Density waves in strongly correlated quantum chains

TL;DR

This paper reviews exact numerical results for one-dimensional quantum systems with half-filled bands, focusing on various models including Peierls transitions, competing interactions, and topological states in fermion and anyon systems.

Contribution

It provides a comprehensive review of numerical results across multiple models of strongly correlated quantum chains, highlighting recent advances and diverse phenomena.

Findings

Detailed characterization of Peierls transitions in various models

Insights into symmetry-protected topological states in fermion and anyon systems

Comparison of competing interactions in one-dimensional quantum chains

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 Heisenberg models with quantum phonons, competing fermion-boson and fermion-fermion interactions, as well as symmetry-protected topological states in fermion and anyon models.

Equations

$${\hat{H}}_{\mathrm{Hol}}=-t\sum _{⟨i,j⟩\sigma }{\hat{c}}_{i\sigma }^{†}{\hat{c}}_{j\sigma }^{}+{\omega }_0\sum _i{\hat{b}}_i^{†}{\hat{b}}_i^{}-g{\omega }_0\sum _i({\hat{b}}_i^{†}+{\hat{b}}_i^{}){\hat{n}}_i.$$

$$K_{\text{c}}=\underset{L\to \infty }{\lim }{K}_{\text{c}}(L),K_{\text{c}}(L)=\pi \frac{S_{\text{c}}(q_1)}{q_1},q_1=\frac{2\pi }{L}$$

$$A(k,\omega )=\frac{1}{Z}\sum _{mn}$$

$$\times \delta [\omega -(E_n-E_m)]$$

$$B(q,\omega )=\frac{1}{Z}$$

$$\times \delta [\omega -(E_n-E_m)];$$

$$S_{\text{c}}(r)$$

$$S_{\text{s}}(r)$$

$$S_{\text{p}}(r)$$

$$\hat{H}=$$

$$-g{\omega }_0\sum _{ir\sigma }\frac{e^{-r/\xi }}{(r^2+1{)}^{3/2}}({\hat{b}}_{i+r}^{†}+{\hat{b}}_{i+r}^{}){\hat{n}}_{i\sigma }$$

$${\hat{H}}_{\text{SSH}}=-t\sum _i{\hat{B}}_i+{\omega }_0\sum _i{\hat{b}}_i^{†}{\hat{b}}_i^{}-g{\omega }_0\sum _i{\hat{B}}_i({\hat{b}}_i^{†}+{\hat{b}}_i^{}).$$

$$\hat{H}=$$

$$-g_{\text{b}}{\omega }_{0,\text{b}}\sum _i{\hat{B}}_i({\hat{b}}_{i,\text{b}}^{†}+{\hat{b}}_{i,\text{b}}^{})$$

$$\hat{H}=-t\sum _{⟨i,j⟩}{\hat{c}}_j^{†}{\hat{c}}_i({\hat{b}}_i^{†}+{\hat{b}}_j)+{\omega }_0\sum _i{\hat{b}}_i^{†}{\hat{b}}_i-\bar{\lambda }\sum _i({\hat{b}}_i^{†}+{\hat{b}}_i).$$

$$A^{\pm }(k,\omega )=\sum _n|⟨{\psi }_n^{\pm }|{\hat{c}}_k^{\pm }|{\psi }_0⟩{|}^2\delta [\omega \mp (E_n^{\pm }-E_0)],$$

$$\hat{H}={\hat{H}}_{\mathrm{Heis}}+{\hat{H}}_{\mathrm{SP}}^{(\mathrm{l},\mathrm{d})}+{\omega }_0\sum _i{\hat{b}}_i^{†}{\hat{b}}_i^{}.$$

$${\hat{H}}_{\mathrm{SP}}^{\mathrm{l}}$$

$${\hat{H}}_{\mathrm{SP}}^{\mathrm{d}}$$

$$\hat{H}={\hat{H}}_{\mathrm{Hol}}+U\sum _i{\hat{n}}_{i\uparrow }{\hat{n}}_{i\downarrow }.$$

$${\Delta }_{{\text{c}}_1}$$

$${\Delta }_{{\text{c}}_2}$$

$${\Delta }_{\text{s}}$$

$${\Delta }_{\text{n}}$$

$$S_{\text{c}}(q,\omega )$$

$$S_{\text{s}}(q,\omega )=\frac{1}{Z}\sum _{mn}{|⟨{\psi }_m|{\hat{S}}_q^z|{\psi }_n⟩|}^2{e}^{-\beta E_m}\delta [\omega -(E_n-E_m)]$$

$$\hat{H}={\hat{H}}_{\text{SSH}}+U\sum _i{\hat{n}}_{i\uparrow }{\hat{n}}_{i\downarrow }+V\sum _i{\hat{n}}_i{\hat{n}}_{i+1},$$

$${\hat{H}}_{\mathrm{EFKM}}=$$

$$+U\sum _i{\hat{c}}_i^{†}{\hat{c}}_i^{}{\hat{f}}_i^{†}{\hat{f}}_i^{}+\frac{D}{2}\sum _i\left({\hat{c}}_i^{†}{\hat{c}}_i^{}-{\hat{f}}_i^{†}{\hat{f}}_i^{}\right),$$

$${\hat{H}}_{\mathrm{EHM}}=-t\sum _{⟨i,j⟩\sigma }{\hat{c}}_{i\sigma }^{†}{\hat{c}}_{j\sigma }^{}+U\sum _i{\hat{n}}_{i\uparrow }{\hat{n}}_{i\downarrow }+V\sum _i{\hat{n}}_i{\hat{n}}_{i+1}.$$

$$\hat{H}={\hat{H}}_{\text{EHM}}+J\sum _{i=1}^{L/2}{\hat{\mathbf{S}}}_{2i-1}\cdot {\hat{\mathbf{S}}}_{2i}.$$

$$\hat{H}={\hat{H}}_{\text{EHM}}-t\delta \sum _{i\sigma }(-1{)}^i({\hat{c}}_{i\sigma }^{†}{\hat{c}}_{i+1\sigma }^{}+H.c.),$$

$$c^{\ast }(L)=\frac{3[S_L(L/2-2)-S_L(L/2)]}{\ln \{\cos [\pi /(L/2)]\}},$$

$${\hat{H}}_{\mathrm{EAHM}}=$$

$$+U\sum _i{\hat{n}}_i({\hat{n}}_i-1)/2+V\sum _i{\hat{n}}_i{\hat{n}}_{i+1},$$

Full text

1 Introduction

The properties of quasi-one-dimensional materials such as conjugated polymers, charge-transfer salts, halogen-bridged or organic superconductors are the result of a subtle interplay of charge, spin, and lattice fluctuations, in addition to the unique effects of one-dimensional (1D) correlated quantum systems. This has stimulated intense research efforts on paradigmatic fermion and fermion-boson models TNYS91 ; BS93 . In particular, the question how a quasi-1D material evolves from a metal—either a Tomonaga-Luttinger liquid (TLL) To50 ; Lu63 or a Luther-Emery liquid (LEL) Lu.Em.74 —to an insulator has remained one of the most heavily debated issues in solid state physics for decades. Apart from band structure Bl29 ; Wi31 and disorder effects An58 , electron-electron and electron-phonon interactions are the driving forces behind the metal-insulator transition in the majority of cases. Coulomb repulsion drives the transition to a Mott insulator (MI) with dominant spin-density-wave (SDW) fluctuations Mo90 , whereas the coupling to the vibrational modes of the crystal triggers the Peierls transition Pe55 to a long-range ordered charge-density-wave (CDW) or bond-order-wave (BOW) insulator Pouget2016332 . If more than one type of interaction is relevant, quantum phase transitions (QPTs) between different insulating phases become possible. Quite generally, retarded boson-mediated interactions are significantly more difficult to describe theoretically than the instantaneous Coulomb repulsion.

