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K. K. Kozlowski

Publications and source records attributed to K. K. Kozlowski.

At least 19 recordsLinked to original sources
arXiv2024

Space-like asymptotics of the thermal two-point functions of the XXZ spin-1/2 chain

This work proposes a closed formula for the leading term of the long-distance and large-time asymptotics in a cone of the space-like regime for the transverse dynamical two-point functions of the XXZ spin 1/2 chain at finite temperatures. The result follows from a simple analysis of the thermal form factor series for dynamical correlation functions. The leading asymptotics we obtain are driven by the Bethe Ansatz data associated with the first sub-leading Eigenvalue of the quantum transfer matrix.

math-ph↗
arXiv2022

Truncated Wiener-Hopf operators with Fisher Hartwig singularities

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 of Toeplitz determinants having Fisher-Hartwig singularities. The Riemann-Hilbert problem approach to these asymptotics yields a systematic although quickly cumbersome way to compute their sub-leading asymptotics.

math.FA↗
arXiv2021

Microscopic approach to a class of 1D quantum critical models

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 approach, in the large-distance critical regime, the local operators of the initial model are represented by well suited vertex operators associated to the free boson model. This provides an effective field theoretic description of the large distance behaviour of correlation functions in 1D quantum critical models. We develop this description starting from the first principles and directly at the microscopic level, namely in terms of the properties of the finite volume matrix elements of local operators.

math-ph↗
arXiv2019

On the thermodynamic limit of form factor expansions of dynamical correlation functions in the massless regime of the XXZ spin $1/2$ chain

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 appropriate way, the thermodynamic limit of the finite volume form factor expansions. The series are structured in way allowing one to identify directly the contributions to the correlator stemming from the conformal-type excitations on the Fermi surface and those issuing from the massive excitations (deep holes, particles and bound states). The obtained form factor series opens up the possibility of a systematic and exact study of asymptotic regimes of dynamical correlation functions in the massless regime of the XXZ spin $1/2$ chain. Furthermore, the assumptions on the microscopic structure of the model's Hilbert space that are necessary so as to write down the series appear to be compatible with any model -- not necessarily integrable -- belonging to the Luttinger liquid universality class. Thus, the present analysis provides also the phenomenological structure of form factor expansions in massless models belonging to this universality class.

math-ph↗
arXiv2019

Riemann-Hilbert approach to a generalised sine kernel

We derive the large distance asymptotics of the Fredholm determinant of the so-called generalised sine kernel at the critical point. This kernel corresponds to a generalisation of the pure sine kernel arising in the theory of random matrices and has potential applications to the analysis of the large-distance asymptotic behaviour of the so-called emptiness formation probability for various quantum integrable models away from their free fermion point.

math-ph↗
arXiv2018

On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain

This work extracts, by means of an exact analysis, the singular behaviour of the dynamical response functions -- the Fourier transforms of dynamical two-point functions -- in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes which describe the local behaviour of the response function near a threshold. The singular behaviour is derived starting from first principle considerations: the method of analysis \textit{does not rely, at any stage}, on some hypothetical correspondence with a field theory or other phenomenological approaches. The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author. It confirms the non-linear Luttinger based predictions relative to the power-law behaviour and of the associated edge exponents which arise in the vicinity of the dispersion relation of one massive excitation (hole, particle or bound state). In addition, the present analysis shows that, due to the lack of strict convexity of the particles dispersion relation and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These origin from multi-particle/hole/bound state excitations maximising the energy at fixed momentum.

math-ph↗
arXiv2018

Solution of Baxter equation for the $q$-Toda and Toda$_2$ chains by NLIE

We construct a basis of solutions of the scalar $\boldsymbol{ \texttt{t} }- \boldsymbol{ \texttt{Q} }$ equation describing the spectrum of the $q$-Toda and Toda$_2$ chains by using auxiliary non-linear integral equations. Our construction allows us to provide quantisation conditions for the spectra of these models in the form of thermodynamic Bethe Ansatz-like equations.

