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3: Correspondence of the upper horizontal line of the 6-vertex model and the first row of the alternating sign matrix.
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This course of lectures on the algebraic Bethe ansatz was given in the Scientific and Educational Center of Steklov Mathematical Institute in Moscow. The course includes both classical well known results and very recent ones.
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MAT037 is an intensive Mathematics course for pre-diploma students (Pre-Commerce) at Universiti Teknologi MARA. It is crucial for the lecturers to have a proper planning on teaching the course. This course includes six chapters, which are arithmetic and algebra (C1), equation and function (C2), index and logarithm (C3), sequence (C4), introduction...
Citations
... There are plenty of introductory lectures and reviews explaining the notion of quantum integrability and these approaches ( see e.g. [1][2][3][4][5]), here we will be mainly focused at structures responsible for classical integrability. ...
... Let us focus on the latter referring reader to the reviews [1][2][3][4][5] for more detailed description of other methods in application to AdS/CFT integrability and spin-chain models. ...
... thermodynamic Bethe ansatz, spectral curve. Although intimately related, these approaches stand beyond the scope of our review and hav been nicely covered in various reviews [1][2][3][4][5]. In this section we focus at integrable non-linear 2d sigma-models on group manifolds and (super-)coset spaces and their continuous Yang-Baxter deformations 3 . ...
This review is a collection of various methods and observations relevant to structures in three-dimensional systems similar to those responsible for integrability of two-dimensional systems. Particular focus is given to Nambu structures and loop variables naturally appearing in membrane dynamics. While reviewing each topic in more details we emphasize connections between them and speculate on possible relations to membrane integrability.
... In Appendix C we give a brief recapitulation of the Quantum Inverse Scattering Method, in which local operators such as σ z m are expressed in terms of entries of the monodromy matrix T = ( A B C D ). For a general overview of this method see [123]. The eigenstates of the XXZ Hamiltonian are given as |{λ} = n k=1 B(λ k )|0 for a set of Bethe roots {λ}, their duals as {λ}| = 0| n k=1 C(λ k ) and we wish to find matrix elements on the form {µ}| i σ a i m i |{λ} , where a i ∈ {z, +, −} and m i = m j for i = j. ...
... For a general overview of transfer matrix formalism, Bethe ansatz and the Quantum Inverse Scattering Method (QISM), see [123]. ...
In this thesis we study non-unitary two-dimensional bulk conformal field theories that appear in the continuum limit of certain critical lattice models, including the Potts and O(n) models. These models have applications to geometrical problems such as percolation and self-avoiding walks (polymers). To handle the technical difficulties caused by the non-unitarity we consider a lattice approach, which includes techniques such as the Bethe ansatz, representation theory of the affine Temperley-Lieb algebra and the usage of a discretization of the Virasoro algebra (the Koo-Saleur generators). We provide a graphical formulation of correlation functions in RSOS minimal models and compare to similar quantities in the Potts model. This allows us explain why an earlier conjecture for the Potts four-point functions, although incorrect, gave results in good agreement with numerical simulations. We detail the connections between RSOS models and anyonic systems, showing that the computation of correlation functions reduce to the evaluation of certain anyonic fusion diagrams. The main part of the thesis is dedicated to investigating what Virasoro modules are present in the six-vertex model, Potts and loop models at generic (irrational) central charge. This is mainly done using the Koo-Saleur generators. We find in the six-vertex model both Verma and co-Verma modules, while in the Potts and loop models the main finding is the presence of logarithmic modules corresponding to rank-two Jordan cells. These Jordan cells are not present at finite size, but appear only in the continuum limit. We also show the convergence of the Koo-Saleur generators to the Virasoro generators in a double limit procedure called the scaling limit, where the order of the two limits is crucial. Finally, knowledge of the logarithmic modules is used to bootstrap the O(n) model at generic complex values of n, using logarithmic conformal blocks in the crossing equations. We numerically compute four-point functions involving the simplest operators, and find that the number of solutions to the crossing equations is consistent with O(n) representation theory.
... Last but not least, one could proceed to explore the application of the Lie bialgebras and Hopf algebras discussed in this thesis in related branches of pure and applied mathematics. For example, in the study of integrable systems and in particular in the quantum inverse scattering method (see [142], [143] and references therein) quantum R-matrices play a vital role which, incidentally, was one of the first motivations to introduce quantum groups. ...
