Marshall-positive<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">SU</mml:mi><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:math>quantum spin systems and classical loop models: A practical strategy to design sign-problem-free spin Hamiltonians

Kaul, Ribhu K.

doi:10.1103/physrevb.91.054413

Cited by 17 publications

(12 citation statements)

References 34 publications

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“…We here also note that the PVBS state can be regarded as a "superposition of superposition states" (locally resonating valence bonds), and this specific aspect of the state seems to be what makes it more difficult to realize a PVBS than a CVBS-at least within the J-Q approach. The broader family of models by Kaul [125], expanding on the J-Q approach, should also be explored more extensively, though our initial attempts have not been successful in producing PVBS states with less than twelve interacting spins.…”

Section: Future Prospectsmentioning

confidence: 99%

Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet

Takahashi

Sandvik

2020

Phys. Rev. Research

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We introduce a quantum spin-1/2 model with many-body correlated Heisenberg-type interactions on the two-dimensional square lattice, designed so that the system can host a fourfold degenerate plaquette valencebond solid (PVBS) ground state that spontaneously breaks Z 4 symmetry. The system is sign-problem free and amenable to large-scale quantum Monte Carlo simulations, thus allowing us to carry out a detailed study of the quantum phase transition between the standard Néel antiferromagnetic (AFM) and PVBS states. We find a first-order transition, in contrast to previously studied continuous transitions from the AFM phase into a columnar valence-bond solid (CVBS) phase. The theory of deconfined quantum criticality predicts generic continuous AFM-CVBS and AFM-PVBS transitions, and, in one version of the theory, the two critical order parameters transform under SO(5) symmetry. Emergent SO(5) symmetry has indeed been observed in studies of the AFM-CVBS transition, and here we show that the first-order AFM-PVBS transition also exhibits SO(5) symmetry at the transition point. Such unexpected symmetry of the coexistence state, which implies a lack of energy barriers between the coexisting phases, has recently been observed at other first-order transitions, but the case presented here is the first example with SO(5) symmetry. The extended symmetry may indicate that the transition is connected to a deconfined critical point. We also discuss the first-order transition in the context of a recent proposal of spinons with fracton properties in the PVBS state, concluding that the fracton scenario is unlikely. Furthermore, we discover a novel type of eightfold degenerate VBS phase, arising when the PVBS state breaks a remaining Z 2 symmetry. This second phase transition, which is continuous, implies that the PVBS phase can be regarded as an intermediate "vestigial" phase, a concept recently introduced to describe multistage phase transitions involving a continuous symmetry. Here we construct a six-dimensional order parameter and also introduce a general graph-theoretic approach to describe the two-stage discrete symmetry breaking. We discuss different ways of breaking the symmetries in one or two stages at zero and finite temperatures. In the latter case, we observe fluctuation-induced first-order transitions, which are hallmarks of vestigial phase transitions. We also mention possible connections of the AFM-PVBS transition to the SO(5) theory of high-T c superconductivity.

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Section: Future Prospectsmentioning

confidence: 99%

Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet

Takahashi

Sandvik

2020

Phys. Rev. Research

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“…We here also note that the PVBS state can be regarded as a "superposition of superposition states" (locally resonating valence bonds), and this specific aspect of the state seems to be what makes it more difficult to realize a PVBS than a CVBS-at least within the J-Q approach. The broader family of models by Kaul [121], expanding on the J-Q approach, should also be explored more extensively, though our initial attempts have not been successful in producing PVBS states with less than twelve interacting spins.…”

