M. Hirayama, Y. Yamaji, T. Misawa, and M. Imada, Ab initioeffective Hamiltonians for cuprate superconductors, Phys. Rev.B 98, 134501 (2018).[28] M. Hirayama, T. Misawa, T. Ohgoe, Y. Yamaji, and M. Imada,Effective Hamiltonian for cuprate superconductors derivedfrom multiscale ab initio scheme with level renormalization,Phys. Rev. B 99, 245155 (2019).[29] T. Ohgoe, M. Hirayama, T. Misawa, K. Ido, Y. Yamaji, and M.Imada, Ab initio study of superconductivity and inhomogene-ity in a Hg-based cuprate superconductor, Phys. Rev. B 101,045124 (2020).[30] J.-B. Morée, M. Hirayama, M. T. Schmid, Y. Yamaji, andM. Imada, Ab initio low-energy effective Hamiltoniansfor the high-temperature superconducting cupratesBi2Sr2CuO6, Bi2Sr2CaCu2O8,HgBa2CuO4, and CaCuO2,Phys. Rev. B 106, 235150 (2022).[31] M. Hirayama, M. T. Schmid, T. Tadano, T. Misawa, and M.Imada, Ab init io material design of Ag-based oxides for high-Tcsuperconductor, arXiv:2207.12595.[32] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.023163 for the complete list ofparameters for the effective Hamiltonians presented in the maintext.[33] X. Zhang, W. H. Lu, and C. K. Ong, Structural changes inHgBa2Ca2Cu3O8+δ under high pressure, Physica C: Supercond.289, 99 (1997).[34] A. R. Armstrong, W. I. F. David, I. Gameson, P. P. Edwards,J. J. Capponi, P. Bordet, and M. Marezio, Crystal structureof HgBa2Ca2Cu3O8+δ at high pressure (to 8.5 GPa) deter-mined by powder neutron diffraction, Phys. Rev. B 52, 15551(1995).[35] B. A. Hunter, J. D. Jorgensen, J. L. Wagner, P. G. Radaelli,D. G. Hinks, H. Shaked, R. L. Hitterman, and R. B. VonDreele, Pressure-induced structural changes in superconductingHgBa2Can−1CunO2n+2+δ (n = 1, 2, 3) compounds, Physica C:Supercond. 221, 1 (1994).[36] J. H. Eggert, J. Z. Hu, H. K. Mao, L. Beauvais, R. L. Meng,and C. W. Chu, Compressibility of the HgBa2Can−1CunO2n+2+δ(n = 1, 2, 3) high-temperature superconductors, Phys. Rev. B49, 15299 (1994).[37] P. Bordet, S. Le Floch, J. J. Capponi, C. Chaillout, M. F.Gorius, M. Marezio, J. L. Tholence, and P. G. Radaelli, Goldsubstitution in mercury cuprate superconductors, Physica C:Supercond. 262, 151 (1996).[38] H. Kotegawa, Y. Tokunaga, K. Ishida, G.-Q. Zheng, Y. Kitaoka,H. Kito, A. Iyo, K. Tokiwa, T. Watanabe, and H. Ihara, Unusualmagnetic and superconducting characteristics in multilayeredhigh-Tc cuprates: 63Cu NMR study, Phys. Rev. B 64, 064515(2001).[39] K. Nakamura, Y. Yoshimoto, Y. Nomura, T. Tadano, M.Kawamura, T. Kosugi, K. Yoshimi, T. Misawa, and Y.Motoyama, RESPACK: An ab initio tool for derivation of ef-fective low-energy model of material, Comput. Phys. Commun.261, 107781 (2021).[40] H. Sakakibara, K. Suzuki, H. Usui, K. Kuroki, R. Arita,D. J. Scalapino, and H. Aoki, Multiorbital analysis of theeffects of uniaxial and hydrostatic pressure on Tc in the single-layered cuprate superconductors, Phys. Rev. B 86, 134520(2012).[41] K. Momma and F. Izumi, VESTA3 for three-dimensional visu-alization of crystal, volumetric and morphology data, J. Appl.Crystallogr. 44, 1272 (2011).[42] T. Misawa, S. Morita, K. Yoshimi, M. Kawamura, Y.Motoyama, K. Ido, T. Ohgoe, M. Imada, and T. Kato, mVMC—open-source software for many-variable variational MonteCarlo method, Comput. Phys. Commun. 235, 447 (2019).[43] D. Wu, R. Rossi, F. Vicentini, N. Astrakhantsev, F. Becca, X.Cao, J. Carrasquilla, F. Ferrari, A. Georges, M. Hibat-Allahet al., Variational benchmarks for quantum many-body prob-lems, arXiv:2302.04919.[44] K. Held, C. Taranto, G. Rohringer, and A. Toschi, Hedin equa-tions, GW, GW+DMFT, and all that, arXiv:1109.3972.[45] F. Krien, A. Kauch, and K. Held, Tiling with triangles: Parquetand GW γ methods unified, Phys. Rev. Res. 3, 013149 (2021).[46] B. Cunningham, M. Grüning, D. Pashov, and M. vanSchilfgaarde, QS GŴ : Quasiparticle self-consistent GW withladder diagrams in W , Phys. Rev. B 108, 165104 (2023).