Nonequilibrium Phase Transitions in Large $N$ Matrix Quantum Mechanics

Abstract

It is believed that the theory of quantum gravity describing our universe is unitary. Nonetheless, if we only have access to a subsystem, its dynamics is described by nonequilibrium physics. Motivated by this, we investigate the planar limit of large $N$ ungauged one-matrix quantum mechanics obeying the Lindblad master equation with dissipative jump terms, focusing on the existence, uniqueness, and properties of steady states that signal nonequilibrium phase transitions. In the first class of examples, where potentials are unbounded from below, we study nonequilibrium critical points above which strong dissipation allows for the existence of normalizable steady states that would otherwise not exist. In the second class of examples, termed matrix quantum optics, we find evidence of nonequilibrium phase transitions analogous to those recently reported in the quantum optics literature for driven-dissipative Kerr resonators. Preliminary results on two-matrix quantum mechanics are also presented. We implement bootstrap methods to obtain concrete and rigorous results for the nonequilibrium steady states of matrix quantum mechanics in the planar limit.

Figures (9)

• Figure 1: Left: Region in the $(\gamma, g)$-plane where SDPA-DD finds SDPsym($L=8$) for the 1-MQM system (\ref{['eqn:doubleWell']}) to be infeasible is colored in red. No normalizable steady state exists in this region. Also shown is the critical value $g_c$ at $\gamma = 0$. Right:SDPsym($L$) upper and lower bounds on ${\langle{\cal N}\rangle \over N^2}$ for $L=6$ (light gray), $L=8$ (gray), and $L=10$ (black) as functions of $\gamma$ at $g = -0.1$ for the 1-MQM system (\ref{['eqn:doubleWell']}), obtained using SDPA-DD. Shaded regions are allowed by bootstrap, while the unshaded regions are excluded. The tightest lower bounds on $\gamma_c(g = -0.1)$ from the infeasibility of SDPsym($L$) at $L=6$, $8$, and $10$ are $\gamma = 1.09$, $1.41$, and $1.48$, shown as blue, orange, and red dotted lines, respectively.
• Figure 2: SDPsym($L=10$) upper and lower bounds on ${\langle{\cal N}\rangle \over N^2}$ (left) and ${\langle H \rangle \over N^2}$ (right) as functions of $\gamma$ for the 1-MQM system (\ref{['eqn:doubleWell']}) at $g = 2$, obtained using MOSEK.
• Figure 3: SDPsym2$(L=6)$ upper and lower bounds on ${\langle{\cal N}\rangle\over N^2}$ for the 2-MQM example (\ref{['eqn:2MQMLind']}) at $g=-1$, obtained using SDPA-DD. The red dotted line indicates $\gamma=2.1$, a lower bound on the nonequilibrium critical point $\gamma_c(g=-1)$.
• Figure 4: SDP($L$) upper and lower bounds on ${\cal P}$ (left) and ${\langle{\cal N}\rangle\over N^2}$ (right) for $L=8$ (light gray), $L=9$ (gray), and $L=10$ (dark gray), at different values of $\gamma$ with $\Delta=-4$, $\chi={1\over2}$, and $\omega=5$ for the matrix quantum optics system (\ref{['eqn:MQO']}), obtained using MOSEK.
• Figure 5: SDP($L=8$) upper and lower bounds on ${\langle{\cal N}\rangle\over N^2}$ (left) and ${\langle H\rangle\over N^2}$ (right) as functions of $\gamma$ at $\Delta=1$, $\chi=1$, and $\omega=1$ for the matrix quantum optics system (\ref{['eqn:MQO']}), obtained by MOSEK.