Bethe Ansatz Method Research Articles

Integrable Deformations of Sine-Liouville Conformal Field Theory and Duality

We study integrable deformations of sine-Liouville conformal field theory. Every integrable perturbation of this model is related to the series of quantum integrals of motion (hierarchy). We construct the factorized scattering matrices for different integrable perturbed conformal field theories. The perturbation theory, Bethe ansatz technique, renormalization group and methods of perturbed conformal field theory are applied to show that all integrable deformations of sine-Liouville model possess non-trivial duality properties.

Exact solutions for a class of quasi-exactly solvable models: A unified treatment

The exact solution of the Schrodinger equation for the four quasi-exactly solvable potentials is presented using the functional Bethe ansatz method. It is shown that all models give rise to the same basic differential equation which is quasi-exactly solvable. The eigenvalues, eigenfunctions and the allowed potential parameters are given for each of the four models in terms of the roots of a set of algebraic Bethe ansatz equations.

Fluctuations of TASEP on a Ring in Relaxation Time Scale

AbstractWe consider the totally asymmetric simple exclusion process on a ring with flat and step initial conditions. We assume that the size of the ring and the number of particles tend to infinity proportionally and evaluate the fluctuations of tagged particles and currents. The crossover from the KPZ dynamics to the equilibrium dynamics occurs when the time is proportional to the 3/2 power of the ring size. We compute the limiting distributions in this relaxation time scale. The analysis is based on an explicit formula of the finite‐time one‐point distribution obtained from the coordinate Bethe ansatz method. © 2017 Wiley Periodicals, Inc.

Solvability of a class of PT-symmetric non-Hermitian Hamiltonians: Bethe ansatz method

We use the Bethe ansatz method to investigate the Schrödinger equation for a class of PT-symmetric non-Hermitian Hamiltonians. Elementary exact solutions for the eigenvalues and the corresponding wave functions are obtained in terms of the roots of a set of algebraic equations. Also, it is shown that the problems possess sl(2) hidden symmetry and then the exact solutions of the problems are obtained by employing the representation theory of sl(2) Lie algebra. It is found that the results of the two methods are the same.

On the solutions of the Zn-Belavin model with arbitrary number of sites

The periodic Zn-Belavin model on a lattice with an arbitrary number of sites N is studied via the off-diagonal Bethe Ansatz method (ODBA). The eigenvalues of the corresponding transfer matrix are given in terms of an unified inhomogeneous T−Q relation. In the special case of N=nl with l being also a positive integer, the resulting T−Q relation recovers the homogeneous one previously obtained via algebraic Bethe Ansatz.

Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces

We introduce two new families of quasi-exactly solvable (QES) extensions of the oscillator in a d-dimensional constant-curvature space. For the first three members of each family, we obtain closed-form expressions of the energies and wavefunctions for some allowed values of the potential parameters using the Bethe ansatz method. We prove that the first member of each family has a hidden sl(2, ℝ) symmetry and is connected with a QES equation of the first or second type, respectively. One-dimensional results are also derived from the d-dimensional ones with d≥2, thereby getting QES extensions of the Mathews-Lakshmanan nonlinear oscillator.

Tail of the two-time height distribution for KPZ growth in one dimension

Obtaining the exact multi-time correlations for one-dimensional growth models described by the Kardar–Parisi–Zhang (KPZ) universality class is presently an outstanding open problem. Here, we study the joint probability distribution function (JPDF) of the height of the KPZ equation with droplet initial conditions, at two different times , in the limit where both times are large and their ratio is fixed. This maps to the JPDF of the free energies of two directed polymers with two different lengths and in the same random potential. Using the replica Bethe ansatz (RBA) method, we obtain the exact tail of the JPDF when one of its argument (the KPZ height at the earlier time t1) is large and positive. Our formula interpolates between two limits where the JPDF decouples: (i) for into a product of two GUE Tracy–Widom (TW) distributions, and (ii) for into a product of a GUE-TW distribution and a Baik–Rains distribution (associated to stationary KPZ evolution). The lowest cumulants of the height at time t2, conditioned on the one at time t1, are expressed analytically as expansions around these limits, and computed numerically for arbitrary . Moreover we compute the connected two-time correlation, conditioned to a large enough value at t1, providing a quantitative prediction for the so-called persistence of correlations (or ergodicity breaking) in the time evolution from the droplet initial condition. Our RBA results are then compared with arguments based on Airy processes, with satisfactory agreement. These predictions are universal for all models in the KPZ class and should be testable in experiments and numerical simulations.

