arXiv subjects

A. Klümper

Publications and source records attributed to A. Klümper.

At least 19 recordsLinked to original sources

arXiv2024

Thermodynamics based on Neural Networks

We present three different neural network algorithms to calculate thermodynamic properties as well as dynamic correlation functions at finite temperatures for quantum lattice models. The first method is based on purification, which allows for the exact calculation of the operator trace. The second one is based on a sampling of the trace using minimally entangled states, whereas the third one makes use of quantum typicality. In the latter case, we approximate a typical infinite-temperature state by wave functions which are given by a product of a projected pair and a neural network part and evolve this typical state in imaginary time.

cond-mat.stat-mech↗

arXiv2023

Canonical formulation for the thermodynamics of $sl_n$-invariant integrable spin chains

Integrable quantum spin chains display distinctive physical properties making them a laboratory to test and assess different states of matter. The study of the finite temperature properties is possible by use of the thermodynamic Bethe ansatz, however at the expense of dealing with non-linear integral equations for, in general, infinitely many auxiliary functions. The definition of an alternative finite set of auxiliary functions allowing for the complete description of their thermodynamic properties at finite temperature and fields has been elusive. Indeed, in the context of $sl_n$-invariant models satisfactory auxiliary functions have been established only for $n \leq 4$. In this paper we take a step further by proposing a systematic approach to generate finite sets of auxiliary functions for $sl_n$-invariant models. We refer to this construction as the canonical formulation. The numerical efficiency is illustrated for $n=5$, for which we present some of the thermodynamic properties of the corresponding spin chain.

math-ph↗

arXiv2022

On the partition function of the $Sp(4)$ integrable vertex model

In this paper we investigate certain fusion relations associated to an integrable vertex model on the square lattice which is invariant under $Sp(4)$ symmetry. We establish a set of functional relations which include a transfer matrix inversion identity. The solution of these relations in the thermodynamic limit allows us to compute the partition function per site of the fundamental $Sp(4)$ representation of the vertex model. As a byproduct we also obtain the partition function per site of a vertex model mixing the four and five dimensional representations of the $Sp(4)$ symmetry.

math-ph↗

arXiv2022

Non-linear Transport by Bethe Bound States

We consider non-linear ballistic spin transport in the XXZ spin chain and derive an analytical result for the non-linear Drude weight $D^{(3)}$ at infinite temperatures. In contrast to the linear Drude weight $D^{(1)}$, we find that the result not only depends on anisotropy but also on the string length of the quasiparticles transporting the spin current. Our result provides further insights into transport by quasiparticles and raises questions about Luttinger liquid universality.

cond-mat.str-el↗

arXiv2021

Reduced qKZ equation: general case

We use the quantum group approach for the investigation of correlation functions of integrable vertex models and spin chains. For the inhomogeneous reduced density matrix in case of an arbitrary simple Lie algebra we find functional equations of the form of the reduced quantum Knizhnik-Zamolodchikov equation. This equation is the starting point for the investigation of correlation functions at arbitrary temperature and notably for the ground state.

math-ph↗

arXiv2020

Critical behavior at the integer quantum Hall transition in a network model on the Kagome lattice

We study a network model on the Kagome lattice (NMKL). This model generalizes the Chalker-Coddington (CC) network model for the integer quantum Hall transition. Unlike random network models we studied earlier, the geometry of the Kagome lattice is regular. Therefore, we expect that the critical behavior of the NMKL should be the same as that of the CC model. We numerically compute the localization length index $ν$ in the NKML. Our result $ν= 2.658 \pm 0.046$ is close to CC model values obtained in a number of recent papers. We also map the NMKL to the Dirac fermions in random potentials and in a fixed periodic curvature background. The background turns out irrelevant at long scales. Our numerical and analytical results confirm our expectation of the universality of critical behavior on regular network models.

