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Annalen der Physik

Quantum Adiabatic Evolution: Symmetry‐Protected Quantum Adiabatic Evolution in Spontaneous Symmetry‐Breaking Transitions (Ann. Phys. 4/2020)

Min Zhuang, Jiahao Huang, Yongguan Ke, Chaohong Lee

Published Apr 1, 2020• 2 citations
Publisher

Abstract

Spontaneous symmetry breaking (SSB) is a fundamental concept in understanding continuous phase transitions. In article number 1900471, Chaohong Lee and co-workers study how to perform a simple and efficient adiabatic evolution in an SSB based on the instantaneous energy spectrum. Based on this adiabatic-parameter-fixed sweeping scheme, an inequality of the final fidelities is analytically obtained.

DOI
10.1002/andp.202070020

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Spontaneous symmetry breaking (SSB) is a fundamental concept in understanding continuous phase transitions. In article number 1900471, Chaohong Lee and co-workers study how to perform a simple and efficient adiabatic evolution in an SSB based on the instantaneous energy spectrum. Based on this adiabatic-parameter-fixed sweeping scheme, an inequality of the final fidelities is analytically obtained.
Abstract Quantum adiabatic evolution describes the dynamical evolution of a slowly driven Hamiltonian. In most systems undergoing spontaneous symmetry‐breaking transitions, the symmetry‐protected quantum adiabatic evolution can still appear, even when the two lowest eigenstates become degenerate. Here, a general derivation to revisit the symmetry‐dependent transition and the symmetry‐dependent adiabatic condition (SDAC) is given. Further, based on the SDAC, an adiabatic‐parameter‐fixed sweeping scheme is used for achieving fast adiabatic evolution, which is more efficient than the linear sweeping scheme. In the limit of small adiabatic parameter, an analytic inequality is obtained for the ground state fidelity only dependent on the adiabatic parameter. The general statements are then demonstrated via two typical systems. Besides, the robustness of the symmetry‐dependent adiabatic evolution against weak symmetry‐breaking sources is studied. The findings can be tested via the techniques in quantum annealing and may provide promising applications in practical quantum technologies.
Spontaneous symmetry breaking occurs in a physical system whenever the ground state does not share the symmetry of the underlying theory, e.g., the Hamiltonian. This mechanism gives rise to massless Nambu-Goldstone modes and massive Anderson-Higgs modes. These modes provide a fundamental understanding of matter in the Universe and appear as collective phase or amplitude excitations of an order parameter in a many-body system. The amplitude excitation plays a crucial role in determining the critical exponents governing universal nonequilibrium dynamics in the Kibble-Zurek mechanism (KZM). Here, we characterize the amplitude excitations in a spin-1 condensate and measure the energy gap for different phases of the quantum phase transition. At the quantum critical point of the transition, finite-size effects lead to a nonzero gap. Our measurements are consistent with this prediction, and furthermore, we demonstrate an adiabatic quench through the phase transition, which is forbidden at the mean field level. This work paves the way toward generating entanglement through an adiabatic phase transition.
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.
Spontaneous symmetry breaking (SSB) is the cornerstone of our understanding of quantum phases of matter. Recent works have generalized this concept to the domain of mixed states in open quantum systems, where symmetries can be realized in two distinct ways dubbed strong and weak. Novel intrinsically mixed phases of quantum matter can then be defined by the spontaneous breaking of strong symmetry down to weak symmetry. However, proposed order parameters for strong-to-weak SSB (based on mixed-state fidelities or purities) seem to require exponentially many copies of the state, raising the question: Is it possible to efficiently detect strong-to-weak SSB, in general? Here, we answer this question negatively in the paradigmatic cases of Z_{2} and U(1) symmetries. We construct ensembles of pseudorandom mixed states that do not break the strong symmetry yet are computationally indistinguishable from states that do. This rules out the existence of efficient state-agnostic protocols to detect strong-to-weak SSB.
We present arguments supporting the use of the Bogoliubov method of quasiaverages for quantum systems. First, we elucidate how it can be used to study phase transitions with spontaneous symmetry breaking (SSB). For this, we consider the example of Bose–Einstein condensation in continuous systems. Analysis of different types of generalized condensations shows that the only physically reliable quantities are those defined by Bogoliubov quasiaverages. In this connection, we also solve the Lieb–Seiringer–Yngvason problem. Second, using the scaled Bogoliubov method of quasiaverages and considering the example of a structural quantum phase transition, we examine a relation between SSB and critical quantum fluctuations. We show that the quasiaverages again provide a tool suitable for describing the algebra of critical quantum fluctuation operators in both the commutative and noncommutative cases.