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Rei Nishinakayama, [Yoshiki J. Sato](https://orcid.org/0000-0002-1750-0373), [Takayoshi Yamanaka](https://orcid.org/0000-0001-5330-0400), [Yoshiteru Maeno](https://orcid.org/0000-0002-3467-9416), [Hiroshi Yaguchi](https://orcid.org/0000-0002-9595-9083), [Naoki Kikugawa](https://orcid.org/0000-0003-3975-4478), [Ryuji Okazaki](https://orcid.org/0000-0001-5234-4110)

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[Negatively enhanced thermopower near a Van Hove singularity in electron-doped Sr2RuO4](https://mdr.nims.go.jp/datasets/4f14b3d7-3d52-46a4-93c3-47942e58f516)

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Negatively enhanced thermopower near a Van Hove singularity in electron-doped ${\rm Sr}_2{\rm RuO}_4$PHYSICAL REVIEW RESEARCH 6, 023222 (2024)Negatively enhanced thermopower near a Van Hove singularity in electron-doped Sr2RuO4Rei Nishinakayama,1,* Yoshiki J. Sato ,1,† Takayoshi Yamanaka ,2 Yoshiteru Maeno ,3 Hiroshi Yaguchi ,1Naoki Kikugawa ,4 and Ryuji Okazaki 1,‡1Department of Physics and Astronomy, Tokyo University of Science, Noda 278-8510, Japan2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan3Toyota Riken-Kyoto University Research Center (TRiKUC), Kyoto 606-8501, Japan4National Institute for Materials Science, Tsukuba 305-0003, Ibaraki, Japan(Received 28 November 2023; revised 14 April 2024; accepted 16 April 2024; published 31 May 2024)The layered perovskite Sr2RuO4 serves as a model material of the two-dimensional (2D) Fermi liquidbut also exhibits various emergent phenomena including the non-Fermi-liquid (NFL) behavior under externalperturbations such as uniaxial pressure and chemical substitutions. Here we present the thermoelectric transportof electron-doped system Sr2−yLayRuO4, in which a filling-induced Lifshitz transition occurs at the Van Hovesingularity (VHS) point of y ≈ 0.2. We find that the sign of the low-temperature thermopower becomes negativeonly near the VHS point, where the NFL behavior has been observed in the earlier study. This observation isincompatible with either a numerical calculation within a constant relaxation-time approximation or a toy-modelcalculation for the 2D Lifshitz transition adopting an elastic carrier scattering. As a promising origin of theobserved negatively enhanced thermopower, we propose a skewed NFL state, in which an inelastic scatteringwith a considerable odd-frequency term plays a crucial role to negatively enhance the thermopower.DOI: 10.1103/PhysRevResearch.6.023222I. INTRODUCTIONBeyond the well-established Fermi-liquid (FL) picture,non-Fermi-liquid (NFL) state has been intensively investi-gated as an essential concept for various quantum phenomena[1–7]. Besides the well-known Tomonaga-Luttinger liquidin one dimension, the prototypical NFL state appears in avicinity of the quantum critical point (QCP), in which athermodynamic phase transition into an ordered state is sup-pressed by tuning external parameters. In the NFL state nearQCP, the finite-temperature properties such as the electronicspecific heat and the electrical resistivity drastically deviatefrom the FL behavior, as widely seen in correlated metalsincluding transition-metal oxides and heavy fermions.A yet unsolved, fundamental issue is how the FL pictureis modified in a vicinity of the Lifshitz transition [8], an elec-tronic topological transition associated with the change in thetopology of the Fermi surfaces. The Lifshitz transition itselfis ubiquitous; it is driven by various parameters such as pres-sure [9–11], magnetic field [12–15], band filling [16–18], and*6222526@alumni.tus.ac.jp†Present address: Graduate School of Science and Engineering,Saitama University, Saitama 338-8570, Japan; yoshikisato@mail.saitama-u.ac.jp‡okazaki@rs.tus.ac.jpPublished 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.even temperature through the temperature-dependent chemi-cal potential [19–22]. In spite of many observed examples, thekinetic properties near the Lifshitz transition are complex andstill controversial owing to a peculiar energy-dependent relax-ation time [23,24], which should become more complicated incorrelated metals.To tackle this problem, here we focus on the quasi-two-dimensional (q-2D) FL material Sr2RuO4 [25–28]. Indeed, itsq-2D Fermi surfaces consisting of hole-like α and electron-like β and γ sheets have been accurately verified by the dHvAand the ARPES experiments [29–32]. The calculated Fermisurfaces of Sr2RuO4 are shown in Fig. 1(a). The normal-statenature in such a q-2D multiband system is well understoodwithin the FL picture [26,33,34], whereas the pairing mech-anism of the superconducting state is still an unsolved issue[35–39]. Significantly, its superconducting transition temper-ature Tc is enhanced to Tc ≈ 3.5 K under compressional stress[40–43], and near the critical compression point in which Tchas a maximum value. Such variation of Tc corresponds to thesharp peak in the density of states (DOS) associated with aVan Hove singularity (VHS) point; such a topology changein the γ band is indeed observed by the ARPES [44]. Mostimportantly, the resistivity clearly deviates from the FL behav-ior near the VHS point [45,46], and also nontrivial electronicstates such as entropic anomaly [47] have been observed nearthe Lifshitz transition [48–50]. Thus, this layered materialoffers a suitable platform to investigate the NFL nature nearthe Lifshitz transition.In this paper, we report a thermopower study of theelectron-doped system Sr2−yLayRuO4 [51–53], in which afilling-induced Lifshitz transition occurs near the critical con-centration yc ≈ 0.2 [31]. The calculated Fermi surfaces for2643-1564/2024/6(2)/023222(10) 023222-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-1750-0373https://orcid.org/0000-0001-5330-0400https://orcid.org/0000-0002-3467-9416https://orcid.org/0000-0002-9595-9083https://orcid.org/0000-0003-3975-4478https://orcid.org/0000-0001-5234-4110https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.6.023222&domain=pdf&date_stamp=2024-05-31https://doi.org/10.1103/PhysRevResearch.6.023222https://creativecommons.org/licenses/by/4.0/REI NISHINAKAYAMA et al. PHYSICAL REVIEW RESEARCH 6, 023222 (2024)FIG. 1. The cross-sectional view of the calculated Fermi surfacesof Sr2−yLayRuO4 at kz = 0 plane for (a) y = 0, (b) y = yc, and(c) y > yc, drawn by using FermiSurfer program [63].y = yc and y > yc are depicted in Figs. 1(b) and 1(c), re-spectively; the topology of the γ sheet changes at y = yc.In Sr2−yLayRuO4, similar electronic features including NFLtransport [51] and the effective mass enhancement [31] havebeen clearly observed near y = 0.2, offering a complementaryapproach toward such an intriguing issue on the Lifshitz tran-sition. We find that the low-temperature thermopower dependson the La content y and that the sign of the thermopowerbecomes negative only near the VHS point. This is in contrastto the results of the Hall effect measurements in which theHall coefficient exhibits no significant anomaly near yc [52].We also show that the present experimental results cannot beexplained either by numerical calculation results within a con-stant relaxation-time approximation or by a simple model forthe 2D neck-disruption-type Lifshitz transition with an elasticcarrier scattering. Instead, we propose a skewed NFL state[54] as a promising explanation for our results, in which anodd-frequency inelastic scattering is considered. The skewedNFL state is indeed a unique state of matter, as it has beenstudied as a nature of strange metal in cuprate superconductorsand may also have a relevance to the transport propertiesin twisted bilayer graphene [54]. This skewed NFL statestrengthens the electron-hole asymmetry owing to the dom-inant odd-frequency term in the scattering rate, providing acrucial role for negatively enhanced thermopower near theLifshitz transition.II. EXPERIMENTALSingle crystals of Sr2−yLayRuO4 were grown by a floating-zone method [51–53]. Typical dimension of the single crystalsis 3 × 1 × 0.1 mm3. The in-plane thermopower was measuredby a steady-state technique using a manganin-constantan dif-ferential thermocouple in a closed-cycle refrigerator [55,56].A typical temperature gradient of 0.5 K/mm, which isadjusted along with the bath temperature, was appliedalong the in-plane direction using a resistive heater andthe distance between the thermocouple contacts is about1 mm. The thermoelectric voltage from the wire leads wassubtracted.III. RESULTS AND DISCUSSIONFigure 2(a) shows the temperature dependence of the in-plane thermopower S of Sr2−yLayRuO4. It is known that Tcdecreases with La substitution and is completely suppressedfor y greater than 0.04 [51]. For the parent compound, theFIG. 