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:cond-mat/0606032 · Physical Review B

Scaling relations in the quasi-two-dimensional Heisenberg antiferromagnet

Antoine Praz, Christopher Mudry, M. B. Hastings

Published Nov 3, 2006• 7 citations
arXivHTMLPDF

Abstract

The large-$N$ expansion of the quasi-two-dimensional quantum nonlinear $\ensuremath{\sigma}$ model is used in order to establish experimentally applicable universal scaling relations for the quasi-two-dimensional Heisenberg antiferromagnet. We show that, at $N=\ensuremath{\infty}$, the renormalized coordination number introduced by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)] is a universal number in the limit of ${J}^{\ensuremath{'}}∕J\ensuremath{\rightarrow}0$. Moreover, similar scaling relations proposed by Hastings and Mudry [Phys. Rev. Lett. 96, 027215 (2006)] are derived at $N=\ensuremath{\infty}$ for the three-dimensional static spin susceptibility at the wave vector $(\ensuremath{\pi},\ensuremath{\pi},0)$, as well as for the instantaneous structure factor at the same wave vector. We then use $1∕N$ corrections to study the relation between interplane coupling, correlation length, and critical temperature, and show that the universal scaling relations lead to logarithmic corrections to previous mean-field results.

DOI
10.1103/physrevb.74.184407

Related papers

Scaling regimes, crossovers, and lattice corrections in two-dimensional Heisenberg antiferromagnets

N. Elstner, Rodney L. Glenister, Rajiv R. P. Singh +1 more

We study scaling behavior in two-dimensional, square lattice, S=1/2 and S=1 Heisenberg antiferromagnets using the data on full q dependences of the equal-time structure factor and the static susceptibility, calculated through high-temperature series expansions. We also carry out comparisons with a model of two coupled S=1/2 Heisenberg planes with the interlayer exchange coupling tuned to the T=0 critical point (two-plane model hereafter). For both S=1/2 and S=1 models, we separately determine the spin-wave velocity c and mass m=c/\ensuremath{\xi}, in addition to the correlation length, \ensuremath{\xi}, and find that c is temperature dependent; only for temperatures below T\ensuremath{\lesssim}JS, where J is the exchange coupling, c approaches its known T=0 values, ${\mathit{c}}_{0}$. This nonuniversal lattice effect is caused by the quantum nature of spin, and is therefore not captured by the quantum nonlinear \ensuremath{\sigma} model.Despite this temperature dependence of the spin-wave velocity, full q and \ensuremath{\omega} dependences of the dynamical susceptibility \ensuremath{\chi}(q,\ensuremath{\omega}) agree with those of the universal scaling function, computable for the \ensuremath{\sigma} model, for temperatures up to ${\mathit{T}}_{0}$\ensuremath{\sim}0.6${\mathit{c}}_{0}$/a, and their further analysis leads us to the inference that below ${\mathit{T}}_{0}$ the S=1 model is in the renormalized classical (RC) regime, the two-plane model is in the quantum critical (QC) regime in agreement with earlier work, and the S=1/2 model exhibits a RC-QC crossover, centered around crossover and for the two-plane model at all T=0.55J. In particular, for the S=1/2 model above the RC-QC temperatures where calculated, the obtained spin-wave mass m=c/\ensuremath{\xi} is in excellent agreement with the known universal QC prediction, m\ensuremath{\simeq}1.04T. In contrast, for the S=1/2 model below the RC-QC crossover, and for the S=1 model at all temperatures, the behavior agrees with the exact RC expression. for all three models, nonuniversal behavior occurs above T\ensuremath{\sim}0.6${\mathit{c}}_{0}$/a. Our results strongly support the conjecture by Chubukov and Sachdev that the S=1/2 model is close enough to the T=0 critical point to exhibit QC behavior at intermediate temperatures.

arXiv:cond-mat/9409037•Physical review. B, Condensed matter•Apr 1, 1995•20 citations

Universal reduction of effective coordination number in the quasi-one-dimensional Ising model

Synge Todo

Critical temperature of quasi-one-dimensional general-spin Ising ferromagnets is investigated by means of the cluster Monte Carlo method performed on infinite-length strips, $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ or $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$. We find that in the weak interchain coupling regime the critical temperature as a function of the interchain coupling is well-described by a chain mean-field formula with a reduced effective coordination number, as the quantum Heisenberg antiferromagnets recently reported by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)]. It is also confirmed that the effective coordination number is independent of the spin size. We show that in the weak interchain coupling limit the effective coordination number is, irrespective of the spin size, rigorously given by the quantum critical point of a spin-$1∕2$ transverse-field Ising model.

