Ultracold atom experiments allow the study of topological insulators, such as the noninteracting Haldane model. In this work we study a generalization of the Haldane model with spin-spin on-site interactions that can be implemented on such experiments. We focus on measuring the winding number, a topological invariant, of the ground state, which we compute using a mean-field calculation that effectively captures long range correlations and a matrix product state computation in a lattice with 64 sites. Our main result is that we show how the topological phases present in the noninteracting model survive until the interactions are comparable to the kinetic energy. We also demonstrate the accuracy of our mean-field approach in efficiently capturing long-range correlations. Based on state-of-the-art ultracold atom experiments, we propose an implementation of our model that can give information about the topological phases.
Topological phases are characterized by their entanglement properties, which is manifest in a direct relation between entanglement spectra and edge states discovered by Li and Haldane. We propose to leverage the power of synthetic quantum systems for measuring entanglement via the Entanglement Hamiltonian to probe this relationship experimentally. This is made possible by exploiting the quasi-local structure of Entanglement Hamiltonians. The feasibility of this proposal is illustrated for two paradigmatic examples realizable with current technology, an integer quantum Hall state of non-interac…
We propose a Haldane-BCS-Hubbard model on a honeycomb lattice, which is composed of two copies of the Haldane model of the quantum anomalous Hall effect, an equal-spin pairing term and an onsite Hubbard interaction term. For any interaction strength, this model is exactly solvable along the symmetric line where the hopping and pairing amplitudes are equal to each other. The ground state of the Haldane-BCS-Hubbard model is a topological superconducting state at weak interaction with two chiral Majorana edge states. A strong interaction drives the system across a topological quantum phase transi…
We study the Haldane model with nearest-neighbor interactions. This model is physically motivated by the associated ultracold atoms implementation. We show that the topological phase of the interacting model can be characterized by a physically observable winding number. The robustness of this number extends well beyond the topological insulator phase towards attractive and repulsive interactions that are comparable to the kinetic energy scale of the model. We identify and characterize the relevant phases of the model.
Motivated by exploring the effect of interactions on Chern insulators, and by recent experiments realizing topological bands for ultracold atoms in synthetic gauge fields, we study the honeycomb lattice Haldane-Hubbard model of spin-$1/2$ fermions. Using an unrestricted mean field approach, we map out the instability of the topological band insulator towards magnetically ordered insulators which emerge with increasing Hubbard repulsion. In addition to the topological Néel phase, we recover various chiral noncoplanar magnetic orders previously identified within a strong coupling approach. We co…
The interplay between antiferromagnetic interaction and hole motion is capable of inducing intriguing conducting topological Haldane phases described by a finite non-local string order parameter. Here we show that these states of matter are captured by the one dimensional $t-J_z$ model which can be experimentally realized with dressed Rydberg atoms trapped onto a one dimensional optical lattice. In the sector with vanishing total magnetization exact Bethe ansatz calculations associated to bosonization technique allow to predict that both metallic and superconducting topological Haldane states …
In this letter we study the Haldane model with on-site repulsive interactions at half-filling. We show that the mean-field Hamiltonian with magnetic order effectively modifies parameters in the Haldane Hamiltonian, such as sublattice energy difference and phase in next nearest hopping. As interaction increases, increasing of magnetic order corresponds to varying these parameters and consequently, drives topological transitions. At the mean-field level, one scenario is that the magnetic order continuously increases, and inevitably, the fermion gap closes at the topological transition point with…
We study the topological properties of the two-body bound states in an interacting Haldane model as a function of interparticle interactions. In particular, we identify topological phases where the two-body edge states have either the same or the opposite chirality as compared to single-particle edge states. We highlight that in the moderately interacting regime, which is relevant for the experimental realization with ultracold atoms, the topological transition is affected by the internal structure of the bound state, and the phase boundaries are consequently deformed.
Neutron scattering played an important role in the experimental exploration of the theoretical ideas of Thouless, Haldane, and Kosterlitz, who won the Nobel Prize in Physics in 2016. This article reviews the signatures of Haldane's predicted topological quantum ground state and their observation in early neutron scattering experiments, which overturned the wisdom of the day.
