scholarly journals Dimensional crossover in the SU(4) Heisenberg model in the six-dimensional antisymmetric self-conjugate representation revealed by quantum Monte Carlo and linear flavor-wave theory

2019 ◽  
Vol 100 (8) ◽  
Author(s):  
Francisco H. Kim ◽  
Fakher F. Assaad ◽  
Karlo Penc ◽  
Frédéric Mila
Keyword(s):  
Monte Carlo ◽  
Quantum Monte Carlo ◽  
Heisenberg Model ◽  
Wave Theory ◽  
Dimensional Crossover

MONTE CARLO STUDY OF THE SPIN-1/2 HEISENBERG MODEL IN 1, 2, AND 3 DIMENSIONS

1991 ◽  
Vol 05 (13) ◽  
pp. 907-914 ◽  
Author(s):  
RICHARD J. CRESWICK ◽  
CYNTHIA J. SISSON
Keyword(s):  
Monte Carlo ◽  
High Temperature ◽  
Spin Wave ◽  
Heisenberg Model ◽  
Transition Probability ◽  
Lattice Models ◽  
Wave Theory ◽  
Temperature Expansion ◽  
High Temperature Expansion ◽  
Good Agreement

The properties of the spin-1/2 Heisenberg model on 1, 2, and 3-dimensional lattices are calculated using the Decoupled Cell Method of Homma et al., and these results are compared with high temperature and spin-wave expansions, and with other numerical approaches. The DCM has advantages over other Monte Carlo methods currently in wide use in that the transition probability is positive definite, there is no need to introduce an additional imaginary time, or Trotter, dimension, and the acceptance rate for transitions is comparable to that of classical lattice models. We find very good agreement between the DCM and the high temperature expansion in the temperature region where the high temperature expansion is valid, and reasonably good agreement at low temperatures with spin wave theory. The DCM fails for temperatures T < Tc which decreases with the size of the cell.

scholarly journals AUXILIARY FIELDS AND THE SIGN PROBLEM

1995 ◽  
Vol 06 (03) ◽  
pp. 427-465 ◽  
Author(s):  
J.H. SAMSON
Keyword(s):  
Monte Carlo ◽  
Correlation Functions ◽  
Quantum Monte Carlo ◽  
Heisenberg Model ◽  
Functional Integral ◽  
Auxiliary Field ◽  
Operator Product ◽  
Phase Problem ◽  
Stochastic Field ◽  
Sign Problem

The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.

scholarly journals QUANTUM EFFECTS AND BROKEN SYMMETRIES IN FRUSTRATED ANTIFERROMAGNETS

2001 ◽  
Vol 15 (12) ◽  
pp. 1799-1842 ◽  
Author(s):  
LUCA CAPRIOTTI
Keyword(s):  
Monte Carlo ◽  
Excitation Spectrum ◽  
Spin Wave ◽  
Ground State ◽  
Quantum Monte Carlo ◽  
Finite Size ◽  
Wave Theory ◽  
Low Energy ◽  
Heisenberg Antiferromagnet ◽  
Néel Order

We investigate the interplay between frustration and zero-point quantum fluctuations in the ground state of the triangular and J1–J2 Heisenberg antiferromagnets, using finite-size spin-wave theory, exact diagonalization, and quantum Monte Carlo methods. In the triangular Heisenberg antiferromagnet, by performing a systematic size-scaling analysis, we have obtained strong evidences for a gapless spectrum and a finite value of the thermodynamic order parameter, thus confirming the existence of long-range Néel order. The good agreement between the finite-size spin-wave results and the exact and quantum Monte Carlo data also supports the reliability of the spin-wave expansion to describe both the ground state and the low-energy spin excitations of the triangular Heisenberg antiferromagnet. In the J1–J2 Heisenberg model, our results indicate the opening of a finite gap in the thermodynamic excitation spectrum at J2/J1≃0.4, marking the melting of the antiferromagnetic Néel order and the onset of a non-magnetic ground state. In order to characterize the nature of the latter quantum-disordered phase we have computed the susceptibilities for the most important crystal symmetry breaking operators. In the ordered phase the effectiveness of the spin-wave theory in reproducing the low-energy excitation spectrum suggests that the uniform spin susceptibility of the model is very close to the linear spin-wave prediction.

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