Magnon-bound-state hierarchy for the two-dimensional transverse-field Ising model in the ordered phase
In the ordered phase for an Ising ferromagnet, the magnons are attractive to form a series of bound states with the mass gaps, . Each ratio (: the single-magnon mass) is expected to be a universal constant in the vicinity of the critical point. In this paper, we devote ourselves to the -dimensional counterpart, for which the universal hierarchical character remains unclear. We employed the exact diagonalization method, which enables us to calculate the dynamical susceptibility via the continued-fraction expansion. Thereby, we observe a variety of signals including , and the spectrum is analyzed with the finite-size-scaling method to estimate the universal mass-gap ratios.
keywords:05.50.+q 05.10.-a 05.70.Jk 64.60.-i
For an Ising ferromagnet in the ordered phase, the magnons are attractive, forming a series of bound states with the mass gaps, . As a matter of fact, in dimensions, where the system is integrable Zamolodchikov98 (), there exist eight types of elementary excitations with the universal mass gaps
(: single-magnon mass gap) in the vicinity of the critical point Delfino04 (); Fonseca03 (). According to the rigorous theory, the elementary magnon is also a composite particle, reflecting a highly non-perturbative character of this problem; in this sense, the underlying physics may lie out of the conventional “magnon” picture. Experimentally, the lowest one (golden ratio) was observed Coldea10 () for a quasi-one-dimensional quantum Ising ferromagnet, CoNbO Lee10 (); Ishimura80 (). Above , there extends a two-magnon continuum, overwhelming fine details of the spectrum; see Fig. 4 E of Ref. Coldea10 (), for instance.
In dimensions, on the contrary, such rigorous information is not available, and details of the bound-state hierarchy are not fully clarified; the role of dimensionality was argued in §5 of Ref. Caselle02 () (see Ref. Rose16 () as well). To the best of author’s knowledge, the second-lowest bound state was detected with the Monte Carlo method Caselle99 (), whereas the lowest one, Dusuel10 (), has been investigated rather extensively so far Caselle99 (); Lee01 (); Caselle02 (); Nishiyama08 (); Nishiyama14 (); Rose16 (). In this paper, we investigate the -dimensional Ising model (1) with the exact diagonalization method, which enables us to calculate the dynamical susceptibility via the continued-fraction expansion Gagliano87 (); note that in the Monte Carlo simulation, one has to resort to the inverse Laplace transform to obtain the spectrum. The spectrum reflects a hierarchical character for . In Fig. 1, we present a schematic drawing for a spectral function within the zero-momentum sector.
It has to be mentioned that the magnon-bound-state hierarchy is relevant to the glueball spectrum (screening masses) for the gauge field theory (Svetitsky-Yaffe conjecture) Svetitsky82 (); Agostini97 (); Fiore03 (). Actually, we show that in the next section, the bound-state hierarchy bears a resemblance to the glueball spectrum for the Z-symmetric gauge field theory Agostini97 (). Here, we dwell on the characterization of the magnon bound states, and the verification of the conjecture itself lies beyond the scope of this paper.
To be specific, we present the Hamiltonian for the two-dimensional spin- transverse-field Ising model
with the quantum spin- operator placed at each square-lattice point . The summations, and , run over all possible nearest-neighbor and next-nearest-neighbor pairs, and , respectively. Correspondingly, () and () are the quadratic (biquadratic) interaction parameters. The symbols, , and , denote the single-ion anisotropy, the transverse- and longitudinal-magnetic fields, respectively. The parameter is tunable, and the phase diagram is presented in Fig. 2. Other interaction parameters are set to
so as to suppress corrections to scaling Nishiyama10 (). The (properly scaled) infinitesimal magnetic field Nishiyama10 () with Hasenbusch10 () resolves the ground-state two-fold degeneracy Fonseca03 (). The -spin model permits us to incorporate extended interactions so as to suppress corrections to scaling Deng03 (). In fact, as demonstrated in Ref. Nishiyama10 (), even for restricted system sizes, the Hamiltonian (1) exhibits suppressed corrections to scaling. Detailed account of the three-dimensional Ising universality is reported in Ref. Hasenbusch10 ().
2 Numerical results
In this section, we present the numerical results for the -dimensional Ising model (1). We employed the exact diagonalization method for the finite-size cluster with spins. We imposed the screw-boundary condition Novotny90 () to treat an arbitrary number of spins . We adopt the simulation algorithm presented in Appendix of Ref. Nishiyama10 (). Because the spins constitute a rectangular cluster, the linear dimension of the cluster is given by the formula
The diagonalization was performed within the zero-momentum subspace.
2.1 Critical behavior of the single-magnon mass gap
In this section, we make a finite-size-scaling analysis for the single-magnon mass gap , which sets a fundamental energy scale in the subsequent scaling analyses. The single-magnon mass corresponds to the first excitation gap above the ground state. (The ground-state two-fold degeneracy is resolved by , as mentioned in Sec. 1.)
