Antiferromagnetism induced by electron-phonon-coupling
AAntiferromagnetism induced by electron-phonon-coupling
Xun Cai, Zi-Xiang Li,
2, 3 and Hong Yao
1, 4, ∗ Institute for Advanced Study, Tsinghua University, Beijing, 100084, China. Department of Physics, University of California, Berkeley, CA 94720, USA. Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China. (Dated: February 22, 2021)Antiferromagnetism (AF) such as Neel ordering is often closely related to Coulomb interactionssuch as Hubbard repulsion in two-dimensional (2D) systems. Whether Neel AF ordering in 2D canbe dominantly induced by electron-phonon couplings (EPC) has not been completely understood.Here, by employing numerically-exact sign-problem-free quantum Monte Carlo (QMC) simulations,we show that optical Su-Schrieffer-Heeger (SSH) phonons with frequency ω and EPC constant λ caninduce AF ordering for a wide range of phonon frequency ω > ω c . For ω < ω c , a valence-bond-solid(VBS) order appears and there is a direct quantum phase transition between VBS and AF phasesat ω c . The phonon mechanism of the AF ordering is related to the fact that SSH phonons directlycouple to electron hopping whose second-order process can induce an effective AF spin exchange.Our results shall shed new lights to understanding AF ordering in correlated quantum materials. Introduction:
Electron-phonon coupling (EPC) ex-ists ubiquitously in quantum materials. Moreover, itplays a crucial role in driving various exotic quantumphenomena, including charge-density wave (CDW) or-der [1, 2], Su-Schrieffer-Heeger topological state [3, 4],and, most notably, BCS superconductivity (SC) [5, 6].Since EPC normally induces an effective attraction be-tween electrons, it has been well understood theoreticallythat EPC induces charge-density-wave, bond-density-wave, or conventional superconductivity in quantum sys-tems, which was further illustrated in recent works [7–17].Nonetheless, the role of EPC in driving antiferromag-netism (AF) and unconventional SC in strongly corre-lated systems has been under debate since it is widely be-lieved that repulsive Coulomb interactions between elec-trons are essential in developing AF and unconventionalSC (including high-temperature SC) [18–23].In the past many years, increasing experimental andtheoretical studies suggest that EPC can be an essen-tial ingredient in understanding high-temperature SC, in-cluding cuprates [24–41] and iron-based superconductors[42–53], and in driving exotic orders such as pair-density-wave [54], raising renewed interests in studying the roleof EPC in correlated quantum systems. Since unconven-tional SC is often closely related to antiferromagnetism[18–23], it is natural to ask whether EPC can play anyessential role in driving AF ordering. Although variousprevious works have studied the competition between AForder dominantly induced by the Hubbard interactionsand other types of orders such as CDW induced by EPC[55–70], whether AF ordering can be induced dominantlyby EPC remains elusive.In this Letter, we fill in the gap by convincingly show-ing that an AF insulator can be dominantly induced byphonons. Specifically, we systematically study the squarelattice Su-Schrieffer-Heeger (SSH) optical phonon modelat half filling by performing large-scale quantum Monte
FIG. 1. The quantum phase diagram of the square latticeoptical SSH model at half filling as a function of dimension-less electron-phonon-coupling (EPC) constant λ and phononfrequency ω . The insets depict AF and VBS orders. Theresults are obtained by large-scale sign-problem-free zero-temperature QMC simulations. Carlo (QMC) simulations [71, 72]. The simulations ofthe model can be rendered sign-problem-free so that wecan access large system sizes to reach reliable results [73–79] (for a recent review of sign-free QMC, see Ref. [80]).Although acoustic and optical SSH phonon models havebeen studied by various theoretical and numerical ap-proaches [81–95], it hasn’t been shown that AF orderingcan be dominantly triggered by SSH phonons. By per-forming the first state-of-the-art zero-temperature QMCsimulation on the 2D SSH model of optical phonons withfrequency ω , we are able to obtain its reliable ground-state phase diagram as a function of ω and EPC constant λ , revealing that the AF ordering emerges in a large por-tion of the phase diagram, as shown in Fig. 1. To thebest of our knowledge, it is the first time that an AFinsulator is shown, in an numerically-exact way, to bedominantly driven by EPC rather than by Coulomb re- a r X i v : . [ c ond - m a t . s t r- e l ] F e b pulsions between electrons. We would like to emphasizethat the phonon mechanism of AF ordering is intimatelyrelated to the fact that SSH phonons couple to electronhopping whose second-order process can induce an effec-tive AF spin exchange and drive an AF ordering, as weshall explain below. Model:
We consider the optical SSH model on thesquare lattice with the following Hamiltonian H = − t (cid:88) (cid:104) ij (cid:105) ( c † iσ c jσ + h.c. ) + (cid:88) (cid:104) ij (cid:105) ˆ P ij M + K X ij + g (cid:88) (cid:104) ij (cid:105) ˆ X ij ( c † iσ c jσ + h.c. ) , (1)where (cid:104) ij (cid:105) refers to the bond between nearest-neighbor(NN) sites i and j , c † iσ creates an electron on site i with spin polarization σ = ↑ / ↓ , ˆ X ij and ˆ P ij are the dis-placement and momentum operators of the optical SSHphonons on the NN bond (cid:104) ij (cid:105) . The chemical potential µ is implicit in the Hamiltonian and hereafter we shall focuson the case of half-filling by setting µ = 0. Here t is theelectron hopping amplitude and SSH phonon frequencyis ω = (cid:112) K/M . The displacement field of SSH phononsis linearly coupled to electron’s NN hopping rather thanto electron density. The strength of EPC can be char-acterized by the dimensionless EPC constant λ ≡ g /KW ,where W = 8 t is the characteristic band width of thesquare lattice. Hereafter, we set t = 1 as energy unit andset K = 1 by appropriately redefining ˆ X ij .It is worth noting that the optical SSH phononmodel at half filling described by Eq. (1) respects theSO(3) ⊗ SO(3) ⊗ Z ⊗ Z symmetry, which is equivalentto O(4) symmetry [96]. Here the first SO(3) refersto spin rotational