2. (a) Temperature dependence of the thermopower S ofSr2−yLayRuO4 (0 � y � 0.27) measured along the ab plane direc-tion. (b) Temperature dependence of dS/dT . The arrows show thecharacteristic temperature T ∗ below which the dS/dT deviates fromthe linear temperature dependence of dS/dT at higher temperatures.The data are shifted vertically for clarity. (c) Temperature depen-dence of S/T . (d) The La content y dependence of S/T measuredat 4 K (solid circles, left axis), T ∗ (open squares, right axis), and TM(open circles, right axis). TM is defined at the peak temperature in themagnetic susceptibility [53]. The blue-dotted line is a guide to the eyeto represent the y dependence of these characteristic temperatures.The vertical-dashed line show the critical La content yc.present results well agree with those of the thermopower inprevious reports [56–60]. The thermopower of Sr2RuO4 hasalso been studied by the dynamical mean-field theory [61]. In023222-2NEGATIVELY ENHANCED THERMOPOWER NEAR A VAN … PHYSICAL REVIEW RESEARCH 6, 023222 (2024)the La-substituted compounds, overall behavior of the ther-mopower is similar to that of the parent crystal.The thermopower in Sr2RuO4 was analyzed in the dif-ferential form dS/dT [Fig. 2(b)] to examine a characteristictemperature, and an anomaly was found near T ∗ ≈ 25 K,below which dS/dT increases with decreasing temperature[57]. Subsequently, through the Seebeck and the Nernst mea-surements, Xu et al. have suggested that the coherence isdeveloped below T ∗ [58]. As displayed in the right axis ofFig. 2(d), T ∗ systematically decreases with the La content y. Itshould be noted that the magnetic susceptibility of Sr2RuO4 isPauli paramagnetic, but the temperature dependence exhibitsa small peak structure at TM ≈ 30 K [26], below which theFL picture is well defined. The La content y dependence ofTM taken from Ref. [53] is also plotted in the right axis ofFig. 2(d). Notably, both TM and T ∗ show similar y depen-dence, indicating that the characteristic temperature belowwhich the coherence is formed in the correlated carriers de-creases with increasing y. This trend is consistent with theNFL behavior near y = 0.2 [51] and also signifies the inherentrole of the carrier scattering at the Lifshitz transition, as willbe discussed later.In Fig. 2(c), we also plot S/T of Sr2−yLayRuO4 as afunction of T . The low-temperature S/T behavior notablydepends on both temperature and the La content y, suggestingthe considerable change in the Fermi surfaces in the presentLa content range as reported in earlier reports [31,51–53]. Itshould be noted that negative thermopower is found at lowtemperatures only for the y = 0.2 crystal, which is near thecritical La content yc [31,51–53]. Figure 2(d) displays the Lacontent y dependence of S/T obtained at 4 K for the left axis.With increasing y, S/T slightly increases and a singularity isclearly observed near y ≈ 0.2. Note that a positive value ofS/T is recovered at the higher content y = 0.27. To explain theobserved La content dependence of S/T , we have examinedthe chemical potential dependence of S/T calculated within aconstant relaxation-time approximation [62] (see Appendix),but the calculated data [Fig. 6(b)] is positively enhanced nearthe VHS points. Obviously, this discrepancy originates fromthe energy dependence of the relaxation time, which is ig-nored in the constant relaxation-time approximation method.Here we discuss the energy dependence of the relaxationtime τ (ε) near the Lifshitz transition. Similar to the caseof Sr2RuO4, the 2D neck-disruption-type Lifshitz transitionfor the cylindrical Fermi surfaces [Fig. 3(a)] has been in-vestigated [64–67]. In this case, the energy dependence ofthe DOS D(ε) shows logarithmic divergence near the criticalenergy Ec at which the Lifshitz transition occurs [Fig. 3(b)].Such logarithmic DOS behavior is also confirmed by thenumerical calculation for Sr2RuO4 [Fig. 6(a) in Appendix].Then, through an elastic impurity scattering, the scatteringprobability acquires a correction of the energy dependence of1/τ (ε) ∝ D(ε) [23,24], which significantly affects the energydependence of the conductivity function σ (ε) � D0v20τ . Notethat the DOS D0 and the velocity v0 in this conductivity func-tion exhibit weak-energy dependence because these mainlycome from the electrons in the regular parts of the Fermisurfaces, which are far from the VHS points [68]. Figure 3(c)shows the calculated electrical conductivity for this simplemodel (see Appendix). The horizontal axis is the chemicalFIG. 