# K. K. KozlowskiEcole normale supérieure de Lyon | ENS Lyon · departement de mathématiques

K. K. Kozlowski

Dr. Hab.

## About

57

Publications

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## Publications

Publications (57)

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...

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 wel...

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 so...

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 fir...

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...

We propose a method for calculating dynamical correlation functions at finite temperature in integrable lattice models of Yang-Baxter type. The method is based on an expansion of the correlation functions as a series over matrix elements of a time-dependent quantum transfer matrix rather than the Hamiltonian. In the infinite Trotter-number limit th...

This work constructs a well-defined and operational form factor expansion in a model having a massless spectrum of excitations. More precisely, the dynamic two-point functions in the massless regime of the XXZ spin-1/2 chain are expressed in terms of properly regularised series of multiple integrals. These series are obtained by taking, in an appro...

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....

A quantisation of the KP equation on a cylinder is proposed that is equivalent to an infinite system of non-relativistic one-dimensional bosons carrying masses $m=1,2,\ldots$ The Hamiltonian is Galilei-invariant and includes the split $\Psi^\dagger_{m_1}\Psi^\dagger_{m_2}\Psi_{m_1+m_2}$ and merge $\Psi^\dagger_{m_1+m_2}\Psi_{m_1}\Psi_{m_2}$ terms f...

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-ope...

We extract the long-distance asymptotic behaviour of two-point correlation
functions in massless quantum integrable models containing multi-species
excitations. For such a purpose, we extend to these models the method of a
large-distance regime re-summation of the form factor expansion of correlation
functions. The key feature of our analysis is a...

This habilitation thesis reviews the progress made by the author respectively
to studying various asymptotic regimes of correlation functions in quantum
integrable models.

I prove that the Bethe roots describing either the ground state or a certain
class of "particle-hole" excited states of the XXZ spin-$1/2$ chain in any
sector with magnetisation $\mathfrak{m} \in [0;1/2]$ exist and form, in the
infinite volume limit, a dense distribution on a subinterval of $\mathbb{R}$.
The results holds for any value of the aniso...

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 obtai...

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...

Starting from the form factor expansion in finite volume, we derive the multidimensional generalization of the so-called Natte series for the time- and distance-dependent reduced density matrix at zero temperature in the non-linear Schrödinger model. This representation allows one to read-off straightforwardly the long-time/large-distance asymptoti...

Starting from the finite volume form factors of local operators, we show how
and under which hypothesis the $c=1$ free boson conformal field theory in
two-dimensions emerges as an effective theory governing the large-distance
regime of multi-point correlation functions in a large class of one dimensional
massless quantum Hamiltonians. In our approa...

We derive the large-N, all order asymptotic expansion for a system of N particles with mean field interactions on top of a Coulomb repulsion at temperature 1/β, under the assumptions that the interactions are analytic, off-critical, and satisfy a local strict convexity assumption. © The Author(s) 2015. Published by Oxford University Press. All righ...

We study the large-volume-$L$ limit of form factors of the longitudinal spin
operators for the XXZ spin-$1/2$ chain in the massive regime. We find that the
individual form factors decay as $L^{-n}$, $n$ being an even integer counting
the number of physical excitations -- the holes -- that constitute the excited
state. Our expression allows us to de...

This paper develops a method to carry out the large-$N$ asymptotic analysis
of a class of $N$-dimensional integrals arising in the context of the so-called
quantum separation of variables method. We push further ideas developed in the
context of random matrices of size $N$, but in the present problem, two scales
$1/N^{\alpha}$ and $1/N$ naturally o...

The purpose of this paper is to push forward the theory of operator-valued
Riemann Hilbert problems and demonstrate their effectiveness in respect to the
implementation of a non-linear steepest descent method \textit{\'{a} la}
Deift-Zhou. In the present paper, we demonstrate that the operator-valued
Riemann--Hilbert problem arising in the character...

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...

We give a basic framework to study generalizations of beta ensembles, with an
arbitrary interaction between eigenvalues (not only pairwise), but assuming
pairwise repulsion at short distance approximated by the Coulomb interaction
already present in beta ensembles, unicity of the equilibrium measure and local
strict convexity of the energy function...

