arXiv subjects

V. E. Korepin

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

At least 19 recordsLinked to original sources

arXiv2020

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion

We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, \texttt{arXiv:1104.1004}, for the reduced density of states of two disjoint intervals with lattice sites $P=\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher-Hartwig symbol using the the Riemann--Hilbert method.

math-ph↗

arXiv2017

Haldane Topological Orders in Motzkin Spin Chains

Motzkin spin chains are frustration-free models whose ground-state is a combination of Motzkin paths. The weight of such path contributions can be controlled by a deformation parameter t. As a function of the latter these models, beside the formation of domain wall structures, exhibit a Berezinskii-Kosterlitz-Thouless phase transition for t=1 and gapped Haldane topological orders with constant decay of the string order parameters for t < 1. By means of numerical calculations we show that the topological properties of the Haldane phases depend on the spin value. This allows to classify different kinds of hidden antiferromagnetic Haldane gapped regimes associated to nontrivial features like symmetry-protected topological order. Our results from one side allow to clarify the physical properties of Motzkin frustration-free chains and from the other suggest them as a new interesting and paradigmatic class of local spin Hamiltonians.

cond-mat.stat-mech↗

arXiv2016

Violation of Cluster Decomposition and Absence of Light-Cones in Local Integer and Half-Integer Spin Chains

We compute the ground state correlation functions of an exactly solvable chain of integer spins, recently introduced in [R. Movassagh and P. W. Shor, arXiv:1408.1657], whose ground-state can be expressed in terms of a uniform superposition of all colored Motzkin paths. Our analytical results show that for spin s$\ge$2 there is a violation of the cluster decomposition property. This has to be contrasted with s=1, where the cluster property holds. Correspondingly, for s=1 one gets a light-cone profile in the propagation of excitations after a local quench, while the cone is absent for s=2, as shown by time dependent density-matrix-renormalization-group. Moreover, we introduce an original solvable model of half-integer spins which we refer to as Fredkin spin chain, whose ground-state can be expressed in terms of superposition of all Dyck paths. For this model we exactly calculate the magnetization and correlation functions, finding that for s=1/2, a cone-like propagation occurs while for higher spins, s$\ge$3/2, the colors prevent any cone formation and clustering is violated, together with square root deviation from the area law for the entanglement entropy.

cond-mat.str-el↗

arXiv2016

Holographic optical traps for atom-based topological Kondo devices

The topological Kondo (TK) model has been proposed in solid-state quantum devices as a way to realize non-Fermi liquid behaviors in a controllable setting. Another motivation behind the TK model proposal is the demand to demonstrate the quantum dynamical properties of Majorana fermions, which are at the heart of their potential use in topological quantum computation. Here we consider a junction of crossed Tonks-Girardeau gases arranged in a star-geometry (forming a Y -junction), and we perform a theoretical analysis of this system showing that it provides a physical realization of the topological Kondo model in the realm of cold atom systems. Using computer-generated holography, we experimentally implement a Y-junction suitable for atom trapping, with controllable and independent parameters. The junction and the transverse size of the atom waveguides are of the order of 5 micrometers, leading to favorable estimates for the Kondo temperature and for the coupling across the junction. Since our results show that all the required theoretical and experimental ingredients are available, this provides the demonstration of an ultracold atom device that may in principle exhibit the topological Kondo effect.

cond-mat.mes-hall↗

arXiv2015

Influence of boundary conditions on bulk properties of six-vertex model

We study the influence of boundary conditions on the entropy of the six-vertex model. We consider the case of fixed boundary conditions in order to argue that the entropy of the six-vertex model vary continuously from its value for ferroelectric to periodic boundary conditions. This is done by merging the ferroelectric boundary and the Néel boundary.

cond-mat.stat-mech↗

arXiv2015

The entropy of the six-vertex model with variety of different boundary conditions

We study the dependence of entropy [per lattice site] of six-vertex model on boundary conditions. We start with lattices of finite size and then proceed to thermodynamic limit. We argue that the six-vertex model with periodic, anti-periodic and mixed boundary conditions produce the same free-energy in the thermodynamic limit. We have found fixed boundary conditions such that the entropy varies continously from zero to its value for periodic boundary condition. We have also shown that the physical quantities of the six-vertex model at the isotropic point does not change in the case of singular toroidal boundary.

cond-mat.stat-mech↗

arXiv2015

Thermodynamic limit of the six-vertex model with reflecting end

We study the thermodynamic limit of the six-vertex model with domain wall boundary and reflecting end. We evaluated the partition function explicitly in special cases. We calculated the homogeneous limit of the Tsuchiya determinant formula for the partition function. We evaluated the thermodynamic limit and obtain the free energy of the six-vertex model with reflecting end. We determined the free energy in the disordered regime.

cond-mat.stat-mech↗

arXiv2013

Reduction of One Loop Feynman Diagrams in Scalar Field Theory

This is a historical note. In 1979 we wrote a paper in a Russian Journal called Vestnik Leingradskogo Gosudarstvennogo Universiteta. We considered massive scalar quantum filed theory. One loop Feynman diagrams were evaluated. Theorem was proved that one loop diagram with many internal lines [more then dimension of space-time] can be expressed in terms of one loop diagram with number of internal lines equal to the dimension of space-time [multiplied by tree diagrams]. This is translation in English.

hep-th↗

arXiv2012

Quantum network teleportation for quantum information distribution and concentration

