arXiv subjects

V. Korepin

Publications and source records attributed to V. Korepin.

12 recordsLinked to original sources

arXiv2016

Negativity in the Generalized Valence Bond Solid State

Using a graphical presentation of the spin $S$ one dimensional Valence Bond Solid (VBS) state, based on the representation theory of the $SU(2)$ Lie-algebra of spins, we compute the spectrum of a mixed state reduced density matrix. This mixed state of two blocks of spins $A$ and $B$ is obtained by tracing out the spins outside $A$ and $B$, in the pure VBS state density matrix. We find in particular that the negativity of the mixed state is non-zero only for adjacent subsystems. The method introduced here can be generalized to the computation of entanglement properties in Levin-Wen models, that possess a similar algebraic structure to the VBS state in the groundstate.

quant-ph↗

arXiv2011

Negativity for two blocks in the one dimensional Spin 1 AKLT model

In this paper we compute the entanglement, as quantified by negativity, between two blocks of length $L_A$ and $L_B$, separated by $L$ sites in the one dimensional spin-1 AKLT model. We took the model with two different boundary conditions. We consider the case of $N$ spins 1 in the bulk and one spin 1/2 at each boundary which constitute an unique ground state, and the case of just spins 1, even at the end of the chain, where the degeneracy of the ground state is four. In both scenarios we made a partition consisting of two blocks $A$ and $B$, containing $L_A$ and $L_B$ sites respectively. The separation of these two blocks is $L$. In both cases we explicitly obtain the reduced density matrix of the blocks $A$ and $B$. We prove that the negativity in the first case vanishes identically for $L\geq 1$ while in the second scenario it may approach a constant value $N=1/2$ for each degenerate eigenstate depending on the way one constructs these eigenstates. However, as there is some freedom in constructing these eigenstates, vanishing entanglement is also possible in the latter case. Additionally, we also compute the entanglement between non-complementary blocks in the case of periodic boundary conditions for the spin-1 AKLT model for which there is a unique ground state. Even in this case, we find that the negativity of separated blocks of spins is zero.

quant-ph↗

arXiv2009

Temperature Correlation of Quantum Spins

This is a historical note. In 1993 we calculated space, time and temperature dependent correlation function in isotropic version of one dimensional XY spin chain. The correlation function decays exponentially with time and space separation. The rate of exponential decay was evaluated explicitly. Since that time similar results were obtained in other models: Bose gas with delta interaction, Ising model and strongly correlated electrons.

quant-ph↗

arXiv2001

Six - Vertex Model with Domain wall boundary conditions. Variable inhomogeneities

We consider the six-vertex model with domain wall boundary conditions. We choose the inhomogeneities as solutions of the Bethe Ansatz equations. The Bethe Ansatz equations have many solutions, so we can consider a wide variety of inhomogeneities. For certain choices of the inhomogeneities we study arrow correlation functions on the horizontal line going through the centre. In particular we obtain a multiple integral representation for the emptiness formation probability that generalizes the known formulæfor XXZ antiferromagnets.

math-ph↗

arXiv2000

Inhomogeneous Six-Vertex Model with Domain Wall Boundary Conditions and Bethe Ansatz

In this note, we consider the six-vertex model with domain wall boundary conditions, defined on a $M\times M$ lattice, in the inhomogeneous case where the partition function depends on 2M inhomogeneities $λ_j$ and $μ_k$. For a particular choice of the set of $λ_j$ we find a new determinant representation for the partition function, which allows evaluation of the bulk free energy in the thermodynamic limit. This provides a new connection between two types of determinant formulae. We also show in a special case that spin correlations on the horizontal line going through the center coincide with the ones for periodic boundary conditions.

nlin.SI↗

arXiv2000

Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions

We address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model. Here we compare the bulk free energy for periodic and domain wall boundary conditions. Using a determinant representation for the partition function with domain wall boundary conditions, we derive Toda differential equations and solve them asymptotically in order to extract the bulk free energy. We find that it is different and bears no simple relation with the free energy for periodic boundary conditions. The six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics (alternating sign matrices). This implies new results for the weighted counting for large size alternating sign matrices. Finally we comment on the interpretation of our results, in particular in connection with domino tilings (dimers on a square lattice).

cond-mat.stat-mech↗

arXiv1997

Maxwell-Bloch equation and Correlation function for penetrable Bose gas

We consider the quantum nonlinear Schrödinger equation in one space and one time dimension. We are interested in the non-free-fermionic case. We consider static temperature-dependent correlation functions. The determinant representation for correlation functions simplifies in the small mass limit of the Bose particle. In this limit we describe the correlation functions by the vacuum expectation value of a boson-valued solution for Maxwell-Bloch differential equation. We evaluate long-distance asymptotics of correlation functions in the small mass limit.

hep-th↗

arXiv1997

Determinant representation for dynamical correlation functions of the Quantum nonlinear Schrödinger equation

The foundation for the theory of correlation functions of exactly solvable models is determinant representation. Determinant representation permit to describe correlation functions by classical completely integrable differential equations [Barough, McCoy, Wu]. In this paper we show that determinant represents works not only for free fermionic models. We obtained determinant representation for the correlation function $<ψ(0,0)ψ^\dagger(x,t)>$ of the quantum nonlinear Schrödinger equation, out of free fermionic point. In the forthcoming publications we shall derive completely integrable equation and asymptotic for the quantum correlation function of this model of interacting fermions.

hep-th↗

arXiv1997

Completely Integrable Equation for the Quantum Correlation Function of Nonlinear Schrödinger Eqaution

Correlation functions of exactly solvable models can be described by differential equation [Barough, McCoy, Wu]. In this paper we show that for non free fermionic case differential equations should be replaced by integro-differential equations. We derive an integro-differential equation, which describes time and temperature dependent correlation function $<ψ(0,0)ψ^\dagger(x,t)>_T$ of penetrable Bose gas. The integro-differential equation turns out be the continuum generalization of classical nonlinear Schrödinger equation.

hep-th↗

arXiv1995

Probability of phase separation for the Bose gas with delta interactions

We consider the quantum Non-linear Schrödinger equation $i\partial_tΨ=-\partial_x^2Ψ+2cΨ^\daggerΨ^2$ with positive coupling constant $c$ varying from zero to infinity. We study quantum correlation functions of this model using the determinant representation of these correlation functions. We consider the case of a finite density ground state and evaluate, using the Riemann Hilbert problem, the asymptotics of the probability that no particles are present in the space interval $[0,x]$ in the large $x$ limit. We call this the probability of phase separation or alternately the emptiness formation probability.

cond-mat↗

arXiv1992

Fredholm Determinant Representation for Correlation Functions in XXO Heisenberg Chain

Space and time dependent temepreture correlation fucntions in the Hiesenberg XXO chain are evaluated in the magnetic field. The other name of the model is isotropic xy model in the transverse magnetic field. In the thermodynamic limit correlations in the model are represented as Fredhom determinanat. We expect this to to solve the problem of evaluation of asymptotics of temperature correlations.

cond-mat↗

arXiv1992

Determinant Representation for time and temperature dependent correlation functions in the isotropic XY model in the transverse magnetic field

Astymptotics of temperature correlations is the most dificult problem of stat. mech. Recently it was solved for the impenetrable Bose Gas. The idea is to represent correlation function as $τ$ function of calssical completely integrable differential equation. Now we start to apply this program to isotropic XY model in the trasnsverse magnetic field. In this paper we are making first step. We representing correlation function as detrminanat of an integral operator.

cond-mat↗