Andreas Klümper

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).

Density matrices for finite segments of Heisenberg chains of arbitrary length

We derive a multiple integral representing the ground-state density matrix of a segment of length m of the XXZ spin chain on L lattice sites, which depends on L only parametrically. This allows us to treat chains of arbitrary finite length. Specializing to the isotropic limit of the XXX chain we show for small m that the multiple integrals factorize. We conjecture that this property holds for arbitrary m and suggest an exponential formula for the density matrix which involves only a double Cauchy type integral in the exponent. We demonstrate the efficiency of our formula by computing the next-to-nearest neighbour zz-correlation function for chain lengths ranging from two to macroscopic numbers.

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 Uq(L(sl2)). We give a complete set of the functional relations correcting inexactitudes in the previous considerations. We especially attend to the interrelation of the representations used to construct the universal transfer operators and Q-operators.

On the universal R-matrix for the Izergin-Korepin model

We continue our exercises with the universal

Oscillator versus prefundamental representations

R

Non-dissipative Thermal Transport and Magnetothermal Effect for the Spin-1/2 Heisenberg Chain

-matrix based on the Khoroshkin and Tolstoy formula. Here we present our results for the case of the twisted affine Kac--Moody Lie algebra of type

Oscillator versus prefundamental representations. II. Arbitrary higher ranks

A^{(2)}_2

Theory of microwave absorption by the spin-1/2 Heisenberg-Ising magnet.

. Our interest in this case is inspired by the fact that the Tzitzeica equation is associated with

Универсальные объекты интегрируемости@@@Universal integrability objects

A^{(2)}_2

The One-Dimensional Hubbard Model: The Yangian symmetry of the Hubbard model

in a similar way as the sine-Gordon equation is related to