Density waves in strongly correlated quantum chains
EEPJ manuscript No. (will be inserted by the editor)
Density waves in strongly correlated quantum chains
Martin Hohenadler and Holger Fehske Institut f¨ur Theoretische Physik und Astrophysik, Universit¨at W¨urzburg, 97074 W¨urzburg, Germany Institut f¨ur Theoretische Physik, Ernst-Moritz-Arndt-Universit¨at Greifswald, 17487 Greifswald, GermanyReceived: date / Revised version: date
Abstract.
We review exact numerical results for one-dimensional quantum systems with half-filled bands.The topics covered include Peierls transitions in Holstein, Fr¨ohlich, Su-Schrieffer-Heeger, and Heisenbergmodels with quantum phonons, competing fermion-boson and fermion-fermion interactions, as well assymmetry-protected topological states in fermion and anyon models.
The properties of quasi-one-dimensional materials such asconjugated polymers, charge-transfer salts, halogen-brid-ged or organic superconductors are the result of a subtleinterplay of charge, spin, and lattice fluctuations, in ad-dition to the unique effects of one-dimensional (1D) cor-related quantum systems. This has stimulated intense re-search efforts on paradigmatic fermion and fermion-bosonmodels [1, 2]. In particular, the question how a quasi-1Dmaterial evolves from a metal—either a Tomonaga-Luttin-ger liquid (TLL) [3, 4] or a Luther-Emery liquid (LEL)[5]—to an insulator has remained one of the most heav-ily debated issues in solid state physics for decades. Apartfrom band structure [6, 7] and disorder effects [8], electron-electron and electron-phonon interactions are the drivingforces behind the metal-insulator transition in the major-ity of cases. Coulomb repulsion drives the transition toa Mott insulator (MI) with dominant spin-density-wave(SDW) fluctuations [9], whereas the coupling to the vi-brational modes of the crystal triggers the Peierls tran-sition [10] to a long-range ordered charge-density-wave(CDW) or bond-order-wave (BOW) insulator [11]. If morethan one type of interaction is relevant, quantum phasetransitions (QPTs) between different insulating phases be-come possible. Quite generally, retarded boson-mediatedinteractions are significantly more difficult to describe the-oretically than the instantaneous Coulomb repulsion.More recently, QPTs between topologically trivial andnontrivial states have come into the focus of attention [12,13]. Topological phases possess characteristic zero-energyedge excitations that reflect the topological features of thebulk [14] and may either arise from topological band struc-tures or from interactions [15]. The topological propertiesare protected by certain symmetries (e.g., inversion, time-reversal or dihedral symmetry [16, 17]). Such symmetry-protected topological (SPT) states have short-range quan-tum entanglement [18] and may displace more conven-tional CDW, BOW, or SDW phases. Examples include dimerized Su-Schrieffer-Heeger (SSH) models [19] and theHaldane insulator [20].While the basic mechanisms underlying metal–insulatorand insulator–insulator QPTs are well known, their de-tailed understanding in microscopic models remains a chal-lenging and active field of research. Convincing evidencefor TLL–insulator QPTs has been obtained for the t - V model [21], the spinless Holstein and SSH models (Secs. 2.1and 2.2), as well as the Edwards fermion-boson model(Sec. 2.4). Minimal settings for LEL–insulator QPTs arethe spinful Holstein and the extended Hubbard model(Secs. 2.1 and 3.4). Insulator–metal–insulator or directinsulator–insulator QPTs have been explored in the ex-tended Hubbard model (Sec. 3.4), the Holstein-Hubbardmodel (Sec. 3.1), the Holstein-SSH model with competingbond and site couplings (Sec. 2.3), and the SSH modelwith additional Coulomb interaction (Sec. 3.2). ExtendedFalicov-Kimball models (Sec. 3.3) exhibit QPTs betweensemimetals or semiconductors and excitonic insulators.Extended Hubbard models with either an additional alter-nating ferromagnetic spin interaction or a bond dimeriza-tion have topologically trivial density-wave (DW) statesbut also SPT phases (Sec. 4.1). Finally, more exotic bosonicor even anyonic models that can be realized, in particu-lar, with highly tunable cold atoms in optical lattices [22]exhibit superfluid, MI, CDW, and SPT states (Sec. 4.2).In this contribution, we review the physics of a varietyof lattice models for quasi-1D strongly correlated parti-cle systems. Focusing on results from numerically exactmethods such as Lanczos exact diagonalization [23, 24],the density matrix renormalization group (DMRG) [25–28], and continuous-time quantum Monte Carlo (QMC)[29–34], we discuss ground-state and spectral propertiesand relate them to the corresponding 1D low-energy the-ories [21, 35]. Given the enormous literature, we mostlyrestrict the scope to half-filled bands for which umklappscattering can give rise to QPTs. Section 2 is devoted tothe effects of fermion-boson coupling, whereas Coulomb a r X i v : . [ c ond - m a t . s t r- e l ] J un Martin Hohenadler, Holger Fehske: Density waves in strongly correlated quantum chains interaction will be discussed in Sec. 3. In Sec. 4 we reviewrecent work on SPT states. Finally, we conclude in Sec. 5.