In Fig. 3, we present the scaling plot, -, for the number of spins, , , , and . Here, the scaling parameters, and , are taken from the existing literatures, Refs. Nishiyama10 () and Hasenbusch10 (), respectively. That is, there is no adjustable fitting parameter in the scaling analysis. We observe that the data in Fig. 3 collapse into a scaling curve for a considerably wide range of the parameter . The simulation data already enter the scaling regime. Encouraged by this finding, we turn to the analysis of the spectral properties.
Last, we address a few remarks. First, the data in Fig. 3 indicate that the gap closes beside the critical point, and the gap itself seems to be rather large. These confusing features are due to the infinitesimal (scaled) longitudinal magnetic field as mentioned in Introduction. This infinitesimal magnetic field resolves the ground-state degeneracy, stabilizing the magnetic excitations Nishiyama10 (). Last, right at the critical point, the excitation hierarchy becomes smeared out. The smearing-out regime extends within a bound , which shrinks in the thermodynamic limit.
2.2 Finite-size-scaling analysis of the dynamical susceptibility : Magnon-bound-state hierarchy
In this section, we investigate the magnon-bound-state hierarchy. For that purpose, we introduce the dynamical susceptibility
with the perturbation operator
and the projection operator . Here, the symbols, and (), denote the energy-resolution parameter and the ground-state energy (vector), respectively. The dynamical susceptibility was calculated with the continued-fraction expansion Gagliano87 (). Rather technically, the continued-fraction expansion requires the iteration sequences less than those of the Lanczos diagonalization for and ; namely, at least for drawing the spectra such as Fig. 4-7, the results converge rapidly possibly owing to the plausible choice of . The dynamical susceptibility obeys Podolsky11 () the scaling formula
with a scaling function . Afterward, the power-law singularity as well as the physical content of are considered.
In Fig. 4, we present the scaling plot, -, with fixed and for various (dotted), (solid), and (dashed). The scaling parameters, and , are the same as those of Fig. 3. The curves in Fig. 4 collapse into a scaling function , Eq. (6), indicating that the simulation data already enter the scaling regime.
The peaks in Fig. 4 reflect a universal character of the magnon excitations in proximity to the critical point; see a schematic drawing, Fig. 1, as well. The peaks at , , , and correspond to the , Caselle99 (); Dusuel10 (); Caselle02 (); Nishiyama08 (); Nishiyama14 (); Rose16 (), (two-magnon-continuum threshold), Caselle99 (), and excitations, respectively. The -peak may be attributed to either a yet higher bound state or a composite particle made of and .
It has to be mentioned that these excitations correspond to the glueball spectrum (screening masses) for the gauge field theory (Svetitsky-Yaffe conjecture Svetitsky82 ()). For instance, according to the Monte Carlo simulation of the lattice gauge-field theory Agostini97 (), there appear a variety of particles (the so-called glueballs), , , and , quite reminiscent of the above-mentioned , respectively; here we follow the notation of Ref. Agostini97 (). Our simulation result supports the Svetitsky-Yaffe conjecture at least up to .
A few remarks are in order. First, we consider the power-law singularity of , Eq.(6). By analogy to the random-walk translation distance, the singularity of the perturbation operator , Eq. (5), is counted as . On the one hand, the reciprocal energy gap should exhibit the singularity, . These formulas lead to . As a matter of fact, there also appears a non-trivial term ; this term is less singular than the above , and it was dropped in Eq. (6). Here, a key ingredient is that an infinitesimal -direction magnetic field is perpendicular to , and hence, the scaling dimension of is a half of . Second, we make an overview of the dynamical susceptibilities appearing in literature. So far, there have been utilized two types of perturbations, namely, off-diagonal and diagonal ones, in the course of study for both - and -dimensional systems. The former was employed in Refs. Seabra13 (); Kjall11 (); Robinson14 (); Wang15 (). Our choice (5) is based on these elaborated studies. We found that the duplicated operations create the particles more efficiently than a mere , at least, for the -dimensional counterpart. The latter was implemented in Refs. Robinson14 (); Wang15 (); Delfino95 (). This choice leads to the ordinary uniform magnetic AC susceptibility, and it is of experimental significance. Last, we mention the parameter setting of the peak-broadening factor . In this paper, aiming to avoid ad hoc adjustment of , we fixed the value of throughout the study. The continuum spectrum in Fig. 7 seems to be well reproduced by this setting.
2.3 Universality of the bound-state hierarchy
In the above section, through the probe , we resolved the signals out of the two-magnon continuum. In this section, we examine the universality of each mass-gap ratio with respect to the variation of .
In Fig. 5, we present the scaling plot, -, with and for various values of the scaling argument, (dashed), (solid), and (dotted); here, the scaling parameters, and , are the same as those of Fig. 3. Note that the curves do not necessarily overlap, because the scaling argument is ranging; see Eq. (6). In fact, the -peak heights do not coincide each other, whereas the peak position, Dusuel10 (), appears to be kept invariant with varied. Our result supports the universality of Dusuel10 (), which has been established in the course of study Caselle99 (); Caselle02 (); Nishiyama08 (); Nishiyama14 (); Rose16 ().
