Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2201.09904 · Physical review. B./Physical review. B

Quantum orders in the frustrated Ising model on the bathroom tile lattice

Sumner N. Hearth, Siddhardh C. Morampudi, Chris R. Laumann

Published May 3, 2022• 2 citations
arXivHTMLPDF

Abstract

We determine the zero- and finite temperature phase diagram of the fully frustrated quantum Ising model on the bathroom tile (4-8) lattice. The phase diagram exhibits a wealth of $2+1\mathrm{d}$ physics, including (1) classical Coulomb dimer liquids of both square and triangular lattice types; (2) quantum order-by-disorder induced phases breaking ${\mathbb{Z}}_{4}, {\mathbb{Z}}_{6}$, and ${\mathbb{Z}}_{8}$ symmetries; (3) finite temperature Kosterlitz-Thouless phases floating over the ${\mathbb{Z}}_{6}$ and ${\mathbb{Z}}_{8}$ orders; and (4) staircases of (in)-commensurate symmetry-breaking phases at intermediate coupling. We establish this elaborate phase diagram using a combination of dimer model mapping, perturbation theory, Landau analysis and Stochastic Series Expansion Quantum Monte Carlo. Our results provide a baseline for studying frustrated magnetism with D-Wave architecture annealers, where the 4-8 lattice can be embedded naturally without ``cloning,'' reducing the number of competing energy scales. Simulations with the D-Wave 2000Q demonstrate qualitative agreement with the high-temperature portion of the phase diagram, but are unable to access the low-temperature phases.

DOI
10.1103/physrevb.105.195101

Related papers

Ordering of geometrically frustrated classical and quantum triangular Ising magnets

Ying Yan Jiang, Thorsten Emig A

A systematic study of both classical and quantum geometric frustrated Ising models with a competing ordering mechanism is reported in this paper. The ordering comes in the classical case from a coupling of two-dimensional (2D) layers and in the quantum model from the quantum dynamics induced by a transverse field. By mapping the Ising models on a triangular lattice to elastic lattices of noncrossing strings, we derive an exact relation between the spin variables and the displacement field of the strings. Using this map for both the classical $(2+1)\mathrm{D}$ stacked model and the quantum frustrated 2D system, we obtain a microscopic derivation of an effective Hamiltonian which was proposed before on phenomenological grounds within a Landau-Ginzburg-Wilson approach. In contrast to the latter approach, our derivation provides the coupling constants and hence the entire transverse field-versus-temperature phase diagram can be deduced, including the universality classes of both the quantum and the finite-temperature transitions. The structure of the ordered phase is obtained from a detailed entropy argument. We compare our predictions to recent simulations of the quantum system and find good agreement. We also analyze the connections to a dimer model on the hexagonal lattice and its height profile representation, providing a simple derivation of the continuum free energy and a physical explanation for the universality of the stiffness of the height profile for anisotropic couplings.

arXiv:cond-mat/0508637•Physical Review B•Mar 31, 2006•23 citations

Ising models of quantum frustration

Roderich Moessner, Shivaji L. Sondhi

We report on a systematic study of two-dimensional, periodic, frustrated Ising models with quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagom\'e lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet, as well as two quasi-one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at $T=0$ in the absence of the transverse field, which enters as a singular perturbation. We analyze these systems with a combination of a variational method at weak fields, a perturbative Landau-Ginzburg-Wilson approach from large fields, as well as quantum Monte Carlo simulations utilizing a cluster algorithm. Our results include instances of quantum order arising from classical criticality (triangular lattice) or classical disorder (pentagonal and probably hexagonal) as well as notable instances of quantum disorder arising from classical disorder (kagom\'e). We also discuss the effect of finite temperature, as well as the interplay between longitudinal and transverse fields---in the kagom\'e problem the latter gives rise to a nontrivial phase diagram with bond-ordered and bond-critical phases in addition to the disordered phase. We also note connections to quantum-dimer models and thereby to the physics of Heisenberg antiferromagnets in short-ranged resonating valence-bond phases that have been invoked in discussions of high-temperature superconductivity.

arXiv:cond-mat/0011250•Physical review. B, Condensed matter•May 4, 2001•396 citations

Phase diagrams of two-dimensional frustrated spin systems

Ansgar Kalz

In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.