The Haldane model on the honeycomb lattice is a paradigmatic example of a Hamiltonian featuring topologically distinct phases of matter. It describes a mechanism through which a quantum Hall effect can appear as an intrinsic property of a band-structure, rather than being caused by an external magnetic field. Although an implementation in a material was considered unlikely, it has provided the conceptual basis for theoretical and experimental research exploring topological insulators and superconductors. Here we report on the experimental realisation of the Haldane model and the characterisati…
We have simulated a half-filled $1D$ $p$-wave periodic Anderson model with numerically exact projector quantum Monte Carlo technique, and the system is indeed located in the Haldane-like state as detected in previous works on the $p$-wave Kondo lattice model, though the soluble non-interacting limit corresponds to the conventional $Z_{2}$ topological insulator. The site-resolved magnetization in an open boundary system and strange correlator for the periodic boundary have been used to identify the mentioned topological states. Interestingly, the edge magnetization in the Haldane-like state is …
Using the density matrix renormalization-group technique, we calculate the impurity energy levels for two different effective models of off-chain doping for quasi-one-dimensional Heisenberg chain compound Y$_2$ Ba Ni O$_5$: ferromagnetic bond doping and antiferromagnetic site spin-1/2 doping. Thresholds of the impurity strength for the appearance of localized states are found for both models. However, the ground-state and low-energy excitations for weak impurity strength are different for these two models and the difference can be detected by experiments.
We have studied the one-dimensional $p$-wave periodic Anderson model at finite temperature with the help of the numerically exact determinant quantum Monte Carlo simulation. It is found that the topological Haldane phase established for ground-state is still stable against small thermal fluctuation and its characteristic edge magnetization develops at low temperature. Moreover, the saturated low-$T$ spin structure factor and the $\frac{1}{T}$-law of susceptibility are useful to detect the free edge spin moment, which may be relevant for experimental explorations. We have also comparatively stu…
We investigate the groundstate properties of a recently proposed model for a topological Kondo insulator in one dimension (i.e., the $p$-wave Kondo-Heisenberg lattice model) by means of the Density Matrix Renormalization Group method. The non-standard Kondo interaction in this model is different from the usual (i.e., local) Kondo interaction in that the localized spins couple to the "$p$-wave" spin density of conduction electrons, inducing a topologically non-trivial insulating groundstate. Based on the analysis of the charge- and spin-excitation gaps, the string order parameter, and the spin …
Haldane model is a noninteracting model for spinless fermions showing nontrivial topological properties. Effect of the electron-electron interaction on the topological phase poses an intriguing question. By means of the Hartree-Fock mean field, the exact diagonalization and the constrained-path Monte Carlo methods we mapped out the phase diagram of the interacting Haldane model. It is found that interaction breaks down the topological phase and drives the system into the charge-density-wave state. Sequence of the two transitions depends on the strength of next-nearest-neighbor hopping. Many-bo…
We study the spinful fermionic Haldane-Hubbard model at half filling using a combination of quantum cluster methods: cluster perturbation theory (CPT), the variational cluster approximation (VCA), and cluster dynamical mean-field theory (CDMFT). We explore possible zero-temperature phases of the model as a function of on-site repulsive interaction strength and next-nearest-neighbor hopping amplitude and phase. Our approach allows us to access the regime of intermediate interaction strength, where charge fluctuations are significant and effective spin model descriptions may not be justified. Ou…
We study the half-filled Haldane model with Anderson and binary disorder and determine its phase diagram, as a function of the Haldane flux and staggered sub-lattice potential, for increasing disorder strength. We establish that disorder stabilizes topologically nontrivial phases in regions of the phase diagram where the clean limit is topologically trivial. At small disorder strength, our results agree with analytical predictions obtained using a first order self-consistent Born approximation, and extend to the intermediate and large disorder values where this perturbative approach fails. We …
We reveal a proximity effect between a topological band (Chern) insulator described by a Haldane model and spin-polarized Dirac particles of a graphene layer. Coupling weakly the two systems through a tunneling term in the bulk, the topological Chern insulator induces a gap and an opposite Chern number on the Dirac particles at half-filling resulting in a sign flip of the Berry curvature at one Dirac point. We study different aspects of the bulk-edge correspondence and present protocols to observe the evolution of the Berry curvature as well as two counter-propagating (protected) edge modes wi…
We present a functional renormalization group approach to interacting topological Green function invariants with a focus on the nature of transitions. The method is applied to chiral symmetric fermion chains in the Mott limit that can be driven into a Haldane phase. We explicitly show that the transition to this phase is accompanied by a zero of the fermion Green function. Our results for the phase boundary are quantitatively benchmarked against DMRG data.
