Searcharxiv⌕ Search

arXiv subjects

V. E. Korepin

Publications and source records attributed to V. E. Korepin.

At least 37 records · Page 2Linked to original sources
arXiv2006

Entropy of XY Spin Chain and Block Toeplitz Determinants

We consider entanglement in the ground state of the XY spin model on infinite chain. We use von Neumann entropy of a sub-system as a measure of entanglement. The entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We prove rigorously expression for limiting entropy which was published before. We observe that the entropy reaches minimum at product states but increases boundlessly at phase transitions.

quant-ph↗
arXiv2006

Analysis of entropy of XY Spin Chain

Entanglement in the ground state of the XY model on the infinite chain can be measured by the von Neumann entropy of a block of neighboring spins. We study a double scaling limit: the size of the block is much larger then 1 but much smaller then the length of the whole chain. In this limit, the entropy of the block approaches a constant. The limiting entropy is a function of the anisotropy and of the magnetic field. The entropy reaches minima at product states and increases boundlessly at phase transitions.

quant-ph↗
arXiv2004

Quantum Communication Through a Spin-Ring with Twisted Boundary Conditions

We investigate quantum communication between the sites of a spin-ring with twisted boundary conditions. Such boundary conditions can be achieved by a flux through the ring. We find that a non-zero twist can improve communication through finite odd numbered rings and enable high fidelity multi-party quantum communication through spin rings (working near perfectly for rings of 5 and 7 spins). We show that in certain cases, the twist results in the complete blockage of quantum information flow to a certain site of the ring. This effect can be exploited to interface and entangle a flux qubit and a spin qubit without embedding the latter in a magnetic field.

quant-ph↗
arXiv2004

Entanglement in XY Spin Chain

We consider the ground state of the XY model on an infinite chain at zero temperature. Following Bennett, Bernstein, Popescu, and Schumacher we use entropy of a sub-system as a measure of entanglement. Vidal, Latorre, Rico and Kitaev conjectured that von Neumann entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We evaluated this limiting entropy as a function of anisotropy and transverse magnetic field. We used the methods based on integrable Fredholm operators and Riemann-Hilbert problem. The entropy is singular at phase transitions.

quant-ph↗
arXiv2004

Correlation Functions in Spin Chains and Information Theory

Antiferromagnetic spin chains play an important role in condensed matter physics and statistical mechanics. Recently XXX spin chain was discussed in relation to the information theory. We consider here localizable entanglement, introduced recently by F.Verstraete, M.Popp and J.I.Cirac. That is how much entanglement can be localized on two spins on average by performing local measurements on the other individual spins in a system of many interacting spins. We consider the ground state in antiferromagnetic spin chains and study localizable entanglement between two spins as a function of the distance. We start with isotropic spin chain. Then we study effect of anisotropy and magnetic field. We conclude that anisotropy increases localizable entanglement. We found an explicit dependence of critical exponents in XXZ spin chain on magnetic field. We discovered that the cases of high symmetry corresponds to high sensitivity of magnetic field. We also calculated the concurrence before the measurement to illustrate that the measurment raises the concurrence.

quant-ph↗
arXiv2003

Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture

We consider one-dimensional quantum spin chain, which is called XX model, XX0 model or isotropic XY model in a transverse magnetic field. We study the model on the infinite lattice at zero temperature. We are interested in the entropy of a subsystem [a block of L neighboring spins]. It describes entanglement of the block with the rest of the ground state. G. Vidal, J.I. Latorre, E. Rico, and A. Kitaev showed that for large blocks the entropy scales logarithmically. We prove the logarithmic formula for the leading term and calculate the next term. We discovered that the dependence on the magnetic field interacting with spins is very simple: the magnetic field effectively reduce the size of the subsystem. We also calculate entropy of a subsystem of a small size. We also evaluated Renyi and Tsallis entropies of the subsystem. We represented the entropy in terms of a Toeplitz determinant and calculated the asymptotic analytically.

quant-ph↗
arXiv2003

Connecting lattice and relativistic models via conformal field theory

We consider the quantum group invariant XXZ-model. In infrared limit it describes Conformal Field Theory with modified energy-momentum tensor. The correlation functions are related to solutions of level -4 of qKZ equations. We describe these solutions relating them to level 0 solutions. We further consider general matrix elements (form factors) containing local operators and asymptotic states. We explain that the formulae for solutions of qKZ equations suggest a decomposition of these matrix elements with respect to states of corresponding Conformal Field Theory .

math-ph↗
arXiv2002

Asymptotic Behavior of the Emptiness Formation Probability in the Critical Phase of XXZ Spin Chain

We study the Emptiness Formation Probability (EFP) for the spin 1/2 XXZ spin chain. EFP P(n) detects a formation of ferromagnetic string of the length n in the ground state. It is expected that EFP decays in a Gaussian way for large strings P(n) ~ n^{-gamma} C^{-n^2}. Here, we propose the explicit expressions for the rate of Gaussian decay C as well as for the exponent gamma. In order to confirm the validity of our formulas, we employed an ab initio simulation technique of the density-matrix renormalization group to simulate XXZ spin chain of sufficient length. Furthermore, we performed Monte-Carlo integration of the Jimbo-Miwa multiple integral for P(n). Those numerical results for P(n) support our formulas fairly definitely.