More recently, QPTs between topologically trivial and nontrivial states have come into the focus of attention Mo10 ; HK10 . Topological phases possess characteristic zero-energy edge excitations that reflect the topological features of the bulk Ha82b and may either arise from topological band structures or from interactions 0953-8984-25-14-143201 . The topological properties are protected by certain symmetries (e.g., inversion, time-reversal or dihedral symmetry GW09 ; PBTO12 ). Such symmetry-protected topological (SPT) states have short-range quantum entanglement CGZLW13 and may displace more conventional CDW, BOW, or SDW phases. Examples include dimerized Su-Schrieffer-Heeger (SSH) models PhysRevLett.89.077002 and the Haldane insulator Hal83 .

While the basic mechanisms underlying metal–insulator and insulator–insulator QPTs are well known, their detailed understanding in microscopic models remains a challenging and active field of research. Convincing evidence for TLL–insulator QPTs has been obtained for the $t$-$V$ model Giamarchi , the spinless Holstein and SSH models (Secs. 2.1 and 2.2), as well as the Edwards fermion-boson model (Sec. 2.4). Minimal settings for LEL–insulator QPTs are the spinful Holstein and the extended Hubbard model (Secs. 2.1 and 3.4). Insulator–metal–insulator or direct insulator–insulator QPTs have been explored in the extended Hubbard model (Sec. 3.4), the Holstein-Hubbard model (Sec. 3.1), the Holstein-SSH model with competing bond and site couplings (Sec. 2.3), and the SSH model with additional Coulomb interaction (Sec. 3.2). Extended Falicov-Kimball models (Sec. 3.3) exhibit QPTs between semimetals or semiconductors and excitonic insulators. Extended Hubbard models with either an additional alternating ferromagnetic spin interaction or a bond dimerization have topologically trivial density-wave (DW) states but also SPT phases (Sec. 4.1). Finally, more exotic bosonic or even anyonic models that can be realized, in particular, with highly tunable cold atoms in optical lattices BDZ08 exhibit superfluid, MI, CDW, and SPT states (Sec. 4.2).

In this contribution, we review the physics of a variety of lattice models for quasi-1D strongly correlated particle systems. Focusing on results from numerically exact methods such as Lanczos exact diagonalization CW85 ; WF08a , the density matrix renormalization group (DMRG) Wh92 ; Sc05 ; JF07 ; McC08 , and continuous-time quantum Monte Carlo (QMC) Rubtsov05 ; Assaad07 ; Gull_rev ; Sandvik02 ; SandvikPhononsa ; CH05 , we discuss ground-state and spectral properties and relate them to the corresponding 1D low-energy theories Voit94 ; Giamarchi . Given the enormous literature, we mostly restrict the scope to half-filled bands for which umklapp scattering can give rise to QPTs. Section 2 is devoted to the effects of fermion-boson coupling, whereas Coulomb interaction will be discussed in Sec. 3. In Sec. 4 we review recent work on SPT states. Finally, we conclude in Sec. 5.

2 Density waves from fermion-boson coupling

2.1 Holstein and Fröhlich-type models

Perhaps the simplest example of a quantum system of coupled fermions and bosons are charge carriers interacting with lattice vibrations, as described by the Holstein Hamiltonian Ho59a ($\hbar=1$)

[TABLE]

It accounts for a single tight-binding electron band emerging from nearest-neighbor hopping, quantum phonons in the harmonic approximation, and a local density-displacement electron-phonon coupling. Here, $\hat{n}_{i}=\sum_{\sigma}\hat{n}_{i\sigma}$ and $\hat{n}_{i\sigma}=\hat{c}^{\dagger}_{i\sigma}\hat{c}_{i\sigma}$, where $\hat{c}^{\dagger}_{i\sigma}$ ($\hat{c}_{i\sigma}$) creates (annihilates) a spin-$\sigma$ electron at site $i$ of a 1D lattice with $L$ sites. Similarly, $\hat{b}^{\dagger}_{i}$ ($\hat{b}_{i}$) creates (annihilates) a dispersionless optical (Einstein) phonon of frequency $\omega_{0}$. Half-filling corresponds to $n=\langle\hat{n}_{i}\rangle=1$ ($n=1/2$) for spinful (spinless) fermions.

The physics of the Holstein model is governed by the competition between the itinerancy of the electrons and the tendency of the electron-phonon coupling to “immobilize” them. Importantly, the interaction is retarded in nature, as described by the adiabaticity ratio $\omega_{0}/t$. Throughout this article, we use $t$ as the energy unit. The electron-phonon coupling is often parameterized by $\lambda=\varepsilon_{\text{p}}/2t$ in the adiabatic regime ($\omega_{0}/t\ll 1$), and by $g^{2}=\varepsilon_{\text{p}}/\omega_{0}$ in the anti-adiabatic regime ($\omega_{0}/t\gg 1$) WF98a ; WF98b ; PhysRevB.56.4484 . For the single-particle case, where $\varepsilon_{\text{p}}$ is the polaron binding energy, the Holstein model has provided important insight into the notoriously difficult problem of polaron formation and self-trapping Ho59a ; AFT10 . The half-filled case considered here provides a framework to investigate the even more intricate problem of the Peierls metal-insulator QPT of spinless Hirsch83a ; CB84 ; ZFA89 ; WF98b ; MHM96 ; BMH98 ; PhysRevB.71.235118 ; Hohenadler06 ; arXiv:1704.07913 or spinful fermions Hirsch83a ; Guinea83 ; CB84 ; JeZhWh99 ; WFWB00 ; PhysRevB.67.081102 ; CH05 ; HC07 ; PhysRevB.71.205113 ; Ba.Bo.07 ; EF09a ; FHJ08 ; EF10 ; Barkim2015 .

The spinless Holstein model is obtained from Eq. (1) by dropping spin sums and indices. Figure 1(a) shows the corresponding ground-state phase diagram from fermion-boson pseudo-site DMRG calculations JF07 . At a critical coupling $g_{c}(\omega_{0})$, a QPT from a TLL to a CDW insulator with long-range $q=2k_{\text{F}}=\pi$ order (alternating occupied and empty sites) and a $2k_{\text{F}}$ Peierls lattice distortion Pe55 takes place. The insulating state can be classified as a traditional band insulator in the adiabatic regime, and as a polaronic superlattice in the anti-adiabatic regime FHW00 ; Hohenadler06 . Numerical evidence for the Kosterlitz-Thouless KT73 transition expected from the low-energy TLL description and the mapping to the $t$-$V$ model at strong coupling Hirsch83a comes from, e.g., XXZ-model physics for large $\omega_{0}$ Hohenadler06 and a cusp in the fidelity susceptibility arXiv:1704.07913 .

