Integrable Quantum Spin Chains Research Articles

The R-mAtrIx Net

We provide a novel neural network architecture that can: i) output R-matrix for a given quantum integrable spin chain, ii) search for an integrable Hamiltonian and the corresponding R-matrix under assumptions of certain symmetries or other restrictions, iii) explore the space of Hamiltonians around already learned models and reconstruct the family of integrable spin chains which they belong to. The neural network training is done by minimizing loss functions encoding Yang–Baxter equation, regularity and other model-specific restrictions such as hermiticity. Holomorphy is implemented via the choice of activation functions. We demonstrate the work of our neural network on the spin chains of difference form with two-dimensional local space. In particular, we reconstruct the R-matrices for all 14 classes. We also demonstrate its utility as an Explorer, scanning a certain subspace of Hamiltonians and identifying integrable classes after clusterisation. The last strategy can be used in future to carve out the map of integrable spin chains with higher dimensional local space and in more general settings where no analytical methods are available.

Canonical formulation for the thermodynamics of sln-invariant integrable spin chains

Integrable quantum spin chains display distinctive physical properties making them a laboratory to test and assess different states of matter. The study of the finite temperature properties is possible by use of the thermodynamic Bethe ansatz, however at the expense of dealing with non-linear integral equations for, in general, infinitely many auxiliary functions. The definition of an alternative finite set of auxiliary functions allowing for the complete description of their thermodynamic properties at finite temperature and fields has been elusive. Indeed, in the context of sln-invariant models satisfactory auxiliary functions have been established only for n≤4. In this paper we take a step further by proposing a systematic approach to generate finite sets of auxiliary functions for sln-invariant models. We refer to this construction as the canonical formulation. The numerical efficiency is illustrated for n=5, for which we present some of the thermodynamic properties of the corresponding spin chain.

The Zoo of Opers and Dualities

Abstract We investigate various spaces of $SL(r+1)$-opers and their deformations. For each type of such opers, we study the quantum/classical duality, which relates quantum integrable spin chains with classical solvable many body systems. In this context, quantum/classical dualities serve as an interplay between two different coordinate systems on the space of opers. We also establish correspondences between the underlying oper spaces, which recently had multiple incarnations in symplectic duality and bispectral duality.

Integrable crosscap states: from spin chains to 1D Bose gas

The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.

Exact quench dynamics from algebraic geometry.

We develop a systematic approach to compute physical observables of integrable spin chains with finite length. Our method is based on the Bethe ansatz solution of the integrable spin chain and computational algebraic geometry. The final results are analytic and no longer depend on Bethe roots. The computation is purely algebraic and does not rely on further assumptions or numerics. This method can be applied to compute a broad family of physical quantities in integrable quantum spin chains. We demonstrate the power of the method by computing two important quantities in quench dynamics-the diagonal entropy and the Loschmidt echo-and obtain analytic results.

Robustness of Kardar-Parisi-Zhang scaling in a classical integrable spin chain with broken integrability

Recent investigations have observed superdiffusion in integrable classical and quantum spin chains. An intriguing connection between these spin chains and Kardar-Parisi-Zhang (KPZ) universality class has emerged. Theoretical developments (e.g. generalized hydrodynamics) have highlighted the role of integrability as well as spin-symmetry in KPZ behaviour. However understanding their precise role on superdiffusive transport still remains a challenging task. The widely used quantum spin chain platform comes with severe numerical limitations. To circumvent this barrier, we focus on a classical integrable spin chain which was shown to have deep analogy with the quantum spin-$\frac{1}{2}$ Heisenberg chain. Remarkably, we find that KPZ behaviour prevails even when one considers integrability-breaking but spin-symmetry preserving terms, strongly indicating that spin-symmetry plays a central role even in the non-perturbative regime. On the other hand, in the non-perturbative regime, we find that energy correlations exhibit clear diffusive behaviour. We also study the classical analog of out-of-time-ordered correlator (OTOC) and Lyapunov exponents. We find significant presence of chaos for the integrability-broken cases even though KPZ behaviour remains robust. The robustness of KPZ behaviour is demonstrated for a wide class of spin-symmetry preserving integrability-breaking terms.