math-ph↗
arXiv2018

Baxter operator and Baxter equation for $q$-Toda and Toda$_2$ chains

We construct the Baxter operator $\boldsymbol{ \texttt{Q} }(λ)$ for the $q$-Toda chain and the Toda$_2$ chain (the Toda chain in the second Hamiltonian structure). Our construction builds on the relation between the Baxter operator and Bäcklund transformations that were unravelled in {\cite{GaPa92}}. We construct a number of quantum intertwiners ensuring the commutativity of $\boldsymbol{ \texttt{Q} }(λ)$ with the transfer matrix of the models and the one of $\boldsymbol{ \texttt{Q} }$'s between each other. Most importantly, $\boldsymbol{ \texttt{Q} }(λ)$ is modular invariant in the sense of Faddeev. We derive the Baxter equation for the eigenvalues $q(λ)$ of $\boldsymbol{ \texttt{Q} }(λ)$ and show that these are entire functions of $λ$. This last property will ultimately lead to the quantisation of the spectrum for the considered Toda chains, in a subsequent publication.

math-ph↗
arXiv2018

The Toda$_2$ chain

We show that a natural discretisation of Virasoro algebra yields a quantum integrable model which is the Toda chain in the second Hamiltonian structure.

math-ph↗
arXiv2017

On condensation properties of Bethe roots associated with the XXZ chain

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 anisotropy $Δ\geq -1 $. In fact, I establish an even stronger result, namely the existence of an all order asymptotic expansion of the counting function associated with such roots. As a corollary, these results allow one to prove the existence and form of the infinite volume limit of various observables attached to the model -the excitation energy, momentum, the zero temperature correlation functions, so as to name a few- that were argued earlier in the literature.

math-ph↗
arXiv2016

Asymptotic expansion of a partition function related to the sinh-model

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 $1/N^α$ and $1/N$ naturally occur. In our case, the equilibrium measure is $N^α$-dependent and characterised by means of the solution to a $2\times 2$ Riemann--Hilbert problem, whose large-$N$ behavior is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-$N$ behavior of the free energy explicitly up to $o(1)$. The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales $1/N^α$ and $1/N$, thus waiving the analyticity assumptions often used in random matrix theory.

math-ph↗
arXiv2016

Asymptotic behaviour of two-point functions in multi-species models

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 technical hypothesis on the large-volume behaviour of the form factors of local operators in such models. We check the validity of this hypothesis on the example of the $SU(3)$-invariant XXX magnet by means of the determinant representations for the form factors of local operators in this model. Our approach confirms the structure of the critical exponents obtained previously for numerous models solvable by the nested Bethe Ansatz.

nlin.SI↗
arXiv2015

On form-factor expansions for the XXZ chain in the massive regime

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 derive the form-factor expansion of two-point spin-spin correlation functions in the thermodynamic limit $L\rightarrow +\infty$. The staggered magnetisation appears naturally as the first term in this expansion. We show that all other contributions to the two-point correlation function are exponentially small in the large-distance regime.

cond-mat.stat-mech↗
arXiv2015

Unitarity of the SoV transform for the Toda chain

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 technique which allows one to prove the unitarity of this map in the case of the quantum Toda chain. Our proof solely builds on objects and relations naturally arising in the framework of the so-called quantum inverse scattering method. Hence, with minor modifications, it should be readily transposable to other quantum integrable models solvable by the quantum separation of variables method. As such, it provides an important alternative to the proof of the map's unitarity based on the group theoretical interpretation of the quantum Toda chain, which is absent for more complex quantum integrable models.

math-ph↗
arXiv2015

Aspects of the inverse problem for the Toda chain

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 schemes discussed in this paper appear to be universal and thus, in principle, applicable to many models solvable through the quantum separation of variables.

nlin.SI↗
arXiv2015

Low-$T$ asymptotic expansion of the solution to the Yang-Yang equation

We prove that the unique solution to the Yang-Yang equation arising in the context of the thermodynamics of the so-called non-linear Schrödinger 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 that were argued previously in various works related to the low-temperature behavior of integrable models.

math-ph↗