3. 2D neck-disruption-type Lifshitz transitions and thephysical properties. (a) The Fermi surface shape and (b) the DOSare symmetric around the critical energy Ec at which the Lifshitztransition occurs. (c) The conductivity and (d) the thermopoweras a function of the chemical potential, which corresponds to theamount of electron doping by La substitution, for several tempera-tures. HT and LT represent high and low temperatures, respectively.The vertical-dotted lines show Ec and the horizontal-dashed line in(d) represents the contributions from the regular part of the Fermisurfaces, which is far from the VHS points.potential and corresponds to the amount of electron doping byLa substitution. As a consequence, the electrical conductivityσ decreases near Ec and such a modification leads to a NFL-like resistivity of ρ(T ) = ρ0 + AT n with n < 2 for Sr2RuO4[46], as observed near the VHS point for both La-substituted[51] and uniaxially compressed [45] cases.In this model, however, as shown in Fig. 3(d), the ther-mopower should exhibit positive and negative peaks with thesame magnitudes below and above Ec, respectively [64–66],as indicated from the Mott formula [69]ST∝ − 1σ∂σ∂ε∼ − 1τ∂τ∂ε, (1)where the energy dependence of τ is crucial as similar to thecase of the conductivity as mentioned before. It should benoted that such thermopower behavior with positive and neg-ative peaks is also obtained in the numerical calculations forSr2RuO4 in the elastic impurity scattering regime [46]. In con-trast, for Sr2−yLayRuO4, the low-temperature thermopowerseems to be enhanced only negatively near the critical con-tent yc ≈ 0.2 [Fig. 2(d)], which is difficult to explain withsuch a conventional neck disruption case. Also, as indicatedin Fig. 2(d), an inelastic electron-electron scattering, not in-cluded in the model of Fig. 3, should be crucial near yc.To discuss the origin of the negatively enhanced ther-mopower near the VHS point, we next consider a phe-nomenological model adopting a skewed NFL state [54],in which an asymmetric inelastic scattering rate 1/τin ∝(πT )νg(ω/T ) characterized by a noneven scaling func-tion g(x) = |
(z)|2 cosh(x/2)/[cosh(α/2)
{(1 + ν)/2}2] be-comes essential, where 
(z) is the 
 function, z = (1 +ν)/2 + i(x + α)/2π , ω = ε − μ is a relative energy from thechemical potential μ, ν (� 1) is an exponent, and α is anparameter to induce the asymmetry in the scattering rate.Figure 4 shows the examples of a scaling function g(ω/T )with ν = 1 for a symmetric (with an asymmetry parameter023222-3REI NISHINAKAYAMA et al. PHYSICAL REVIEW RESEARCH 6, 023222 (2024)FIG. 4. Scaling function g(ω/T ) for an exponent ν = 1 and anasymmetry parameter α = 0 and 1.5 [54]. The inelastic scatteringrate 1/τin is given as 1/τin ∝ (πT )νg(ω/T ).α = 0) case and an asymmetric (α = 1.5) case [54]. In theasymmetric case, owing to the odd-frequency term of thescattering rate, the contribution of either electrons or holesbecomes stronger, and most importantly, the thermopower, asensitive probe to the electron-hole asymmetry, is enhancedeither negatively or positively. We infer that, along with theobservation of the NFL resistivity near the VHS point [51], theobserved negative sign of the thermopower may be a hallmarkof such a skewed NFL state. Moreover, such unexpected signchange in the thermopower has also been discussed in theNFL regime of cuprate superconductors [70]. Note that thethermopower is also known as a sensitive probe for the en-tropic properties such as magnetic fluctuations but the presentnonmagnetic system may not be adapted to such situation. Atthis stage, it is not easy to make a more quantitative calcula-tion in this model, and further theoretical study is necessaryto