The TLL charge parameter $K_{\text{c}}$—determining the decay of correlation functions Voit94 ; Giamarchi —from a finite-size scaling of the long-wavelength limit of the charge structure factor $S_{\text{c}}(q)=\frac{1}{L}\sum_{j,l}e^{iq(j-l)}\langle\hat{n}_{j}\hat{n}_{l}\rangle$ according to Giamarchi ; EF09a

[TABLE]

is shown in Fig. 1(b). Contrary to earlier numerical results FHW00 ; WF98b , the TLL turns out to be repulsive ($K_{\text{c}}<1$) for any $\omega_{0}$ EF09a . Accordingly, charge correlations ($\sim r^{-2K_{\text{c}}}$) dominate over pairing correlations ($\sim r^{-2/K_{\text{c}}}$) throughout the TLL phase and show a crossover from weak to strong $2k_{\text{F}}$ power-law correlations with increasing coupling PhysRevB.92.245132 ; arXiv:1704.07913 . As shown in Fig. 1, $K_{\text{c}}=1/2$ at the critical point, as expected for the umklapp-driven Mott transition in a spinless TLL Giamarchi .

Figure 2 shows excitation spectra from QMC simulations Hohenadler10a ; PhysRevB.94.245138 , namely the single-particle spectral function

[TABLE]

and the phonon spectral function

[TABLE]

$E_{n}$ is the eigenvalue for $\left|\psi_{n}\right\rangle$, $Z$ the partition function.

In the adiabatic regime, the single-particle spectrum in the TLL phase [Fig. 2(a)] is gapless but significantly modified by the hybridization of charge and phonon modes Meden94 ; Hohenadler06 ; Hohenadler10a . In the CDW phase, it exhibits a Peierls gap and backfolded shadow bands Vo.Pe.Zw.Be.Ma.Gr.Ho.Gr.00 ; FeWeHaWeBi03 ; Hohenadler06 ; Hohenadler10a [Fig. 2(b)]. Near the critical point, soliton excitations PhysRevLett.42.1698 can be observed Hohenadler10a . The phonon spectrum PhysRevB.94.245138 reveals the renormalization of the phonon mode due to electron-phonon coupling. In the adiabatic regime, the mode softens at the zone boundary in the TLL phase [Fig. 2(c)], becomes completely soft for $q=2k_{\text{F}}$ at the critical point [Fig. 2(d)], and hardens again in the CDW phase CrSaCa05 ; SyHuBeWeFe04 ; Hohenadler10a ; PhysRevB.94.245138 . In contrast, for $\omega_{0}\gg t$, the phonon mode hardens in the metallic phase and a central mode appears at $\lambda_{c}$ Hohenadler06 . These findings are consistent with a soft-mode transition for $\omega_{0}\ll t$ and a central-peak transition for $\omega_{0}\gg t$ Hohenadler06 .

A complete picture of the physics of the spinful Holstein model (1) has only emerged recently. Whereas early work Hirsch83a ; PhysRevB.71.205113 ; Ba.Bo.07 suggested the absence of a metallic phase, the existence of the latter has since been confirmed JeZhWh99 ; CH05 ; HC07 ; FHJ08 ; EF10 ; Barkim2015 ; for a detailed review see PhysRevB.92.245132 . In terms of g-ology Giamarchi , the attractive umklapp scattering arising from the Holstein coupling remains irrelevant for $\lambda<\lambda_{c}(\omega_{0})$. However, for any $\lambda>0$, attractive backscattering opens a spin gap PhysRevB87.075149 . Therefore, the metallic phase is in fact a 1D spin-gapped metal—also known as an LEL Lu.Em.74 . Using the notation C$x$S$y$ for a system with $x$ ($y$) gapless charge (spin) modes PhysRevB.53.12133 , the LEL has C1S0. For $\lambda>\lambda_{c}$, umklapp scattering is relevant and the ground state is a $2k_{\text{F}}$ CDW insulator (alternating doubly occupied and empty sites) with C0S0. Estimates for $\lambda_{c}$ are contained in the phase diagram of the Holstein-Hubbard model in Fig. 11 in the limit $U=0$.

LEL physics and the Peierls QPT are also revealed by the real-space correlation functions

[TABLE]

measuring charge, spin, and s-wave pairing correlations. As in the spinless case, charge correlations dominate over pairing in the metallic phase PhysRevB.84.165123 ; PhysRevB87.075149 ; PhysRevB.92.245132 , see Fig. 3. Such behavior necessarily requires a spin gap Voit98 ($K_{\text{s}}=0$) and repulsive interactions ($K_{\text{c}}<1$). The spin gap complicates the determination of $K_{\text{c}}$ PhysRevB.92.245132 but the correlation functions in Fig. 3 clearly rule out claims of dominant pairing CH05 . Spectral properties of the spinful Holstein model have also been calculated ZhJeWh99 ; NiZhWuLi05 ; Ho.As.Fe.12 ; PhysRevB87.075149 ; WeAsHo15I . Most notably, the single-particle spectrum is gapped even in the metallic phase (although the spin gap—not taken into account in Meden94 —is difficult to detect numerically at weak coupling), and the phonon spectrum reveals a soft-mode transition similar to the spinless case for $\omega_{0}/t<1$ WeAsHo15I .

A local electron-phonon interaction as in the Holstein model (1) is a priori not justified for materials with incomplete screening. Results for nonlocal interactions in the empty-band limit reveal significantly reduced polaron and bipolaron masses 0034-4885-72-6-066501 . More recently, numerical results for half-filling were obtained Ho.As.Fe.12 . As a function of the screening length $\xi$, the Hamiltonian Ho.As.Fe.12

[TABLE]

interpolates between a local Holstein and a long-range Fröhlich-type coupling AlKo99 . As shown in Fig. 4(a), for small to intermediate $\xi$, the same LEL and CDW phases are found, but $\lambda_{c}$ is enhanced with increasing $\xi$. For large $\xi$ and strong coupling, the nonlocal interaction gives rise to multipolaron droplets and phase separation (PS) Ho.As.Fe.12 , as detected from the $q=0$ divergence of the charge structure factor [Fig. 4(b)] that implies $K_{\text{c}}=\infty$ and hence a divergent compressibility Giamarchi . The CDW–PS QPT appears to be of first order Ho.As.Fe.12 . Increasing the interaction range at a fixed $\lambda$ drives a CDW–LEL QPT. The concomitant suppression of CDW order gives rise to degenerate pairing and charge correlations in the Fröhlich limit $\xi\to\infty$.

2.2 Su-Schrieffer-Heeger model

The SSH model of polyacetylene captures fluctuations of the carbon-carbon bond lengths and their effect on the electronic hopping integral PhysRevLett.42.1698 (for related earlier work see PhysRevLett.25.919 ). It has a coupling term of the form $\sum_{i}\hat{B}_{i}(\hat{Q}_{i+1}-\hat{Q}_{i})$, where $\hat{Q}_{i}\sim\hat{b}^{\dagger}_{i}+\hat{b}^{\phantom{{\dagger}}}_{i}$ is the displacement of atom $i$ from its equilibrium position and $\hat{B}_{i}=\sum_{\sigma}(\hat{c}^{\dagger}_{i\sigma}\hat{c}^{\phantom{{\dagger}}}_{i+1\sigma}+\text{H.c.})$. The phonons have an acoustic dispersion $\omega_{q}=\omega_{0}\sin(q/2)$. Reviews in the context of conjugated polymers were given in Baeriswylreview92 ; Barfordbook . Theoretical arguments PhysRevB.27.1680 ; PhysRevB.91.245147 and exact numerical results PhysRevB.91.245147 suggest that at half-filling the SSH model is equivalent to the simpler optical SSH model PhysRevB.67.245103

[TABLE]

Here, $\hat{b}^{\dagger}_{i}$ and $\hat{b}^{\phantom{{\dagger}}}_{i}$ are associated with an optical phonon mode describing fluctuations of the bond lengths. Figure 5 illustrates the quantitative agreement of the two models for the single-particle Green function and the dynamic bond structure factor, which can be attributed to the inherent dominance of $q=2k_{\text{F}}=\pi$ order at half-filling PhysRevB.91.245147 .

