Frank Göhmann

The One-Dimensional Hubbard Model: Index

The description of a solid at a microscopic level is complex, involving the interaction of a huge number of its constituents, such as ions or electrons. It is impossible to solve the corresponding many-body problems analytically or numerically, although much insight can be gained from the analysis of simplified models. An important example is the Hubbard model, which describes interacting electrons in narrow energy bands, and which has been applied to problems as diverse as high-Tc superconductivity, band magnetism and the metalinsulator transition. Remarkably, the one-dimensional Hubbard model can be solved exactly using the Bethe ansatz method. The resulting solution has become a laboratory for theoretical studies of non-perturbative effects in strongly correlated electron systems. Many methods devised to analyse such effects have been applied to this model, both to provide complementary insight into what is known from the exact solution and as an ultimate test of their quality. This book presents a coherent, self-contained account of the exact solution of the Hubbard model in one dimension. The early chapters develop a self-contained introduction to Bethe’s ansatz and its application to the one-dimensional Hubbard model, and will be accessible to beginning graduate students with a basic knowledge of quantum mechanics and statistical mechanics. The later chapters address more advanced topics, and are intended as a guide for researchers to some of the more recent scientific results in the field of integrable models. The authors are distinguished researchers in the field of condensed matter physics and integrable systems, and have contributed significantly to the present understanding of the one-dimensional Hubbard model. Fabian Essler is a University Lecturer in Condensed Matter Theory at Oxford University. Holger Frahm is Professor of Theoretical Physics at the University of Hannover. Frank Göhmann is a Lecturer at Wuppertal University, Germany. Andreas Klümper is Professor of Theoretical Physics at Wuppertal University. Vladimir Korepin is Professor at the Yang Institute for Theoretical Physics, State University of New York at Stony Brook, and author of Quantum Inverse Scattering Method and Correlation Functions (Cambridge, 1993).

Integral representations for correlation functions of the XXZ chain at finite temperature

We derive a novel multiple integral representation for a generating function of the σz–σz correlation functions of the spin- XXZ chain at finite temperature and finite, longitudinal magnetic field. Our work combines algebraic Bethe ansatz techniques for the calculation of matrix elements with the quantum transfer matrix approach to thermodynamics.

Solution of the quantum inverse problem

We derive a formula that expresses the local spin and field operators of fundamental graded models in terms of the elements of the monodromy matrix. This formula is a quantum analogue of the classical inverse scattering transform. It applies to fundamental spin chains, such as the XYZ chain, and to a number of important exactly solvable models of strongly correlated electrons, such as the supersymmetric t-J model or the the EKS model.

Exercises with the universal R-matrix

Using the formula for the universal R-matrix proposed by Khoroshkin and Tolstoy, we give a detailed derivation of L-operators for the quantum groups associated with the generalized Cartan matrices A(1)1 and A(1)2.

Integral representation of the density matrix of the XXZ chain at finite temperatures

We present an integral formula for the density matrix of a finite segment of the infinitely long spin- XXZ chain. This formula is valid for any temperature and any longitudinal magnetic field.

Thermodynamics and excitations of the one-dimensional Hubbard model

Abstract We review fundamental issues arising in the exact solution of the one-dimensional Hubbard model. We perform a careful analysis of the Lieb–Wu equations, paying particular attention to the so-called ‘string solutions’. Two kinds of string solutions occur: Λ strings, related to spin degrees of freedom and k – Λ strings, describing spinless bound states of electrons. Whereas Λ strings were thoroughly studied in the literature, less is known about k – Λ strings. We carry out a thorough analytical and numerical analysis of k – Λ strings. We further review two different approaches to the thermodynamics of the Hubbard model, the Yang–Yang approach and the quantum transfer matrix approach, respectively. The Yang–Yang approach is based on strings, the quantum transfer matrix approach is not. We compare the results of both methods and show that they agree. Finally, we obtain the dispersion curves of all elementary excitations at zero magnetic field for the less than half-filled band by considering the zero-temperature limit of the Yang–Yang approach.

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 (5l 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.

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

It was recently shown by Jimbo et al (2008 arXiv:0811.0439) that the correlation functions of a generalized XXZ chain associated with an inhomogeneous six-vertex model with a 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 et al (2008 arXiv:0811.0439) 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.

A note on the spin-½ XXZ chain concerning its relation to the Bose gas

By considering the one-particle and two-particle scattering data for the spin-½ 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 to the Heisenberg chain we obtain expressions for the correlation functions of the Bose gas at arbitrary temperatures.

Short-distance thermal correlations in the massive XXZ chain

AbstractWe 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 = ∞). 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.