arXiv subjects

Andreas Klümper

Publications and source records attributed to Andreas Klümper.

At least 19 recordsLinked to original sources

arXiv2026

Spin Quantum Hall Effect: the Critical Exponents

The spin quantum Hall effect (SQHE) provides one of the few examples of an Anderson localization transition for which exact critical exponents are known, making it an important testing ground for theories of disordered topological systems and conformal field theory. The corresponding network model, obtained by replacing the random $U(1)$ phases of the Chalker--Coddington model with random $SU(2)$ matrices, belongs to symmetry class C of the Altland--Zirnbauer classification and is believed to describe quasiparticle transport in two-dimensional disordered superconductors with broken time-reversal symmetry. In this work, we perform high-precision numerical calculations of the localization-length exponent ($ν$) and the boundary critical exponent ($μ$) for the SQHE network model using the recently developed $S$-matrix approach to random networks.

cond-mat.dis-nn↗

arXiv2026

The integer quantum Hall transition: an $S$-matrix approach to random networks

In this paper we propose a new $S$-matrix approach to numerical simulations of network models and apply it to random networks that we proposed in a previous work 10.1103/PhysRevB.95.125414. Random networks are modifications of the Chalker-Coddington (CC) model for the integer quantum Hall transition that more faithfully capture the physics of electrons moving in a strong magnetic field and a smooth disorder potential. The new method has considerable advantages compared to the transfer matrix approach, and gives the value $ν\approx 2.4$ for the critical exponent of the localization length in a random network. This finding confirms our previous result and is surprisingly close to the experimental value $ν_{\text{exp}} \approx 2.38$ observed at the integer quantum Hall transition but substantially different from the CC value $ν_\text{CC} \approx 2.6$.

cond-mat.dis-nn↗

arXiv2025

Ballistic particle transport and Drude weight in gases

Owing to the fact that the particle current operator in non-relativistic gases is proportional to the total momentum operator, the particle transport in such systems is always ballistic and fully characterized by a Drude weight $Δ$. The Drude weight can be calculated within linear response theory. It is given by the formula $Δ= 2 πD$, where $D$ is the density of the gas. This holds in any dimension and for every equilibrium ensemble, in particular for generalized Gibbs ensembles that describe possible equilibrium states of isolated integrable quantum systems. In the canonical ensemble case, the Drude weight can be equivalently obtained from a generalized susceptibility related to the fluctuations of the conserved particle current. Such susceptibility can be rigorously calculated for the integrable Lieb-Liniger Bose gas in any generalized Gibbs ensemble using a generalized Yang-Yang thermodynamic formalism. The resulting expression agrees with a prediction made within the context of generalized hydrodynamics. It also allows us to see explicitly that, within truly generalized Gibbs ensembles, the conductivity related with the particle current is not determined by the corresponding current-current auto-correlation function.

cond-mat.quant-gas↗

arXiv2025

Chiral eigenbases of the XX and XY quantum spin chains

We calculate the values of observables in chiral eigenstates of the XX quantum spin chain that were introduced in previous work and compare the form of the result with the respective expressions obtained in the more familiar eigenbasis of states with fixed magnetization in the $z$ direction. We carry out the diagonalization of the XY spin chain in the chiral basis. We calculate the norm of the chiral XY eigenstates, and the values of the one-point functions and some neighbor two-point correlation functions. We interpret the spectrum and the particle content of the XY chain in terms of scattering states of an even number of kink and antikink excitations that are created over a reduced Brillouin zone.

cond-mat.stat-mech↗

arXiv2025

Harris-Luck criterion in the plateau transition of the Integer Quantum Hall Effect

The Harris criterion imposes a constraint on the critical behavior of a system upon introduction of new disorder, based on its dimension $d$ and localization length exponent $ν$. It states that the new disorder can be relevant only if $d ν< 2$. We analyze the applicability of the Harris criterion to the GKNS network disorder formulated in the paper [I. A. Gruzberg, A. Klümper, W. Nuding, and A. Sedrakyan, Phys. Rev. B 95, 125414 (2017)] and show that the fluctuations of the geometry are relevant despite $d ν> 2$, implying that Harris criterion should be modified. We have observed that the fluctuations of the critical point in different quenched configurations of disordered network blocks is of order $L^0$, i.e.~it does not depend on block size $L$ in contrast to the expectation based on the Harris criterion that they should decrease as $L^{-d/2}$ according to the central limit theorem. Since $L^0 > (x-x_c)$ is always satisfied near the critical point, the mentioned network disorder is relevant and the critical indices of the system can be changed. We have also shown that the GKNS disordered network is fundamentally different from Voronoi-Delaunay and dynamically triangulated random lattices: the probability of higher connectivity in the GKNS network decreases in a power law as opposed to an exponential, indicating that we are dealing with a ``scale free" network, such as the Internet, protein-protein interactions, etc.