We study the phase diagram of the interacting spin-$1/2$ Haldane model with chiral phase $ϕ= π/2$ at half-filling. Both on-site and long-range Coulomb repulsive interactions (Haldane-Hubbard-Coulomb model) are considered. The problem with on-site interaction $U$ alone was addressed in the past by a variety of approximate and finite size methods that produced results in disagreement with each other both quantitatively and qualitatively. Here we employ the Diagrammatic Monte Carlo technique to accurately locate phase transition points to the topologically nontrivial phases in the $(Δ, U)$-plane,…
We study the topological edge states of the Haldane's graphene model with zigzag/armchair lattice edges. The Harper equation for solving the energies of the edge states is derived. The results show that there are two edge states in the bulk energy gap, corresponding to the two zero points of the Bloch function on the complex-energy Riemann surface. The edge-state energy loops move around the hole of the Riemann surface in appropriate system parameter regimes. The quantized Hall conductance can be expressed by the winding numbers of the edge states, which reflects the topological feature of the…
Motivated by experiments on ultracold atoms which have realized the Haldane model for a Chern insulator, we consider its strongly correlated Mott limit with spin-$1/2$ fermions. We find that slave rotor mean field theory yields gapped or gapless chiral spin liquid Mott insulators. To study competing magnetic orders, we consider the strong coupling effective spin Hamiltonian which includes chiral three-spin exchange. We obtain its classical phase diagram, uncovering various chiral magnetic orders including tetrahedral, cone, and noncoplanar spiral states which can compete with putative chiral q…
We study, by means of the density-matrix renormalization group (DMRG) technique, the evolution of the ground state in a one-dimensional topological insulator, from the non-interacting to the strongly-interacting limit, where the system can be mapped onto a topological Kondo-insulator model. We focus on a toy model Hamiltonian (i.e., the interacting "$sp$-ladder" model), which could be experimentally realized in optical lattices with higher orbitals loaded with ultra-cold fermionic atoms. Our goal is to shed light on the emergence of the strongly-interacting ground state and its topological cla…
We study the interplay of topological band structure and conventional magnetic long-range order in spinful Haldane model with onsite repulsive interaction. Using the dynamical cluster approximation with clusters of up to 24 sites we find evidence of a first order phase transition from a Chern insulator at weak coupling to a topologically trivial antiferromagnetic insulator at strong coupling. These results call into question a previously found intermediate state with coexisting topological character and antiferromagnetic long-range order. Experimentally measurable signatures of the first order…
Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The paradigmatic model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain. Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterised by vanishing two-point spin correlations, gapped excitations, and a characteristic non-local order parameter. More recently it was understood that the Haldane chain forms a specific example o…
We consider the Haldane model, a 2D topological insulator whose phase is defined by the Chern number. We study its phases as temperature varies by means of the Uhlmann number, a finite temperature generalization of the Chern number. Because of the relation between the Uhlmann number and the dynamical transverse conductivity of the system, we evaluate also the conductivity of the model. This analysis does not show any sign of a phase transition induced by the temperature, nonetheless it gives a better understanding of the fate of the topological phase with the increase of the temperature, and i…
We consider the one dimensional, periodic spin chain with $N$ sites, similar to the one studied by Haldane \cite{hal}, however in the opposite limit of very large anisotropy and small nearest neighbour, anti-ferromagnetic exchange coupling between the spins, which are of large magnitude $s$. For a chain with an even number of sites we show that actually the ground state is non degenerate and given by a superposition of the two Néel states, due to quantum spin tunnelling. With an odd number of sites, the Néel state must necessarily contain a soliton. The position of the soliton is arbitrary thu…
We study the Haldane-Hubbard model by exact Renormalization Group techniques. We construct the topological phase diagram, for weak interactions. We predict that many-body interactions induce a shift of the transition line. The presence of new intermediate phases, absent in the non interacting case, is rigorously excluded at weak coupling. Despite the nontrivial renormalization of the wave function and of the Fermi velocity, the conductivity is universal: at the renormalized critical line, both the discontinuity of the transverse conductivity and the longitudinal conductivity do not depend on t…