Perhaps the simplest example of a quantum system ofcoupled fermions and bosons are charge carriers interact-ing with lattice vibrations, as described by the HolsteinHamiltonian [36] ( (cid:126) = 1)ˆ H Hol = − t (cid:88) (cid:104) i,j (cid:105) σ ˆ c † iσ ˆ c jσ + ω (cid:88) i ˆ b † i ˆ b i − gω (cid:88) i (ˆ b † i + ˆ b i )ˆ n i . (1)It accounts for a single tight-binding electron band emerg-ing from nearest-neighbor hopping, quantum phonons inthe harmonic approximation, and a local density-displace-ment electron-phonon coupling. Here, ˆ n i = (cid:80) σ ˆ n iσ andˆ n iσ = ˆ c † iσ ˆ c iσ , where ˆ c † iσ (ˆ c iσ ) creates (annihilates) a spin- σ electron at site i of a 1D lattice with L sites. Similarly,ˆ b † i (ˆ b i ) creates (annihilates) a dispersionless optical (Ein-stein) phonon of frequency ω . Half-filling corresponds to n = (cid:104) ˆ n i (cid:105) = 1 ( n = 1 /
2) for spinful (spinless) fermions.The physics of the Holstein model is governed by thecompetition between the itinerancy of the electrons andthe tendency of the electron-phonon coupling to “immobi-lize” them. Importantly, the interaction is retarded in na-ture, as described by the adiabaticity ratio ω /t . Through-out this article, we use t as the energy unit. The electron-phonon coupling is often parameterized by λ = ε p / t inthe adiabatic regime ( ω /t (cid:28) g = ε p /ω inthe anti-adiabatic regime ( ω /t (cid:29)
1) [37–39]. For thesingle-particle case, where ε p is the polaron binding en-ergy, the Holstein model has provided important insightinto the notoriously difficult problem of polaron formationand self-trapping [36, 40]. The half-filled case consideredhere provides a framework to investigate the even moreintricate problem of the Peierls metal-insulator QPT ofspinless [38, 41–48] or spinful fermions [34, 41, 42, 49–59].The spinless Holstein model is obtained from Eq. (1)by dropping spin sums and indices. Figure 1(a) shows thecorresponding ground-state phase diagram from fermion-boson pseudo-site DMRG calculations [27]. At a criticalcoupling g c ( ω ), a QPT from a TLL to a CDW insulatorwith long-range q = 2 k F = π order (alternating occupiedand empty sites) and a 2 k F Peierls lattice distortion [10]takes place. The insulating state can be classified as atraditional band insulator in the adiabatic regime, and asa polaronic superlattice in the anti-adiabatic regime [47,60]. Numerical evidence for the Kosterlitz-Thouless [61]transition expected from the low-energy TLL descriptionand the mapping to the t - V model at strong coupling [41]comes from, e.g., XXZ-model physics for large ω [47] anda cusp in the fidelity susceptibility [48].The TLL charge parameter K c —determining the de-cay of correlation functions [21, 35]—from a finite-size scal-ing of the long-wavelength limit of the charge structure t / ω g (a) K c g ω /t =1 . ω / t = ω /t =0 . (b) . . . . . . TLL CDW bandinsulatorpolaronicsuperlattice . . . . . . . . . . . Fig. 1. (a) Phase diagram and (b) TLL parameter K c of thespinless Holstein model from DMRG calculations [56]. factor S c ( q ) = L (cid:80) j,l e iq ( j − l ) (cid:104) ˆ n j ˆ n l (cid:105) according to [21, 56] K c = lim L →∞ K c ( L ) , K c ( L ) = π S c ( q ) q , q = 2 πL (2)is shown in Fig. 1(b). Contrary to earlier numerical re-sults [38, 60], the TLL turns out to be repulsive ( K c < ω [56]. Accordingly, charge correlations ( ∼ r − K c )dominate over pairing correlations ( ∼ r − /K c ) through-out the TLL phase and show a crossover from weak tostrong 2 k F power-law correlations with increasing cou-pling [48, 62]. As shown in Fig. 1, K c = 1 / A ( k, ω ) = 1 Z (cid:88) mn | (cid:104) ψ m | ˆ c k | ψ n (cid:105) | ( e − βE m + e − βE n ) × δ [ ω − ( E n − E m )] (3)and the phonon spectral function B ( q, ω ) = 1 Z √ M ω (cid:88) mn |(cid:104) ψ m | ˆ b † q + ˆ b − q | ψ n (cid:105)| e − βE m × δ [ ω − ( E n − E m )] ; (4) E n is the eigenvalue for | ψ n (cid:105) , Z the partition function.In the adiabatic regime, the single-particle spectrum inthe TLL phase [Fig. 2(a)] is gapless but significantly mod-ified by the hybridization of charge and phonon modes[47, 63, 65]. In the CDW phase, it exhibits a Peierls gapand backfolded shadow bands [47, 63, 66, 67] [Fig. 2(b)].Near the critical point, soliton excitations [68] can be ob-served [63]. The phonon spectrum [64] reveals the renor-malization of the phonon mode due to electron-phononcoupling. In the adiabatic regime, the mode softens at thezone boundary in the TLL phase [Fig. 2(c)], becomes com-pletely soft for q = 2 k F at the critical point [Fig. 2(d)], andhardens again in the CDW phase [63, 64, 69, 70]. In con-trast, for ω (cid:29) t , the phonon mode hardens in the metallicphase and a central mode appears at λ c [47]. These find-ings are consistent with a soft-mode transition for ω (cid:28) t and a central-peak transition for ω (cid:29) t [47]. artin Hohenadler, Holger Fehske: Density waves in strongly correlated quantum chains 3 − − π/ π ω / t k − − π/ π k . . . . . π/ π ω / t q . . . . . π/ π q . . (a) A ( k, ω ) , λ = 0 . . . (b) A ( k, ω ) , λ = 1 . . (c) B ( q, ω ) , λ = 0 . . (d) B ( q, ω ) , λ = 0 . Fig. 2.
Single-particle [(a),(b)] and phonon [(c),(d)] spectraof the spinless Holstein model from QMC [63, 64]. Dashed linesindicate E F = 0 and ω = ± ω ( ω /t = 0 . A complete picture of the physics of the spinful
Hol-stein model (1) has only emerged recently. Whereas earlywork [41, 54, 55] suggested the absence of a metallic phase,the existence of the latter has since been confirmed [34,50, 53, 57–59]; for a detailed review see [62]. In terms of g-ology [21], the attractive umklapp scattering arising fromthe Holstein coupling remains irrelevant for λ < λ c ( ω ).However, for any λ >
0, attractive backscattering opensa spin gap [71]. Therefore, the metallic phase is in fact a1D spin-gapped metal—also known as an LEL [5]. Usingthe notation C x S y for a system with x ( y ) gapless charge(spin) modes [72], the LEL has C1S0. For λ > λ c , umk-lapp scattering is relevant and the ground state is a 2 k F CDW insulator (alternating doubly occupied and emptysites) with C0S0. Estimates for λ c are contained in thephase diagram of the Holstein-Hubbard model in Fig. 11in the limit U = 0.LEL physics and the Peierls QPT are also revealed bythe real-space correlation functions S c ( r ) = (cid:104) (ˆ n r − n )(ˆ n − n ) (cid:105) , (5) S s ( r ) = (cid:104) ˆ S xr ˆ S x (cid:105) ,S p ( r ) = (cid:104) ˆ ∆ † r ˆ ∆ (cid:105) ( ˆ ∆ r = ˆ c † r ↑ ˆ c † r ↓ ) , measuring charge, spin, and s-wave pairing correlations.As in the spinless case, charge correlations dominate overpairing in the metallic phase [62, 71, 73], see Fig. 3. Suchbehavior necessarily requires a spin gap [74] ( K s = 0)and repulsive interactions ( K c < K c [62] but the correlationfunctions in Fig. 3 clearly rule out claims of dominantpairing [34]. Spectral properties of the spinful Holsteinmodel have also been calculated [71, 75–78]. Most notably,the single-particle spectrum is gapped even in the metallicphase (although the spin gap—not taken into account in[65]—is difficult to detect numerically at weak coupling),and the phonon spectrum reveals a soft-mode transitionsimilar to the spinless case for ω /t < − − − − − x (a) . S c ( x ) − − − − − x (b) S p ( x ) − − − − − x (c) . S s ( x ) λ = 0 . λ = 0 . λ = 0 . λ = 0 . λ = 0 . Fig. 3.
