arXiv subjects

Zoran Ristivojevic

Publications and source records attributed to Zoran Ristivojevic.

At least 19 recordsLinked to original sources

arXiv2026

Bethe-ansatz study of the Bose-Fermi mixture

We consider a one-dimensional mixture of bosons and spinless fermions with contact interactions. In this system, the elementary excitations at low energies are described by four linearly dispersing modes characterized by two excitation velocities. Here we study the velocities in a system with equal interaction strengths and equal masses of bosons and fermions. The resulting model is integrable and admits an exact Bethe-ansatz solution. We analyze it and analytically derive various exact results, which include the Drude weight matrix. We show that the excitation velocities can be calculated from the knowledge of the matrices of compressibility and the Drude weights, as their squares are the eigenvalues of the product of the two matrices. The elements of the Drude weight matrix obey certain sum rules as a consequence of Galilean invariance. Our results are consistent with the presence of a momentum-momentum coupling term between the two subsystems of bosons and fermions in the effective low-energy Hamiltonian. The analytical method developed in the present study can be extended to other models that possess a nested Bethe-ansatz structure.

cond-mat.quant-gas↗

arXiv2026

The Lieb-Liniger model

The Lieb-Liniger model describes one-dimensional bosons with contact interactions. This many-body system admits an exact solution in terms of the Bethe ansatz. Some of the exact and perturbative results for this model are reviewed. Particular attention is devoted to the explicit evaluation, in terms of the interaction parameter, of physical quantities that can be formally exactly extracted from the Bethe ansatz solution. Another goal of this review is to stress exact relations between various quantities. The technical developments are explained in detail. The most relevant experimental realisations of the studied problems are eventually discussed. This review also contains several new results such as the study of convergence of the ground-state energy series at strong interactions, the excitation spectrum at high energies, and the evaluation of the boundary energy.

cond-mat.quant-gas↗

arXiv2025

The exchange coupling of a Wigner dimer

We study the exchange coupling in small Wigner crystals confined to one-dimensional space. In particular we concentrate on the simplest nontrivial case of two electrons in a box potential and calculate analytically the energy splitting between the lowest spatially symmetric and antisymmetric states, which is a relevant energy scale for the magnetic properties of the system. In the approximation of a fixed center of mass coordinate, the splitting decays exponentially with the square root of the distance between the electrons at the leading order. We show that the subleading exponential correction significantly increases the splitting and thus becomes crucial in order to describe correctly the exact numerical data for system sizes that are not astronomically large. Two methods of calculation of the energy splitting are developed. The first is based on the analysis of the exact solution that can be expressed in terms of the Whittaker functions. It applies at all values of the short-distance cutoff played by the width of one-dimensional wire that regularizes the Coulomb potential. The second method is based on the quasiclassical (or Wentzel-Kramers-Brillouin) approximation, which applies only for sufficiently large values of the cutoff. The two methods give identical result in the overlapping region. As a side result, our study gives the energy splitting in a triangular double well potential of inverse ``M'' shape.

cond-mat.str-el↗

arXiv2025

Exactly solvable models for universal operator growth

Quantum observables of generic many-body systems exhibit a universal pattern of growth in the Krylov space of operators. This pattern becomes particularly manifest in the Lanczos basis, where the evolution superoperator assumes the tridiagonal form. According to the universal operator growth hypothesis, the nonzero elements of the superoperator, known as Lanczos coefficients, grow asymptotically linearly. We introduce and explore broad families of Lanczos coefficients that are consistent with the universal operator growth and lead to the exactly solvable dynamics. Within these families, the subleading terms of asymptotic expansion of the Lanczos sequence can be controlled and fine-tuned to produce diverse dynamical patterns. For one of the families, the Krylov complexity is computed exactly.

