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Karol K. Kozlowski

Publications and source records attributed to Karol K. Kozlowski.

35 records · Page 2Linked to original sources
arXiv2019

High-temperature analysis of the transverse dynamical two-point correlation function of the XX quantum-spin chain

We analyse the transverse dynamical two-point correlation function of the XX chain by means of a thermal form factor series. The series is rewritten in terms of the resolvent and the Fredholm determinant of an integrable integral operator. This connects it with a matrix Riemann-Hilbert problem. We express the correlation function in terms of the solution of the matrix Riemann-Hilbert problem. The matrix Riemann-Hilbert problem is then solved asymptotically in the high-temperature limit. This allows us to obtain the leading high-temperature contribution to the two-point correlation function at any fixed space-time separation.

math-ph↗
arXiv2019

Equilibrium dynamics of the XX chain

The equilibrium dynamics of the spin-1/2 XX chain is re-examined within a recently developed formalism based on the quantum transfer matrix and a thermal form factor expansion. The transversal correlation function is evaluated in real time and space. The high-accuracy calculation reproduces several exact results in limiting cases as well as the well-known asymptotic formulas obtained by the matrix Riemann-Hilbert approach. Furthermore, comparisons to numerical data based on a direct evaluation of the Pfaffian as well as to asymptotic formulas obtained within non-linear Luttinger liquid theory are presented.

cond-mat.stat-mech↗
arXiv2019

Late-time large-distance asymptotics of the transverse correlation functions of the XX chain in the space-like regime

We derive an explicit expression for the leading term in the late-time, large-distance asymptotic expansion of a transverse dynamical two-point function of the XX chain in the spacelike regime. This expression is valid for all non-zero finite temperatures and for all magnetic fields below the saturation threshold. It is obtained here by means of a straightforward term-by-term analysis of a thermal form factor series, derived in previous work, and demonstrates the usefulness of the latter.

cond-mat.stat-mech↗
arXiv2019

Long-distance and large-time asymptotic behaviour of dynamic correlation functions in the massless regime of the XXZ spin-1/2 chain

Starting from the massless form factor expansion for the two-point dynamical correlation functions obtained recently, I extract the long-distance and large-time asymptotics of these correlators. The analysis yields the critical exponents and associated amplitudes characterising the asymptotics. The results are obtained on the basis of exact and first principle based considerations: they \textit{do not rely, at any stage}, on some hypothetical correspondence with a field theory or the use of any other phenomenological approach. Being based on form factor expansion, the method allows one to clearly identify which contributions to the asymptotics issues from which class of excited states. All this permits to settle the long-standing question of the contribution of bound states to the asymptotics of two-point functions. For instance, when considering the long-distance $m$ behaviour of equal-time correlators, the analysis shows that while, \textit{in fine}, the bound states only produce contributions that are exponentially small in $m$, they also play a key role in cancelling out certain power-law contributions which, should they be present, would break explicitly the universality structure of the long-distance behaviour.

math-ph↗
arXiv2019

Full counting statistics of energy transfers in inhomogeneous nonequilibrium states of (1+1)D CFT

Employing the conformal welding technique, we find an exact expression for the Full Counting Statistics of energy transfers in a class of inhomogeneous nonequilibrium states of a (1+1)-dimensional unitary Conformal Field Theory. The expression involves the Schwarzian action of a complex field obtained by solving a Riemann-Hilbert type problem related to conformal welding of infinite cylinders. On the way, we obtain a formula for the extension of characters of unitary positive-energy representations of the Virasoro algebra to 1-parameter groups of circle diffeomorphisms and we develop techniques, based on the analysis of certain classes of Fredholm operators, that allow to control the leading asymptotics of such extensions for small real part of the modular parameter τ

math-ph↗
arXiv2019

Thermodynamics of the spin-$1/2$ Heisenberg-Ising chain at high temperatures: a rigorous approach

