Searcharxiv⌕ Search

arXiv subjects

Frank Göhmann

Publications and source records attributed to Frank Göhmann.

At least 55 records · Page 3Linked to original sources
arXiv2010

Thermodynamics and short-range correlations of the XXZ chain close to its triple point

The XXZ quantum spin chain has a triple point in its ground state $h$-$1/Δ$ phase diagram. This first order critical point is located at the joint end point of the two second order phase transition lines marking the transition from the gapless phase to the fully polarized phase and to the Néel ordered phase, respectively. We explore the magnetization and the short-range correlation functions in its vicinity using the exact solution of the model. In the critical regime above the triple point we observe a strong variation of all physical quantities on a low energy scale of order $1/Δ$ induced by the transversal quantum fluctuations. We interpret this phenomenon starting from a strong-coupling perturbation theory about the highly degenerate ground state of the Ising chain at the triple point. From the perturbation theory we identify the relevant scaling of the magnetic field and of the temperature. Applying the scaling to the exact solutions we obtain explicit formulae for the magnetization and short-range correlation functions at low temperatures.

cond-mat.stat-mech↗
arXiv2009

On the physical part of the factorized correlation functions of the XXZ chain

It was recently shown by Jimbo, Miwa and Smirnov that the correlation functions of a generalized XXZ chain associated with an inhomogeneous six-vertex model with disorder parameter $α$ and with arbitrary inhomogeneities on the horizontal lines factorize and can all be expressed in terms of only two functions $ρ$ and $ω$. Here we approach the description of the same correlation functions and, in particular, of the function $ω$ from a different direction. We start from a novel multiple integral representation for the density matrix of a finite chain segment of length $m$ in the presence of a disorder field $α$. We explicitly factorize the integrals for $m=2$. Based on this we present an alternative description of the function $ω$ in terms of the solutions of certain linear and nonlinear integral equations. We then prove directly that the two definitions of $ω$ describe the same function. The definition in the work of Jimbo, Miwa and Smirnov was crucial for the proof of the factorization. The definition given here together with the known description of $ρ$ in terms of the solutions of nonlinear integral equations is useful for performing e.g.\ the Trotter limit in the finite temperature case, or for obtaining numerical results for the correlation functions at short distances. We also address the issue of the construction of an exponential form of the density matrix for finite $α$.

math-ph↗
arXiv2009

Short-distance thermal correlations in the massive XXZ chain

We explore short-distance static correlation functions in the infinite XXZ chain using previously derived formulae which represent the correlation functions in factorized form. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field in the massive regime $Δ>1$, extending our previous results to the full parameter plane of the antiferromagnetic chain ($Δ> -1$ and arbitrary field $h$). The factorized formulae are numerically efficient and allow for taking the isotropic limit ($Δ= 1$) and the Ising limit ($Δ= \infty$). At the critical field separating the fully polarized phase from the Néel phase, the Ising chain possesses exponentially many ground states. The residual entropy is lifted by quantum fluctuations for large but finite $Δ$ inducing unexpected crossover phenomena in the correlations.

cond-mat.str-el↗
arXiv2007

From multiple integrals to Fredholm determinants

We consider a multiple integral representation for the finite temperature density-density correlation functions of the one-dimensional Bose gas with delta function interaction in the limits of infinite and vanishing repulsion. In the former case a known Fredholm determinant is recovered. In the latter case a similar expression appears with permanents replacing determinants.

math-ph↗
arXiv2007

A note on the spin-1/2 XXZ chain concerning its relation to the Bose gas

By considering the one-particle and two-particle scattering data of the spin-1/2 Heisenberg chain at T=0 we derive a continuum limit relating the spin chain to the 1D Bose gas. Applying this limit to the quantum transfer matrix approach of the Heisenberg chain we obtain expressions for the correlation functions of the Bose gas at arbitrary temperatures.

cond-mat.stat-mech↗
arXiv2007

Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field

We present a conjecture for the density matrix of a finite segment of the XXZ chain coupled to a heat bath and to a constant longitudinal magnetic field. It states that the inhomogeneous density matrix, conceived as a map which associates with every local operator its thermal expectation value, can be written as the trace of the exponential of an operator constructed from weighted traces of the elements of certain monodromy matrices related to $U_q (\hat{\mathfrak{sl}}_2)$ and only two transcendental functions pertaining to the one-point function and the neighbour correlators, respectively. Our conjecture implies that all static correlation functions of the XXZ chain are polynomials in these two functions and their derivatives with coefficients of purely algebraic origin.

hep-th↗
arXiv2006

Factorization of multiple integrals representing the density matrix of a finite segment of the Heisenberg spin chain

