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arXiv:2412.12738 · Physical review. B./Physical review. B

Strong and weak symmetries and their spontaneous symmetry breaking in mixed states emerging from the quantum Ising model under multiple decoherence

Takahiro Orito, Yoshihito Kuno, Ikuo Ichinose

Published Feb 10, 2025• 13 citations
arXivHTMLPDF

Abstract

Discovering and categorizing quantum orders in mixed many-body systems are currently one of the most important problems. Specific types of decoherence applied to typical quantum many-body states can induce a novel kind of mixed state accompanying characteristic symmetry orders, which has no counterparts in pure many-body states. We study phenomena generated by the interplay between two types of decoherence applied to the one-dimensional transverse field Ising model (TFIM). We show that in the doubled Hilbert space formalism, the decoherence can be described by a filtering operation applied to matrix product states (MPS) defined in the doubled Hilbert system. The filtering operation induces specific deformation of the MPS, which approximates the ground state of a certain parent Hamiltonian in the doubled Hilbert space. In the present case, such a parent Hamiltonian is the quantum Ashkin-Teller model, having a rich phase diagram with critical lines and quantum phase transitions. By investigating the deformed MPS, we find various types of mixed states emergent from the ground states of the TFIM and clarify phase transitions between them. In that study, strong and weak ${Z}_{2}$ symmetries play an important role, for which we introduce efficient order parameters, such as R\'enyi-2 correlators, entanglement entropy, etc., in the doubled Hilbert space.