quant-ph↗

arXiv2025

Local Properties of the Rapidity Distribution in the Lieb-Liniger Model

We study the rapidity distribution in the Lieb-Liniger model and derive exact relations for its derivatives at the Fermi level. The latter enables us to treat analytically the free energy of the system at low temperatures and arbitrary interactions. We calculated the leading-order correction to the well-known result obtained using conformal field theory. In contrast to the leading-order term controlled by the sound velocity or the Luttinger liquid parameter, the new term is controlled by an additional dimensionless parameter. We calculated its series expansions in the limiting cases of weak and strong interactions. Our results are generalized to other Galilean-invariant integrable systems.

cond-mat.stat-mech↗

arXiv2023

The density of a one-dimensional Bose gas far from an impurity

We consider an impurity in a one-dimensional weakly-interacting Bose gas and analytically calculate the density profile of the Bose gas. Within the mean-field approximation, by increasing the distance from the impurity, the Bose gas density saturates exponentially fast to its mean thermodynamic-limit value at distances beyond the healing length. The effect of quantum fluctuations drastically changes this behavior, leading to a power law decay of the density deviation from the mean density. At distances longer than the healing length and shorter than a new length scale proportional to the impurity coupling strength, the power-law exponent is $2$, while at longest distances the corresponding exponent becomes $3$. The latter crossover does not exist in two special cases. The first one is realized for infinitely strongly coupled impurity; then the density deviation always decays with the exponent $2$. The second special case occurs when the new length scale is smaller than the healing length, i.e., at weak impurity coupling; then the density deviation always decays with the exponent $3$. The obtained results are exact in the impurity coupling strength and account for the leading order in the interaction between the particles of the Bose gas.

cond-mat.quant-gas↗

arXiv2023

Dissipative dynamics of a heavy impurity in a Bose gas in the strong coupling regime

We study the motion of a heavy impurity in a one-dimensional Bose gas. The impurity experiences the friction force due to scattering off thermally excited quasiparticles. We present detailed analysis of an arbitrarily strong impurity-boson coupling in a wide range of temperatures within a microscopic theory. Focusing mostly on weakly interacting bosons, we derive an analytical result for the friction force and uncover new regimes of the impurity dynamics. Particularly interesting is the low-temperature $T^2$ dependence of the friction force obtained for a strongly coupled impurity, which should be contrasted with the expected $T^4$ scaling. This new regime applies to systems of bosons with an arbitrary repulsion strength. We finally study the evolution of the impurity with a given initial momentum. We evaluate analytically its non-stationary momentum distribution function. The impurity relaxation towards the equilibrium is a realization of the Ornstein-Uhlenbeck process in momentum space.

cond-mat.quant-gas↗

arXiv2023

Exact Results for the Moments of the Rapidity Distribution in Galilean-Invariant Integrable Models

We study a class of Galilean-invariant one-dimensional Bethe ansatz solvable models in the thermodynamic limit. Their rapidity distribution obeys an integral equation with a difference kernel over a finite interval, which does not admit a closed-form solution. We develop a general formalism enabling one to study the moments of the rapidity distribution, showing that they satisfy a difference-differential equation. The derived equation is explicitly analyzed in the case of the Lieb-Liniger model and the moments are analytically calculated. In addition, we obtained the exact information about the ground-state energy at weak repulsion. The obtained results directly enter a number of physically relevant quantities.

cond-mat.stat-mech↗

arXiv2022

Method of difference-differential equations for some Bethe ansatz solvable models

In studies of one-dimensional Bethe ansatz solvable models, a Fredholm integral equation of the second kind with a difference kernel on a finite interval often appears. This equation does not generally admit a closed-form solution and hence its analysis is quite complicated. Here we study a family of such equations concentrating on their moments. We find exact relations between the moments in the form of difference-differential equations. The latter results significantly advance the analysis, enabling one to practically determine all the moments from the explicit knowledge of the lowest one. As applications, first we study the moments of the quasimomentum distribution in the Lieb-Liniger model and find explicit analytical results. The latter moments determine several basic quantities, e.g., the $N$-body local correlation functions. We prove the equivalence between different expressions found in the literature for the three-body local correlation functions and find an exact result for the four-body local correlation function in terms of the moments of the quasimomentum distributions. We eventually find the analytical results for the three- and four-body correlation functions in the form of asymptotic series in the regimes of weak and strong interactions. Next, we study the exact form of the low-energy spectrum of a magnon (a polaron) excitation in the two-component Bose gas described by the Yang-Gaudin model. We find its explicit form, which depends on the moments of the quasimomentum distributions of the Lieb-Liniger model. Then, we address a seemingly unrelated problem of capacitance of a circular capacitor and express the exact result for the capacitance in the parametric form. In the most interesting case of short plate separations, the parametric form has a single logarithmic term. This should be contrasted with the explicit result that has a complicated structure of logarithms.

