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Magnetization process of the spin-1/2 XXZ models on square and cubic lattices

The magnetization process of the spin-1/2 antiferromagnetic XXZ model with Ising-like anisotropy in the ground state is investigated. We show numerically that the Ising-like XXZ models on square and cubic lattices show a first-order phase transition at some critical magnetic field. We estimate the value of the critical field and the magnetization jump on the basis of the Maxwell construction. The magnetization jump in the Ising-limit is investigated by means of perturbation theory. Based on our numerical results, we briefly discuss the phase diagram of the extended Bose-Hubbard model in the hard-core limit.


I. INTRODUCTION
Néel [1] predicted the first-order transition of anisotropic antiferromagnets in the presence of a magnetic field in 1936. He pointed out that the spins will abruptly change directions from parallel to perpendicular with respect to the c-axis (easy axis of sublattice magnetization) at some value of the external magnetic field, when the magnetic field is applied to the direction parallel to the c-axis. His prediction was confirmed experimentally [2], and this first-order phase transition is now known as the spin-flopping process. Thirty years later than the discovery of the spin-flopping process, C.N. Yang and C.P. Yang showed by the Bethe ansatz that the one-dimensional (1d) spin-1/2 XXZ model with Ising-like anisotropy exhibits a second-order transition in the presence of a magnetic field [3]. Thus, for one-dimensional Ising-like antiferromagnets, quantum fluctuations, which are neglected in the mean-field approximation, play an essential role. In this way, quantum fluctuations may drastically modify the classical behavior depending on the dimensionality. Hence, we investigate the magnetization process of the spin-1/2 Ising-like XXZ (I-XXZ) models in two and three dimensions in order to see how quantum fluctuations modify the classical behavior of the magnetization process of Ising-like antiferromagnets.
Also, the spin-1/2 XXZ model can be translated into the hard-core boson model with nearest neighbor repulsion [4]. This model corresponds to a special case of the extended Bose-Hubbard model which is considered to be relevant for low-temperature properties of liquid helium on a periodic substrate and also for Josephson junction arrays [5][6][7]. From the theoretical point of view, a lot of attention has been paid to the Bose-Hubbard model as the simplest model to describe the superfluid-insulator transition [5][6][7][8][9]. We can obtain information about the superfluid-insulator transition occurring in the extended Bose-Hubbard model through the investigation of the spin-1/2 I-XXZ model. This paper is organized as follows: In Sec. II, the XXZ model is defined. The details of the numerical calculations are presented. In Sec. III, we review the classical Ising-like XXZ model. Numerical results on the magnetization curve of the spin-1/2 I-XXZ models in two and three dimensions are shown in Sec. IV. In Sec. V, the behavior of the magnetization curve in the Ising-limit is investigated by means of perturbation theory. In Sec. VI, we briefly discuss the superfluid-insulator transition in the extended Bose-Hubbard model in the hard-core limit based on the numerical results of the spin-1/2 I-XXZ model. Section VII is devoted to summary.

II. MODEL AND METHOD
In the present paper, we consider the XXZ model defined by the following Hamiltonian: where S x(y,z) i denote the x(y, z) components of the spin operator at site i. Here i, j denotes nearest neighbors. The anisotropic coupling constant is denoted by λ. For λ = 1, the isotropic Heisenberg model is recovered. We investigate the spin-1/2 XXZ models on square and cubic lattices in the ground state in the canonical ensemble. Namely, we measure the In order to see finite-size effects, we show the size-dependence of the energy gap ∆ g (≡ E(M = 1) − E(M = 0)), the critical field H c and the magnetization jump M s in Fig.1. As shown in this figure, the size-dependence is very small. As a check of numerical accuracy, we compare the energy gap ∆ g obtained by this method with those obtained by other methods as shown in Fig.2. Our result is quite consistent with those of third-order spin-wave theory [11] and the series expansion around the Ising-limit [12]. Hence, we consider that the inverse temperature β and the width of the Trotter slice ∆τ are sufficient. [See also Fig.4.] For three dimensions, we report the 64-site results.

