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Symmetry in mixed quantum states can manifest in two distinct forms: , where each individual pure state in the quantum ensemble is symmetric with the same charge, and , which applies only to the entire ensemble. This paper explores a novel type of spontaneous symmetry breaking (SSB) where a strong symmetry is broken to a weak one. While the SSB of a weak symmetry is measured by the long-ranged two-point correlation function, the strong-to-weak SSB (SWSSB) is measured by the . We prove that SWSSB is a universal property of mixed-state quantum phases, in the sense that the phenomenon of SWSSB is robust against symmetric low-depth local quantum channels. We also show that the symmetry breaking is “spontaneous” in the sense that the effect of a local symmetry-breaking measurement cannot be recovered locally. We argue that a thermal state at a nonzero temperature in the canonical ensemble (with fixed symmetry charge) should have spontaneously broken strong symmetry. Additionally, we study nonthermal scenarios where decoherence induces SWSSB, leading to phase transitions described by classical statistical models with bond randomness. In particular, the SWSSB transition of a decohered Ising model can be viewed as the “ungauged” version of the celebrated toric-code decodability transition. We confirm that, in the decohered Ising model, the SWSSB transition defined by the fidelity correlator is the only physical transition in terms of channel recoverability. We also comment on other (inequivalent) definitions of SWSSB, through correlation functions with higher Rényi indices. Published by the American Physical Society2025
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Discrete symmetry breaking defines the Mott quartic fixed point
Because Fermi liquids are inherently non-interacting states of matter, all electronic levels below the chemical potential are doubly occupied. Consequently, the simplest way of breaking the Fermi-liquid theory is to engineer a model in which some of those states are singly occupied, keeping time-reversal invariance intact. We show that breaking an overlooked1 local-in-momentum space ℤ2 symmetry of a Fermi liquid does precisely this. As a result, although the Mott transition from a Fermi liquid is correctly believed to arise without breaking any continuous symmetry, a discrete symmetry is broken. This symmetry breaking serves as an organizing principle for Mott physics whether it arises from the tractable Hatsugai–Kohmoto model or the intractable Hubbard model. Through a renormalization-group analysis, we establish that both are controlled by the same fixed point. An experimental manifestation of this fixed point is the onset of particle–hole asymmetry, a widely observed2,3,4,5,6,7,8,9,10 phenomenon in strongly correlated systems. Theoretically, the singly occupied region of the spectrum gives rise to a surface of zeros of the single-particle Green function, denoted as the Luttinger surface. Using K-homology, we show that the Bott topological invariant guarantees the stability of this surface to local perturbations. Our proof demonstrates that the strongly coupled fixed point only corresponds to those Luttinger surfaces with co-dimension p + 1 with odd p. We conclude that both Hubbard and Hatsugai–Kohmoto models lie in the same high-temperature universality class and are controlled by a quartic fixed point with broken ℤ2 symmetry.
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- Award ID(s):
- 2111379
- PAR ID:
- 10322180
- Date Published:
- Journal Name:
- Nature physics
- ISSN:
- 1745-2473
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1038/s41567-022-01529-8