This work develops a rigorous setting allowing one to prove several features related to the behaviour of the Heisenberg-Ising (or XXZ) spin-$1/2$ chain at finite temperature $T$. Within the quantum inverse scattering method the physically pertinent observables at finite $T$, such as the \textit{per}-site free energy or the correlation length, have been argued to admit integral representations whose integrands are expressed in terms of solutions to auxiliary non-linear integral equations. The derivation of such representations was based on numerous conjectures: the possibility to exchange the infinite volume and the infinite Trotter number limits, the existence of a real, non-degenerate, maximal in modulus Eigenvalue of the quantum transfer matrix, the existence and uniqueness of solutions to the auxiliary non-linear integral equations, as well as the possibility to take the infinite Trotter number limit on their level. We rigorously prove all these conjectures for temperatures large enough. As a by product of our analysis, we obtain the large-$T$ asymptotic expansion for a subset of sub-dominant Eigenvalues of the quantum transfer matrix and thus of the associated correlation lengths. This result was never obtained previously, not even on heuristic grounds.

math-ph↗
arXiv2018

On the separation of variables for the modular XXZ magnet and the lattice Sinh-Gordon models

We construct the generalised Eigenfunctions of the entries of the monodromy matrix of the $N$-site modular XXZ magnet and show, in each case, that these form a complete orthogonal system in $L^2(\mathbb{R}^N)$. In particular, we develop a new and simple technique, allowing one to prove the completeness of such systems. As a corollary of out analysis, we prove the Bystko-Teschner conjecture relative to the structure of the spectrum of the $\boldsymbol{ \texttt{B} }(\la)$-operator for the odd length lattice Sinh-Gordon model.

math-ph↗
arXiv2017

Thermal form factor approach to the ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime

We use the form factors of the quantum transfer matrix in the zero-temperature limit in order to study the two-point ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime. We obtain novel form factor series representations of the correlation functions which differ from those derived either from the q-vertex-operator approach or from the algebraic Bethe Ansatz approach to the usual transfer matrix. We advocate that our novel representations are numerically more efficient and allow for a straightforward calculation of the large-distance asymptotic behaviour of the two-point functions. Keeping control over the temperature corrections to the two-point functions we see that these are of order $T^\infty$ in the whole antiferromagnetic massive regime. The isotropic limit of our result yields a novel form factor series representation for the two-point correlation functions of the XXX chain at zero magnetic field.

cond-mat.stat-mech↗
arXiv2016

Form factors of bound states in the XXZ chain

This work focuses on the calculation of the large-volume behaviour of form factors of local operators in the XXZ spin-$1/2$ chain taken between the ground state and an excited state containing bound states. The analysis is rigorous and builds on various fine properties of the string solutions to the Bethe equations and certain technical hypotheses. These technical hypotheses are satisfied for a generic excited state. The results obtained in this work pave the way for extracting, starting from the first principles, the large-distance and long-time asymptotic behaviour of the XXZ chain's two-point functions just as the so-called edge singularities of their Fourier transforms.

math-ph↗
arXiv2015

Asymptotics of correlation functions of the Heisenberg-Ising chain in the easy-axis regime

We analyze the long-time large-distance asymptotics of the longitudinal correlation functions of the Heisenberg-Ising chain in the easy-axis regime. We show that in this regime the leading asymptotics of the dynamical two-point functions is entirely determined by the two-spinon contribution to their form factor expansion. Its explicit form is obtained from a saddle-point analysis of the corresponding double integral. It describes the propagation of a wave front with velocity $v_{c_1}$ which is found to be the maximal possible group velocity. Like in wave propagation in dispersive media the wave front is preceded by a precursor running ahead with velocity $v_{c_2}$. As a special case we obtain the explicit form of the asymptotics of the auto-correlation function.

cond-mat.str-el↗
arXiv2015

Low-temperature spectrum of correlation lengths of the XXZ chain in the antiferromagnetic massive regime