Exact solution of the relativistic quantum Toda chain

The relativistic quantum Toda chain model is studied with the generalized algebraic Bethe Ansatz method. By employing a set of local gauge transformations, proper local vacuum states can be obtained for this model. The exact spectrum and eigenstates of the model are thus constructed simultaneously.

Exactly solvable interacting two-particle quantum graphs

We construct models of exactly solvable two-particle quantum graphs with certain non-local two-particle interactions, establishing appropriate boundary conditions via suitable self-adjoint realisations of the two-particle Laplacian. Showing compatibility with the Bethe ansatz method, we calculate quantisation conditions in the form of secular equations from which the spectra can be deduced. We compare spectral statistics of some examples to well known results in random matrix theory, analysing the chaotic properties of their classical counterparts.

The quasi-exactly solvable problems for two dimensional quantum systems

In this paper, we study the quasi-exactly solvable problems for two dimensional quantum systems. By using the Bethe ansatz method, we obtain the general form of the quasi-exactly solvable potential. Then, we present several examples to give the specific forms of quasi-exactly solvable potentials. In the examples, some physical models of quasi-exactly solvable problems are re-exhibited.

Lie Symmetry and the Bethe Ansatz Solution of a New Quasi-Exactly Solvable Double-Well Potential

We study the Schrödinger equation with a new quasi-exactly solvable double-well potential. Exact expressions for the energies, the corresponding wave functions, and the allowed values of the potential parameters are obtained using two different methods, the Bethe ansatz method and the Lie algebraic approach. Some numerical results are reported and it is shown that the results are in good agreement with each other and with those obtained previously via a different method.

Exact Solutions of a Class of Double-Well Potentials: Algebraic Bethe Ansatz

Applying the Bethe ansatz method, we investigate the Schrödinger equation for the three quasi-exactly solvable double-well potentials, namely, the generalized Manning potential, the Razavy bistable potential, and the hyperbolic Shifman potential. General exact expressions for the energies and the associated wave functions are obtained in terms of the roots of a set of algebraic equations. Also, we solve the same problems using the Lie algebraic approach of quasi-exact solvability through the sl(2) algebraization and show that the results are the same. The numerical evaluation of the energy spectrum is reported to display explicitly the energy levels splitting.

Bethe ansatz solutions of the τ 2-model with arbitrary boundary fields

The quantum $\tau_2$-model with generic site-dependent inhomogeneity and arbitrary boundary fields is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix are given in terms of an inhomogeneous T-Q relation, which is based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices. Moreover, the associated Bethe Ansatz equations are also obtained.

Solutions of the Schrödinger equation under topological defects space-times and generalized uncertainty principle

We study the behavior of non-relativistic quantum particles interacting with the linear potential plus the Coulomb potential (LCP) and Kratzer potential under the generalized uncertainty principle (GUP) and the cosmic string space-time. By using the functional Bethe ansatz method, we give closed-form solutions for the energies and the wave functions. The allowed potential parameters have been discussed and the corresponding constraint equations have been obtained.

Exact Bethe Ansatz Spectrum of a Tight-Binding Chain with Dephasing Noise.

We construct an exact map between a tight-binding model on any bipartite lattice in the presence of dephasing noise and a Hubbard model with imaginary interaction strength. In one dimension, the exact many-body Liouvillian spectrum can be obtained by application of the Bethe ansatz method. We find that both the nonequilibrium steady state and the leading decay modes describing the relaxation at late times are related to the η-pairing symmetry of the Hubbard model. We show that there is a remarkable relation between the time evolution of an arbitrary k-point correlation function in the dissipative system and k-particle states of the corresponding Hubbard model.

