Universal scaled Higgs-mass gap for the bilayer Heisenberg model in the ordered phase
The spectral properties for the bilayer quantum Heisenberg model were investigated with the numerical diagonalization method. In the ordered phase, there appears the massive Higgs excitation embedded in the continuum of the Goldstone excitations. Recently, it was claimed that the properly scaled Higgs mass is a universal constant in proximity to the critical point. Diagonalizing the finite-size cluster with spins, we calculated the dynamical scalar susceptibility , which is rather insensitive to the Goldstone mode. A finite-size-scaling analysis of is made, and the universal (properly scaled) Higgs mass is estimated.
pacs:75.10.Jm Quantized spin models and 05.70.Jk Critical point phenomena and 75.40.Mg Numerical simulation studies and 05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
In the spontaneous-symmetry-breaking phase, there appear the Goldstone and Higgs excitations in the low-energy spectrum. The former (latter) excitation is massless (massive), and the continuum of the former overwhelms the latter dispersion branch. [In regard to the wine-bottle bottom potential, the former (latter) excitation corresponds to the azimuthal (radial) modulation of the order parameter concerned.] Recently, the Higgs-excitation spectral peak was observed for the the two-dimensional ultra-cold atom Endres12 (). Here, a key ingredient is that the external disturbance, namely, the trap-potential modulation, retains the U (gauge) symmetry, and it is rather insensitive to the (low-lying) Goldstone modes. Moreover, the experiment revealed a gradual closure of the Higgs mass in proximity to the critical point between the superfluid and Mott-insulator phases; the criticality belongs to the -dimensional O universality class, and the singularity lies out of the scope of the Ginzburg-Landau theory. Such O- [equivalently, U-] symmetric system is ubiquitous in nature, and the underlying physics is common to a wide variety of substances; we refer readers to Ref. Pekker15 () for a review.
In this paper, we investigate the O-symmetric counterpart, namely, the bilayer quantum Heisenberg model Matsushita97 (); Troyer98 (); Sommer01 (), by means of the numerical diagonalization method. Our aim is to estimate the scaled Higgs mass (universal amplitude ratio) (: the excitation gap in the adjacent paramagnetic phase); technical details are addressed in Sec. 2. The scaled Higgs mass has been estimated as Gazit13a () and Lohofer15 () with the (quantum) Monte Carlo method. On the one hand, via the elaborated renormalization-group analyses, the scaled Higgs mass was estimated as Rose15 () and Katan15 (). An advantage of the numerical diagonalization approach is that the spectral property is accessible directly Gagliano87 () without resorting to the inverse Laplace transformation (see Appendix B of Ref. Gazit13b ()). It has to be mentioned that the scaled Higgs mass has been investigated extensively as for the O-symmetric case, Gazit13a (); Rose15 (); Katan15 (); Gazit13b (); Chen13 (); Rancon14 (); Nishiyama15 (). According to the study Rancon14 () of the O-symmetric system with generic , the Higgs-excitation peak should get broadened for large .
Here, the spin- operator is placed at each square-lattice point () within each layer () The summation runs over all possible nearest neighbor pairs within each layer. The parameter  denotes the intra- (inter-) layer ferromagnetic (antiferromagnetic) Heisenberg interaction; hereafter, we consider as the unit of energy (). The phase diagram Matsushita97 () is presented in Fig. 1. At Troyer98 (), there occurs a phase transition, separating the paramagnetic () and ordered () phases; the phase transition belongs to the three-dimensional O universality class Troyer98 (). The criticality of the spectral function in the ordered phase is our concern. It has to be mentioned that the recent quantum Monte Carlo simulation Lohofer15 () also treats the bilayer Heisenberg model (1), albeit with an antiferromagnetic intra-layer interaction, . The setting of the interaction parameter may be arranged suitably for each methodology. Nevertheless, details of magnetism should not influence the criticality of .