cond-mat.dis-nn↗

arXiv2025

Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields

The XXX spin-$\frac{1}{2}$ Heisenberg chain with non-diagonal boundary fields represents a cornerstone model in the study of integrable systems with open boundaries. Despite its significance, solving this model exactly has remained a formidable challenge due to the breaking of $U(1)$ symmetry. Building on the off-diagonal Bethe Ansatz (ODBA), we derive a set of nonlinear integral equations (NLIEs) that encapsulate the exact spectrum of the model. For $U(1)$ symmetric spin-$\frac{1}{2}$ chains such NLIEs involve two functions $a(x)$ and $\bar{a}(x)$ coupled by an integration kernel with short-ranged elements. The solution functions show characteristic features for arguments at some length scale which grows logarithmically with system size $N$. For the non $U(1)$ symmetric case, the equations involve a novel third function $c(x)$, which captures the inhomogeneous contributions of the $T$-$Q$ relation. The kernel elements coupling this function to the standard ones are long-ranged and lead for the ground-state to a winding phenomenon. In $\log(1+a(x))$ and $\log(1+\bar a(x))$ we observe a sudden change by $2π$i at a characteristic scale $x_1$ of the argument. Other features appear at a value $x_0$ which is of order $\log N$. These two length scales, $x_1$ and $x_0$, are independent: their ratio $x_1/x_0$ is large for small $N$ and small for large $N$. Explicit solutions to the NLIEs are obtained numerically for these limiting cases, though intermediate cases ($x_1/x_0 \sim 1$) present computational challenges. This work lays the foundation for studying finite-size corrections and conformal properties of other integrable spin chains with non-diagonal boundaries, opening new avenues for exploring boundary effects in quantum integrable systems.

cond-mat.str-el↗

arXiv2025

Modular covariant torus partition functions of dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models

Yang-Baxter integrable dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models are considered on the torus in their simplest physical regimes. A combination of boundary conditions $(h,v)$ is applied in the horizontal and vertical directions with $h,v=0$ and $1$ for periodic and antiperiodic boundary conditions respectively. The fugacities of non-contractible and contractible loops are denoted by $α$ and $β$ respectively where $β$ is simply related to the crossing parameter $λ$. At roots of unity, when $λ/π\in\mathbb Q$, these models are the dense ${\cal LM}(p,p')$ and dilute ${\cal DLM}(p,p')$ logarithmic minimal models with $p,p'$ coprime integers. We conjecture the scaling limits of the transfer matrix traces in the standard modules with $d$ defects and deduce the conformal partition functions ${\cal Z}_{\textrm{dense}}^{(h,v)}(α)$ and ${\cal Z}_{\textrm{dilute}}^{(h,v)}(α)$ using Markov traces. These are expressed in terms of functions ${\cal Z}_{m,m'}(g)$ known from the Coulomb gas arguments of Di Francesco, Saleur and Zuber and subsequently as sesquilinear forms in Verma characters. Crucially, we find that the partition functions are identical for the dense and dilute models. The coincidence of these conformal partition functions provides compelling evidence that, for given $(p,p')$, these dense and dilute theories lie in the same universality class. In root of unity cases with $α=2$, the $(h,v)$ modular covariant partition functions are also expressed as sesquilinear forms in affine $u(1)$ characters involving generalized Bezout conjugates. These also give the modular covariant partition functions for the 6-vertex and Izergin-Korepin 19-vertex models in the corresponding regimes.

math-ph↗

arXiv2024

Managing Singular Kernels and Logarithmic Corrections in the Staggered Six-Vertex Model

In this paper, we investigate the spectral properties of the staggered six-vertex model with ${\cal Z}_2$ symmetry for arbitrary system sizes $L$ using non-linear integral equations (NLIEs). Our study is motivated by two key questions: what is the accuracy of results based on the ODE/IQFT correspondence in the asymptotic regime of large system sizes, and what is the optimal approach based on NLIE for analyzing the staggered six-vertex model? We demonstrate that the quantization conditions for low-lying primary and descendant states, derived from the ODE/IQFT approach in the scaling limit, are impressively accurate even for relatively small system sizes. Specifically, in the anisotropy parameter range $π/4 < γ< π/2$, the difference between NLIE and ODE/IQFT results for energy and quasi-momentum eigenvalues is of order $\mathcal{O}(L^{-2})$. Furthermore, we present a unifying framework for NLIEs, distinguishing between versions with singular and regular kernels. We provide a compact derivation of NLIE with a singular kernel, followed by an equivalent set with a regular kernel. We address the stability issues in numerical treatments and offer solutions to achieve high-accuracy results, validating our approach for system sizes ranging from $L=2$ to $L=10^{24}$. Our findings not only validate the ODE/IQFT approach for finite system sizes but also enhance the understanding of NLIEs in the context of the staggered six-vertex model. We hope the insights gained from this study have significant implications for resolving the spectral problem of other lattice systems with emergent non-compact degrees of freedom and provide a foundation for future research in this domain.

