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arXiv:2307.07003

Antiunitary symmetry breaking and a hierarchy of purification transitions in Floquet non-unitary circuits

Carolyn Zhang, Etienne Granet

Published Jul 13, 2023• 0 citations
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Abstract

We consider how a maximally mixed state evolves under $(1+1)D$ Floquet non-unitary circuits with an antiunitary symmetry that squares to identity, that serves as a generalized $\mathcal{PT}$ symmetry. Upon tuning a parameter, the effective Hamiltonian of the Floquet operator demonstrates a symmetry breaking transition. We show that this symmetry breaking transition coincides with different kinds of purification transitions. Gaussian non-unitary circuits are mixed (not purifying) on both sides of the symmetry breaking transition, while interacting but integrable non-unitary circuits are mixed on the symmetric side and ``weakly purifying" on the symmetry breaking side. In the weakly purifying phase, the initial mixed state purifies on a time scale proportional to the system size. We obtain numerically the critical exponents associated with the divergence of the purification time at the purification transition, which depend continuously on the parameters of the model. Upon adding a symmetric perturbation that breaks integrability, the weakly purifying phase becomes strongly purifying, purifying in a time independent of the system size, for sufficiently large system size. Our models have an extra $U(1)$ symmetry that divides the Hilbert space into different magnetization sectors, some of which demonstrate logarithmic scaling of entanglement in the weakly purifying phase.