2 Numerical results
In this section, we present the numerical results. We employed the numerical diagonalization method for the finite-size cluster with spins. We implemented the screw-boundary condition (Appendix) Novotny90 () in order to treat a variety of systematically. Because the spins constitute the bilayer cluster, the linear dimension of the cluster is given by
2.1 Finite-size-scaling analysis of the Goldstone mass
As a preliminary survey, we analyze the Goldstone mass with the finite-size-scaling method. The Goldstone mass is identified as the triplet (magnon) excitation gap; hence, the simulation was performed within the subspace specified by the longitudinal total magnetization, either or .
In Fig. 2, we present the scaled Goldstone mass for various and system sizes, (), () () and (). The data suggest that a phase transition takes place at ; note that the intersection point of the curves indicates the location of the critical point. In Ref. Troyer98 (), the critical point was estimated as ; our result agrees with this estimate. The Goldstone mass appears to close, , in the ordered phase as (thermodynamic limit). On the contrary, in the paramagnetic phase , a finite mass gap opens; in the quantum-magnetism language, the mass gap is interpreted as the spin (magnon) gap for the spin-liquid phase. The spin gap [see Eq. (5)] sets a fundamental energy scale for the subsequent scaling analyses.
In Fig. 3, we present the scaling plot for the Goldstone mass, -, for various and system sizes, (), () () and (). Here, the scaling parameters, and , are taken from Refs. Troyer98 () and Hasenbusch01 (); Campostrini02 (), respectively; namely, there are no adjustable parameters involved in the scaling analysis. From Fig. 3, we see that the data collapse into a scaling curve satisfactorily for a considerably wide range of . Such a feature indicates that the simulation data already enter the scaling regime. Encouraged by this finding, we turn to the analysis of the spectral properties.
2.2 Spectral function (dynamical scalar susceptibility)
for various with fixed and . The energy-resolution parameter is set to (solid) and (dotted). Here, the symbol () denotes the ground-state vector (energy), and the operator is given by with the projection operator . The spectral function is sensitive (less sensitive) to the Higgs (Goldstone) mode, because the external perturbation retains the O symmetry. An advantage of the numerical diagonalization approach is that the resolvent is accessible directly via the continued-fraction expansion Gagliano87 (). Actually, the continued-fraction-expansion method is essentially the same as that of the Lanczos diagonalization algorithm (tri-diagonalization sequence), and computationally less demanding. The external perturbation is seemingly different from the conventional ones (implemented in the Monte Carlo simulations, for instance). However, as far as the symmetry is concerned, those choices are all equivalent, yielding an identical critical behavior as to . Here, we employed the Hamiltonian itself as for , which turned out to be less influenced by corrections to scaling.
In Fig. 4 (solid), we observe a Higgs-excitation peak with the mass (excitation gap), . As mentioned above, the signal from the Higgs excitation comes up, because the scalar susceptibility is a good probe specific to it Podolsky11 (); actually, there should exist low-lying () Goldstone and its continuum modes, as illustrated in Sec. 2.1. Above the threshold , a tail background extends. As mentioned afterward, the present simulation was performed so as to examine the main (Higgs) peak, and such high-lying spectral intensities are beyond the scope of the present analysis.
As a reference, we also presented a high-resolution result [Fig. 4 (dotted)], which reveals fine details of the spectral function, namely, the series of the constituent -function subpeaks. The Higgs peak splits into the primary and secondary subpeaks, which locate at and , respectively. As demonstrated in the next section, these fine structures (finite-size artifacts) have to be smeared out by an adequate in order to attain plausible finite-size-scaling behaviors.
Last, we address a number of remarks. First, as mentioned above, the Higgs peak consists of two subpeaks, and hence, it has an appreciable peak width. Such feature agrees with the claim Rancon14 () that the Higgs peak gets broadened for the O-symmetric model with large . Last, rather technically, the continued-fraction expansion Gagliano87 () was iterated until the above-mentioned secondary subpeak converges. The computational effort is comparable to that of the evaluation of .