cond-mat.dis-nn↗

arXiv2019

Correlation functions of the integrable $SU(n)$ spin chain

We study the correlation functions of $SU(n)$ $n>2$ invariant spin chains in the thermodynamic limit. We formulate a consistent framework for the computation of short-range correlation functions via functional equations which hold even at finite temperature. We give the explicit solution for two- and three-site correlations for the $SU(3)$ case at zero temperature. The correlators do not seem to be of factorizable form. From the two-sites result we see that the correlation functions are given in terms of Hurwitz' zeta function, which differs from the $SU(2)$ case where the correlations are expressed in terms of Riemann's zeta function of odd arguments.

math-ph↗

arXiv2018

The spin Drude weight of the XXZ chain and generalized hydrodynamics

Based on a generalized free energy we derive exact thermodynamic Bethe ansatz formulas for the expectation value of the spin current, the spin current-charge, charge-charge correlators, and consequently the Drude weight. These formulas agree with recent conjectures within the generalized hydrodynamics formalism. They follow, however, directly from a proper treatment of the operator expression of the spin current. The result for the Drude weight is identical to the one obtained 20 years ago based on the Kohn formula and TBA. We numerically evaluate the Drude weight for anisotropies $Δ=\cos(γ)$ with $γ= nπ/m$, $n\leq m$ integer and coprime. We prove, furthermore, that the high-temperature asymptotics for general $γ=πn/m$---obtained by analysis of the quantum transfer matrix eigenvalues---agrees with the bound which has been obtained by the construction of quasi-local charges.

cond-mat.stat-mech↗

arXiv2017

Oscillator versus prefundamental representations II. Arbitrary higher ranks

We find the $\ell$-weights and the $\ell$-weight vectors for the highest $\ell$-weight $q$-oscillator representations of the positive Borel subalgebra of the quantum loop algebra $U_q(\mathcal L(\mathfrak{sl}_{l+1}))$ for arbitrary values of $l$. Having this, we establish the explicit relationship between the $q$-oscillator and prefundamental representations. Our consideration allows us to conclude that the prefundamental representations can be obtained by tensoring $q$-oscillator representations.

math-ph↗

arXiv2017

Anisotropic magnetic interactions and spin dynamics in the spin-chain compound Cu(py)$_2$Br$_2$: An experimental and theoretical study

We compare theoretical results for electron spin resonance (ESR) properties of the Heisenberg-Ising Hamiltonian with ESR experiments on the quasi-one-dimensional magnet Cu(py)$_2$Br$_2$ (CPB). Our measurements were performed over a wide frequency and temperature range giving insight into spin dynamics, spin structure, and magnetic anisotropy of this compound. By analyzing the angular dependence of ESR parameters (resonance shift and linewidth) at room temperature we show that the two weakly coupled inequivalent spin chain types inside the compound are well described by Heisenberg-Ising chains with their magnetic anisotropy axes perpendicular to the chain direction and almost perpendicular to each other. We further determine the full $g$-tensor from these data. In addition, the angular dependence of the linewidth at high temperatures gives us access to the exponent of the algebraic decay of a dynamical correlation function of the isotropic Heisenberg chain. From the temperature dependence of static susceptibilities we extract the strength of the exchange coupling ($J/k_B = 52.0\,\text{K}$) and the anisotropy parameter ($δ\approx -0.02$) of the model Hamiltonian. An independent compatible value of $δ$ is obtained by comparing the exact prediction for the resonance shift at low temperatures with high-frequency ESR data recorded at $4\,\text{K}$. The spin structure in the ordered state implied by the two (almost) perpendicular anisotropy axes is in accordance with the propagation vector determined from neutron scattering experiments. In addition to undoped samples we study the impact of partial substitution of Br by Cl ions on spin dynamics. From the dependence of the ESR linewidth on doping level we infer an effective decoupling of the anisotropic component $Jδ$ from the isotropic exchange $J$ in these systems.

cond-mat.str-el↗

arXiv2016

Correlation functions of the integrable spin-s chain

We study the correlation functions of su(2) invariant spin-s chains in the thermodynamic limit. We derive non-linear integral equations for an auxiliary correlation function $ω$ for any spin s and finite temperature T. For the spin-3/2 chain for arbitrary temperature and zero magnetic field we obtain algebraic expressions for the reduced density matrix of two-sites. In the zero temperature limit, the density matrix elements are evaluated analytically and appear to be given in terms of Riemann's zeta function values of even and odd arguments.

