Quantum Transfer Matrix Research Articles

Algebraic Bethe ansatz for the gl(1|2) generalized model: II. the three gradings

The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same $R$-matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe ansatz for all models with $9 \times 9$, rational, gl(1$|$2)-invariant $R$-matrix and all three possibilities of choosing the grading. Our Bethe ansatz solution applies, for instance, to the supersymmetric t-J model, the supersymmetric $U$ model and a number of interesting impurity models. It may be extended to obtain the quantum transfer matrix spectrum for this class of models. The properties of a specific model enter the Bethe ansatz solution (i.e. the expression for the transfer matrix eigenvalue and the Bethe ansatz equations) through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model.

Exact results for the thermal and magnetic properties of strong coupling ladder compounds.

We investigate the thermal and magnetic properties of the integrable su(4) ladder model by means of the quantum transfer matrix method. The magnetic susceptibility, specific heat, magnetic entropy, and high field magnetization are evaluated from the free energy derived via the recently proposed method of high temperature expansion for exactly solved models. We show that the integrable model can be used to describe the physics of the strong coupling ladder compounds. Excellent agreement is seen between the theoretical results and the experimental data for the known ladder compounds (5IAP)2CuBr4.2H(2)O, Cu2(C5H12N2)2Cl4, etc.

Application of the density matrix renormalization group method to finite temperatures and two-dimensional systems

The density matrix renormalization group (DMRG) method and its applications to finite temperatures and two-dimensional systems are reviewed. The basic idea of the original DMRG method, which allows precise study of the ground-state properties and low-energy excitations, is presented for models which include long-range interactions. The DMRG scheme is then applied to the diagonalization of the quantum transfer matrix for one-dimensional systems, and a reliable algorithm at finite temperatures is formulated. Dynamic correlation functions at finite temperatures are calculated from the eigenvectors of the quantum transfer matrix with analytical continuation to the real frequency axis. An application of the DMRG method to two-dimensional quantum systems in a magnetic field is demonstrated and reliable results for quantum Hall systems are presented.

The semimetallic S = 1/2 antiferromagnetic chain Yb4As3: quantum transfer‐matrix simulations and experimental field‐dependent specific‐heat data

AbstractThe S=1/2 antiferromagnetic Heisenberg model with the Dzyaloshinsky–Moriya interaction is applied to Yb4As3. In this paper we calculate the specific heat of this model, mapped onto the isotropic Heisenberg model with both uniform field Bx and staggered field Bys, by the numerical quantum transfer‐matrix (QTM) method and we present new experimental results for the polydomain sample in the applied field.

Nonlinear integral equations for thermodynamics of thesl(r 1) Uimin Sutherland model

We derive traditional thermodynamic Bethe ansatz (TBA) equations for the sl(r + 1) Uimin–Sutherland model from the T-system of the quantum transfer matrix. These TBA equations are identical to the those from the string hypothesis. Next we derive a new family of nonlinear integral equations (NLIEs). In particular, a subset of these NLIEs forms a system of NLIEs which contains only a finite number of unknown functions. For r = 1, this subset of NLIEs reduces to Takahashi's NLIE for the XXX spin chain. A relation between the traditional TBA equations and our new NLIEs is clarified. Based on our new NLIEs, we also calculate the high-temperature expansion of the free energy.

Quantum transfer-matrix method and thermo-quantum dynamics

The quantum transfer-matrix (QTM) method is reviewed and thermo-quantum dynamics is formulated on the basis of the QTM. The free energy of the relevant quantum system (e.g. the one-dimensional Heisenberg model) is expressed only in terms of the maximum eigenvalue λ max of the QTM. Even the correlation length ξ( T) is expressed in terms of the ratio of λ max over the next largest eigenvalue λ 2 as ξ( T) −1=log( λ max/ λ 2). This new formulation has the great merit that the thermal average 〈 Q〉 for any local observable Q in the thermodynamic limit is expressed as an expectation value over a temperature-dependent state vector in the single (conjugate) Hilbert space in contrast to the usage of the double Hilbert space in the thermo-field dynamics.

Direct calculation of thermodynamic quantities for the Heisenberg model

The XXX Heisenberg model is studied at finite temperature. The free energy is derived without recourse to thermal Bethe ansatz method and quantum transfer matrix method. The result perfectly agrees with the free energy derived by thermal Bethe ansatz method. An explicit expression of the cluster expansion coefficient in arbitrary order is presented for the first time.

Integral equations for thermodynamics of the osp(1|2s) integrable spin chain

We propose a system of nonlinear integral equations (NLIE), which gives the free energy of the osp(1|2s) integrable spin chain at finite temperatures. In contrast with usual thermodynamic Bethe ansatz equations, our new NLIE contain only a finite number of unknown functions. On deriving NLIE, we use our osp(1|2s) version of the T-system and the quantum transfer matrix method. Based on our NLIE, we also calculate the high temperature expansion of the free energy and the specific heat.

A NOTE ON THE osp(1|2s) THERMODYNAMIC BETHE ANSATZ EQUATION

A Bethe ansatz equation associated with the Lie superalgebra osp(1|2s) is studied. A thermodynamic Bethe ansatz (TBA) equation is derived by the string hypothesis. The high temperature limit of the entropy density is expressed in terms of the solution of the osp(1|2s) version of the Q-system. In particular for the fundamental representation case, we also derive a TBA equation from the osp(1|2s) version of the T-system and the quantum transfer matrix method. This TBA equation is identical to the one from the string hypothesis. The central charge is expressed by the Rogers dilogarithmic function and identified to s.

