Hierarchy of the low-lying excitations for the $(2+1)$-dimensional $q=3$ Potts model in the ordered phase

0.19 MB·

by  Yoshihiro Nishiyama

Hierarchy of the low-lying excitations for the
7 (2 + 1)-dimensional q = 3 Potts model in the ordered
1
phase
0
2

n
Yoshihiro Nishiyama
a
J Department ofPhysics, Faculty of Science,OkayamaUniversity,Okayama700-8530, Japan

6
1

]
h
c
e Abstract
m
The (2+1)-dimensionalq =3 Pottsmodelwassimulatedwith the exactdiago-
-
t
a nalizationmethod. Inthe orderedphase,the elementaryexcitations(magnons)
t
s
. areattractive,forminga seriesofbound statesin the low-energyspectrum. We
t
a
investigate the low-lying spectrum through a dynamical susceptibility, which
m
- is readily tractable with the exact diagonalization method via the continued-
d
n fraction expansion. As a result, we estimate the series of (scaled) mass gaps,
o
c m2,3,4/m1 (m1: single-magnon mass), in proximity to the transition point.
[
Keywords:
1
v 75.10.Jm75.40.Mg 05.50.+q , 05.70.Jk
2
3
2
1. Introduction
4
0
1. In the ordered phase for an Ising ferromagnet, the elementary excitations
0
(magnons)areattractive,formingaseriesofboundstates(composite particles)
7
1 with the mass gaps, m <m <... (m : single-magnonmass). In fact, for the
1 2 1
:
v
(1+1)-dimensionalIsing model[1,2, 3], there appeareighttypes ofexcitations
i
X
with the (scaled) mass gaps
r
a
m /m = 2cosπ/5 (1)
2 1
m /m = 2cosπ/30 (2)
3 1
m /m = 2cos7π/30 (3)
4 2
m /m = 2cos2π/15 (4)
5 2
Preprint submitted toNuclear Physics B January 17, 2017
m /m = 2cosπ/30 (5)
6 2
m /m = 4cosπ/5cos7π/30 (6)
7 2
m /m = 4cosπ/5cos2π/15, (7)
8 2
underaproperlyscaledmagneticfieldsoastopreservetheintegrability;namely,
thescaledgapratio,m /m ,displaysauniversalhierarchicalcharacter. For
2,3,... 1
a quasi-one-dimensional ferromagnet, CoNb O , the primary one, m /m =
2 6 2 1
1.61...(goldenratio),wasconfirmedby means ofthe inelastic neutronscatter-
ing[4]. Forthe(2+1)-dimensionalcounterpart,suchrigorousinformationisnot
available, and various approaches have been made so as to fix the hierarchical
structure m /m [5, 6, 7, 8, 9, 10, 11, 12]. Meanwhile, it turned out that
2,3,... 1
the spectrum for the three-state (q = 3) Potts model exhibits a hierarchical
characteraswell[13,14,15,16,17,18]. Relatedresultsarerecalledafterwards.
AccordingtotheMonte Carlosimulationsin(2+1)dimensions[18,19], the
hierarchym ofthePottsmodelandthatofthepuregaugetheorylookalike.
1,2,...
Actually, for the Z case in (2+1) dimensions, a duality relation [20, 21, 22]
2
does hold, ensuring the correspondence between them. Generically [23], the
SU(N) gauge theory displays a global Z symmetry (center of SU(N)), which
N
immediately establishes a relationship between them; for N 3, the transition
≥
would not be critical, and the universality idea does not apply nonetheless.
Meanwhile, an extensive lattice-gauge-theory simulation reveals the “weak N
dependence”[24]ofthe SU(N)theory,suggestingarobustnessofthe hierarchy,
m /m . On the one hand, the q =3 Potts model exhibits an “approximate
2,3,... 1
universality” [25], even though the phase transition is of first order. Hence, it
is expected that the hierarchyshould display a model-independent characterto
some extent.
In this paper, we investigate the (2+1)-dimensional q =3 Potts model [26,
27]bymeansofthenumericaldiagonalizationmethod. Themethodallowsusto
evaluate the dynamical susceptibilities, Eqs. (12) and (17), via the continued-
fraction-expansion method [28]. In Fig. 1, we present a schematic drawing for
a spectral function.
2
To be specific, we presentthe Hamiltonianforthe (2+1)-dimensionalq =3
Potts model
N N
2 2π 2 2πL
= cos (L L ) λ (R++R−) H cos i. (8)
H − 3 3 i− j − i i − 3 3
Xhiji (cid:18) (cid:19) Xi=1 Xi=1
Here, the operator
0 0 0
Li = 0 1 0 , (9)
 
