Integrable Lattice Models Research Articles

On different approaches to integrable lattice models

Interaction-round the face (IRF) models are two-dimensional lattice models of statistical mechanics defined by an affine Lie algebra and admissibility conditions depending on a choice of representation of that affine Lie algebra. Integrable IRF models, i.e. the models the Boltzmann weights of which satisfy the quantum Yang–Baxter equation, are of particular interest. In this paper, we investigate trigonometric Boltzmann weights of integrable IRF models. By using an ansatz proposed by one of the authors in some previous works, the Boltzmann weights of the restricted IRF models based on the affine Lie algebras su(2)k and su(3)k are computed for fundamental and adjoint representations for some fixed levels k. New solutions for the Boltzmann weights are obtained. We also study the vertex-IRF correspondence in the context of an unrestricted IRF model based on su(3)k (for general k) and discuss how it can be used to find Boltzmann weights in terms of the quantum R^ matrix when the adjoint representation defines the admissibility conditions.

TT¯ Deformation: A Lattice Approach

Integrable quantum field theories can be regularized on the lattice while preserving integrability. The resulting theories on the lattice are integrable lattice models. A prototype of such a regularization is the correspondence between a sine-Gordon model and a six-vertex model on a light-cone lattice. We propose an integrable deformation of the light-cone lattice model such that in the continuum limit we obtain the TT¯-deformed sine-Gordon model. Under this deformation, the cut-off momentum becomes energy dependent and the underlying Yang–Baxter integrability is preserved. Therefore, this deformation is integrable but non-local: similar to the TT¯ deformation of quantum field theory.

Bailey pairs for the q-hypergeometric integral pentagon identity

In this work, we construct new Bailey pairs for the integral pentagon identity in terms of q-hypergeometric functions. The pentagon identity considered here represents the equality of the partition functions of certain three-dimensional supersymmetric dual theories. It can be also interpreted as the star-triangle relation for the Ising-type integrable lattice model.

Thermal form-factor expansion of the dynamical two-point functions of local operators in integrable quantum chains

Evaluating a lattice path integral in terms of spectral data and matrix elements pertaining to a suitably defined quantum transfer matrix, we derive form-factor series expansions for the dynamical two-point functions of arbitrary local operators in fundamental Yang–Baxter integrable lattice models at finite temperature. The summands in the series are parameterised by solutions of the Bethe Ansatz equations associated with the eigenvalue problem of the quantum transfer matrix. We elaborate on the example of the XXZ chain for which the solutions of the Bethe Ansatz equations are sufficiently well understood in certain limiting cases. We work out in detail the case of the spin-zero operators in the antiferromagnetic massive regime at zero temperature. In this case the thermal form-factor series turn into series of multiple integrals with fully explicit integrands. These integrands factorize into an operator-dependent part, determined by the so-called Fermionic basis, and a part which we call the universal weight as it is the same for all spin-zero operators. The universal weight can be inferred from our previous work. The operator-dependent part is rather simple for the most interesting short-range operators. It is determined by two functions ρ and ω for which we obtain explicit expressions in the considered case. As an application we rederive the known explicit form-factor series for the two-point function of the magnetization operator and obtain analogous expressions for the magnetic current and the energy operators.

On Bailey pairs for mathcal{N} = 2 supersymmetric gauge theories on {S}_b^3/{mathbb{Z}}_r

We study Bailey pairs construction for hyperbolic hypergeometric integral identities acquired via the duality of lens partitions functions for the three-dimensional mathcal{N} = 2 supersymmetric gauge theories on {S}_b^3/{mathbb{Z}}_r . The novel Bailey pairs are constructed for the star-triangle relation, the star-star relation, and the pentagon identity. The first two of them are integrability conditions for the Ising-type integrable lattice models. The last one corresponds to the representation of the basic 2 − 3 Pachner move for triangulated 3-manifolds.

Integrable 3D lattice model in M-theory

It is argued that the supersymmetric index of a certain system of branes in M-theory is equal to the partition function of an integrable three-dimensional lattice model. The local Boltzmann weights of the lattice model satisfy a generalization of Zamolodchikov’s tetrahedron equation. In a special case the model is described by a solution of the tetrahedron equation discovered by Kapranov and Voevodsky and by Bazhanov and Sergeev.