Real-space correlation functions of the spinful Holsteinmodel (1) for (a) charge, (b) pairing, and (c) spin from QMCsimulations [62]. Here, ω /t = 0 . x = L sin ( πr/L ) is theconformal distance [79], and the solid line indicates 1 /x . A local electron-phonon interaction as in the Holsteinmodel (1) is a priori not justified for materials with in-complete screening. Results for nonlocal interactions inthe empty-band limit reveal significantly reduced polaronand bipolaron masses [80]. More recently, numerical re-sults for half-filling were obtained [77]. As a function ofthe screening length ξ , the Hamiltonian [77]ˆ H = − t (cid:88) (cid:104) i,j (cid:105) σ ˆ c † iσ ˆ c jσ + ω (cid:88) i ˆ b † i ˆ b i (6) − gω (cid:88) irσ e − r/ξ ( r + 1) / (ˆ b † i + r + ˆ b i + r )ˆ n iσ interpolates between a local Holstein and a long-rangeFr¨ohlich-type coupling [81]. As shown in Fig. 4(a), forsmall to intermediate ξ , the same LEL and CDW phasesare found, but λ c is enhanced with increasing ξ . For large ξ and strong coupling, the nonlocal interaction gives rise tomultipolaron droplets and phase separation (PS) [77], asdetected from the q = 0 divergence of the charge structurefactor [Fig. 4(b)] that implies K c = ∞ and hence a diver-gent compressibility [21]. The CDW–PS QPT appears tobe of first order [77]. Increasing the interaction range at afixed λ drives a CDW–LEL QPT. The concomitant sup-pression of CDW order gives rise to degenerate pairingand charge correlations in the Fr¨ohlich limit ξ → ∞ . The SSH model of polyacetylene captures fluctuations ofthe carbon-carbon bond lengths and their effect on theelectronic hopping integral [68] (for related earlier work see[82]). It has a coupling term of the form (cid:80) i ˆ B i ( ˆ Q i +1 − ˆ Q i ),where ˆ Q i ∼ ˆ b † i + ˆ b i is the displacement of atom i from itsequilibrium position and ˆ B i = (cid:80) σ (ˆ c † iσ ˆ c i +1 σ + H.c.). Thephonons have an acoustic dispersion ω q = ω sin( q/ Martin Hohenadler, Holger Fehske: Density waves in strongly correlated quantum chains S c ( q ) q (b) λ = 0 .
20 (LEL) λ = 0 .
45 (CDW) λ = 0 .
55 (PS) π π Fig. 4. (a) Phase diagram and (b) charge structure factorin the three phases of the nonlocal electron-phonon model (6)from QMC [77]. Here, ω /t = 0 . is equivalent to the simpler optical SSH model [87]ˆ H SSH = − t (cid:88) i ˆ B i + ω (cid:88) i ˆ b † i ˆ b i − gω (cid:88) i ˆ B i (ˆ b † i + ˆ b i ) . (7)Here, ˆ b † i and ˆ b i are associated with an optical phononmode describing fluctuations of the bond lengths. Figure 5illustrates the quantitative agreement of the two modelsfor the single-particle Green function and the dynamicbond structure factor, which can be attributed to the in-herent dominance of q = 2 k F = π order at half-filling [86].For spinless fermions, the model (7) has a repulsiveTLL ground state with dominant BOW correlations belowa critical coupling λ c (for SSH models λ = g ω / t ), andan insulating Peierls ground state with a long-range 2 k F BOW (alternating weak and strong bonds) for λ > λ c ( ω )[55, 85, 86]. The transition from power-law to long-rangeBOW correlations can be seen in Fig. 5(b) from S b ( r ) = (cid:104) ( ˆ B r − (cid:104) ˆ B r (cid:105) )( ˆ B − (cid:104) ˆ B (cid:105) ) (cid:105) . At the critical point, the cor-relation functions are consistent with K c = 1 /
2. In theadiabatic regime, the phase transition is again of the soft-mode type [86]. Apart from the interchange of the rolesof charge and bond degrees of freedom, the spinless SSHmodel is in many respects similar to the spinless Holsteinmodel, including spectral and thermodynamic properties[86, 88]. However, subtle differences arise due to the differ-ent symmetries of the two models (class BDI of the generalclassification [89] for the SSH model, class AI with brokenparticle-hole and chiral symmetry for the Holstein model)[88]. Note that the name SSH model is often used to referto the mean-field approximation of the true SSH model,i.e., a fermionic Hamiltonian with dimerized hopping butno phonons (see also Sec. 4.1).In contrast to the spinful Holstein model, the spin-ful SSH model does not have a metallic phase. Althoughquantum fluctuations significantly reduce the dimeriza-tion compared to the mean-field solution [90], the groundstate is an insulating BOW-Peierls state (C0S0) for any λ > ω [55, 59, 85–87, 91–93]. Direct nu-merical evidence for this conclusion is shown in Fig. 5(c).Because Eq. (7) is symmetric under the transformationˆ c i ↓ (cid:55)→ ( − i ˆ c † i ↓ that interchanges spin and charge opera-tors, spin and charge correlators are exactly equal. There- − − − − τ t (a) ω /t = 0 . G (0 , τ ) S b ( π, τ ) − − − −
10 20 50 S b ( x ) x (b) ω /t = 0 . . . . .
00 0 .
04 0 . K c ( L ) , K s ( L ) /L (c) ω /t = 0 . λ = 0 . λ = 0 . λ = 0 . λ = 1 . λ = 1 . Fig. 5. (a) Single-particle Green function and dynamic bondstructure factor for the spinful optical SSH model (7) and theoriginal SSH model. (b) Real-space bond correlations of thespinless optical SSH model. The full (dashed) line correspondsto 1 /x (1 /x ). (c) Finite-size estimates of the TLL parameters K c and K s . All results are from QMC simulations [86]. fore, the finite-size estimate of K s ( L ) < K s = K c = 0 by symmetry and hence aninsulating state [86]. The spinful SSH model hence pro-vides an example where Peierls’ theorem [10] holds evenfor quantum phonons. This property can be traced backto the fact that forward scattering vanishes whereas umk-lapp scattering is repulsive (rather than attractive, as inthe Holstein model) and hence always relevant [59]. Exci-tation spectra for the spinful SSH model closely resemblethose of the spinless model in the ordered phase [86]. Whereas Holstein and SSH models have been studied in-tensely, the even more complex problem of competing siteand bond couplings—which in principle coexist in mostmaterials—has been addressed by QMC only recently [94]using the Holstein-SSH Hamiltonianˆ H = − t (cid:88) i ˆ B i + (cid:88) iα ω ,α ˆ b † i,α ˆ b i,α − g s ω , s (cid:88) i ˆ n i (ˆ b † i, s + ˆ b i, s ) − g b ω , b (cid:88) i ˆ B i (ˆ b † i, b + ˆ b i, b ) (8)with independent site ( α = s) and bond ( α = b) phononmodes as well as corresponding coupling constants λ α .Of particular interest is the question if the metallicphase of the Holstein model is stable with respect to theSSH coupling, or if metallic behavior is entirely absentas in the SSH model (7). In terms of g-ology, both cou-plings produce negative backscattering matrix elementsthat give rise to a spin gap. On the other hand, the umk-lapp matrix elements have opposite sign and can thereforecompensate, allowing for an extended LEL (C1S0) metal-lic region. This picture is confirmed by QMC data [94]summarized in the qualitative phase diagram in Fig. 6. artin Hohenadler, Holger Fehske: Density waves in strongly correlated quantum chains 5 Fig. 6.