 0 0 2 
 
 
is placed at each square-lattice point i = 1,2,...,N; namely, the base l
i
| i
(l =0,1,2)satisfies L l =l l . The summation runs over all possible
i i| ii i| ii hiji
nearest neighbor pairs hiji. The operator Ri± inducPes the transition Ri±|lii =
l 1mod3 ,andtheparameterλdenotesthecorrespondingcouplingconstant.
i
| ± i
This model exhibits the first-order phase transition at λ = λ = 0.8758(14)
c
(H = 0) [25], which separates the ordered (λ < λ ) and disordered (λ > λ )
c c
phases. AninfinitesimalmagneticfieldH =20/L2.5(L: lineardimensionofthe
finite-size cluster) stabilizes [3] the groundstate, and the power2.5 comes from
the putative scaling theory [26] for the (2+1)-dimensional q = 3 Potts model.
Notethatthefirst-orderphasetransitionalsoobeystheremediedscalingtheory
[29, 30, 31, 32].
Therestofthispaperisorganizedasfollows. Inthenextsection,wepresent
the numericalresults. The simulation algorithmis presentedas well. In Sec. 3,
we address the summary and discussions.
2. Numerical results
In this section, we presentthe numericalresults for the Potts model (8). To
begin with, we explain the simulation algorithm.
2.1. Numerical algorithm
We employedthe exactdiagonalizationmethod to simulatethe Pottsmodel
(8) for a rectangular cluster with N 22 spins. In order to treat a variety of
≤
3
N = 16,18,... systematically, we implemented the screw-boundary condition
[33]. According to Ref. [33], an alignment of spins l with the first- and
i
{ }
√N-th-neighbor interactions reduces to a rectangular cluster under the screw-
boundary condition. Based on this idea, we express the Hamiltonian as
N N
3 2πL
=H (1)+H √N λ (R++R−) H cos i. (10)
H D D − i i − 2 3
(cid:16) (cid:17) Xi=1 Xi=1
Here, the diagonal matrix H (v) denotes the v-th-neighbor interaction for an
D
alignment l ; that is, the diagonal element is given by l H (v) l =
i i D i
{ } h{ }| |{ }i
l TPv l with the translationoperator, P l = l , and the Potts
i i i i+1
h{ }| |{ }i |{ }i |{ }i
interaction, h{li}|T|{li′}i = − Ni=1δli,l′i. The above formulae are mathemati-
cally closed; however, for an effiPcient simulation, Eqs. (9) and (10) of Ref. [34]
may be of use.
WeperformedthenumericaldiagonalizationfortheHamiltonianmatrix(10)
by means of the Lanczos method. The single-magnon mass gap m is given by
1
m =E E , (11)
1 1 0
−
with the ground-state (E ) and first-excited (E ) energy levels within the zero-
0 1
momentum sector. Because the N spins form a rectangular cluster, the linear
dimension is given by L=√N, which sets a foundermental length scale in the
subsequent scaling analyses.
2.2. Single-magnon mass gap m
1
In this section, we investigate the single-magnon mass gap m (11) with
1
the scaling theory [26, 31]. The first-order phase transition obeys the properly
remedied scaling theory.
InFig. 2,wepresentthescalingplot,(λ λ )L1/ν-m /m ,for(+)N =16,
c 1 1c
−
( ) 18, ( ) 20, and ((cid:3)) 22. Here, the symbol m denotes m =m [31],
× ∗ 1c 1c 1|λ=λc
and the scaling parameters, λ =0.8758 and ν =0.5, are taken from Refs. [25]
c
and [26, 31], respectively. We stress that there is no ad hoc fitting parameter
involved in the scaling analysis.
4
The data in Fig. 2 seem to collapse into a scaling curve, indicating that the
simulation data already enter the scaling regime. Encouraged by this finding,
we turn to the analysis of the spectral properties.
2.3. Hierarchical spectral peaks m via χ′′(ω)
2,3,... A
Based on the finite-size scaling [26, 31] demonstrated in the preceding sec-
tion, we analyze the dynamical susceptibility
χ′′(ω)= 0A†(ω+E +iη)−1A0 , (12)
A −ℑh | 0−H | i
with the ground-state energy (vector) E (0 ) and the energy-resolution pa-
0
| i
rameter η. Here, the perturbation operator is set to
2
N
A= J , (13)
i
P !
i=1
X
with
0 i i
Ji = i 0 0 , (14)
−
 
