Nonanaliticities and ergodicity breaking in noninteracting many-body dynamics via stochastic resetting and global measurements

TL;DR

The paper investigates conditional stochastic resetting in a noninteracting spin system, showing that global magnetization measurements followed by majority-rule resets induce collective, nonanalytic stationary states in the thermodynamic limit. The authors map the dynamics to a Markov chain on reset states, revealing absorbing-state driven ergodicity breaking and, for integer spins, multicritical behavior with coexisting first- and second-order singularities. Time-reversal symmetry underpins the qualitative difference between half-integer and integer spins, determining whether nonanalyticities are solely first-order or multicritical. The framework provides a robust, interaction-free route to engineer complex stationary states via global monitoring, with potential applications in quantum sensing and quantum hardware platforms capable of global reset and measurement operations.

Abstract

Stochastic resetting generates nonequilibrium steady states by interspersing unitary quantum dynamics with resets at random times. When the state to which the system is reset is chosen conditionally on the outcome of a global and spatially resolved measurement, the steady state can feature collective behavior similar to what is typically observed at phase transitions. Here we investigate such conditional reset protocol in a system of noninteracting spins, where the reset state is chosen as a magnetization eigenstate, that is selected (conditioned) on the outcome of a previous magnetization measurement. The stationary states that emerge from this protocol are characterized by the density of spins in a given magnetization eigenstate, which is the analogue of the order parameter. The resulting stationary phase diagram features multiple nonanalytic points. They are of first-order type for half-integer spin, while multicritical behavior, signalled by both first and second-order discontinuities, is found for integer spin. We also show that the associated dynamics is nonergodic, i.e., which stationary state the system ultimately assumes is determined be the initial state. Interestingly, the mechanism underlying these phenomena does not rely on interactions, but the emergent nonlinear behavior is solely a consequence of correlations induced by the measurement.

Figures (13)

