Bond Order Wave Phase Research Articles

Frustrated mixed-spin ladders: Evidence for a bond-order wave phase between rung-singlet and Haldane phases

In frustrated spin ladders the interplay of frustration and correlations leads to the familiar Haldane (H) and rung-singlet (RS) phases. The nature of the transition between these two phases is still under debate. In this paper we tackle this issue using tools of quantum information theory. We consider frustrated mixed-spin-(1, 1/2) ladders with antiferromagnetic leg, rung and diagonal couplings, and calculate various quantities, such as the entanglement entropy (EE), the Schmidt gap, and the level degeneracy of the entanglement spectrum (ES). We use two numerical techniques, the infinite time-evolving block decimation (iTEBD) and the density matrix renormalization group (DMRG). We demonstrate that there exists an intermediate phase in which the ES levels do not exhibit the characteristic degeneracies of the H and RS phases. To understand the underlying physics in this phase, we investigate short-range spin correlations along legs, rungs and diagonals and show that in this intermediate phase long-wavelength modulations occur, akin to bond order waves.

Revealing the topological nature of the bond order wave in a strongly correlated quantum system

We investigate the topological properties of the bond order wave phase arising in the extended Fermi-Hubbard model. In particular, we uncover a topological sector, which remained elusive in previous finite-size numerical studies due to boundary effects. We first show that, for an infinite system, the bond order wave regime is characterized by two degenerate bulk states corresponding to the trivial and topological sectors. The latter turns out to be indeed characterized by an even degeneracy of the entanglement spectrum and long-range order of a string correlation function. For finite-size systems, we show that the topological sector can be stabilized by imposing a suitable border potential. This therefore provides a concrete protocol for the observation of topologically protected degenerate edge modes in finite-size systems. Furthermore, we show that the bulk of the system is characterized by exotic solitonic solutions interpolating between the trivial and topological sectors. Finally, we propose an implementation and detection scheme of this strongly correlated topological phase in a quantum simulator based on dipolar Fermi gases in optical lattices.Received 13 January 2022Revised 19 April 2022Accepted 14 June 2022DOI:https://doi.org/10.1103/PhysRevResearch.4.L032005Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCold gases in optical latticesDipolar gasesSymmetry protected topological statesTechniquesDensity matrix renormalization groupExtended Hubbard modelAtomic, Molecular & Optical

Competing orders and unconventional criticality in the Su-Schrieffer-Heeger model

The phase diagram of the one-dimensional Su-Schrieffer-Heeger model of spinless fermions coupled to quantum phonons is determined by quantum Monte Carlo simulations. It differs significantly from previous work. In addition to Luttinger liquid and bond-order-wave (BOW) phases, we find an extended charge-density-wave (CDW) phase. Because of different broken symmetries, BOW and CDW phases are connected by a retardation-driven phase transition. Our results are consistent with the theory of the frustrated $XXZ$ chain, including unconventional power-law exponents at criticality, and an interpretation in terms of deconfined quantum criticality via proliferation of solitons.

Spin-gapped phases in the frustrated Hubbard chain with transverse spin anisotropy

At weak coupling, we investigate the frustrated Hubbard chain including nearest-neighbor and next-nearest-neighbor antiferromagnetic exchange with easy-plane anisotropy J1xy and J2xy. At half filling, we obtain three different phases in the J1xy–J2xy plane: a transverse spin density wave phase, a bond order wave phase and a Luther–Emery metallic phase characterized by the coexistence of singlet superconductivity and charge density wave correlations. The frustrating exchange J2xy accounts for the spin-gapped phases.

Probing the Bond Order Wave Phase Transitions of the Ionic Hubbard Model by Superlattice Modulation Spectroscopy.

An exotic phase, the bond order wave, characterized by the spontaneous dimerization of the hopping, has been predicted to exist sandwiched between the band and Mott insulators in systems described by the ionic Hubbard model. Despite growing theoretical evidence, this phase still evades experimental detection. Given the recent realization of the ionic Hubbard model in ultracold atomic gases, we propose here to detect the bond order wave using superlattice modulation spectroscopy. We demonstrate, with the help of time-dependent density-matrix renormalization group and bosonization, that this spectroscopic approach reveals characteristics of both the Ising and Kosterlitz-Thouless transitions signaling the presence of the bond order wave phase. This scheme also provides insights into the excitation spectra of both the band and Mott insulators.