symmetry, the second SO(3) pseu-dospin symmetry [97], the first Z the usual particle-hole symmetry for both spin-up and spin-down elec-trons ( c iσ → ( − i c † iσ ), and the second Z the particle-hole symmetry for spin-down electrons ( c i ↓ → ( − i c † i ↓ ).The pseudospin rotation can transform the CDW order N (cid:80) i ( − i (cid:104) c † iσ c iσ (cid:105) to the SC order N (cid:80) i (cid:104) c † i ↑ c † i ↓ (cid:105) [97],where N = L × L is the system size. The spin-downparticle-hole symmetry can transform the usual AF or-dering into pseudospin-AF ordering (pseudospin-AF re-ferring to CDW/SC) [98] so that AF and pseudospin-AForder parameters are degenerate. The Hubbard interac-tion H U = U (cid:80) i ( n i ↑ − )( n i ↓ − ), which breaks thesecond Z symmetry explicitly, can lift the degeneracybetween the AF and pseudospin-AF ordering; AF order-ing is favored over pseudospin-AF ordering by any finite(even infinitesimal) Hubbard repulsion U >
0. Hereafter,we implicitly assume that a weak Hubbard repulsion U is present to break the degeneracy between AF andpseudospin-AF at half-filling. The model ˜ H = H + H U is dubbed as the Su-Schrieffer-Heeger-Hubbard model.The optical SSH model in Eq. (1) is sign-problem-free FIG. 2. The QMC results of AF correlations in the anti-adiabatic limit ( ω = ∞ ). (a) The AF correlation ratio R S AF as a function of dimensionless EPC constant λ for different L . (b) The extrapolated AF order parameter M AF = |(cid:104) S i (cid:105)| to the thermodynamic limit ( L → ∞ ) as a function of λ . Inthe anti-adiabatic limit, AF ordering occurs for any λ > so that we can perform large-scale projector QMC sim-ulations to investigate its ground-state phase diagramby varying phonon frequency ω and EPC constant λ .The projector QMC is numerically-exact and is able tostudy the zero-temperature properties directly. Detailsof the projector QMC method can be found in the Sup-plemental Material (SM). We emphasize that the simu-lations here are free from the notorious sign problem [80]so that we can study large system size. To investigatevarious possible symmetry-breaking orders, we computethe structure factor S ( q , L ) = N (cid:80) i,j e i q · ( r i − r j ) (cid:104) ˆ O i ˆ O j (cid:105) of the corresponding order O and evaluate the RG-invariant ratio of the structure factor, namely the cor-relation ratio, R S ( L ) = 1 − S ( Q + δq ,L ) S ( Q ,L ) , where Q refersto the ordering momentum and δq = ( πL , πL ) is a min-imal momentum shift from Q . For both Neel AF andstaggered VBS ordering, Q = ( π, π ). In the thermody-namic limit ( L → ∞ ), an ordered phase is recognized by R S → R S →
0. ForAF ordering, we further compute the susceptibility ratio R χ ( L ) = 1 − χ ( Q + δq ,L ) χ ( Q ,L ) , where χ ( q ) represents magneticsusceptibility at momentum q , as it has smaller finite-sizecorrections than the correlation ratio [99] (see the SM fortechnical details of evaluating susceptibilities in QMC). Results in adiabatic and anti-adiabatic limit:
In-tegrating out phonons with finite frequency yields an re-tarded interactions between electrons. The retardationeffect of EPC plays a central role in driving various novelphysics, including SC. The retardation is usually char-acterized by the ratio between the phonon frequency ω and the Fermi energy or band width W . Before per-forming systematic QMC simulations on the SSH modelat a generic finite phonon frequency, we first study theground-state properties at the adiabatic limit ( ω = 0)and anti-adiabatic limit ( ω = ∞ ), respectively.In the adiabatic limit ( ω = 0), the phonon is static atzero temperature and the exact solution can be obtainedby treating the phonon displacement configuration X ij asvariational parameters. As electron’s bare Fermi surface FIG. 3. The QMC results of VBS correlations as a function of ω for λ ≈ .
25 ( g = 1 . R S VBS for different L implies that the VBS transitionoccurs at ω c ≈ .
5. (b) The VBS transition at ω c ≈ . x -bond correlations and y -bond correlations. features a perfect nesting vector Q = ( π, π ), it is natu-ral to expect that the Fermi surface is unstable towardsstaggered VBS ordering for any finite EPC constant λ .Indeed, our calculations show that the expectation valueof electron hopping on NN bonds alternates in staggeredpattern (see the inset of Fig. 1), which breaks the latticetranslational symmetry as well as C rotational symme-try (see the SM for details of the calculations).In the anti-adiabatic (AA) limit ( ω = ∞ ), the effec-tive electronic interaction mediated by phonons becomesinstantaneous, which is proportional to the square of hop-ping on NN bonds. Consequently, in the AA limit, theoriginal optical SSH model can be reduced to the follow-ing effective Hamiltonian H AA = − t (cid:88) (cid:104) ij (cid:105) ( c † iσ c jσ + h.c. ) + J (cid:88) (cid:104) ij (cid:105) ( S i · S j + ˜ S i · ˜ S j ) , (2)where J = g /K is the strength of instantaneous inter-actions mediated by optical phonons in the AA limit, S i and ˜ S i are spin and pseudospin operators on site i , re-spectively. Specifically, S i = c † i σ c i and ˜ S i = ˜ c † i σ ˜ c i ,where c † i = ( c † i ↑ , c † i ↓ ), ˜ c † i = ( c † i ↑ , ( − i c i ↓ ), and σ repre-sents the vector of Pauli matrices. The phonon-mediatedinteractions include antiferromagnetic spin-exchange in-teraction, repulsive density-density interaction, and pairhopping terms. It is worth to emphasize that antiferro-magnetic ( J >
0) spin exchange interactions are gener-ated by EPC of SSH phonons, yielding the possibility ofAF ordering at half-filling. By performing QMC simu-lations on H AA , we obtained the results of AF correla-tion ratio and AF order parameter as a function of EPCconstant λ = J/W , as shown in Fig. 2. The AF corre-lation ratio R S AF monotonically increases with the size L for all studied λ , as shown in Fig. 2(a), indicating thatAF ordering occurs for all λ >
0. Furthermore, we ob-tained the AF order parameter by finite-size scaling tothe thermodynamical limit, as shown in Fig. 2(b), whichreveals that AF order induced by SSH phonons increaseswith λ . Consequently, we conclude that the ground-state FIG. 4. (a) The QMC results of the AF correlation ratio R S AF as the function of ω for λ ≈ .
