Universal scaled Higgs-mass gap for the bilayer Heisenberg model in the ordered phase

0.16 MB·

by  Yoshihiro Nishiyama

EPJ manuscript No.
(will be inserted by the editor)
Universal scaled Higgs-mass gap for the bilayer Heisenberg
model in the ordered phase
6
1
0
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Yoshihiro Nishiyama

n
a
J Department of Physics, Faculty of Science, OkayamaUniversity,Okayama700-8530, Japan

9
1
Received: date/ Revised version: date

]
h
c
e Abstract. The spectral properties for the bilayer quantum Heisenberg model were investigated with the
m
numerical diagonalization method. In the ordered phase, there appears the massive Higgs excitation em-
-
t
a
bedded in the continuum of the Goldstone excitations. Recently, it was claimed that the properly scaled
t
s
.
t Higgs mass is a universal constant in proximity to the critical point. Diagonalizing the finite-size cluster
a
m
with N ≤36 spins, we calculated the dynamical scalar susceptibility χ′′(ω),which is rather insensitive to
s
-
d the Goldstone mode. A finite-size-scaling analysis of χ′′(ω) is made, and the universal (properly scaled)
n s
o
Higgs mass is estimated.
c
[

1
v
PACS. 75.10.Jm Quantized spin models – 05.70.Jk Critical point phenomena – 75.40.Mg Numerical
2
7
simulation studies – 05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
7
4
0
. 1 Introduction eter concerned.] Recently, the Higgs-excitation spectral
1
0
peak was observed for the the two-dimensional ultra-cold
6
1
In the spontaneous-symmetry-breaking phase, there ap-
: atom [1]. Here, a key ingredient is that the external dis-
v
i peartheGoldstoneandHiggsexcitationsinthelow-energy
X turbance, namely, the trap-potential modulation, retains
r
a spectrum.Theformer(latter)excitationismassless(mas- the U(1) (gauge) symmetry, and it is rather insensitive
sive), and the continuum of the former overwhelms the
to the (low-lying)Goldstone modes.Moreover,the exper-
latterdispersionbranch.[Inregardtothewine-bottlebot-
iment revealed a gradual closure of the Higgs mass m
H
tom potential, the former (latter) excitation corresponds
in proximity to the critical point between the superfluid
to the azimuthal (radial) modulation of the order param-
and Mott-insulator phases; the criticality belongs to the
Send offprint requests to: (2+1)-dimensionalO(2)universalityclass,andthesingu-
2 Yoshihiro Nishiyama: Universal scaled Higgs-mass gap for thebilayer Heisenberg model in theordered phase
larityliesoutofthescopeoftheGinzburg-Landautheory. Here, the spin-S = 1/2 operator S is placed at each
ai
SuchO(2)-[equivalently,U(1)-]symmetricsystemisubiq- square-latticepointi (i=1,2,...,N/2)withineachlayer
uitous in nature, and the underlying physics is common a (a = 1,2) The summation runs over all possi-
hiji
P
to a wide variety of substances; we refer readers to Ref. ble nearest neighbor pairs ij within each layer. The pa-
h i
[2] for a review. rameter J(> 0) [J′(> 0)] denotes the intra- (inter-) layer
ferromagnetic(antiferromagnetic)Heisenberginteraction;
Inthispaper,weinvestigatetheO(3)-symmetriccoun-
hereafter, we consider J′ as the unit of energy (J′ = 1).
terpart,namely,thebilayerquantumHeisenbergmodel[3,
Thephasediagram[3]ispresentedinFig.1.AtJ =0.435
c
4,5], by means of the numerical diagonalization method.
[4],thereoccursaphasetransition,separatingtheparam-
Our aim is to estimate the scaled Higgs mass (univer-
agnetic (J < J ) and ordered (J > J ) phases; the phase
c c
sal amplitude ratio) m /∆ (∆: the excitation gap in the
H
transition belongs to the three-dimensional O(3) univer-
adjacent paramagnetic phase); technical details are ad-
sality class [4]. The criticality of the spectral function in
dressed in Sec. 2. The scaled Higgs mass has been es-
the ordered phase is our concern. It has to be mentioned
timated as m /∆ = 2.2(3) [6] and 2.6(4) [7] with the
H
that the recent quantum Monte Carlo simulation [7] also
(quantum) Monte Carlo method. On the one hand, via
treats the bilayer Heisenberg model (1), albeit with an
the elaboratedrenormalization-groupanalyses,the scaled
antiferromagnetic intra-layer interaction, J < 0. The set-
Higgsmasswasestimatedasm /∆=2.7[8]and1.64[9].
H
tingoftheinteractionparametermaybearrangedsuitably
Anadvantageofthenumericaldiagonalizationapproachis
for each methodology. Nevertheless, details of magnetism
that the spectral property is accessible directly [10] with-
should not influence the criticality of m .
H
out resorting to the inverse Laplace transformation (see
Therestofthispaperisorganizedasfollows.InSec.2,
