XXZ Spin Chain Research Articles

A complete proof that square ice entropy is

AbstractIn this text, we provide a fully rigorous and complete proof of E.H. Lieb’s statement that (topological) entropy of square ice (or six-vertex model, XXZ spin chain for anisotropy parameter $\Delta =1/2$ ) is equal to $\tfrac 32\log _{2} (4/3)$ .

Exact time evolution formulae in the XXZ spin chain with domain wall initial state

We study the time evolution of the spin-1/2 XXZ chain initialized in a domain wall state, where all spins to the left of the origin are up, all spins to its right are down. The focus is on exact formulae, which hold for arbitrary finite (real or imaginary) time. In particular, we compute the amplitudes corresponding to the process where all but k spins come back to their initial orientation, as a k-fold contour integral. These results are obtained using a correspondence with the six vertex model, and taking a somewhat complicated Hamiltonian/Trotter-type limit. Several simple applications are studied and also discussed in a broader context.

Spectrum of the quantum integrable {D}_2^{(2)} spin chain with generic boundary fields

Exact solution of the quantum integrable {D}_2^{(2)} spin chain with generic integrable boundary fields is constructed. It is found that the transfer matrix of this model can be factorized as the product of those of two open staggered anisotropic XXZ spin chains. Based on this identity, the eigenvalues and Bethe ansatz equations of the {D}_2^{(2)} model are derived via off-diagonal Bethe ansatz.

Dualities in quantum integrable many-body systems and integrable probabilities. Part I

In this study we map the dualities observed in the framework of integrable probabilities into the dualities familiar in a realm of integrable many-body systems. The dualities between the pairs of stochastic processes involve one representative from Macdonald-Schur family, while the second representative is from stochastic higher spin six-vertex model of TASEP family. We argue that these dualities are counterparts and generalizations of the familiar quantum-quantum (QQ) dualities between pairs of integrable systems. One integrable system from QQ dual pair belongs to the family of inhomogeneous XXZ spin chains, while the second to the Calogero-Moser-Ruijsenaars-Schneider (CM-RS) family. The wave functions of the Hamiltonian system from CM-RS family are known to be related to solutions to (q)KZ equations at the inhomogeneous spin chain side. When the wave function gets substituted by the measure, bilinear in wave functions, a similar correspondence holds true. As an example, we have elaborated in some details a new duality between the discrete-time inhomogeneous multispecies TASEP model on the circle and the quantum Goldfish model from the RS family. We present the precise map of the inhomogeneous multispecies TASEP and 5-vertex model to the trigonometric and rational Goldfish models respectively, where the TASEP local jump rates get identified as the coordinates in the Goldfish model. Some comments concerning the relation of dualities in the stochastic processes with the dualities in SUSY gauge models with surface operators included are made.

Krylov localization and suppression of complexity

Quantum complexity, suitably defined, has been suggested as an important probe of late-time dynamics of black holes, particularly in the context of AdS/CFT. A notion of quantum complexity can be effectively captured by quantifying the spread of an operator in Krylov space as a consequence of time evolution. Complexity is expected to behave differently in chaotic many-body systems, as compared to integrable ones. In this paper we investigate Krylov complexity for the case of interacting integrable models at finite size and find that complexity saturation is suppressed as compared to chaotic systems. We associate this behavior with a novel localization phenomenon on the Krylov chain by mapping the theory of complexity growth and spread to an Anderson localization hopping model with off-diagonal disorder, and find that localization is enhanced in the integrable case due to a stronger disorder in the hopping amplitudes, inducing an effective suppression of Krylov complexity. We demonstrate this behavior for an interacting integrable model, the XXZ spin chain, and show that the same behavior results from a phenomenological model that we define: this model captures the essential features of our analysis and is able to reproduce the behaviors we observe for chaotic and integrable systems via an adjustable disorder parameter.

Identifying correlation clusters in many-body localized systems

We introduce techniques for analysing the structure of quantum states of many-body localized (MBL) spin chains by identifying correlation clusters from pairwise correlations. These techniques proceed by interpreting pairwise correlations in the state as a weighted graph, which we analyse using an established graph theoretic clustering algorithm. We validate our approach by studying the eigenstates of a disordered XXZ spin chain across the MBL to ergodic transition, as well as the non-equilibrium dyanmics in the MBL phase following a global quantum quench. We successfully reproduce theoretical predictions about the MBL transition obtained from renormalization group schemes. Furthermore, we identify a clear signature of many-body dynamics analogous to the logarithmic growth of entanglement. The techniques that we introduce are computationally inexpensive and in combination with matrix product state methods allow for the study of large scale localized systems. Moreover, the correlation functions we use are directly accessible in a range of experimental settings including cold atoms.

