Self-Induced Glassy Phase in Multimodal Cavity Quantum Electrodynamics

We provide strong evidence that the effective spin-spin interaction in a multimodal confocal optical cavity gives rise to a self-induced glassy phase, which emerges exclusively from the peculiar Euclidean correlations and is not related to the presence of disorder as in standard spin glasses. As recently shown, this spin-spin effective interaction is both nonlocal and nontranslational invariant, and randomness in the atoms’ positions produces a spin glass phase. Here we consider the simplest feasible disorder-free setting, where atoms form a one-dimensional regular chain and we study the thermodynamics of the resulting effective Ising model. We present extensive results showing that the system has a low-temperature glassy phase. The model depends on the adimensional parameter $\alpha =(a/w_0{)}^2$, $a$ being a lattice spacing and $w_0$ an interaction length scale. Notably, for rational values of $\alpha =p/q$, the number of metastable states at low temperature grows exponentially with $q$ and the problem of finding the ground state rapidly becomes computationally intractable, suggesting that the system develops high-energy barriers and ergodicity breaking occurs.

Figure 1

Collective variables, reduced interaction matrix, and proliferation of metastable states increasing $q$. (a) Identification of the collective variables to study the partition function analytically. Since the interaction is periodic, it is natural to group spins lying on the sublattices ${\mathrm{\Lambda }}_r=\{iq+r,i=0,1,\dots ,2\ell -1\}$. In this way, we can define $q$ mesoscopic magnetizations $m_r$ with $r=0,1,\dots ,q-1$ and reduce the partition function to an integral over these $q$ variables. (b) Reduced $q\times q$ interaction matrix for two different values of $\alpha =p/q$. Removing the first row and column, the resulting $(q-1)\times (q-1)$ matrix displays a reflection symmetry around the antidiagonal, which allows us to further reduce the dimensionality of the free-energy manifold from $q$ to $\lfloor q/2+1\rfloor$. (c) Schematic representation by hierarchical clustering of the energy landscape at the two different values of $\alpha =13/21,21/34$. We obtain this representation by displaying the local minima of the energy in Eq. (2) as the leaves of a dendrogram. Distance is increasing along the vertical axis and pairs of distinct clusters at distance $d$ (measured from their center) are iteratively joined together in a new node at height $d$. As $q$ increases, we observe a proliferation of local minima.

Figure 2

Exponential number of metastable states increasing $q$. (a) Scaling of the number of local minima of the energy in Eq. (2) as a function of $q$ (different values of $p$ are shown for each value of $q$; see Supplemental Material [33]). To check whether at a given $q$ the estimate of the number of metastable states is reliable, in the insets we display the number of distinct minima found as a function of the number of gradient descents attempted for three representative values of $q$. The closer we are to saturation, the more accurate is the estimate. Above $q\sim 60$ the estimate is not reliable, since we are missing a large number of minima (see right inset). The log-linear scale of the main plot highlights the exponential growth below $q\sim 60$, where the estimate is reliable (see left insets). (b) The average linear size of the basins of attraction at fixed $\alpha =p/q$ grows algebraically with $q$ as $q^{0.37}$. Error bars denote one sample variance. (c) The linear size of a given basin of attraction is measured by collecting all the initial conditions that flow to a given minimum and keeping as the linear size the distance between the minimum and the farthest initial condition from it.

Figure 3

The run time for the ground-state search problem scales exponentially with $q$. Scaling of the run time of the ground-state (GS) search performed using a state-of-the-art optimizer (IBM CPLEX [44, 45]) as a function of $q$ (again, different values of $p$ are shown for each value of $q$). Up to $q=45$, the run time scales exponentially with $q$, suggesting that the GS search problem is a worst-case instance of nonpositive-definite quadratic programming.