Thermodynamic Bethe ansatz equations of one-dimensional Hubbard model and high-temperature expansion

A numerical method to calculate thermodynmic Bethe ansatz equations is proposed based on Newton's method. Thermodynamic quantities of the one-dimensional Hubbard model are numerically calculated and compared with high-temperature expansion and numerical results of the quantum transfer matrix method by J\uttner, Kl\umper, and Suzuki. The coincidence is surprisingly good. We obtain the high-temperature expansion of the grand potential up to ${\ensuremath{\beta}}^{6}.$

Matrix Product State Approximation for the Maximum-Eigenvalue Eigenstate of the Quantum Transfer Matrix

We propose a matrix product state (MPS) formulation to calculate thermodynamic quantities of one dimensional (1D) quantum systems. The maximum-eigenvalue eigenstate of the quantum transfer matrix is represented as the product of local matrices, which are obtained by the DMRG method for the two dimensional (2D) classical system mapped from the original 1D quantum system. This MPS formulation is successfully applied to the S = ½ XY model.

Bethe ansatz solvable multi-chain quantum systems

In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (‘basic’ model) of this class of models: the antiferromagnetic two-chain spin- 1 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate–incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the ‘quantum transfer matrix’) for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.

Finite-temperature correlations for theUq(𝔰𝔩(2|1))-invariant generalized Hubbard model

We study an integrable model of one-dimensional strongly correlated electrons at finite temperature by explicit calculation of the correlation lengths of various correlation functions. The model is invariant with respect to the quantum superalgebra U_q(sl(2|1)) and characterized by the Hubbard interaction, correlated hopping and pair-hopping terms. Using the integrability, the graded quantum transfer matrix is constructed. From the analyticity of its eigenvalues, a closed set of non-linear integral equations is derived which describe the thermodynamical quantities and the finite temperature correlations. The results show a crossover from a regime with dominating density-density correlations to a regime with dominating superconducting pair correlations. Analytical calculations in the low temperature limit are also discussed.

FermionicR-Operator and Algebraic Structure of 1D Hubbard Model: Its Application to Quantum Transfer Matrix

The algebraic structure of the 1D Hubbard model is studied by means of the fermionic R -operator approach. This approach treats the fermion models directly in the framework of the quantum inverse scattering method. Compared with the graded approach, this approach has several advantages. First, the global properties of the Hamiltonian are naturally reflected in the algebraic properties of the fermionic R -operator which is a local operator acting on fermion Fock spaces. In particular, S O (4) symmetry and the invariance under the partial particle-hole transformation are shown. Second, a genuinely fermionic quantum transfer matrix (QTM) can be constructed in terms of the fermionic R -operator. By use of the algebraic Bethe ansatz, the fermionic QTM is diagonalized and its properties are discussed.

Equivalence of TBA and QTM

The traditional thermodynamic Bethe ansatz equations for the XXZ model at |Δ|≥1 are derived within the quantum transfer matrix method. This provides further evidence of the equivalence of both methods. Most importantly, we derive an integral equation for the free energy formulated for just one unknown function. This integral equation differs in physical and mathematical aspects to the established ones. The single integral equation is analytically continued to the regime |Δ|<1.

Intra- and inter-chain dynamic response functions in quasi-one-dimensional conductors

Intra- and inter-chain excitation spectra are studied in a spinless fermion model on a two-leg ladder at half filling. Numerical results are obtained by application of the density matrix renormalization group technique to the quantum transfer matrix for infinite systems. Near the metal-insulator transition, the intra-chain Coulomb repulsion is found to affect local current correlation more sensitively in the rung direction than in the leg direction.

Antiferromagnetic zigzag spin chain in magnetic fields at finite temperatures

We study thermodynamic behaviors of the antiferromagnetic zigzag spin chain in magnetic fields, using the density-matrix renormalization group method for the quantum transfer matrix. We focus on the thermodynamics of the system near the critical fields in the ground-state magnetization process($M$-$H$ curve): the saturation field, the lower critical field associated with excitation gap, and the field at the middle-field cusp singularity. We calculate magnetization, susceptibility and specific heat of the zigzag chain in magnetic fields at finite temperatures, and then discuss how the calculated quantities reflect the low-lying excitations of the system related with the critical behaviors in the $M$-$H$ curve.

THERMODYNAMIC BETHE ANSATZ EQUATION FROM FUSION HIERARCHY OF osp(1|2) INTEGRABLE SPIN CHAIN

The thermodynamic Bethe ansatz (TBA) and the excited state TBA equations for an integrable spin chain related to the Lie superalgebra osp (1|2) are proposed by the quantum transfer matrix (QTM) method. We introduce the fusion hierarchy of the QTM and derive the functional relations among them (T-system) and their certain combinations (Y-system). Their analytical property leads to the nonlinear integral equations which describe the free energy and the correlation length at any finite temperatures. With regard to the free energy, they coincide with the TBA equation based on the string hypothesis.

Susceptibility Behaviour of CuGeO3: Comparison between Experiment and the Quantum Transfer Matrix Approach

Susceptibility Behaviour of CuGeO₃: Comparison between Experiment and the Quantum Transfer Matrix Approach

Static spin structure factor of CuGeO 3 above spin-Peierls transition temperature

We study, by means of the quantum transfer matrix method, the finite-temperature behavior of the S=1/2 antiferromagnetic chain with nearest neighbor and competing next nearest neighbor interactions, which is a model for the spin-Peierls material, CuGeO 3. In particular, the static spin structure factor is found to display an interesting phenomenon as a consequence of the competition: a characteristic wave number for the maximum value of the structure factor takes a commensurate or an incommensurate value, depending on temperature. The boundary, called a Lifshitz temperature, can be found in this material. We discuss such behavior in connection with the experimental result of the neutron scattering in CuGeO 3 above the spin-Peierls transition temperature.