We study the continuum limit of the quantum nonlinear \ensuremath{\sigma} model in 2+1 dimensions and at finite temperature T using both Monte Carlo simulation on large-size lattices ( ${100}^{2}$ \ifmmode\times\else\texttimes\fi{}8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point ${g}_{c}$ that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). We calculate the model's renormalization group \ensuremath{\beta} function close to the critical point. Using the \ensuremath{\beta} function, we rescale the correlation lengths calculated at various values of the coupling constant (spin stiffness) and temperature and find that they all collapse on the same curve \ensuremath{\xi}/${a}_{\ensuremath{\sigma}}$=f(T/${T}_{\ensuremath{\sigma}}$). Even though the lattice spacing vanishes, a finite unit of length ${a}_{\ensuremath{\sigma}}$ and a temperature scale ${T}_{\ensuremath{\sigma}}$ is generated via dimensional transmutation. Assuming that the nonlinear \ensuremath{\sigma} model and the spin-(1/2 antiferromagnetic (AF) Heisenberg model are equivalent at low temperature, we relate the units ${a}_{\ensuremath{\sigma}}$ and ${T}_{\ensuremath{\sigma}}$ to the lattice spacing ${a}_{H}$ and the AF coupling J of the Heisenberg model so that the correlation lengths obtained from the simulation of the two models agree. In order to achieve this agreement we find that (a) the spin-(1/2 AF Heisenberg model should order at T=0 and (b) the relationship between the scales ${a}_{\ensuremath{\sigma}}$,${T}_{\ensuremath{\sigma}}$ and ${a}_{H}$ , J is obtained, and f(T/${T}_{\ensuremath{\sigma}}$) can be accurately approximated by an exponential of ${T}_{\ensuremath{\sigma}}$/T below ${g}_{c}$ . We obtain a reasonable fit to the neutron scattering data of the insulator ${\mathrm{La}}_{2}$ ${\mathrm{CuO}}_{4}$ by taking J=1270 K, a value close to that reported by Raman scattering experiments.We study the continuum limit of the quantum nonlinear \ensuremath{\sigma} model in 2+1 dimensions and at finite temperature T using both Monte Carlo simulation on large-size lattices (${100}^{2}$\ifmmode\times\else\texttimes\fi{}8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point ${g}_{c}$ that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). We calculate the model's renormalization group \ensuremath{\beta} function close to the critical point. Using the \ensuremath{\beta} function, we rescale the correlation lengths calculated at various values of the coupling constant (spin stiffness) and temperature and find that they all collapse on the same curve \ensuremath{\xi}/${a}_{\ensuremath{\sigma}}$=f(T/${T}_{\ensuremath{\sigma}}$). Even though the lattice spacing vanishes, a finite unit of length ${a}_{\ensuremath{\sigma}}$ and a temperature scale ${T}_{\ensuremath{\sigma}}$ is generated via dimensional transmutation. Assuming that the nonlinear \ensuremath{\sigma} model and the spin-(1/2 antiferromagnetic (AF) Heisenberg model are equivalent at low temperature, we relate the units ${a}_{\ensuremath{\sigma}}$ and ${T}_{\ensuremath{\sigma}}$ to the lattice spacing ${a}_{H}$ and the AF coupling J of the Heisenberg model so that the correlation lengths obtained from the simulation of the two models agree. In order to achieve this agreement we find that (a) the spin-(1/2 AF Heisenberg model should order at T=0 and (b) the relationship between the scales ${a}_{\ensuremath{\sigma}}$,${T}_{\ensuremath{\sigma}}$ and ${a}_{H}$, J is obtained, and f(T/${T}_{\ensuremath{\sigma}}$) can be accurately approximated by an exponential of ${T}_{\ensuremath{\sigma}}$/T below ${g}_{c}$. We obtain a reasonable fit to the neutron scattering data of the insulator ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$ by taking J=1270 K, a value close to that reported by Raman scattering experiments.

Physical review. B, Condensed matterAug 1, 198931 citations

On the high temperature susceptibility of the Heisenberg model

David W. Wood

Physics LettersFeb 1, 19656 citations

Finite-size scaling study of the three-dimensional classical Heisenberg model

Christian Holm, Wolfhard Janke E

arXiv:hep-lat/9209017Physics Letters AJan 1, 199353 citations

Study of Surface Effects in Quantum Heisenberg and XY Antiferromagnets

J. Cabral Neto, J. Ricardo de Sousa

Using a finite-cluster mean-field approximation, we study the characteristics of phase transitions for the semi-infinite simple cubic quantum spin-1/2 anisotropic Heisenberg model with ferromagnetic and antiferromagnetic interactions. The dependence of the critical temperature on the parameter Δ is obtained, where Δ = Js/Jb – 1, Js andJb are the nearest-neighbour coupling constants at the surface and bulk, respectively. Particular attention has been drawn to the difference between the criticality for the F (ferromagnetic) and AF (antiferromagnetic) models. It is shown that quantum influence modifies the criticality of the anisotropic Heisenberg model with Tc (F case) < TN (AF case) and Tc > TN in the bulk-paramagnetic and surface paramagnetic critical lines, respectively. On the other hand, the quantum XY model presents the same criticality for the two types of interactions (F and AF).Using a finite-cluster mean-field approximation, we study the characteristics of phase transitions for the semi-infinite simple cubic quantum spin-1/2 anisotropic Heisenberg model with ferromagnetic and antiferromagnetic interactions. The dependence of the critical temperature on the parameter Δ is obtained, where Δ = Js/Jb – 1, Js andJb are the nearest-neighbour coupling constants at the surface and bulk, respectively. Particular attention has been drawn to the difference between the criticality for the F (ferromagnetic) and AF (antiferromagnetic) models. It is shown that quantum influence modifies the criticality of the anisotropic Heisenberg model with Tc (F case) < TN (AF case) and Tc > TN in the bulk-paramagnetic and surface paramagnetic critical lines, respectively. On the other hand, the quantum XY model presents the same criticality for the two types of interactions (F and AF).

