Search Results (18)

Let us consider a two-sided multi-species stochastic particle model with finitely many particles on $\mathbb{Z}$, defined as follows. Suppose that each particle is labelled by a positive integer l, and waits a random time exponentially distributed with rate 1. It then chooses the right direction to jump with probability p, or the left direction with probability $q=1-p$. If the particle chooses the right direction, it jumps to the nearest site occupied by a particle $l^{\prime }<l$ (with the convention that an empty site is considered as a particle with labelled 0). If the particle chooses the left direction, it jumps to the next site on the left only if that site is either empty or occupied by a particle $l^{\prime }<l$, and in the latter case, particles l and $l^{\prime }$ swap their positions. We show that this model is integrable, and provide the exact formula of the transition probability using the Bethe ansatz. Full article

Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a sinh-Gordon-type model with an infinite number of states but with a real parameter q. This model is the subject of Sklyanin’s Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: (1) Construction of eigenstates of an off-diagonal element of a monodromy matrix. The most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. (2) Separately, we discuss the Clebsh-Gordan coefficients, as well as the Wigner’s 6j symbols, in details. The later are rather well known in the theory of $3D$ indices. Thus, the Sklyanin basis of the quantum separation of variables is constructed. The matrix elements of an eigenstate of the auxiliary transfer matrix in this basis are products of functions satisfying the Baxter equation. Such functions are called usually the Q-operators. We investigate the Baxter equation and Q-operators from two points of view. (3) In the model considered the most convenient Bethe-type variables are the zeros of a Wronskian of two well defined particular solutions of the Baxter equation. This approach works perfectly in the thermodynamic limit. We calculate the distribution of these roots in the thermodynamic limit, and so we reproduce in this way the partition function of the model. (4) The real parameter q, which is the standard quantum group parameter, plays the role of the absolute temperature in the model considered. Expansion with respect to q (tropical expansion) gives an alternative way to establish the structure of the eigenstates. In this way we classify the elementary excitations over the ground state. Full article

Integrable quantum field theories can be regularized on the lattice while preserving integrability. The resulting theories on the lattice are integrable lattice models. A prototype of such a regularization is the correspondence between a sine-Gordon model and a six-vertex model on a light-cone lattice. We propose an integrable deformation of the light-cone lattice model such that in the continuum limit we obtain the $T\overline{T}$-deformed sine-Gordon model. Under this deformation, the cut-off momentum becomes energy dependent and the underlying Yang–Baxter integrability is preserved. Therefore, this deformation is integrable but non-local: similar to the $T\overline{T}$ deformation of quantum field theory. Full article

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux $Y(s_1,\dots ,s_k)$ with $k\ge 2$ rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a $k=2$ Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic $sp\left(2k\right)$ algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the $su(1,1)$ algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for $Y(s_1,s_2)$ is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin $(s_1,s_2)$ is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin $(1,1)$ is obtained. Full article

We study a gas of bosonic dipolar atoms in the presence of a transverse harmonic trapping potential by using an improved variational Bethe ansatz, which includes the transverse width of the atomic cloud as a variational parameter. Our calculations show that the system behavior evolves from quasi-one dimensional to a strictly one-dimensional one by changing the atom–atom interaction, or the axial density, or the frequency of the transverse confinement. Quite remarkably, in the droplet phase induced by the attractive dipolar interaction the system becomes sub-one dimensional when the transverse width is smaller than the characteristic length of the transverse harmonic confinement. Full article

We studied the Gaudin models with $\mathfrak{gl}\left(1\right|1)$ symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation $\mathfrak{gl}\left(1\right|1\left)\right[t]$-modules. Namely, we gave an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation $\mathfrak{gl}\left(1\right|1\left)\right[t]$-modules and showed that a bijection exists between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also gave dimensions of the generalized eigenspaces. Full article

We study the quantum dynamics of the opened three-level su(1, 1) bosonic model. The effective non-Hermitian Hamiltonians describing the system of the Lindblad equation in the short time limit are constructed. The obtained non-Hermitian Hamiltonians are exactly solvable by the Algebraic Bethe Ansatz. This approach allows representing biorthogonal and nonorthogonal bases of the system. We analyze the biorthogonal expectation values of a number of particles in the zero mode and represent it in the determinantal form. The time-dependent density matrix satisfying the Lindblad master equation is found in terms of the nonorthogonal basis. Full article