quantitatively account for the the sign change. Also, thefrequency-dependent experiments such as the optical conduc-tivity measurement will be crucial to examine the skewed NFLstate.Lifshitz transitions in correlated matter thus bring an in-triguing issue on the NFL state. In this context, Sr2RuO4,in which a variety of electronic and magnetic states emergeunder external perturbations [71–75], is of peculiar interestbecause it also exhibits the NFL behavior in Ti-substitutedsystem [76]. Moreover, recent experimental and theoreticalstudies have revealed the appearance of enigmatic NFL statessuch as a Planckian metal characterized by a linear temper-ature dependence of the resistivity [77,78] and the quantumcritical phase in frustrated materials [79,80], deepening un-derlying physics of the NFL state of matter.IV. SUMMARYIn summary, we have measured the thermopower ofthe electron-doped system Sr2−yLayRuO4 and observed anunusual sign change in the thermopower near the Lifshitz tran-sition at the VHS point yc ≈ 0.2. We discuss the thermopowerin a skewed NFL state, in which an asymmetric frequencydependence of the inelastic scattering rate is crucial, possiblyleading to qualitative explanation for the negative sign of thethermopower near yc. The present results thus offer a fascinat-ing playground to investigate a variety of quantum phenomenaof the correlated electrons near the Lifshitz transition.ACKNOWLEDGMENTSWe appreciate Y. Fukumoto and R. Kurihara for discussionand R. Otsuki, H. Shiina, and R. Taira for the assistance. Thiswork was partly supported by JSPS KAKENHI Grants No.17H06136, No. 21H01033, No. 22H01166, No. 22H01168,and No. 22K19093.APPENDIX1. First-principles calculationsIn order to investigate the thermopower theoretically,we performed first-principles calculations based on den-sity functional theory (DFT) using Quantum Espresso[81–83]. We used the projector-augmented-wave pseu-dopotentials with the Perdew-Burke-Ernzerhof generalized-gradient-approximation (PBE-GGA) exchange-correlationfunctional. The cut-off energies for plane waves and chargedensities were set to 70 and 560 Ry, respectively, and thek-point mesh was set to 20 × 20 × 20 uniform grid to ensurethe convergence. Using the obtained eigenvalues of the nthband at k point En,k, the DOS D(ε) = ∑n,k δ(ε − En,k) wasobtained using the optimized tetrahedron method [84], whereδ is the delta function. We used on-site Coulomb energyU = 3.5 eV and exchange parameter J = 0.6 eV for Ru ions[85], and performed full relativistic calculations with spin-orbit coupling (DFT + U + SOC).Figure 5(a) shows the calculated electronic band structurenear the Fermi energy EF of the parent material, which wellcoincides with the results in earlier studies [86–89]. The de-picted k path is shown in Fig. 5(b). The β and γ bands at thehigh-symmetry points 
 and Z split owing to the inclusionof SOC [88], and the eg bands are shifted upward due to theon-site U . The calculated DOS is depicted in Fig. 6(a), andthe Van Hove singularity (VHS) point of the γ band to showthe cusp anomaly is shifted slightly from E ≈ 50 meV to E ≈30 meV by including U + SOC. Such a trend is consistentwith the results of ARPES experiment [31,32].2. Calculated transport properties within a constantrelaxation-time approximationTo examine the experimentally observed singular behav-ior in S/T , we have calculated the thermopower from theelectronic band structure within the constant relaxation-timeapproximation. Note that the correlation effect of 4d elec-trons is considerable in Sr2RuO4 and the calculation resultswith local-density approximation are quantitatively differentfrom the experimental observations [61]. On the other hand,the thermopower behavior in correlated metals is well de-scribed within a Fermi-liquid picture, where the correlationeffect is included in the carrier effective mass [69]. Also, in023222-4NEGATIVELY ENHANCED THERMOPOWER NEAR A VAN … PHYSICAL REVIEW RESEARCH 6, 023222 (2024)FIG. 5. (a) Calculated band structure of Sr2RuO4 with DFT +U + SOC scheme (solid curves). The dashed curves represent theresults of scalar relativistic calculations and U is not included. TheVan Hove singularity is at the M point, at which the γ band is slightlyabove EF. (b) High-symmetry