Schematic phase diagram of the Holstein-SSHmodel (8) based on QMC simulations for ω /t = 0 . If the SSH coupling dominates, the system is a BOW in-sulator just like the SSH model. If the Holstein couplingdominates, a CDW ground state exists. Both states areof type C0S0. If the couplings are comparable, the com-petition between the two orders results in a metallic LELphase. Starting in the CDW phase and increasing λ b , thecorrelation functions in Fig. 7 reveal a suppression (en-hancement) of CDW (BOW) order and a QPT to the LELphase with power-law correlations. At stronger SSH cou-plings, long-range BOW order emerges. For all parame-ters, spin correlations remain exponential due to the spingap. The QPT between the two different Peierls statesis found to be continuous, and in the adiabatic regimeinvolves two soft-mode transitions for the site and bondphonon modes, respectively [94]. The single-particle gapis minimal but finite at the QPT [94]. These numericalresults contradict earlier approximate results suggestinga first-order BOW–CDW transition [95] or a ferroelectricphase with coexistence of BOW and CDW order [96]. The discussion so far has revealed that the coupling tothe lattice can modify the transport properties of low-dimensional systems to the point of insulating behavior.Quantum transport in general takes place in some “back-ground”, which may consist of lattice but also spin or or-bital degrees of freedom. For instance, a key problem in thewidely studied high- T c cuprates [97] and colossal magne-toresistance manganites [98] is that of (doped) holes mov-ing in an ordered magnetic insulator [99]. As the holes − − − − − x (a) S b ( x ) − − − − − x (b) S c ( x ) − − − − − x (c) S p ( x ) − − − − − x (d) S s ( x ) λ b = 0 . λ b = 0 . λ b = 0 . λ b = 0 . λ b = 0 . Fig. 7. (a) Bond, (b) charge, (c) pairing, and (d) spin corre-lations of the Holstein-SSH model (8) from QMC simulations[94]. Here, ω /t = 0 .
5. The solid line indicates 1 /x . t / ω / ¯ λ (a) ω / t ¯ λ ω / t n (b) . . . . . . . TLL CDW .
00 0 .
04 0 . TLL . . . . . . . . . . TLL(rep.)TLL(attr.) TLL(attr.)PS PS
Fig. 8. (a) DMRG phase diagram of the Edwards model (9)at half-filling [103]. (b) Ground states as a function of bandfilling and ω /t in the slow-boson regime at fixed ¯ λ/t = 0 . move, they disrupt the order of the background which,conversely, hinders hole motion. Coherent motion may stilloccur, albeit on a reduced energy scale determined by thefluctuations and correlations in the background.A fermion-boson model by Edwards describes the in-teraction of particles with the background in terms of acoupling to bosonic degrees of freedom [100, 101]:ˆ H = − t (cid:88) (cid:104) i,j (cid:105) ˆ c † j ˆ c i (ˆ b † i + ˆ b j ) + ω (cid:88) i ˆ b † i ˆ b i − ¯ λ (cid:88) i (ˆ b † i + ˆ b i ) . (9)In this model, every hop of a (spinless fermionic) chargecarrier along a 1D transport path either creates a (localbosonic) excitation with energy ω in the background atthe site it leaves, or annihilates an existing excitation atthe site it enters. The fermion-boson coupling in Eq. (9)differs significantly from the Holstein and SSH couplingsdiscussed before. In particular, no static distortion arisesin the limit ω →
0. Furthermore, spontaneous bosoncreation and annihilation processes are possible, i.e., thebackground distortions can relax with a relaxation rate¯ λ , for example due to quantum fluctuations. Any particlemotion is affected by the background and vice versa. Infact, the Edwards model describes three different regimes:quasi-free, diffusive, and boson-assisted transport [101]. Inthe latter case, excitations of the background are energeti-cally costly ( ω /t >
1) and the background relaxation rateis small (¯ λ/t (cid:28) K c <
1) [103].Remarkably, particle motion is possible even for ¯ λ = 0, inlowest order by a vacuum-restoring six-step process where3 bosons are excited in steps 1-3 and afterwards consumedin steps 4-6 with the particle moving two sites [101]. Incontrast to the spinless Holstein model, the CDW state of Martin Hohenadler, Holger Fehske: Density waves in strongly correlated quantum chains k ( ω − E F ) /t (a) ω /t = 10 , ¯ λ = 0 . ( ω − E F ) /t ω /t = 10 , ¯ λ = 0 . (b) ( ω − E F ) /t ω /t = 1 , ¯ λ = 0 . (c) π π − π π − π π − − π π − Fig. 9.
Single-particle spectral function of the Edwardsmodel (9) at half-filling from the dynamical DMRG [106]. In-set: dispersion of the absorption/emission maximum. the Edwards model is a few-boson state [102]. As shownin Fig. 8(b), at low and high band filling n the attractiveinteraction mediated by the slow bosons becomes strongenough to give rise to first an attractive TLL ( K c > A ± ( k, ω ) = (cid:88) n |(cid:104) ψ ± n | ˆ c ± k | ψ (cid:105)| δ [ ω ∓ ( E ± n − E )] , (10)where ˆ c + k = ˆ c † k , ˆ c − k = ˆ c k , | ψ (cid:105) is the ground state for N e particles and | ψ ± n (cid:105) the n -th excited state with N e ± A ( k, ω ) = A − ( k, ω ) + A + ( k, ω ) inthe regime where background excitations have a large en-ergy and the bosons strongly affect particle transport. Thequasiparticle mass is significantly enhanced and a renor-malized band structure appears. However, if ¯ λ is suffi-ciently large, the system remains metallic, as indicated bya finite spectral weight at the Fermi energy E F [Fig. 9(a)].As the possibility of relaxation reduces a gap opens at k F = π/ O ( t /ω ), while a doped particle can move bya two-step process of order O ( t /ω ) [102]. By decreasing ω at fixed ¯ λ the fluctuations overcome the correlationsand the system returns to a metallic state. However, thelatter differs from the state we started with. In particular, A ( k, ω ) in Fig. 9(c) shows sharp absorption features onlynear k F and “overdamping” at the zone boundaries wherethe spectrum is dominated by bosonic excitations. The Peierls (dimerization) instability triggered by the lat-tice degrees of freedom can be observed not only in quasi-1D itinerant electron systems but also in spin chains withmagneto-elastic couplings. Experimentally, such behaviorwas first seen in the 1970s for organic compounds of theTTF and TCNQ family [107]. Interest in the subject re-vived after the discovery of the first inorganic spin-Peierlscompound CuGeO in 1993 [108], in particular due tothe fact that the displacive spin-Peierls transition in thismaterial does not involve phonon softening. Instead, thePeierls-active optical phonon modes with frequencies ω , (cid:39) J and ω , (cid:39) J ( J being the exchange coupling be-tween neighboring Cu ions that form well separatedspin- chains) harden by about 5% at the transition whichtherefore occurs at very strong spin-phonon coupling [109].Phonon hardening for experimentally relevant parameterswas demonstrated for the magnetorestrictive XY modelby calculating the dynamic structure factor [110]. Thephysics of CuGeO reveals that the canonical adiabatictreatment of the lattice [111, 112] is inadequate for thismaterial [113]. Instead, the application of numerical meth-ods to paradigmatic quantum models yields key informa-tion about the nature of the phase transition and the cor-rect models for inorganic spin-Peierls materials.The simplest model containing all important featuresof a spin-Peierls system is an antiferromagnetic Heisen-berg chain, ˆ H Heis = J (cid:80) i ˆ S i · ˆ S i +1 ( J <
0, ˆ S i is a spin- operator at site i ), coupled to Einstein quantum phonons:ˆ H = ˆ H Heis + ˆ H (l , d)SP + ω (cid:88) i ˆ b † i ˆ b i . (11)Here we consider two different spin-phonon couplings,ˆ H lSP = gω (cid:88) i (ˆ b † i + ˆ b i ) ˆ S i · ˆ S i +1 , (12)ˆ H dSP = gω (cid:88) i (ˆ b † i +1 + ˆ b i +1 − ˆ b † i − ˆ b i ) ˆ S i · ˆ S i +1 . (13)The local coupling ˆ H lSP captures the modification of thespin exchange by a local lattice degree of freedom (model-ing, e.g., side-group effects) [114, 115]. The difference cou-pling ˆ H dSP describes a linear dependence of the spin ex-change on the difference between the phonon amplitudesat sites i and i + 1 [116]. A first insight into these mod-els can be gained by integrating out the phonons in theanti-adiabatic limit ω (cid:29) J to obtain an effective Heisen-berg model with longer-ranged interactions that give riseto frustration. The spin Hamiltonian ˆ H = J (cid:80) i ( ˆ S i · ˆ S i +1 + α ˆ S i · ˆ S i +2 ) has a dimerized ground state (alternating longand short bonds) for α ≥ α c = 0 .