The $T=0$ excitation spectra of the antiferromagnetic ($J>0$) anisotropic Heisenberg chain of spins 1/2 are studied using the Bethe Ansatz equations for $\Delta =cos(\pi /n)$, $n=3,4$ and 5. The number of unknown functions is $n-1$ for $\Delta =cos(\pi /n)$ and can be solved numerically for a finite external field. The low-energy excitations form a Luttinger liquid parametrized by a conformal field theory with conformal charge of $c=1$. For higher energy excitations, the spectral functions display deviations from the Luttinger behavior arising from the curvature in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this difference. The “impurity” is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-spinwave particle and hole Green’s function correctly. Full article

We introduce a quantum version for the statistical complexity measure, in the context of quantum information theory, and use it as a signaling function of quantum order–disorder transitions. We discuss the possibility for such transitions to characterize interesting physical phenomena, as quantum phase transitions, or abrupt variations in correlation distributions. We apply our measure on two exactly solvable Hamiltonian models: the $1D$-Quantum Ising Model (in the single-particle reduced state), and on Heisenberg XXZ spin-$1/2$ chain (in the two-particle reduced state). We analyze its behavior across quantum phase transitions for finite system sizes, as well as in the thermodynamic limit by using Bethe Ansatz technique. Full article

In this Communication, we consider a generalised Tavis–Cummings model when the damping process is taken into account. We show that the quantum dynamics governed by a non-Hermitian Hamiltonian is exactly solvable using the Quantum Inverse Scattering Method, and the Algebraic Bethe Ansatz. The leakage of photons is described by a Lindblad-type master equation. The non-Hermitian Hamiltonian is diagonalised by state vectors, which are elementary symmetric functions parametrised by the solutions of the Bethe equations. The time evolution of the photon annihilation operator is defined via a corresponding determinant representation. Full article

Assume that each species l has its own jump rate $b_l$ in the multi-species totally asymmetric simple exclusion process. We show that this model is integrable in the sense that the Bethe ansatz method is applicable to obtain the transition probabilities for all possible N-particle systems with up to N different species. Full article

In this work, we present a unified approach to the thermodynamics of hadron–quark–gluon matter at finite temperatures on the basis of a quark cluster expansion in the form of a generalized Beth–Uhlenbeck approach with a generic ansatz for the hadronic phase shifts that fulfills the Levinson theorem. The change in the composition of the system from a hadron resonance gas to a quark–gluon plasma takes place in the narrow temperature interval of 150–190 MeV, where the Mott dissociation of hadrons is triggered by the dropping quark mass as a result of the restoration of chiral symmetry. The deconfinement of quark and gluon degrees of freedom is regulated by the Polyakov loop variable that signals the breaking of the $Z\left(3\right)$ center symmetry of the color $SU\left(3\right)$ group of QCD. We suggest a Polyakov-loop quark–gluon plasma model with $\mathcal{O}\left({\alpha }_s\right)$ virial correction and solve the stationarity condition of the thermodynamic potential (gap equation) for the Polyakov loop. The resulting pressure is in excellent agreement with lattice QCD simulations up to high temperatures. Full article

In this paper we deal with the trigonometric Gaudin model, generalized using a nontrivial triangular reflection matrix (corresponding to non-periodic boundary conditions in the case of anisotropic XXZ Heisenberg spin-chain). In order to obtain the generating function of the Gaudin Hamiltonians with boundary terms we follow an approach based on Sklyanin’s derivation in the periodic case. Once we have the generating function, we obtain the corresponding Gaudin Hamiltonians with boundary terms by taking its residues at the poles. As the main result, we find the generic form of the Bethe vectors such that the off-shell action of the generating function becomes exceedingly compact and simple. In this way—by obtaining Bethe equations and the spectrum of the generating function—we fully implement the algebraic Bethe ansatz for the generalized trigonometric Gaudin model. Full article

The one-dimensional gas of bosons interacting via a repulsive contact potential was solved long ago via Bethe’s ansatz by Lieb and Liniger (Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State). The low energy excitation spectrum is a Luttinger liquid parametrized by a conformal field theory with conformal charge $c=1$ . For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The “impurity” term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole Green’s function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function. Full article

We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of the initial hypergeometric integrals. In some cases, we interpret the p-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field Fp. This interpretation is similar to the classical interpretation by Yu. I. Manin of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. We discuss the associated Bethe ansatz. Full article