Preparing exact eigenstates of the open XXZ chain on a quantum computer

The open spin-1/2 XXZ spin chain with diagonal boundary magnetic fields is the paradigmatic example of a quantum integrable model with open boundary conditions. We formulate a quantum algorithm for preparing Bethe states of this model, corresponding to real solutions of the Bethe equations. The algorithm is probabilistic, with a success probability that decreases with the number of down spins. For a Bethe state of L spins with M down spins, which contains a total of terms, the algorithm requires L + M 2 + 2M qubits.

Open quantum dynamics theory for a complex subenvironment system with a quantum thermostat: Application to a spin heat bath.

Complex environments, such as molecular matrices and biological material, play a fundamental role in many important dynamic processes in condensed phases. Because it is extremely difficult to conduct full quantum dynamics simulations on such environments due to their many degrees of freedom, here, we treat in detail the environment only around the main system of interest (the subenvironment), while the other degrees of freedom needed to maintain the equilibrium temperature are described by a simple harmonic bath, which we call a quantum thermostat. The noise generated by the subenvironment is spatially non-local and non-Gaussian and cannot be characterized by the fluctuation-dissipation theorem. We describe this model by simulating the dynamics of a two-level system (TLS) that interacts with a subenvironment consisting of a one-dimensional XXZ spin chain. The hierarchical Schrödinger equations of motion are employed to describe the quantum thermostat, allowing for time-irreversible simulations of the dynamics at arbitrary temperature. To see the effects of a quantum phase transition of the subenvironment, we investigate the decoherence and relaxation processes of the TLS at zero and finite temperatures for various values of the spin anisotropy. We observed the decoherence of the TLS at finite temperature even when the anisotropy of the XXZ model is enormous. We also found that the population-relaxation dynamics of the TLS changed in a complex manner with the change in the anisotropy and the ferromagnetic or antiferromagnetic orders of spins.

Scattering matrix of elementary excitations in the antiperiodic XXZ spin chain with [formula omitted

We study the thermodynamic limit of the antiperiodic XXZ spin chain with the anisotropic parameter η=πi3. We parameterize eigenvalues of the transfer matrix by their zero points instead of Bethe roots. We obtain patterns of the distribution of zero points. Based on them, we calculate the ground state energy and the elementary excitations in the thermodynamic limit. We also obtain the two-body scattering matrix of elementary excitations. Two types of elementary excitations and three types of scattering processes are discussed in detailed.

Preparing Bethe Ansatz Eigenstates on a Quantum Computer

Several quantum many-body models in one dimension possess exact solutions via the Bethe ansatz method, which has been highly successful for understanding their behavior. Nevertheless, there remain physical properties of such models for which analytic results are unavailable, and which are also not well-described by approximate numerical methods. Preparing Bethe ansatz eigenstates directly on a quantum computer would allow straightforward extraction of these quantities via measurement. We present a quantum algorithm for preparing Bethe ansatz eigenstates of the spin-1/2 XXZ spin chain that correspond to real-valued solutions of the Bethe equations. The algorithm is polynomial in the number of T gates and circuit depth, with modest constant prefactors. Although the algorithm is probabilistic, with a success rate that decreases with increasing eigenstate energy, we employ amplitude amplification to boost the success probability. The resource requirements for our approach are lower than other state-of-the-art quantum simulation algorithms for small error-corrected devices, and thus may offer an alternative and computationally less-demanding demonstration of quantum advantage for physically relevant problems.

Dynamical obstruction to localization in a disordered spin chain.

We analyze a one-dimensional XXZ spin chain in a disordered magnetic field. As the main probes of the system's behavior, we use the sensitivity of eigenstates to adiabatic transformations, as expressed through the fidelity susceptibility, in conjunction with the low-frequency asymptotes of the spectral function. We identify a region of maximal chaos-with exponentially enhanced susceptibility-which separates the many-body localized phase from the diffusive ergodic phase. This regime is characterized by slow transport, and we argue that the presence of such slow dynamics highly constrains any possible localization transition in the thermodynamic limit. Rather, the results are more consistent with absence of the localized phase.

Root patterns and energy spectra of quantum integrable systems without U(1) symmetry: the antiperiodic XXZ spin chain

Finding out root patterns of quantum integrable models is an important step to study their physical properties in the thermodynamic limit. Especially for models without U(1) symmetry, their spectra are usually given by inhomogeneous T − Q relations and the Bethe root patterns are still unclear. In this paper with the antiperiodic XXZ spin chain as an example, an analytic method to derive both the Bethe root patterns and the transfer-matrix root patterns in the thermodynamic limit is proposed. Based on them the ground state energy and elementary excitations in the gapped regime are derived. The present method provides an universal procedure to compute physical properties of quantum integrable models in the thermodynamic limit.