cond-mat.quant-gas↗

arXiv2022

Dispersion relation of a polaron in the Yang-Gaudin Bose gas

We study a one-dimensional Bose gas with two internal states described by the Yang-Gaudin model and calculate analytically the dispersion relation of a polaron quasiparticle, which is the lowest excitation branch. We study the dispersion in the thermodynamic limit in the regimes of weak and strong interaction without limitations on the momentum. At weak interaction the polaron dispersion is in the vicinity of the dark soliton one; we calculate the leading deviation in the parametric form. At strong interaction we find an ansatz for the explicit form of the polaron dispersion. It has the form of a power series of the sine function of the momentum with interaction-dependent coefficients. By increasing the power of the series, the corresponding coefficients show faster decay and thus one practically needs only a few of them; we give the results for the first three. The coefficients of the series are connected to the Maclaurin series of the polaron dispersion and thus it is sufficient to calculate latter quantity to automatically find the power series result for the dispersion at all momenta. The derived results for the dispersion can be used to obtain explicit expressions for the exponents of the power-law singularities in the response functions at the spectral edge.

cond-mat.quant-gas↗

arXiv2021

Exact result for the polaron mass in a one-dimensional Bose gas

We study the polaron quasiparticle in a one-dimensional Bose gas. In the integrable case described by the Yang-Gaudin model, we derive an exact result for the polaron mass in the thermodynamic limit. It is expressed in terms of the derivative with respect to the density of the ground-state energy per particle of the Bose gas without the polaron. This enables us to find high-order power series for the polaron mass in the regimes of weak and strong interaction.

cond-mat.quant-gas↗

arXiv2021

Mediated interaction between polarons in a one-dimensional Bose gas

We study a weakly-interacting one-dimensional Bose gas with two impurities coupled locally to the boson density. We derive analytical results for the induced interaction between the impurities at arbitrary coupling and separation $r$. At $r\lesssim ξ$, where $ξ$ denotes the healing length of the Bose gas, the interaction is well described by the mean-field contribution. Its form changes as the coupling is increased, approaching a linear function of $r$ at short distances in the regime of strong coupling. The mean-field contribution decays exponentially at arbitrary coupling for $r\ggξ$. At such long distances, however, the effect of quantum fluctuations becomes important, giving rise to a long-ranged quantum contribution to the induced interaction. At longest distances it behaves as $1/r^3$, while at strong coupling we find an intermediate distance regime with a slower decay, $1/r$. The quantum contribution in the crossover regime is also calculated. The induced interaction between impurities (i.e., polarons) is attractive and leads to the formation of their bound state, known as bipolaron. We discuss its binding energy.

cond-mat.quant-gas↗

arXiv2021

Quasiparticle energy relaxation in a gas of one-dimensional fermions with Coulomb interaction

We consider a system of charged one-dimensional spin-$\frac{1}{2}$ fermions at low temperature. We study how the energy of a highly-excited quasiparticle (or hole) relaxes toward the chemical potential in the regime of weak interactions. The dominant relaxation processes involve collisions with two other fermions. We find a dramatic enhancement of the relaxation rate at low energies, with the rate scaling as the inverse sixth power of the excitation energy. This behavior is caused by the long-range nature of the Coulomb interaction.

cond-mat.str-el↗

arXiv2020

Density profile of a semi-infinite one-dimensional Bose gas and bound states of the impurity