Rational $Q$-systems, Higgsing and mirror symmetry

The rational QQ-system is an efficient method to solve Bethe ansatz equations for quantum integrable spin chains. We construct the rational QQ-systems for generic Bethe ansatz equations described by an A_{\ell-1}Aℓ−1 quiver, which include models with multiple momentum carrying nodes, generic inhomogeneities, generic diagonal twists and qq-deformation. The rational QQ-system thus constructed is specified by two partitions. Under Bethe/Gauge correspondence, the rational QQ-system is in a one-to-one correspondence with a 3d \mathcal{N}=4𝒩=4 quiver gauge theory of the type {T}_{{\rho}}^{{\sigma}}[\mathrm{SU}(n)]Tρσ[SU(n)], which is also specified by the same partitions. This shows that the rational QQ-system is a natural language for the Bethe/Gauge correspondence, because known features of the {T}_{{\rho}}^{{\sigma}}[\mathrm{SU}(n)]Tρσ[SU(n)] theories readily translate. For instance, we show that the Higgs and Coulomb branch Higgsing correspond to modifying one of the partitions in the rational QQ-system while keeping the other untouched. Similarly, mirror symmetry is realized in terms of the rational QQ-system by simply swapping the two partitions - exactly as for {T}_{{\rho}}^{{\sigma}}[\mathrm{SU}(n)]Tρσ[SU(n)]. We exemplify the computational efficiency of the rational QQ-system by evaluating topologically twisted indices for 3d \mathcal{N}=4𝒩=4\mathrm{U}(n)U(n) SQCD theories with n=1,\ldots,5n=1,…,5.

Integrable crosscap states in mathfrak{gl} (N) spin chains

We study the integrable crosscap states of the integrable quantum spin chains and we classify them for the mathfrak{gl} (N) symmetric models. We also give a derivation for the exact overlaps between the integrable crosscap states and the Bethe states. The first part of the derivation is to calculate sum formula for the off-shell overlap. Using this formula we prove that the normalized overlaps of the multi-particle states are ratios of the Gaudin-like determinants. Furthermore we collect the integrable crosscap states which can be relevant in the AdS/CFT correspondence.

$\mathrm{T}\overline{\mathrm{T}}$-deformed 1d Bose gas

\mathrm{T}\bar{\mathrm{T}}TT‾ deformation was originally proposed as an irrelevant solvable deformation for 2d relativistic quantum field theories (QFTs). The same family of deformations can also be defined for integrable quantum spin chains which was first studied in the context of integrability in AdS/CFT. In this paper, we construct such deformations for yet another type of models, which describe a collection of particles moving in 1d and interacting in an integrable manner. The prototype of such models is the Lieb-Liniger model. This shows that such deformations can be defined for a very wide range of systems. We study the finite volume spectrum and thermodynamics of the \mathrm{T}\overline{\mathrm{T}}TT¯-deformed Lieb-Liniger model. We find that for one sign of the deformation parameter (\lambda<0)(λ<0), the deformed spectrum becomes complex when the volume of the system is smaller than certain critical value, signifying the break down of UV physics. For the other sign (\lambda>0)(λ>0), there exists an upper bound for the temperature, similar to the Hagedorn behavior of the \mathrm{T}\overline{\mathrm{T}}TT¯ deformed QFTs. Both behaviors can be attributed to the fact that \mathrm{T}\overline{\mathrm{T}}TT¯ deformation changes the size the particles. We show that for \lambda>0λ>0, the deformation increases the spaces between particles which effectively increases the volume of the system. For \lambda<0λ<0, \mathrm{T}\overline{\mathrm{T}}TT¯ deformation fattens point particles to finite size hard rods. This is similar to the observation that the action of \mathrm{T}\overline{\mathrm{T}}TT¯-deformed free boson is the Nambu-Goto action, which describes bosonic strings — also an extended object with finite size.

Exact ground state and elementary excitations of a competing spin chain with twisted boundary condition

A novel Bethe ansatz scheme is proposed to investigate the exact physical properties of an integrable anisotropic quantum spin chain with competing interactions among the nearest, next nearest neighbor and chiral three spin couplings, where the boundary condition is the twisted one. The eigenvalue of the transfer matrix is characterized by its zero roots instead of the traditional Bethe roots. Based on the exact solution, the conserved momentum and charge operators of this U(1)-symmetry broken system are obtained. The ground state energy and density of rapidities are calculated. It is found that there exist three kinds of elementary excitations and the corresponding dispersion relations are obtained, which gives a different picture from that with periodic boundary condition.

Spin chains with boundary inhomogeneities

We investigate the effect of introducing a boundary inhomogeneity in the transfer matrix of an integrable open quantum spin chain. We find that it is possible to construct a local Hamiltonian, and to have quantum group symmetry. The boundary inhomogeneity has a profound effect on the Bethe ansatz solution.

Exact solution of an anisotropic J 1–J 2 model with the Dzyloshinsky–Moriya interactions at boundaries

We propose a method to construct new quantum integrable models. As an example, we construct an integrable anisotropic quantum spin chain which includes the nearest-neighbor, next-nearest-neighbor and chiral three-spin couplings. It is shown that the boundary fields can enhance the anisotropy of the first and last bonds, and can induce the Dzyloshinsky–Moriya interactions along the z-direction at the boundaries. By using the algebraic Bethe ansatz, we obtain the exact solution of the system. The energy spectrum of the system and the associated Bethe ansatz equations are given explicitly. The method provided in this paper is universal and can be applied to constructing other exactly solvable models with certain interesting interactions.

