Finite-size scaling for a first-order transition where a continuous symmetry is broken: The spin-flop transition in the three-dimensional $XXZ$ Heisenberg antiferromagnet

Finite-size scaling for a first-order phase transition where a continuous symmetry is broken is developed using an approximation of Gaussian probability distributions with a phenomenological “degeneracy” factor included. Predictions are compared with data from Monte Carlo simulations of the three-dimensional, $XXZ$ Heisenberg antiferromagnet in a field in order to study the finite-size behavior on a $L\times L\times L$ simple cubic lattice for the first-order “spin-flop” transition between the Ising-like antiferromagnetic state and the canted, $XY$-like state. Our theory predicts that for large linear dimension $L$ the field dependence of all moments of the order parameters as well as the fourth-order cumulants exhibit universal intersections. Corrections to leading order should scale as the inverse volume. The values of these intersections at the spin-flop transition point can be expressed in terms of a factor $q$ that characterizes the relative degeneracy of the ordered phases. Our theory yields $q=\pi$, and we present numerical evidence that is compatible with this prediction. The agreement between the theory and simulation implies a heretofore unknown universality can be invoked for first-order phase transitions.

Figure 1

Spin configurations for different phases in the anisotropic Heisenberg model in an applied field. Illustrated are spin configurations of the two sublattices in the (a) antiferromagnetic (AF) and (b) spin-flop (SF) phases. ${\theta }_{\mathrm{SF}}$ is the angle that the spins make with respect to the applied field.

Figure 2

Phase diagram for the simple cubic, anisotropic Heisenberg model in an applied magnetic field, $H$, near the bicritical point [17]. Both the field and the temperature, $T$, ($k_B$ is Boltzmann's constant) are normalized by the exchange constant $J$. The order parameter for the antiferromagnetic phase is $\widetilde{m_z}$ and for the spin-flop phase is $\vec{\psi }$. The $z$ component of the uniform magnetization is $m_z$.

Figure 3

Schematic variation of thermodynamic quantities with field $H$ for $T<T_b$. Top: Free energy $F(T,H)$. The absolute magnitude of the free-energy differences $\mathrm{\Delta }{F}_{\mathrm{AF}}$, $\mathrm{\Delta }{F}_{\mathrm{SF}}$ in the antiferromagnetic (AF) and spin-flop (SF) phases are relative to the free energy (per spin) at the transition $\left(F_t\right)$. Near the transition field $H^t$ these free-energy differences vary linearly with $H$. Middle: Magnetization $m_{z,\infty }$ along the field direction. At $H^t$ a jump occurs from $m_{z,\infty }^{\mathrm{AF}}$ (in the thermodynamic limit, $L\to \infty$) in the AF phase to $m_{z,\infty }^{\mathrm{SF}}$ in the SF phase. Near $H^t$ the variation of $m_{z,\infty }$ with $H$ is linear. Bottom: Absolute value of the order parameter of the SF phase, ${\psi }_{\infty }$. Since the AF-SF transition is first order, ${\psi }_{\infty }$ jumps discontinuously from zero to a nonzero value when $H=H^t$.

Figure 4

Probability distribution (unnormalized) of the energy at the transition field $H_L^t$ for different lattice sizes $L$.

Figure 5

Probability distribution (unnormalized) of the $z$ component of the magnetization at the transition field $H_L^t$ for different lattice sizes $L$.

Figure 6

Extrapolation of the locations of the transition fields determined from the “equal height” rule for peaks in the probability distribution of the energy and from the minimum of the fourth-order cumulant of the energy vs the inverse volume of the system.

Figure 7

Size dependence of the finite-size lattice transition field $H_L^t$ determined from the locations of the maxima of multiple susceptibilities vs $L^{-3}$ for lattices sizes from $L=30$ to $L=100$. The solid lines show extrapolations to $L=\infty$ for $L\ge 50$.

Figure 8

Contours for the order parameter distribution $P_L\left(\vec{\psi }\right)$ with $\vec{\psi }=({\psi }_x,{\psi }_y$) being the two component order parameter comprising the $xy$ components of the staggered magnetization in the spin-flop phase for an $L=60$ lattice at $H=H_L^t$. Different colors (in the electronic version) denote the magnitude of the probability (from the center outward the probability first decreases and then increases again).

Figure 9

Variation of the fourth-order cumulant of the $z$ component of the magnetization vs $H$ for different lattice sizes.

Figure 10

Variations of the fourth-order cumulant of the energy for different lattice sizes vs applied field $H$.

Figure 11

Probability distribution (unnormalized) of the SF order parameter ${\psi }_x$, with ${\psi }_y=0$ for different lattice sizes.

Figure 12

Probability distribution (unnormalized) of the AF order parameter $\widetilde{m_z}$ for different lattice sizes.

Figure 13

Top: Variation of the fourth-order cumulant of the AF order parameter vs magnetic field for different lattice sizes. Bottom: Same as above but on a finer scale.

Figure 14

Top: Variation of the fourth-order cumulant of the SF order parameter vs magnetic field for different lattice sizes. Bottom: Same as above but on a finer scale.

Figure 15

Variation of the minima in the fourth-order cumulants of the AF and SF order parameters, the energy $E$, and the uniform magnetization $m_z$ for different lattice sizes.

Figure 16

Top: Variation of the SF order parameter vs magnetic field for different lattice sizes. Bottom: Variation of the square of the $xy$ component of the order parameter.

Figure 17

Top: Variation of the AF order parameter vs magnetic field for different lattice sizes. Bottom: Variation of the square of the $z$ component of the order parameter.

Figure 18

Top: Variation of the $z$-component of the magnetization vs magnetic field for different lattice sizes. Bottom: Variation of the square of the $z$ component of the magnetization.

Figure 19

Variation of the “normalized” slope of ${\langle {\psi }^2\rangle }_L$ with respect to $H$ at the transition field $H^t$ vs the inverse volume $L^{-3}$ of the system. The straight line is a linear extrapolation to the thermodynamic limit.

Figure 20

Finite-size scaling plot for ${\langle {\psi }^2\rangle }_L$. The heavy, solid curve shows the asymptotic theoretical behavior in the limit $L=\infty$.

Figure 21

Phase diagram near the bicritical point showing crossover (shaded regions) between the bicritical and spin-flop regions. (Crossover regions between the bicritical point and other phase boundaries are omitted for clarity.)