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Klaus Fabricius

Publications and source records attributed to Klaus Fabricius.

At least 19 recordsLinked to original sources
arXiv2011

Properties of the String Operator in the Eight-Vertex Model

The construction of creation operators of exact strings in eigenvectors of the eight vertex model at elliptic roots of unity of the crossing parameter which allow the generation of the complete set of degenerate eigenstates is based on the conjecture that the 'naive' string operator vanishes. In this note we present a proof of this conjecture. Furthermore we show that for chains of odd length the string operator is either proportional to the symmetry operator $S$ or vanishes depending on the precise form of the crossing parameter.

cond-mat.stat-mech↗
arXiv2008

New Q matrices and their functional equations for the eight vertex model at elliptic roots of unity

The Q matrix invented by Baxter in 1972 to solve the eight vertex model at roots of unity exists for all values of N, the number of sites in the chain, but only for a subset of roots of unity. We show in this paper that a new Q matrix, which has recently been introduced and is non zero only for N even, exists for all roots of unity. In addition we consider the relations between all of the known Q matrices of the eight vertex model and conjecture functional equations for them.

cond-mat.stat-mech↗
arXiv2007

The Eight Vertex Model.New results

Whereas the tools to determine the eigenvalues of the eight-vertex transfer matrix T are well known there has been until recently incomplete knowledge about the eigenvectors of T. We describe the construction of eigenvectors of T corresponding to degenerate eigenvalues and discuss the related hidden elliptic symmetry.

cond-mat.stat-mech↗
arXiv2007

A new Q-matrix in the Eight-Vertex Model

We construct a $Q$-matrix for the eight-vertex model at roots of unity for crossing parameter $η=2mK/L$ with odd $L$, a case for which the existing constructions do not work. The new $Q$-matrix $\Q$ depends as usual on the spectral parameter and also on a free parameter $t$. For $t=0$ $\Q$ has the standard properties. For $t\neq 0$, however, it does not commute with the operator $S$ and not with itself for different values of the spectral parameter. We show that the six-vertex limit of $\Q(v,t=iK'/2)$ exists.

cond-mat.stat-mech↗
arXiv2007

The TQ equation of the 8 vertex model for complex elliptic roots of unity

We extend our studies of the TQ equation introduced by Baxter in his 1972 solution of the 8 vertex model with parameter $η$ given by $2Lη=2m_1K+im_2K'$ from $m_2=0$ to the more general case of complex $η.$ We find that there are several different cases depending on the parity of $m_1$ and $m_2$.

cond-mat.stat-mech↗
arXiv2007

New Developments in the Eight Vertex Model II. Chains of odd length

We study the transfer matrix of the 8 vertex model with an odd number of lattice sites $N.$ For systems at the root of unity points $η=mK/L$ with $m$ odd the transfer matrix is known to satisfy the famous ``$TQ$'' equation where ${\bf Q}(v)$ is a specifically known matrix. We demonstrate that the location of the zeroes of this ${\bf Q}(v)$ matrix is qualitatively different from the case of even $N$ and in particular they satisfy a previously unknown equation which is more general than what is often called ``Bethe's equation.'' For the case of even $m$ where no ${\bf Q}(v)$ matrix is known we demonstrate that there are many states which are not obtained from the formalism of the SOS model but which do satisfy the $TQ$ equation. The ground state for the particular case of $η=2K/3$ and $N$ odd is investigated in detail.

cond-mat.stat-mech↗
arXiv2006

An elliptic current operator for the 8 vertex model

We compute the operator which creates the missing degenerate states in the algebraic Bethe ansatz of the 8 vertex model at roots of unity and relate it to the concept of an elliptic current operator. We find that in sharp contrast with the corresponding formalism in the six-vertex model at roots of unity the current operator is not nilpotent with the consequence that in the construction of degenerate eigenstates of the transfer matrix an arbitrary number of exact strings can be added to the set of regular Bethe roots. Thus the original set of free parameters {s,t} of an eigenvector of T is enlarged to become {s,t,λ_{c,1}, ..., λ_{c,n}\} with arbitrary string centers λ_{c,j} and arbitrary n.

cond-mat.stat-mech↗
arXiv2004

Root of unity symmetries in the 8 and 6 vertex models

We review the recently discovered symmetries of the 8 and 6 vertex models which exist at roots of unity and present their relation with representation theory of affine Lie algebras, Drinfeld polynomials and Bethe vectors.