cond-mat.stat-mech↗

arXiv2024

Chiral basis for qubits and spin-helix decay

We propose a qubit basis composed of transverse spin helices with kinks. Unlike the usual computational basis, this chiral basis is well suited for describing quantum states with nontrivial topology. Choosing appropriate parameters the operators of the transverse spin components, $σ_n^x$ and $σ_n^y$, become diagonal in the chiral basis, which facilitates the study of problems focused on transverse spin components. As an application, we study the temporal decay of the transverse polarization of a spin helix in the XX model that has been measured in recent cold atom experiments. We obtain an explicit universal function describing the relaxation of helices of arbitrary wavelength.

quant-ph↗

arXiv2024

A Pedestrian's Way to Baxter's Bethe Ansatz for the Periodic XYZ Chain

A chiral coordinate Bethe ansatz method is developed to study the periodic XYZ chain. We construct a set of chiral vectors with fixed number of kinks. All vectors are factorized and have simple structures. Under roots of unity conditions, the Hilbert space has an invariant subspace and our vectors form a basis of this subspace. We propose a Bethe ansatz solely based on the action of the Hamiltonian on the chiral vectors, avoiding the use of transfer matrix techniques. This allows to parameterize the expansion coefficients and derive the homogeneous Bethe ansatz equations whose solutions give the exact energies and eigenstates. Our analytic results agree with earlier approaches, notably by Baxter, and are supported by numerical calculations.

cond-mat.stat-mech↗

arXiv2023

Critical site percolation on the triangular lattice: From integrability to conformal partition functions

Critical site percolation on the triangular lattice is described by the Yang-Baxter solvable dilute $A_2^{(2)}$ loop model with crossing parameter specialized to $λ=\frac\pi3$, corresponding to the contractible loop fugacity $β=-2\cos4λ=1$. We study the functional relations satisfied by the commuting transfer matrices of this model and the associated Bethe ansatz equations. The single and double row transfer matrices are respectively endowed with strip and periodic boundary conditions, and are elements of the ordinary and periodic dilute Temperley-Lieb algebras. The standard modules for these algebras are labeled by the number of defects $d$ and, in the latter case, also by the twist $e^{iγ}$. Nonlinear integral equation techniques are used to analytically solve the Bethe ansatz functional equations in the scaling limit for the central charge $c=0$ and conformal weights $Δ,\barΔ$. For the groundstates, we find $Δ=Δ_{1,d+1}$ for strip boundary conditions and $(Δ,\barΔ)=(Δ_{γ/π,d/2},Δ_{γ/π,-d/2})$ for periodic boundary conditions, where $Δ_{r,s}=\frac1{24}((3r-2s)^2-1)$. We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For $d\le8$, these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading $20$ or more conformal eigenenergies. With these conjectures, we apply the Markov traces to obtain the conformal partition functions on the cylinder and torus. These precisely coincide with our previous results for critical bond percolation on the square lattice described by the dense $A_1^{(1)}$ loop model with $λ=\frac\pi3$. The concurrence of all this conformal data provides compelling evidence supporting a strong form of universality between these two stochastic models as logarithmic CFTs.

math-ph↗

arXiv2022

Invariant subspaces and explicit Bethe vectors in the integrable open spin $1/2$ $\XYZ$ chain

We derive a criterion under which splitting of all eigenstates of an open $\XYZ$ Hamiltonian with boundary fields into two invariant subspaces, spanned by chiral shock states, occurs. The splitting is governed by an integer number, which has the geometrical meaning of the maximal number of kinks in the basis states. We describe the generic structure of the respective Bethe vectors. We obtain explicit expressions for Bethe vectors, in the absence of Bethe roots, and those generated by one Bethe root, and investigate the \multiplet. We also describe in detail an elliptic analogue of the spin-helix state, appearing in both the periodic and the open $\XYZ$ model, and derive the eigenstate condition. The elliptic analogue of the spin-helix state is characterized by a quasi-periodic modulation of the magnetization profile, governed by Jacobi elliptic functions.

cond-mat.stat-mech↗

arXiv2021

Short-distance thermal correlations in the XXZ chain

Recent studies have revealed much of the mathematical structure of the static correlation functions of the XXZ chain. Here we use the results of those studies in order to work out explicit examples of short-distance correlation functions in the infinite chain. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field for various anisotropies in the massless regime $- 1 < Δ< 1$. It turns out that the new formulae are numerically efficient and allow us to obtain the correlations functions over the full parameter range with arbitrary precision.