Exact solution of an integrable quantum spin chain with competing interactions* *Project supported by the National Key Research and Development Program of China (Grant Nos. 2016YFA0300600 and 2016YFA0302104), the National Natural Science Foundation of China (Grant Nos. 12074410, 11934015, 11975183, 11947301, and 11774397), the Major Basic Research Program of Natural Science of Shaanxi Province, China (Grant Nos. 2017KCT-12 and 2017ZDJC-32),

We construct an integrable quantum spin chain that includes the nearest-neighbor, next-nearest-neighbor, chiral three-spin couplings, Dzyloshinsky–Moriya interactions and unparallel boundary magnetic fields. Although the interactions in bulk materials are isotropic, the spins nearby the boundary fields are polarized, which induce the anisotropic exchanging interactions of the first and last bonds. The U(1) symmetry of the system is broken because of the off-diagonal boundary reflections. Using the off-diagonal Bethe ansatz, we obtain an exact solution to the system. The inhomogeneous T–Q relation and Bethe ansatz equations are given explicitly. We also calculate the ground state energy. The method given in this paper provides a general way to construct new integrable models with certain interesting interactions.

Quantum Gaudin model, spin chains, and universal characters

In this paper, we will construct the connection between the quantum Gaudin model and the universal character hierarchy by a newly defined coupled master T-operator. The coupled master T-operator of the quantum Gaudin model satisfies the bilinear identity of the universal character hierarchy, which is an extension of the Kadomtsev–Petviashvili hierarchy. In addition, for the generalized quantum integrable spin chains with rational GL(N)-invariant R-matrices, we construct its coupled master T-operator, which represents a generating function for two-folds commuting quantum transfer matrices. The functional relations for the transfer matrices are equivalent to an infinite set of Hirota bilinear equations of the modified universal character hierarchy.

The Am(1) Q-system

We propose a Q-system for the [Formula: see text] quantum integrable spin chain. We also find compact determinant expressions for all the Q-functions, both for the rational and trigonometric cases.

Towards the solution of an integrable spin chain

Two branches of integrable open quantum-group invariant quantum spin chains are known. For one branch (), a complete Bethe ansatz solution has been proposed. However, the other branch () has so far resisted solution. In an effort to address this problem, we consider here the simplest case n = 1. We propose a Bethe ansatz solution, which however is not complete, as it describes only the transfer-matrix eigenvalues with odd degeneracy. We also consider a proposal for the missing eigenvalues.

Off-diagonal Bethe Ansatz on the so(5) spin chain

The so(5) (i.e., B2) quantum integrable spin chains with both periodic and non-diagonal boundaries are studied via the off-diagonal Bethe Ansatz method. By using the fusion technique, sufficient operator product identities (comparing to those in [1]) to determine the spectrum of the transfer matrices are derived. For the periodic case, we recover the results obtained in [1], while for the non-diagonal boundary case, a new inhomogeneous T−Q relation is constructed. The present method can be directly generalized to deal with the so(2n+1) (i.e., Bn) quantum integrable spin chains with general boundaries.

Canonical Drude Weight for Non-integrable Quantum Spin Chains

The Drude weight is a central quantity for the transport properties of quantum spin chains. The canonical definition of Drude weight is directly related to Kubo formula of conductivity. However, the difficulty in the evaluation of such expression has led to several alternative formulations, accessible to different methods. In particular, the Euclidean, or imaginary-time, Drude weight can be studied via rigorous renormalization group. As a result, in the past years several universality results have been proven for such quantity at zero temperature; remarkably the proof works for both for integrable and non-integrable quantum spin chains. Here we establish the equivalence of Euclidean and canonical Drude weights at zero temperature. Our proof is based on rigorous renormalization group methods, Ward identities, and complex analytic ideas.

Surveying the quantum group symmetries of integrable open spin chains

Using anisotropic R-matrices associated with affine Lie algebras gˆ (specifically, A2n(2), A2n−1(2), Bn(1), Cn(1), Dn(1)) and suitable corresponding K-matrices, we construct families of integrable open quantum spin chains of finite length, whose transfer matrices are invariant under the quantum group corresponding to removing one node from the Dynkin diagram of gˆ. We show that these transfer matrices also have a duality symmetry (for the cases Cn(1) and Dn(1)) and additional Z2 symmetries that map complex representations to their conjugates (for the cases A2n−1(2), Bn(1) and Dn(1)). A key simplification is achieved by working in a certain “unitary” gauge, in which only the unbroken symmetry generators appear. The proofs of these symmetries rely on some new properties of the R-matrices. We use these symmetries to explain the degeneracies of the transfer matrices.

Action of Clifford Algebra on the Space of Sequences of Transfer Operators

We deduce from a determinant identity on quantum transfer matrices of generalized quantum integrable spin chain model their generating functions. We construct the isomorphism of Clifford algebra modules of sequences of transfer matrices and the boson space of symmetric functions. As an application, tau-functions of transfer matrices immediately arise from classical tau-functions of symmetric functions.