•Jan 1, 2012•0 citations

Phase diagram of the fully frustrated transverse-field Ising model on the honeycomb lattice

Tommaso Coletta, Jean-David Picon, S. E. Korshunov +1 more

Motivated by the current interest in the quantum dimer model on the triangular lattice, we investigate the phase diagram of the closely related fully frustrated transverse-field Ising model on the honeycomb lattice using classical and semiclassical approximations. We show that, in addition to the fully polarized phase at a large field, the classical model possesses a multitude of phases that break the translational symmetry which, in the dimer language, correspond to a plaquette phase and a columnar phase separated by an infinite cascade of mixed phases. The modification of the phase diagram by quantum fluctuations has been investigated in the context of linear spin-wave theory. The extrapolation of the semiclassical energies suggests that the plaquette phase extends down to zero field for spin 1/2, in agreement with the $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ phase of the quantum dimer model on the triangular lattice with only kinetic energy.

arXiv:1006.5287•Physical Review B•Feb 3, 2011•15 citations

Primary and secondary order parameters in the fully frustrated transverse-field Ising model on the square lattice

Gabe Schumm, Hui Shao, Wenan Guo +2 more

Using quantum Monte Carlo simulations and field-theory arguments, we study the fully frustrated transverse-field Ising model on the square lattice for the purpose of quantitatively relating two different order parameters to each other. We consider a ``primary'' spin order parameter and a ``secondary'' dimer order parameter, which both lead to the same phase diagram but detect ${\mathbb{Z}}_{8}$ and ${\mathbb{Z}}_{4}$ symmetry breaking, respectively. While at $T>0$ their scaling exponents are simply related to each other, as explained by a mapping to a height model, we show that at $T=0$ they correspond to different charge sectors of the O(2) model in $2+1$ dimensions with nontrivial exponents that are not simply related to each other. Our insights are transferrable to a broad class of Ising models whose low-energy physics involves dimer degrees of freedom, and also serve as a guide to treating primary and secondary order parameters more generally.

arXiv:2309.02407•Physical review. B./Physical review. B•Apr 26, 2024•3 citations

Phase diagram of the quantum Ising model on a triangular lattice under external field

Yuan Da Liao, Han Li, Zheng Yan +4 more

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 ${C}_{3}$ 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 $q=3$ 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 ${\mathrm{TmMgGaO}}_{4}$ and programmable quantum simulators are also discussed.