241 167 [117]. Accord-ingly, the spin-phonon coupling must be larger than anonzero critical value g c ( ω ) for the spin-Peierls instabil-ity to occur [115, 116, 118]. This is similar to the Holsteinmodel (1) but in contrast to the static limit ω /J = 0.Figure 10 shows the phase diagram of the model (11)for either the coupling (12) or (13) from two-block [116] artin Hohenadler, Holger Fehske: Density waves in strongly correlated quantum chains 7 J / ω g J / ω g . . . . . (b) spin liquid Peierls . . . . . spin liquid Peierls (a) Fig. 10.
DMRG phase diagrams of the Heisenberg spin-Peierlsmodel (11) with (a) a local coupling ˆ H lSP [119] and (b) a dif-ference coupling ˆ H dSP [116]. and four-block [119] DMRG calculations, respectively. TheQPT from the the gapless spin-liquid state to the gappeddimerized phase was detected using the well-establishedcriterion of a level crossing between the first singlet andthe first triplet excitation; the latter was derived for thefrustrated spin chain [116, 120, 121]. For finite systems, thesinglet lies above the triplet excitation in the spin liq-uid, and the two levels become degenerate with the singletground state as L → ∞ . In the symmetry-broken gappedphase, the lowest singlet state becomes degenerate withthe ground state. The Heisenberg spin-Peierls model withquantum phonons is in the same universality class as thefrustrated spin chain [116]. The phonon spectral functionhas been analyzed in [122]. Related spin-boson models ex-hibiting TLL and CDW phases have also been investigatedin the context of dissipative quantum systems [123]. From the 1D Hubbard model [124], it is well establishedthat a local Coulomb repulsion favors a correlated MI withdominant 2 k F SDW fluctuations [21]. In contrast to CDWand BOW order, the continuous SU(2) spin symmetrycannot be spontaneously broken [125]. Instead, the SDWcorrelations are critical ( ∼ /r ) [21, 126]. Of key interest isthe interplay or competition of retarded electron-phononand instantaneous electron-electron interactions that de-termines if the ground state is a CDW/BOW, SDW, orLEL state. A minimal but rich model capturing this in-terplay is the Holstein-Hubbard model with Hamiltonianˆ H = ˆ H Hol + U (cid:88) i ˆ n i ↑ ˆ n i ↓ . (14)The ground-state phase diagram of Eq. (14) was thesubject of intense debate. Even after early claims [41] (un-founded [62] but supported by RG calculations [54, 55]) ofthe absence of metallic behavior in the spinful Holsteinmodel were contradicted by DMRG results [50], numeri-cal work on the Holstein-Hubbard model initially focused λ U/ t (a) ω /t = 0 . U/ t (b) ω /t = 5 . . . . . . . . . . . . CDWLEL SDW . . . . . . . . CDW LEL SDW . . λ U/t = 4
Fig. 11.
DMRG phase diagram of the Holstein-Hubbardmodel (14) in (a) the adiabatic and (b) the anti-adiabaticregime [57]. Dashed (solid) lines are CDW–LEL (LEL–SDW)critical values from QMC [34], the dotted line is U = 4 λt .Squares (triangles) indicate the CDW Peierls (LEL) phase. Theinset shows the one-particle (circles), two-particle (diamonds),and spin (stars) excitation gaps in the thermodynamic limit. on strong couplings where a direct SDW–CDW QPT isobserved. Evidence for an intermediate metallic phase—expected from the adiabatic connection to the Holsteinmodel as U → ∆ c = E ( L + 1 , ) + E ( L − , ) − E ( L, ,∆ c = E ( L + 2 ,
0) + E ( L − , − E ( L, ,∆ s = E ( L, − E ( L, ,∆ n = E ( L, − E ( L, . (15) E ( N e , S z tot ) [ E ( N e , S z tot )] is the energy of the ground-state (first excited state) of a system with L sites, N e electrons and total spin- z S z tot . The CDW state has C0S0( ∆ c1 > ∆ s > ∆ c1 > ∆ s = 0). The different nature of excitations inthese phases is clearly visible in the spectra in Fig. 12 (forprevious work see [67, 74, 78]). The single-particle spec-tral function in Figs. 12(a) and (b) has a gap at E F inboth phases, but distinct soliton excitations and back-folded shadow bands only in the CDW phase. Spin-chargeseparation [21] can be observed for strong interactions[76, 130]. The dynamic charge structure factor S c ( q, ω ) = 1 Z (cid:88) mn | (cid:104) ψ m | ˆ ρ q | ψ n (cid:105) | e − βE m δ [ ω − ( E n − E m )](16)with ˆ ρ q = (cid:80) r e iqr (ˆ n r − n ) / √ L in Figs. 12(c),(d) reveals a q = 0 charge gap in both insulating phases and the renor- Martin Hohenadler, Holger Fehske: Density waves in strongly correlated quantum chains − − π/ π ω / t k − − π/ π k π/ π ω / t q π/ π q π/ π ω / t q π/ π q . . (a) A ( k, ω ) (CDW) . . (b) A ( k, ω ) (SDW) . . (c) S c ( q, ω ) (CDW) . . (d) S c ( q, ω ) (SDW) . . (e) S s ( q, ω ) (CDW) . . (f) S s ( q, ω ) (SDW) Fig. 12.
