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arXiv:2502.16252 · Physical review. B./Physical review. B

Boundary perturbation effects in quantum systems with conserved energy and continuous symmetry

Qu-Cheng Gao, Xiao Chen

Published Feb 23, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate one-dimensional systems with both energy conservation and a continuous symmetry, focusing on the impact of a boundary perturbation that breaks the continuous symmetry. Our study reveals two distinct dynamical phases: one in which the corresponding charge exhibits extensive fluctuations, and another where the charge remains conserved. These phases appear in both free and interacting models. We interpret this behavior through a boundary-induced pumping mechanism, which estimates the amplitude connecting two degenerate states from different charge sectors via a local charge-non-conserving operator. In the Floquet setting, we show that the frozen phase can survive at high driving frequencies but vanishes at low frequencies. This phenomenon is exact in free-fermion systems in the thermodynamic limit, but in interacting systems it appears only at finite system size. The emergence of the charge-frozen phase is attributed to effective energy conservation, and we demonstrate that this phase disappears when effective energy conservation is broken or replaced by other symmetries.

DOI
10.1103/5j2h-ftvc

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We investigate one-dimensional systems with both energy conservation and a continuous symmetry, focusing on the impact of a boundary perturbation that breaks the continuous symmetry. Our study reveals two distinct dynamical phases: one in which the corresponding charge exhibits extensive fluctuations, and another where the charge remains conserved. These phases appear in both free and interacting models. We interpret this behavior through a boundary-induced pumping mechanism, which estimates the amplitude connecting two degenerate states from different charge sectors via a local charge-non-conserving operator. In the Floquet setting, we show that the frozen phase can survive at high driving frequencies but vanishes at low frequencies. This phenomenon is exact in free-fermion systems in the thermodynamic limit, but in interacting systems it appears only at finite system size. The emergence of the charge-frozen phase is attributed to effective energy conservation, and we demonstrate that this phase disappears when effective energy conservation is broken or replaced by other symmetries.
We present a theory characterizing the phases emerging as a consequence of continuous symmetry-breaking in quantum and classical systems. In symmetry-breaking phases, dynamics is restricted due to the existence of a set of conserved charges derived from the order parameter of the phase transition. Their expectation values are determined by the privileged direction appearing in the ordered phase as a consequence of symmetry breaking, and thus they can be used to determine whether this direction is well defined or has quantum fluctuations. Our theory is numerically exemplified via the two-dimensional limit of the vibron model, a fully connected system invariant under a rotation operator which generates the continuous symmetry-breaking.
Quantized charge pumping in quantum Hall systems is a bulk topological property, typically manifesting on gapless boundaries as spectral flow through chiral edge modes upon flux insertion. When the boundary is anomaly-free, it can be symmetrically gapped by symmetric mass generation (SMG) interactions. However, the effects on quantized charge pumping at an interacting gapped boundary are unclear. We address this using density-matrix renormalization group simulations of a four-leg ladder quasi-(2+1)d realization of the (1+1)d 3-4-5-0 lattice chiral fermion model with six-fermion interactions at one edge. We find gapless modes near the interacting edge for both weak and strong couplings, but the edge can be in a fully gapped SMG phase at intermediate coupling. We also identify spontaneous breaking of the model's Z2\mathbb{Z}_2Z2​ symmetry above a critical interaction strength. Across all three boundary phases, we find quantized charge pumping with the same pumped charge per flux quantum. The spatial distribution of the pumped charge varies between phases. Besides charge pumping, we study how a single injected fermion propagates under open boundary conditions. In the SMG phase, an interaction-induced flavor-mixing current develops at the endpoints of the gapped edge, which is absent in the weak- and strong-coupling limits. This current reveals a charge-redistribution mechanism unique to the SMG phase. Additionally, we propose an analogy between charge pumping in this model and a topological leptogenesis scenario in the (3+1)d Standard Model. The anomaly-free chiral U(1) symmetries of the 3-4-5-0 model play a role analogous to the baryon-minus-lepton symmetry of the Standard Model. In both settings, charge pumping can be associated with a continuous deformation along a classically defined but quantum-anomalous direction, while a discrete subgroup remains an exact quantum symmetry.