We study the effect of the boundary on a system of weakly interacting bosons in one dimension. It strongly influences the boson density which is completely suppressed at the boundary position. Away from it, the density is depleted over the distances on the order of the healing length at the mean-field level. Quantum fluctuations modify the density profile considerably. The local density approaches the average one as an inverse square of the distance from the boundary. We calculate an analytic expression for the density profile at arbitrary separations from the boundary. We then consider the problem of localization of a foreign quantum particle (impurity) in the potential created by the inhomogeneous boson density. At the mean-field level, we find exact results for the energy spectrum of the bound states, the corresponding wave functions, and the condition for interaction-induced localization. The quantum contribution to the boson density gives rise to small corrections of the bound state energy levels. However, it is fundamentally important for the existence of a long-range Casimir-like interaction between the impurity and the boundary.

cond-mat.quant-gas↗

arXiv2020

Relaxation of the degenerate one-dimensional Fermi gas

We study how a system of one-dimensional spin-1/2 fermions at temperatures well below the Fermi energy approaches thermal equilibrium. The interactions between fermions are assumed to be weak and are accounted for within the perturbation theory. In the absence of an external magnetic field, spin degeneracy strongly affects relaxation of the Fermi gas. For sufficiently short-range interactions, the rate of relaxation scales linearly with temperature. Focusing on the case of the system near equilibrium, we linearize the collision integral and find exact solution of the resulting relaxation problem. We discuss the application of our results to the evaluation of the transport coefficients of the one-dimensional Fermi gas.

cond-mat.mes-hall↗

arXiv2020

Damping of elementary excitations in one-dimensional dipolar Bose gases

In the presence of dipolar interactions the excitation spectrum of a Bose gas can acquire a local minimum. The corresponding quasiparticles are known as rotons. They are gaped and do not decay at zero temperature. Here we study the decay of rotons in one-dimensional Bose gases at low temperatures. It predominantly occurs due to the backscattering of thermal phonons on rotons. The resulting rate scales with the third power of temperature and is inversely proportional to the sixth power of the roton gap near the solidification phase transition. The hydrodynamic approach used here enables us to find the decay rate for quasiparticles at practically any momenta, with minimal assumptions on the exact form of the interparticle interactions. Our results are an essential prerequisite for the description of all the dissipative phenomena in dipolar gases and have direct experimental relevance.

cond-mat.quant-gas↗

arXiv2020

Analytical results for the capacitance of a circular plate capacitor

We study the classic problem of the capacitance of a circular parallel plate capacitor. At small separations between the plates, it is initially considered in 19th century by Kirchhoff who found the leading and the subleading term in the capacitance. Despite a large interest in the problem, one and a half century later, analytically was found only the second subleading term. Using the recent advances in the asymptotic analysis of Fredholm integral equations of the second kind with finite support, here we study the one governing the circular capacitor, known as the Love equation. We found analytically many new subleading terms in the capacitance at small separations. We also calculated the asymptotic expansion at large separations, thus providing the two simple expressions which practically describe the capacitance at all distances. The approach described here could be used to find exact analytical expansions for the capacitance to an arbitrary number of terms in both regimes of small and large separations.

math-ph↗

arXiv2019

Fluctuation-induced potential for an impurity in a semi-infinite one-dimensional Bose gas

We consider an impurity in a semi-infinite one-dimensional system of weakly-interacting bosons. We calculate the interaction potential for the impurity due to the end of the system, i.e., the wall. For local repulsive (attractive) interaction between the impurity and the Bose gas, the interaction potential is attractive (repulsive). At short distances from the wall it decays exponentially crossing over into a universal $1/r^2$ behavior at separations $r$ above the healing length. Our results can also be interpreted as a Casimir-like interaction between two impurities, where one of them is infinitely strongly coupled to the Bose gas. We discuss various scenarios for the induced interaction between the impurities using the scattering approach. We finally address the phenomenon of localization of the impurity near the wall. In the paper we mainly study the case of a static impurity, however the universal $1/r^2$ interaction also holds for a slowly moving impurity.

cond-mat.quant-gas↗