Eigenvalues Of Transfer Matrix Research Articles

Space-like asymptotics of the thermal two-point functions of the XXZ spin-1/2 chain

This work proposes a closed formula for the leading term of the large-distance and long-time asymptotics in a cone of the space-like regime for the transverse dynamical two-point functions of the XXZ spin 1/2 chain at finite temperatures. The result follows from a simple analysis of the thermal form factor series for dynamical correlation functions. The obtained leading asymptotics are driven by the Bethe Ansatz data associated with the first sub-leading eigenvalue of the quantum transfer matrix.

Localization as a consequence of quasiperiodic bulk-bulk correspondence

We report on a direct connection between quasi-periodic topology and the Almost Mathieu (Andre-Aubry) metal insulator transition (MIT). By constructing quasi-periodic transfer matrix equations from the limit of rational approximate projected Green's functions, we relate results from $\text{SL}(2,\mathbb{R})$ co-cycle theory (transfer matrix eigenvalue scaling) to consequences of rational band theory. This reduction links the eigenfunction localization of the MIT to the chiral edge modes of the Hofstadter Hamiltonian, implying the localized phase roots in a topological "bulk-bulk" correspondence, a bulk-boundary correspondence between the 1D AAH system (boundary) and its 2D parent Hamiltonian (bulk). This differentiates quasi-periodic localization from Anderson localization in disordered systems. Our results are widely applicable to systems beyond this paradigmatic model.

Surface energy of the one-dimensional supersymmetric [formula omitted] model with general integrable boundary terms in the antiferromagnetic sector

In this paper, we study the surface energy of the one-dimensional supersymmetric t−J model with nonparallel boundary magnetic fields, which is a typical U(1)-symmetry broken quantum integrable strongly correlated electron system. It is shown that at the ground state, the contribution of inhomogeneous term in the Bethe ansatz solution of eigenvalues of transfer matrix satisfies the finite size scaling law Lβ where β<0. Based on it, the physical quantities of the system in the thermodynamic limit are calculated. We obtain the patterns of Bethe roots, root densities, ground state energy and surface energy. We also find that there exist the stable boundary bound states if the boundary fields satisfy some constraints.

Mutual information and breakdown of the Perron-Frobenius scenario in zero-temperature triangular Ising antiferromagnets on cylinders.

A nominally two-dimensional spin model wrapped onto a cylinder can profitably be viewed, especially for long cylinders, as a one-dimensional chain. Each site of such a chain is a ring of spins with a complex state space. Traditional correlation functions are inadequate for the study of correlations in such a system and need to be replaced with something like mutual information. Being induced purely by frustration, the disorder of a cylindrical zero-temperature triangular Ising antiferromagnet (TIAFM) and attendant correlations have a chance of evading the consequences of the Perron-Frobenius theorem which describes and constrains correlations in thermally disordered one-dimensional systems. Correlations in such TIAFM systems and the aforementioned evasion are studied here through a fermionic representation. For cylindrical TIAFM models with open boundary conditions, we explain and derive the following characteristics of end-to-end mutual information: period-three oscillation of the decay length, halving of the decay length compared to what Perron-Frobenius predicts on the basis of transfer matrix eigenvalues, and subexponential decay-inverse square in the length-for certain systems.

On scalar products and form factors by separation of variables: the antiperiodic XXZ model

We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by separation of variables, and the eigenstates can be constructed in terms of Q-functions, solution of a Baxter TQ-equation, which have double periodicity compared to the periodic case. We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations with rows and columns labelled by the roots of the Q-functions of the corresponding separate states, as in the periodic case, although the form of the determinant are here slightly different. We also propose alternative types of determinant representations written directly in terms of the transfer matrix eigenvalues.