cond-mat.str-el↗

arXiv2021

Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains

We demonstrate the existence of special phantom excitations for open and periodically closed integrable systems at the example of the $XXZ$ Heisenberg spin chain. The phantom excitations do not contribute to the energy of the Bethe state and correspond to special solutions to the Bethe Ansatz equations with infinite "phantom" Bethe roots. The phantom Bethe roots lead to degeneracies between different magnetization sectors in the periodic case and to the appearance of spin helix states (SHS), i.e. periodically modulated states of chiral nature in both open and closed systems. For the periodic chain, phantom Bethe root (PBR) solutions appear for anisotropies $\De=\coshη$ with $\exp(η)$ being a root of unity, thus restricting the phenomenon to the critical region $|\De|<1$. For the open chain, PBR solutions appear for any value of anisotropy, both in the critical and in the non-critical region, provided that the boundary fields satisfy a criterion which we derive in this paper. There exist PBR solutions with all Bethe roots being phantom, and PBR solutions that consist of phantom roots as well as regular (finite) roots. Implications of our results for an experiment are discussed.

cond-mat.stat-mech↗

arXiv2021

Analytical results for the low-temperature Drude weight of the XXZ spin chain

The spin-$1/2$ XXZ chain is an integrable lattice model and parts of its spin current can be protected by local conservation laws for anisotropies $-1<Δ<1$. In this case, the Drude weight $D(T)$ is non-zero at finite temperatures $T$. Here we obtain analytical results for $D(T)$ at low temperatures for zero external magnetic field and anisotropies $Δ=\cos(nπ/m)$ with $n,m$ coprime integers, using the thermodynamic Bethe ansatz. We show that to leading orders $D(T)=D(0)-a(Δ)T^{2K-2}-b_1(Δ)T^2$ where $K$ is the Luttinger parameter and the prefactor $a(Δ)$, obtained in closed form, has a fractal structure as function of anisotropy $Δ$. The prefactor $b_1(Δ)$, on the other hand, does not have a fractal structure and can be obtained in a standard field-theoretical approach. Including both temperature corrections, we obtain an analytic result for the low-temperature asymptotics of the Drude weight in the entire regime $-1<Δ=\cos(nπ/m)<1$.

cond-mat.stat-mech↗

arXiv2021

Phantom Bethe roots in the integrable open spin $1/2$ $XXZ$ chain

We investigate special solutions to the Bethe Ansatz equations (BAE) for open integrable $XXZ$ Heisenberg spin chains containing phantom (infinite) Bethe roots. The phantom Bethe roots do not contribute to the energy of the Bethe state, so the energy is determined exclusively by the remaining regular excitations. We rederive the phantom Bethe roots criterion and focus on BAE solutions for mixtures of phantom roots and regular (finite) Bethe roots. We prove that in the presence of phantom Bethe roots, all eigenstates are split between two invariant subspaces, spanned by chiral shock states. Bethe eigenstates are described by two complementary sets of Bethe Ansatz equations for regular roots, one for each invariant subspace. The respective "semi-phantom" Bethe vectors are states of chiral nature, with chirality properties getting less pronounced when more regular Bethe roots are added. For the easy plane case "semi-phantom" Bethe states carry nonzero magnetic current, and are characterized by quasi-periodic modulation of the magnetization profile, the most prominent example being the spin helix states (SHS). We illustrate our results investigating "semi-phantom" Bethe states generated by one regular Bethe root (the other Bethe roots being phantom), with simple structure of the invariant subspace, in all details. We obtain the explicit expressions for Bethe vectors, and calculate the simplest correlation functions, including the spin-current for all the states in the single particle multiplet.

cond-mat.stat-mech↗

arXiv2021

Groundstate finite-size corrections and dilogarithm identities for the twisted $A_1^{(1)}$, $A_2^{(1)}$ and $A_2^{(2)}$ models

We consider the $Y$-systems satisfied by the $A_1^{(1)}$, $A_2^{(1)}$, $A_2^{(2)}$ vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin-Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley-Lieb loop models respectively. For all three models, our focus is on roots of unity values of $e^{iλ}$ with the crossing parameter $λ$ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of Thermodynamic Bethe Ansatz (TBA) equations, we solve the $Y$-systems for the finite-size $\frac 1N$ corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for $c-24Δ$, where $c$ is the central charge and $Δ$ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods and are new for the dual series of the $A_2^{(1)}$ model.

math-ph↗

arXiv2020

Thermal form-factor approach to dynamical correlation functions of integrable lattice models

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 the matrix elements become time independent and turn into the thermal form factors studied previously in the context of static correlation functions. We make this explicit with the example of the XXZ model. We show how the form factors can be summed utilizing certain auxiliary functions solving finite sets of nonlinear integral equations. The case of the XX model is worked out in more detail leading to a novel form-factor series representation of the dynamical transverse two-point function.

cond-mat.stat-mech↗