III. REVIEW OF THE CLASSICAL SPIN CASE
Before investigating the spin-1/2 XXZ models, we briefly review the magnetization process of the classical I-XXZ model in the ground state [1,13]. The ground state energy of the classical I-XXZ model may be written in the following form: where S x(y,z) A(B) represent the x(y, z) components of the spin operators at a site in the A(B) sublattices. The length of the spin and the coordination number are denoted by S and z(= 2d), respectively. The angles θ and φ are defined as in Fig.3 The Zeeman term E Z is written in the following form: By minimizing the total energy (E tot ≡ E C−XXZ + E Z ) with respect to θ and φ, one finds the following stable states ( Fig.3(b)): whereJ andH are defined asJ ≡ JN s zS 2 /2 andH ≡ HN s S/2. The magnetization curve of the classical I-XXZ model is shown in Fig.3(c). The transition from the state (i) to the state (ii) is known as the spin-flopping process [1,13]. The critical field H c is defined as the magnetic field above which the ground state has non-zero magnetization. The λ dependence of the critical field H c and that of the magnetization jump M s are obtained as where H max = JSz(λ + 1) and M max = N s S. We show the λ dependence of the critical field H c in Fig.6. The critical field H c is suppressed by quantum fluctuations. In order to see how large the critical field H c is suppressed by quantum fluctuations, we have tried to fit the numerical data as H c /H max = ( λ−1 λ+1 ) α , analogously with the classical result α = 0.5 (eq. (3.3)). We estimate α to be α = 0.64 ± 0.01 for 2d and α = 0.57 ± 0.01 for 3d. Note that the λ dependence of the critical field H c in two and three dimensions is quite different from the one-dimensional case, where the gap Here, we mention the relation between the critical field H c and the energy gap ∆ g . It is expected that the energy gap ∆ g is larger than the critical field H c , if a first-order transition occurs in the presence of a magnetic field. The reason is as follows. The ground state of M = 1 is considered to be the one-magnon state, which may be described by spin-wave theory. Hence, the gap ∆ g corresponds to the excitation energy of one magnon from the ground state of M = 0. On the other hand, phase separation occurs, because magnons gain energy by interacting attractively with each other. The critical field H c would be determined by the effective attractive interactions between the macroscopic number of magnons. As a result, if phase separation occurs, the gap ∆ g is expected to be larger than the critical field H c , i.e.
where M is assumed to be a macroscopic number when the limit M/M max → +0 is taken.
We compare the gap ∆ g and the critical field H c of the spin-1/2 I-XXZ model on a square lattice in Fig.7. The gap ∆ g is always larger than the critical field H c as expected. It is interesting to contrast this behavior with the one-dimensional result. For one dimension, the transition is of second order [3], and the following relation is satisfied: . This is considered to be due to effective repulsive interactions.
The λ dependence of the magnetization jump M s is shown in Fig.8. We estimate the critical value of λ, where M s vanishes, as λ c = 1.00 ± 0.02 by extrapolating the data in Fig.8. This confirms that the spin-1/2 I-XXZ models on square and cubic lattices show a first-order transition at some critical field for any value of the anisotropic coupling constant larger than one (λ > 1). The λ dependence of the magnetization jump M s is remarkably different from the classical result, especially in the large λ regime.

V. ISING-LIMIT
In this section, we discuss the magnetization process of the spin-1/2 XXZ model in the Ising-limit. In Fig.8, the value of the magnetization jump M s in the Ising-limit (λ → ∞) does not coincide with that of the Ising model (λ = ∞)(M s (λ = ∞) = M max ): This can be explained by means of perturbation theory as follows. We rewrite the spin-1/2 XXZ model as whereJ and ǫ are defined asJ ≡ Jλ and ǫ ≡ 1/λ. We consider the XY -term as the perturbation. At M = 0, the unperturbed ground states are the two-degenerate Néel states.
The leading perturbation energy is of order ǫ 2 . On the other hand, in the limit of M → M max , the leading perturbation energy is of order ǫ and proportional to M − M max . Hence it is expected that phase separation occurs for the magnetization smaller than some value M s (< M max ) in the Ising-limit. We numerically estimate the value of M s in the Ising-limit with first-order perturbation theory in the following way. The first-order perturbation energy E 1 is obtained as where |α and |β denote unperturbed ground states of the Ising model in the subspace of fixed magnetization. We generate |α 's randomly and measure E 1 using Monte Carlo technique. The value of M s is determined based on the Maxwell construction. Figure 9 shows the first-order perturbation energy E 1 and the value of M s in the Ising-limit on hypercubic lattices in dimensions up to six. We extrapolate the data in Fig.9(b) and estimate the inverse dimensionality, where M s coincides with M max , as 1/d = 0.01 ± 0.02. This confirms that the value of M s in the Ising-limit (λ → ∞) does not coincide with that of the Ising model (λ = ∞) in finite spatial dimensions.

VI. RELATION TO THE BOSE-HUBBARD MODEL
In this section, we briefly discuss the phase diagram of the hard-core boson model with nearest-neighbor repulsion based on the numerical results of the spin-1/2 XXZ models. The hard-core boson model with nearest-neighbor repulsion can be obtained from the following extended Bose-Hubbard model by taking the on-site repulsion U infinity: in the region of V > t/2. This transition is a first-order transition, which is consistent with recent investigation of the Bose-Hubbard model [9]. Phase separation does not occur for the density ρ smaller than ρ c , for finite This ρ c approaches zero as the spatial dimensionality d goes to infinity ( Fig.9(b)).

VII. SUMMARY
In summary, numerical results on the magnetization process of the spin-1/2 Ising-like XXZ models have been reported. The spin-1/2 XXZ models on square and cubic lattices show a first-order phase transition at some critical magnetic field for the anisotropic coupling constant larger than one (λ > 1). The critical field H c and the magnetization jump M s are estimated on the basis of the Maxwell construction. The critical field H c is suppressed by quantum fluctuations (Fig.6). We have demonstrated that the energy gap ∆ g is larger than the critical field H c (Fig.7). The anisotropy λ dependence of the magnetization jump M s is remarkably different from the classical result (Fig.8). It is strongly suggested that the value of M s in the Ising-limit (λ → ∞) does not coincide with that of the Ising model (λ = ∞) in finite spatial dimensions due to quantum effects (Fig.9).
than those in a homogeneous state because of macroscopic fluctuations. The system sizes examined in this paper may be insufficient to obtain an accurate value for the energy of the phase-separated state. Therefore we use the Maxwell construction to determine it. [