25 ( g = 1 . L indicates that the AF transitionoccurs at ω = ω c ≈ .
5. (b) The finite size scaling of the spingap for ω near the criticality ω c . (c) For fixed ω = 1 .
0, theAF correlation ratio R S AF as the function of λ and the AFtransition occurs at λ c ≈ .
18. (d) The schematic picture ofthe second-order process of EPC which generates an effectiveretarded antiferromagnetic spin-exchange interactions. of the optical SSH model in the AA limit possesses AFlong-range order for any λ >
0. Moreover, it is an AFinsulator as its Fermi surface is fully gapped by AF order.
Antiferromagnetism at finite frequency:
We nowstudy the quantum phase diagram of the SSH model ofoptical phonons with a generic finite frequency (0 < ω < ∞ ). Since the ground-state is AF in the AA limit ( ω = ∞ )and VBS in the adiabatic limit ( ω = 0), there must be atleast one quantum phase transition (QPT) between VBSand AF phases when ω is varied from 0 to ∞ . Indeed,for a given λ , our QMC simulations show that there is adirect QPT between AF and VBS phases by varying ω .For λ ≈ .
25 ( g = 1 . L implies that the VBS or-der persists from ω = 0 to a critical frequency ω c ≈ . x or y bonds, as shown in Fig. 3(b), we further ver-ified that the VBS ordering pattern for 0 < ω < ω c is astaggered VBS breaking the lattice C symmetry, similarto the one observed in the adiabatic limit.More interestingly, our QMC simulations show thatthe long-range AF order emerges for ω > ω c . Here thecritical frequency ω c can be accurately extracted from thecrossing of AF susceptibility ratio R χ AFM ( L ) for different L . For λ ≈ .
25 ( g = 1 . R χ AFM ( L ) displays good crossing near ω ≈ .
5, as shownin Fig. 4(a), which implies that AF order develops for ω >ω c with ω c ≈ .
5. To further verify the AF phase withspontaneous spin-SU(2) rotational symmetry breaking,we compute the spin gap for ω around ω c ≈ .
5, as shownin Fig. 4(c). The spin gap is finite in the VBS regime,but it is extrapolated to zero in the AF regime ω > ω c ,indicating the emergence of gapless spin-wave excitationsas Goldstone modes of spin SU(2) symmetry breaking inthe AF phase. Taken together, these results convincinglyshow that the occurrence of phonon-induced AF long-range order for ω > ω c , where ω c depends on λ .Evidences of AF ordering at ω > ω c ( λ ) are also ob-tained for various other EPC dimensionless parameters λ , from weak to strong, as plotted in Fig. 1. As thecritical frequency ω c ( λ ) increases monotonically with in-creasing λ , for a fixed frequency it is expected that theAF phases should emerge in the regime of λ < λ c where λ c is the critical EPC constant. Indeed, for the fixed fre-quency ω = 1 .
0, AF ordering is observed in the regimeof λ < λ c ≈ .
18 from the crossing of the AF suscepti-bility ratio for different L , as shown in Fig. 4(b). The( π, π ) AF ordering fully gaps out the Fermi surface suchthat the ground state is an AF insulator for λ < λ c . Asmentioned earlier, the optical SSH model at half-fillingrespects the O(4) symmetry, giving rise to the degeneracybetween spin AF and pseudospin AF (namely CDW/SC).The degeneracy can be lifted and spin AF is more favoredby turning on a weak repulsive Hubbard interaction, asshown in the QMC simulations of models with a weakHubbard interaction (see the SM for details).It is worth to understand heuristically why AF order-ing emerges for small λ . For sufficiently small λ , one cantreat electron-phonon coupling term g as a weak pertur-bation and the second-order process in g would generatea spin exchange process when the spin polarizations inthe NN sites are opposite, as shown in Fig. 4(d). If thetwo spins on NN sites are parallel (namely forming atriplet), the exchange process is not allowed. Since thissecond-order spin-exchange process can gain energy, thespin-exchange interaction is antiferromagnetic. This pro-vides a phonon mechanism to drive AF ordering, which isqualitatively different from the usual AF exchange mech-anism of strong Hubbard Coulomb interaction.Note that AF ordering was not observed in an earlierQMC study of the 2D optical SSH phonon model [85].There the absence of AF ordering is possibly due to thefact that the QMC simulations were at finite tempera-ture and spin-SU(2) rotational symmetry in 2D cannotbe spontaneously broken at any finite temperature. Incontrast, we performed zero-temperature QMC simula-tions which can directly access