We turn to the analysis of the second-lowest bound state, . In Fig. 5, we observe that the peak position is kept invariant with varied. In closer look (), the peak positions read , , and for , , and , respectively. Namely, the condition is an optimal one in the sense that the peak position takes a stable (extremal) value
Our result agrees with the Monte Carlo result Caselle99 ().
Additionally, we are able to appreciate the universal intrinsic peak width for . The particle may decay into a pair of , and eventually, the peak acquires a finite life time, namely, a reciprocal intrinsic peak width; because the -dimensional system is not integrable, it is natural that the spectral intensities are diffused Robinson14 (). In Fig. 6, we present the scaling plot, -, with and for (dotted), (solid), and (dashed); the scaling parameters, and , are the same as those of Fig. 3. In this scaling regime, , the peak splits into two sub-peaks at and (). The distance between these sub-peaks provides an indicator for the intrinsic peak width
The peak width is smaller by one order of magnitude than the mass (7), indicating that the particle is a stable collective mode, and it would be observable experimentally. We consider that the error margin for , Eq. (7), would not exceed a half of this peak width; namely, the uncertainty for , Eq. (7), is bounded by .
Last, we consider the signal. From Fig. 5, we estimate the mass-gap ratio as . The peak position appears to drift, indicating that this peak consists of diffused intensities. In fact, there are a number of decay modes such as , and , giving rise to a considerably broadened line shape as to . Further details are not fixed by the available result.
We address a remark on the -dependence of the spectrum. As mentioned above, the peak position takes a stable (extremal) value at an optimal condition . In a closer look, however, there are irregularities caused by the level crossing. A level-crossing point locates at the above-mentioned point , and another one occurs around , where the data of and are not very influenced. In this sense, these intermittent irregularities are due to the finite-size artifact, reflecting the fact that the bound state is embedded within the continuum. As mentioned above, the level crossing provides information how the spectral intensity is distributed over the discrete levels, and hence, it indicates the intrinsic peak width of the spectral peak concerned. Nevertheless, apart from these intermittent irregularities, the peak position is kept invariant for a considerably wide range of , and the slight wavy deviation appears to be bounded by the above-mentioned error margin.
2.4 Finite-size-scaling analysis of the dynamical susceptibility : Two-magnon continuum
In this section, we investigate the dynamical susceptibility , Eq. (9); here, the perturbation operator (10) is identical to the term of the Hamiltonian (1), and such a dynamical susceptibility is called the scalar susceptibility Podolsky11 () in literature.
In Fig. 7, we present the scaling plot, -, with and for (dotted), (solid), and (dashed); the scaling parameters, and , are the same as those of Fig. 3. The dynamical susceptibility is defined by the formula
with the perturbation operator
The dynamical susceptibility obeys the scaling law Podolsky11 () with a scaling function .
The data in Fig. 7, indicate that this probe is sensitive to the two-magnon continuum, particularly, its threshold . Actually, in contrast to Fig. 4, the bound-state hierarchy becomes invisible instead. Our simulation result suggests that the choice of the perturbation operator is vital to the detection of the bound-state hierarchy.
We address a remark. The scatter of the scaling curves in Fig. 7 should be attributed to the regular part (non-singular contribution) of . Because the leading singularity of is rather weak, it is contaminated by such residual scaling corrections.
3 Summary and discussions
The universal character of the magnon-bound-state hierarchy for the two-dimensional quantum Ising model (1) in the ordered phase was investigated. So far, the lowest bound state has been investigated rather extensively Caselle99 (); Dusuel10 (); Lee01 (); Caselle02 (); Nishiyama08 (); Nishiyama14 (); Rose16 (). We employed the exact diagonalization method, which yields the dynamical susceptibilities via the continued-fraction expansion Gagliano87 (). The dynamical susceptibility resolves the bound-state hierarchy out of the two-magnon continuum. Making the finite-size-scaling analysis of , we estimated the mass-gap ratios. The lowest one agrees with the preceding result Dusuel10 (), which has been established in the course of study Caselle99 (); Caselle02 (); Nishiyama08 (); Nishiyama14 (); Rose16 (). For the second-lowest bound state, we estimated the mass-gap ratio, . Our result agrees with the Monte Carlo result Caselle99 (). Additionally, we appreciated its intrinsic peak width, ; the bound state exhibits a rather long life time as a stable collective mode, though it is embedded within the continuum. A signature for the third-lowest bound state is also detected around . In contrast to , the probe is sensitive to the two-magnon continuum, which smears out the signals eventually. The choice of the perturbation field may be significant for resolving the magnon bound states out of the continuum.
As demonstrated in Sec. 2.3, the bound state hierarchy bears a resemblance to the glueball spectrum Agostini97 (), supporting the Svetitsky-Yaffe conjecture Svetitsky82 (). It is tempting to apply the similar approach to the -state Potts model in order to examine the validity of the conjecture. This problem 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).
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