This thesis is devoted to the study of Lie bialgebra and Hopf algebra structures related to certain versions of non-commutative geometry constructed on infinite-dimensional Lie algebras that arise in the context of asymptotic symmetries of spacetime. We prove a number of theorems about cohomology groups that aid the classification of the Lie bialgebras and explicitly construct and analyze selected Hopf algebras. Particularly interesting behavior was found by studying the contraction limit of spacetimes with cosmological constant and the inclusion of central charges on the level of Lie bialgebras and Hopf algebras. Phenomenological consequences, like deformed in-vacuo dispersion relations, known from the study of $\kappa$-Poincar\'e quantum groups, are investigated. Furthermore, we examine how a new proposal in the context of the black hole information loss paradox and counterarguments against it are affected.
... These conditions are fulfilled by stronglyinteracting systems that are integrable via the Bethe Ansatz. In fact, in the framework of the algebraic Bethe Ansatz it is possible to show that there exist extensively many commuting conserved charges that can be retrieved from the series expansion in the spectral parameter of the transfer matrix [45,54,55]. However, these symmetries do not necessarily imply an extensive number of degeneracies in the spectrum of the Hamiltonian: rather, they enforce an intricate block-diagonal structure in which eigenstates carry several quantum numbers corresponding to different conserved charges (even though the underlying global symmetry might not be immediately apparent and the charges might not even be local). ...
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 increasing the amount of disorder effectively suppresses 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.
... In appendix B we give a brief recapitulation of the Quantum Inverse Scattering Method, in which local operators such as σ z m are expressed in terms of entries of the monodromy matrix T = A B C D . For a general overview of this method see [51]. The eigenstates of the XXZ Hamiltonian are given as |{λ} = n k=1 B(λ k )|0 for a set of Bethe roots {λ}, their duals as {λ}| = 0| n k=1 C(λ k ) and we wish to find matrix elements on the form ...
... For a general overview of transfer matrix formalism, Bethe ansatz and the Quantum Inverse Scattering Method (QISM), see [51]. ...
A bstract
We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules W j, 1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights ( h r,s , h r,−s ) and ( h r,−s , h r,s ) that give rise to dual structures: Verma or co-Verma modules. The limit of $$ {W}_{0,{\mathfrak{q}}^{\pm 2}} $$ W 0 , q ± 2 contains diagonal fields ( h r, 1 , h r, 1 ) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in $$ {\mathfrak{q}}^{\pm 2} $$ q ± 2 . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.
... The method is analogous to a general method for the derivation of asymptotics of equilibrium correlation functions in Bethe Ansatz solvable models [2][3][4][5][6][7][8][9] ( [10] for a recent review). The main challenge in this type of problems is that, even though all of the ingredients that are necessary for a formal expression of the observable of interest are known, at least in some implicit form (Bethe states and Bethe roots, matrix elements of the observables in the Bethe state basis), expanding in a thermodynamically large eigenstate basis and extracting the asymptotics of the resulting sum is very difficult. ...
... Luckily also this problem can be circumvented, using essentially the same complex analysis trick used in the Tonks-Girardeau case [1]: The sum over eigenstates can be still transformed into a (multiple) contour integral over complex rapidities by introducing a suitable function of the rapidities that has simple poles of residue 1/(2πi) precisely at the roots of Bethe equations, and distinguishes between odd and even integer sectors for each rapidity λ j through the corresponding discrete index s j . Such a function is indeed possible to construct, despite the fact that the Bethe roots are only implicitly known [2][3][4]10]. As we explain in detail in App. ...
... Recall now the multivariable version of the residue theorem [1,10], which states that, given a function F ...
We continue our study of the emergence of Non-Equilibrium Steady States in quantum integrable models focusing on the expansion of a Lieb-Liniger gas for arbitrary repulsive interaction. As a first step towards the derivation of the asymptotics of observables in the thermodynamic and large distance and time limit, we derive an exact multiple integral representation of the time evolved many-body wave-function. Starting from the known but complicated expression for the overlaps of the initial state of a geometric quench, which are derived from the Slavnov formula for scalar products of Bethe states, we eliminate the awkward dependence on the system size and distinguish the Bethe states into convenient sectors. These steps allow us to express the rather impractical sum over Bethe states as a multiple rapidity integral in various alternative forms. Moreover, we examine the singularities of the obtained integrand and calculate the contribution of the multivariable kinematical poles, which is essential information for the derivation of the asymptotics of interest.