properties of the groundstate of the 2D phonon model and observe a spontaneousspin-SU(2) symmetry breaking.We have shown evidences of a direct QPT between theAF and VBS phases. It is natural to ask if the directQPT between AF and VBS phases here is first order orcontinuous. Since AF and VBS phases break totally dif-ferent symmetries, the QPT between them is putativelyfirst-order in the Landau paradigm although it would be intriguing to explore if a deconfined quantum criti-cal point (DQCP) [100, 101] occurs in this case. Thephenomena of DQCP have been extensively studied forQPTs between Neel AF and columnar VBS [102–112].More recently, it has been argued from duality relationsthat, at such transition point, the SO(5) symmetry mightemerge at low energy [113–118]. However, the VBS or-der in the optical SSH phonon model studied here is thestaggered one, for which the VBS Z vortex is featureless,namely not carrying a spinon [119, 120]. Consequently,a (possibly weak) first-order transition instead of DQCP[121] would be expected for the QPT between AF andstaggered VBS phases in the phonon model under study. Conclusions and discussions:
We have systemat-ically explored the ground-state phase diagram of the2D optical SSH model taking account of full quantumphonon dynamics by zero-temperature QMC simulations.Remarkably, from the state-of-the-art numerically-exactsimulations, we have shown that the optical SSH phononscan induce a Neel AF order when the phonon frequencyis larger than a critical value ( ω > ω c ) or the EPC con-stant is smaller than a critical value ( λ < λ c ). The crit-ical frequency ω c can be much smaller than the bandwidth W for weak or moderate EPC constant λ , whichmakes the phonon mechanism of AF ordering practicallyfeasible in realistic quantum materials. For instance, forthe optical SSH model on the square lattice, we obtained ω c /W ∼ . λ ≈ . Acknowledgement .—We would like to thank SteveKivelson, Dung-Hai Lee, and Yoni Schattner for help-ful discussions. This work is supported in part bythe NSFC under Grant No. 11825404 (XC and HY),the MOSTC under Grant Nos. 2016YFA0301001 and2018YFA0305604 (HY), the CAS Strategic Priority Re-search Program under Grant No. XDB28000000 (HY),Beijing Municipal Science and Technology Commissionunder Grant No. Z181100004218001 (HY), and the Gor-don and Betty Moore Foundation’s EPiQS under GrantNo. GBMF4545 (ZXL). ∗ [email protected][1] R. E. Peierls, Quantum Theory of Solids (Oxford Uni-versity, New York/London, 1955).[2] G. Gr¨uner, Rev. Mod. Phys. , 1129 (1988).[3] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev.Lett. , 1698 (1979).[4] A. J. Heeger, S. A. Kivelson, J. R. Schrieffer, and W. P.Su, Rev. Mod. Phys. , 781 (1988).[5] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys.Rev. , 1175 (1957).[6] J. R. Schrieffer, Theory of Superconductivity (W. A.Benjamin, 1964).[7] I. Esterlis, B. Nosarzewski, E. W. Huang, B. Moritz,T. P. Devereaux, D. J. Scalapino, and S. A. Kivelson,Phys. Rev. B , 140501 (2018).[8] I. Esterlis, S. A. Kivelson, and D. J. Scalapino, Phys.Rev. B , 174516 (2019).[9] Z.-X. Li, M. L. Cohen, and D.-H. Lee, Phys. Rev. B , 245105 (2019).[10] N. C. Costa, T. Blommel, W.-T. Chiu, G. Batrouni, andR. T. Scalettar, Phys. Rev. Lett. , 187003 (2018).[11] Y.-X. Zhang, W.-T. Chiu, N. C. Costa, G. G. Batrouni,and R. T. Scalettar, Phys. Rev. Lett. , 077602(2019).[12] C. Chen, X. Y. Xu, Z. Y. Meng, and M. Hohenadler,Phys. Rev. Lett. , 077601 (2019).[13] G. G. Batrouni and R. T. Scalettar, Phys. Rev. B ,035114 (2019).[14] B. Cohen-Stead, K. Barros, Z. Meng, C. Chen, R. T.Scalettar, and G. G. Batrouni, Phys. Rev. B ,161108 (2020).[15] C. Feng and R. T. Scalettar, arXiv:2009.05595.[16] Z. Li, G. Antonius, M. Wu, F. H. da Jornada, and S. G.Louie, Phys. Rev. Lett. , 186402 (2019).[17] M. Gao, X.-W. Yan, Z.-Y. Lu, and T. Xiang, Phys. Rev.B , 094501 (2020).[18] P. W. Anderson, Science , 1196 (1987).[19] S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan,J. M. Tranquada, A. Kapitulnik, and C. Howald, Rev.Mod. Phys. , 1201 (2003).[20] P. W. Anderson, P. A. Lee, M. Randeria, T. M. Rice,N. Trivedi, and F. C. Zhang, Journal of Physics: Con-densed Matter , R755 (2004).[21] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. , 17 (2006).[22] D. J. Scalapino, Rev. Mod. Phys. , 1383 (2012).[23] J. C. S. Davis and D.-H. Lee, Proceedings of the Na-tional Academy of Sciences , 17623 (2013).