Coexistence of diffusive and ballistic transport in integrable quantum lattice models

We investigate the high-temperature dynamical conductivity $\sigma(\omega)$ in two one-dimensional integrable quantum lattice models: the anisotropic XXZ spin chain and the Hubbard chain. The emphasis is on the metallic regime of both models, where besides the ballistic component, the regular part of conductivity might reveal a diffusive-like transport. To resolve the low-frequency dynamics, we upgrade the microcanonical Lanczos method enabling studies of finite-size systems with up to $L\leq 32$ sites for XXZ spin model with the frequency resolution $\delta \omega \sim 10^{-3} J$. Results for the XXZ chain reveal a fine structure of $\sigma(\omega)$ spectra, which originates from the discontinuous variation of the stiffness, previously found at commensurate values of the anisotropy parameter $\Delta$. Still, we do not find a clear evidence for a diffusive component, at least not for commensurate values of $\Delta$, particularly for $\Delta =0.5$, as well as for $\Delta \to 0$. Similar is the conclusion for the Hubbard model away from half-filling, where the spectra reveal more universal behavior.

Comments on Slavnov products, Temperley-Lieb open spin chains, and KP tau functions

We study Slavnov's inner products for open Temperley-Lieb chains and their relations with the KP hierarchy. We show that when s=1/2 the quantum group invariant XXZ spin chain has Slavnov products given by the quotient of tau functions. The Schur polynomials expansion and a matrix construction for the Baker-Akhiezer functions are also briefly considered.

Hydrodynamic nonlinear response of interacting integrable systems

We develop a formalism for computing the nonlinear response of interacting integrable systems. Our results are asymptotically exact in the hydrodynamic limit where perturbing fields vary sufficiently slowly in space and time. We show that spatially resolved nonlinear response distinguishes interacting integrable systems from noninteracting ones, exemplifying this for the Lieb-Liniger gas. We give a prescription for computing finite-temperature Drude weights of arbitrary order, which is in excellent agreement with numerical evaluation of the third-order response of the XXZ spin chain. We identify intrinsically nonperturbative regimes of the nonlinear response of integrable systems.

Closing of the Haldane gap in a spin-1 XXZ chain

Closing of the Haldane gap in a spin-1 XXZ chain

Quantum K-theory of quiver varieties and many-body systems

We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.

Scalar product for the XXZ spin chain with general integrable boundaries * *Dedicated to the memory of Omar Foda.

We calculate the scalar product of Bethe states of the XXZ spin- chain with general integrable boundary conditions. The off-shell equations satisfied by the transfer matrix and the off-shell Bethe vectors allow one to derive a linear system for the scalar product of off-shell and on-shell Bethe states. We show that this linear system can be solved in terms of a compact determinant formula that involves the Jacobian of the transfer matrix eigenvalue and certain q-Pochhammer polynomials of the boundary couplings.

The Yangian relations of Heisenberg spin chain model

In this paper, we investigate the Yangian relations of Heisenberg spin chain systems. Firstly, we consider the closed XXZ spin chain model, through the Heisenberg spin XXZ model, we found the Hamiltonians for one kind system of three adjacent partial particles interaction systems. The model’s constitution rules of energy levels and energy states which expand from the few-particle system to multi-particle system have good regularity. In this system, we found Yangian’s law and illustrate it through graphs. Secondly, we further consider the closed XXZ spin chain’s generalization of other three neighboring particles interaction systems from few-particle system to multi-particle system. Finally, we also discussed the laws of the three adjacent particles system of some models, they are the XXZ model with twist boundary condition, the open XXZ spin chain model and the XXZ model containing the next neighbor. In addition, not only XXZ model, XXX model, XY model and Ising model, but the relevant laws of spin-1 systems of these models were also discussed, they have similar rules to the XXZ model. Through calculation and research, the eigensystems of these models all have good Yangian and constitution laws.

The relevant excitations for the one-body function in the Lieb–Liniger model

We study the ground state one-body correlation function in the Lieb–Liniger model. In the spectral representation, correlations are built from contributions stemming from different excited states of the model. We aim to understand which excited states carry significant contributions, specifically focusing on the small energy–momentum part of the dynamic one-body function. We conjecture that relevant excitations take form similar to two-spinon states known from XXZ spin chain. We validate this hypothesis by numerical evaluation of the correlator with ABACUS algorithm and by analytical computations in the strongly interacting regime.