 i 0 0 
 − 
 
and the projection operator = 1 0 0. We calculated the dynamical sus-
P −| ih |
ceptibility (12) with the continued-fraction expansion [28]. The dynamical sus-
ceptibility (spectral function) obeys the scaling formula
χ′′ L5f(ω/m ,(λ λ )L1/ν), (15)
A ∼ 1 − c
with a certain scaling function f [35, 12].
In Fig. 4, we present the scaling plot, ω/m -L−5χ′′(ω), with fixed (λ
1 A −
λ )L1/ν = 4 and η = 0.1m for various N = 18 (dotted), 20 (solid), and 22
c 1
−
(dashed); here, the scaling parameters, λ and ν, are the same as those of Fig.
c
2. The data appear to collapse into a scaling curve satisfactorily.
Each signal in Fig. 3 is interpreted by the diagram in Fig. 1. That is,
the peaks around ω/m 1.8, 2.5 and 3 correspond to the m excitations,
1 2,3,4
≈
respectively. The shoulder peak around ω/m 2 should be the two-magnon-
1
≈
spectrum threshold. The signal ω/m 3.5 may be either the m particle or a
1 5
≈
5
composite one consisting of m and m . The ratios m /m are estimated in
1 2 2,3,4 1
the next section more in detail.
Last, we mention the choice of the perturbation operator A (13). In a
preliminary stage, we surveyed various types of the perturbation operators,
aiming to create the m particles effectively. Actually, neither the first
2,3,...
or the second term of the Hamiltonian (8) commutes with A; otherwise, the
susceptibilityreducestoamerespecificheat. Akeyingredientisthatthe exact
diagonalizationmethod permits us to treat any off-diagonal operators.
2.4. Universality of the scaled masses m /m
2,3,4 1
In this section, we devote ourselves to the analysis of the scaled masses
m /m ; it is expected that each ratio takes a constant value in proximity to
2,3,4 1
the transition point [25].
In Fig. 4, we present the scaling plot, ω/m -L−5χ′′(ω), with fixed N = 22
1 A
and η = 0.1m for various values of the scaling parameter, (λ λ )L1/ν = 3
1 c
− −
(dotted), 4 (solid), and 5 (dashes); the scaling parameters, λ and ν, are
c
− −
the same as those of Fig. 2. Note that these curves do not necessarily overlap,
becausethescalingparameter(λ λ )L1/ν isnotaconstantvalue;seeEq. (15)
c
−
for the scaling formula. Each peak position seems to be kept invariant, albeit
with the scaling parameter varied. As a result, we estimate the scaled mass
gaps as
(m /m ,m /m ,m /m )=[1.80(3),2.5(1),3.05(25)]. (16)
2 1 3 1 4 1
Here, each error margin was determined from the finite-size drift between N =
16 and 20; a dominant source of the error margin comes from the oscillatory
deviation (an artifact due to the screw-boundary condition), which depends on
the condition whether the system size is a quadratic number N 9,16,... or
∼
not N 12( 3.52),20( 4.52),....
∼ ≈ ≈
Each particle m may possess a finite life time, because it is embedded
3,4
within the two-particle spectrum. As a matter of fact, the data for (λ
−
λ )L1/ν = 3 (dotted) in Fig. 4 exhibit split peaks around ω/m 2.5, in-
c 1
− ≈
dicating that the bound state m has an appreciable peak width, ∆m /m =
3 3 1
6
0.3. Similarly, we observed ∆m /m = 0.35 for the data with N = 18 and
4 1
(λ λ )L1/ν = 6. To summarize, we estimate the intrinsic peak widths (re-
c
− −
ciprocal life time) as ∆m /m =0.3 and ∆m /m =0.35. Each peak width is
3 1 4 1
about one tenth of the corresponding mass gap.
This is a good position to address an overview of the related studies. First,
for the classical three-dimensional q =3 Potts model, an estimate m /m =
2+ 0+
2.43(10)[18]wasreported;thenotation(symmetryindex)istakenfromtheorig-
inal paper. This result may correspond to the present result, m /m = 2.5(1),
3 1
Eq. (16), supporting an “approximate universality” [25] for the q = 3 Potts
model. Second, as for the Z [36] and SU(2) [37] gauge field theories, the re-
2
sults, (m(0+)′/m0+,m(0+)′′/m0+,m(2+)′/m0+) = [1.88(2),2.59(4),3.23(7)] and
(m(0+)′/m0+,m(0+)′′/m0+,m2+/m0+) = [1.89(16),2.35(10),3.36(40)], respec-
tively, were obtained. The hierarchicalstructures are quite reminiscent of ours,
Eq. (16). As a matter of fact, for the gauge field theory, the “week N depen-
dence” of the gauge group SU(N) was reported [24], indicating a robustness of
thehierarchym /m . Forthe SU(3)gaugefield[19],aglueballmass,either
2,3,... 1
m /m =3.214(64)or 3.172(65),was estimated. The result may correspond
2+ 0+
