Spontaneous symmetry breaking in open quantum systems: strong, weak, and strong-to-weak

TL;DR

This work analyzes spontaneous symmetry breaking in open quantum systems under Markovian dynamics, identifying two distinct symmetry forms: strong and weak. It constructs concrete Liouvillian models for each case and demonstrates that strong symmetry inevitably breaks to the corresponding weak symmetry, with strong-to-weak breaking producing gapless Goldstone-like diffusion modes for the conserved charge and, in some regimes, long-range order and two Goldstone modes. A field-theoretic (Keldysh) treatment reveals lower critical dimensions and emergent hydrodynamics, including a Bose-surface–to–long-range-order transition tied to filling. The authors further connect strong-to-weak symmetry breaking with an enhanced Lieb–Schultz–Mattis theorem, showing gapless Liouvillian spectra in translationally invariant systems independent of filling, and outline future directions toward non-abelian symmetries and rigorous general proofs.

Abstract

Depending on the coupling to the environment, symmetries of open quantum systems manifest in two distinct forms, the strong and the weak. We study the spontaneous symmetry breaking among phases with different symmetries. Concrete Liouvillian models with strong and weak symmetry are constructed, and different scenarios of symmetry-breaking transitions are investigated from complementary approaches. It is demonstrated that strong symmetry always spontaneously breaks into the corresponding weak symmetry. For strong $U(1)$ symmetry, we show that strong-to-weak symmetry breaking leads to gapless Goldstone modes dictating diffusion of the symmetry charge in translational invariant systems. We conjecture that this relation among strong-to-weak symmetry breaking, gapless modes, and symmetry-charge diffusion is general for continuous symmetries. It can be interpreted as an "enhanced Lieb-Schultz-Mattis (LSM) theorem" for open quantum systems, according to which the gapless spectrum does not require non-integer filling. We also investigate the scenario where the strong symmetry breaks completely. In the symmetry-broken phase, we identify an effective Keldysh action with two Goldstone modes, describing fluctuations of the order parameter and diffusive hydrodynamics of the symmetry charge, respectively. For a particular model studied here, we uncover a transition from a symmetric phase with a "Bose surface" to a symmetry-broken phase with long-range order induced by tuning the filling. It is also shown that the long-range order of $U(1)$ symmetry breaking is possible in spatial dimension $d\geq 3$, in both weak and strong symmetry cases. Our work outline the typical scenarios of spontaneous symmetry breaking in open quantum systems, and highlights their physical consequences.

Figures (3)

• Figure 1: Different spontaneous symmetry breaking processes are discussed in this work, including weak symmetry breaking, strong-to-weak symmetry breaking, and complete breaking of strong symmetry. Model I with weak $U(1)$ symmetry breaking is studied in Sec. \ref{['sec:weak']}. Model II with strong and strong-to-weak $U(1)$ symmetry breaking is studied in Sec. \ref{['sec:strong']}. In Sec. \ref{['sec:s-to-w']}, we further discuss strong-to-weak symmetry breaking and its connection to the diffusion of the symmetry charge, with model III as an example.
• Figure 2: (a) We consider square lattice case, where the $B$ sublattice is still a square lattice. (b) the Bose surface on the $B$ sublattice.
• Figure 3: (a) In the dark states $|\psi_d\rangle$'s Eq. (\ref{['eq:dark']}), $B$ sites are freely occupiable, whereas an $A$ site can only be occupied if its neighboring $B$ sites are also occupied. (b) Perturbatively, the Hamiltonian connects different $|\psi_d\rangle$ configurations. Bosons can hop between the $B$ sublattice sites via second-order perturbation of $H$. On the $A$ sublattice, bosons can only move if the neighboring $B$ sites form a connected cluster of occupation.