Frustration-driven singlet superconductivity in the one-dimensional [formula omitted] model with positive interactions

Motivated by the discovery of iron-based superconductors, we analytically investigate the one-dimensional correlated electron system described by the t−U−J1−J2 model at weak coupling and at half filling. The low-energy field theory analysis combining the bosonization and renormalization-group scenario helps us to derive the quantum phase diagram, which includes a spin density wave phase, a bond order wave phase and a superconducting phase characterized by the coexistence of singlet superconductivity and charge density wave instabilities. The frustration J2 accounts for the realization of the superconductivity in the regime of positive interactions.

Phase diagram of interacting spinless fermions on the honeycomb lattice: A comprehensive exact diagonalization study

We investigate the phase diagram of spinless fermions with nearest and next-nearest neighbour density-density interactions on the honeycomb lattice at half-filling. Using Exact Diagonalization techniques of the full Hamiltonian and constrained subspaces, combined with a careful choice of finite-size clusters, we determine the different charge orderings that occur for large interactions. In this regime we find a two-sublattice N\'eel-like state, a charge modulated state with a tripling of the unit cell, a zig-zag phase and a novel charge ordered states with a 12 site unit cells we call N\'eel domain wall crystal, as well as a region of phase separation for attractive interactions. A sizeable region of the phase diagram is classically degenerate, but it remains unclear whether an order-by-disorder mechanism will lift the degeneracy. For intermediate repulsion we find evidence for a Kekul\'e or plaquette bond-order wave phase. We also investigate the possibility of a spontaneous Chern insulator phase (dubbed topological Mott insulator), as previously put forward by several mean-field studies. Although we are unable to detect convincing evidence for this phase based on energy spectra and order parameters, we find an enhancement of current-current correlations with the expected spatial structure compared to the non-interacting situation. While for the studied $t{-}V_1{-}V_2$ model the phase transition to the putative topological Mott insulator is preempted by the phase transitions to the various ordered states, our findings might hint at the possibility for a topological Mott insulator in an enlarged Hamiltonian parameter space, where the competing phases are suppressed.

Quantum Phase Diagram of One-Dimensional Spin and Hubbard Models with Transitions to Bond Order Wave Phases

Similar quantum phase diagrams and transitions are found for three classes of one-dimensional models with equally spaced sites, singlet ground states (GS), inversion symmetry at sites and a bond order wave (BOW) phase in some sectors. The models are frustrated spin-1/2 chains with variable range exchange, half-filled Hubbard models with spin-independent interactions and modified Hubbard models with site energies for describing organic charge transfer salts. In some range of parameters, the models have a first order quantum transition at which the GS expectation value of the sublattice spin of odd or even-numbered sites is discontinuous. There is an intermediate BOW phase for other model parameters that lead to two continuous quantum transitions with continuous . Exact diagonalization of finite systems and symmetry arguments provide a unified picture of familiar 1D models that have appeared separately in widely different contexts.

Magnetic susceptibility of alkali-tetracyanoquinodimethane salts and extended Hubbard models with bond order and charge density wave phases

The molar spin susceptibilities χ(T) of Na-tetracyanoquinodimethane (TCNQ), K-TCNQ, and Rb-TCNQ(II) are fit quantitatively to 450 K in terms of half-filled bands of three one-dimensional Hubbard models with extended interactions using exact results for finite systems. All three models have bond order wave (BOW) and charge density wave (CDW) phases with boundary V = V(c)(U) for nearest-neighbor interaction V and on-site repulsion U. At high T, all three salts have regular stacks of TCNQ(-) anion radicals. The χ(T) fits place Na and K in the CDW phase and Rb(II) in the BOW phase with V ≈ V(c). The Na and K salts have dimerized stacks at T < T(d) while Rb(II) has regular stacks at 100 K. The χ(T) analysis extends to dimerized stacks and to dimerization fluctuations in Rb(II). The three models yield consistent values of U, V, and transfer integrals t for closely related TCNQ(-) stacks. Model parameters based on χ(T) are smaller than those from optical data that in turn are considerably reduced by electronic polarization from quantum chemical calculation of U, V, and t of adjacent TCNQ(-) ions. The χ(T) analysis shows that fully relaxed states have reduced model parameters compared to optical or vibration spectra of dimerized or regular TCNQ(-) stacks.

