KPZ physics and phase transition in a classical single random walker under continuous measurement

Abstract

We introduce and study a new model consisting of a single classical random walker undergoing continuous monitoring at rate $\gamma$ on a discrete lattice. Although such a continuous measurement cannot affect physical observables, it has a non-trivial effect on the probability distribution of the random walker. At small $\gamma$, we show analytically that the time-evolution of the latter can be mapped to the Stochastic Heat Equation (SHE). In this limit, the width of the log probability thus follows a Family-Vicsek scaling law, $N^{\alpha }f(t/N^{\alpha /\beta })$, with roughness and growth exponents corresponding to the Kardar-Parisi-Zhang (KPZ) universality class, i.e ${\alpha }_{\mathrm{KPZ}}^{1D}=1/2$ and ${\beta }_{\mathrm{KPZ}}^{1D}=1/3$ respectively. When $\gamma$ is increased outside this regime, we find numerically in 1D a crossover from the KPZ class to a new universality class characterized by exponents ${\alpha }_{\text{M}}^{1\mathrm{D}}\approx 1$ and ${\beta }_{\text{M}}^{1\mathrm{D}}\approx 1.4$. In 3D, varying $\gamma$ beyond a critical value ${\gamma }_{\mathrm{M}}^c$ leads to a phase transition from a smooth phase that we identify as the Edwards-Wilkinson (EW) class to a new universality class with ${\alpha }_{\text{M}}^{3\mathrm{D}}\approx 1$.

9 pages, 3 figures

References in corpus (2)

• A Straightforward Introduction to Continuous Quantum Measurement
• Critical properties of the measurement-induced transition in random quantum circuits

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• Measurement-Induced Entanglement Phase Transition in Free Fermion Systems
• Generalizing measurement-induced phase transitions to information exchange symmetry breaking
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• Measurement-Induced Phase Transitions in Informational Active Matter
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• Multifractality in monitored single-particle dynamics
• Dynamic scaling and Family-Vicsek universality in $SU(N)$ quantum spin chains
• Entanglement phases and phase transitions in monitored free fermion system due to localizations