Single-particle [(a),(b)], density [(c),(d)], and spin[(e),(f)] excitation spectra of Holstein-Hubbard model (14)from QMC [71] in the CDW ( ω /t = 0 . U/t = 0 . λ = 0 . ω /t = 5, U/t = 4, λ = 0 . malized phonon frequency in the CDW phase [Fig. 12(c)].Finally, the dynamic spin structure factor S s ( q, ω ) = 1 Z (cid:88) mn | (cid:104) ψ m | ˆ S zq | ψ n (cid:105) | e − βE m δ [ ω − ( E n − E m )](17)shows a clear spin gap in the CDW phase [Fig. 12(e)]whereas the SDW phase has ∆ s = 0 and strong 2 k F = π fluctuations [Fig. 12(f)].Similar to the Holstein model, the intermediate phaseis a spin-gapped LEL (C1S0, ∆ c1 > ∆ s >
0, but ∆ c2 =0) [53, 57, 59, 71]. In the anti-adiabatic regime [Fig. 11(b)],where retardation effects are small, the LEL–SDW QPToccurs close to the value U = 4 λt expected from an ef-fective Hubbard model. Whereas DMRG and QMC re-sults agree quite well for the LEL–SDW QPT, the LEL–CDW QPT line is not completely settled. The exponentialopening of the charge gap is nontrivial to detect with theDMRG, and the charge susceptibility used in QMC [34]is problematic due to the spin gap [62]. The latter alsocomplicates the calculation of TLL parameters [62]. Theintermediate LEL phase has K s = 0, so that the low-energy theory is that of bosonic pairs (bipolarons). K c as extracted from the electronic density structure factorgives K c > K c = 1 at the LEL–CDW QPT. An inter-esting open problem is to reconcile the vanishing of the λ ω /t (a) U/t = 2 . α U/t (b) ω /t = 1 . . . . . . . . . . . BOW SDW . . . . .
12 0 2 4 6 8 10
BOW SDW
Fig. 13.
Phase diagrams of the SSH- UV model (18) fromQMC simulations for V = U/
4. Data taken from [87]. (bipolaron) binding energy in parts of the LEL phase [58]with the nonzero spin gap. Finally, CDW and SDW statesof the Holstein-Hubbard model have been studied numeri-cally in the context of pump-probe experiments [131–137].
U V model
The competition between electron-phonon and electron-electron interaction has also been studied in the frame-work of the SSH-
U V
Hamiltonianˆ H = ˆ H SSH + U (cid:88) i ˆ n i ↑ ˆ n i ↓ + V (cid:88) i ˆ n i ˆ n i +1 , (18)which is directly relevant for conjugated polymers [83,138]. The phase diagram from QMC simulations [87] isshown in Fig. 13. A key difference to the Holstein-Hubbardmodel is that no metallic phase results from the compet-ing interactions. Instead, for ω > U > V , the ground state is a MI with critical SDW cor-relations (C0S1) for λ < λ c , and a BOW Peierls state(C0S0) for λ > λ c [42, 86, 87, 139]; in contrast to the con-tinuous suppression of CDW correlations by the Hubbardrepulsion in the Holstein-Hubbard model, the amplitudeof BOW correlations is enhanced by the Coulomb repul-sion in the SSH- U V model [86, 140–143]. Finally, for large U , the SSH- U V model is closely related to the spin-Peierlsmodels discussed in Sec. 2.5 [87].
CDW, BOW, SDW or orbital DW states can also arisepurely from the Coulomb interaction, so that extensionsof the Hubbard Hamiltonian may be regarded as minimaltheoretical models. An important example is the asym-metric Hubbard model with spin-dependent band energies ε kσ = E σ − t σ cos k , where E σ defines the center of thespin- σ band and t σ is the nearest-neighbor hopping am-plitude [144, 145]. For E ↑ < E ↓ and t ↑ t ↓ < t ↑ t ↓ >
0) adirect (indirect) band gap is realized. The σ -electron den-sity n σ = L (cid:80) k (cid:104) ˆ c † kσ ˆ c kσ (cid:105) , with n ↑ + n ↓ = 1 at half-filling. artin Hohenadler, Holger Fehske: Density waves in strongly correlated quantum chains 9 U/t c D / t c SOOEI BI t f /t c = − . U/t V / t CDW (∆ c >
0, ∆ s > c >
0, ∆ s > c >
0, ∆ s = 0) XY ( c = 1) (cid:52) tricritical point (cid:53) critical endpointfirst order Fig. 14.
DMRG phase diagrams of the extended Falicov-Kimball model (19) (left) [155] and the extended Hubbardmodel (20) (right) [156].
The asymmetric Hubbard model has been used to inves-tigate various many-body effects in (mixed/intermediate-valence) rare-earth and transition-metal compounds, in-cluding the DW–PS QPT [146], electronic ferroelectric-ity [147, 148] and (pressure-induced) exciton condensation[149], as well as multiorbital correlation physics in coldatoms [150]. Regarding σ as an orbital flavor, the asym-metric Hubbard model is equivalent to the extended Falicov-Kimball model (EFKM) [144, 145, 147, 151]ˆ H EFKM = − t c (cid:88) (cid:104) i,j (cid:105) ˆ c † i ˆ c j − t f (cid:88) (cid:104) i,j (cid:105) ˆ f † i ˆ f j (19)+ U (cid:88) i ˆ c † i ˆ c i ˆ f † i ˆ f i + D (cid:88) i (cid:16) ˆ c † i ˆ c i − ˆ f † i ˆ f i (cid:17) , describing two species of spinless fermions, namely, light c (or d ) electrons and heavy f electrons. A finite f -bandwidthallows for f - c electron coherence, which will take accountof a mixed-valence situation as well as of c -electron f -hole(exciton) bound-state formation and condensation [151,152]. By contrast, t f = 0 in the original FKM [153], sothat the number of f -electrons is strictly conserved andno coherence between f and c electrons can arise [154].The left panel of Fig. 14 shows the DMRG phase dia-gram at half-filling. Depending on the orbital level split-ting there exist staggered orbital ordered (SOO) or bandinsulator (BI) phases, separated by a critical excitonic in-sulator (EI) [157]. In the absence of true long-range or-der in one dimension, the characteristic signatures of anexcitonic Bose-Einstein condensate are a power-law de-cay of the correlator (cid:104) ˆ X † i ˆ X j (cid:105) with ˆ X † i = ˆ c † i ˆ f i and a di-vergence of the excitonic momentum distribution N ( q ) = (cid:104) ˆ X † q ˆ X q (cid:105) with ˆ X † q = √ L (cid:80) k ˆ c † k + q ˆ f k for the lowest-energystate (which has q = 0 for the case of a direct gap). Thecriticality of the EI can also be detected from the von Neu-mann entropy and the central charge [ c ∗ ( L ) (cid:39) U yields clear evidence for a BCS–BEC crossover[155]. The addition of an electron-phonon coupling termto the EFKM leads to a competition between an “exci- tonic” CDW and a “phononic” CDW, while an additionalHund’s coupling promotes an excitonic SDW state [158]. Another important and intensely investigated purely elec-tronic model is the extended Hubbard model (EHM)ˆ H EHM = − t (cid:88) (cid:104) i,j (cid:105) σ ˆ c † iσ ˆ c jσ + U (cid:88) i ˆ n i ↑ ˆ n i ↓ + V (cid:88) i ˆ n i ˆ n i +1 . (20)It describes the competition between a local Hubbard re-pulsion U and a nonlocal (nearest-neighbor) repulsion V .The phase diagram at half-filling has been determinedby analytical [159–162] and numerical [163–166] methods.While there is agreement that for U (cid:46) V ( U (cid:38) V ) theground state has long-range (critical) 2 k F CDW (SDW)correlations, the criticality of the QPTs and the possibilityof an intermediate BOW phase remain under debate. Theright panel of Fig. 14 shows the currently perhaps mostaccurate DMRG phase diagram [156]. The CDW phaseis of type C0S0, whereas the SDW phase has C0S1. Be-low a critical end point, they are separated by a narrowC0S0 phase with long-range BOW order [156, 162, 165].Exactly on the CDW–BOW critical line ∆ c = 0 but ∆ c , ∆ s >
0, corresponding to an LEL (C1S0) [156, 159,164, 165]. The CDW–BOW QPT changes from continuous(XY universality, central charge c = 1) to first order atthe tricritical point ( U t /t, V t /t ) (cid:39) (5 . , .