Bond-order wave phase of the extended Hubbard model: Electronic solitons, paramagnetism, and coupling to Peierls and Holstein phonons

The bond order wave (BOW) phase of the extended Hubbard model (EHM) in one dimension (1D) is characterized at intermediate correlation $U = 4t$ by exact treatment of $N$-site systems. Linear coupling to lattice (Peierls) phonons and molecular (Holstein) vibrations are treated in the adiabatic approximation. The molar magnetic susceptibility $\chi_M(T)$ is obtained directly up to $N = 10$. The goal is to find the consequences of a doubly degenerate ground state (gs) and finite magnetic gap $E_m$ in a regular array. Degenerate gs with broken inversion symmetry are constructed for finite $N$ for a range of $V$ near the charge density wave (CDW) boundary at $V \approx 2.18t$ where $E_m \approx 0.5t$ is large. The electronic amplitude $B(V)$ of the BOW in the regular array is shown to mimic a tight-binding band with small effective dimerization $\delta_{eff}$. Electronic spin and charge solitons are elementary excitations of the BOW phase and also resemble topological solitons with small $\delta_{eff}$. Strong infrared intensity of coupled molecular vibrations in dimerized 1D systems is shown to extend to the regular BOW phase, while its temperature dependence is related to spin solitons. The Peierls instability to dimerization has novel aspects for degenerate gs and substantial $E_m$ that suppresses thermal excitations. Finite $E_m$ implies exponentially small $\chi_M(T)$ at low temperature followed by an almost linear increase with $T$. The EHM with $U = 4t$ is representative of intermediate correlations in quasi-1D systems such as conjugated polymers or organic ion-radical and charge-transfer salts. The vibronic and thermal properties of correlated models with BOW phases are needed to identify possible physical realizations.

Bond-order wave phase, spin solitons, and thermodynamics of a frustrated linear spin-12Heisenberg antiferromagnet

The linear spin-1/2 Heisenberg antiferromagnet with exchanges $J_1$, $J_2$ between first and second neighbors has a bond-order wave (BOW) phase that starts at the fluid-dimer transition at $J_2/J_1 = 0.2411$ and is particularly simple at $J_2/J_1 = 1/2$. The BOW phase has a doubly degenerate singlet ground state, broken inversion symmetry and a finite energy gap $E_m$ to the lowest triplet state. The interval $0.4<J_2/J_1<1.0$ has large $E_m$ and small finite size corrections. Exact solutions are presented up to $N=28$ spins with either periodic or open boundary conditions and for thermodynamics up to $N=18$. The elementary excitations of the BOW phase with large $E_m$ are topological spin-1/2 solitons that separate BOWs with opposite phase in a regular array of spins. The molar spin susceptibility $\chi_M(T)$ is exponentially small for $T \ll E_m$ and increases nearly linearly with $T$ to a broad maximum. $J_1$, $J_2$ spin chains approximate the magnetic properties of the BOW phase of Hubbard-type models and provide a starting point for modeling alkali-TCNQ salts.

1:1 Alkali-TCNQ salts and the bond order wave (BOW) phase of half-filled linear Hubbard-type models

The bond order wave (BOW) phase of half-filled linear Hubbard-type models is narrow and difficult to characterize aside from a few ground state properties. The BOW phase of a frustrated Heisenberg spin chain is wide and tractable. It has broken inversion symmetry C i in a regular array and finite gap E m to the lowest triplet state. The spin-BOW is exact in finite systems at a special point. Its elementary excitations are spin-1/2 solitons that connect BOWs with opposite phase. The same patterns of spin densities and bond orders appear in the BOW phase of Hubbard-type models. Infrared (IR) active lattice phonons or molecular vibrations are derivatives of P, the polarization along the stack. Molecular vibrations that are forbidden in regular arrays become IR active when C i symmetry is broken. 1:1 Alkali-TCNQ salts contain half-filled regular TCNQ − stacks at high temperature, down to 100 K in the Rb-TCNQ(II) polymorph whose magnetic susceptibility and polarized IR spectra indicate a BOW phase. More complete modeling will require explicit electronic coupling to phonons and molecular vibrations.

Realization of the bond order wave (BOW) phase of extended Hubbard models in Rb–TCNQ(II)

Rb–TCNQ(II) is shown to be a realization of the BOW phase of half-filled extended Hubbard models. The BOW phase has a regular array of sites with inversion (C i) symmetry, a finite magnetic gap E m and broken electronic C i symmetry. The phase is conditionally stable against dimerization for linear electron–phonon (e–ph) coupling. At 100 K, Rb–TCNQ(II) crystals have regular TCNQ − stacks at C i centers, negligible spin susceptibility that indicates finite E m, and infrared (ir) spectra that indicate broken C i symmetry. The 100 and 295 K crystal structures rule out a dimerization transition around 220 K that had previously been inferred from magnetic and ir data.