The alternating central extension of the Onsager Lie algebra

The Onsager Lie algebra O is often used to study integrable lattice models. The universal enveloping algebra of O admits a q-deformation Oq called the q-Onsager algebra. Recently, an algebra was introduced called the alternating central extension of Oq . We introduce a Lie algebra that is roughly described by the following two analogies: (i) is to O as is to Oq ; (ii) Oq is to O as is to We call the alternating central extension of O. We give a comprehensive description of

Equivalence of spatial and particle entanglement growth after a quantum quench

We analyze fermions after an interaction quantum quench in one spatial dimension and study the growth of the steady state entanglement entropy density under either a spatial mode or particle bipartition. For integrable lattice models, we find excellent agreement between the increase of spatial and particle entanglement entropy, and for chaotic models, an examination of two further neighbor interaction strengths suggests similar correspondence. This result highlights the generality of the dynamical conversion of entanglement to thermodynamic entropy under time evolution that underlies our current framework of quantum statistical mechanics.

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.

Conjectures on hidden Onsager algebra symmetries in interacting quantum lattice models

We conjecture the existence of hidden Onsager algebra symmetries in two interacting quantum integrable lattice models, i.e. spin-1/2 XXZ model and spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy. The conjectures relate the Onsager generators to the conserved charges obtained from semi-cyclic transfer matrices. The conjectures are motivated by two examples which are spin-1/2 XX model and spin-1 U(1)-invariant clock model. A novel construction of the semi-cyclic transfer matrices of spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy is carried out via the transfer matrix fusion procedure.

Anisotropic Landau-Lifshitz model in discrete space-time

We construct an integrable lattice model of classical interacting spins in discrete space-time, representing a discrete-time analogue of the lattice Landau-Lifshitz ferromagnet with uniaxial anisotropy. As an application we use this explicit discrete symplectic integration scheme to compute the spin Drude weight and diffusion constant as functions of anisotropy and chemical potential. We demonstrate qualitatively different behavior in the easy-axis and the easy-plane regimes in the non-magnetized sector. Upon approaching the isotropic point we also find an algebraic divergence of the diffusion constant, signaling a crossover to spin superdiffusion.

Integrable models and K-theoretic pushforward of Grothendieck classes

We show that a multiple commutation relation of the Yang-Baxter algebra of integrable lattice models derived by Shigechi and Uchiyama can be used to connect two types of Grothendieck classes by the K-theoretic pushforward from the Grothendieck group of Grassmann bundles to the Grothendieck group of a nonsingular variety. Using the commutation relation, we show that two types of partition functions of an integrable five-vertex model, which can be explicitly described using skew Grothendieck polynomials, and can be viewed as Grothendieck classes, are directly connected by the K-theoretic pushforward. We show that special cases of the pushforward formula which correspond to the nonskew version are also special cases of the formulas derived by Buch. We also present a skew generalization of an identity for the Grothendieck polynomials by Guo and Sun, which is an extension of the one for Schur polynomials by Fehér, Némethi and Rimányi. We also show an application of the pushforward formula and derive an integration formula for the Grothendieck polynomials.

Ballistic transport in integrable quantum lattice models with degenerate spectra

We study the ballistic transport in integrable lattice models, i.e., the spin XXZ and Hubbard chains, close to the noninteracting limit. The stiffnesses of spin and charge currents reveal, at high temperatures, a discontinuous reduction (jump) when the interaction is introduced. We show that the jumps are related to the large degeneracy of the parent noninteracting models. These degeneracies are properly captured by the degenerate perturbation calculations which may be performed for large systems. We find that the discontinuities and the quasilocality of the conserved current in this limit can be traced back to the nonlocal character of an effective interaction. From the latter observation we identify a class of observables which show discontinuities in both models. We also argue that the known local conserved quantities are insufficient to explain the stiffnesses in the Hubbard chain in the regime of weak interaction.

Analytical results for the low-temperature Drude weight of the XXZ spin chain

The spin-$1/2$ XXZ chain is an integrable lattice model and parts of its spin current can be protected by local conservation laws for anisotropies $-1<\Delta<1$. In this case, the Drude weight $D(T)$ is non-zero at finite temperatures $T$. Here we obtain analytical results for $D(T)$ at low temperatures for zero external magnetic field and anisotropies $\Delta=\cos(n\pi/m)$ with $n,m$ coprime integers, using the thermodynamic Bethe ansatz. We show that to leading orders $D(T)=D(0)-a(\Delta)T^{2K-2}-b_1(\Delta)T^2$ where $K$ is the Luttinger parameter and the prefactor $a(\Delta)$, obtained in closed form, has a fractal structure as function of anisotropy $\Delta$. The prefactor $b_1(\Delta)$, on the other hand, does not have a fractal structure and can be obtained in a standard field-theoretical approach. Including both temperature corrections, we obtain an analytic result for the low-temperature asymptotics of the Drude weight in the entire regime $-1<\Delta=\cos(n\pi/m)<1$.