We investigate the schemes of quantum network teleportation for quantum information distribution and concentration which are essential in quantum cloud computation and quantum internet. In those schemes, the cloud can send simultaneously identical unknown quantum states to clients located in different places by a network like teleportation with a prior shared multipartite entangled state resource. The cloud first perform the quantum operation, each client can recover their quantum state locally by using the classical information announced by the cloud about the measurement result. The number of clients can be beyond the number of identical quantum states intentionally being sent, this quantum network teleportation can make sure that the retrieved quantum state is optimal. Furthermore, we present a scheme to realize its reverse process, which concentrates the states from the clients to reconstruct the original state of the cloud. These schemes facilitate the quantum information distribution and concentration in quantum networks in the framework of quantum cloud computation. Potential applications in time synchronization are discussed.

quant-ph↗

arXiv2011

Higher conservation laws for the quantum non-linear Schroedinger equation

Quantum non-linear SCHROEDINGER equation is equivalent to Lieb-Liniger model. It has non-trivial conservation laws. Recently these conservation laws were used for evaluation of the three-body recombination rate for interacting gas of quantum bosons. These conservations laws were known already in 1989. Submitted text is retyping of the preprint of Centre for Mathematical Analysis of Australian National University CMA-R33-89. It was discussed in Leningrad Branch of the V.A. Steklov Mathematical Institute at that time. A copy of the original preprint can be found in the section Quantum Inverse Scattering Method of the web-page http://insti.physics.sunysb.edu/~korepin/

math-ph↗

arXiv2011

Entanglement Entropy for Disjoint Subsystems in XX Spin Chain

Fisher-Hartwig formula has been successful applied to describe the von Neumann and Rényi entropies of a block of spins in the ground state of XX spin chain. It was based on a determinant representation. In this paper, we generalize the free fermion method to obtain an exact formulation for the entropy of any finite subsystem in XX spin chain. Based on this, we derive a determinant representation of the entropy of multiple disjoint intervals in the ground state of $XX$ model.

quant-ph↗

arXiv2010

Entanglement Spectrum for the XY Model in One Dimension

We consider the reduced density matrix of a large block of consecutive spins in the ground states of the XY spin chain on an infinite lattice. We derive the spectrum of the density matrix using the expression of the Renyi entropy in terms of modular functions. The eigenvalues λ_n form an exact geometric sequence. For example, for strong magnetic field λ_n = C \exp{(-πτ_0 n)}, here τ_0>0 and C > 0 depend on the anisotropy and the magnetic field. Different eigenvalues are degenerated differently. The largest eigenvalue is unique, but the degeneracy g_n increases sub-exponentially as eigenvalues diminish: g_n \sim \exp{(π\sqrt{n/3})}. For weak magnetic field expressions are similar.

quant-ph↗

arXiv2009

The Fisher-Hartwig Formula and Generalized Entropies in XY Spin Chain

Toeplitz matrices have applications to different problems of statistical mechanics. Recently they were used for calculation of entanglement entropy in spin chains. We use the Fisher-Hartwig formula to calculate entanglement entropy of large block of spins in the ground state of XY spin chain. We also calculate Renyi entropy and prove that the spectrum of the density matrix of a block of spins is exact geometric sequence [also different eigenvalues are degenerated differently].

math-ph↗

arXiv2009

A lattice model related to the nonlinear Schroedinger equation

This is a historical note. In 1981 we constructed a discrete version of quantum nonlinear Schroedinger equation. This led to our discovery of quantum determinant: it appeared in construction of anti-pod (11). Later these became important in quantum groups: it describes the center of Yang-Baxter algebra. Our paper was published in Doklady Akademii Nauk vol 259, page 76 (July l981) in Russian language.

math.QA↗

arXiv2009

Cancellation of ultra-violet infinities in one loop gravity

This is a historical note. In 1974 I was an undergraduate student of L.D. Faddeev. I was working on quantum gravity [without matter] in one loop approximation. I discovered [simultaneously with G. t'Hooft M. Veltman] that on mass shell ultra-violet divergences cancel. The submission is a translation of the Diploma.

gr-qc↗

arXiv2009

Extraction of Pure Entangled States from Many Body Systems by Distant Local Projections

We study the feasibility of extracting a pure entangled state of non-complementary, and potentially well separated, regions of a quantum many-body system. It is shown that this can indeed be accomplished in non-equilibrium scenarios as well as the ground state of the considered spin chain models when one locally measures observables such as magnetization in separated blocks of spins. A general procedure is presented, which can search for the optimal way to extract a pure entangled state through local projections. Our results indicate a connection of the projective extraction of entanglement to good quantum numbers of the underlying Hamiltonian.

quant-ph↗

arXiv2007

Renyi Entropy of the XY Spin Chain

We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power $α$ of the density matrix of the block. We calculate the asymptotic for $L \to \infty$ analytically in terms of Klein's elliptic $λ$ - function. We study the limiting entropy as a function of its parameter $α$. We show that up to the trivial addition terms and multiplicative factors, and after a proper re-scaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map $α\to α^{-1}$ and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when $α$ is a integer power of 2, this includes the purity $tr ρ^2$. We also analyze the behavior of the entropy as $α\to 0$ and $\infty$ and at the critical magnetic field and in the isotropic limit [XX model].

quant-ph↗

arXiv2007

Ellipses of Constant Entropy in the 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. The entropy of the block has an asymptotic limit. We study this limiting entropy as a function of the anisotropy and of the magnetic field. We identify its minima at product states and its divergencies at the quantum phase transitions. We find that the curves of constant entropy are ellipses and hyperbolas and that they all meet at one point (essential critical point). Depending on the approach to the essential critical point the entropy can take any value between 0 and infinity. In the vicinity of this point small changes in the parameters cause large change of the entropy.

quant-ph↗