arXiv:cond-mat/0606790•Physical Review B•Sep 22, 2006•6 citations

High-precision finite-size scaling analysis of the quantum-critical point of S=1∕2 Heisenberg antiferromagnetic bilayers

Ling Wang, K. S. D. Beach, Anders W. Sandvik

We use quantum Monte Carlo (stochastic series expansion) and finite-size scaling to study the quantum critical points of two $S=1∕2$ Heisenberg antiferromagnets in two dimensions: a bilayer and a Kondo-lattice-like system (incomplete bilayer), each with intraplane and interplane couplings $J$ and ${J}_{\ensuremath{\perp}}$. We discuss the ground-state finite-size scaling properties of three different quantities---the Binder moment ratio, the spin stiffness, and the long-wavelength magnetic susceptibility---which we use to extract the critical value of the coupling ratio $g={J}_{\ensuremath{\perp}}∕J$. The individual estimates of ${g}_{c}$ are consistent provided that subleading finite-size corrections are properly taken into account. For both models, we find that the spin stiffness has the smallest subleading finite-size corrections; in the case of the incomplete bilayer we find that the first subleading correction vanishes or is extremely small. In agreement with predictions, we find that at the critical point the Binder ratio has a universal value and the product of the spin stiffness and the long-wavelength susceptibility scales as $1∕{L}^{2}$ with a universal prefactor. Our results for the critical coupling ratios are ${g}_{c}=2.5220(1)$ (full bilayer) and ${g}_{c}=1.3888(1)$ (incomplete bilayer), which represent improvements of more than an order of magnitude over the previous best estimates. For the correlation length exponent we obtain $\ensuremath{\nu}=0.7106(9)$, consistent with the expected three-dimensional Heisenberg universality class.

arXiv:cond-mat/0509747•Physical Review B•Jan 27, 2006•175 citations

Equivalence between the nonlinearσmodel and the spin-(1/2 antiferromagnetic Heisenberg model: Spin correlations in La2 CuO4

Efstratios Manousakis, Román Salvador

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 matter•Aug 1, 1989•31 citations

Universal relations in the finite-size correction terms of two-dimensional Ising models.

Y Okabe, N Kawashima

Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.

Chinese Journal of Natural Medicines•Sep 1, 2001•12 citations

Universal relations in the finite-size correction terms of two-dimensional Ising models

Yutaka Okabe, Naoki Kawashima

Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.