cond-mat.stat-mech↗
arXiv2003

Functional Equations and Fusion Matrices for the Eight Vertex Model

We derive sets of functional equations for the eight vertex model by exploiting an analogy with the functional equations of the chiral Potts model. From these equations we show that the fusion matrices have special reductions at certain roots of unity. We explicitly exhibit these reductions for the 3,4 and 5 order fusion matrices and compare our formulation with the algebra of Sklyanin.

cond-mat.stat-mech↗
arXiv2002

New Developments in the Eight Vertex Model

We demonstrate that the Q matrix introduced in Baxter's 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for Q(v) which both contains the Bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues.

cond-mat.stat-mech↗
arXiv2001

Evaluation parameters and Bethe roots for the six vertex model at roots of unity

We propose an expression for the current form of the lowering operator of the $ {sl}_2$ loop algebra symmetry of the six vertex model (XXZ spin chain) at roots of unity. This operator has poles which correspond to the evaluation parameters of representation theory which are given as the roots of the Drinfeld polynomial. We explicitly compute these polynomials in terms of the Bethe roots which characterize the highest weight states for all values of $S^z$. From these polynomials we find that the Bethe roots satisfy sum rules for each value of S^z.

cond-mat.stat-mech↗
arXiv2001

Completing Bethe's equations at roots of unity

In a previous paper we demonstrated that Bethe's equations are not sufficient to specify the eigenvectors of the XXZ model at roots of unity for states where the Hamiltonian has degenerate eigenvalues. We here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the $sl_2$ loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by Bethe's ansatz.

cond-mat.stat-mech↗
arXiv2000

Bethe's equation is incomplete for the XXZ model at roots of unity

We demonstrate for the six vertex and XXZ model parameterized by $Δ=-(q+q^{-1})/2\neq \pm 1$ that when q^{2N}=1 for integer $N\geq 2$ the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl_2 loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl_2 loop algebra can be used to complete this specification.

cond-mat.stat-mech↗
arXiv2000

The sl_2 loop algebra symmetry of the six-vertex model at roots of unity

We demonstrate that the six vertex model (XXZ spin chain) with $Δ=(q+q^{-1})/2$ and $q^{2N}=1$ has an invariance under the loop algebra of $sl_2$ which produces a special set of degenerate eigenvalues. For $Δ=0$ we compute the multiplicity of the degeneracies using Jordan Wigner techniques

cond-mat.stat-mech↗
arXiv1998

Temperature dependent spatial oscillations in the correlations of the XXZ spin chain

We study the correlation $<σ^z_0σ^z_n>$ for the XXZ chain in the massless attractive (ferromagnetic) region at positive temperatures by means of a numerical study of the quantum transfer matrix. We find that there is a range of temperature where the behavior of the correlation for large separations is oscillatory with an incommensurate period which depends on temperature.

cond-mat.stat-mech↗
arXiv1998

Quantum-classical crossover in the spin 1/2 XXZ chain

We compute, by means of exact diagonalization of systems of N=16 and 18 spins, the correlation function <σ^z_0σ^z_n> at nonzero temperature for the XXZ model with anisotropy Δ. In the gapless ferromagnetic region -1<Δ<0 for fixed separation the temperature can always be made sufficiently low so that the correlation is always negative for n \neq 0. However we find that for sufficiently large temperatures and fixed separation or for fixed temperature greater than some T0(\DElta) and sufficiently large separations the correlations are always positive. This sign changing effect has not been previously seen and we interpret it as a crossover from quantum to classical behavior.

cond-mat.stat-mech↗
arXiv1997

Spin diffusion and the spin 1/2 XXZ chain at $T=\infty$ from exact diagonalization

We study the long time behavior of the zz and xx time dependent autocorrelation function of the spin 1/2 XXZ chain at $T=\infty$ by exact diagonalizations on a chain of 16 sites. We find that the numerical results for the zz correlation are very well fit by the formula $t^{-d}[A+Be^{-γ(t-t_0)}\cos Ω(t-t_0)].$ From this we estimate $d$ as a function of the anisotropy of the chain and study the crossover from ballistic to diffusive behavior.

cond-mat↗