[24] A. Lanzara, P. V. Bogdanov, X. J. Zhou, S. A. Kellar,D. L. Feng, E. D. Lu, T. Yoshida, H. Eisaki, A. Fuji-mori, K. Kishio, J.-I. Shimoyama, T. Noda, S. Uchida,Z. Hussain, and Z.-X. Shen, Nature , 510 (2001).[25] Z.-X. Shen, A. Lanzara, S. Ishihara, and N. Na-gaosa, Philosophical Magazine B , 1349 (2002),https://doi.org/10.1080/13642810208220725.[26] T. Cuk, F. Baumberger, D. H. Lu, N. Ingle, X. J.Zhou, H. Eisaki, N. Kaneko, Z. Hussain, T. P. Dev-ereaux, N. Nagaosa, and Z.-X. Shen, Phys. Rev. Lett. , 117003 (2004).[27] A. S. Mishchenko and N. Nagaosa, Phys. Rev. Lett. ,036402 (2004). [28] X. J. Zhou, J. Shi, T. Yoshida, T. Cuk, W. L. Yang,V. Brouet, J. Nakamura, N. Mannella, S. Komiya,Y. Ando, F. Zhou, W. X. Ti, J. W. Xiong, Z. X.Zhao, T. Sasagawa, T. Kakeshita, H. Eisaki, S. Uchida,A. Fujimori, Z. Zhang, E. W. Plummer, R. B. Laughlin,Z. Hussain, and Z.-X. Shen, Phys. Rev. Lett. , 117001(2005).[29] T. Cuk, D. H. Lu, X. J. Zhou, Z.-X. Shen, T. P. Dev-ereaux, and N. Nagaosa, Physica Status Solidi B ,11 (2005).[30] O. R¨osch, O. Gunnarsson, X. J. Zhou, T. Yoshida,T. Sasagawa, A. Fujimori, Z. Hussain, Z.-X. Shen, andS. Uchida, Phys. Rev. Lett. , 227002 (2005).[31] J. L. Tallon, R. S. Islam, J. Storey, G. V. M. Williams,and J. R. Cooper, Phys. Rev. Lett. , 237002 (2005).[32] J. Lee, K. Fujita, K. McElroy, J. A. Slezak, M. Wang,Y. Aiura, H. Bando, M. Ishikado, T. Masui, J.-X. Zhu,A. V. Balatsky, H. Eisaki, S. Uchida, and J. C. Davis,Nature , 546 (2006).[33] S. Johnston, F. Vernay, B. Moritz, Z.-X. Shen, N. Na-gaosa, J. Zaanen, and T. P. Devereaux, Phys. Rev. B , 064513 (2010).[34] S. Gerber, S.-L. Yang, D. Zhu, H. Soifer, J. A. Sobota,S. Rebec, J. J. Lee, T. Jia, B. Moritz, C. Jia, A. Gau-thier, Y. Li, D. Leuenberger, Y. Zhang, L. Chaix, W. Li,H. Jang, J.-S. Lee, M. Yi, G. L. Dakovski, S. Song, J. M.Glownia, S. Nelson, K. W. Kim, Y.-D. Chuang, Z. Hus-sain, R. G. Moore, T. P. Devereaux, W.-S. Lee, P. S.Kirchmann, and Z.-X. Shen, Science , 71 (2017).[35] Y. He, M. Hashimoto, D. Song, S.-D. Chen, J. He, I. M.Vishik, B. Moritz, D.-H. Lee, N. Nagaosa, J. Zaanen,T. P. Devereaux, Y. Yoshida, H. Eisaki, D. H. Lu, andZ.-X. Shen, Science , 62 (2018).[36] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida,and J. Zaanen, Nature , 179 (2015).[37] Y.-H. Liu, R. M. Konik, T. M. Rice, and F.-C. Zhang,Nature Communications , 10378 (2016).[38] Y. Zhong, Y. Wang, S. Han, Y.-F. Lv, W.-L. Wang,D. Zhang, H. Ding, Y.-M. Zhang, L. Wang, K. He,R. Zhong, J. A. Schneeloch, G.-D. Gu, C.-L. Song, X.-C.Ma, and Q.-K. Xue, Science Bulletin , 1239 (2016).[39] J.-Y. Chen, S. A. Kivelson, and X.-Q. Sun, Phys. Rev.Lett. , 167601 (2020).[40] C. Gadermaier, A. S. Alexandrov, V. V. Kabanov,P. Kusar, T. Mertelj, X. Yao, C. Manzoni, D. Brida,G. Cerullo, and D. Mihailovic, Phys. Rev. Lett. ,257001 (2010).[41] Y. He, S. Wu, Y. Song, W.-S. Lee, A. H. Said, A. Alatas,A. Bosak, A. Girard, S. M. Souliou, A. Ruiz, M. Hep-ting, M. Bluschke, E. Schierle, E. Weschke, J.-S. Lee,H. Jang, H. Huang, M. Hashimoto, D.-H. Lu, D. Song,Y. Yoshida, H. Eisaki, Z.-X. Shen, R. J. Birgeneau,M. Yi, and A. Frano, Phys. Rev. B , 035102 (2018).[42] Q.-Y. Wang, Z. Li, W.-H. Zhang, Z.-C. Zhang, J.-S.Zhang, W. Li, H. Ding, Y.-B. Ou, P. Deng, K. Chang,J. Wen, C.-L. Song, K. He, J.-F. Jia, S.-H. Ji, Y.-Y.Wang, L.-L. Wang, X. Chen, X.-C. Ma, and Q.-K. Xue,Chinese Physics Letters , 037402 (2012).[43] J. J. Lee, F. T. Schmitt, R. G. Moore, S. Johnston, Y.-T.Cui, W. Li, M. Yi, Z. K. Liu, M. Hashimoto, Y. Zhang,D. H. Lu, T. P. Devereaux, D.-H. Lee, and Z.-X. Shen,Nature , 245 (2014).[44] Z.-X. Li, F. Wang, H. Yao, and D.-H. Lee, Science Bul-letin , 925 (2016). [45] Y. Wang, A. Linscheid, T. Berlijn, and S. Johnston,Phys. Rev. B , 134513 (2016).[46] Q. Song, T. L. Yu, X. Lou, B. P. Xie, H. C. Xu, C. H. P.Wen, Q. Yao, S. Y. Zhang, X. T. Zhu, J. D. Guo,R. Peng, and D. L. Feng, Nature Communications ,758 (2019).[47] S. Zhang, T. Wei, J. Guan, Q. Zhu, W. Qin, W. Wang,J. Zhang, E. W. Plummer, X. Zhu, Z. Zhang, andJ. Guo, Phys. Rev. Lett. , 066802 (2019).[48] Y. Zhou and A. J. Millis, Phys. Rev. B , 054516(2017).[49] W. Zhao, M. Li, C.-Z. Chang, J. Jiang, L. Wu, C. Liu,J. S. Moodera, Y. Zhu, and M. H. W. Chan, ScienceAdvances , eaao2682 (2018).[50] Z.-X. Li, T. P. Devereaux, and D.-H. Lee, Phys. Rev. B , 241101 (2019).[51] R. Peng, K. Zou, M. G. Han, S. D. Albright, H. Hong,C. Lau, H. C. Xu, Y. Zhu, F. J. Walker, and C. H. Ahn,Science Advances , eaay4517 (2020).