Appendix B of Ref. [11]). It has to be mentioned that
weanalyzethespectralpropertiesfortheHamiltonian(1)
the scaled Higgs mass m /∆ has been investigated ex-
H
by means of the numerical diagonalization method. The
tensivelyas forthe O(2)-symmetriccase,[6,8,9,11,12,13,
simulation algorithm is presented in Appendix. In Sec. 3,
14]. According to the study [13] of the O(N)-symmetric
we address the summary and discussions.
system with generic N, the Higgs-excitation peak should
get broadened for large N.
2 Numerical results
To be specific, we present the Hamiltonian for the bi-
In this section, we present the numerical results. We em-
layer Heisenberg model [3,4,5]
ployedthenumericaldiagonalizationmethodforthefinite-
2 N/2 sizeclusterwithN 36spins.Weimplementedthescrew-
≤
= J Sai Saj +J′ S1i S2i. (1)
H − · ·
Xa=1Xhiji Xi=1 boundary condition (Appendix) [15] in order to treat a
Yoshihiro Nishiyama: Universal scaled Higgs-mass gap for thebilayer Heisenberg model in theordered phase 3
variety of N = 30,32,... systematically. Because the N N = 30 (+), 32 ( ) 34 ( ) and 36 (✷). Here, the scaling
× ∗
spins constitute the bilayer cluster, the linear dimension parameters, J = 0.435 and ν = 0.7112, are taken from
c
of the cluster is given by Refs.[4]and[16,17],respectively;namely,therearenoad-
justableparametersinvolvedinthescalinganalysis.From
L= N/2. (2)
p
Fig. 3, we see that the data collapse into a scaling curve
2.1 Finite-size-scaling analysis of the Goldstone mass satisfactorily for a considerably wide range of J. Such a
m feature indicates that the simulation data already enter
G
thescalingregime.Encouragedbythisfinding,weturnto
As a preliminary survey, we analyze the Goldstone mass
the analysis of the spectral properties.
m with the finite-size-scaling method. The Goldstone
G
mass m is identified as the triplet (magnon) excitation
G
2.2 Spectral function (dynamical scalar susceptibility)
gap;hence, the simulationwas performedwithin the sub-
χ′′(ω)
s
space specified by the longitudinal total magnetization,
either Stzot =0 or 1. In Fig. 4, we present the spectral function (dynamical
In Fig. 2, we present the scaled Goldstone mass LmG scalar susceptibility) [18]
for various J and system sizes, N(= 2L2) = 30 (+), 32
1 1
χ′′(ω)= Im g † g , (3)
( ) 34 ( ) and 36 (✷). The data suggest that a phase s −N (cid:28) (cid:12)(cid:12)E H−Eg −ω+iδE(cid:12)(cid:12) (cid:29)
× ∗ (cid:12) (cid:12)
(cid:12) (cid:12)
transition takes place at J 0.4; note that the inter- for various ω with fixed J = 0.8(> J ) and N = 36.
c
≈
section point of the curves indicates the location of the The energy-resolution parameter is set to δ = 1.4 (solid)
critical point. In Ref. [4], the critical point was estimated and 0.3 (dotted). Here, the symbol g (E ) denotes the
g
| i
as J = 0.435; our result agrees with this estimate. The ground-state vector (energy), and the operator is given
c
E
Goldstone mass appears to close, mG[< O(L−1)] → 0, by E = PH|J=Jc with the projection operator P = 1−
in the ordered phase J > J as L (thermodynamic g g . The spectral function χ′′ is sensitive (less sensi-
c → ∞ | ih | s
limit).Onthecontrary,intheparamagneticphaseJ <J , tive) to the Higgs (Goldstone) mode, because the exter-
c
a finite mass gap opens; in the quantum-magnetism lan- nal perturbation retains the O(3) symmetry. An ad-
E
guage, the mass gap is interpreted as the spin (magnon) vantageofthe numericaldiagonalizationapproachis that
gap for the spin-liquid phase. The spin gap [see Eq. (5)] the resolvent f(z) = g †( z)−1 g is accessible di-
h |E H− E| i
setsafundamentalenergyscaleforthesubsequentscaling rectly via the continued-fractionexpansion[10]. Actually,
analyses. thecontinued-fraction-expansionmethodisessentiallythe
InFig.3,wepresentthescalingplotfortheGoldstone sameasthatoftheLanczosdiagonalizationalgorithm(tri-
mass, (J J )L1/ν-Lm , for various J and system sizes, diagonalization sequence), and computationally less de-
c G
−
4 Yoshihiro Nishiyama: Universal scaled Higgs-mass gap for thebilayer Heisenberg model in theordered phase
manding.Theexternalperturbation is seeminglydiffer- ture agrees with the claim [13] that the Higgs peak gets
E
entfromtheconventionalones(implementedintheMonte broadened for the O(N)-symmetric model with large N.
Carlo simulations, for instance). However, as far as the Last, rather technically, the continued-fraction expansion
symmetry is concerned, those choices are all equivalent, [10]wasiterateduntiltheabove-mentionedsecondarysub-
yielding an identical critical behavior as to m . Here, we peakconverges.Thecomputationaleffortiscomparableto