We study the dynamical spin excitations of the one-dimensional $S=1$ Heisenberg antiferromagnetic chain with single-ion anisotropy by using quantum Monte Carlo simulations and stochastic analytic continuation of imaginary-time correlation function. Using the transverse dynamic spin structure factor, we observe the quantum phase transition with a critical point between the topological Haldane gapped phase and the trivial phase. At the quantum critical point, we find a broad continuum characterized by the Tomonaga-Luttinger liquid similar to a $S=1/2$ Heisenberg antiferromagnetic chain. We furth…
In condensed matter physics many features can be understood in terms of their topological properties. Here we report evidence of a topological quantum transition driven by the charge-phonon coupling in the spinless Haldane model on a honeycomb lattice, a well-known prototypical model of Chern insulator. Starting from parameters describing the topological phase in the bare Haldane model, we show that the increasing of the strength of the charge lattice coupling drives the system towards a trivial insulator. The average number of fermions in the Dirac point, characterized by the lowest gap, exhi…
Using mainly numerical methods, we investigate the ground-state phase diagram of the S=2 quantum spin chain described by $H = \sum_j (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y + ΔS_j^z S_{j+1}^z) + D \sum_j (S_j^z)^2$, where $Δ$ denotes the $XXZ$ anisotropy parameter of the nearest-neighbor interactions and $D$ the on-site anisotropy parameter. We restrict ourselves to the case with $Δ\ge 0$ and $D \ge 0$ for simplicity. Each of the phase boundary lines is determined by the level spectroscopy or the phenomenological renormalization analysis of numerical results of exact-diagonalization calculations. T…
We propose to realize an exotic Haldane superfluid (HSF) phase in an extended Bose-Hubbard model on the two-leg ladder (i.e., a two-species mixture of interacting bosons). The proposal is confirmed by means of large-scale quantum Monte Carlo simulations, with a significant part of the ground-state phase diagram being revealed. Most remarkably, the newly discovered HSF phase features both superfluidity and the non-local topological Haldane order. The effects induced by varying the number of legs are furthermore explored. Our results shed light on how topological superfluid emerges in bosonic sy…
A Heisenberg spin-$s$ chain with alternating ferromagnetic ($-J_1^F<0$) and antiferromagnetic ($J_1^A>0$) nearest-neighbor (NN) interactions, exhibits the Dimer and spin-$2s$ Haldane phases in the limits $J_1^F/J_1^A \rightarrow 0$ and $J_1^F/J_1^A \rightarrow \infty$ respectively. These two phases are understood to be topologically equivalent. Induction of the frustration through the next nearest-neighbor ferromagnetic interaction ($-J_2^F<0$) produces a very rich quantum phase diagram. With frustration, the whole phase diagram is divided into a ferromagnetic (FM) and a nonmagnetic (NM) phase…
We elaborate the first theoretical realization of two dimensional itinerant topological magnons, based on the quarter filled Haldane-Hubbard model with a nearly-flat electron band. By using the exact diagonalization method with a projection onto this band, we obtain the spin wave excitations over the itinerant ferromagnetic ground state. In the flatband limit, the excitation exhibits similar dispersion to the free electron band with Dirac magnons. The nonflatness of the electron band opens a topological gap at Dirac points and leads to an acoustic magnon band with a nonzero Chern number. We fu…
We present a reentrant phase transition in a bridging model between two different topological models: Kitaev and Haldane chains. This model is activated by introducing a bond alternation into the Kitaev chain [A. Yu Kitaev, Phys.-Usp. {\bf 44} 131 (2001)]. Without the bond alternation, finite pairing potential induces a topological state defined by zero-energy Majorana edge mode, while finite bond alternation without the pairing potential makes a different topological state similar to the Haldane state, which is defined by local Berry phase in the bulk. The topologically-ordered state correspo…
Institute of Theoretical Physics, Utrecht University（乌得勒支大学理论物理研究所）；Laboratoire d’Acoustique de l’Université du Mans, Unite Mixte de Recherche 6613, Centre National de la Recherche Scientifique（勒芒大学声学实验室，联合研究单位6613，法国国家科学研究中心）
We use linked-cluster expansions to analyze the quantum phase transitions between symmetry unbroken trivial and topological Haldane phases in two different spin-one chains. The first model is the spin-one Heisenberg chain in the presence of a single-ion anisotropy while the second one is the dimerized spin-one Heisenberg chain. For both models we determine the ground-state energy and the one-particle gap inside the non-topological phase as a high-order series using perturbative continuous unitary transformations. Extrapolations of the gap series are applied to locate the quantum critical point…
We calculate the Haldane gap of the $\mathrm{SU}(3)$ spin $[3~0~0]$ Heisenberg model using variational uniform fully symmetric $\mathrm{SU}(3)$ matrix product states, and find that the minimal gap $Δ/J = 0.0263 $ is obtained in the $[2~1~0]$ sector at momentum $2π/3$. We also discuss the symmetry protected topological order of the ground state, and determine the full dispersion relation of the elementary excitations and the correlation lengths of the system.
