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Cambridge University Press eBooks

The Tower of States from Exact Diagonalisation

Aron J. Beekman, Louk Rademaker, Jasper van Wezel

Published Sep 3, 2026• 0 citations
Publisher

Abstract

This chapter introduces the method of exact diagonalisation and applies it to symmetry breaking in a Heisenberg antiferromagnet. Both the ground state and the tower of states enabling spontaneous symmetry breaking are identified and their roles elucidated.
DOI
10.1017/9781009575782.021

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Tower of States analysis is a powerful tool for investigating phase transitions in condensed matter systems. Spontaneous symmetry breaking implies a specific structure of the energy eigenvalues and their corresponding quantum numbers on finite systems. In these lecture notes we explain the group representation theory used to derive the spectral structure for several scenarios of symmetry breaking. We give numerous examples to compute quantum numbers of the degenerate groundstates, including translational symmetry breaking or spin rotational symmetry breaking in Heisenberg antiferromagnets. These results are then compared to actual numerical data from Exact Diagonalization.
Exact diagonalization is a powerful numerical technique for analysing static and dynamical quantities in quantum many-body lattice models. It provides unbiased information concerning quantum numbers, energies and wavefunctions of the low-energy eigenstates for almost any kind of microscopic model. The information about energies and quantum numbers is particularly useful to detect possible spontaneous symmetry breaking at T = 0. We review some of the advances in the field of frustrated quantum magnets which have been possible thanks to detailed symmetry analysis of exact diagonalization spectra. New results concerning the kagome and star lattice Heisenberg antiferromagnets are presented.
The roots of the modern understanding of symmetries in physics can be traced to the work of Sophus Lie on transformations of differential equations in the nineteenth century. In this chapter, the history of the subject is used to emphasize the distinction between the symmetry of the dynamics of a physical system and the symmetry of its specific state. The central concept of spontaneous symmetry breaking is then introduced as the discrepancy between the two notions of symmetry. It is shown how the presence of spontaneous symmetry breaking can be detected through the sensitivity of measurements on the physical state to external perturbations. This naturally leads to the concepts of the order parameter and the vacuum manifold. The rest of the chapter is devoted to highlighting some subtle features of spontaneous symmetry breaking. These include the properties of the ground state and the tower of low-lying excited states in a finite volume, and the absence of well-defined unitary operators representing broken symmetry in the thermodynamic limit.
We present a mathematically simple procedure to explain spontaneous symmetry breaking in quantum systems. The procedure is applicable to a wide range of models and can be easily used to explain the existence of a symmetry broken state in crystals, antiferromagnets, and even superconductors. It has the advantage that it automatically brings to the fore the main players in spontaneous symmetry breaking: the symmetry-breaking field, the thermodynamic limit, and the global excitations of a “thin” spectrum.