Kazumitsu Sakai

Spectrum of a multi-species asymmetric simple exclusion process on a ring

The spectrum of the Hamiltonian (Markov matrix) of a multi-species asymmetric simple exclusion process on a ring is studied. The dynamical exponent concerning the relaxation time is found to coincide with the one-species case. It implies that the system belongs to the Kardar–Parisi–Zhang or Edwards–Wilkinson universality classes depending on whether the hopping rate is asymmetric or symmetric, respectively. Our derivation exploits a poset structure of the particle sectors, leading to a new spectral duality and inclusion relations. The Bethe ansatz integrability is also demonstrated.

Continued fraction TBA and functional relations in XXZ model at root of unity

Abstract Thermodynamics of the spin 1 2 XXZ model is studied in the critical regime using the quantum transfer matrix (QTM) approach. We find functional relations indexed by the Takahashi-Suzuki numbers among the fusion hierarchy of the QTMs ( T -system) and their certain combinations ( Y -system). By investigating analyticity of the latter, we derive a closed set of nonlinear integral equations which characterize the free energy and the correlation lengths for both 〈 σ j + σ i − 〉 and 〈 σ j z σ i z 〉 at any finite temperatures. Concerning the free energy, they exactly coincide with Takahashi-Suzukis TBA equations based on the string hypothesis. By solving the integral equations numerically the correlation lengths are determined, which agrees with the earlier results in the low temperature limit.

Next nearest neighbor correlation functions of the spin 1/2 XXZ chain at critical region

The correlation functions of the spin-1/2 XXZ chain in the ground state are expressed in the form of multiple integrals. For −1 < Δ < 1, they were obtained by Jimbo and Miwa in 1996. In particular, the next-nearest-neighbour correlation functions are given as certain three-dimensional integrals. We show that these can be reduced to one-dimensional integrals and thereby we evaluate the values of the next-nearest-neighbour correlation functions. We have also found that the remaining one-dimensional integrals can be evaluated analytically, when ν = cos−1(Δ)/π is a rational number.

Vertex models, TASEP and Grothendieck polynomials

We examine the wavefunctions and their scalar products of a one-parameter family of integrable five-vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system—the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on-/off-shell wavefunctions of the five-vertex models are represented as a certain determinant form. Up to some normalization factors, we find that the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for the Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of the Grothendieck polynomials.

Multiple Schramm–Loewner evolutions for conformal field theories with Lie algebra symmetries

Abstract We provide multiple Schramm–Loewner evolutions (SLEs) to describe the scaling limit of multiple interfaces in critical lattice models possessing Lie algebra symmetries. The critical behavior of the models is described by Wess–Zumino–Witten (WZW) models. Introducing a multiple Brownian motion on a Lie group as well as that on the real line, we construct the multiple SLE with additional Lie algebra symmetries. The connection between the resultant SLE and the WZW model can be understood via SLE martingales satisfied by the correlation functions in the WZW model. Due to interactions among SLE traces, these Brownian motions have drift terms which are determined by partition functions for the corresponding WZW model. As a concrete example, we apply the formula to the su ˆ ( 2 ) k -WZW model. Utilizing the fusion rules in the model, we conjecture that there exists a one-to-one correspondence between the partition functions and the topologically inequivalent configurations of the SLE traces. Furthermore, solving the Knizhnik–Zamolodchikov equation, we exactly compute the probabilities of occurrence for certain configurations (i.e. crossing probabilities) of traces for the triple SLE.

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.

Non-dissipative thermal transport in the massive regimes of the XXZ chain

We present exact results on the thermal conductivity of the one-dimensional spin-1/2 XXZ model in the massive antiferromagnetic and ferromagnetic regimes. The thermal Drude weight is calculated by a lattice path integral formulation. Numerical results for wide ranges of temperature and anisotropy as well as analytical results in the low- and high-temperature limits are presented. At finite temperature, the thermal Drude weight is finite and hence there is non-dissipative thermal transport even in the massive regime. At low temperature, the thermal Drude weight behaves as where δ is the one-spinon (respectively one-magnon) excitation energy for the antiferromagnetic (respectively ferromagnetic) regime.

Third-neighbor correlators of a one-dimensional spin-1/2 Heisenberg antiferromagnet.

We exactly evaluate the third-neighbor correlator S(z)(j)S(z)(j+3) and all the possible nonzero correlators S(alpha)(j)S(beta)(j+1)S(gamma;)(j+2)S(delta)(j+3) of the one-dimensional spin-1/2 Heisenberg XXX antiferromagnet in the ground state without magnetic field. All the correlators are expressed in terms of certain combinations of logarithm ln 2, the Riemann zeta function zeta(3), zeta(5) with rational coefficients. The results accurately coincide with the numerical ones obtained by the density-matrix renormalization group method and the numerical diagonalization.

COMMUTING QUANTUM TRANSFER-MATRIX APPROACH TO INTRINSIC FERMION SYSTEM : CORRELATION LENGTH OF A SPINLESS FERMION MODEL

The quantum transfer-matrix approach to integrable lattice fermion systems is presented. As a simple case we treat the spinless fermion model with repulsive interaction in critical regime. We derive a set of nonlinear integral equations which characterize the free energy and the correlation length of

Dynamical correlation functions of the XXZ model at finite temperature

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