Dressed energy of the XXZ chain in the complex plane

We consider the dressed energy varepsilon of the XXZ chain in the massless antiferromagnetic parameter regime at 0< Delta < 1 and at finite magnetic field. This function is defined as a solution of a Fredholm integral equation of the second kind. Conceived as a real function over the real numbers, it describes the energy of particle–hole excitations over the ground state at fixed magnetic field. The extension of the dressed energy to the complex plane determines the solutions to the Bethe Ansatz equations for the eigenvalue problem of the quantum transfer matrix of the model in the low-temperature limit. At low temperatures, the Bethe roots that parametrize the dominant eigenvalue of the quantum transfer matrix come close to the curve mathrm{Re}, varepsilon (lambda ) = 0. We describe this curve and give lower bounds to the function mathrm{Re}, varepsilon in regions of the complex plane, where it is positive.

Formula omitted] relation and spectrum of supersymmetric [formula omitted] model without [formula omitted] symmetry

We propose a novel characterization of the exact solution of the one-dimensional supersymmetric t−J model with generic non-diagonal boundary conditions. Instead of using Bethe roots, we parameterize the eigenvalues of transfer matrix in the spin sector with their zero-roots. By constructing the t−W relation, we obtain the explicit forms of the eigenvalues of the system and the homogeneous Bethe ansatz equations. This scheme is universal and can be applied to other quantum integrable systems with or without U(1) symmetry.

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.

A time-domain stability analysis method for paralleled LLC resonant converter system based on Floquet theory

In this paper, a stability analysis method based on Floquet theory is firstly proposed to provide an effective stability analysis method for paralleled LLC resonant converter system. Firstly, the time-domain model of a paralleled LLC resonant converter system is established, then, the stable range of the paralleled LLC resonant converter system can be determined by analysing the eigenvalues of the state transfer matrix based on Floquet theory, which is obtained from the established time-domain model. Finally, a two-stage paralleled LLC resonant converter system controlled by average current mode is taken as an example, the correctness and effectiveness of the proposed stability analysis method based on Floquet Theory is verified by simulation. The proposed stability analysis method can avoid the complex derivation of transfer function and impedance in frequency domain analysis methods.

Separation of variables bases for integrable $gl_{\mathcal{M}|\mathcal{N}}$ and Hubbard models

We construct quantum Separation of Variables (SoV) bases for both the fundamental inhomogeneous % gl_{\mathcal{M}|\mathcal{N}}glℳ|𝒩 supersymmetric integrable models and for the inhomogeneous Hubbard model both defined with quasi-periodic twisted boundary conditions given by twist matrices having simple spectrum. The SoV bases are obtained by using the integrable structure of these quantum models, i.e. the associated commuting transfer matrices, following the general scheme introduced in [1]; namely, they are given by set of states generated by the multiple actions of the transfer matrices on a generic co-vector. The existence of such SoV bases implies that the corresponding transfer matrices have non-degenerate spectrum and that they are diagonalizable with simple spectrum if the twist matrices defining the quasi-periodic boundary conditions have that property. Moreover, in these SoV bases the resolution of the transfer matrix eigenvalue problem leads to the resolution of the full spectral problem, i.e. both eigenvalues and eigenvectors. Indeed, to any eigenvalue is associated the unique (up to a trivial overall normalization) eigenvector whose wave-function in the SoV bases is factorized into products of the corresponding transfer matrix eigenvalue computed on the spectrum of the separated variables. As an application, we characterize completely the transfer matrix spectrum in our SoV framework for the fundamental gl_{1|2}gl1|2 supersymmetric integrable model associated to a special class of twist matrices. From these results we also prove the completeness of the Bethe Ansatz for that case. The complete solution of the spectral problem for fundamental inhomogeneous gl_{\mathcal{M}|\mathcal{N}}glℳ|𝒩 supersymmetric integrable models and for the inhomogeneous Hubbard model under the general twisted boundary conditions will be addressed in a future publication.

Wronskian-type formula for inhomogeneous $$TQ$$ equations

The transfer-matrix eigenvalues of the isotropic open Heisenberg quantum spin-1/2 chain with non-diagonal boundary magnetic fields are known to satisfy a TQ-equation with an inhomogeneous term. We derive here a discrete Wronskian-type formula relating a solution of this inhomogeneous TQ-equation to the corresponding solution of a dual inhomogeneous TQ-equation.

