Quantum Breakdown Condensate as a Disorder-Free Quantum Glass

Abstract

We study the phase diagram of a one-dimensional spin quantum breakdown model, which has an exponential $U(1)$ symmetry with charge unit decaying as $2^{-j}$ with site position $j$. By exact diagonalization (ED), we show that the model with spin $S\ge2$ exhibits an exponential $U(1)$ spontaneous symmetry breaking (SSB) phase dubbed a quantum breakdown condensate. It exhibits a bulk gap violating the Goldstone theorem, and an edge mode only on the left edge if in open boundary condition. In a length $L$ lattice, the condensate has $\mathcal{O}(2^L)$ number of SSB ground states originating from the $\mathcal{O}(2^L)$ number of exponential $U(1)$ charge sectors, leading to a finite entropy density $\ln 2$. This enforces a first order SSB phase transition into this phase, as observed in ED and verified in the large $S$ limit on an exactly solvable Rokhsar-Kivelson line. The condensate has an SSB order parameter being the local in-plane spin, which points in angles related by the chaotic Bernoulli (dyadic) map and thus is effectively random. Moreover, we show the condensate exhibits non-decaying local autocorrelations, and does not have an off-diagonal long-range order. The quantum breakdown condensate thus behaves as a disorder-free quantum glass and is beyond the existing classifications of phases of matter.

Figures (4)

• Figure 1: (a) Illustration of the spin breakdown model in Eq. \ref{['eq:spin_ham']}. (b) The in-plane spin directions on different sites for the quantum breakdown condensate under PBC with order parameter $\langle S_j^+\rangle=\alpha_S e^{i2^{L-j}\theta}$. We take $L=16$ and $\theta=\frac{2\pi}{2^L-1}k$ with representative $k$ values labeled in the plot.
• Figure 2: ED calculations for $\lambda=h=1$. (a-d) The ED spectra (subtracting the global ground state energy $E_\mathrm{gs}$) versus charge sector $Q$ for $S=5$, $L=6$, in the PM phase ($J=0.1$) in (a-b) and the SSB quantum breakdown condensate phase ($J=1$) in (c-d). (a) and (c) are calculated with PBC, (b) and (d) are calculated with OBC. (e-f) The expectation $\braket{\hat{S}_j^z}$ versus site $j$ for the PM ground state (triangle markers) and the SSB global ground state (square markers), calculated (e) with PBC (for parameters in (a) and (c)), and (f) with OBC (for parameters in (b) and (d)). Dot markers in (f) show $\braket{\hat{S}_j^z}$ of the OBC SSB ground state $|\psi_Q\rangle$ in charge sectors $Q\neq Q_0$, where $Q_0$ is the charge sector of the global SSB ground state; the (f) inset shows $\delta S_j^z\equiv\braket{\psi_Q|\hat{S}_j^z|\psi_Q}-\braket{\psi_{Q_0}|\hat{S}_j^z|\psi_{Q_0}}$ decays exponentially in $j$. (g) The excitation gap $\Delta$ and (h) bandwidth $W$ of the SSB ground-state manifold with PBC for different $S$ and $L$, where we fix $J=h=\lambda=1$.
• Figure 3: (a) The phase diagram for Gibbs states. (b) The heat capacity $C_v$ for $\lambda=h=1$, $S=5$, $L=6$ with PBC. (c-d) The specific heat $C_v/L$ versus $J$ at temperature (c) $T=0.4$ and (d) $T=0.05$, for parameters in (b). The inset of (d) shows the entropy density $\mathcal{S}/L$ at $T=0.05$. The legend in (c) labels $L$ for (c-d).
• Figure 4: (a) Autocorrelation $C_{XX}(t)=\braket{\psi_Q|\hat{S}_L^x(t)\hat{S}_L^x(0)|\psi_Q}$ for PBC ground state $\ket{\psi_Q}$ of three representative charge sectors $Q$, where $J=\lambda=h=1$, $S=5$ and $L=5$. These states are highlighted in (b), which plots the energy levels in the PBC ground-state manifold (the dashed line indicates the bandwidth $W=0.00241$).