arXiv:cond-mat/0107514•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 27, 2001•6 citations
The large-NNN expansion of the quasi-two-dimensional quantum nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model is used in order to establish experimentally applicable universal scaling relations for the quasi-two-dimensional Heisenberg antiferromagnet. We show that, at N=\ensuremath∞N=\ensuremath{\infty}N=\ensuremath∞, the renormalized coordination number introduced by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)] is a universal number in the limit of J\ensuremath′∕J\ensuremath→0{J}^{\ensuremath{'}}∕J\ensuremath{\rightarrow}0J\ensuremath′∕J\ensuremath→0. Moreover, similar scaling relations proposed by Hastings and Mudry [Phys. Rev. Lett. 96, 027215 (2006)] are derived at N=\ensuremath∞N=\ensuremath{\infty}N=\ensuremath∞ for the three-dimensional static spin susceptibility at the wave vector (\ensuremathπ,\ensuremathπ,0)(\ensuremath{\pi},\ensuremath{\pi},0)(\ensuremathπ,\ensuremathπ,0), as well as for the instantaneous structure factor at the same wave vector. We then use 1∕N1∕N1∕N corrections to study the relation between interplane coupling, correlation length, and critical temperature, and show that the universal scaling relations lead to logarithmic corrections to previous mean-field results.
We study scaling behavior in two-dimensional, square lattice, S=1/2 and S=1 Heisenberg antiferromagnets using the data on full q dependences of the equal-time structure factor and the static susceptibility, calculated through high-temperature series expansions. We also carry out comparisons with a model of two coupled S=1/2 Heisenberg planes with the interlayer exchange coupling tuned to the T=0 critical point (two-plane model hereafter). For both S=1/2 and S=1 models, we separately determine the spin-wave velocity c and mass m=c/\ensuremath{\xi}, in addition to the correlation length, \ensuremath{\xi}, and find that c is temperature dependent; only for temperatures below T\ensuremath{\lesssim}JS, where J is the exchange coupling, c approaches its known T=0 values, c0{\mathit{c}}_{0}c0​. This nonuniversal lattice effect is caused by the quantum nature of spin, and is therefore not captured by the quantum nonlinear \ensuremath{\sigma} model.Despite this temperature dependence of the spin-wave velocity, full q and \ensuremath{\omega} dependences of the dynamical susceptibility \ensuremath{\chi}(q,\ensuremath{\omega}) agree with those of the universal scaling function, computable for the \ensuremath{\sigma} model, for temperatures up to T0{\mathit{T}}_{0}T0​\ensuremath{\sim}0.6c0{\mathit{c}}_{0}c0​/a, and their further analysis leads us to the inference that below T0{\mathit{T}}_{0}T0​ the S=1 model is in the renormalized classical (RC) regime, the two-plane model is in the quantum critical (QC) regime in agreement with earlier work, and the S=1/2 model exhibits a RC-QC crossover, centered around crossover and for the two-plane model at all T=0.55J. In particular, for the S=1/2 model above the RC-QC temperatures where calculated, the obtained spin-wave mass m=c/\ensuremath{\xi} is in excellent agreement with the known universal QC prediction, m\ensuremath{\simeq}1.04T. In contrast, for the S=1/2 model below the RC-QC crossover, and for the S=1 model at all temperatures, the behavior agrees with the exact RC expression. for all three models, nonuniversal behavior occurs above T\ensuremath{\sim}0.6c0{\mathit{c}}_{0}c0​/a. Our results strongly support the conjecture by Chubukov and Sachdev that the S=1/2 model is close enough to the T=0 critical point to exhibit QC behavior at intermediate temperatures.
Critical temperature of quasi-one-dimensional general-spin Ising ferromagnets is investigated by means of the cluster Monte Carlo method performed on infinite-length strips, $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ or $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$. We find that in the weak interchain coupling regime the critical temperature as a function of the interchain coupling is well-described by a chain mean-field formula with a reduced effective coordination number, as the quantum Heisenberg antiferromagnets recently reported by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)]. It is also confirmed that the effective coordination number is independent of the spin size. We show that in the weak interchain coupling limit the effective coordination number is, irrespective of the spin size, rigorously given by the quantum critical point of a spin-1∕21∕21∕2 transverse-field Ising model.
We use quantum Monte Carlo (stochastic series expansion) and finite-size scaling to study the quantum critical points of two S=1∕2S=1∕2S=1∕2 Heisenberg antiferromagnets in two dimensions: a bilayer and a Kondo-lattice-like system (incomplete bilayer), each with intraplane and interplane couplings JJJ and ${J}_{\ensuremath{\perp}}$. We discuss the ground-state finite-size scaling properties of three different quantities---the Binder moment ratio, the spin stiffness, and the long-wavelength magnetic susceptibility---which we use to extract the critical value of the coupling ratio $g={J}_{\ensuremath{\perp}}∕J$. The individual estimates of gc{g}_{c}gc​ are consistent provided that subleading finite-size corrections are properly taken into account. For both models, we find that the spin stiffness has the smallest subleading finite-size corrections; in the case of the incomplete bilayer we find that the first subleading correction vanishes or is extremely small. In agreement with predictions, we find that at the critical point the Binder ratio has a universal value and the product of the spin stiffness and the long-wavelength susceptibility scales as 1∕L21∕{L}^{2}1∕L2 with a universal prefactor. Our results for the critical coupling ratios are gc=2.5220(1){g}_{c}=2.5220(1)gc​=2.5220(1) (full bilayer) and gc=1.3888(1){g}_{c}=1.3888(1)gc​=1.3888(1) (incomplete bilayer), which represent improvements of more than an order of magnitude over the previous best estimates. For the correlation length exponent we obtain $\ensuremath{\nu}=0.7106(9)$, consistent with the expected three-dimensional Heisenberg universality class.
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 (1002{100}^{2}1002\ifmmode\times\else\texttimes\fi{}8 is our largest-size lattice) and saddle-point approximation. At zero temperature, we find the critical point gc{g}_{c}gc​ 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 aH{a}_{H}aH​ 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 aH{a}_{H}aH​, J is obtained, and f(T/${T}_{\ensuremath{\sigma}}$) can be accurately approximated by an exponential of ${T}_{\ensuremath{\sigma}}$/T below gc{g}_{c}gc​. We obtain a reasonable fit to the neutron scattering data of the insulator La2{\mathrm{La}}_{2}La2​CuO4{\mathrm{CuO}}_{4}CuO4​ by taking J=1270 K, a value close to that reported by Raman scattering experiments.
Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.
Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.