points in the Brillouin zone.Sr2−yLayRuO4, the electron-doping effect by La substitutionis well explained within a rigid band picture [31,51]. We thusexamine how the band structure affects the thermopower.The transport coefficients were calculated based on the lin-earized Boltzmann equations under constant relaxation-timeapproximation [62]. The transport distribution function tensorLi j (ε) is calculated asLi j (ε) =∑nLni j (ε) =∑n∑kviv jτδ(ε − En,k ), (A1)where Lni j (ε) is the partial transport distribution function ten-sor of the n-th band, vi is the i-th component of the bandvelocity v = 1h̄∇kEn,k, and τ is the relaxation time. We calcu-lated the partial electrical conductivity tensor of the n-th bandof σ ni j (μ) = e2∫ ∞−∞ dε(− ∂ f0∂ε)Lni j , where e is the elementarycharge and f0 is the Fermi-Dirac distribution function for thechemical potential μ and temperature T . The total electricalconductivity tensor is σi j (μ) = ∑n σ ni j (μ). Similarly, the par-tial Peltier conductivity tensor of nth band Pni j (μ) = [σS]ni j (μ)isPni j (μ) = − eT∫ ∞−∞dε(−∂ f0∂ε)(ε − μ)Lni j, (A2)where Si j is the thermopower tensor. The total Peltier conduc-tivity is given as Pi j (μ) = ∑n Pni j (μ). Hereafter we considerthe in-plane component (i j = aa) only and the subscript willbe omitted. The thermopower of the n-th band is then obtainedas Sn = Pn/σ n, and the total in-plane thermopower is given asS = P/σ = ∑n Pn/∑n σ n as is generally seen in multibandsystems.FIG. 6. (a) Calculated total and partial density of states withDFT + U + SOC scheme (solid-filled curves). The dashed curvesrepresent the results of scalar relativistic calculations and U is notincluded. (b) Thermopower divided by temperature, S/T , calculatedfor several temperatures within the relaxation-time approximation.The horizontal axis shows the chemical potential measured from theFermi energy of the parent compound. The black curves represent thetotal S/T and the red, blue, and green curves show the band-resolveddata Sn/T for the α, β, and γ bands, respectively. Varying μ acrossthe VHS results in the enhancement of the positive S/T .Figure 6(b) depicts the thermopower divided by temper-ature, S/T , calculated for several temperatures within theconstant relaxation-time approximation. The black curves rep-resent the total S/T and the red, blue, and green curvesshow the band-resolved Sn/T for n = α, β, γ , respec-tively. The horizontal axis is the chemical potential measuredfrom the Fermi energy of the parent compound and corre-sponds to the amount of electron doping by La substitution.Note that the calculated values of S/T are significantly smallerthan the experimental data because the electron correlationeffect is not accurately included in this scheme and shouldbe modified by using more precise methods such as the dy-namical mean field theory [61]. Nevertheless, characteristicfeatures reflecting the Lifshitz transition in Sr2RuO4 may beobserved in the present calculations; the band-resolved Sn/Tshows divergent behavior at low temperatures near 0.3 eV (βband) and 30 meV (γ band) corresponding to the VHS pointsof DOS for each band [Fig. 6(a)], while Sn/T exhibits almostno temperature dependence for the α band like a simple metal.023222-5REI NISHINAKAYAMA et al. PHYSICAL REVIEW RESEARCH 6, 023222 (2024)FIG. 7. Two types of the Lifshitz transitions for 3D case and thephysical properties: (a)–(d) void formation and (e)–(h) neck disrup-tion. For each panel, the dashed lines represent the contributions fromthe regular part of the Fermi surfaces. The vertical-dotted lines showthe critical energy Ec at which the Lifshitz transition occurs. HT andLT represent high and low temperatures, respectively.In general, the VHS points of DOS strongly affect the ther-mopower [90]. It is now important that the experimental dataof S/T seems to negatively diverge near the critical concen-tration [Fig. 2(d) in the main text], while the calculated datais positively enhanced near the VHS points [Fig. 6(b)]. Thisdiscrepancy obviously originates from the energy dependenceof the relaxation time ignored in the constant relaxation-timeapproximation.3. Peculiar energy-dependent relaxation time and thermopowernear the Lifshitz transitionHere we briefly review