Robust symmetry breaking in gapless quantum magnets

Abstract

We prove the existence of spontaneous symmetry breaking in suitably low-energy eigenstates of certain gapless and frustrated many-body quantum systems, namely symmetric quantum perturbations to classical models which exhibit spontaneous symmetry breaking of a finite group at some positive temperature. Additionally, the classical model need not be local in space, as long as it satisfies a quantum analogue of the Peierls condition. As an example of our technique, we establish robust ferromagnetism in random-bond Ising models in $d= 2$ dimensions with sufficiently biased random couplings, with weak transverse field. Our mathematical technique is based on establishing quantum bottlenecks, similar to a "many-body WKB" method for evaluating tunneling rates. Using these same methods, we provide new proofs of metastability and the slow decay of the false vacuum, applicable to gapless metastable states. Our work represents a first step towards a rigorous classification of stable gapless quantum phases.

Figures (3)

• Figure 1: (a) Domain walls specify excitations about a ground state; the number of distinct domain walls of length $\ell$ passing through a given vertex on the square lattice is $<3^\ell$, as illustrated by the blue step (there were 3 possible places for the domain wall to grow). (b) The classical Peierls condition implies a bottleneck in the configuration space -- long domain walls are exceedingly hard to find at low temperature in the classical Gibbs ensemble. (c) The quantum Peierls condition implies that a low-energy many-body wave function is very unlikely to be found in a state where all of the bonds along a large closed loop are flipped. In red we illustrate a fixed loop $\gamma$, while blue edges represent examples of wave functions that exist in each sector. The dashed blue line denotes one possible way that such an excitation can exist (which we do not keep track of in \ref{['eq:cleverbasis']}). $\Delta E$ is the expectation of $H-E_0$ in the sector.
• Figure 2: Illustration of a quantum bottleneck structure. Here $\mathcal{B}_{\rm out}$ is not required for the basic Definition \ref{['def:q_bottle']}, but appears in the global version Definition \ref{['def:QPC_global']}.
• Figure 3: (a) Illustration of a new bottleneck indicator in the proof of Proposition \ref{['prop:Ising_OP']} that detects $\ge \frac{1}{3}L_0$$0$s and $\frac{1}{3}L_0$$1$s in a row, based on the DW-loops (red) that cross the row. Although the number of crossings $p$ may be small, the total length $L'\ge \frac{1}{3}L_0$ of the DW-loops has to be large. (b) In the proof of Theorem \ref{['thm:transla']}, we defined a truncated translation operator $\mathcal{T}_A$ along the $x$ direction of a region $A$ corresponding to a bottleneck indicator $B$ (red loop).

Theorems & Definitions (55)

• Definition 1.1: Configuration space
• Definition 1.2: Bottleneck and Peierls condition
• Definition 1.3: Thermal ensemble/Gibbs state
• Proposition 1.4: Peierls argument
• proof
• Proposition 1.5
• proof
• Example 1.6: Ising models
• Proposition 1.7: Peierls condition for 2d Ising models
• proof