Exact solution of the Floquet-PXP cellular automaton

Abstract

We study the dynamics of a bulk deterministic Floquet model, the Rule 201 synchronous onedimensional reversible cellular automaton (RCA201). The system corresponds to a deterministic, reversible, and discrete version of the PXP model, whereby a site flips only if both its nearest neighbours are unexcited. We show that the RCA201/Floquet-PXP model exhibits ballistic propagation of interacting quasiparticles -or solitons -corresponding to the domain walls between non-trivial three-fold vacuum states. Starting from the quasiparticle picture, we find the exact matrix product state form of the non-equilibrium stationary state for a range of boundary conditions, including both periodic and stochastic. We discuss further implications of the integrability of the model. I.

Exact solution of the Rule 150 reversible cellular automaton

“…The Rule 150 RCA is similar in many respects to other recently studied RCA, namely, Rule 54 [2][3][4][5][6][7][8][9][10][11][12][13] (see Ref. [14] for a review) and Rule 201 [15,16]: (i) its dynamics is defined in terms of local reversible gates applied periodically (in this sense it can be thought of as a driven Floquet system); (ii) the local dynamical rules impose kinetic constraints similar to those of known stochastic kinetically constrained models (KCMs [17][18][19]), the XOR-Fredrickson-Andersen [20] (or XOR-FA) model in the case of Rule 150, and the FA [21] and PXP [22] (or 2spin-facilitated-FA [17]) models for Rule 54 [5] and Rule 201 [16], respectively; and (iii) Rule 150 is an integrable model [23], but, in contrast to Rules 54 and 201, its quasiparticles are noninteracting [8].…”

“…[14,44]) and, similarly, for Rule 201 (see Ref. [16]). Explicitly, the probability of finding the virtual sites at the right and left boundaries in the states n 5 and n 0 , respectively, given that the pairs of adjacent spins are in the configurations (n 3 , n 4 ) and (n 1 , n 2 ), that is R n3,n4,n5 and L n0,n1,n2 is equivalent to the conditional probability of finding the three sites in the configurations (n 3 , n 4 , n 5 ) and (n 0 , n 1 , n 2 ), given the states of the sites (n 3 , n 4 ) and (n 1 , n 2 ).…”

“…Properties (i) and (ii) mean that the Rule 150 RCA can alternatively be called the "Floquet-XOR-FA" model, as Rules 54 and 201 can, respectively, be called the Floquet-FA [7] and Floquet-PXP [16]. Property (iii) means that we can readily solve exactly the Rule 150 RCA, with its noninteracting nature making the solution simpler than those for Rule 54 and Rule 201.…”

Constructing quantum many-body scar Hamiltonians from Floquet automata

“…Floquet automata are unitary circuits that effect permutations of computational basis states on a chain of qubits. They are naturally associated with systems with state space [29][30][31] or kinetic constraints [32][33][34][35] and can exhibit non-trivial dynamical features in both the classical [36] and quantum settings [37][38][39][40]. The Hilbert space is fragmented into disjoint subspaces of computational basis states which are cycled through with successive applications of the automata circuit, reviving at fixed time intervals.…”

Measurement catastrophe and ballistic spread of charge density with vanishing current

“…In each sector the model is almost free, with a diagonal scattering matrix in which only one diagonal element is different from −1 and equal to −e iq , where q is the difference between the momenta of the scattering particles, in a remarkable analogy with the scattering phase modi-fication factor in hard-rod deformed models (see, e.g., Refs [16,17]). A striking feature found in fragmented models, such as the folded XXZ one, is the existence of an exponentially large sector consisting of jammed states: particles are stuck and can not move [18][19][20][21][22][23]. In Ref.…”