We show how the trajectories of d-dimensional cellular automata (CA) can be used to systematically construct the ground states of (d+1)-dimensional classical spin models, and we characterize their quantum phase transition, when in the presence of a transverse magnetic field. For each of the 256 one-dimensional elementary CA we explicitly construct the simplest local two-dimensional classical spin model associated to the given CA, and we also describe this method for d>1 through selected examples. We illustrate our general observations with detailed studies of: (i) the d=1 CA Rule 150 and its d=2 four-body plaquette spin model, (ii) the d=2 CA whose associated model is the d=3 square-pyramid plaquette model, and (iii) two counterpropagating d=1 Rule 60 CA that correspond to the two-dimensional Baxter-Wu spin model. For the quantum spin models, we show that the connection to CAs implies a sensitivity on the approach to the thermodynamic limit via finite-size scaling for their quantum phase transitions.
We show how the trajectories of d-dimensional cellular automata (CA) can be used to systematically construct the ground states of (d+1)-dimensional classical spin models, and we characterize their quantum phase transition, when in the presence of a transverse magnetic field. For each of the 256 one-dimensional elementary CA we explicitly construct the simplest local two-dimensional classical spin model associated to the given CA, and we also describe this method for d>1 through selected examples. We illustrate our general observations with detailed studies of: (i) the d=1 CA Rule 150 and its d=2 four-body plaquette spin model, (ii) the d=2 CA whose associated model is the d=3 square-pyramid plaquette model, and (iii) two counterpropagating d=1 Rule 60 CA that correspond to the two-dimensional Baxter-Wu spin model. For the quantum spin models, we show that the connection to CAs implies a sensitivity on the approach to the thermodynamic limit via finite-size scaling for their quantum phase transitions.
We study classical and quantum spin models derived from one-dimensional cellular automata (CA) with nonlinear update rules, focusing on rules 30, 54 and 201. We argue that the classical models, defined such that their ground states correspond to allowed trajectories of the CA, are frustrated and can be described in terms of local defect variables. Including quantum fluctuations through the addition of a transverse field, we study their ground state phase diagram and quantum phase transitions. We show that the nonlinearity of the CA rule leads to a quantum order-by-disorder mechanism, which selects a particular (rule-dependent) spatial structure for small transverse fields, with spontaneous breaking of the translation symmetry in some cases. Using numerical results for larger fields, we also observe a first-order quantum phase transition into a quantum paramagnet, as in previous studies of spin models based on linear CA rules.
Abstract We present a class of two-dimensional randomized plaquette models, where the multi-spin interaction term, referred to as the plaquette term, is replaced by a single-site spin term with a probability of 1 − p . By varying p, we observe a ground state phase transition, or equivalently, a phase transition of the symmetry operator. We find that as we vary p, the symmetry operator changes from being extensive to being localized in space. These models can be equivalently understood as 1+1D randomized cellular automaton dynamics, allowing the 2D transition to be interpreted as a 1+1D dynamical absorbing phase transition. In this paper, our primary focus is on the plaquette term with three or five-body interactions, where we explore the universality classes of the transitions. Specifically, for the model with five-body interaction, we demonstrate that it belongs to the same universality class as the measurement-induced entanglement phase transition observed in 1+1D Clifford dynamics, as well as the boundary entanglement transition of the 2D cluster state induced by random bulk Pauli measurements. This work establishes a connection between transitions in classical spin models, cellular automata, and hybrid random circuits.
This thesis studies a selection of time-independent phase transitions in lattice spin models and of dynamical phase transitions in stochastic dynamics. First, we explore the quantum phase transitions of some old and some newly introduced plaquette spin models. The classical zero-temperature ground states of these plaquette spin models can be equally constructed through a mapping to the dynamical evolution of the respective cellular automata, thus manifesting a relation between dynamical and static processes. We describe in detail the quantum phase transitions of those plaquette models whose cellular automaton constraint is linear when in the presence of quantum fluctuation fields. We also introduce and study paradigmatic models from the nonlinear cellular automaton class. We study frustration effects and effective lattice gauge theories for extensions of the models with linear constraints, thus relating disparate areas of classical and quantum many-body theory with gauge theories, classical satisfiability problems, and glasses. At the same time, we study kinetically constrained spin models whose constraints are implemented dynamically. Specifically, we focus on relating concepts of quantum dualities with classical biased stochastic models and present whole classes with multicritical points and non-invertible symmetries.
Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and link quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, remains challenging: although a topological action yields a Hamiltonian with commuting terms that realizes the desired ground state, it does not by itself specify an automorphism of the full operator algebra. We develop a unified algebraic construction that extends such Hamiltonians to complete separator-flipper algebras on the full tensor-product Hilbert space, thereby defining the corresponding QCAs microscopically. In 3 spatial dimensions, our framework unifies all previously known QCA constructions associated with the Z8×Z2 subgroup of the Witt group, including the U(1)2 and U(1)4 QCAs. The same algebraic structure yields new infinite families of generalized U(1)2 and U(1)4 non-Clifford QCAs in spatial dimensions d=4k−1. We also reformulate the 4-dimensional w2w3 QCA and use it to construct QCAs from TQFTs associated with arbitrary products of Stiefel--Whitney classes. For example, we explicitly construct two infinite families: w2nw3m QCAs in dimension d=2n+3m−1 and w2w4k−1 QCAs in dimension d=4k. For the 5-dimensional w32 and w23 QCAs, our construction gives finite-depth quantum circuits, consistent with the cobordism classification. More generally, we show that our construction yields a finite-depth quantum circuit whenever the corresponding Stiefel-Whitney number vanishes on every closed oriented manifold. Overall, these results convert invertible TQFTs into microscopic QCAs, provide higher-dimensional constructions beyond the Clifford setting, and open a systematic route to characterizing their stable structures and boundary anomalies.
We study the triangular plaquette model (TPM), also known as the Newman-Moore model, in the presence of a transverse magnetic field on a lattice with periodic boundaries in both spatial dimensions. We consider specifically the approach to the ground state phase transition of this quantum TPM (QTPM), or quantum Newman-Moore model, as a function of the system size and type of boundary conditions. Using methods based on cellular automata, we obtain a characterization of the minimum energy configurations of the TPM for numerically accessible tori sizes. For the QTPM, we use these cycle patterns to obtain the symmetries of the model which, we argue, indicate the nature of its quantum phase transition: we always identify it as a first-order phase transition, with the addition of spontaneous symmetry breaking for system sizes which have degenerate classical ground states. For sizes accessible to numerics, we corroborate our findings with exact diagonalization, matrix product states, and quantum Monte Carlo simulations.
Probabilistic cellular automata provide a simple framework for exploring classical nonequilibrium processes. Recently, quantum cellular automata have been proposed that rely on the propagation of a one-dimensional quantum state along a fictitious discrete time dimension via the sequential application of quantum gates. The resulting (1+1)-dimensional space-time structure makes these automata special cases of recurrent quantum neural networks which can implement broad classes of classical nonequilibrium processes. Here, we present a general prescription by which these models can be extended into genuinely quantum nonequilibrium models via the systematic inclusion of asynchronism. This is illustrated for the classical contact process, where the resulting model is closely linked to the quantum contact process (QCP), developed in the framework of open quantum systems. Studying the mean-field behavior of the model, we find evidence of an "asynchronism transition," i.e., a sudden qualitative change in the phase transition behavior once a certain degree of asynchronicity is surpassed, a phenomenon we link to observations in the QCP.