2.3 Finite-size-scaling analysis of
In this section, we analyze the finite-size-scaling behavior for in the ordered phase, .
The spectral function obeys the finite-size-scaling formula Podolsky12 ()
with the critical point , the correlation-length critical exponent , a certain scaling function and the excitation gap
reflected as to the critical point ; note that the Goldstone mass was considered in Sec. 2.1. In other words, the Goldstone mode (in ) and the fundamental energy scale (in ) continue adiabatically.
In Fig. 5, we present the scaling plot, -, for (dotted), (solid) and (dashed) with fixed and . Here, the scaling parameters, Troyer98 () and Campostrini02 (), are the same as those of Fig. 3; that is, there are no adjustable parameters in the scaling analysis. The scaled-spectral-function curves collapse into a scaling function satisfactorily. From Fig. 5, we notice that the (properly scaled) Higgs mass takes a universal value . The universality (stability) of with respect to the variation of the scaling argument is examined in the next section.
2.4 Universal character of the scaled Higgs peak
In the above section, we investigated the universal behavior of (4) at a particular scaling argument, , and observed a scaled Higgs mass . In this section, we vary in order to survey the universal character of the Higgs peak, particularly, the scaled Higgs mass.
In Fig. 6, we present the scaling plot, -, for various (dotted), (solid) and (dashed) with fixed and ; here, the scaling parameters, and , are the same as those of Fig. 3. The data in Fig. 6 illustrate that the Higgs-peak position, , is kept invariant with respect to the variation of . On the contrary, the Higgs-peak height seems to be scattered; note that according to Eq. (4), the Higgs peaks do not necessarily overlap, because a scaling argument is no longer a constant value. In our preliminary survey, scanning the parameter space , we observed the following tendency. For , the scaled Higgs mass is kept invariant. In closer look, however, for , the Higgs peak drifts to the high-energy side gradually possibly because of the finite-size artifact (limitation of the tractable system size). The microscopic origin of the drift is as follows. The spectral weight transfers from the primary subpeak [see Fig. 4 (dotted)] to the secondary (and even ternary…) one(s) for , and the Higgs peak drifts (and gets broadened); for exceedingly large , eventually, the simulation data may get out of the scaling regime. On the one hand, in the side, the Higgs mass acquires a significant enhancement. This narrow regime is not physically relevant, because the regime shrinks in the raw-parameter scale [like ] as . To summarize, at least for the available system sizes , the scaling regime is optimal in the sense that the scaled Higgs mass takes a stable minimal value
As mentioned above, the Higgs peak consists of two subpeaks. As a byproduct, we are able to estimate the intrinsic peak width. For and , these subpeaks locate at and with almost identical spectral weights; hence, the center locates at . The distance, , between these subpeaks may be a good indicator as to the intrinsic width of the Higgs peak, . It has been claimed Gazit13b () that the Higgs peak for the O-symmetric model should be broadened significantly. Our result supports this claim.
3 Summary and Discussions
The criticality of the Higgs-excitation spectrum [Eq. (3)] for the bilayer Heisenberg model (1) was investigated by means of the numerical diagonalization method; the spectral function is accessible directly via the continued-fraction expansion Gagliano87 (). The spectral function appears to obey the scaling formula (4) satisfactorily, indicating that the simulation data already enter the scaling regime. As a result, we estimated the scaled Higgs mass with the peak width . So far, with the (quantum) Monte Carlo method, the scaled Higgs mass has been estimated as Gazit13a () and Lohofer15 (). According to the normalization-group analysis, the scaled Higgs mass was estimated as Rose15 () and Katan15 (). Our result agrees with these preceding estimates Gazit13a (); Lohofer15 (); Rose15 (); the error margin of our estimate should be bounded by half a peak width, .