We review the basic properties of the Haldane phase in spin-1 Heisenberg antiferromagnetic chains, including its persistence in quasi-one-dimensional geometries. Using large-scale numerical simulations, we map out the phase diagram for a realistic model applicable to experimental Haldane compounds. We also investigate the effect of different chain coupling geometries and confirm a general mean field universality of the critical coupling times the coordination number of the lattice. Inspired by the recent development of characterization of symmetry protected topological states, of which the Hal…
For the Haldane phase, the magnetic field usually tends to break the symmetry and drives the system into a topologically trivial phase. Here, we report a novel reentrance of the Haldane phase at zero temperature in the spin-1 antiferromagnetic Heisenberg model on sawtooth chain. A partial Haldane phase is induced by the magnetic field, which is the combination of the Haldane state in one sublattice and a ferromagnetically ordered state in the other sublattice. Such a partial topological order is a result of the zero-temperature entropy due to quantum fluctuations caused by geometrical frustrat…
Anfuso and Rosch [Phys. Rev. B 75, 144420 (2007)] showed that the "topological" Haldane phase in a fermionic spin-1/2 ladder can be continuously deformed into a "trivial" phase without explicitly breaking symmetries when local charge fluctuations are taken into account. Within the framework of symmetry protected topological phases, we revisit the model and demonstrate how the Haldane phase can be adiabatically connected to a trivial phase due to charge fluctuations. Furthermore, we show that the Haldane phase remains stable as long as the system is symmetric under particular reflection symmetr…
We investigate the nonequilibrium dynamics of the spinless Haldane model with nearest-neighbor interactions on the honeycomb lattice by employing an unbiased numerical method. In this system, a first-order transition from the Chern insulator (CI) at weak coupling to the charge-density-wave (CDW) phase at strong coupling can be characterized by a level crossing of the lowest energy levels. Here we show that adiabatically following the eigenstates across this level crossing, their Chern numbers are preserved, leading to the identification of a topologically-nontrivial low-energy excited state in…
We investigate the nature of the Mott-insulating phases of half-filled 2N-component fermionic cold atoms loaded into a one-dimensional optical lattice. By means of conformal field theory techniques and large-scale DMRG calculations, we show that the phase diagram strongly depends on the parity of $N$. First, we single out charged, spin-singlet, degrees of freedom, that carry a pseudo-spin ${\cal S}=N/2$ allowing to formulate a Haldane conjecture: for attractive interactions, we establish the emergence of Haldane insulating phases when $N$ is even, whereas a metallic behavior is found when $N$ …
Gapped quantum spin chains with symmetry PSU(N)=SU(N)/Z(N) are known to possess N distinct symmetry protected topological phases. Besides the trivial phase, there are N-1 Haldane phases which are distinguished by the occurrence of massless boundary spins. Motivated by the potential realization in alkaline-earth atomic Fermi gases, we explicitly construct previously unknown Hamiltonians for two classes of chiral AKLT states and we discuss their physical properties. We also point out a deep connection between symmetry protection in gapped and gapless 1D quantum spin systems and its implications …
Deconfined quantum phase transition from Néel phase to Valence bond crystal state in 2D Heisenberg model is under debate nowadays. One crucial issue is the suppression of Haldane's instanton on quantum critical point which drives the spinon deconfined. In this paper, by making use of the $ϕ$-mapping topological current theory, we reexamine the Haldane's instanton in an alternative way along the direction of topology. We find that the the monopole events are space-time singularities of Néel field $\vec{n}$, the corresponding topological charges are the wrapping number of $\vec{n}$ around the si…
We formulate the dynamics of local order parameters by extending the recently developed adiabatic spinwave theory involving the Berry curvature, and derive a formula showing explicitly the role of the Berry phase in determining the spectral form of the low-lying collective modes. For antiferromagnetic spin chains, the Berry phase becomes a topological invariant known as the Chern number. Our theory predicts the existence of the Haldane gap for a topologically trivial ground state, and a linear dispersion of low-lying excitations for a non-trivial ground state.
We study the flat-band ferromagnetic phase of the Haldane-Hubbard model on a honeycomb lattice within a bosonization scheme for flat-band Chern insulators, focusing on the calculation of the spin-wave excitation spectrum. We consider the Haldane-Hubbard model with the noninteracting lower bands in a nearly-flat band limit, previously determined for the spinless model, and at 1/4-filling of its corresponding noninteracting limit. Within the bosonization scheme, the Haldane-Hubbard model is mapped into an effective interacting boson model, whose quadratic term allows us to determine the spin-wav…
We propose to realize one-dimensional topological phases protected by SU($N$) symmetry using alkali or alkaline-earth atoms loaded into a bichromatic optical lattice. We derive a realistic model for this system and investigate it theoretically. Depending on the parity of $N$, two different classes of symmetry-protected topological (SPT) phases are stabilized at half-filling for physical parameters of the model. For even $N$, the celebrated spin-1 Haldane phase and its generalization to SU($N$) are obtained with no local symmetry breaking. In stark contrast, at least for $N=3$, a new class of S…