We provide a microscopic model setting that allows us to readily access to
the large-distance asymptotic behaviour of multi-point correlation functions in
massless, one-dimensional, quantum models. The method of analysis we propose is
based on the form factor expansion of the correlation functions and does not
build on any field theory reasonings....

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 mag...

By using Riemann--Hilbert problem based techniques, we obtain the asymptotic
expansion of lacunary Toeplitz determinants $\det_N\big[ c_{\ell_a-m_b}[f]
\big]$ generated by holomorhpic symbols, where $\ell_a=a$ (resp. $m_b=b$)
except for a finite subset of indices $a=h_1,\dots, h_n$ (resp. $b=t_1,\dots,
t_r$). In addition to the usual Szeg\"{o} asym...

We generalize Babelon's approach to equations in dual variables so as to be able to treat new types of operators which we build out of the sub-constituents of the model's monodromy matrix. Further, we also apply Sklyanin's recent monodromy matrix identities so as to obtain equations in dual variables for yet other operators. The sche...

The quantum separation of variables method consists in mapping the original
Hilbert space where a spectral problem is formulated onto one where the
spectral problem takes a simpler "separated" form. In order to realise such a
program, one should construct the map explicitly and then show that it is
unitary.
In the present paper, we develop a techni...

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...

Integrable integral operator can be studied by means of a matrix Riemann-Hilbert problem. However, in the case of so-called integrable operators with shifts, the associated Riemann-Hilbert problem becomes operator-valued and this complicates strongly the analysis. In this note, we show how to circumvent, in a very simple way, the use of such a sett...

We develop a form factor approach to the study of dynamical correlation
functions of quantum integrable models in the critical regime. As an example,
we consider the quantum non-linear Schr\"odinger model. We derive
long-distance/long-time asymptotic behavior of various two-point functions of
this model. We also compute edge exponents and amplitude...

A generalization of the Bethe ansatz equations is studied, where a scalar
two-particle S-matrix has several zeroes and poles in the complex plane, as
opposed to the ordinary single pole/zero case. For the repulsive case (no
complex roots), the main result is the enumeration of all distinct solutions to
the Bethe equations in terms of the Fuss-Catal...

We study the boundary free energy of the XXZ spin-1/2
chain subject to diagonal boundary fields. We first show that the representation for its
finite Trotter number approximant obtained by Göhmann, Bortz and Frahm is related to the
partition function of the six-vertex model with reflecting ends. Building on the Tsuchiya
determinant representation...

We prove that the unique solution to the Yang-Yang equation arising in the
context of the thermodynamics of the so-called non-linear Schr\"{o}dinger model
admits a low-temperature expansion to all orders. Our approach provides a
rigorous justification, for a certain class of non-linear integral equations,
of the low-temperature asymptotic expansion...

We propose a form factor approach for the computation of the large distance
asymptotic behavior of correlation functions in quantum critical (integrable)
models. In the large distance regime we reduce the summation over all excited
states to one over the particle/hole excitations lying on the Fermi surface in
the thermodynamic limit. We compute the...

We present a new method allowing one to derive the long-time and large-distance asymptotic behavior of the correlation functions of quantum integrable models from their exact representations. Starting from the form factor expansion of the correlation functions in finite volume, we explain how to reduce the complexity of the computation in the so-ca...

Izergin-Korepin's lattice discretization of the nonlinear Schrödinger model along with Oota's inverse problem provides one with determinant representations for the form factors of the lattice discretized conjugated field operator. We prove that these form factors converge, in the zero lattice spacing limit, to those of the conjugated field operator...

Izergin-Korepin's lattice discretization of the non-linear Schr\"odinger
model along with Oota's inverse problem provides one with determinant
representations for the form factors of the lattice discretized conjugated
field operator. We prove that these form factors converge, in the zero lattice
spacing limit, to those of the conjugated field opera...

We consider the low-temperature limit of the long-distance asymptotic
behavior of the finite temperature density-density correlation function in the
one-dimensional Bose gas derived recently in the algebraic Bethe ansatz
framework. Our results confirm the predictions based on the Luttinger liquid
and conformal field theory approaches. We also demon...

