Quantum integrable systems

Introduction

Quantum integrable systems are a class of quantum mechanical systems that are exactly solvable due to the presence of a large number of conserved quantities. These systems are of significant interest in theoretical physics and mathematical physics due to their rich structure and the insights they provide into non-perturbative phenomena. Unlike classical integrable systems, which are well-understood through the framework of Hamiltonian mechanics, quantum integrable systems require sophisticated techniques from quantum mechanics and algebraic structures.

Historical Background

The study of integrable systems dates back to the 19th century with the discovery of the KdV equation and the inverse scattering transform. However, the extension of these ideas to quantum systems began in earnest in the mid-20th century. Key milestones include the Bethe ansatz for solving the Heisenberg model and the development of the Quantum Inverse Scattering Method (QISM).

Fundamental Concepts

Integrability

In the context of quantum mechanics, a system is considered integrable if it possesses a sufficient number of commuting observables that can be used to diagonalize the Hamiltonian. These observables are often related to the symmetries of the system. The presence of these conserved quantities implies that the system can be solved exactly, often through algebraic methods.

Bethe Ansatz

The Bethe ansatz is a method for finding the exact eigenstates and eigenvalues of certain quantum integrable systems. It was originally developed by Hans Bethe for the one-dimensional Heisenberg spin chain. The ansatz involves expressing the wavefunction as a superposition of plane waves and solving the resulting algebraic equations.

Quantum Inverse Scattering Method

The Quantum Inverse Scattering Method (QISM) is a powerful technique for solving quantum integrable systems. It generalizes the classical inverse scattering transform to the quantum domain. The method involves the use of Lax pairs and R-matrices to construct the conserved quantities and solve the system.

Key Models

Heisenberg Spin Chain

The Heisenberg spin chain is one of the most studied quantum integrable systems. It consists of spins arranged in a one-dimensional lattice, interacting through nearest-neighbor exchange interactions. The Hamiltonian for the Heisenberg model is given by: \[ H = -J \sum_{i} \mathbf{S}_i \cdot \mathbf{S}_{i+1} \] where \( \mathbf{S}_i \) are the spin operators and \( J \) is the exchange constant.

Lieb-Liniger Model

The Lieb-Liniger model describes a system of bosons in one dimension with delta-function interactions. The Hamiltonian is given by: \[ H = -\frac{\hbar^2}{2m} \sum_{i} \frac{\partial^2}{\partial x_i^2} + g \sum_{i < j} \delta(x_i - x_j) \] where \( g \) is the interaction strength. This model can be solved exactly using the Bethe ansatz.

Hubbard Model

The Hubbard model is a fundamental model for studying strongly correlated electron systems. The Hamiltonian is given by: \[ H = -t \sum_{\langle i,j \rangle, \sigma} (c_{i\sigma}^\dagger c_{j\sigma} + \text{h.c.}) + U \sum_{i} n_{i\uparrow} n_{i\downarrow} \] where \( t \) is the hopping parameter, \( U \) is the on-site interaction, and \( c_{i\sigma}^\dagger \) and \( c_{i\sigma} \) are the creation and annihilation operators, respectively.

Algebraic Structures

Yang-Baxter Equation

The Yang-Baxter equation is a consistency condition that arises in the study of quantum integrable systems. It ensures the commutativity of transfer matrices and is central to the construction of R-matrices. The equation is given by: \[ R_{12}(\lambda) R_{13}(\lambda + \mu) R_{23}(\mu) = R_{23}(\mu) R_{13}(\lambda + \mu) R_{12}(\lambda) \] where \( R_{ij} \) are the R-matrices acting on the tensor product of vector spaces.

Quantum Groups

Quantum groups are algebraic structures that generalize classical Lie algebras and play a crucial role in the study of quantum integrable systems. They are defined by deformations of the universal enveloping algebra of a Lie algebra and are characterized by the presence of a Hopf algebra structure.

Applications

Quantum integrable systems have applications in various fields of physics, including condensed matter physics, statistical mechanics, and quantum field theory. They provide exact solutions for models of strongly correlated electrons, spin chains, and low-dimensional quantum gases. These solutions are valuable for understanding phenomena such as quantum phase transitions, quantum entanglement, and topological order.

Advanced Topics

Quantum Knizhnik-Zamolodchikov Equation

The Quantum Knizhnik-Zamolodchikov (qKZ) equation is a difference equation that arises in the study of correlation functions in quantum integrable systems. It generalizes the classical Knizhnik-Zamolodchikov equation to the quantum domain and is closely related to the representation theory of quantum groups.

Algebraic Bethe Ansatz

The algebraic Bethe ansatz is a method for solving quantum integrable systems using the algebraic structures of quantum groups. It involves the construction of eigenstates of the transfer matrix using creation and annihilation operators that satisfy certain commutation relations.

Thermodynamic Bethe Ansatz

The thermodynamic Bethe ansatz (TBA) is a method for studying the thermodynamic properties of quantum integrable systems. It involves solving a set of coupled integral equations that describe the distribution of rapidities in the system. The TBA provides insights into the finite-temperature behavior of integrable models.

References