Section: Future Prospectsmentioning

confidence: 99%

Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet

Takahashi,

Sandvik

2020

Preprint

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We introduce a quantum spin-1/2 model with many-body correlated Heisenberg-type interactions on the two-dimensional square lattice, designed so that the system can host a four-fold degenerate plaquette valence-bond solid (PVBS) ground state that spontaneously breaks Z4 symmetry. The system is sign-problem free and amenable to large-scale quantum Monte Carlo simulations, thus allowing us to carry out a detailed study of the quantum phase transition between the standard Néel antiferromagnetic (AFM) and PVBS states. We find a first-order transition, in contrast to previously studied continuous transitions from the AFM phase into a columnar valence-bond solid (CVBS) phase. The theory of deconfined quantum criticality predicts generic continuous AFM-CVBS and AFM-PVBS transitions, and, in one version of the theory, the two critical order parameters transform under SO(5) symmetry. Emergent SO(5) symmetry has indeed been observed in studies of the AFM-CVBS transition, and here we show that the first-order AFM-PVBS transition also is associated with SO(5) symmetry. Such unexpected symmetry of the coexistence state, which implies a lack of energy barriers between the coexisting phases, has recently been observed at other first-order transitions, but the case presented here is the first example with SO(5) symmetry. The extended symmetry may indicate that the transition is connected to a deconfined critical point. We also discuss the first-order transition in the context of a recent proposal of spinons with fracton properties in the PVBS state, concluding that the fracton scenario is unlikely. Furthermore, we discover a novel type of eight-fold degenerate VBS phase, arising when the PVBS state breaks a remaining Z2 symmetry. This second phase transition, which is continuous, implies that the PVBS phase can be regarded as an intermediate "vestigial" phase, a concept recently introduced to describe multi-stage phase transitions involving a continuous symmetry. Here we construct a six-dimensional order parameter and also introduce a general graph-theoretic approach to describe the two-stage discrete symmetry breaking. We discuss different ways of breaking the symmetries in one or two stages at zero and finite temperatures. In the latter case we observe fluctuation-induced first-order transitions, which are hallmarks of vestigial phase transitions. We also mention possible connections of the AFM-PVBS transition to the SO(5) theory of high-Tc superconductivity.

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“…This naturally calls for design principles. For bosonic and quantum spin systems a valuable guiding principle is the Marshall sign rule [6,7] which ensures nonnegative weight for all configurations. The design of the sign-problem-free fermionic Hamiltonians is harder.…”

mentioning

confidence: 99%

“…where the matrix (Λ σ ij ) lm = σλ(δ li δ mj + δ lj δ mi ) according to the exponential factor of Eq. (7). Equation ( 8) is in the general form of Eq.…”

mentioning

confidence: 99%

Split Orthogonal Group: A Guiding Principle for Sign-Problem-Free Fermionic Simulations

et al. 2015

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We present a guiding principle for designing fermionic Hamiltonians and quantum Monte Carlo (QMC) methods that are free from the infamous sign problem by exploiting the Lie groups and Lie algebras that appear naturally in the Monte Carlo weight of fermionic QMC simulations. Specifically, rigorous mathematical constraints on the determinants involving matrices that lie in the split orthogonal group provide a guideline for sign-free simulations of fermionic models on bipartite lattices. This guiding principle not only unifies the recent solutions of the sign problem based on the continuous-time quantum Monte Carlo methods and the Majorana representation, but also suggests new efficient algorithms to simulate physical systems that were previously prohibitive because of the sign problem.PACS numbers: 02.70. Ss, 71.10.Fd, 02.20.Tw One of the biggest challenges to classical simulation of quantum systems is the infamous fermion sign problem of quantum Monte Carlo (QMC) simulations. It appears when the weights of configurations in a QMC simulation may become negative and therefore cannot be directly interpreted as probabilities [1]. In the presence of a sign problem, the simulation effort typically grows exponentially with system size and inverse temperature.While the sign problem is nondeterministic polynomial (NP) hard [2], implying that there is little hope of finding a generic solution, this does not exclude ad hoc solutions to the sign problem for specific models. For example, one can sometimes exploit symmetries to design appropriate sign-problem-free QMC algorithms for a restricted class of models [3]. However, it is unclear how broad these classes are and it is in general hard to foresee whether a given physical model would have a sign problem in any QMC simulations. The situation is not dissimilar to the study of many intriguing problems in the NP complexity class, where a seemingly infeasible problem might turn out to have a polynomial-time solution surprisingly [4].A fruitful approach in pursuing such specific solutions is to design Hamiltonians that capture the right low energy physics and allow sign-problem-free QMC simulations at the same time, called "designer" Hamiltonians [5]. This naturally calls for design principles. For bosonic and quantum spin systems a valuable guiding principle is the Marshall sign rule [6,7] which ensures nonnegative weight for all configurations. The design of the sign-problem-free fermionic Hamiltonians is harder. The method of choice for fermionic QMC simulations are the determinantal QMC approaches, including traditional discrete-time [8] and new continuous-time approaches [9][10][11][12][13]. Both approaches map the original interacting system to free fermions with an imaginary-time dependent Hamiltonian. The partition function is then written as a weighted sum of matrix determinants after tracing out the fermions [8,9,12]:where f C is a c-number and H C (τ ) is an imaginary-time dependent single-particle Hamiltonian matrix (whose matrix elements denote hopping amplitude...

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Marshall-positiveSU(N)quantum spin systems and classical loop models: A practical strategy to design sign-problem-free spin Hamiltonians

Cited by 17 publications

References 34 publications

Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet

Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet

Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet

Split Orthogonal Group: A Guiding Principle for Sign-Problem-Free Fermionic Simulations

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