Boundary Perturbation Effects in Quantum Systems with Conserved Energy and Continuous Symmetry

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.

Figures (14)

• Figure 1: Free-fermion chain. (a) The density of charge variance $\delta N^2/L$ as a function of $\mu_0$ for different system sizes. (b) The charge variance $\delta N^2$ as a function of $\mu_0$ for different system sizes on a log-lin scale. Here, $t_0=\Delta=1$, $\nu=0.5$, and the results are averaged over $1000$ samples. (c) The charge difference $\delta N$ as a function of $\mu_0$ for different filling factors. Note that when the initial state is at half-filling, the charge difference cannot detect the phase transition. Here, $t_0=\Delta=1$, $L=100$, and the results are averaged over $1000$ samples.
• Figure 2: Interacting fermionic chain. (a) The density of charge fluctuation $\delta N^2/L$ as a function of $\mu_0$ for different system sizes. (b) The charge variance $\delta N^2$ as a function of $\mu_0$ for different system sizes on a log-lin scale. Here, $t_0=\Delta=1$, $U=2$, $\nu=0.5$, and the results are averaged over $200$ samples. [(c) and (d)] Same as (b) and (c), but with $\mu_0=4$ fixed and $U$ varied. (e) Two-dimensional phase diagram of $\mu_0$ and $U$. The blue region represents extensive charge fluctuation and is labeled as "Fluctuating charge". The gray region represents negligible charge fluctuation and is labeled as "Frozen charge".
• Figure 3: Interacting spin chain. (a) The density of charge fluctuation $\delta {S^z}^2/L$ as a function of $h$ for different system sizes. (b) The charge variance $\delta {S^z}^2$ as a function of $h$ for different system sizes on a log-lin scale. Here, $J^\perp=\Delta=1$, $J^z=1$, and the results are averaged over $200$ samples. [(c) and (d)] Same as (b) and (c), but with $h=7$ fixed and $J^z$ varied. (e) Two-dimensional phase diagram of $h$ and $J^z$. The blue region represents extensive charge fluctuation and is labeled as "Fluctuating charge". The gray region represents negligible charge fluctuation and is labeled as "Frozen charge".
• Figure 4: Part of the many-body spectrum of $\hat{H}_0$ of free-fermion chain. The shaded gray area represents eigenstates that share the same labeled particle number: $N-2$, $N$, or $N+2$. $E_n^0$ is one particular eigenvalue of $\hat{H}_0$.
• Figure 5: The structure of $\hat{P}\hat{H}\hat{P}$. Interacting fermionic chain: (a) The absolute value of the overlap between $\hat{c}_1^{\dagger}\hat{c}_2^{\dagger}\left| N \right\rangle$ and $\left| N' \right\rangle$ for different values of $\mu_0$. Here, $L=12$, $t_0=\Delta=1$, $U=2$, and the results are averaged over $1000$ pairs of eigenstates with an energy difference smaller than $0.1$, satisfying $|N'-N-2|<10^{-5}$. [Corresponding to Fig. \ref{['fig2']}(a) and Fig. \ref{['fig2']}(b)]. (b) Same as (a), but with $\mu_0=4$ fixed and $U$ varied. [Corresponding to Fig. \ref{['fig2']}(c) and Fig. \ref{['fig2']}(d)]. Interacting spin chain: (c) The absolute value of the overlap between $\hat{H}_B\left| S^z \right\rangle$ and $\left| {S^z}' \right\rangle$ for different values of $h$. Here, $L=12$, $J^\perp=\Delta=1$, $J^z=1$, and the results are averaged over $1000$ pairs of eigenstates with an energy difference smaller than $0.3$, satisfying $|{S^z}'-S^z-1|<0.005$ or $|{S^z}'-S^z+1|<0.005$. [Corresponding to Fig. \ref{['fig3']}(a) and Fig. \ref{['fig3']}(b)]. (d) Same as (c), but with $h=7$ fixed and $J^z$ varied. [Corresponding to Fig. \ref{['fig3']}(c) and Fig. \ref{['fig3']}(d)].

Theorems & Definitions (6)

• Lemma 1: Generic 1D short-range
• proof
• Lemma 2: Finite-size refinement in 1D
• proof
• Lemma 3: Free (quadratic) $H_0$
• proof