For spinless fermions, the model (7) has a repulsive TLL ground state with dominant BOW correlations below a critical coupling $\lambda_{c}$ (for SSH models $\lambda=g^{2}\omega_{0}/2t$), and an insulating Peierls ground state with a long-range $2k_{\text{F}}$ BOW (alternating weak and strong bonds) for $\lambda>\lambda_{c}(\omega_{0})$ PhysRevB.27.1680 ; Ba.Bo.07 ; PhysRevB.91.245147 . The transition from power-law to long-range BOW correlations can be seen in Fig. 5(b) from $S_{\text{b}}(r)=\langle(\hat{B}_{r}-\langle\hat{B}_{r}\rangle)(\hat{B}_{0}-\langle\hat{B}_{0}\rangle)\rangle$. At the critical point, the correlation functions are consistent with $K_{\text{c}}=1/2$. In the adiabatic regime, the phase transition is again of the soft-mode type PhysRevB.91.245147 . Apart from the interchange of the roles of charge and bond degrees of freedom, the spinless SSH model is in many respects similar to the spinless Holstein model, including spectral and thermodynamic properties PhysRevB.91.245147 ; PhysRevB.94.155150 . However, subtle differences arise due to the different symmetries of the two models (class BDI of the general classification PhysRevB.78.195125 for the SSH model, class AI with broken particle-hole and chiral symmetry for the Holstein model) PhysRevB.94.155150 . Note that the name SSH model is often used to refer to the mean-field approximation of the true SSH model, i.e., a fermionic Hamiltonian with dimerized hopping but no phonons (see also Sec. 4.1).

In contrast to the spinful Holstein model, the spinful SSH model does not have a metallic phase. Although quantum fluctuations significantly reduce the dimerization compared to the mean-field solution PhysRevB.25.7789 , the ground state is an insulating BOW-Peierls state (C0S0) for any $\lambda>0$ irrespective of $\omega_{0}$ Su1982497 ; PhysRevB.27.1680 ; PhysRevLett.60.2089 ; PhysRevB.67.245103 ; PhysRevB.73.045106 ; Ba.Bo.07 ; Barkim2015 ; PhysRevB.91.245147 . Direct numerical evidence for this conclusion is shown in Fig. 5(c). Because Eq. (7) is symmetric under the transformation $\hat{c}^{\phantom{{\dagger}}}_{i\downarrow}\mapsto(-1)^{i}\hat{c}^{\dagger}_{i\downarrow}$ that interchanges spin and charge operators, spin and charge correlators are exactly equal. Therefore, the finite-size estimate of $K_{\text{s}}(L)<1$ (a reliable indicator of a spin gap for models with SU(2) spin symmetry HC07 ) implies $K_{\text{s}}=K_{\text{c}}=0$ by symmetry and hence an insulating state PhysRevB.91.245147 . The spinful SSH model hence provides an example where Peierls’ theorem Pe55 holds even for quantum phonons. This property can be traced back to the fact that forward scattering vanishes whereas umklapp scattering is repulsive (rather than attractive, as in the Holstein model) and hence always relevant Barkim2015 . Excitation spectra for the spinful SSH model closely resemble those of the spinless model in the ordered phase PhysRevB.91.245147 .

2.3 Holstein-Su-Schrieffer-Heeger model

Whereas Holstein and SSH models have been studied intensely, the even more complex problem of competing site and bond couplings—which in principle coexist in most materials—has been addressed by QMC only recently PhysRevLett.117.206404 using the Holstein-SSH Hamiltonian

[TABLE]

with independent site ($\alpha=\text{s}$) and bond ($\alpha=\text{b}$) phonon modes as well as corresponding coupling constants $\lambda_{\alpha}$.

Of particular interest is the question if the metallic phase of the Holstein model is stable with respect to the SSH coupling, or if metallic behavior is entirely absent as in the SSH model (7). In terms of g-ology, both couplings produce negative backscattering matrix elements that give rise to a spin gap. On the other hand, the umklapp matrix elements have opposite sign and can therefore compensate, allowing for an extended LEL (C1S0) metallic region. This picture is confirmed by QMC data PhysRevLett.117.206404 summarized in the qualitative phase diagram in Fig. 6. If the SSH coupling dominates, the system is a BOW insulator just like the SSH model. If the Holstein coupling dominates, a CDW ground state exists. Both states are of type C0S0. If the couplings are comparable, the competition between the two orders results in a metallic LEL phase. Starting in the CDW phase and increasing $\lambda_{\text{b}}$, the correlation functions in Fig. 7 reveal a suppression (enhancement) of CDW (BOW) order and a QPT to the LEL phase with power-law correlations. At stronger SSH couplings, long-range BOW order emerges. For all parameters, spin correlations remain exponential due to the spin gap. The QPT between the two different Peierls states is found to be continuous, and in the adiabatic regime involves two soft-mode transitions for the site and bond phonon modes, respectively PhysRevLett.117.206404 . The single-particle gap is minimal but finite at the QPT PhysRevLett.117.206404 . These numerical results contradict earlier approximate results suggesting a first-order BOW–CDW transition PhysRevB.28.4833 or a ferroelectric phase with coexistence of BOW and CDW order 0953-8984-2-16-003 .

2.4 Edwards model

The discussion so far has revealed that the coupling to the lattice can modify the transport properties of low-dimensional systems to the point of insulating behavior. Quantum transport in general takes place in some “background”, which may consist of lattice but also spin or orbital degrees of freedom. For instance, a key problem in the widely studied high-$T_{c}$ cuprates BM86 and colossal magnetoresistance manganites JS50 is that of (doped) holes moving in an ordered magnetic insulator Be09 . As the holes move, they disrupt the order of the background which, conversely, hinders hole motion. Coherent motion may still occur, albeit on a reduced energy scale determined by the fluctuations and correlations in the background.

A fermion-boson model by Edwards describes the interaction of particles with the background in terms of a coupling to bosonic degrees of freedom Ed06 ; AEF07 :

[TABLE]

In this model, every hop of a (spinless fermionic) charge carrier along a 1D transport path either creates a (local bosonic) excitation with energy $\omega_{0}$ in the background at the site it leaves, or annihilates an existing excitation at the site it enters. The fermion-boson coupling in Eq. (9) differs significantly from the Holstein and SSH couplings discussed before. In particular, no static distortion arises in the limit $\omega_{0}\to 0$. Furthermore, spontaneous boson creation and annihilation processes are possible, i.e., the background distortions can relax with a relaxation rate $\bar{\lambda}$, for example due to quantum fluctuations. Any particle motion is affected by the background and vice versa. In fact, the Edwards model describes three different regimes: quasi-free, diffusive, and boson-assisted transport AEF07 . In the latter case, excitations of the background are energetically costly ($\omega_{0}/t>1$) and the background relaxation rate is small ($\bar{\lambda}/t\ll 1$), i.e., the background is “stiff”. Then, for a half-filled band, strong correlations can develop and even drive the system into an insulating state by establishing long-range CDW order WFAE08 ; EHF09 ; EF09b .