Unitarity and symmetries of the multicomponent scattering matrix

We present a theoretical procedure, which is fundamental for unitarity preservation in multicomponent–multiband systems, for a synchronous mixed-particle quantum transport. This study focuses the problem of (N×N) interacting components (with N≥2), in the framework of the envelope function approximation (EFA), and the standard unitary properties of the (N=1) scattering matrix are recovered. Rather arbitrary conditions to the basis-set and/or to the output scattering coefficients, are not longer required to preserve flux conservation, if the eigen-functions are orthonormalized in both the configuration and the spinorial spaces. We predict that the present approach, is valid for different kind of multiband–multicomponent physical systems of coupled charge–spin carriers, within the EFA, with small transformations if any. We foretell the interplay for the state-vector transfer matrix eigen-values, together with the large rates of its condition number, as novel complementary tools for a more accurate definition of the threshold for tunneling channels in a scattering experiment.

Chordal Graphs and Partial Positive Passivity Assessment

This paper presents a novel approach to passivity assessment of multiport frequency-dependent rational models. The strategy exploits the interlacing properties relating the eigenvalues of a full transfer matrix and those of its submatrices plus some additional concepts from graph theory. Contrary to many existing approaches in which passivity assessment applies only to the full system, this novel scheme aims at identifying passive subsystems embedded into a non-passive system so that a partial positive system could be specified. This enables the identification of parameters contributing to passivity violations and how these are distributed across the system ports. We also propose an index to quantify how passivity violations permeate into the system and take as cases studies actual transformer data to demostrate its applicability.

Formula omitted]–symmetric tight-binding model with asymmetric couplings

We study spectral and transport properties of one-dimensional (1D) tight-binding PT–symmetric chains with non-equal consecutive couplings, i.e. asymmetric couplings. Based on the transfer matrix method, we obtain analytical expressions for the transmission and reflection coefficients for any values of the model parameters. These expressions, given in a very compact form, separately imbed the generic energy dependence, valid for any periodic structure, as well as specific properties of the dimers composing the 1D scattering chains. Our main interest is in the specific properties of the left/right reflections which are due to the PT–symmetric structure of the model. We have found that for the asymmetric coupling, there is no one-to-one correspondence between the degeneracy of the transfer matrix eigenvalues and PT–symmetric transport properties, such as the unidirectional reflectivity. We show that this correspondence, however, holds in the S−matrix approach, both for symmetric and asymmetric couplings.

Thermodynamics of the Spin-1/2 Heisenberg–Ising Chain at High Temperatures: a Rigorous Approach

This work develops a rigorous setting allowing one to prove several features related to the behaviour of the Heisenberg–Ising (or XXZ) spin-1/2 chain at finite temperature T. Within the quantum inverse scattering method the physically pertinent observables at finite T, such as the per-site free energy or the correlation length, have been argued to admit integral representations whose integrands are expressed in terms of solutions to auxiliary non-linear integral equations. The derivation of such representations was based on numerous conjectures: the possibility to exchange the infinite volume and the infinite Trotter number limits, the existence of a real, non-degenerate, maximal in modulus Eigenvalue of the quantum transfer matrix, the existence and uniqueness of solutions to the auxiliary non-linear integral equations, as well as the possibility to take the infinite Trotter number limit on their level. We rigorously prove all these conjectures for temperatures large enough. As a by product of our analysis, we obtain the large-T asymptotic expansion for a subset of sub-dominant Eigenvalues of the quantum transfer matrix and thus of the associated correlation lengths. This result was never obtained previously, not even on heuristic grounds.

On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions

The transfer matrix of the square-lattice eight-vertex model on a strip with vertical lines and open boundary conditions is investigated. It is shown that for vertex weights that obey the relation and appropriately chosen K-matrices this transfer matrix possesses the remarkably simple, non-degenerate eigenvalue . For positive vertex weights, is shown to be the largest transfer-matrix eigenvalue. The corresponding eigenspace is equal to the space of the ground states of the Hamiltonian of a related XYZ spin chain. An essential ingredient in the proofs is the supersymmetry of this Hamiltonian.

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.