physica status solidi (b)Apr 1, 19996 citations

Antiferromagnetic transitions of Dirac fermions in three dimensions

Yiqun Huang, Huaiming Guo, Joseph Maciejko +2 more

We use determinant quantum Monte Carlo simulations to study the role of electron-electron interactions on three-dimensional (3D) Dirac fermions based on the $\ensuremath{\pi}$-flux model on a cubic lattice. We show that the Hubbard interaction drives the 3D Dirac semimetal to an antiferromagnetic (AF) insulator only above a finite critical interaction strength and the long-range AF order persists up to a finite temperature. We evaluate the critical interaction strength and temperatures using finite-size scaling of the spin structure factor. The critical behaviors are consistent with the (3+1)-dimensional Gross-Neveu universality class for the quantum critical point and 3D Heisenberg universality class for the thermal phase transitions. We further investigate correlation effects in the birefringent Dirac fermion system. It is found that the critical interaction strength ${U}_{c}$ is decreased by reducing the velocity of the Dirac cone, quantifying the effect of velocity on the critical interaction strength in 3D Dirac fermion systems. Our findings unambiguously uncover correlation effects in 3D Dirac fermions and may be observed using ultracold atoms in an optical lattice.We use determinant quantum Monte Carlo simulations to study the role of electron-electron interactions on three-dimensional (3D) Dirac fermions based on the $\ensuremath{\pi}$-flux model on a cubic lattice. We show that the Hubbard interaction drives the 3D Dirac semimetal to an antiferromagnetic (AF) insulator only above a finite critical interaction strength and the long-range AF order persists up to a finite temperature. We evaluate the critical interaction strength and temperatures using finite-size scaling of the spin structure factor. The critical behaviors are consistent with the (3+1)-dimensional Gross-Neveu universality class for the quantum critical point and 3D Heisenberg universality class for the thermal phase transitions. We further investigate correlation effects in the birefringent Dirac fermion system. It is found that the critical interaction strength ${U}_{c}$ is decreased by reducing the velocity of the Dirac cone, quantifying the effect of velocity on the critical interaction strength in 3D Dirac fermion systems. Our findings unambiguously uncover correlation effects in 3D Dirac fermions and may be observed using ultracold atoms in an optical lattice.

arXiv:2007.15175Physical review. B./Physical review. BOct 30, 202013 citations

Critical Correlations and Finite Thickness Effects in the Heisenberg Model .

Douglas S. Ritchie

Jan 1, 19710 citations

We study the continuum limit of the quantum nonlinear \ensuremath{\sigma} model in 2+1 dimensions and at finite temperature T using both Monte Carlo simulation on large-size lattices (

{100}^{2}

\ifmmode\times\else\texttimes\fi{}8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point

{g}_{c}

that separates the quantum disordered phase from the phase with spontaneous symmetry breaking (nonzero staggered magnetization). We calculate the model's renormalization group \ensuremath{\beta} function close to the critical point. Using the \ensuremath{\beta} function, we rescale the correlation lengths calculated at various values of the coupling constant (spin stiffness) and temperature and find that they all collapse on the same curve \ensuremath{\xi}/${a}_{\ensuremath{\sigma}}$=f(T/${T}_{\ensuremath{\sigma}}$). Even though the lattice spacing vanishes, a finite unit of length ${a}_{\ensuremath{\sigma}}$ and a temperature scale ${T}_{\ensuremath{\sigma}}$ is generated via dimensional transmutation. Assuming that the nonlinear \ensuremath{\sigma} model and the spin-(1/2 antiferromagnetic (AF) Heisenberg model are equivalent at low temperature, we relate the units ${a}_{\ensuremath{\sigma}}$ and ${T}_{\ensuremath{\sigma}}$ to the lattice spacing

{a}_{H}

and the AF coupling J of the Heisenberg model so that the correlation lengths obtained from the simulation of the two models agree. In order to achieve this agreement we find that (a) the spin-(1/2 AF Heisenberg model should order at T=0 and (b) the relationship between the scales ${a}_{\ensuremath{\sigma}}$,${T}_{\ensuremath{\sigma}}$ and

{a}_{H}

, J is obtained, and f(T/${T}_{\ensuremath{\sigma}}$) can be accurately approximated by an exponential of ${T}_{\ensuremath{\sigma}}$/T below

{g}_{c}

. We obtain a reasonable fit to the neutron scattering data of the insulator

{\mathrm{La}}_{2}

{\mathrm{CuO}}_{4}

by taking J=1270 K, a value close to that reported by Raman scattering experiments.

{U}_{c}

is decreased by reducing the velocity of the Dirac cone, quantifying the effect of velocity on the critical interaction strength in 3D Dirac fermion systems. Our findings unambiguously uncover correlation effects in 3D Dirac fermions and may be observed using ultracold atoms in an optical lattice.