Dynamics in the presence of local symmetry-breaking impurities

TL;DR

The paper develops a unified framework to study how local symmetry-breaking impurities perturb the late-time dynamics of symmetric quantum systems, using Brownian circuits and a superoperator formulation. By mapping conserved and approximately conserved quantities to ground and low-lying states of a perturbed super-Hamiltonian, it shows that impurities are RG-relevant in one dimension and modify hydrodynamic modes, yielding diffusion-to-boundary-absorbing transitions for U$(1)$ and subdiffusion-to-boundary transitions for dipole-conserving systems. In strongly fragmented models, impurities lift massive ground-state degeneracies and induce exponentially slow relaxation governed by SLIOMs, producing long prethermal plateaus in both boundary and bulk correlations; two-boundary impurities can alter this dramatically, potentially restoring ergodicity on polynomial scales. The results offer a systematic route to predict how local impurities affect relaxation in a broad class of symmetric, non-integrable systems and suggest extensions to higher dimensions and more complex symmetries. Overall, the work provides precise, analytically tractable descriptions of impurity-driven relaxation and highlights the universality of impurity effects across Brownian-circuit models and cellular automata.

Abstract

Continuous symmetries lead to universal slow relaxation of correlation functions in quantum many-body systems. In this work, we study how local symmetry-breaking impurities affect the dynamics of these correlation functions using Brownian quantum circuits, which we expect to apply to generic non-integrable systems with the same symmetries. While explicitly breaking the symmetry is generally expected to lead to eventual restoration of full ergodicity, we find that approximately conserved quantities that survive under such circumstances can still induce slow relaxation. This can be understood using a super-Hamiltonian formulation, where low-lying excitations determine the late-time dynamics and exact ground states correspond to conserved quantities. We show that in one dimension, symmetry-breaking impurities modify diffusive and subdiffusive behaviors associated with U$(1)$ and dipole conservation at late-times, e.g., by increasing power-law decay exponents of the decay of autocorrelation functions. This stems from the fact that for these symmetries, impurities are relevant in the renormalization group sense, e.g., bulk impurities effectively disconnect the system, completely modifying both temporal and spatial correlations. On the other hand, for an impurity that disrupts strong Hilbert space fragmentation, the super-Hamiltonian only acquires an exponentially small gap, leading to prethermal plateaus in autocorrelation functions which extend for times that scale exponentially with the distance to the impurity. Overall, our approach systematically characterizes how symmetry breaking impurities affect relaxation dynamics in symmetric systems.

Figures (17)