The Ginzburg-Landau theory (based on the wine-bottle-bottom potential) yields the critical amplitude ratio . Clearly, the Ginzburg-landau theory fails in describing the spectral property for the O universality class. In other words, such a spectral property reflects a character of each universality class rather sensitively. As a matter of fact, as for the “deconfined critical” phenomenon Huh13 (), an exotic spectral property was predicted. A consideration toward this direction is left for the future study.
This work was supported by a Grant-in-Aid for Scientific Research (C) from Japan Society for the Promotion of Science (Grant No. 25400402).
Appendix A Numerical algorithm: Screw-boundary condition Novotny90 ()
In this Appendix, we explain the simulation algorithm to diagonalize the Hamiltonian matrix for the bilayer Heisenberg model (1). We implemented the screw-boundary condition Novotny90 (), with which one is able to treat a variety of system sizes (: the number of constituent spins) systematically. According to Ref. Novotny90 (), an alignment of spins () with both nearest- and th-neighbor interactions is equivalent to a two-dimensional cluster under the screw-boundary condition; here, the periodical boundary condition as to the spin alignment, namely, , is imposed. Based on this idea, we express the Hamiltonian matrix
with the translation operator (by one lattice spacing) Novotny90 (); namely, a relation holds. We diagonalized the above Hamiltonian matrix (7) with the Lanczos method so as to evaluate the ground-state vector (energy) (). The above expression (7) is mathematically closed. However, as for an efficient simulation, a formula (11) of Ref. Nishiyama08 () may be of use in order to cope with the operation .
- (1) M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P. Schauß, C. Gross, E. Demler, S. Kuhrand and I. Bloch, Nature 487 (2012) 454.
- (2) D. Pekker and C.M. Varma, Annual Rev. Condens. Matter Phys. 6 (2015) 269.
- (3) Y. Matsushita, M. P. Gelfand and C. Ishii, J. Phys. Soc. Japan 66 (1997) 3648.
- (4) M. Troyer and S. Sachdev, Phys. Rev. Lett. 81 (1998) 5418.
- (5) T. Sommer, M. Vojta and K. W. Becker, Eur. Phys. J. B 23 (2001) 329.
- (6) S. Gazit, D. Podolsky and A. Auerbach, Phys. Rev. Lett. 110 (2013) 140401.
- (7) M. Lohöfer, T. Coletta, D. G. Joshi, F. F. Assaad, M. Vojta, S. Wessel and F. Mila, arXiv:1508.07816.
- (8) F. Rose, F. Léonard and N. Dupuis, Phys. Rev. B 91 (2015) 224501.
- (9) Y. T. Katan and D. Podolsky, Phys. Rev. B 91 (2015) 075132.
- (10) E. R. Gagliano and C. A. Balseiro: Phys. Rev. Lett. 59 (1987) 2999.
- (11) S. Gazit, D. Podolsky, A. Auerbach and D. P. Arovas, Phys. Rev. B 88 (2013) 235108.
- (12) K. Chen, L. Liu, Y. Deng, L. Pollet and N. Prokof’ev, Phys. Rev. Lett. 110 (2013) 170403.
- (13) A. Rançon and N. Dupuis, Phys. Rev. B 89 (2014) 180501.
- (14) Y. Nishiyama, Nucl. Phys. B 897 (2015) 555.
- (15) M.A. Novotny, J. Appl. Phys. 67 (1990) 5448.
- (16) M. Hasenbusch, J. Phys. A: Mathematical and General 34 (2001) 8221.
- (17) M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi and E. Vicari, Phys. Rev. B 65 (2002) 144520.
- (18) D. Podolsky, A. Auerbach, and D. P. Arovas, Phys. Rev. B 84 (2011) 174522.
- (19) D. Podolsky and S. Sachdev, Phys. Rev. B 86 (2012) 054508.
- (20) Y. Huh, P. Strack and S. Sachdev, Phys. Rev. Lett. 111 (2013) 166401.
- (21) Y. Nishiyama, Phys. Rev. E 78 (2008) 021135.