Finite-temperature transport in one-dimensional quantum lattice models

The last decade has witnessed an impressive progress in the theoretical understanding of transport properties of clean, one-dimensional quantum lattice systems. Many physically relevant models in one dimension are Bethe-ansatz integrable, including the anisotropic spin-1/2 Heisenberg (also called spin-1/2 XXZ chain) and the Fermi-Hubbard model. Nevertheless, practical computations of, for instance, correlation functions and transport coefficients pose hard problems from both the conceptual and technical point of view. Only due to recent progress in the theory of integrable systems on the one hand and due to the development of numerical methods on the other hand has it become possible to compute their finite temperature and nonequilibrium transport properties quantitatively. Most importantly, due to the discovery of a novel class of quasilocal conserved quantities, there is now a qualitative understanding of the origin of ballistic finite-temperature transport, and even diffusive or super-diffusive subleading corrections, in integrable lattice models. We shall review the current understanding of transport in one-dimensional lattice models, in particular, in the paradigmatic example of the spin-1/2 XXZ and Fermi-Hubbard models, and we elaborate on state-of-the-art theoretical methods, including both analytical and computational approaches. Among other novel techniques, we discuss matrix-product-states based simulation methods, dynamical typicality, and, in particular, generalized hydrodynamics. We will discuss the close and fruitful connection between theoretical models and recent experiments, with examples from both the realm of quantum magnets and ultracold quantum gases in optical lattices.

One-dimensional model for deconfined criticality with Z3×Z3 symmetry

We continue recent efforts to discover examples of deconfined quantum criticality in one-dimensional models. In this work we investigate the transition between a $\mathbb{Z}_3$ ferromagnet and a phase with valence bond solid (VBS) order in a spin chain with $\mathbb{Z}_3\times\mathbb{Z}_3$ global symmetry. We study a model with alternating projective representations on the sites of the two sublattices, allowing the Hamiltonian to connect to an exactly solvable point having VBS order with the character of SU(3)-invariant singlets. Such a model does not admit a Lieb-Schultz-Mattis theorem typical of systems realizing deconfined critical points. Nevertheless, we find evidence for a direct transition from the VBS phase to a $\mathbb{Z}_3$ ferromagnet. Finite-entanglement scaling data are consistent with a second-order or weakly first-order transition. We find in our parameter space an integrable lattice model apparently describing the phase transition, with a very long, finite, correlation length of 190878 lattice spacings. Based on exact results for this model, we propose that the transition is extremely weakly first order, and is part of a family of DQCP described by walking of renormalization group flows.

Interaction-round-a-face and consistency-around-a-face-centered-cube

There is a correspondence between integrable lattice models of statistical mechanics and discrete integrable equations that satisfy multidimensional consistency, where the latter may be found in a quasi-classical expansion of the former. This paper extends this correspondence to interaction-round-a-face (IRF) models, resulting in a new formulation of the consistency-around-a-cube integrability condition applicable to five-point equations in the square lattice. Multidimensional consistency for these equations is formulated as consistency-around-a-face-centered-cube (CAFCC), which, namely, involves satisfying an overdetermined system of 14 five-point lattice equations for eight unknown variables on the face-centered cubic unit cell. From the quasi-classical limit of IRF models, which are constructed from the continuous spin solutions of the star–triangle relations associated with the Adler–Bobenko–Suris list, 15 sets of equations are obtained, which satisfy CAFCC.

Microscopic conservation laws for integrable lattice models

We consider two discrete completely integrable evolutions: the Toda Lattice and the Ablowitz–Ladik system. The principal thrust of the paper is the development of microscopic conservation laws that witness the conservation of the perturbation determinant under these dynamics. In this way, we obtain discrete analogues of objects that we found essential in our recent analyses of KdV, NLS, and mKdV. In concert with this, we revisit the classical topic of microscopic conservation laws attendant to the (renormalized) trace of the Green’s function.

On quantum separation of variables beyond fundamental representations

We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate bases in which their spectral problem is separated, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called “non-fundamental” models we construct two different types of SoV bases. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin’s approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic Y(gl_{2})Y(gl2) Yang-Baxter algebra. Our SoV approach also leads to the construction of a QQ-operator in terms of the fused transfer matrices. Finally, we show that the QQ-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases.

Integrable lattice models and holography

We study four-dimensional Chern-Simons theory on D × ℂ (where D is a disk), which is understood to describe rational solutions of the Yang-Baxter equation from the work of Costello, Witten and Yamazaki. We find that the theory is dual to a boundary theory, that is a three-dimensional analogue of the two-dimensional chiral WZW model. This boundary theory gives rise to a current algebra that turns out to be an “analytically-continued” toroidal Lie algebra. In addition, we show how certain bulk correlation functions of two and three Wilson lines can be captured by boundary correlation functions of local operators in the three-dimensional WZW model. In particular, we reproduce the leading and subleading nontrivial contributions to the rational R-matrix purely from the boundary theory.