tom /m =3.05(25),Eq. (16). Last,fortheIsingmodel[5,6,7,8,9,10,11,12],
4 1
estimates, m /m = 1.82(2) [11] and m /m = 2.45(10) [5], were reported.
2 1 3 1
These results resemble to ours, Eq. (16), supporting a model-independence on
the hierarchy.
Last,weaddressaremark. Becausethephasetransitionisdiscontinuous,the
continuumlimit cannotbe takenproperly. The aboveestimates suchas the life
time are specific to a lattice realization, although seemingly preferable scaling
behaviorwasobserved. However,inanapproximatesense,thesimulationresults
seem to be comparable with the related ones, as claimed by the preceeding
studies [18, 19, 25].
2.5. Continuum-threshold peak via χ′′(ω)
B
As a comparison, we present a simulation result for χ′′ (17), aiming to see
B
to what extent the spectral weight is affected by the choice of the perturbation
7
operator. The dynamical susceptibility χ′′ is defined by
B
χ′′(ω)= 0B†(ω+E +iη)−1B 0 , (17)
B −ℑh | 0−H | i
with a perturbation operator
N
B = (R++R−). (18)
P i i
i=1
X
NotethattheoperatorB coincideswiththesecondtermoftheHamiltonian(8).
Hence, it exhibits the specific-heat-type singularity χ′′ L2/ν−1 with ν = 0.5
B ∼
[26, 31] right at the transition point.
In Fig. 5, we present the scaling plot, ω/m -L−3χ′′(ω), with fixed (λ
1 B −
λ )L1/ν = 4 and η = 0.1m for various N = 18 (dots), 20 (solid), and 22
c 1
−
(dashed); the scaling parameters, λ and ν, are the same as those of Fig. 2. In
c
contrast with χ′′ in Fig. 3, the susceptibility χ′′ detects the m signals and
A B 1,2
the two-magnon-spectrum-threshold peak ω/m 2; instead, the bound-state
1
≈
hierarchy m becomes hardly observable.
3,4
The result indicates that a naive external disturbance such as the specific-
heat-type perturbation B does not create the bound states higher than m
1,2
very efficiently. It is significant to set up the perturbation operator, which
does not commute with any terms of the Hamiltonian. Note that the exact
diagonalization method allows us to survey various types of the (off-diagonal)
perturbation operators so as to observe m clearly.
3,4,...
3. Summary and discussions
The hierarchy m /m for the (2+1)-dimensional q =3 Potts model (8)
2,3,4 1
was investigated with the numerical diagonalization method. The method al-
lows us to calculate the dynamical susceptibilities χ′′ , Eqs. (12) and (17),
A,B
via the continued-fraction expansion [28]. Through the probe χ′′, we obtained
A
(m /m ,m /m ,m /m ) = [1.80(3),2.5(1),3.05(25)]. The particles m ac-
2 1 3 1 4 1 3,4
quire intrinsic peak widths, ∆m /m = 0.3 and ∆m /m = 0.35, respectively;
3 1 4 1
these spectra are embedded within the two-magnon spectrum. According to
8
Refs. [18, 19], the hierarchy m /m of the Potts model and that of the
2,3,4 1
pure gaugetheory are alike. For instance, as for the Z -symmetric gauge group
2
[36], there was reported a hierarchy,(m(0+)′/m0+,m(0+)′′/m0+,m(2+)′/m0+)=
[1.88(2),2.59(4),3.23(7)],quite reminiscent of ours, Eq. (16).
As a reference, we calculated χ′′ (17); here, the operator B coincides with
B
the second term of the Hamiltonian (8), and hence, it would be relevant to
the experimental study. It turned out that the probe χ′′ is insensitive to the
B
hierarchy m , indicating that the choice of the perturbation operator is
3,4,...
vital to observe m . In this sense, the exact diagonalizationmethod has an
3,4,...
advantage in that we are able to treat various perturbation operators so as to
observe the hierarchy m clearly.
3,4,...
Acknowledgment
ThisworkwassupportedbyaGrant-in-AidforScientific Research(C)from
Japan Society for the Promotion of Science (Grant No. 25400402).
References
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χ
’’(ω)
(k=0)
lowest
two-magnon
bound state
elementary second-lowest
magnon two-magnon third-lowest
bound state two-magnon
bound state
two-magnon
spectrum
ω
0 m1 m2 2m1 m3 m4
Figure 1: A schematic drawing of a dynamical susceptibility for the Potts model (8) in
the ordered phase within the zero-momentum (k = 0) sector is presented. There appear
hierarchicalpeakswiththemassgapsm1,2,...: Theelementaryexcitationm1 correspondsto
thesinglemagnon, whichformsaseriesofbound states m2,3,.... Thetwo-magnon spectrum
extendsaboveω>2m1. Theexcitationsm1,2,...mayhavearelevancetotheglueballspectrum
forthe(pure)gaugefieldtheory[18,19].
10