[52] D. Huang and J. E. Hoffman, Annual Review of Con-densed Matter Physics , 311 (2017).[53] D.-H. Lee, Annual Review of Condensed Matter Physics , 261 (2018).[54] Z. Han, S. A. Kivelson, and H. Yao, Phys. Rev. Lett. , 167001 (2020).[55] F. F. Assaad, M. Imada, and D. J. Scalapino, Phys.Rev. Lett. , 4592 (1996).[56] F. F. Assaad, M. Imada, and D. J. Scalapino, Phys.Rev. B , 15001 (1997).[57] A. Macridin, G. A. Sawatzky, and M. Jarrell, Phys. Rev.B , 245111 (2004).[58] P. Werner and A. J. Millis, Phys. Rev. Lett. , 146404(2007).[59] S. Johnston, E. A. Nowadnick, Y. F. Kung, B. Moritz,R. T. Scalettar, and T. P. Devereaux, Phys. Rev. B ,235133 (2013).[60] T. Ohgoe and M. Imada, Phys. Rev. Lett. , 197001(2017).[61] N. C. Costa, K. Seki, S. Yunoki, and S. Sorella,arXiv:1910.01146.[62] P. Sengupta, A. W. Sandvik, and D. K. Campbell, Phys.Rev. B , 245103 (2003).[63] N. C. Costa, K. Seki, and S. Sorella, arXiv:2009.05586.[64] M. Hohenadler and G. G. Batrouni, Phys. Rev. B ,165114 (2019).[65] C. Wang, Y. Schattner, and S. A. Kivelson,arXiv:2010.12588.[66] C. Honerkamp, H. C. Fu, and D.-H. Lee, Phys. Rev. B , 014503 (2007).[67] J. P. Hague, P. E. Kornilovitch, J. H. Samson, and A. S.Alexandrov, Phys. Rev. Lett. , 037002 (2007).[68] D. Wang, W.-S. Wang, and Q.-H. Wang, Phys. Rev. B , 195102 (2015).[69] Y. Wang, I. Esterlis, T. Shi, J. I. Cirac, and E. Demler,Phys. Rev. Research , 043258 (2020).[70] J. Lee, S. Zhang, and D. R. Reichman,arXiv:2012.13473.[71] R. Blankenbecler, D. J. Scalapino, and R. L. Sugar,Phys. Rev. D , 2278 (1981).[72] F. Assaad and H. Evertz, World-line and determinantalquantum monte carlo methods for spins, phonons andelectrons, in Computational Many-Particle Physics,edited by H. Fehske, R. Schneider, and A. Weiße(Springer Berlin Heidelberg, Berlin, Heidelberg, 2008) pp. 277–356.[73] Z.-X. Li, Y.-F. Jiang, and H. Yao, Phys. Rev. B ,241117 (2015).[74] Z.-X. Li, Y.-F. Jiang, and H. Yao, Phys. Rev. Lett. ,267002 (2016).[75] Z. C. Wei, C. Wu, Y. Li, S. Zhang, and T. Xiang, Phys.Rev. Lett. , 250601 (2016).[76] E. Berg, M. A. Metlitski, and S. Sachdev, Science ,1606 (2012).[77] C. Wu and S.-C. Zhang, Phys. Rev. B , 155115 (2005).[78] M. Troyer and U.-J. Wiese, Phys. Rev. Lett. , 170201(2005).[79] L. Wang, Y.-H. Liu, M. Iazzi, M. Troyer, and G. Harcos,Phys. Rev. Lett. , 250601 (2015).[80] Z.-X. Li and H. Yao, Annual Review of Condensed Mat-ter Physics , 337 (2019).[81] J. K. Freericks and E. H. Lieb, Phys. Rev. B , 2812(1995).[82] Y. Ono and T. Hamano, Journal of the Physical Societyof Japan , 1769 (2000).[83] S. Beyl, F. Goth, and F. F. Assaad, Phys. Rev. B ,085144 (2018).[84] S. Li and S. Johnston, npj Quantum Materials , 40(2020).[85] B. Xing, W.-T. Chiu, D. Poletti, R. T. Scalettar, andG. Batrouni, Phys. Rev. Lett. , 017601 (2021).[86] E. Fradkin and J. E. Hirsch, Phys. Rev. B , 1680(1983).[87] P. Sengupta, A. W. Sandvik, and D. K. Campbell, Phys.Rev. B , 245103 (2003).[88] H. Bakrim and C. Bourbonnais, Phys. Rev. B ,195115 (2007).[89] D. J. J. Marchand, G. De Filippis, V. Cataudella,M. Berciu, N. Nagaosa, N. V. Prokof’ev, A. S.Mishchenko, and P. C. E. Stamp, Phys. Rev. Lett. ,266605 (2010).[90] M. Hohenadler, F. F. Assaad, and H. Fehske, Phys. Rev.Lett. , 116407 (2012).[91] A. Nocera, J. Sous, A. E Feiguin, and M. Berciu,arXiv:2008.03304.[92] J. Sous, M. Chakraborty, R. V. Krems, and M. Berciu,Phys. Rev. Lett. , 247001 (2018).[93] M. Weber, F. F. Assaad, and M. Hohenadler, Phys.Rev. B , 245147 (2015).[94] M. Weber, Phys. Rev. B , L041105 (2021).[95] M. Weber, F. Parisen Toldin, and M. Hohenadler, Phys.Rev. Research , 023013 (2020).[96] C. N. Yang and S.-C. Zhang, Mod. Phys. Lett. B , 959(1990).[97] S.-C. Zhang, Phys. Rev. Lett. , 120 (1990).[98] E. Fradkin, Field Theories of Condensed Matter Physics,2nd ed. (Cambridge University Press, 2013).[99] F. Parisen Toldin, M. Hohenadler, F. F. Assaad, andI. F. Herbut, Phys. Rev. B , 165108 (2015).[100] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, andM. P. A. Fisher, Science , 1490 (2004).[101] T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, andM. P. A. Fisher, Phys. Rev. B , 144407 (2004).[102] A. W. Sandvik, Phys. Rev. Lett. , 227202 (2007).[103] R. G. Melko and R. K. Kaul, Phys. Rev. Lett. ,017203 (2008).[104] S. Pujari, K. Damle, and F. Alet, Phys. Rev. Lett. ,087203 (2013).[105] H. Shao, W. Guo, and A. W. Sandvik, Science , 213 (2016).