We present a new method allowing us to derive the long-time and
large-distance asymptotic behavior of the correlations functions of quantum
integrable models from their exact representations. Starting from the form
factor expansion of the correlation functions in finite volume, we explain how
to reduce the complexity of the computation in the so-ca...

We derive the leading asymptotic behavior and build a new series
representation for the Fredholm determinant of integrable integral operators
appearing in the representation of the time and distance dependent correlation
functions of integrable models described by a six-vertex R-matrix. This series
representation opens a systematic way for the comp...

We describe a Bethe ansatz based method to derive, starting from a multiple
integral representation, the long-distance asymptotic behavior at finite
temperature of the density-density correlation function in the interacting
one-dimensional Bose gas. We compute the correlation lengths in terms of
solutions of non-linear integral equations of the the...

We give a direct derivation of a proposal of Nekrasov-Shatashvili concerning the quantization conditions of the Toda chain. The quantization conditions are formulated in terms of solutions to a nonlinear integral equation similar to the ones coming from the thermodynamic Bethe ansatz. This is equivalent to extremizing a certain function called Yang...

We study the thermodynamic limit of the particle-hole form factors of the XXZ
Heisenberg chain in the massless regime. We show that, in this limit, such form
factors decrease as an explicitly computed power-law in the system-size.
Moreover, the corresponding amplitudes can be obtained as a product of a
"smooth" and a "discrete" part: the former dep...

The asymptotic expansion of $n$-dimensional cyclic integrals was expressed as
a series of functionals acting on the symmetric function involved in the cyclic
integral. In this article, we give an explicit formula for the action of these
functionals on a specific class of symmetric functions. These results are
necessary for the computation of the O(...

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 c...

We consider the problem of computing form factors of the massless XXZ Heisenberg spin-1/2 chain in a magnetic ﬁeld in the (thermodynamic) limit where the size M of the chain becomes large. For that purpose, we take the particular example of the matrix element of the third component of spin between the ground state and an excited state with one part...

We describe a method to derive, from first principles, the long-distance asymptotic behavior of correlation functions of integrable models in the framework of the algebraic Bethe ansatz. We apply this approach to the longitudinal spin- spin correlation function of the XXZ Heisenberg spin-1/2 chain (with magnetic field) in the disordered regime as w...

We derive compact multiple integral formulae for several physical spin correlation functions in the
semi-infinite XXZ
chain with a longitudinal boundary magnetic field. Our formulae follow from several
effective resummations of the multiple integral representation for the elementary blocks
obtained in our previous paper (I). In the free fermion po...

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 symbo...

We derive the asymptotic behavior of determinants of truncated Wiener-Hopf operators generated by symbols having Fisher-Hartwig singularities. This task is achieved thanks to an asymptotic resolution of the Riemann-Hilbert problem associated to some generalized sine kernel. As a byproduct, we give yet another derivation of the asymptotic behavior o...

This paper is devoted to the study of the emptiness formation probability
τ(m) of the open
XXZ chain. We derive
a closed form for τ(m)
at Δ = 1/2
when the boundary field vanishes. Moreover we obtain its leading asymptotics for an arbitrary
boundary field at the free fermion point. Finally, we compute the first term of the asymptotics of
ln(τ(m))...

This paper is devoted to the study of the emptiness formation probability $\tau\pa{m}$ of the open XXZ chain. We derive a closed form for $\tau\pa{m}$ at $\Delta=\tf{1}{2}$ when the boundary field vanishes. Moreover we obtain its leading asymptotics for an arbitrary boundary field at the free fermion point. Finally, we compute the first term of the...

We consider the XXZ spin chain with diagonal boundary conditions in the framework of algebraic Bethe Ansatz. Using the explicit computation of the scalar products of Bethe states and a revisited version of the bulk inverse problem, we calculate the elementary building blocks for the correlation functions. In the limit of half-infinite chain, they a...

We derive an analogue of the master equation, obtained recently for correlation functions of the
XXZ
chain, for a wide class of quantum integrable systems described by the
R-matrix of the six-vertex model, including in particular continuum models. This
generalized master equation allows us to obtain multiple integral representations for
the correl...