We investigate how symmetries and conserved quantities relate to the occurrence of the boundary time crystal (BTC) phase in a generalized spin model with Lindblad dissipation. BTCs are a nonequilibrium phase of matter in which the system, coupled to an external environment, breaks the continuous time translational invariance. We perform a detailed mean-field study aided by a finite-size analysis of the quantum model of a p,qp,qp,q-spin-interaction system, a generalized ppp-spin-interaction system, which can be implemented in fully connected spin-1/2 ensembles. We find the following conditions for the observation of the BTC phase. First, the BTC appears when the discrete symmetry held by the Hamiltonian, Z2{\mathbb{Z}}_{2}Z2​ in the considered models, is explicitly broken by the Lindblad jump operators. Second, the system must be coupled uniformly to the same bath in order to preserve the total angular momentum during the time evolution. If these conditions are not satisfied, any oscillatory behavior appears only as a transient in the dynamics and a time-independent stationary state is eventually reached. Our results suggest that these two elements may be general requirements for the observation of a stable BTC phase relating symmetries and conserved quantities in arbitrary spin models.
“Deep thermalization” describes the emergence of universal wave-function distributions in quantum many-body dynamics, appearing on a local subsystem upon measurement of its environment. In this work, we study in detail the effect of continuous internal symmetries and associated conservation laws on deep thermalization. Concretely, we consider quantum spin systems with a U ( 1 ) symmetry associated with the conservation of magnetization (or “charge”) and analyze how the choice of initial states (specifically, their degree of charge fluctuations) and the choice of measurement basis (specifically, whether or not it can reveal information about the local charge density) determine the ensuing universal wave-function distributions. We put forth a universal ansatz for the limiting form of the projected ensemble, motivated by maximum-entropy principles rooted in statistical physics and quantum information theory. This limiting form depends on a polynomial amount of data on the initial state and measurement basis, a “coarse graining” that is an essential feature of thermodynamic ensembles. We support our ansatz with three complementary approaches: (i) a rigorous proof in the simplest case of no charge fluctuations in either the initial state or the measurement basis; (ii) analytical calculations using a “replica-limit” approach, applicable when charge fluctuations are allowed in either the input state or the measurement basis but not both; and (iii) extensive numerical simulations of finite-sized systems in the most general case. Our findings demonstrate a rich interplay between symmetries and the information extracted by measurements, which allows deep thermalization to exhibit a range of universal behaviors far beyond regular thermalization.
We demonstrate the existence of a universal phase diagram of quantum chains with range- k interactions subject to the conservation of a total charge and its dipole moment. These systems exhibit “freezing” transitions between strongly and weakly Hilbert-space-fragmented phases as the charge filling ν is varied. We show that these continuous phase transitions occur at a critical charge filling of ν c = ( k − 2 ) − 1 of the on-site Hilbert-space dimension d . To this end, we analytically prove that, for any d , any state with ν < ν c hosts a finite density of sites belonging to “blockages,” which we define as subregions of the chain across which transport of charge and dipole moment cannot occur. Some blockages arise from sequences of frozen sites, i.e., sites with an unchanging on-site charge, while others do not involve frozen sites at all. We prove that the presence of blockages implies strong fragmentation of typical symmetry sectors into Krylov subspaces, each of which forms an exponentially vanishing fraction of the total sector. By studying the distribution of blockages we analytically characterize how typical states are subdivided into dynamically disconnected local “active bubbles” and prove that typical eigenstates at these charge fillings exhibit area-law entanglement entropy, while there exist rare eigenstates featuring non-area-law scaling. We also numerically show that for ν > ν c and arbitrary d , typical symmetry sectors are weakly fragmented, with their dominant Krylov sectors constituted of states that are free of blockages. We analytically derive some critical exponents characterizing the transition and numerically determine the density of blockages at ν = ν c , with clear implications for transport at the critical point. Finally, we investigate the properties of special-case models for which no phase transitions occur.
We study the dynamics of a class of integrable non-Hermitian free-fermionic models driven periodically using a continuous drive protocol characterized by an amplitude g1{g}_{1}g1​ and frequency ${\ensuremath{\omega}}_{D}$. We derive an analytic, albeit perturbative, Floquet Hamiltonian for describing such systems using Floquet perturbation theory with ${g}_{1}^{\ensuremath{-}1}$ being the perturbation parameter. Our analysis indicates the existence of special drive frequencies at which an approximately conserved quantity emerges. The presence of such an almost conserved quantity is reflected in the dynamics of the fidelity, the correlation functions, and the half-chain entanglement entropy of the driven system. In addition, it also controls the nature of the steady state of the system. We show that one-dimensional transverse field Ising model, with an imaginary component of the transverse field, serves as an experimentally relevant example of this phenomenon. In this case, the transverse magnetization is approximately conserved; this conservation leads to complete suppression of oscillatory features in the transient dynamics of fidelity, magnetization, and entanglement of the driven chain at special drive frequencies. We discuss the nature of the steady state of the Ising chain near and away from these special frequencies, demonstrate the protocol independence of this phenomenon by showing its existence for discrete drive protocols, and suggest experiments which can test our theory.