• Figure 1: Collective behavior in the conditional resetting protocol of spins. (a) In the conditional reset protocol for a number $N$ of spin $S$ spins, there are $j=1,2,\dots 2S+1$ reset states $\ket{j}_N$\ref{['eq:reset_states']}. The system undergoes unitary time evolution according to the Hamiltonian \ref{['eq:Hamiltonian']} from the reset state $\ket{j}_N$ for a random time $\tau$. The next reset state $\ket{k}_N$ is chosen conditionally on the outcome of a projective measurement of the total magnetization $S^z$. The majority rule is adopted, so that one chooses the state $\ket{j}_N$ corresponding to the magnetization value $j$ measured for the majority of spins. (b) A Markov chain between reset states $\ket{j}_N$ and $\ket{k}_N$ can then be associated with the dynamics. As $N\to \infty$, the Markov chain model (in the inset of the figure) is reducible into smaller Markov chains containing absorbing states. In the cartoon plot, we exemplify this mechanism for $S=1$, with three reset states. For $\tilde{\Omega}<\tilde{\Omega}_c$, the Markov chain contains two absorbing reset states, namely $\ket{1}_N$ and $\ket{3}_N$. A Critical point $\tilde{\Omega}_c$ emerges when transitions between reset states acquire nonzero probability (red arrows). For $\tilde{\Omega}<\tilde{\Omega}_c$, the states $\ket{1}_N$ and $\ket{3}_N$ are no longer absorbing. The associated stationary density $p^{\mathrm{NESS}}_j$ for state $j$ displays collective behavior at $\tilde{\Omega}_c$. This manifests in first-order jump discontinuities ($j=1$). For integer spin $S$, additionally, second-order discontinuities in the first derivative are present ($j=2$).
• Figure 3: Markov chain of reset states in ergodic spin-$1/2$ system. In the thermodynamic limit $N\to \infty$ when $\tilde{\Omega}>\tilde{\Omega}_c$, sequences of resets from state $\ket{1}_N$ to $\ket{2}_N$ (and viceversa) become possible. Accordingly, $R_{12}=R_{21}\neq 0$ and the Markov chain becomes ergodic. The definition of the critical value $\tilde{\Omega}_c$ of the ratio $\tilde{\Omega}=\Omega/\Delta$ is reported in Eq. \ref{['eq: def of om_c']} below. Note that both transition probabilities become simultaneously nonzero as a consequence of the symmetry \ref{['eq:symmetry_R']}.
• Figure 4: Reset-free evolution and Markov transition probabilities for spin $1/2$. In panels (a) and (b), we plot reset-free single-particle occupation probabilities $p^F_{is}(t)$ as a function of the time $t$ after a reset to state $\ket{1}_N$, with $s=1,2$. In panel (a), one has $\tilde{\Omega}=0.9<\tilde{\Omega}_c=1$. It then holds at $p^F_{11}(t)>p^F_{12}(t)$. The state $\ket{1}_N$ is therefore an absorbing reset state and all subsequent resets will be to this state. In panel (b), it is $\tilde{\Omega}=1.3>\tilde{\Omega}_c$. Resets to the state $\ket{2}_N$ are now allowed. In panel (c), we plot the probability $W^{(\infty)}_{12}(t)$ of resetting to state $2$ at time $t$ after the last reset to state $1$ for $\tilde{\Omega}=1.3$ as in panel (b). In the thermodynamic limit $N\to \infty$, $W^{(\infty)}_{12}(t)$ is fully determined by the mean occupation probabilities $p^F_{1s}(t)$. We have that, $W^{(\infty)}_{12}(t)=1$ for all the times $t$ such that $p^F_{12}(t)>p^F_{11}(t)$ holds. Otherwise, it is $W^{(\infty)}_{ij}(t)=0$. In the figure we set $\Delta =1$.
• Figure 5: Markov chain of reset states in a non-ergodic spin-$3/2$ system. The reset states $\ket{1}_N$ and $\ket{4}_N$ are absorbing. The Markov chain is reducible into three noncommunicating classes. The first one is formed by the reset state $\ket{1}_N$, the second by the states $\ket{2}_N$ and $\ket{3}_N$, and the third by $\ket{4}_N$. The reset states $\ket{2}_N$ and $\ket{3}_N$ form an ergodic subsystem. Due to the symmetry \ref{['eq:symmetry_R']}, the transition probabilities obey $R_{11}=R_{44}$, $R_{22}=R_{33}$ and $R_{23}=R_{32}$.
• Figure 6: Stationary occupation probabilities for noninteracting spins $S=1/2$ under conditional resetting. Plot of the stationary occupation probabilities $p_{1,2}^{\mathrm{NESS}}$ from Eq. \ref{['eq: excitation density S=1/2']} as a function of $\tilde{\Omega}=\Omega/\Delta$. The system is initialized in the state $\ket{1}_N$. For $\tilde{\Omega}<1$, the Markov chain of reset states is nonergodic and the system is always reset to $\ket{1}_N$. The curves of $p_{1,2}^{\mathrm{NESS}}$ therefore coincide with the ones obtained for unconditional resetting to state $\ket{1}_N$. At $\tilde{\Omega}=\tilde{\Omega}_c=1$, both curves have a jump discontinuity, since for $\tilde{\Omega} >1$, the resetting Markov chain becomes ergodic $R_{12}=R_{21}\neq 0$. In particular, $p_1^{\mathrm{NESS}}=p_2^{\mathrm{NESS}}=1/2$ for $\tilde{\Omega}>1$ independently on the initial state. The dashed curves are obtained by implementing the unconditional reset protocol to state $\ket{1}_N$ for comparison. No discontinuity is present within this protocol, and the stationary probabilities are analytic functions of $\tilde{\Omega}$. Note that the same plot also holds for initializing the system in $\ket{2}_N$ simply swapping the roles of $\ket{1}$ and $\ket{2}$. In the figure, we set $\gamma=\Delta=1$.