cond-mat.stat-mech↗

arXiv2015

Oscillator versus prefundamental representations

For the case of quantum loop algebras $\mathrm U_q(\mathcal L(\mathfrak{sl}_{l + 1}))$ with $l = 1, 2$ we find the $\ell$-weights and the corresponding $\ell$-weight vectors for the representations obtained via Jimbo's homomorphism, known also as evaluation representations. Then we find the $\ell$-weights and the $\ell$-weight vectors for the $q$-oscillator representations of Borel subalgebras of the same quantum loop algebras. This allows, in particular, to relate $q$-oscillator and prefundamental representations.

math-ph↗

arXiv2015

Universal R-matrix and functional relations

We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group $U_q(\mathcal L(\mathfrak{sl}_2))$ related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.

math-ph↗

arXiv2015

Quantum groups and functional relations for higher rank

A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_3))$ is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described.

math-ph↗

arXiv2014

Localization length index in a Chalker-Coddington model: a numerical study

We calculated numerically the localization length index $ν$ for the Chalker-Coddington model of the plateau-plateau transitions in the quantum Hall effect. By taking into account finite size effects we have obtained $ν= 2.593 \pm 0.0297$. The calculations were carried out by two different programs that produced close results, each one within the error bars of the other. We also checked the possibility of logarithmic corrections to finite size effects and found, that they come with much larger error bars for $ν$.

cond-mat.mes-hall↗

arXiv2012

Universal probes for antiferromagnetic correlations and entropy in cold fermions on optical lattices

We determine antiferromagnetic (AF) signatures in the half-filled Hubbard model at strong coupling on a cubic lattice and in lower dimensions. Upon cooling, the transition from the charge-excitation regime to the AF Heisenberg regime is signaled by a universal minimum of the double occupancy at entropy s=S/(N k_B)=s*=ln(2) per particle and a linear increase of the next-nearest neighbor (NNN) spin correlation function for s<s*. This crossover, driven by a gain in kinetic exchange energy, appears as the essential AF physics relevant for current cold-atom experiments. The onset of long-range AF order (at low s on cubic lattices) is hardly visible in nearest-neighbor spin correlations versus s, but could be detected in spin correlations at or beyond NNN distances.

cond-mat.quant-gas↗

arXiv2012

Universal integrability objects

We discuss the main points of the quantum group approach in the theory of quantum integrable systems and illustrate them for the case of the quantum group $U_q(\mathcal L(\mathfrak{sl}_2))$. We give a complete set of the functional relations correcting inexactitudes of the previous considerations. A special attention is given to the connection of the representations used to construct the universal transfer operators and $Q$-operators.

math-ph↗

arXiv2012

On the absorption of microwaves by the one-dimensional spin-1/2 Heisenberg-Ising magnet

We analyze the absorption of microwaves by the Heisenberg-Ising chain combining exact calculations, based on the integrability of the model, with numerical calculations. Within linear response theory the absorbed intensity is determined by the imaginary part of the dynamical susceptibility. The moments of the normalized intensity can be used to define the shift of the resonance frequency induced by the interactions and the line width independently of the shape of the spectral line. These moments can be calculated exactly as functions of temperature and strength of an external magnetic field, as long as the field is directed along the symmetry axis of the chain. This allows us to discuss the line width and the resonance shift for a given magnetic field in the full range of possible anisotropy parameters. For the interpretation of these data we need a qualitative knowledge of the line shape which we obtain from fully numerical calculations for finite chains. Exact analytical results on the line shape are out of reach of current theories. From our numerical work we could extract, however, an empirical parameter-free model of the line shape at high temperatures which is extremely accurate over a wide range of anisotropy parameters and is exact at the free fermion point and at the isotropic point. Another prediction of the line shape can be made in the zero-temperature and zero magnetic field limit, where the sufficiently anisotropic model shows strong absorption. For anisotropy parameters in the massive phase we derive the exact two-spinon contribution to the spectral line. From the intensity sum rule it can be estimated that this contribution accounts for more than 80% of the spectral weight if the anisotropy parameter is moderately above its value at the isotropic point.

cond-mat.str-el↗