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arXiv:2301.02826 · Physical review. B./Physical review. B

Boundary conditions dependence of the phase transition in the quantum Newman-Moore model

Konstantinos Sfairopoulos, Luke Causer, Jamie F. Mair, Juan P. Garrahan

Published Nov 22, 2023• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.108.174107

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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.
We study the quantum Newman-Moore model, or quantum triangular plaquette model (qTPM), in the presence of a longitudinal field (qTPMz). We present evidence that indicates that the ground state phase diagram of the qTPMz includes various frustrated phases breaking translational symmetries, dependent on the specific sequence of system sizes used to take the large-size limit. This phase diagram includes the known first-order phase transition of the qTPM, but also additional first-order transitions due to the frustrated phases. Using the average longitudinal magnetization as an order parameter, we analyze the magnetization plateaus that characterize the ground state phases, describe their degeneracies, and obtain the qTPMz phase diagram using classical transfer matrix and quantum matrix product state techniques. We identify a region of the parameter space, which can be effectively described by a Rydberg blockade model on the triangular lattice. At the same time, it can be reformulated as an effective lattice gauge theory but also a model of quantum trimers. In this same region, we provide a variational ground state wave function similar to a resonating valence bond solid state, which accurately describes the ground state wave function of the system. Lastly, we fail to find indications of a phase with Z 2 topological order connecting the quantum paramagnetic and classical frustrated phases.
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.
Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $\ensuremath{\alpha}$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\ensuremath{\alpha}$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,\ensuremath{-}\frac{M}{2},\ensuremath{-}\frac{M}{2})$ with unsaturated M<1M<1M<1 in a process known as ``order by disorder'' similar to the nearest-neighbor antiferromagnet on the triangular lattice. Connecting the known limiting cases of nearest-neighbor and infinite-range interactions, a semiquantitative phase diagram is obtained. Magnetization curves for the ferromagnet for experimentally relevant system sizes and with open boundary conditions are presented.
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.
The one-dimensional transverse field Ising model with boundary fields is studied analytically and numerically. The phase diagram in the ordered state is obtained. We find that there exist two types of phase transitions. The longitudinal boundary fields applied to the left and right ends of the Ising chain are hL{h}_{L}hL​ and hR{h}_{R}hR​, respectively. For $|{h}_{L}|,|{h}_{R}|<\sqrt{1\ensuremath{-}g}$, where ggg is the transverse field and hLhR<0{h}_{L}{h}_{R}<0hL​hR​<0, a first-order phase transition occurs with the changing hL{h}_{L}hL​ or hR{h}_{R}hR​. The energy gap and boundary magnetization are solved exactly. The analytical expressions of the finite-size scaling for the first-order phase transition are obtained. For $|{h}_{R}|>\sqrt{1\ensuremath{-}g}$ and hLhR<0{h}_{L}{h}_{R}<0hL​hR​<0, there exists a continuous phase transition with changing hL{h}_{L}hL​ and vice versa. This transition is identified as a quantum wetting transition. The singularity of the boundary magnetization in this phase transition is explicitly shown. A simple computational procedure with high accuracy and efficiency is proposed to calculate the magnetization. The magnetization profile, correlation functions, and wetting layer thickness are studied numerically.
Quantum Ising model on a triangular lattice hosts a finite temperature Berezinskii-Kosterlitz-Thouless (BKT) phase with emergent U(1) symmetry, and it will transit into an up-up-down (UUD) phase with C3{C}_{3}C3​ symmetry breaking upon an infinitesimal external field along the longitudinal direction, but the overall phase diagram spanned by the axes of external field and temperature remains opaque due to the lack of systematic investigations with controlled methodologies. By means of quantum Monte Carlo at finite temperature and ground state density matrix renormalization group simulations, we map out the phase diagram of triangular quantum Ising model. Starting from the upper BKT temperature at zero field, we obtain the phase boundary between the UUD and paramagnetic phases with its 2D q=3q=3q=3 Potts universality at weak field and weakly first order transition at strong field. Originated from the lower BKT temperature at zero field, we analyze the low temperature phase boundary between the clock phase and the UUD phase with Ising symmetry breaking at weak fields and the quantum phase transition between the UUD and fully polarized phases at strong fields. The accurate many-body numerical results are consistent with our field theoretical analysis. The experimental relevance towards the BKT magnet TmMgGaO4{\mathrm{TmMgGaO}}_{4}TmMgGaO4​ and programmable quantum simulators are also discussed.