cond-mat.stat-mech↗
arXiv2002

Emptiness Formation Probability and Quantum Knizhnik-Zamolodchikov Equation

We consider the one-dimensional XXX spin 1/2 Heisenberg antiferromagnet at zero temperature and zero magnetic field. We are interested in a probability of formation of a ferromagnetic string in the antiferromagnetic ground-state. We call it emptiness formation probability [EFP]. We suggest a new technique for computation of EFP in the inhomogeneous case. It is based on quantum Knizhnik-Zamolodchikov equation. We evalauted EFP for strings of the length six in the inhomogeneous case. The homogeneous limit confirms our hypothesis about the relation of quantum correlations to number theory. We also make a conjecture about a general structure of EFP for arbitrary lenght of the string \.

hep-th↗
arXiv2002

Quantum Correlations and Number Theory

We study spin-1/2 Heisenberg XXX antiferromagnet. The spectrum of the Hamiltonian was found by Hans Bethe in 1931. We study the probability of formation of ferromagnetic string in the antiferromagnetic ground state, which we call emptiness formation probability P(n). This is the most fundamental correlation function. We prove that for the short strings it can be expressed in terms of the Riemann zeta function with odd arguments, logarithm ln 2 and rational coefficients. This adds yet another link between statistical mechanics and number theory. We have obtained an analytical formula for P(5) for the first time. We have also calculated P(n) numerically by the Density Matrix Renormalization Group. The results agree quite well with the analytical ones. Furthermore we study asymptotic behavior of P(n) at finite temperature by Quantum Monte-Carlo simulation. It also agrees with our previous analytical results.

cond-mat.stat-mech↗
arXiv2001

On the asymptotic expansion of the solutions of the separated nonlinear Schroedinger equation

Nonlinear Schrödinger equation (with the Schwarzian initial data) is important in nonlinear optics, Bose condensation and in the theory of strongly correlated electrons. The asymptotic solutions in the region $x/t={\cal O}(1)$, $t\to\infty$, can be represented as a double series in $t^{-1}$ and $\ln t$. Our current purpose is the description of the asymptotics of the coefficients of the series.

nlin.SI↗
arXiv2001

Quantum Spin Chains and Riemann Zeta Function with Odd Arguments

Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We study XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in thermodynamics limit. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments.

hep-th↗
arXiv1999

Solution of the quantum inverse problem

We derive a formula that expresses the local spin and field operators of fundamental graded models in terms of the elements of the monodromy matrix. This formula is a quantum analogue of the classical inverse scattering transform. It applies to fundamental spin chains, such as the XYZ chain, and to a number of important exactly solvable models of strongly correlated electrons, such as the supersymmetric t-J model or the the EKS model.

hep-th↗
arXiv1999

The Hubbard chain: Lieb-Wu equations and norm of the eigenfunctions

We argue that the square of the norm of the Hubbard wave function is proportional to the determinant of a matrix, which is obtained by linearization of the Lieb-Wu equations around a solution. This means that in the vicinity of a solution the Lieb-Wu equations are non-degenerate, if the corresponding wave function is non-zero. We further derive an action that generates the Lieb-Wu equations and express our determinant formula for the square of the norm in terms of the Hessian determinant of this action.

cond-mat.str-el↗
arXiv1999

Universal correlations of one-dimensional interacting electrons in the gas phase

We consider dynamical correlation functions of short range interacting electrons in one dimension at finite temperature. Below a critical value of the chemical potential there is no Fermi surface anymore, and the system can no longer be described as a Luttinger liquid. Its low temperature thermodynamics is that of an ideal gas. We identify the impenetrable electron gas model as a universal model for the gas phase and present exact and explicit expressions for the asymptotics of correlation functions at small temperatures, in the presence of a magnetic field.

cond-mat.str-el↗
arXiv1999

Thermodynamics and excitations of the one-dimensional Hubbard model

We review fundamental issues arising in the exact solution of the one-dimensional Hubbard model. We perform a careful analysis of the Lieb-Wu equations, paying particular attention to so-called `string solutions'. Two kinds of string solutions occur: $Λ$ strings, related to spin degrees of freedom and $k-Λ$ strings, describing spinless bound states of electrons. Whereas $Λ$ strings were thoroughly studied in the literature, less is known about $k-Λ$ strings. We carry out a thorough analytical and numerical analysis of $k-Λ$ strings. We further review two different approaches to the thermodynamics of the Hubbard model, the Yang-Yang approach and the quantum transfer matrix approach, respectively. The Yang-Yang approach is based on strings, the quantum transfer matrix approach is not. We compare the results of both methods and show that they agree. Finally, we obtain the dispersion curves of all elementary excitations at zero magnetic field for the less than half-filled band by considering the zero temperature limit of the Yang-Yang approach.

cond-mat.str-el↗
arXiv1998

The form factors in the finite volume

The form factors of integrable models in finite volume are studied. We construct the explicite representations for the form factors in terms of determinants.

math-ph↗