arXiv:2101.11610•Physical review. B./Physical review. B•Mar 11, 2021•26 citations
We determine the zero- and finite temperature phase diagram of the fully frustrated quantum Ising model on the bathroom tile (4-8) lattice. The phase diagram exhibits a wealth of 2+1d2+1\mathrm{d}2+1d physics, including (1) classical Coulomb dimer liquids of both square and triangular lattice types; (2) quantum order-by-disorder induced phases breaking Z4,Z6{\mathbb{Z}}_{4}, {\mathbb{Z}}_{6}Z4​,Z6​, and Z8{\mathbb{Z}}_{8}Z8​ symmetries; (3) finite temperature Kosterlitz-Thouless phases floating over the Z6{\mathbb{Z}}_{6}Z6​ and Z8{\mathbb{Z}}_{8}Z8​ orders; and (4) staircases of (in)-commensurate symmetry-breaking phases at intermediate coupling. We establish this elaborate phase diagram using a combination of dimer model mapping, perturbation theory, Landau analysis and Stochastic Series Expansion Quantum Monte Carlo. Our results provide a baseline for studying frustrated magnetism with D-Wave architecture annealers, where the 4-8 lattice can be embedded naturally without ``cloning,'' reducing the number of competing energy scales. Simulations with the D-Wave 2000Q demonstrate qualitative agreement with the high-temperature portion of the phase diagram, but are unable to access the low-temperature phases.
A systematic study of both classical and quantum geometric frustrated Ising models with a competing ordering mechanism is reported in this paper. The ordering comes in the classical case from a coupling of two-dimensional (2D) layers and in the quantum model from the quantum dynamics induced by a transverse field. By mapping the Ising models on a triangular lattice to elastic lattices of noncrossing strings, we derive an exact relation between the spin variables and the displacement field of the strings. Using this map for both the classical (2+1)D(2+1)\mathrm{D}(2+1)D stacked model and the quantum frustrated 2D system, we obtain a microscopic derivation of an effective Hamiltonian which was proposed before on phenomenological grounds within a Landau-Ginzburg-Wilson approach. In contrast to the latter approach, our derivation provides the coupling constants and hence the entire transverse field-versus-temperature phase diagram can be deduced, including the universality classes of both the quantum and the finite-temperature transitions. The structure of the ordered phase is obtained from a detailed entropy argument. We compare our predictions to recent simulations of the quantum system and find good agreement. We also analyze the connections to a dimer model on the hexagonal lattice and its height profile representation, providing a simple derivation of the continuum free energy and a physical explanation for the universality of the stiffness of the height profile for anisotropic couplings.
We report on a systematic study of two-dimensional, periodic, frustrated Ising models with quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagom\'e lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet, as well as two quasi-one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at T=0T=0T=0 in the absence of the transverse field, which enters as a singular perturbation. We analyze these systems with a combination of a variational method at weak fields, a perturbative Landau-Ginzburg-Wilson approach from large fields, as well as quantum Monte Carlo simulations utilizing a cluster algorithm. Our results include instances of quantum order arising from classical criticality (triangular lattice) or classical disorder (pentagonal and probably hexagonal) as well as notable instances of quantum disorder arising from classical disorder (kagom\'e). We also discuss the effect of finite temperature, as well as the interplay between longitudinal and transverse fields---in the kagom\'e problem the latter gives rise to a nontrivial phase diagram with bond-ordered and bond-critical phases in addition to the disordered phase. We also note connections to quantum-dimer models and thereby to the physics of Heisenberg antiferromagnets in short-ranged resonating valence-bond phases that have been invoked in discussions of high-temperature superconductivity.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
Motivated by the current interest in the quantum dimer model on the triangular lattice, we investigate the phase diagram of the closely related fully frustrated transverse-field Ising model on the honeycomb lattice using classical and semiclassical approximations. We show that, in addition to the fully polarized phase at a large field, the classical model possesses a multitude of phases that break the translational symmetry which, in the dimer language, correspond to a plaquette phase and a columnar phase separated by an infinite cascade of mixed phases. The modification of the phase diagram by quantum fluctuations has been investigated in the context of linear spin-wave theory. The extrapolation of the semiclassical energies suggests that the plaquette phase extends down to zero field for spin 1/2, in agreement with the $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ phase of the quantum dimer model on the triangular lattice with only kinetic energy.
Using quantum Monte Carlo simulations and field-theory arguments, we study the fully frustrated transverse-field Ising model on the square lattice for the purpose of quantitatively relating two different order parameters to each other. We consider a ``primary'' spin order parameter and a ``secondary'' dimer order parameter, which both lead to the same phase diagram but detect Z8{\mathbb{Z}}_{8}Z8​ and Z4{\mathbb{Z}}_{4}Z4​ symmetry breaking, respectively. While at T>0T>0T>0 their scaling exponents are simply related to each other, as explained by a mapping to a height model, we show that at T=0T=0T=0 they correspond to different charge sectors of the O(2) model in 2+12+12+1 dimensions with nontrivial exponents that are not simply related to each other. Our insights are transferrable to a broad class of Ising models whose low-energy physics involves dimer degrees of freedom, and also serve as a guide to treating primary and secondary order parameters more generally.
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.