Low-frequency wave-energy amplification in graded two-dimensional resonator arrays

Energy amplification in square-lattice arrays of C-shaped low-frequency resonators, where the resonator radii are graded with distance, is investigated in the two-dimensional linear acoustics setting for both infinite (in one dimension) and finite arrays. Large amplifications of the incident energy are shown in certain array locations. The phenomenon is analysed using: (i) band diagrams for doubly-periodic arrays; (ii) numerical simulations for infinite and finite arrays; and (iii) eigenvalue analysis of transfer matrices operating over individual columns of the array. It is shown that the locations of the large amplifications are predicted by propagation cut-offs in the modes associated with the transfer-matrix eigenvalues. For the infinite array, the eigenvalues form a countable set, and for the low frequencies considered, only a single propagating mode exists for a given incident wave, which cuts off within the array, leading to predictive capabilities for the amplification location. For the finite array, it is shown that (in addition to a continuous spectrum of modes) multiple discrete propagating modes can be excited, with the grading generating new modes, as well as cutting others off, leading to complicated amplification patterns. The numerical simulations reveal that the largest amplifications are achieved for a single row array, with amplifications an order of magnitude smaller for the corresponding infinite array. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.

Complete spectrum of quantum integrable lattice models associated to by separation of variables

In this paper we apply our new separation of variables approach to completely characterize the transfer matrix spectrum for quantum integrable lattice models associated to fundamental evaluation representations of with quasi-periodic boundary conditions. We consider here the case of generic deformations associated to a parameter q which is not a root of unity. The separation of variables (SoV) basis for the transfer matrix spectral problem is generated by using the action of the transfer matrix itself on a generic co-vector of the Hilbert space, following the general procedure described in our paper (Maillet and Niccoli 2018 J. Math. Phys. 59 091417). Such a SoV construction allows to prove that for general values of the parameters defining the model the transfer matrix is diagonalizable and with simple spectrum for any twist matrix which is also diagonalizable with simple spectrum. Then, using together the knowledge of such a SoV basis and of the fusion relations satisfied by the hierarchy of transfer matrices, we derive a complete characterization of the transfer matrix eigenvalues and eigenvectors as solutions of a system of polynomial equations of order n + 1. Moreover, we show that such a SoV discrete spectrum characterization is equivalently reformulated in terms of a finite difference functional equation, the quantum spectral curve equation, under a proper choice of the set of its solutions. A construction of the associated Q-operator induced by our SoV approach is also presented.

Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables

We apply our new approach of quantum Separation of Variables (SoV) to the complete characterization of the transfer matrix spectrum of quantum integrable lattice models associated to \bm{gl_n}𝐠𝐥𝐧-invariant \bm{R}𝐑-matrices in the fundamental representations. We consider lattices with \bm{N}𝐍-sites and general quasi-periodic boundary conditions associated to an arbitrary twist matrix \bm{K}𝐊 having simple spectrum (but not necessarily diagonalizable). In our approach the SoV basis is constructed in an universal manner starting from the direct use of the conserved charges of the models, e.g. from the commuting family of transfer matrices. Using the integrable structure of the models, incarnated in the hierarchy of transfer matrices fusion relations, we prove that our SoV basis indeed separates the spectrum of the corresponding transfer matrices. Moreover, the combined use of the fusion rules, of the known analytic properties of the transfer matrices and of the SoV basis allows us to obtain the complete characterization of the transfer matrix spectrum and to prove its simplicity. Any transfer matrix eigenvalue is completely characterized as a solution of a so-called quantum spectral curve equation that we obtain as a difference functional equation of order \bm{n}𝐧. Namely, any eigenvalue satisfies this equation and any solution of this equation having prescribed properties that we give leads to an eigenvalue. We construct the associated eigenvector, unique up to normalization, of the transfer matrices by computing its decomposition on the SoV basis that is of a factorized form written in terms of the powers of the corresponding eigenvalues. Finally, if the twist matrix \bm{K}𝐊 is diagonalizable with simple spectrum we prove that the transfer matrix is also diagonalizable with simple spectrum. In that case, we give a construction of the Baxter \bm{Q}𝐐-operator and show that it satisfies a \bm{T}𝐓-\bm{Q}𝐐 equation of order \bm{n}𝐧, the quantum spectral curve equation, involving the hierarchy of the fused transfer matrices.