Spontaneous continuous-symmetry breaking and tower of states in a comb chain

TL;DR

This work demonstrates spontaneous continuous symmetry breaking in a one-dimensional spin-1/2 Heisenberg model on a comb lattice, a setting that satisfies the Marshall–Lieb–Mattis ferrimagnetism conditions and the Shen–Qiu–Tian connection to long-range order. Using stochastic series expansion quantum Monte Carlo, spin-wave theory, and renormalization-group analysis, the authors establish ground-state degeneracy, finite long-range order, and a consistent low-energy description. Crucially, they reveal a tower of states in a genuine 1D short-range system, with TOS levels in the k=0 sector reflecting O(3)→O(2) symmetry breaking. An effective ferromagnetic spin-1/2 chain with dispersion $2 S J_{eff} (1-\,\cos k)$ (with $J_{eff}=2/9$ for the chosen parameters) accounts for the observed quadratic low-energy mode and harmonizes the results with known 1D symmetry-breaking constraints.

Abstract

Based on the study of a one-dimensional (1D) antiferromagnetic Heisenberg model on a comb lattice, this work identifies an example of spontaneous continuous symmetry breaking in a 1D system with short-range interactions. When a symmetry-preserving relevant perturbation is applied to the system, we find that this model can always be described by the Marshall-Lieb-Mattis theorem. The Shen-Qiu-Tian theorem establishes a direct connection between the Marshall-Lieb-Mattis theorem (in the case of bipartite lattices with unequal numbers of sites in the two sublattices) and the breaking of continuous symmetry. Moreover, although previous studies have suggested that the presence of a tower of states (TOS) serves as an important numerical diagnostic of a system's tendency toward spontaneous symmetry breaking, these investigations have primarily focused on 2D systems. In 1D systems, however, the presence of long-range order does not automatically imply the emergence of a TOS. Here, we observe the existence of a TOS in a 1D realistic ferrimagnetic lattice system with short-range interactions.

Figures (6)

• Figure 1: A 1D comb lattice. $V=1$ and $V_1 >0$ represent AF Heisenberg interactions between nearest neighbors. It is clear that this model is a bipartite lattice, where the sites of the lattice can be decomposed into two colors. In addition, from the perspective of lattice translation symmetry, this lattice can be divided into three sublattices: A, B, and C. For clarity, we use the term sublattice* when referring to the bipartite structure, and sublattice when discussing the translational lattice structure.
• Figure 2: Correlation $C(L/2)$, Binder cumulant $U_2$ vs the inverse of the system size $1/L$ at $V=V_1=1$ (a–c) . AC, B or ABC indicate that calculated physical quantities belong to the AC or B sublattice or the whole system, as shown in the Fig.\ref{['comb']}. For comparison, we also computed the relevant physical results for a pure 1D AF Heisenberg chain in (a,b). The red points indicate the extrapolated results in the thermodynamic limit.
• Figure 3: Energy spectra a 1D comb lattice. $V=V_1=1$ with system sizes (a) $L=6$ and (b) $L=8$, respectively. The blue dots represent the full energy spectrum, while the orange dots correspond to the spectrum extracted from the $k=0$ momentum subspace. The TOS levels are connected by red lines.
• Figure 4: Dispersions calculated by linear spin wave theory (green, blue and red solid lines), low energy effective model from Kadanoff's approach (dashed line) and SAC (spectral functions marked by color intensity) of the comb chain, where $V_1=V=1$.
• Figure 5: spectrum calculated by quantum Monte-Carlo-stochastic analytic continuation for L=256, where $V=1$, (a) $V_1=0.8$, (b)$V_1=1$ and (c) $V_1=1.2$.