[106] R. K. Kaul, R. G. Melko, and A. W. Sandvik, AnnualReview of Condensed Matter Physics , 179 (2013).[107] L. Wang, Z.-C. Gu, F. Verstraete, and X.-G. Wen, Phys.Rev. B , 075143 (2016).[108] N. Ma, Y.-Z. You, and Z. Y. Meng, Phys. Rev. Lett. , 175701 (2019).[109] Y. Liu, Z. Wang, T. Sato, M. Hohenadler, C. Wang,W. Guo, and F. F. Assaad, Nature Communications ,2658 (2019).[110] Z.-X. Li, S.-K. Jian, and H. Yao, arXiv:1904.10975.[111] R. Ma and C. Wang, Phys. Rev. B , 020407 (2020).[112] A. Nahum, Phys. Rev. B , 201116 (2020).[113] A. Nahum, P. Serna, J. T. Chalker, M. Ortu˜no, andA. M. Somoza, Phys. Rev. Lett. , 267203 (2015).[114] C. Wang and T. Senthil, Phys. Rev. X , 041031 (2015). [115] C. Wang, A. Nahum, M. A. Metlitski, C. Xu, andT. Senthil, Phys. Rev. X , 031051 (2017).[116] M. A. Metlitski and A. Vishwanath, Phys. Rev. B ,245151 (2016).[117] N. Seiberg, T. Senthil, C. Wang, and E. Witten, Annalsof Physics , 395 (2016).[118] Y. Q. Qin, Y.-Y. He, Y.-Z. You, Z.-Y. Lu, A. Sen, A. W.Sandvik, C. Xu, and Z. Y. Meng, Phys. Rev. X , 031052(2017).[119] M. Levin and T. Senthil, Phys. Rev. B , 220403(2004).[120] C. Xu and L. Balents, Phys. Rev. B , 014402 (2011).[121] A. Sen and A. W. Sandvik, Phys. Rev. B , 174428(2010).[122] X. Cai, Z.-X. Li, and H. Yao, in preparation. SUPPLEMENTAL MATERIAL
A. The method of projector quantum Monte Carlo
We employ the method of projector QMC to investigate the ground-state properties of the SSH phonon model atfinite frequency described in Eq. (1) and the effective Hamiltonian described in Eq. (2) in the anti-adiabatic limit.The algorithm is based on the principle that the ground-state expectation value of operator ˆ O in the exact groundstate | ψ G (cid:105) can be computed exactly via projecting a trial wave function | ψ T (cid:105) along the imaginary time axis (cid:104) ˆ O (cid:105) = (cid:104) ψ G | ˆ O | ψ G (cid:105)(cid:104) ψ G | ψ G (cid:105) = lim Θ →∞ (cid:104) ψ T | e − Θ H ˆ O e − Θ H | ψ T (cid:105)(cid:104) ψ T | e − H | ψ T (cid:105) (S1)as long as the trial wave function | ψ T (cid:105) has a finite overlap with the true ground state | ψ G (cid:105) . The algorithm isintrinsically unbiased against the choice of | ψ T (cid:105) assuming that the trial wave function is not orthogonal to the trueground state, namely (cid:104) ψ T | ψ G (cid:105) (cid:54) = 0, which is generically satisfied for a quantum many-body system with finite size.In this paper, we choose | ψ T (cid:105) to be the ground-state wave function of the non-interacting part of the model underconsideration. In practical QMC simulations, we set the projection parameter Θ to a finite but sufficiently large valueso that the expectation value (cid:104) ˆ O (cid:105) converges when larger Θ is considered. We set Θ = 34 /t, /t, /t, /t, /t for L = 6 , , , ,
14 accordingly, each of which has been checked to be large enough for convergence. Similar to finitetemperature algorithm, Trotter decomposition is implemented by discretizing Θ in spacing ∆ τ = Θ /L τ . The Trottererror scales as ∆ τ . In this work we choose ∆ τ = 0 . /t . The convergence of the discretization has also been checkedby comparing results with smaller ∆ τ .The SSH model in Eq. (1) becomes quadratic in electron operators for a specific space-time phonon configuration.We compute the electron’s Green’s function for each phonon configuration, and sample the phonon configuration byMonte Carlo. As for the effective model in the anti-adiabatic limit in Eq. (2), auxiliary fields are introduced into theHamiltonian via a SU(2) symmetric Hubbard-Stratonovich decomposition [72]. The AF and VBS order parametersfor a finite system size N = L are given byˆ O AF ( q ) = 1 L (cid:88) j e i q · R j ˆ S zj (S2)ˆ O VBS ( q ) = 1 L (cid:88) j e i q · R j (cid:16) ˆ B j, ˆ x + i ˆ B j, ˆ y (cid:17) (S3)where ˆ B j,δ = c † j,σ c j + δ,σ + h.c. is the kinetic operator on δ = ˆ x, ˆ y bonds. VBS breaks the lattice Z symmetry while AFbreaks spin SU(2) rotational symmetry. The structure factors for each order parameter with momentum Q is definedas S ( Q ) = (cid:104)| ˆ O ( Q ) | (cid:105) . For both AF and staggered VBS, the peaked momentum is Q = ( π, π ). The AF susceptibilitycan be computed as an integration over the imaginary-time interval τ M χ AFM ( q ) = (cid:90) Θ+ τ M / − τ M / d τ (cid:104) ˆ O AFM ( q , τ ) ˆ O AFM ( q , (cid:105) . (S4) FIG. S1. The QMC-computed AF order parameter as a function of inverse phonon frequency at λ ≈ .