DOI
10.48550/arxiv.2307.07003

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We consider how a maximally mixed state evolves under (1+1)D(1+1)D(1+1)D Floquet non-unitary circuits with an antiunitary symmetry that squares to identity, that serves as a generalized PT\mathcal{PT}PT symmetry. Upon tuning a parameter, the effective Hamiltonian of the Floquet operator demonstrates a symmetry breaking transition. We show that this symmetry breaking transition coincides with different kinds of purification transitions. Gaussian non-unitary circuits are mixed (not purifying) on both sides of the symmetry breaking transition, while interacting but integrable non-unitary circuits are mixed on the symmetric side and ``weakly purifying" on the symmetry breaking side. In the weakly purifying phase, the initial mixed state purifies on a time scale proportional to the system size. We obtain numerically the critical exponents associated with the divergence of the purification time at the purification transition, which depend continuously on the parameters of the model. Upon adding a symmetric perturbation that breaks integrability, the weakly purifying phase becomes strongly purifying, purifying in a time independent of the system size, for sufficiently large system size. Our models have an extra U(1)U(1)U(1) symmetry that divides the Hilbert space into different magnetization sectors, some of which demonstrate logarithmic scaling of entanglement in the weakly purifying phase.
We explore the effect of decoherence on many-body mixed states from a purification perspective. Here, quantum channels map to unitary transformations acting on a purified state defined within an extended Hilbert space. First, we exploit this approach to analyze the phenomenon of spontaneous strong-to-weak symmetry breaking (sw-SSB) triggered by strongly symmetric local quantum channels. We find that sw-SSB for mixed states relates to proximity to a symmetry-protected topological (SPT) order in the purified state. Remarkably, the measurement-induced long-range order in the purified SPT state, as characterized by the R\'enyi-2 correlator, mirrors the ensuing long-range order in the mixed state due to sw-SSB. Moreover, we establish a 1-to-1 correspondence between various sw-SSB order parameters for the mixed state, and strange correlators in the purification, which signify the SPT order. This purification perspective is further extended to explore intrinsic mixed-state topological order and decoherent symmetry-protected topological phases.
We study purification dynamics in monitored quantum processes governed by ensembles of quantum circuits in different random-matrix symmetry classes. We analyze the universal aspects that emerge away from the measurement induced phase transition and inside the volume/weak measurement phase and in the scaling limit of large time and Hilbert space dimension. We present two toy models that reveal two complementary visions and provide quantitative access to universal scaling: i) a discrete-time dynamic in which each time step corresponds to multiplication by a Gaussian random matrix; ii) weak continuous-time monitoring that induces a Dyson brownian motion of the eigenvalues of the density matrix. The first approach provides an algebraic characterization based on rotational invariance emerging in Kraus's operator space, focusing in particular on the unitary and orthogonal cases, respectively β=2β=2β=2 and β=1β=1β=1, with βββ the Dyson random-matrix index. The second approach, on the other hand, allows for a unified treatment for any βββ, thanks to the mapping of the Fokker-Planck evolution of eigenvalues onto the Calogero-Sutherland integrable Hamiltonian diagonalized in terms of Jack polynomials. We provide explicit expressions for the universal decrease of Rényi entropies. We show that, approaching the universal scaling limit, numerical simulations of different models agree with each other and with our theoretical predictions. Our results clarify the existence of different classes of universality for the purification process in hybrid quantum systems, accessible in random circuit architectures and weak measurement protocols.
Open quantum systems host mixed-state phases that go beyond the symmetry-protected topological and spontaneous symmetry-breaking paradigms established for closed, pure-state systems. Developing a unified and physically transparent classification of such phases remains a central challenge. In this work, we introduce a purification-based framework that systematically characterizes all mixed-state phases in one-dimensional systems with Z2σ×Z2τ\mathbb{Z}_2^σ \times \mathbb{Z}_2^τZ2σ​×Z2τ​ symmetry. By introducing an ancillary κκκ chain and employing decorated domain-wall constructions, we derive eight purified fixed-point Hamiltonians labeled by topological indices (μστ,μτκ,μκσ)∈{±1}3(μ_{στ},μ_{τκ},μ_{κσ}) \in \{\pm1\}^3(μστ​,μτκ​,μκσ​)∈{±1}3. Tracing out the ancilla recovers the full structure of mixed-state phases, including symmetric, strong-to-weak spontaneous symmetry breaking, average symmetry-protected topological phases, and their nontrivial combinations. Interpolations between the eight fixed points naturally define a three-dimensional phase diagram with a cube geometry. The edges correspond to elementary transitions associated with single topological indices, while the faces host intermediate phases arising from competing domain-wall decorations. Along the edges, we identify a class of critical behavior that connects distinct strong-to-weak symmetry-breaking patterns associated with distinct strong subgroups, highlighting a mechanism unique to mixed-state settings. Large-scale tensor-network simulations reveal a rich phase structure, including pyramid-shaped symmetry-breaking regions and a fully symmetry-broken phase at the cube center. Overall, our purification approach provides a geometrically transparent and physically complete classification of mixed-state phases, unified with a single Z2σ×Z2τ×Z2κ\mathbb{Z}_2^σ \times \mathbb{Z}_2^τ \times \mathbb{Z}_2^κZ2σ​×Z2τ​×Z2κ​ model.
Time evolution of quantum many-body systems typically leads to a state with maximal entanglement allowed by symmetries. Two distinct routes to impede entanglement growth are inducing localization via spatial disorder, or subjecting the system to non-unitary evolution, e.g., via projective measurements. Here we employ the idea of space-time rotation of a circuit to explore the relation between systems that fall into these two classes. In particular, by space-time rotating unitary Floquet circuits that display a localization transition, we construct non-unitary circuits that display a rich variety of entanglement scaling and phase transitions. One outcome of our approach is a non-unitary circuit for free fermions in 1d that exhibits an entanglement transition from logarithmic scaling to volume-law scaling. This transition is accompanied by a 'purification transition' analogous to that seen in hybrid projective-unitary circuits. We follow a similar strategy to construct a non-unitary 2d Clifford circuit that shows a transition from area to volume-law entanglement scaling. Similarly, we space-time rotate a 1d spin chain that hosts many-body localization to obtain a non-unitary circuit that exhibits an entanglement transition. Finally, we introduce an unconventional correlator and argue that if a unitary circuit hosts a many-body localization transition, then the correlator is expected to be singular in its non-unitary counterpart as well.
A quantum system subject to continuous measurement and postselection evolves according to a non-Hermitian Hamiltonian. We show that, as one increases the strength of postselection, this non-Hermitian Hamiltonian can undergo a spectral phase transition. On one side of this phase transition (for weak postselection), an initially mixed density matrix remains mixed at all times, and an initially unentangled state develops volume-law entanglement; on the other side, an arbitrary initial state approaches a unique pure state with low entanglement. We identify this transition with an exceptional point in the spectrum of the non-Hermitian Hamiltonian, at which PT symmetry is spontaneously broken. We characterize the transition as well as the nontrivial steady state that emerges at late times in the mixed phase using exact diagonalization and an approximate, analytically tractable mean-field theory; these methods yield consistent conclusions.
We investigate the crucial role played by a global symmetry in the purification timescales and the phase transitions of monitored free fermionic systems separating a mixed and a pure phase. Concretely, we study Majorana and Dirac circuits with Z2{\mathbb{Z}}_{2}Z2​ and U(1) symmetries, respectively. In the first case, we demonstrate the mixed phase of LLL sites has a purification timescale that scales as ${\ensuremath{\tau}}_{P}\ensuremath{\sim}LlnL$. At $1\ensuremath{\ll}t\ensuremath{\ll}{\ensuremath{\tau}}_{P}$ the system attains a finite residual entropy, that we use to unveil the critical properties of the purification transition. In contrast, free fermions with U(1) manifest a sublinear purification timescale at any measurement rate and an apparent Berezinskii-Kosterlitz-Thouless criticality. We find the mixed phase is characterized by ${\ensuremath{\tau}}_{P}\ensuremath{\sim}{L}^{\ensuremath{\alpha}(p)}$, with a continuously varying exponent $\ensuremath{\alpha}(p)<1$.