Exact solution of theD3non-Abelian anyon chain

Commuting transfer matrices for linear chains of interacting non-Abelian anyons from the two-dimensional irreducible representation of the dihedral group $D_3$ (or, equivalently, the integer sector of the $su(2)_4$ spin-$1$ chain) are constructed using the spin-anyon correspondence to a $D_3$-symmetric formulation of the XXZ Heisenberg spin chain. The spectral problem is solved using discrete inversion identities satisfied by these transfer matrices and functional Bethe ansatz methods. The resulting spectrum can be related to that of the XXZ spin-$1/2$ Heisenberg chain with boundary conditions depending on the topological sector of the anyon chain. The properties of this model in the critical regime are studied by finite size analysis of the spectrum. In particular, points in the phase diagram where the anyon chain realizes some of the rational $\mathbb{Z}_2$ orbifold theories are identified.

Spin and magnetothermal transport in the S = 1/2 XXZ chain

We present a temperature and magnetic field dependence study of spin transport and magnetothermal corrections to the thermal conductivity in the spin S = 1/2 easy-plane Heisenberg chain, extending an earlier analysis based on the Bethe ansatz method. We critically discuss the low temperature, weak magnetic field behavior, the effect of magnetothermal corrections in the vicinity of the critical field and their role in recent thermal conductivity experiments in 1D quantum magnets.

Spinon dynamics in quantum integrable antiferromagnets

The excitations of the Heisenberg antiferromagnetic spin chain in zero field are known as spinons. As pairwise-created fractionalized excitations, spinons are important in the understanding of inelastic neutron scattering experiments in (quasi) one-dimensional materials. In the present work, we consider the real space-time dynamics of spinons originating from a local spin flip on the antiferromagnetic ground state of the (an)isotropic Heisenberg spin-1/2 model and the Babujan-Takhtajan spin-1 model. By utilizing algebraic Bethe ansatz methods at finite system size to compute the expectation value of the local magnetization and spin-spin correlations, spinons are visualized as propagating domain walls in the antiferromagnetic spin ordering with anisotropy dependent behavior. The spin-spin correlation after the spin flip displays a light cone, satisfying the Lieb-Robinson bound for the propagation of correlations at the spinon velocity.

Universal low-energy physics in 1D strongly repulsive multi-component Fermi gases

It has been shown (Yang and You 2011 Chin. Phys. Lett. 28 020503) that at zero temperature the ground state of the one-dimensional (1D) w-component Fermi gas coincides with that of the spinless Bose gas in the limit . This behavior was experimentally evidenced through quasi-1D tightly trapping ultracold 173Yb atoms in a recent paper (Pagano et al 2014 Nat. Phys. 10 198). However, understanding of low-temperature behavior of Fermi gases with a repulsive interaction requires spin-charge separated conformal field theories of an effective Tomonaga–Luttinger liquid and an antiferromagnetic SU(w) Heisenberg spin chain. Here we analytically derive universal thermodynamics of 1D strongly repulsive fermionic gases with SU(w) symmetry via the Yang–Yang thermodynamic Bethe ansatz method. The analytical free energy and magnetic properties of the systems at low temperature in a weak magnetic field are obtained through the Wiener–Hopf method. In particular, the free energy essentially manifests the spin-charge separated conformal field theories for high-symmetry systems with arbitrary repulsive interaction strength. We also find that the sound velocity of the Fermi gases in the large w limit coincides with that for the spinless Bose gas, whereas the spin velocity vanishes quickly as w becomes large. This indicates strong suppression of the Fermi exclusion statistics by the commutativity feature among the w-component fermions with different spin states in the Tomonaga–Luttinger liquid phase. Moreover, the equations of state and critical behavior of physical quantities at finite temperature are analytically derived in terms of the polylogarithm functions in the quantum critical region.

Crossing probability for directed polymers in random media. II. Exact tail of the distribution.

We study the probability p ≡ p(η)(t) that two directed polymers in a given random potential η and with fixed and nearby endpoints do not cross until time t. This probability is itself a random variable (over samples η), which, as we show, acquires a very broad probability distribution at large time. In particular, the moments of p are found to be dominated by atypical samples where p is of order unity. Building on a formula established by us in a previous work using nested Bethe ansatz and Macdonald process methods, we obtain analytically the leading large time behavior of all moments p(m) ≃ γ(m)/t. From this, we extract the exact tail ∼ρ(p)/t of the probability distribution of the noncrossing probability at large time. The exact formula is compared to numerical simulations, with excellent agreement.