25 ( g = 1 . To compute χ AF ( q ), the imaginary-time interval τ M should be large enough such that χ is converged, while thecorrelators (cid:104) ˆ O ( q , τ ) ˆ O ( q , (cid:105) should be computed after sufficiently long projection. Thus, in practice we choose τ M ≈ Θ / − τ M / τ M / B. Notes on finite-size analysis
In the main text, we performed the finite-size scaling to obtain the quantities in the thermodynamic limit. Thescaling function we usually use is C ( L ) = C ( L → ∞ ) + a/L b , where C ( L ) stands for the quantity such as AF orderparameter and spin gap computed for system size L . We estimate the order parameter in finite lattice as the squareroot of structure factor M AF = (cid:112) S AF ( π, π ), where factor 3 comes from the equal contribution of x, y, z componentsof the spin structure factor. In Fig. 2(b), we presented the result of AF order parameters as a function of λ in theanti-adiabatic limit. For λ ≈ .
25 and finite phonon frequency (0 < ω < ∞ ), the extrapolated AF order as a functionof ω is shown in Fig. S1. The value at t/ω = 0 comes from the computation at anti-adiabatic limit. The criticalfrequency ω c discussed in the main text is the phonon frequency where M AF vanishes.We compute spin gap ∆ s according to the asymptotic scaling behavior of time-dependent spin-spin correlation: (cid:104) S z ( Q , τ ) S z ( Q , (cid:105) ∼ e − ∆ s τ , where Q = ( π, π ). The spin gap ∆ s ( L ) for different lattice size L can be extracted byfitting the correlation when τ is sufficiently large. Then, ∆( L ) is extrapolated to L → ∞ via a power-law scalingfunction ∆( L ) = ∆ + a/L b to obtain the spin gap in the thermodynamic limit. C. Exact solution in the adiabatic limit
In the adiabatic limit ( ω = 0), phonon displacements X ij become classical quantities without quantum dynamics.At zero-temperature, the phonon displacements are static and the fermions are described by a quadratic Hamiltoniandepending on the phonon displacement X = { X ij } . Consequently, an exact solution of the ground state of the SSHmodel in the adiabatic limit can be obtained in the variational sense. In the adiabatic limit, the SSH model Eq. (1)is reduced to the following quadratic form depending on the phonon configuration X : H [ X ] = (cid:88) i,δ (cid:20) K X i,δ + ( gX i,δ − t ) ( c † i,σ c i + δ,σ + h.c. ) (cid:21) , (S5)where δ = x, y refers to unit vectors on x or y directions and X i,δ = X i,i + δ is the phonon displacement on the NN bond (cid:104) i, i + δ (cid:105) . The variational method is applied here by minimizing the ground-state energy, which yields the followingself-consistent equation: g (cid:68) ( c † i,σ c i + δ,σ + h.c. ) (cid:69) = − KX i,δ . FIG. S2. The finite size scaling of correlation ratio of AF and pseudospin-AF ordering at ω = 3 t and λ ≈ .
25 ( g = 1 . t ) for U = 0 . Here we compute the energy assuming various different ansatz of the phonon displacement configuration X , includingstaggered, staircase, columnar, and plaquette VBS patterns. The ordering vector for staggered and staircase patternsis ( π, π ) while the ordering vector of columnar and plaquette patterns is ( π,
0) or (0 , π ). Since the Fermi surfaceof the non-interacting electrons at half filling is perfectly nested by the wave vector ( π, π ), it is expected that theVBS patterns with ordering vector ( π, π ) is more favored than the ones with (0 , π ) or ( π, C symmetry while the staircase and plaquette ones develop dimerordering on both x and y directions. We assume X i,δ = m δ + ( − i x + i y ∆ stag δ + ( − i δ ∆ col δ , where m δ is the uniformcomponent of the phonon displacements such that the uniform hopping amplitude along δ = x, y direction is givenby t δ = t − gm δ . Our calculations clearly show that the SSH phonons in the adiabatic limit favors the staggered VBSstate with momentum ( π, π ) and C symmetry breaking for any finite EPC constant λ . Besides, when the staggeredVBS order parameter develops spontaneously in x -direction, namely ∆ stag x (cid:54) = 0 while ∆ stag y = 0, the uniform hoppingamplitude t x and t y exhibits the anisotropy t x > t y . Such anisotropy develops so that the shifted Fermi surface − t x cos k x − t y cos k y = 0 does not cross the nodal lines of gap function 2∆ stag x sin k x = 0; namely the Fermi surfacecan be fully gapped to gain energy. D. The effect of repulsive Hubbard interaction
As mentioned in the main text, the SSH model in Eq. (1) respects the O(4) symmetry, which implies the degeneracybetween AF and pseudospin-AF (namely CDW/SC) correlations. However, this degeneracy can be lifted by any finite(even infinitesimal) Hubbard interaction. By turning on a weak repulsive Hubbard interaction H U , AF is more favoredthan pseudospin-AF ordering so that the degeneracy between them is lifted. For instance, in Fig. S2 we present theresults of turning on a weak Hubbard U at ω = 3 t and λ ≈ .
25 ( g = 1 . t ) where the ground state without U isdegenerate between AF and pseudospin-AF (namely CDW/SC) ordering. Once a weak Hubbard U = 0 ..