Critical behavior of the Higgs- and Goldstone-mass gaps for the two-dimensional S=1 XY model

Spectral properties for the two-dimensional quantum S=1 XY model were investigated with the exact diagonalization method. In the symmetry-broken phase, there appear the massive Higgs and massless Goldstone excitations, which correspond to the longitudinal and transverse modes of the spontaneous magnetic moment, respectively. The former excitation branch is embedded in the continuum of the latter, and little attention has been paid to the details, particularly, in proximity to the critical point. The finite-size-scaling behavior is improved by extending the interaction parameters. An analysis of the critical amplitude ratio for these mass gaps is made.


Introduction
In the symmetry-broken phase, the O(2)-symmetric system, such as the XY model, exhibits a massless Goldstone excitation, which corresponds to the transverse modulation of the magnetic moment. On the one hand, the longitudinal mode, namely, the Higgs excitation, is massive, embedded in the continuum of the former; see Ref. [1] for a review. The O(2)-[equivalently, U(1)-] symmetric system is ubiquitous in nature, and such a characteristic spectrum has been observed for a variety of substances [2,3,4,5,6,7,8,9,10,11]. The perturbation field (experimental probe) should retain the O(2) (axial) symmetry [1,12,13,14,15]; otherwise, the contribution from the Goldstone excitations smears out the Higgs-mode branch [16,17,18,19].
(For instance, the chemical-potential modulation for the bosonic system does not conflict with the symmetry.) Recent studies [20,21] shed light on a universal character of the spectrum in proximity to the phase transition, especially, in (2 + 1) dimensions; it would be intriguing that the spectral property is also under the reign of universality. In (3 + 1) dimensions, the criticality is described simply by the Ginzburg-Landau theory (Gaussian fixed point). On the contrary, in (2 + 1) dimensions, the spectral property is non-perturbative by nature. In particular, a universal amplitude ratio for the mass gaps [see Eq. (1) mentioned afterward] is arousing much attention recently.
In this paper, we investigate the two-dimensional quantum S = 1-spin XY model (2) with the exact diagonalization method. The method enables us to calculate the low-lying level indexed by quantum numbers. In order to suppress corrections to scaling, we incorporate various types of interaction parameters in addition to the ordinary nearest-neighbor ferromagnetic interaction J N N . Thereby, we investigate the universality for the critical amplitude ratio with the Higgs mass m H , the Goldstone mass m G , and the reflected gap  [27].
To be specific, we present the Hamiltonian for the S = 1 XY model [28] Here, the quantum S = 1-spin operator S i is placed at each square-lattice point i. The summations, ij , ij , and [ijkl] , run over all possible nearest-neighbor, next-nearest-neighbor, and plaquette spins, respectively.
was adjusted [28] to an IR fixed point with almost eliminated irrelevant interactions; that is, the coupling constants (J * N N , J * N N N , D * ) were determined through an approximative real-space renormalization group, and the remaining one D * was finely tuned via the conventional finite-size-scaling analysis.
As shown in Eq. (2), the S = 1-spin model allows us to incorporate various interactions such as the single-ion anisotropy, with which one is able to realize the XY -paramagnetic phase transition. In this sense, the extention of the magnetic moment to S = 1 is essential in our study.
The rest of this paper is organized as follows. In Sec. 2, we present the simulation results. Technical details are explained as well. In Sec. 3, we address the summary and discussions.

Numerical results
In this section, we present the simulation results. To begin with, we explain the simulation technique.

Simulation algorithm
In this section, we explain the simulation algorithm. As mentioned in Introduction, the XY model (2) was simulated with the exact diagonalization method. We implemented the screw-boundary condition [29] in order to treat a variety of system sizes N = 10, 12, . . . , 22 (N: number of constituent spins) systematically; note that conventionally, the system size N is restricted within N = 9, 16, . . .. We adopt the algorithm presented in Sec. II of Ref. [28]. The linear dimension L is given by L = √ N; note that the N spins constitute a rectangular cluster.
Thereby, we evaluated the mass gaps m H and m G via the following scheme. The exact diagonalization method yields the low-lying energy levels is specified by the momentum k and the perpendicular magnetic moment S z tot ; in practice, the numerical diagonalization was performed within the subspace (k, S z tot ). The Higgs-and Goldstone-mass gaps are characterized by and respectively. The reflected gap ∆ with respect to the critical point J , and hence, the ratio m H /∆ makes sense in the XY phase. The gap ∆ > 0 is interpreted as the insulator gap through regarding the ladder operators S ± i as the bosonic creation-annihilation operators.

Scaling analyses of m H,G
In this section, we investigate the scaling behaviors for the mass gaps m H We address a few remarks. First, the data in Figs. 4 and 5 collapse into the scaling curves satisfactorily. Such a feature indicates that corrections to scaling are almost negligible owing to the fine adjustment [28] of the coupling constants to Eq. (4). Because the tractable system size with the exact diagonalization method is severely restricted, it is significant to accelerate the convergence to the scaling limit. Second, the scaling parameters, J * N N and ν, are taken from the literatures, Refs. [28] and [30], respectively. That is, there are no adjustable ad hoc parameters in the present scaling analyses.
Last, as demonstrated in Figs. 2 and 3, both mass gaps m H,G possess an identical scaling dimension. Hence, the amplitude ratio (1) makes sense, and the criticality is explored in the next section.

Analysis of the amplitude ratio m H /∆
In this section, encouraged by the findings in Sec. 2.2, we turn to the analysis of the amplitude ratio m H /∆, Eq. (1).
In Fig. 6 as an indicator for m H /∆. The amplitude ratio (7) A comment may be in order, the series of data in Fig. 7 appear to be oscillatory; actually, we observe a slight bump around N(= L 2 ) ≈ 16. Such an oscillatory behavior is an artifact of the screw-boundary condition [29], rendering an ambiguity as to the extrapolation to L → ∞. The ambiguity appears to be bounded by the above-mentioned error margin, which is estimated by performing two independent extrapolation schemes.

Summary and discussions
The critical behavior of m H,G was investigated for the two-dimensional quantum S = 1 XY model (2) by means of the numerical diagonalization method [29,28]. The interaction parameters were adjusted to Eq. (3) in order to suppress corrections to scaling [28]. As a consequence, the data (Figs. 4 and 5) collapse into the scaling curves satisfactorily, indicating that the data already enter the scaling regime. Thereby, we confirm a universal character for the mass-gap ratio (Fig. 6), and estimate the amplitude ratio as m H /∆ = 2.1(2).
As mentioned in Introduction, the amplitude ratio has been estimated with the (quantum) Monte Carlo method, m H /∆ = 2.1(3) [22, 23] and 3.3(8) [24], as well as the renormalization-group approaches, 2.4 [25], 2.2 [26], and 1.67 [27]. According to the Ginzburg-Landau (mean-field) theory, the amplitude ratio should be m H /∆ = √ 2. Clearly, the spectral property reveals a notable deviation from that anticipated from the mean-field theory; the Ising counterpart was studied in Ref. [32]. In this respect, detailed analyses of other spectral properties such as the AC conductivity [23,33] would be desirable. A progress toward this direction is left for the future study.      The least-squares fit to the data yields m H /∆ = 2.119(13) in the thermodynamic limit L → ∞. A possible extrapolation error is considered in the text. An oscillatory deviation (slight bump around L 2 ≈ 16) is an artifact of the screw-boundary condition [29].