the three-dimensional (3D) case tosee the significance of the scattering process. In the 3D case,the Lifshitz transition is categorized into two types of topo-logical change in the Fermi surface called void formation andneck disruption, which are schematically shown in Figs. 7(a)and 7(e), respectively [8]. In the void formation case, for ex-ample, the DOS behaves as D(ε) − D0 ∼ |ε − Ec|1/2, whereD0 is the DOS from the regular part of the Fermi surface (largevoid) and Ec is the critical energy above which a new voidappears [Fig. 7(b)]. At first glance, such a small void withalmost zero carrier velocity seems to give no contribution tothe transport properties. Through the scattering process, how-ever, electrons in the regular part exchange the momenta withthose in the singular part (small void) and get virtually into thesingular part [23,24,68]. As a result, according to the goldenrule, the scattering probability acquires a correction of theenergy dependence of 1/τ (ε) ∝ D(ε), and then the electricalconductivity σ ∝ τ decreases above Ec [Fig. 7(c)]. Note thatthe energy dependence of τ is essential here as an approximateform σ (μ) � e2∫ ∞−∞ dε(− ∂ f0∂ε)Dv2τ � e2D0v20τ (μ), becausethe DOS D0 and the velocity v0 in the conductivity mainlycome from the electrons in the large void [68].According to the Mott formula, the thermopower S is givenasST∝ − 1σ∂σ∂ε∼ − 1τ∂τ∂ε, (A3)where the energy dependence of τ is crucial as similar tothe case of the conductivity, and thus it shows a sharp peakstructure at low temperatures [Fig. 7(d)]. The similar situationoccurs in the case of the neck disruption [Figs. 7(e)–7(h)] andthe thermopower is also enhanced positively near the criticalpoint. Note that the singularities in σ and S are smeared withincreasing temperature. These thermoelectric singularities in3D case have been experimentally observed in the Li-Mg alloy[91].4. Calculations of the transport propertiesfor the Lifshitz transitionsHere we show the calculation details for the transportcoefficients near the Lifshitz transition by using the energy-dependent scattering time. The electrical conductivity σ andthe Peltier conductivity P = σS are given as[σP]=[e2− eT] ∫ ∞−∞dε(−∂ f0∂ε)[1ε − μ]L (A4)= e4kBT 2∫ ∞−∞dωcosh2(βω/2)[eT−ω]L, (A5)where ω = ε − μ is the relative energy measured from thechemical potential. Using the energy-dependent scatteringtime, the transport function near the Lifshitz transition isapproximately given asL = D0v20τ (ε), (A6)and the transport coefficients are given as[σP]= eD0v204kBT 2∫ ∞−∞dωcosh2(βω/2)[eT−ω]τ (ω), (A7)where the scattering time is model dependent as describedbelow.For the 3D void formation case in Figs. 7(a)–7(d), thedensity of states is expressed asD(ε) ∼ D0 + a|ε − Ec|1/2θ (ε − Ec), (A8)where a (> 0) is a constant and θ is the Heaviside step func-tion, as is shown in Fig. 7(b). The scattering time is then givenasτ (ε) ∼ D(ε)−1 (A9)∼ D0 − a|ε − Ec|1/2θ (ε − Ec) (A10)= D0 − a|ω + Z|1/2θ (ω + Z ), (A11)023222-6NEGATIVELY ENHANCED THERMOPOWER NEAR A VAN … PHYSICAL REVIEW RESEARCH 6, 023222 (2024)where Z = μ − Ec is the chemical potential measured fromthe critical energy. The transport coefficients are now cal-culated and the thermopower S is give as S = P/σ . Notethat although the regular part D0 also depends on the energy[23,24], the energy dependence of the singular part is muchsignificant. Indeed, the calculation results shown in Fig. 7 aresimilar to the earlier studies.For the 3D neck disruption case [Figs. 7(e)–7(h)], the scat-tering time is given asτ (ε) ∼ D0 + a|Ec − ε|1/2θ (Ec − ε) (A12)= D0 + a| − ω − Z|1/2θ (−ω − Z ), (A13)which is similar to the case of void formation.For the symmetric 2D neck disruption case [Figs. 3(a)–3(d) in the main text], the density of states near the Lifshitztransition is given as a logarithmic form ofD(ε) ∼ lnt|ε − Ec| , (A14)where t (> 0) is a constant as is shown in Fig. 3(b) in the maintext. The scattering time is given asτ (ε) ∼(lnt|ε − Ec|)−1=(lnt|ω + Z|)−1. (A15)[1] J. A. Hertz, Quantum critical phenomena, Phys. Rev. B 14, 1165(1976).[2] A. J. Millis, Effect of a nonzero temperature on quantum criticalpoints in itinerant fermion systems, Phys. Rev. B 48, 7183(1993).[3] G. R. 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