[Paper Review] Studying Continuous Symmetry Breaking using Energy Level Spectroscopy

Alexander Wietek, Michael Schüler+1 more|arXiv (Cornell University)|Apr 27, 2017

Power Transformer Diagnostics and Insulation17 citations

TL;DR

This paper presents a group-theoretical framework for identifying continuous symmetry breaking in quantum many-body systems using energy level spectroscopy, particularly through the tower of states (TOS) structure in finite-size systems. By analyzing quantum numbers and degeneracies in exact diagonalization data, it demonstrates that the TOS scaling ∝ S(S+1)/N and irreducible representations predict spontaneous breaking of continuous symmetries—such as spin-rotation or translational symmetry—offering a robust numerical signature for symmetry-broken phases.

ABSTRACT

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.

Motivation & Objective

• To establish a systematic method for identifying continuous symmetry breaking in quantum systems using energy spectrum analysis.
• To connect the spectral structure of finite systems to the thermodynamic limit via the tower of states (TOS) formalism.
• To apply group representation theory to predict quantum numbers and degeneracies of symmetry-broken ground states.
• To validate theoretical predictions against exact diagonalization data for spin and translational symmetry breaking.
• To extend the TOS framework to higher symmetry groups such as SU(n) and O(2), relevant for ultracold fermions and Bose-Einstein condensates.

Proposed method

• Use of group representation theory to classify irreducible representations (irreps) of space groups and continuous symmetry groups (e.g., SU(2), O(2), SU(n)).
• Derivation of the TOS scaling behavior S(S+1)/N for SU(2) symmetric Heisenberg models, linking energy levels to total spin quantum numbers.
• Application of the Lieb-Mattis model as a solvable toy model to analytically derive the TOS structure and verify its scaling.
• Exact diagonalization of finite lattices (e.g., square, triangular) to compute energy spectra and quantum numbers, comparing results with TOS predictions.
• Characterization of symmetry sectors via irreducible representations (e.g., Γ.A1, K.B1) and multiplicity counting in magnetic and nematic phases.
• Extension of the TOS formalism to SU(n) and O(2) symmetries, with scaling proportional to the quadratic Casimir operator C₂(n)/N.

Experimental results

Research questions

• RQ1 How can the energy spectrum of a finite system reveal the presence of continuous symmetry breaking?
• RQ2 What quantum numbers and degeneracies characterize the tower of states in systems with spontaneously broken spin-rotation or translational symmetry?
• RQ3 How can group representation theory predict the irreducible representations and multiplicities of the groundstate manifold in symmetry-broken phases?
• RQ4 What is the scaling behavior of the TOS in finite systems, and how does it converge to the thermodynamic limit?
• RQ5 Can the TOS formalism be generalized to higher symmetry groups such as SU(n) or O(2), and how do their Casimir operators affect the spectrum?

Key findings

• The tower of states (TOS) structure emerges in finite systems with continuous symmetry breaking, with energy levels scaling as S(S+1)/N, where S is the total spin quantum number.
• For the Heisenberg antiferromagnet on a square lattice, the groundstate is a singlet (S_tot = 0), but the TOS structure reveals the underlying Néel order via degeneracies in higher spin sectors.
• Exact diagonalization results for the 120° Néel phase on the triangular lattice show perfect agreement with TOS predictions: irreducible representations Γ.A1, Γ.B1, K.A1, etc., with multiplicities matching group theory.
• The TOS for the AFQ (valence bond solid) phase exhibits distinct irreducible representations and multiplicities, confirming its distinct symmetry-broken order from the Néel state.
• The TOS scaling is generalized to SU(n) models, where the energy levels scale as C₂(n)/N, with C₂(n) being the quadratic Casimir operator of SU(n), confirming universality beyond SU(2).
• In systems with O(2) symmetry (e.g., Bose-Einstein condensates), the TOS structure persists and can be characterized using similar group-theoretical methods, with universal critical spectra observed in continuous quantum phase transitions.