Figure 8(a) shows the DMRG phase diagram at half-filling. The critical values were determined from the charge gap and the CDW order parameter EHF09 . The metallic phase found below a critical boson frequency and above a critical relaxation rate is a repulsive TLL ($K_{\text{c}}<1$) EHF09 . Remarkably, particle motion is possible even for $\bar{\lambda}=0$, in lowest order by a vacuum-restoring six-step process where 3 bosons are excited in steps 1-3 and afterwards consumed in steps 4-6 with the particle moving two sites AEF07 . In contrast to the spinless Holstein model, the CDW state of the Edwards model is a few-boson state WFAE08 . As shown in Fig. 8(b), at low and high band filling $n$ the attractive interaction mediated by the slow bosons becomes strong enough to give rise to first an attractive TLL ($K_{\text{c}}>1$) and finally electronic phase separation ESBF12 .

The metal–insulator QPT at half-filling is also reflected in the photoemission spectra measured by

[TABLE]

where $\hat{c}^{+}_{k}=\hat{c}^{\dagger}_{k}$, $\hat{c}^{-}_{k}=\hat{c}_{k}$, $|\psi_{0}\rangle$ is the ground state for $N_{\text{e}}$ particles and $|\psi_{n}^{\pm}\rangle$ the $n$-th excited state with $N_{\text{e}}\pm 1$ particles. Figure 9 shows $A(k,\omega)=A^{-}(k,\omega)+A^{+}(k,\omega)$ in the regime where background excitations have a large energy and the bosons strongly affect particle transport. The quasiparticle mass is significantly enhanced and a renormalized band structure appears. However, if $\bar{\lambda}$ is sufficiently large, the system remains metallic, as indicated by a finite spectral weight at the Fermi energy $E_{\text{F}}$ [Fig. 9(a)]. As the possibility of relaxation reduces a gap opens at $k_{\text{F}}=\pi/2$ [Fig. 9(b)], indicating insulating behavior. We note the internal feedback mechanism: The collective boson excitations originate from the motion of the charge carriers and have to persist long enough to finally inhibit particle transport, thereby completely changing the nature of the many-particle ground state. This complex boson-particle dynamics leads to a new, correlation-induced band structure reminiscent of the (extended) Falicov-Kimball model, with a very narrow valence band and a rather broad conduction band [inset of Fig. 9(b)]. The asymmetric masses can be understood by “doping” a perfect CDW state: To restore the CDW order a doped hole has to be transferred by the above-mentioned six-step process of order ${\cal O}(t^{6}/\omega_{0}^{5})$, while a doped particle can move by a two-step process of order ${\cal O}(t^{2}/\omega_{0})$ WFAE08 . By decreasing $\omega_{0}$ at fixed $\bar{\lambda}$ the fluctuations overcome the correlations and the system returns to a metallic state. However, the latter differs from the state we started with. In particular, $A(k,\omega)$ in Fig. 9(c) shows sharp absorption features only near $k_{\text{F}}$ and “overdamping” at the zone boundaries where the spectrum is dominated by bosonic excitations.

2.5 Heisenberg spin-Peierls models

The Peierls (dimerization) instability triggered by the lattice degrees of freedom can be observed not only in quasi-1D itinerant electron systems but also in spin chains with magneto-elastic couplings. Experimentally, such behavior was first seen in the 1970s for organic compounds of the TTF and TCNQ family Brea75 . Interest in the subject revived after the discovery of the first inorganic spin-Peierls compound $\rm CuGeO_{3}$ in 1993 HTU93 , in particular due to the fact that the displacive spin-Peierls transition in this material does not involve phonon softening. Instead, the Peierls-active optical phonon modes with frequencies $\omega_{0,1}\simeq J$ and $\omega_{0,2}\simeq 2J$ ($J$ being the exchange coupling between neighboring $\rm Cu^{2+}$ ions that form well separated spin-$\frac{1}{2}$ chains) harden by about 5% at the transition which therefore occurs at very strong spin-phonon coupling BHRDR98 . Phonon hardening for experimentally relevant parameters was demonstrated for the magnetorestrictive XY model by calculating the dynamic structure factor HFW01 . The physics of $\rm CuGeO_{3}$ reveals that the canonical adiabatic treatment of the lattice Py74 ; CF79 is inadequate for this material BFKW99 . Instead, the application of numerical methods to paradigmatic quantum models yields key information about the nature of the phase transition and the correct models for inorganic spin-Peierls materials.

The simplest model containing all important features of a spin-Peierls system is an antiferromagnetic Heisenberg chain, $\hat{H}_{\rm Heis}=J\sum_{i}\hat{{\bm{S}}}_{i}\cdot\hat{{\bm{S}}}_{i+1}$ ($J<0$, $\hat{{\bm{S}}}_{i}$ is a spin-$\frac{1}{2}$ operator at site $i$), coupled to Einstein quantum phonons:

[TABLE]

Here we consider two different spin-phonon couplings,

[TABLE]

The local coupling $\hat{H}_{\rm SP}^{\rm l}$ captures the modification of the spin exchange by a local lattice degree of freedom (modeling, e.g., side-group effects) WFK98 ; WWF99 . The difference coupling $\hat{H}_{\rm SP}^{\rm d}$ describes a linear dependence of the spin exchange on the difference between the phonon amplitudes at sites $i$ and $i+1$ BMH99 . A first insight into these models can be gained by integrating out the phonons in the anti-adiabatic limit $\omega_{0}\gg J$ to obtain an effective Heisenberg model with longer-ranged interactions that give rise to frustration. The spin Hamiltonian $\hat{H}=J\sum_{i}(\hat{{\bm{S}}}_{i}\cdot\hat{{\bm{S}}}_{i+1}+\alpha\hat{{\bm{S}}}_{i}\cdot\hat{{\bm{S}}}_{i+2})$ has a dimerized ground state (alternating long and short bonds) for $\alpha\geq\alpha_{c}=0.241\,167$ ON92 . Accordingly, the spin-phonon coupling must be larger than a nonzero critical value $g_{c}(\omega_{0})$ for the spin-Peierls instability to occur Uh98 ; WWF99 ; BMH99 . This is similar to the Holstein model (1) but in contrast to the static limit $\omega_{0}/J=0$.

Figure 10 shows the phase diagram of the model (11) for either the coupling (12) or (13) from two-block BMH99 and four-block WHBF06 DMRG calculations, respectively. The QPT from the the gapless spin-liquid state to the gapped dimerized phase was detected using the well-established criterion of a level crossing between the first singlet and the first triplet excitation; the latter was derived for the frustrated spin chain AGSZ89 ; BMH99 ; HWWJF07 . For finite systems, the singlet lies above the triplet excitation in the spin liquid, and the two levels become degenerate with the singlet ground state as $L\to\infty$. In the symmetry-broken gapped phase, the lowest singlet state becomes degenerate with the ground state. The Heisenberg spin-Peierls model with quantum phonons is in the same universality class as the frustrated spin chain BMH99 . The phonon spectral function has been analyzed in arXiv:0705.0799 . Related spin-boson models exhibiting TLL and CDW phases have also been investigated in the context of dissipative quantum systems PhysRevLett.113.260403 .