We consider the spectrum of correlation lengths of the spin-$\frac{1}{2}$ XXZ chain in the antiferromagnetic massive regime. These are given as ratios of eigenvalues of the quantum transfer matrix of the model. The eigenvalues are determined by integrals over certain auxiliary functions and by their zeros. The auxiliary functions satisfy nonlinear integral equations. We analyse these nonlinear integral equations in the low-temperature limit. In this limit we can determine the auxiliary functions and the expressions for the eigenvalues as functions of a finite number of parameters which satisfy finite sets of algebraic equations, the so-called higher-level Bethe Ansatz equations. The behaviour of these equations, if we send the temperature $T$ to zero, is different for zero and non-zero magnetic field $h$. If $h$ is zero the situation is much like in the case of the usual transfer matrix. Non-trivial higher-level Bethe Ansatz equations remain which determine certain complex excitation parameters as functions of hole parameters which are free on a line segment in the complex plane. If $h$ is non-zero, on the other hand, a remarkable restructuring occurs, and all parameters which enter the description of the quantum transfer matrix eigenvalues can be interpreted entirely in terms of particles and holes which are freely located on two curves when $T$ goes to zero.

cond-mat.stat-mech↗
arXiv2014

Functions Characterizing the Ground State of the XXZ Spin-1/2 Chain in the Thermodynamic Limit

We establish several properties of the solutions to the linear integral equations describing the infinite volume properties of the XXZ spin-1/2 chain in the disordered regime. In particular, we obtain lower and upper bounds for the dressed energy, dressed charge and density of Bethe roots. Furthermore, we establish that given a fixed external magnetic field (or a fixed magnetization) there exists a unique value of the boundary of the Fermi zone.

math-ph↗
arXiv2014

Low-temperature large-distance asymptotics of the transversal two-point functions of the XXZ chain

We derive the low-temperature large-distance asymptotics of the transversal two-point functions of the XXZ chain by summing up the asymptotically dominant terms of their expansion into form factors of the quantum transfer matrix. Our asymptotic formulae are numerically efficient and match well with known results for vanishing magnetic field and for short distances and magnetic fields below the saturation field.

cond-mat.stat-mech↗
arXiv2013

Thermal form factors of the XXZ chain and the large-distance asymptotics of its temperature dependent correlation functions

We derive expressions for the form factors of the quantum transfer matrix of the spin-1/2 XXZ chain which are suitable for taking the infinite Trotter number limit. These form factors determine the finitely many amplitudes in the leading asymptotics of the finite-temperature correlation functions of the model. We consider form-factor expansions of the longitudinal and transversal two-point functions. Remarkably, the formulae for the amplitudes are in both cases of the same form. We also explain how to adapt our formulae to the description of ground state correlation functions of the finite chain. The usefulness of our novel formulae is demonstrated by working out explicit results in the high- and low-temperature limits. We obtain, in particular, the large-distance asymptotics of the longitudinal two-point functions for small temperatures by summing up the asymptotically most relevant terms in the form factor expansion of a generating function of the longitudinal correlation functions. As expected the leading term in the expansion of the corresponding two-point functions is in accordance with conformal field theory predictions. Here it is obtained for the first time by a direct calculation.

cond-mat.stat-mech↗
arXiv2011

Riemann-Hilbert approach to a generalized sine kernel and applications

We investigate the asymptotic behavior of a generalized sine kernel acting on a finite size interval [-q,q]. We determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its Fredholm determinant. Further, we apply our results to build the resolvent of truncated Wiener--Hopf operators generated by holomorphic symbols. Finally, the leading asymptotics of the Fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models.

math-ph↗
arXiv2009

On 4-point correlation functions in simple polymer models

We derive an exact formula for the covariance of cartesian distances in two simple polymer models, the freely-jointed chain and a discrete flexible model with nearest-neighbor interaction. We show that even in the interaction-free case correlations exist as long as the two distances at least partially share the same segments. For the interacting case, we demonstrate that the naive expectation of increasing correlations with increasing interaction strength only holds in a finite range of values. Some suggestions for future single-molecule experiments are made.

cond-mat.soft↗