Tuning the bond-order wave phase in the half-filled extended Hubbard model

Theoretical and computational studies of the quantum phase diagram of the one-dimensional half-filled extended Hubbard model (EHM) indicate a narrow bond-order wave (BOW) phase with finite magnetic gap E-m for on-site repulsion U < U-*, the critical point, and nearest-neighbor interaction V-c approximate to U/2 near the boundary of the charge-density wave (CDW) phase. Potentials with more extended interactions that retain the EHM symmetry are shown to have a less cooperative CDW transition with higher U-* and wider BOW phase. Density-matrix renormalization group is used to obtain E-m directly as the singlet-triplet gap, with finite E-m marking the BOW boundary V-s(U). The BOW/CDW boundary V-c(U) is obtained from exact finite-size calculations that are consistent with previous EHM determinations. The kinetic energy or bond order provides a convenient new estimate of U-* based on a metallic point at V-c(U) for U < U-*. Tuning the BOW phase of half-filled Hubbard models with different intersite potentials indicates a ground state with large charge fluctuations and magnetic frustration. The possibility of physical realizations of a BOW phase is raised for Coulomb interactions.

Phase Diagram of the One-Dimensional Half-Filled Extended Hubbard Model

We determine the ground-state phase diagram of the one-dimensional half-filled Hubbard model with on-site (nearest-neighbor) repulsive interaction U (V) and nearest-neighbor hopping t using the density-matrix renormalization group technique. Based on the results of the excitation gaps, Luttinger-liquid exponents, and bond-order-wave (BOW) order parameter, we confirm that the BOW phase appears in a substantial region between the charge-density-wave (CDW) and spin-density-wave phases. Each phase boundary is determined by multiple means and it allows us to make a cross-check on the validity of our estimations. We also find that the BOW-CDW transition changes from continuous to first order at the tricritical point (U(t),V(t)) approximately (5.89 t,3.10 t) and the BOW phase shrinks to zero at the critical end point (U(c),V(c)) approximately (9.25 t,4.76 t).

Quantum phase transition in the one-dimensional extended Peierls-Hubbard model

We consider the one-dimensional extended Hubbard model in the presence of an explicit dimerization $\delta$. For a sufficiently strong nearest neighbour repulsion we establish the existence of a quantum phase transition between a mixed bond-order wave and charge-density wave phase from a pure bond-order wave phase. This phase transition is in the universality class of the two-dimensional Ising model.

Study of the ionic Peierls-Hubbard model using density matrix renormalization group methods

Density matrix renormalization group methods are used to investigate the quantum phase diagram of a one-dimensional half-filled ionic Hubbard model with bond-charge attraction, which can be mapped from the Su-Schrieffer-Heeger-type electron-phonon coupling at the antiadiabatic limit. A bond order wave (dimerized) phase which separates the band insulator from the Mott insulator always exists as long as electron-phonon coupling is present. This is qualitatively different from that at the adiabatic limit. Our results indicate that electron-electron interaction, ionic potential and quantum phonon fluctuations combine in the formation of the bond-order wave phase.

Jeckelmann Replies:

In their Comment (see cond-mat/0301237), Sandvik, Sengupta, and Campbell present some numerical evidences to support the existence of an extended bond-order-wave (BOW) phase at couplings (U,V) weaker than a tricritical point (U_t,V_t) in the ground state phase diagram of the one-dimensional half-filled U-V Hubbard model. They claim that their results do not agree with the phase diagram proposed in my Letter (cond-mat/0204244), which shows a BOW phase for couplings stronger than the critical point only. However, I argue here that their results are not conclusive and do not refute the phase diagram described in the Letter.

Inducement of bond-order wave due to electron correlation in one dimension

Density-matrix renormalization-group methods are used to investigate the quantum phase transitions in a Hubbard chain with an alternating on-site energy $\ensuremath{\Delta}$ at half filling. Our results support the existence of a bond-order wave phase, which separates the band insulator from the Mott insulator even in the absence of electron-phonon coupling. The critical points are determined accurately at fixed staggered potentials. Our results indicate a slightly smaller size of the bond-order wave phase than that of previous results.

Influence of the anion potential on the charge ordering in quasi-one-dimensional charge-transfer salts

We examine the various instabilities of quarter-filled strongly correlated electronic chains in the presence of a coupling to the underlying lattice. To mimic the physics of the tetramethyltetrathiafulvene $(\mathrm{TMTTF}{)}_{2}X$ Bechgaard-Fabre salts we also include electrostatic effects of intercalated anions. We show that small displacements of the anion can stabilize new mixed charged-density wave--bond order wave phases in which central symmetry centers are suppressed. This finding is discussed in the context of recent experiments. We suggest that the recently observed charge ordering is due to a cooperative effect between the Coulomb interaction and the coupling of the electronic stacks to the anions. On the other hand, the Spin-Peierls instability at lower temperature requires a Peierls-like lattice coupling.