3 Density waves and Coulomb interaction

3.1 Holstein-Hubbard model

From the 1D Hubbard model Hu63 , it is well established that a local Coulomb repulsion favors a correlated MI with dominant $2k_{\text{F}}$ SDW fluctuations Giamarchi . In contrast to CDW and BOW order, the continuous SU(2) spin symmetry cannot be spontaneously broken PhysRevLett.17.1133 . Instead, the SDW correlations are critical ($\sim 1/r$) Schulz90 ; Giamarchi . Of key interest is the interplay or competition of retarded electron-phonon and instantaneous electron-electron interactions that determines if the ground state is a CDW/BOW, SDW, or LEL state. A minimal but rich model capturing this interplay is the Holstein-Hubbard model with Hamiltonian

[TABLE]

The ground-state phase diagram of Eq. (14) was the subject of intense debate. Even after early claims Hirsch83a (unfounded PhysRevB.92.245132 but supported by RG calculations PhysRevB.71.205113 ; Ba.Bo.07 ) of the absence of metallic behavior in the spinful Holstein model were contradicted by DMRG results JeZhWh99 , numerical work on the Holstein-Hubbard model initially focused on strong couplings where a direct SDW–CDW QPT is observed. Evidence for an intermediate metallic phase—expected from the adiabatic connection to the Holstein model as $U\to 0$—at weaker couplings was obtained with a variety of different methods TC03 ; TAA05 ; TAA07 ; CH05 ; HC07 ; FHJ08 ; EF10 . Very recently, it was shown that the absence of such a phase in RG calculations PhysRevB.71.205113 ; Ba.Bo.07 is due to the neglected momentum dependence of the interaction Barkim2015 . A detailed discussion of these contradictory findings was given in PhysRevB.92.245132 .

The currently most reliable phase boundaries come from DMRG FHJ08 ; EF10 and QMC CH05 ; HC07 calculations. The DMRG critical values shown in Fig. 11 were obtained from a finite-size scaling of the single-particle, charge-, spin-, and neutral gaps defined as

[TABLE]

$E_{0}(N_{\text{e}},S^{z}_{\text{tot}})$ [$E_{1}(N_{\text{e}},S^{z}_{\text{tot}})$] is the energy of the ground-state (first excited state) of a system with $L$ sites, $N_{\text{e}}$ electrons and total spin-$z$ $S_{\text{tot}}^{z}$. The CDW state has C0S0 ($\Delta_{\text{c}1}>0$, $\Delta_{\text{s}}>0$), whereas the SDW state has C0S1 ($\Delta_{\text{c}1}>0$, $\Delta_{\text{s}}=0$). The different nature of excitations in these phases is clearly visible in the spectra in Fig. 12 (for previous work see Voit98 ; FeWeHaWeBi03 ; WeAsHo15I ). The single-particle spectral function in Figs. 12(a) and (b) has a gap at $E_{\text{F}}$ in both phases, but distinct soliton excitations and backfolded shadow bands only in the CDW phase. Spin-charge separation Giamarchi can be observed for strong interactions PhysRevB.75.205128 ; NiZhWuLi05 . The dynamic charge structure factor

[TABLE]

with $\hat{\rho}_{q}=\sum_{r}e^{iqr}(\hat{n}_{r}-n)/\sqrt{L}$ in Figs. 12(c),(d) reveals a $q=0$ charge gap in both insulating phases and the renormalized phonon frequency in the CDW phase [Fig. 12(c)]. Finally, the dynamic spin structure factor

[TABLE]

shows a clear spin gap in the CDW phase [Fig. 12(e)] whereas the SDW phase has $\Delta_{\text{s}}=0$ and strong $2k_{\text{F}}=\pi$ fluctuations [Fig. 12(f)].

Similar to the Holstein model, the intermediate phase is a spin-gapped LEL (C1S0, $\Delta_{\text{c}1}>0$, $\Delta_{\text{s}}>0$, but $\Delta_{\text{c}2}=0$) HC07 ; FHJ08 ; PhysRevB87.075149 ; Barkim2015 . In the anti-adiabatic regime [Fig. 11(b)], where retardation effects are small, the LEL–SDW QPT occurs close to the value $U=4\lambda t$ expected from an effective Hubbard model. Whereas DMRG and QMC results agree quite well for the LEL–SDW QPT, the LEL–CDW QPT line is not completely settled. The exponential opening of the charge gap is nontrivial to detect with the DMRG, and the charge susceptibility used in QMC CH05 is problematic due to the spin gap PhysRevB.92.245132 . The latter also complicates the calculation of TLL parameters PhysRevB.92.245132 . The intermediate LEL phase has $K_{\text{s}}=0$, so that the low-energy theory is that of bosonic pairs (bipolarons). $K_{\text{c}}$ as extracted from the electronic density structure factor gives $K_{\text{c}}>1$ even though pairing correlations are always subdominant Barkim2015 (see also Fig. 3). Moreover, in contrast to the extended Hubbard model, there is no symmetry argument for $K_{\text{c}}=1$ at the LEL–CDW QPT. An interesting open problem is to reconcile the vanishing of the (bipolaron) binding energy in parts of the LEL phase EF10 with the nonzero spin gap. Finally, CDW and SDW states of the Holstein-Hubbard model have been studied numerically in the context of pump-probe experiments JPSJ.81.013701 ; PhysRevB.79.125118 ; PhysRevB.84.195109 ; PhysRevB.88.064303 ; PhysRevLett.109.246404 ; PhysRevLett.109.176402 ; PhysRevLett.116.086401 .

3.2 SSH-$UV$ model

The competition between electron-phonon and electron-electron interaction has also been studied in the framework of the SSH-$UV$ Hamiltonian

[TABLE]

which is directly relevant for conjugated polymers Baeriswylreview92 ; PhysRevB.57.11838 . The phase diagram from QMC simulations PhysRevB.67.245103 is shown in Fig. 13. A key difference to the Holstein-Hubbard model is that no metallic phase results from the competing interactions. Instead, for $\omega_{0}>0$ and realistic values $U>2V$, the ground state is a MI with critical SDW correlations (C0S1) for $\lambda<\lambda_{c}$, and a BOW Peierls state (C0S0) for $\lambda>\lambda_{c}$ CB84 ; PhysRevB.67.245103 ; PhysRevB.83.195105 ; PhysRevB.91.245147 ; in contrast to the continuous suppression of CDW correlations by the Hubbard repulsion in the Holstein-Hubbard model, the amplitude of BOW correlations is enhanced by the Coulomb repulsion in the SSH-$UV$ model PhysRevB.26.4278 ; PhysRevLett.51.292 ; PhysRevLett.51.296 ; PhysRevLett.62.1053 ; PhysRevB.91.245147 . Finally, for large $U$, the SSH-$UV$ model is closely related to the spin-Peierls models discussed in Sec. 2.5 PhysRevB.67.245103 .