10) [156]. TheSDW–BOW QPT is characterized by the opening of thespin gap. A detailed discussion of the low-energy theoryand correlation functions has been given in [159]. Opti-cal excitation spectra were calculated in [167]. While itdoes not account for retardation effects, the EHM sharesmany of the features of the Holstein-Hubbard, Holstein-SSH, and SSH-
U V models discussed in Sec. 2. Material-specific EHMs such as H¨uckel-Hubbard-Ohno and Peierls-Hubbard-Ohno models have been studied in detail withthe DMRG method [168, 169]. Finally, a TLL to 4 k F -CDWQPT as a function of the Coulomb interaction range canbe observed at quarter-filling [170]. We now explore the competition between traditional DWinsulators and SPT insulators (SPTIs). A prominent rep-resentative of an SPTI is the Haldane insulator (HI) phaseof the spin-1 Heisenberg chain [20]. Recently, it has beendemonstrated that an SPT state also exists in the EHMwith an additional ferromagnetic spin interaction (
J < H = ˆ H EHM + J L/ (cid:88) i =1 ˆ S i − · ˆ S i . (21) U/t V / t CDWSPTI δ/t = 0 . δ tricritical Isingpoint ( c = 7 / c = 1 /
2) first order0 5 100246
J/t = − . CDW SPTI J Fig. 15.
Phase diagram of the dimerized extended Hubbardmodel [Eq. (22)] from infinite-size DMRG. The CDW–SPTIQPT is continuous with c = 1 / c = 7 /
10 [175]. Dashed anddotted lines are the CDW–BOW–SDW phase boundaries of thepure EHM (cf. Fig. 14). Inset: phase diagram of the extendedHubbard model with an alternating ferromagnetic Heisenberginteraction [Eq. (21)] from infinite-size DMRG [171, 172, 176].
Here, ˆ S i = (cid:80) σσ (cid:48) ˆ c † iσ σ σσ (cid:48) ˆ c iσ (cid:48) . Since the EHM behaveslike a spin- chain for large U/V , the Heisenberg term inEq. (21) promotes the formation of spin-1 moments fromneighboring spins. The resulting effective antiferromag-netic spin-1 chain supports an HI phase with zero-energyedge excitations, similar to the spin-1 XXZ chain [17]. TheSPTI replaces the SDW and BOW phases of the EHM(Sec. 3.4) and reduces the extent of the CDW phase (seeinset of Fig. 15). In the SPTI, the entanglement spectrumshows a characteristic double-degeneracy of levels that isabsent in the topologically trivial CDW phase [171].A similar scenario emerges in the context of carrier-lattice coupling. The half-filled EHM with a staggeredbond dimerization δ [172–175],ˆ H = ˆ H EHM − tδ (cid:88) iσ ( − i (ˆ c † iσ ˆ c i +1 σ + H . c . ) , (22)describes the formation of an SPT phase as a result ofa Peierls instability. The bond dimerization in Eq. (22) isequivalent to mean-field BOW order in the SSH model (7).The phase diagram of the dimerized EHM (22) for δ =0 . U = 0 and small V /t ,which confirms previous RG results [172] and leads to a re-duction of the CDW phase at weak couplings. The criticalline of the continuous Ising QPT terminates at a tricriticalpoint, above which the CDW–SPTI QPT becomes first or-der. The same holds for the EHM with ferromagnetic spinexchange [Eq. (21)].The various excitation gaps are shown in Fig. 16. Forthe pure EHM both ∆ c and ∆ n vanish at the continu-ous CDW–BOW QPT. For a nonzero dimerization δ , theneutral gap closes whereas ∆ c remains finite, indicatingthat the CDW–SPTI QPT belongs to the Ising universal- ity class. At strong coupling, all gaps remain finite acrossthe QPT. The jump in the spin gap ∆ s indicates a first-order transition. At very large U , the low-lying excitationsof Eq. (22) are related to the chargeless singlet and tripletexcitations of an effective spin-Peierls Hamiltonian.At criticality, the central charge c can easily be ex-tracted from the entanglement entropy [177, 178]. For peri-odic boundary conditions, conformal field theory predictsthe von Neumann entropy to be S L ( (cid:96) ) = c ln (cid:2) Lπ sin (cid:0) π(cid:96)L (cid:1)(cid:3) + s where s is a nonuniversal constant [179]. A finite-sizeestimate for the central charge is then obtained via [180] c ∗ ( L ) = 3[ S L ( L/ − − S L ( L/ { cos[ π/ ( L/ } , (23)taking the doubled unit cell of the SPTI into account. Thebottom panel of Fig. 16 gives c ∗ ( L ), calculated along thePI-CDW QPT line by varying U and V simultaneouslyat fixed dimerization. With increasing U there is clearevidence for a crossover from c ∗ ( L ) (cid:39) / c ∗ ( L ) (cid:39) /
10, signaling Ising tricriticality. A bosonization-basedfield-theory analysis of the power-law (exponential) decayof the CDW, SDW, and BOW correlations confirms theuniversality class of the tricritical Ising model [175].