• Figure 1: Effect of symmetry-breaking impurities. (a) A one-dimensional chain with a local symmetry-breaking impurity (red triangle) at the boundary. (b) Energy spectra of the super-Hamiltonians. Without any impurity, the super-Hamiltonian $\hat{\mathcal{P}}$ has (possibly highly) degenerate zero-energy ground states corresponding to (exact) conserved quantities. With an impurity, the ground state degeneracy is lifted, modifying the low-lying spectrum of $\hat{\mathcal{P}}_{\mathrm{imp}}$, which in turn alters the long-time behaviors. (c) Nature of energy spectra and autocorrelation functions $C_{Z_j}(t)$ at site $j$ for charge- or dipole-conserving systems with fully symmetry-breaking impurities. The energy spectrum of relevant eigenstates that contribute to correlation functions, which correspond to approximately conserved quantities (blue dots for unperturbed eigenstates and red crosses for eigenstates under the perturbation). The energy gap scales as $\mathcal{O}(L^{-z})$ with or without impurity, but the structure of the approximately conserved quantities is modified, with $z$ the corresponding dynamical exponent. The autocorrelation function $C_{Z_j}(t)$ (shown in log-log scale) exhibits three regimes: (i) At $t \lesssim \mathcal{O}(j^z)$ before the impurity takes effect, it decays as $t^{-1/z}$. (ii) In the time window $\mathcal{O}(j^z) \lesssim t \lesssim \mathcal{O}(L^z)$, the impurity has taken effect and it decays as $j^z t^{-(1+1/z)}$. (iii) At timescales $t\gtrsim \mathcal{O}(L^z)$, the correlation decays exponentially as $j^z L^{-(z+1)} e^{-t/L^z}$ due to the finiteness of the system. (d) Nature of the energy spectrum and autocorrelation functions of strongly fragmented systems with fully symmetry-breaking impurities. The exponentially large unperturbed ground state degeneracy ($\dim \mathcal{C}~\sim \mathcal{O}(e^{L})$) is lifted in the presence of symmetry-breaking impurities, opening up an energy gap $\mathcal{O}(e^{-f(L)})$, where $f(x)$ depends on the specific model. The gray area indicates a continuous spectrum with a polynomial gap $L^{-\beta}$ (distance between the lowest black solid line and the lower part of the gray area). The autocorrelation function then exhibits the following regimes: (i) At $t\lesssim O(j^\alpha)$, it decays to a prethermal plateau whose value is mainly determined by the SLIOMs localized around the evaluation point $j$. (ii) The prethermal plateau lasts exponentially long for a time determined by the distance $j$ to the impurity, i.e., for $O(j^\alpha) \lesssim t \lesssim \mathcal{O}(e^{f(j)})$. This is exponentially long in system size for $j = x L$ keeping $x$ fixed making the flat plateau window sharply defined. (iii) Finally, at the longest times the autocorrelation function decays exponentially at times $t \gtrsim \mathcal{O}(e^{f(L)})$ when all approximately conserved quantities have decayed. The possible emergent "hydrodynamic" behaviors connecting these different regimes depends on the specifics of the strongly-fragmented models and lies beyond the scope of this work.
• Figure 2: Correlation functions $C_{Z_j}$ of U($1$)-symmetric systems with charge-breaking impurity at the boundary $j_s=1$. (a) Correlation functions with single particle Hamiltonian Eq. \ref{['eq:U1_Heff_imp']} for $L=10000$ and $g=1$. For $j-j_s \geq \mathcal{O}(\sqrt{Dt})$, the correlation functions first decays as $t^{-1/2}$, and then transits to $t^{-3/2}$ when the impurity takes effect. The transition time $t_{\text{tran}}$ for correlation function $C_{Z_j}$ scales as $t_{\text{tran}} \sim (j-j_s)^2$, which is shown in the inset. The grey (black) dashed lines indicate the $t^{-1/2} (t^{-3/2})$ scalings. (b) Correlation functions obtained from stochastic cellular automata for $L=100$. Each curve is averaged over $10^8$ random realizations. The correlation functions show similar behaviors as given by the single-particle Hamiltonian, which decay as $t^{-1/2}$ and then transit to $t^{-3/2}$. This verifies the analytic calculations.
• Figure 3: Eigenstates $\phi_{k_n}(x)$ of dipole-conserving systems with or without impurities. Comparison of solutions of the continuum equations with boundary conditions (solid lines) and eigenstates of the single-particle Hamiltonian for the hydro-mode in the super-Hamiltonian approach (circles) for dipole-conserving systems with system size $L=200$ (upper panel) and $L=1000$ (lower panel), for the cases (a) and (d) without impurity, (b) and (e) with charge-preserving impurity on the left boundary, and (c) and (f) with fully symmetry-breaking impurity. The impurity strength of the single-particle Hamiltonian is $g=1$. For the case of eigenstates on the lattice, we only plot the value of $\phi_{k_n}(x)$ for $25$ equispaced data points. The solutions of the continuum equation with boundary conditions in Eq. \ref{['eq:Dipole_boundary_no_imp']} for the case without impurity, Eq. \ref{['eq:Dipole_boundary_breakP']} for charge-preserving impurity, and Eq. \ref{['eq:Dipole_boundary_breakPQ']} for fully symmetry-breaking impurity are compatible with eigenstates of the single-particle Hamiltonian, validating the choice of the boundary conditions for the continuum equations.
• Figure 4: Bulk autocorrelation functions of the dipole-conserving system with a local dipole-breaking but charge-preserving impurity at the left boundary $j_s = 1$. (a) Correlation functions obtained from the single particle Hamiltonian Eq. \ref{['eq:H_breakP']} with dipole-breaking but charge-conserving impurity, for system size $L=10000$ and $g=1$ The correlation functions then decay as $t^{-1/4}$ (grey dashed lines) at long times before saturation (due to charge conservation). In addition, a front appears at intermediate times, as the particles reflect back from the boundary due to particle conservation. The inset shows that the front appears in $C_{Z_j}$ at time $t_{\text{front}}$ that scales as $t_{\mathrm{front}} \sim (j-j_s)^4$, which is compatible with the reflective boundary prediction. (b) Correlation functions from stochastic dynamics with spin-$1/2$ and $4$- and $5$-local dipole-conserving terms, with system size $L=100$. The correlation functions show a front at intermediate times and $t^{-1/4}$ scaling at long times, similar to the results from the single particle Hamiltonian.
• Figure 5: Bulk autocorrelation functions of the dipole-conserving system with fully symmetry-breaking impurities at the left boundary on three consecutive sites $j_s = 1$, $j_s+1$, $j_s+2$. (a) Correlation functions obtained from hydro-mode single particle Hamiltonian for the spin-$1/2$ system described by Eq. \ref{['eq:H_breakPQ']}. The system size is $L=10000$ with $g=1$. The correlation functions decay as $t^{-1/4}$ (grey dashed line) at early times until the boundary impurity takes effect, then decay as $t^{-5/4}$ (black dashed line) at long times. (b) Correlation functions from stochastic dynamics with $4$-local and $5$-local dipole-conserving gates and fully symmetry-breaking local impurities (three-site spin flips at the boundary), for system size $L=100$ and spin-$1/2$. The stochastic dynamics show similar behaviors as the single particle Hamiltonian. Each curve is averaged over more than $10^8$ random samples.