3.3 Extended Falicov-Kimball model

CDW, BOW, SDW or orbital DW states can also arise purely from the Coulomb interaction, so that extensions of the Hubbard Hamiltonian may be regarded as minimal theoretical models. An important example is the asymmetric Hubbard model with spin-dependent band energies $\varepsilon_{k\sigma}=E_{\sigma}-2t_{\sigma}\cos k$, where $E_{\sigma}$ defines the center of the spin-$\sigma$ band and $t_{\sigma}$ is the nearest-neighbor hopping amplitude LD94 ; Fa08 . For $E_{\uparrow}<E_{\downarrow}$ and $t_{\uparrow}t_{\downarrow}<0$ ($t_{\uparrow}t_{\downarrow}>0$) a direct (indirect) band gap is realized. The $\sigma$-electron density $n_{\sigma}=\frac{1}{L}\sum_{k}\langle\hat{c}_{k\sigma}^{\dagger}\hat{c}_{k\sigma}\rangle$, with $n_{\uparrow}+n_{\downarrow}=1$ at half-filling. The asymmetric Hubbard model has been used to investigate various many-body effects in (mixed/intermediate-valence) rare-earth and transition-metal compounds, including the DW–PS QPT WCG75 , electronic ferroelectricity Ba02b ; YMDLH03 and (pressure-induced) exciton condensation IPBBF08 , as well as multiorbital correlation physics in cold atoms WCR12 . Regarding $\sigma$ as an orbital flavor, the asymmetric Hubbard model is equivalent to the extended Falicov-Kimball model (EFKM) LD94 ; Fa08 ; Ba02b ; ZIBF12

[TABLE]

describing two species of spinless fermions, namely, light $c$ (or $d$) electrons and heavy $f$ electrons. A finite $f$-bandwidth allows for $f$-$c$ electron coherence, which will take account of a mixed-valence situation as well as of $c$-electron $f$-hole (exciton) bound-state formation and condensation ZIBF12 ; Ku15 . By contrast, $t_{f}=0$ in the original FKM FK69 , so that the number of $f$-electrons is strictly conserved and no coherence between $f$ and $c$ electrons can arise SB88 .

The left panel of Fig. 14 shows the DMRG phase diagram at half-filling. Depending on the orbital level splitting there exist staggered orbital ordered (SOO) or band insulator (BI) phases, separated by a critical excitonic insulator (EI) JRK67 . In the absence of true long-range order in one dimension, the characteristic signatures of an excitonic Bose-Einstein condensate are a power-law decay of the correlator $\langle\hat{X}_{i}^{\dagger}\hat{X}_{j}^{\phantom{\dagger}}\rangle$ with $\hat{X}_{i}^{\dagger}=\hat{c}_{i}^{\dagger}\hat{f}_{i}^{\phantom{\dagger}}$ and a divergence of the excitonic momentum distribution $N(q)=\langle\hat{X}_{q}^{\dagger}\hat{X}_{q}^{\phantom{\dagger}}\rangle$ with $\hat{X}_{q}^{\dagger}=\frac{1}{\sqrt{L}}\sum_{k}\hat{c}_{k+q}^{\dagger}\hat{f}_{k}^{\phantom{\dagger}}$ for the lowest-energy state (which has $q=0$ for the case of a direct gap). The criticality of the EI can also be detected from the von Neumann entropy and the central charge [$c^{*}(L)\simeq 1$]. Monitoring the coherence length and binding energy with increasing $U$ yields clear evidence for a BCS–BEC crossover EKOF13 . The addition of an electron-phonon coupling term to the EFKM leads to a competition between an “excitonic” CDW and a “phononic” CDW, while an additional Hund’s coupling promotes an excitonic SDW state KZFO15 .

3.4 Extended Hubbard model

Another important and intensely investigated purely electronic model is the extended Hubbard model (EHM)

[TABLE]

It describes the competition between a local Hubbard repulsion $U$ and a nonlocal (nearest-neighbor) repulsion $V$. The phase diagram at half-filling has been determined by analytical PhysRevB.45.4027 ; Na99 ; TF02 ; TTC06 and numerical Hi84 ; Je02 ; SBC04 ; PhysRevB.70.235107 methods. While there is agreement that for $U\lesssim 2V$ ($U\gtrsim 2V$) the ground state has long-range (critical) $2k_{\text{F}}$ CDW (SDW) correlations, the criticality of the QPTs and the possibility of an intermediate BOW phase remain under debate. The right panel of Fig. 14 shows the currently perhaps most accurate DMRG phase diagram EN07 . The CDW phase is of type C0S0, whereas the SDW phase has C0S1. Below a critical end point, they are separated by a narrow C0S0 phase with long-range BOW order SBC04 ; TTC06 ; EN07 . Exactly on the CDW–BOW critical line $\Delta_{\text{c}_{2}}=0$ but $\Delta_{\text{c}_{1}},\Delta_{\text{s}}>0$, corresponding to an LEL (C1S0) PhysRevB.45.4027 ; Je02 ; SBC04 ; EN07 . The CDW–BOW QPT changes from continuous (XY universality, central charge $c=1$) to first order at the tricritical point $(U_{\text{t}}/t,V_{\text{t}}/t)\simeq(5.89,3.10)$ EN07 . The SDW–BOW QPT is characterized by the opening of the spin gap. A detailed discussion of the low-energy theory and correlation functions has been given in PhysRevB.45.4027 . Optical excitation spectra were calculated in PhysRevB.67.075106 . While it does not account for retardation effects, the EHM shares many of the features of the Holstein-Hubbard, Holstein-SSH, and SSH-$UV$ models discussed in Sec. 2. Material-specific EHMs such as Hückel-Hubbard-Ohno and Peierls-Hubbard-Ohno models have been studied in detail with the DMRG method PhysRevB.87.245116 ; PhysRevB.95.085150 . Finally, a TLL to $4k_{\text{F}}$-CDW QPT as a function of the Coulomb interaction range can be observed at quarter-filling PhysRevB.85.195115 .

4 Density waves and symmetry protection

4.1 Dimerized Extended Hubbard model

We now explore the competition between traditional DW insulators and SPT insulators (SPTIs). A prominent representative of an SPTI is the Haldane insulator (HI) phase of the spin-$1$ Heisenberg chain Hal83 . Recently, it has been demonstrated that an SPT state also exists in the EHM with an additional ferromagnetic spin interaction ($J<0$) on every other bond LEF15 ,

[TABLE]

Here, \hat{\bm{S}}_{i}=\mbox{\frac{1}{2}}\sum_{\sigma\sigma^{\prime}}\hat{c}_{i\sigma^{\phantom{\prime}}}^{\dagger}{\bm{\sigma}}_{\sigma\sigma^{\prime}}^{\phantom{\dagger}}\hat{c}_{i\sigma^{\prime}}^{\phantom{\dagger}}. Since the EHM behaves like a spin-$\frac{1}{2}$ chain for large $U/V$, the Heisenberg term in Eq. (21) promotes the formation of spin-$1$ moments from neighboring spins. The resulting effective antiferromagnetic spin-$1$ chain supports an HI phase with zero-energy edge excitations, similar to the spin-$1$ XXZ chain PBTO12 . The SPTI replaces the SDW and BOW phases of the EHM (Sec. 3.4) and reduces the extent of the CDW phase (see inset of Fig. 15). In the SPTI, the entanglement spectrum shows a characteristic double-degeneracy of levels that is absent in the topologically trivial CDW phase LEF15 .

A similar scenario emerges in the context of carrier-lattice coupling. The half-filled EHM with a staggered bond dimerization $\delta$ TF04 ; BEG06 ; PhysRevB.86.205119 ; EELF16 ,

[TABLE]

describes the formation of an SPT phase as a result of a Peierls instability. The bond dimerization in Eq. (22) is equivalent to mean-field BOW order in the SSH model (7).

The phase diagram of the dimerized EHM (22) for $\delta=0.2$ was determined with the infinite-size DMRG method and is shown in Fig. 15 EELF16 . The SDW and BOW phases of the pure EHM are entirely replaced by the SPTI. The latter also has the lowest energy for $U=0$ and small $V/t$, which confirms previous RG results TF04 and leads to a reduction of the CDW phase at weak couplings. The critical line of the continuous Ising QPT terminates at a tricritical point, above which the CDW–SPTI QPT becomes first order. The same holds for the EHM with ferromagnetic spin exchange [Eq. (21)].

