Beyond the freshman's dream: Classical fractal spin liquids from matrix cellular automata in three-dimensional lattice models

Biswas, Sounak; Kwan, Yves H.; Parameswaran, S. A.

doi:10.1103/physrevb.105.224410

Cited by 6 publications

(7 citation statements)

References 64 publications

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“…These dynamical constraints lead to an activated relaxation similar to that of the East model [24], with relaxation times growing as the exponential of the inverse temperature squared [2] (a super-Arrhenius form known as the "parabolic law" [25]). Similar glassy behaviour is seen in generalisations of the TPM with odd plaquette interactions [8,26]. The TPM has also been considered in the presence of a (longitudinal) magnetic field [27] and in the related case of coupled replicas [28], and the TPM with open boundary conditions was studied in Ref.…”

Section: A Classicalmentioning

confidence: 70%

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

Sfairopoulos¹,

Causer²,

Mair³

et al. 2023

Preprint

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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 cellular automata methods, we obtain a full characterization of the minimum energy configurations of the TPM for arbitrary tori sizes. For the QTPM, we use these cycle patterns to obtain the symmetries of the model, which we argue determine its quantum phase transition: we find it to be 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 also find that this classification is consistent with exact diagonalization, Matrix Product States and Quantum Monte Carlo simulations.

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Section: A Classicalmentioning

confidence: 70%

“…The physics of the TPM can also be generalised to three dimensions, for example in the five-spin interaction square-pyramid model [7], or maintaining the triangular interactions in the models of Ref. [8]. A three-dimensional generalisation of the TPM with non-commuting terms [9] actually started what is now the field of fractons [10,11].…”

Section: Introductionmentioning

confidence: 99%

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

Sfairopoulos¹,

Causer²,

Mair³

et al. 2023

Preprint

View full text Add to dashboard Cite

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“…Their C 3 -rotated variants coarse grain to a linear combination of (𝜕 𝑥 , −𝜕 𝑦 ) and (𝜕 𝑦 , 𝜕 𝑥 ). Hence, as shown in Appendix D, the equations that define which functions are conserved by the Hamiltonian are precisely discrete versions of the Cauchy-Riemann equations (45), whose long-wavelength solutions will give rise to (approximate) holomorphic conserved charges. In a similar manner to Sec.…”

Section: Lattice Hamiltonianmentioning

confidence: 99%

“…In practice, most works implicitly assume an underlying lattice. When lattice symmetries are treated seriously, the problem is almost always formulated on a square or cubic lattice, with rare exceptions (e.g., [45]). However, given the wide range of crystal structures, it is natural to wonder what happens if we formulate the problem on an arbitrary crystal structure that is not a hypercubic lattice.…”

mentioning

confidence: 99%

Multipole groups and fracton phenomena on arbitrary crystalline lattices

Bulmash¹,

Hart²,

Nandkishore³

2023

Preprint

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Multipole symmetries are of interest in multiple contexts, from the study of fracton phases, to nonergodic quantum dynamics, to the exploration of new hydrodynamic universality classes. However, prior explorations have focused on continuum systems or hypercubic lattices. In this work, we systematically explore multipole symmetries on arbitrary crystal lattices. We explain how, given a crystal structure (specified by a space group and the occupied Wyckoff positions), one may systematically construct all consistent multipole groups. We focus on two-dimensional crystal structures for simplicity, although our methods are general and extend straightforwardly to three dimensions. We classify the possible multipole groups on all two-dimensional Bravais lattices, and on the kagome and breathing kagome crystal structures to illustrate the procedure on general crystal lattices. Using Wyckoff positions, we provide an in-principle classification of all possible multipole groups in any space group. We explain how, given a valid multipole group, one may construct an effective Hamiltonian and a low-energy field theory. We then explore the physical consequences, beginning by generalizing certain results originally obtained on hypercubic lattices to arbitrary crystal structures. Next, we identify two seemingly novel phenomena, including an emergent, robust subsystem symmetry on the triangular lattice, and an exact multipolar symmetry on the breathing kagome lattice that does not include conservation of charge (monopole), but instead conserves a vector charge. This makes clear that there is new physics to be found by exploring the consequences of multipolar symmetries on arbitrary lattices, and this work provides the map for the exploration thereof, as well as guiding the search for emergent multipolar symmetries and the attendant exotic phenomena in real materials based on nonhypercubic lattices. CONTENTS 3. Discussion D. Vector conserved quantities 1. Lattice Hamiltonian 2. Discussion V. Conclusions A. Group theory A.1. Irreducible representations of D 𝑀 A.2. Sorting polynomials into irreps A.3. Clebsch-Gordon coefficients for D 𝑀 B. Vector charge theory C. Discrete vs continuous translations D. From discrete derivatives to spin Hamiltonians D.1. Bravais lattice D.2. Introducing a basis D.3. Additional symmetries D.4. More general Hamiltonians E. Haar-random circuits and automaton dynamics References I. INTRODUCTIONMultipole symmetries have become a major topic of interest in condensed matter physics, quantum dynamics, and quantum information. This began with the work of Pretko [1] identifying conserved multipole moments as underlying the

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“…When lattice symmetries are treated seriously, the problem is almost always formulated on a square or cubic lattice, with rare exceptions (e.g., Ref. [46]). However, given the wide range of crystal structures, it is natural to wonder what happens if we formulate the problem on an arbitrary crystal structure that is not a hypercubic lattice.…”

Section: Introductionmentioning

confidence: 99%

Multipole groups and fracton phenomena on arbitrary crystalline lattices

Bulmash,

Hart,

Nandkishore

2023

SciPost Phys.

View full text Add to dashboard Cite

Multipole symmetries are of interest in multiple contexts, from the study of fracton phases, to nonergodic quantum dynamics, to the exploration of new hydrodynamic universality classes. However, prior explorations have focused on continuum systems or hypercubic lattices. In this work, we systematically explore multipole symmetries on arbitrary crystal lattices. We explain how, given a crystal structure (specified by a space group and the occupied Wyckoff positions), one may systematically construct all consistent multipole groups. We focus on two-dimensional crystal structures for simplicity, although our methods are general and extend straightforwardly to three dimensions. We classify the possible multipole groups on all two-dimensional Bravais lattices, and on the Kagome and breathing Kagome crystal structures to illustrate the procedure on general crystal lattices. Using Wyckoff positions, we provide an in-principle classification of all possible multipole groups in any space group. We explain how, given a valid multipole group, one may construct a consistent lattice Hamiltonian and a low-energy field theory. We then explore the physical consequences, beginning by generalizing certain results originally obtained on hypercubic lattices to arbitrary crystal structures. Next, we identify two apparently novel phenomena: an emergent, robust subsystem symmetry on the triangular lattice, and an exact multipolar symmetry on the breathing Kagome lattice that does not include conservation of charge (monopole), but instead conserves a vector charge. This makes clear that there is new physics to be found by exploring the consequences of multipolar symmetries on arbitrary lattices, and this work provides the map for the exploration thereof, as well as guiding the search for emergent multipolar symmetries and the attendant exotic phenomena in real materials based on nonhypercubic lattices.

show abstract

Beyond the freshman's dream: Classical fractal spin liquids from matrix cellular automata in three-dimensional lattice models

Cited by 6 publications

References 64 publications

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

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

Multipole groups and fracton phenomena on arbitrary crystalline lattices

Multipole groups and fracton phenomena on arbitrary crystalline lattices

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