Ultracold atomic gases in optical lattices provide the pos-sibility to study not only fermions or bosons but alsoanyons. Exchanges of the latter result in a phase factor e i θ in the many-body wave function. The statistical pa-rameter θ can take on any value between 0 and π , sothat anyons interpolate between bosons and fermions [182,183]. With Haldane’s generalized Pauli principle [184], theanyon concept becomes important also in 1D systems. Afascinating question is if the HI phase observed, e.g., in theextended Bose-Hubbard model (EBHM) [178, 185] also ex-ists in the extended anyon-Hubbard model (EAHM).After a fractional Jordan-Wigner transformation of theanyon operators, ˆ a i (cid:55)→ ˆ b i e i θ (cid:80) i − l =1 ˆ n l [186], the Hamiltonianof the EAHM takes the form [181]ˆ H EAHM = − t (cid:88) i (ˆ b † i ˆ b i +1 e i θ ˆ n i + e − i θ ˆ n i ˆ b † i +1 ˆ b j )+ U (cid:88) i ˆ n i (ˆ n i − / V (cid:88) i ˆ n i ˆ n i +1 , (24)where ˆ b † i (ˆ b i ) is a bosonic creation (annihilation) operator,and ˆ n i = ˆ b † i ˆ b i = ˆ a † i ˆ a i . A boson hopping from site i + 1to site i acquires an occupation-dependent phase. Notethat anyons on the same site behave as ordinary bosons.Anyons with θ = π represent so-called “pseudofermions”,namely, they are fermions offsite but bosons onsite. If themaximum number of particles per site is restricted to n p =2, the EBHM—the θ → θ = π/ θ = 0) and n p = 2 are shown in the top panel of artin Hohenadler, Holger Fehske: Density waves in strongly correlated quantum chains 11 . . . V /t S E ( χ ) ∆ / t S E ( χ ) U/t = 4 ∆ n ∆ s ∆ c .
18 6 .
19 6 . V /t
U/t = 12 ∆ n ∆ s ∆ c .
19 6 . . . ∆ s / t δ s . . L = 30 L = 70 J/t = − . . . . . U/t c ∗ ( L ) L =32 L =48 L =64 δ/t = 0 . Fig. 16.
Top: DMRG charge ( ∆ c ), spin ( ∆ s ), and neutral( ∆ n ) gaps for the extended Hubbard model with bond dimer-ization [Eq. (22)]. Here, δ/t = 0 . U/t = 4 (left) and
U/t = 12(right). The SPTI (CDW) phase is marked in gray (white).The spin gap exhibits a jump δ s ≡ ∆ s ( V +c ) − ∆ s ( V − c ) at V c /t .Bottom: Central charge c ∗ ( L ) along the CDW–SPTI transitionline from DMRG calculations. The data indicate Ising univer-sality ( c = 1 /
2) for
U < U t and, most notably, a tricriticalIsing point with c = 7 /
10 at U t (red dotted line) [175]. Theinset shows results for the extended Hubbard model with anadditional spin-spin interaction [Eq. (21)] [171]. Fig. 17. Most notably, the HI—located between MI andDW insulating phases in the EBHM—survives for any fi-nite fractional phase, i.e., in the anyonic case [181]. Like-wise, the superfluid (SF) appears for very weak coupling.The critical values for the MI–HI QPT (squares) and theHI–DW QPT (circles) were determined from a divergenceof the correlation length ξ χ with increasing DMRG bond-dimension χ ; the model becomes critical with central charge c = 1 and c = 1 /
2, respectively.The HI may naively be expected to disappear in theEAHM with θ > T ) norinversion (ˆ I ) symmetry. However, it has been shown thatthere exists a nontrivial topological phase protected by thecombination of ˆ R z = e i π (cid:80) j (ˆ n i − and ˆ K = e i θ (cid:80) i ˆ n i (ˆ n i − / ˆ I ˆ T [181]. A nonlocal order parameter O can be con-structed that discriminates between states that are sym-metric under both ˆ K and ˆ R z and states that are not. Themiddle panel of Fig. 17 demonstrates that O can be usedto distinguish the topologically trivial MI and DW phases( O = 1) from the topologically nontrivial HI ( O = − θ = 0 θ = 0 θ = π MI HI DW c = c = / SF V /t U / t . . . . −
101 MI HI DW θ = π/ V /t O . . θ = 0 V /t (cid:15) α (cid:15) (cid:15) (cid:15) (cid:15) Fig. 17.
Top: Phase diagram of the extended anyon-Hubbardmodel (24) with n = 1, n p = 2, θ = π/ θ = 0) [177]. The dashed-dotted line with tri-angles up marks the first-order MI–DW QPT for θ = π . Mid-dle: Order parameter O (see text) for the EAHM with U/t = 5.Bottom: DMRG data for the entanglement spectrum ξ α of theextended bosonic Hubbard model with U/t = 5.
Valuable information about topological phases is alsoprovided by the entanglement spectrum { ξ α } [188]. Theconcept of entanglement is inherent in any DMRG algo-rithm based on matrix-product states. Dividing the sys-tem into two subsystems, ξ α = − λ α is determined bythe singular values λ α of the reduced density matrix [171].The lower panel of Fig. 17 shows the entanglement spec-trum of the EBHM with U/t = 5. In the HI phase theentanglement spectrum is expected to be at least four-fold degenerate, reflecting the broken Z × Z symmetry.This is clearly seen in the HI phase. By contrast, in thetrivial MI and DW phases, the lowest entanglement levelis always nondegenerate. The 1D correlated quantum systems reviewed here exhibita remarkably rich variety of physical properties that can be studied and understood in particular by powerful nu-merical methods. For the case of half-filled bands consid-ered, metallic phases are either spinless TLLs or spinfulLELs of repulsive nature, i.e., with dominant CDW orBOW correlations. The insulating phases fall into threecategories: (i) long-range ordered with a spontaneouslybroken Ising symmetry (BOW, CDW), (ii) critical withno symmetry breaking (SDW, EI), (iii) topologically non-trivial with short-range entanglement (HI). While the ex-istence of these phases and the phase transitions can inprinciple be inferred from the low-energy field theory, thedetails for a given microscopic model typically require nu-merical solutions. In particular, mean-field, variational oreven bosonization/RG approaches are in general not suf-ficient, especially for problems with retarded interactions.Despite the significant advances reviewed here, 1D cor-related quantum systems remain an active, rewarding andchallenging topic of condensed matter physics. Even withthe physics of the most fundamental models now unrav-eled, there remain many future problems of importance inrelation to experiment. The list of topics includes the effectof Jahn-Teller coupling at finite band filling, competinglong-range interactions, thermodynamics, time-dependentor nonequilibrium phenomena, as well as the coupling toa substrate or other chains.
Acknowledgments
MH was supported by the German Research Foundation(DFG) through SFB 1170 ToCoTronics and FOR 1807(
Advanced Computational Methods for Strongly CorrelatedQuantum Systems ), HF by SFB 652 und the priority pro-gramme 1648 (
Software for Exascale Computing ). We aregrateful to A. Alvermann, F. F. Assaad, D. M. Edwards,S. Ejima, F. H. L. Essler, G. Hager, E. Jeckelmann, F.Lange, M. Weber, A. Weiße, and G. Wellein for fruitfulcollaboration. We also thank S. Ejima for preparing someof the figures.
Author contributions
Both authors contributed equally to the preparation ofthe manuscript and approved it in its final form.
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