The various excitation gaps are shown in Fig. 16. For the pure EHM both $\Delta_{\text{c}_{2}}$ and $\Delta_{\text{n}}$ vanish at the continuous CDW–BOW QPT. For a nonzero dimerization $\delta$, the neutral gap closes whereas $\Delta_{\text{c}_{2}}$ remains finite, indicating that the CDW–SPTI QPT belongs to the Ising universality class. At strong coupling, all gaps remain finite across the QPT. The jump in the spin gap $\Delta_{\text{s}}$ indicates a first-order transition. At very large $U$, the low-lying excitations of Eq. (22) are related to the chargeless singlet and triplet excitations of an effective spin-Peierls Hamiltonian.

At criticality, the central charge $c$ can easily be extracted from the entanglement entropy ELF14 ; EF15 . For periodic boundary conditions, conformal field theory predicts the von Neumann entropy to be $S_{L}(\ell)=\frac{c}{3}\ln\left[\frac{L}{\pi}\sin\left(\frac{\pi\ell}{L}\right)\right]+s_{1}$ where $s_{1}$ is a nonuniversal constant CC04 . A finite-size estimate for the central charge is then obtained via Ni11

[TABLE]

taking the doubled unit cell of the SPTI into account. The bottom panel of Fig. 16 gives $c^{\ast}(L)$, calculated along the PI-CDW QPT line by varying $U$ and $V$ simultaneously at fixed dimerization. With increasing $U$ there is clear evidence for a crossover from $c^{\ast}(L)\simeq 1/2$ to $c^{\ast}(L)\simeq 7/10$, signaling Ising tricriticality. A bosonization-based field-theory analysis of the power-law (exponential) decay of the CDW, SDW, and BOW correlations confirms the universality class of the tricritical Ising model EELF16 .

4.2 Extended anyon-Hubbard model

Ultracold atomic gases in optical lattices provide the possibility to study not only fermions or bosons but also anyons. Exchanges of the latter result in a phase factor $e^{\mathrm{i}\theta}$ in the many-body wave function. The statistical parameter $\theta$ can take on any value between [math] and $\pi$, so that anyons interpolate between bosons and fermions LM77 ; Wi82 . With Haldane’s generalized Pauli principle Ha91 , the anyon concept becomes important also in 1D systems. A fascinating question is if the HI phase observed, e.g., in the extended Bose-Hubbard model (EBHM) TBA06 ; EF15 also exists in the extended anyon-Hubbard model (EAHM).

After a fractional Jordan-Wigner transformation of the anyon operators, $\hat{a}_{i}\mapsto\hat{b}_{i}e^{\mathrm{i}\theta\sum_{l=1}^{i-1}\hat{n}_{l}}$ KLMR11 , the Hamiltonian of the EAHM takes the form LEF17

[TABLE]

where $\hat{b}_{i}^{\dagger}$ ($\hat{b}_{i}^{\phantom{\dagger}}$) is a bosonic creation (annihilation) operator, and $\hat{n}_{i}=\hat{b}_{i}^{\dagger}\hat{b}_{i}^{\phantom{\dagger}}=\hat{a}_{i}^{\dagger}\hat{a}_{i}^{\phantom{\dagger}}$. A boson hopping from site $i+1$ to site $i$ acquires an occupation-dependent phase. Note that anyons on the same site behave as ordinary bosons. Anyons with $\theta=\pi$ represent so-called “pseudofermions”, namely, they are fermions offsite but bosons onsite. If the maximum number of particles per site is restricted to $n_{\text{p}}=2$, the EBHM—the $\theta\to 0$ limit of the EAHM (24)—maps to an effective XXZ spin-$1$ chain BTGA08 .

The phase boundaries of the EAHM (EBHM) with $\theta=\pi/4$ ($\theta=0$) and $n_{\text{p}}=2$ are shown in the top panel of Fig. 17. Most notably, the HI—located between MI and DW insulating phases in the EBHM—survives for any finite fractional phase, i.e., in the anyonic case LEF17 . Likewise, the superfluid (SF) appears for very weak coupling. The critical values for the MI–HI QPT (squares) and the HI–DW QPT (circles) were determined from a divergence of the correlation length $\xi_{\chi}$ with increasing DMRG bond-dimension $\chi$; the model becomes critical with central charge $c=1$ and $c=1/2$, respectively.

The HI may naively be expected to disappear in the EAHM with $\theta>0$ which has neither time reversal ($\hat{\cal T}$) nor inversion ($\hat{\cal I}$) symmetry. However, it has been shown that there exists a nontrivial topological phase protected by the combination of $\hat{\cal R}^{z}=e^{\mathrm{i}\pi\sum_{j}(\hat{n}_{i}-1)}$ and $\hat{\cal K}=e^{\mathrm{i}\theta\sum_{i}\hat{n}_{i}(\hat{n}_{i}-1)/2}$ $\hat{\cal I}\,\hat{\cal T}$ LEF17 . A nonlocal order parameter ${O}$ can be constructed that discriminates between states that are symmetric under both $\hat{\cal K}$ and $\hat{{\cal R}^{z}}$ and states that are not. The middle panel of Fig. 17 demonstrates that ${O}$ can be used to distinguish the topologically trivial MI and DW phases (${O}=1$) from the topologically nontrivial HI (${O}=-1$).

Valuable information about topological phases is also provided by the entanglement spectrum $\{\xi_{\alpha}\}$ LH08 . The concept of entanglement is inherent in any DMRG algorithm based on matrix-product states. Dividing the system into two subsystems, $\xi_{\alpha}=-2\ln\lambda_{\alpha}$ is determined by the singular values $\lambda_{\alpha}$ of the reduced density matrix LEF15 . The lower panel of Fig. 17 shows the entanglement spectrum of the EBHM with $U/t=5$. In the HI phase the entanglement spectrum is expected to be at least four-fold degenerate, reflecting the broken $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry. This is clearly seen in the HI phase. By contrast, in the trivial MI and DW phases, the lowest entanglement level is always nondegenerate.

5 Conclusions

The 1D correlated quantum systems reviewed here exhibit a remarkably rich variety of physical properties that can be studied and understood in particular by powerful numerical methods. For the case of half-filled bands considered, metallic phases are either spinless TLLs or spinful LELs of repulsive nature, i.e., with dominant CDW or BOW correlations. The insulating phases fall into three categories: (i) long-range ordered with a spontaneously broken Ising symmetry (BOW, CDW), (ii) critical with no symmetry breaking (SDW, EI), (iii) topologically nontrivial with short-range entanglement (HI). While the existence of these phases and the phase transitions can in principle be inferred from the low-energy field theory, the details for a given microscopic model typically require numerical solutions. In particular, mean-field, variational or even bosonization/RG approaches are in general not sufficient, especially for problems with retarded interactions.

Despite the significant advances reviewed here, 1D correlated quantum systems remain an active, rewarding and challenging topic of condensed matter physics. Even with the physics of the most fundamental models now unraveled, there remain many future problems of importance in relation to experiment. The list of topics includes the effect of Jahn-Teller coupling at finite band filling, competing long-range interactions, thermodynamics, time-dependent or nonequilibrium phenomena, as well as the coupling to a 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 Correlated Quantum Systems), HF by SFB 652 und the priority programme 1648 (Software for Exascale Computing). We are grateful 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 fruitful collaboration. We also thank S. Ejima for preparing some of the figures.

Author contributions

Both authors contributed equally to the preparation of the manuscript and approved it in its final form.