DOI
10.1103/physrevb.111.054106

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Discovering and categorizing quantum orders in mixed many-body systems are currently one of the most important problems. Specific types of decoherence applied to typical quantum many-body states can induce a novel kind of mixed state accompanying characteristic symmetry orders, which has no counterparts in pure many-body states. We study phenomena generated by the interplay between two types of decoherence applied to the one-dimensional transverse field Ising model (TFIM). We show that in the doubled Hilbert space formalism, the decoherence can be described by a filtering operation applied to matrix product states (MPS) defined in the doubled Hilbert system. The filtering operation induces specific deformation of the MPS, which approximates the ground state of a certain parent Hamiltonian in the doubled Hilbert space. In the present case, such a parent Hamiltonian is the quantum Ashkin-Teller model, having a rich phase diagram with critical lines and quantum phase transitions. By investigating the deformed MPS, we find various types of mixed states emergent from the ground states of the TFIM and clarify phase transitions between them. In that study, strong and weak Z2{Z}_{2}Z2​ symmetries play an important role, for which we introduce efficient order parameters, such as R'enyi-2 correlators, entanglement entropy, etc., in the doubled Hilbert space.
Strong-to-Weak Spontaneous Symmetry Breaking in a (2+1)(2+1)(2+1)D Transverse-Field Ising Model under Decoherence
Decoherence in many-body quantum systems can give rise to intrinsically mixed-state phases and phase transitions beyond the pure-state paradigm. Here we study the (2+1)(2+1)(2+1)D transverse-field Ising model subject to a strongly Z2\mathbb{Z}_2Z2​-symmetric decoherence channel, with a focus on strong-to-weak spontaneous symmetry breaking (SWSSB). This problem is challenging because the relevant transitions occur in the strong-decoherence regime, beyond the reach of perturbative expansions around the pure-state limit, while conventional quantum Monte Carlo (QMC) methods are hampered by the need to access nonlinear observables and by the sign problem. We overcome these difficulties by developing a QMC algorithm that efficiently evaluates nonlinear Rényi-2 correlators in higher dimensions, complemented by an effective field-theoretic approach. We show that the decohered state realizes a rich mixed-state phase diagram governed by an effective 2D Ashkin-Teller theory. This theory enables analytical predictions for the mixed-state phases and the universality classes of the phase boundaries, all of which are confirmed by large-scale QMC simulations.
There are two kinds of symmetries in quantum mixed states (QMSs) -- called strong and weak symmetries-- and studying their spontaneous symmetry breaking (SSB) pattern helps classify QMSs, which is an ongoing important issue. The authors study here a decohered cluster model with ${Z}_{2}\ensuremath{\bigotimes}{Z}_{2}$ symmetry and uncover two kinds of symmetry-protected topological (SPT) mixed states. Using doubled Hilbert formalism, they demonstrate a transition between the two states, named strong and double average SPT states, where the latter features coexisting SPT and SSB.
Symmetry breaking has been a central theme in classifying quantum phases and phase transitions. Recently, this concept has been extended to the mixed states of open systems, attracting considerable attention due to the emergence of novel physics beyond closed systems. In this work, we reveal a new type of phase transition in mixed states, termed strong-to-weak spontaneous symmetry breaking (SWSSB). Using a combination of field theory calculations and large-scale matrix product state simulations, we map out the global phase diagram of the X X Z critical spin chain under local strong symmetry preserving decoherence, which features an SWSSB phase and a trivial Luttinger liquid phase, separated by a straight critical line that belongs to the boundary Berezinskii-Kosterlitz-Thouless universality class with a varying effective central charge. Importantly, we analyze this transition from two complementary perspectives: on one hand, through the behavior of order parameters that characterize the symmetry breaking; on the other hand, from a quantum information viewpoint by studying entropic quantities and the concept of quantum recoverability. This dual approach allows us to provide a more comprehensive understanding of the phase structure and the nature of the transition. Remarkably, the SWSSB transition in our case is in the sense that it can only be driven by tuning the Hamiltonian parameter even under arbitrary decoherence strength, fundamentally distinguishing it from the decoherence-driven SWSSB transitions extensively discussed in previous literature. Importantly, our unified theoretical framework is applicable to a broad class of one-dimensional quantum systems, including spin chains and fermionic systems, whose low-energy physics can be described by Luttinger liquid theory, under arbitrary symmetry-preserving decoherence channels. Finally, we also discuss the experimental relevance of our theory on quantum simulator platforms.
Discovering mixed state quantum orders is an ongoing issue. Recently, it has been recognized that there are (at least) two kinds of symmetries in the mixed state: strong and weak symmetries. Under symmetry-respective decoherence, spontaneous strong-to-weak symmetry breaking (SSSB) can occur. This paper provides a scheme to describe SSSB and other decoherence phenomena in the mixed state by employing the stabilizer formalism and the efficient numerical algorithm of Clifford circuits. We present two systematic numerical studies. In a two-dimensional circuit with a stochastic Ising type decoherence, an SSSB phase transition is clearly observed and its criticality is elucidated by the numerical methods. In particular, we calculate R\'enyi-2 correlations and estimate critical exponents of the SSSB transition. For the second system, we introduce an idea of subgroup SSSB. As an example, we study a system with symmetry-protected-topological order provided by both one-form and zero-form symmetries, and observe how the system evolves under decoherence. After displaying numerical results, we show that viewpoint of percolation is quite useful to understand the SSSB transition, which is applicable for a wide range of decohered states. Finally, we comment on SSSB of one-form-symmetry exemplifying toric code.
Decoherence on quantum many-body systems induces nontrivial mixed states with specific properties that have no counterparts in pure states. The authors clarify here the emergence of ``intrinsic mixed state topological order'' in a toric code state under decoherence, which accompanies exotic anyon proliferation. Furthermore, the authors find that the decoherence-induced phase transition in the stabilizer formalism is described by a percolation picture in two dimensions. The viewpoint of one-form symmetry and its spontaneous symmetry breaking plays an important role in this study.
In mixed quantum states, the notion of symmetry is divided into two types: strong and weak symmetry. While spontaneous symmetry breaking (SSB) for a weak symmetry is detected by two-point correlation functions, SSB for a strong symmetry is characterized by the Renyi-2 correlators. In this work, we present a way to construct various SSB phases for strong symmetries, starting from the ground state phase diagram of lattice gauge theory models. In addition to introducing a new type of mixed-state topological phases, we provide models of the criticalities between them, including those with gapless symmetry-protected topological order. We clarify that the ground states of lattice gauge theories are purified states of the corresponding mixed SSB states. Our construction can be applied to any finite gauge theory and offers a framework to study quantum operations between mixed quantum phases.