... Therefore, to understand the stuff, the reader must possess the basic principles and techniques of the ABA. In addition to the original works mentioned above, they can be found in [13][14][15][16]. ...
... A key equation of the ABA is an RT T -relation [1,2,[13][14][15][16] R(u, v) T (u) ⊗ I I ⊗ T (v) = I ⊗ T (v) T (u) ⊗ I R(u, v). ...
... (1. 16) One would expect that an analog of R trig (u, v) in the case C N ⊗ C N is ...
We give a detailed description of the nested algebraic Bethe ansatz. We consider integrable models with a $\mathfrak{gl}_3$-invariant $R$-matrix as the basic example, however, we also describe possible generalizations. We give recursions and explicit formulas for the Bethe vectors. We also give a representation for the Bethe vectors in the form of a trace formula.
... The monodromy matrix of the continuous model satisfies the differential equation (4.1). In order to formulate a natural recipe for taking the → ∞ limit [7,17,18], let us derive a finite-difference version of this differential equation. First, we extract the matrix σ 3 from the L-operator (2.2), ...
... In the quantum NLS model [2,7,9,17,18], the fields obey the canonical commutation relations ψ (x),ψ(y) = [ψ(x), ψ(y)] = 0, ψ(x),ψ(y) = δ(x − y). ...
In this paper, we show that an operator introduced by A. A. Tsvetkov enjoys all the necessary properties of a Q-operator. It is shown that the Q-operator of the XXX spin chain with spin ℓ turns into Tsvetkov’s operator in the continuous limit as ℓ→∞. Bibliography: 18 titles.
... where the monodromy matrix and the transfer matrix read respectively, using the standard notations of the algebraic Bethe ansatz approach [9], ...
... where the hat B indicates that the corresponding factor is omitted in the product. Although the residues do vanish in case of strings, we have B(λ 1 )...B(λ K )|0 = 0 [9,10], so that the normalized Bethe state is not necessarily an eigenvector. In fact, the solutions to the Bethe equations with exact strings sometimes do yield eigenvalues and eigenvectors of the Hamiltonian, and sometimes not. ...
... Let us briefly explain our reasoning. In presence of exact strings, the residues in (7) still vanish as in the case of non-singular Bethe roots; however, the Bethe state B(λ 1 ) · · · B(λ n )|0 vanishes as well (see [9,10] and Lemma 3), and imposing the TQ relation (3) alone is then non-conclusive. We want to show that one can find a regularization such that the residues in (7) vanish faster than B(λ 1 ) · · · B(λ n )|0 when → 0, if and only if P is a polynomial. ...
We prove that physical solutions to the Heisenberg spin chain Bethe ansatz equations are exactly obtained by imposing two zero-remainder conditions. This bridges the gap between different criteria, and proves the correctness of a recently devised algorithm based on $QQ$ relations.
... The reason for such an interest is their integrability: the Bethe ansatz [1,2,3,4] enables, in principle, to compute energy levels or form factors or spin-spin correlations exactly (see e.g. [5] for a recent review). However, the simple and physically relevant question of the magnetization acquired by the spin chain when put in an external magnetic field is still unsolved [6,7]. ...
We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field $h$ as a series in $\sqrt{h_c-h}$, where $h_c$ is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. The radius of convergence of the series in the full range $0<h\leq h_c$ is studied numerically. To that end we express the free energy at mean magnetization per site $-1/2\leq \langle \sigma^z_i\rangle\leq 1/2$ as a series in $1/2-\langle \sigma^z_i\rangle$ whose coefficients can be similarly recursively computed in closed form. This series converges for all $0\leq \langle \sigma^z_i\rangle\leq 1/2$. The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in $1/2-\langle \sigma^z_i\rangle$. It can also be used to derive the corrections in finite size, that correspond to the spectrum of a free compactified boson whose radius can be expanded as a similar series. The method presumably applies to a large class of models: it also successfully applies to a case where the Bethe roots lie on a curve in the complex plane.