[47] S. Acharya, C. Weber, E. Plekhanov, D. Pashov, A. Taraphder,and M. Van Schilfgaarde, Metal-insulator transition in copperoxides induced by apex displacements, Phys. Rev. X 8, 021038(2018).023163-21https://doi.org/10.7566/JPSJ.85.024711https://doi.org/10.1103/PhysRevLett.105.167002https://doi.org/10.1103/PhysRevB.95.064503https://doi.org/10.1038/362226a0https://doi.org/10.1103/PhysRev.136.B864https://doi.org/10.1103/PhysRev.140.A1133https://doi.org/10.1103/PhysRevB.70.195104https://doi.org/10.1103/PhysRevB.74.125106https://doi.org/10.1143/JPSJ.79.112001https://doi.org/10.1103/PhysRevB.87.195144https://doi.org/10.7566/JPSJ.84.093703https://doi.org/10.1103/PhysRevB.98.134501https://doi.org/10.1103/PhysRevB.99.245155https://doi.org/10.1103/PhysRevB.101.045124https://doi.org/10.1103/PhysRevB.106.235150https://arxiv.org/abs/2207.12595http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.023163https://doi.org/10.1016/S0921-4534(97)01461-5https://doi.org/10.1103/PhysRevB.52.15551https://doi.org/10.1016/0921-4534(94)90659-9https://doi.org/10.1103/PhysRevB.49.15299https://doi.org/10.1016/0921-4534(96)00224-9https://doi.org/10.1103/PhysRevB.64.064515https://doi.org/10.1016/j.cpc.2020.107781https://doi.org/10.1103/PhysRevB.86.134520https://doi.org/10.1107/S0021889811038970https://doi.org/10.1016/j.cpc.2018.08.014https://arxiv.org/abs/2302.04919https://arxiv.org/abs/1109.3972https://doi.org/10.1103/PhysRevResearch.3.013149https://doi.org/10.1103/PhysRevB.108.165104https://doi.org/10.1103/PhysRevX.8.021038MORÉE, YAMAJI, AND IMADA PHYSICAL REVIEW RESEARCH 6, 023163 (2024)[48] S. Acharya, M. I. Katsnelson, and M. van Schilfgaarde, Vertexdominated superconductivity in intercalated FeSe, npj QuantumMater. 8, 24 (2023).[49] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R.Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M.Cococcioni, I. Dabo et al., QUANTUM ESPRESSO: Amodular and open-source software project for quantum sim-ulations of materials, J. Phys.: Condens. Matter 21, 395502(2009).[50] P. Giannozzi Jr, O. Andreussi, T. Brumme, O. Bunau, M. B.Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M.Cococcioni et al., Advanced capabilities for materials mod-elling with QUANTUM ESPRESSO, J. Phys.: Condens. Matter29, 465901 (2017).[51] M. Schlipf and F. Gygi, Optimization algorithm for the genera-tion of ONCV pseudopotentials, Comput. Phys. Commun. 196,36 (2015).[52] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradientapproximation made simple, Phys. Rev. Lett. 77, 3865 (1996).[53] http://www.quantum-espresso.org[54] L. Nordheim, The electron theory of metals, Ann. Phys. 401,607 (1931).[55] N. Marzari and D. Vanderbilt, Maximally localized generalizedWannier functions for composite energy bands, Phys. Rev. B56, 12847 (1997).[56] I. Souza, N. Marzari, and D. Vanderbilt, Maximally localizedWannier functions for entangled energy bands, Phys. Rev. B 65,035109 (2001).[57] T. Miyake, F. Aryasetiawan, and M. Imada, Ab initio procedurefor constructing effective models of correlated materials withentangled band structure, Phys. Rev. B 80, 155134 (2009).[58] T. Misawa and M. Imada, Quantum criticality around metal-insulator transitions of strongly correlated electron systems,Phys. Rev. B 75, 115121 (2007).023163-22https://doi.org/10.1038/s41535-023-00556-9https://doi.org/10.1088/0953-8984/21/39/395502https://doi.org/10.1088/1361-648X/aa8f79https://doi.org/10.1016/j.cpc.2015.05.011https://doi.org/10.1103/PhysRevLett.77.3865http://www.quantum-espresso.orghttps://doi.org/10.1002/andp.19314010507https://doi.org/10.1103/PhysRevB.56.12847https://doi.org/10.1103/PhysRevB.65.035109https://doi.org/10.1103/PhysRevB.80.155134https://doi.org/10.1103/PhysRevB.75.115121