We consider the inhomogeneous generalization of the density matrix of a finite segment of length $m$ of the antiferromagnetic Heisenberg chain. It is a function of the temperature $T$ and the external magnetic field $h$, and further depends on $m$ `spectral parameters' $ξ_j$. For short segments of length 2 and 3 we decompose the known multiple integrals for the elements of the density matrix into finite sums over products of single integrals. This provides new numerically efficient expressions for the two-point functions of the infinite Heisenberg chain at short distances. It further leads us to conjecture an exponential formula for the density matrix involving only a double Cauchy-type integral in the exponent. We expect this formula to hold for arbitrary $m$ and $T$ but zero magnetic field.

hep-th↗
arXiv2005

Finite temperature density matrix and two-point correlations in the antiferromagnetic XXZ chain

We derive finite temperature versions of integral formulae for the two-point correlation functions in the antiferromagnetic XXZ chain. The derivation is based on the summation of density matrix elements characterizing a finite chain segment of length $m$. On this occasion we also supply a proof of the basic integral formula for the density matrix presented in an earlier publication.

cond-mat.stat-mech↗
arXiv2005

Surface free energy for systems with integrable boundary conditions

The surface free energy is the difference between the free energies for a system with open boundary conditions and the same system with periodic boundary conditions. We use the quantum transfer matrix formalism to express the surface free energy in the thermodynamic limit of systems with integrable boundary conditions as a matrix element of certain projection operators. Specializing to the XXZ spin 1/2 chain we introduce a novel `finite temperature boundary operator' which characterizes the thermodynamical properties of surfaces related to integrable boundary conditions.

cond-mat.stat-mech↗
arXiv2004

Algebraic Bethe ansatz for the gl(1$|$2) generalized model II: the three gradings

The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same $R$-matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe ansatz for all models with $9 \times 9$, rational, gl(1$|$2)-invariant $R$-matrix and all three possibilities of choosing the grading. Our Bethe ansatz solution applies, for instance, to the supersymmetric t-J model, the supersymmetric $U$ model and a number of interesting impurity models. It may be extended to obtain the quantum transfer matrix spectrum for this class of models. The properties of a specific model enter the Bethe ansatz solution (i.e. the expression for the transfer matrix eigenvalue and the Bethe ansatz equations) through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model.

cond-mat.stat-mech↗
arXiv2003

A note on the Bethe ansatz solution of the supersymmetric t-J model

The three different sets of Bethe ansatz equations describing the Bethe ansatz solution of the supersymmetric t-J model are known to be equivalent. Here we give a new, simplified proof of this fact which relies on the properties of certain polynomials. We also show that the corresponding transfer matrix eigenvalues agree.

cond-mat.str-el↗
arXiv2002

Algebraic Bethe ansatz for the gl(1|2) generalized model and Lieb-Wu equations

We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang's model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by the gl(1|2) invariant R-matrix.

cond-mat.stat-mech↗
arXiv1998

Fermionic representations of integrable lattice systems

We develop a general scheme for the use of Fermi operators within the framework of integrable systems. This enables us to read off a fermionic Hamiltonian from a given solution of the Yang-Baxter equation and to express the corresponding $L$-matrix and the generators of symmetries in terms of Fermi operators. We illustrate our approach through a number of examples. Our main example is the algebraic Bethe ansatz solution of the Hubbard model in the infinite coupling limit.

cond-mat↗
arXiv1997

Algebraic Solution of the Hubbard Model on the Infinite Interval

We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite line at zero density. This enables us to diagonalize the Hamiltonian algebraically. The eigenstates can be classified as scattering states of particles, bound pairs of particles and bound states of pairs. We obtain the corresponding creation and annihilation operators and calculate the S-matrix. The Hamiltonian on the infinite line is invariant under the Yangian quantum group Y(su(2)). We show that the n-particle scattering states transform like n-fold tensor products of fundamental representations of Y(su(2)) and that the bound states are Yangian singlet.

cond-mat.stat-mech↗
arXiv1996

Algebraic and Analytic Properties of the One-Dimensional Hubbard Model

We reconsider the quantum inverse scattering approach to the one-dimensional Hubbard model and work out some of its basic features so far omitted in the literature. It is our aim to show that $R$-matrix and monodromy matrix of the Hubbard model, which are known since ten years now, have good elementary properties. We provide a meromorphic parametrization of the transfer matrix in terms of elliptic functions. We identify the momentum operator for lattice fermions in the expansion of the transfer matrix with respect to the spectral parameter and thereby show the locality and translational invariance of all higher conserved quantities. We work out the transformation properties of the monodromy matrix under the su(2) Lie algebra of rotations and under the $\h$-pairing su(2) Lie algebra. Our results imply su(2)$\oplus$su(2) invariance of the transfer matrix for the model on a chain with an even number of sites.

cond-mat.stat-mech↗