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arXiv:2412.02733 · Physical review. B./Physical review. B

Mixed-state phase transitions in spin-Holstein models

Brett Min, Yuxuan Zhang, Yuxuan Guo, Dvira Segal, Yuto Ashida

Published Mar 10, 2025• 7 citations
arXivHTMLPDF

Abstract

Understanding coupled electron-phonon systems is one of the fundamental issues in strongly correlated systems. In this paper, we aim to extend the notion of mixed-state phases to the realm of coupled electron/spin-phonon systems. Specifically, we consider a two-dimensional cluster Hamiltonian locally coupled to a set of single bosonic modes with arbitrary coupling strength. First, we adopt a pure-state framework and examine whether a ground-state phase transition out of the symmetry-protected topological phase can be captured using the standard polaron unitary transformation. This approach involves restricting the analysis to the low-energy manifold of the phonon degrees of freedom. We find that the pure-state approach fails to detect the anticipated transition to a topologically trivial phase at strong spin-phonon coupling. Next, we turn to a mixed-state picture. Here, we analyze mixed states of the model obtained by tracing out the phonon degrees of freedom. We employ two distinct diagnostics for mixed-state phase transitions: (i) the von Neumann conditional mutual information (CMI) and (ii) the R\'enyi-2 CMI. We argue that both measures detect signatures of mixed-state phase transitions, albeit at different critical spin-phonon coupling strengths, each corresponding to a subtly distinct notion of the mixed-state phases.

DOI
10.1103/physrevb.111.115123

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Understanding coupled electron-phonon systems is one of the fundamental issues in strongly correlated systems. In this paper, we aim to extend the notion of mixed-state phases to the realm of coupled electron/spin-phonon systems. Specifically, we consider a two-dimensional cluster Hamiltonian locally coupled to a set of single bosonic modes with arbitrary coupling strength. First, we adopt a pure-state framework and examine whether a ground-state phase transition out of the symmetry-protected topological phase can be captured using the standard polaron unitary transformation. This approach involves restricting the analysis to the low-energy manifold of the phonon degrees of freedom. We find that the pure-state approach fails to detect the anticipated transition to a topologically trivial phase at strong spin-phonon coupling. Next, we turn to a mixed-state picture. Here, we analyze mixed states of the model obtained by tracing out the phonon degrees of freedom. We employ two distinct diagnostics for mixed-state phase transitions: (i) the von Neumann conditional mutual information (CMI) and (ii) the R'enyi-2 CMI. We argue that both measures detect signatures of mixed-state phase transitions, albeit at different critical spin-phonon coupling strengths, each corresponding to a subtly distinct notion of the mixed-state phases.
We investigate the influence of local decoherence on a symmetry-protected topological (SPT) phase of the two-dimensional (2D) cluster state. Mapping the 2D cluster state under decoherence to a classical spin model, we show a topological phase transition of a ${\mathbb{Z}}_{2}^{(0)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{(1)}$ SPT phase into a trivial phase occurring at a finite decoherence strength. To characterize the phase transition, we employ three distinct diagnostic methods, namely, the relative entropy between two decohered SPT states with different topological edge states, the strange correlation function of Z2(1){\mathbb{Z}}_{2}^{(1)}Z2(1)​ charge, and the multipartite negativity of the mixed state on a disk. All the diagnostics can be obtained as certain thermodynamic quantities in the corresponding classical model, and the results of the three diagnostic tests are consistent with each other. Given that the 2D cluster state possesses universal computational capabilities in the context of measurement-based quantum computation, the topological phase transition found here can also be interpreted as a transition in the computational power.
The polaron concept captures physical situations involving an itinerant quantum particle (excitation) that interacts strongly with bosonic degrees of freedom and becomes heavily boson-dressed. While the Gerlach-L\"owen theorem rules out the occurrence of nonanalyticities of ground-state-related quantities for a broad class of polaron models, examples were found in recent years of sharp transitions pertaining to strongly momentum-dependent interactions of an excitation with dispersionless (zero-dimensional) phonons. On the example of a lattice model with Peierls-type excitation-phonon interaction, such level-crossing-type small-polaron transitions are analyzed here through the prism of the entanglement spectrum of the excitation-phonon system. By evaluating this spectrum in a numerically exact fashion it is demonstrated that the behavior of the entanglement entropy in the vicinity of the critical excitation-phonon coupling strength chiefly originates from one specific entanglement-spectrum eigenvalue, namely the smallest one. While the discrete translational symmetry of the system implies that those eigenvalues can be labeled by the bare-excitation quasimomentum quantum numbers, here it is shown numerically that they are predominantly associated with the quasimomenta 0 and $\ensuremath{\pi}$, including cases where a transition between the two takes place deeply in the strong-coupling regime.
We investigate the extension of pure-state symmetry protected topological phases to mixed-state regime with a strong U(1) and a weak Z2\mathbb{Z}_2Z2​ symmetries in one-dimensional spin systems by the concept of quantum channels. We propose a corresponding topological phase order parameter for short-range entangled mixed states by showing that it is quantized and its distinct values can be realized by concrete spin systems with disorders, sharply signaling phase transitions among them. We also give a model-independent way to generate two distinct phases by various types of translation and reflection transformations. These results on the short-range entangled mixed states further enable us to generalize the conventional Lieb-Schultz-Mattis theorem to mixed states, even without the concept of spectral gaps and lattice Hamiltonians.
We analyze the global ground-state (quantum) phase diagram of the one-dimensional spinful Holstein model at half filling as a function of the strength of the electron-phonon coupling (represented by the strength of the phonon-induced attraction, UUU) and the phonon frequency ${\ensuremath{\omega}}_{0}$. In addition to reanalyzing the various asymptotic regimes, we carry out density-matrix renormalization group simulations to correct previous inferences concerning the antiadiabatic (large ${\ensuremath{\omega}}_{0}$) and strong-coupling (large UUU) regimes. There are two distinct phases---a fully gapped commensurate charge-density-wave phase and a spin-gapped Luther-Emery phase with a gapless charge mode---separated by a phase boundary, with a shape that reflects different microscopic physics in the weak- and strong-coupling limits.
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.
We study quantum many-body mixed states with a symmetry from the perspective of , i.e., whether a mixed state can be expressed as an ensemble of short-range-entangled symmetric pure states. We provide evidence for “symmetry-enforced separability transitions” in a variety of states, where in one regime the mixed state is expressible as a convex sum of symmetric short-range-entangled pure states, while in the other regime, such a representation is not feasible. We first discuss the Gibbs state of Hamiltonians that exhibit spontaneous breaking of a discrete symmetry, and argue that the associated thermal phase transition can be thought of as a symmetry-enforced separability transition. Next we study cluster states in various dimensions subjected to local decoherence, and identify several distinct mixed-state phases and associated separability phase transitions, which also provides an alternative perspective on recently discussed “average symmetry-protected topological order.” We also study decohered p + i p superconductors, and find that if the decoherence breaks the fermion parity explicitly, then the resulting mixed state can be expressed as a convex sum of nonchiral states, while a fermion parity–preserving decoherence results in a phase transition at a nonzero threshold that corresponds to spontaneous breaking of fermion parity. Finally, we briefly discuss systems that satisfy the no low-energy trivial state property, such as the recently discovered good low-density parity-check codes, and argue that the Gibbs state of such systems exhibits a temperature-tuned separability transition. Published by the American Physical Society 2024