H
employed the Hamiltonian itself as for , which turned that of the evaluation of g .
E | i
out to be less influenced by corrections to scaling.
2.3 Finite-size-scaling analysis of χ′′
In Fig. 4 (solid), we observe a Higgs-excitation peak s
with the mass (excitation gap), m = 2.2. As mentioned
H In this section, we analyze the finite-size-scaling behavior
above,the signal fromthe Higgs excitation comes up, be-
for χ′′ in the ordered phase, J >J .
s c
cause the scalar susceptibility χ′′ is a good probe specific
s The spectral function obeys the finite-size-scaling for-
to it [18]; actually, there should exist low-lying (0 < ω <
mula [19]
2.2)Goldstone andits continuummodes, as illustratedin
χ′′(ω)=L2/ν−3f[ω/∆,(J J )L1/ν], (4)
Sec. 2.1. Above the threshold ω > 2.2, a tail background s − c
extends. As mentioned afterward, the present simulation with the critical point J , the correlation-length critical
c
was performed so as to examine the main (Higgs) peak, exponentν,acertainscalingfunctionf andtheexcitation
and such high-lying spectral intensities are beyond the gap
scope of the present analysis. ∆(J)=m (2J J), (5)
G c
−
As a reference, we also presented a high-resolutionre-
reflected as to the critical point J ; note that the Gold-
c
sult [Fig. 4 (dotted)], which reveals fine details of the
stonemassm wasconsideredinSec.2.1.Inotherwords,
G
spectral function, namely, the series of the constituent δ-
the Goldstone mode (in J > J ) and the fundamental
c
functionsubpeaks.TheHiggspeaksplitsintotheprimary
energy scale ∆ (in J <J ) continue adiabatically.
c
andsecondarysubpeaks,whichlocate atω =1.9 and3.5, InFig.5,wepresentthescalingplot,ω/∆-L3−2/νχ′′(ω),
s
respectively. As demonstrated in the next section, these
forN =32(dotted),34(solid)and36(dashed)withfixed
fine structures (finite-size artifacts) have to be smeared δ =1.7∆and(J J )L1/ν =2.5.Here,thescalingparam-
c
−
out by an adequate δ in order to attain plausible finite-
eters, J =0.435 [4] and ν =0.7112 [17], are the same as
c
size-scaling behaviors.
thoseofFig.3;thatis,therearenoadjustableparameters
Last, we address a number of remarks. First, as men- inthescalinganalysis.Thescaled-spectral-functioncurves
tioned above, the Higgs peak consists of two subpeaks, collapse into a scaling function f satisfactorily.From Fig.
and hence, it has an appreciable peak width. Such fea- 5, we notice that the (properly scaled) Higgs mass takes
Yoshihiro Nishiyama: Universal scaled Higgs-mass gap for thebilayer Heisenberg model in theordered phase 5
a universal value m /∆ = 2.7. The universality (stabil- (dotted)]to the secondary(andeventernary...)one(s)for
H
ity) of m /∆ with respect to the variation of the scaling (J J )L1/ν > 3.5, and the Higgs peak drifts (and gets
H c
−
argument (J J )L1/ν is examined in the next section. broadened);forexceedinglylarge(J J )L1/ν,eventually,
c c
− −
thesimulationdatamaygetoutofthe scalingregime.On
2.4 Universal character of the scaled Higgs peak theonehand,inthe(J J )L1/ν <2side,theHiggsmass
c
−
acquires a significant enhancement. This narrow regime
In the above section, we investigated the universal be-
is not physically relevant, because the regime shrinks in
havior of χ′′ (4) at a particular scaling argument, (J
s − the raw-parameter scale [like J J (< 2/L1/ν) 0] as
c
J )L1/ν =2.5,andobservedascaledHiggsmassm /∆= − →
c H
L . To summarize, at least for the available system
2.7.Inthissection,wevary(J J )L1/ν inordertosurvey → ∞
c
− sizes N 36, the scaling regime (J J )L1/ν 2.5 is
c
≤ − ≈
theuniversalcharacteroftheHiggspeak,particularly,the
optimal in the sense that the scaled Higgs mass takes a
scaled Higgs mass.
stable minimal value
InFig.6,wepresentthescalingplot,ω/∆-L3−2/νχ′′(ω),
s
for various (J Jc)L1/ν = 2 (dotted), 2.5 (solid) and mH/∆=2.7. (6)
−
3 (dashed) with fixed δ = 1.7∆ and N = 36; here, the
As mentioned above, the Higgs peak consists of two
scaling parameters, J and ν, are the same as those of
c
subpeaks.As a byproduct,weare ableto estimate the in-
Fig. 3. The data in Fig. 6 illustrate that the Higgs-peak
trinsic peak width. For (J J )L1/ν = 4.5 and N = 36,
c
−
position, m /∆ = 2.7, is kept invariant with respect to
H
these subpeaks locate at ω/∆ 2 and 3.5 with almost
≈
the variationof (J J )L1/ν. Onthe contrary,the Higgs-
c
− identical spectral weights; hence, the center locates at
peak height seems to be scattered; note that according
ω/∆ 2.75. The distance, 1.5, between these subpeaks
≈
to Eq. (4), the Higgs peaks do not necessarily overlap,
may be a good indicator as to the intrinsic width of the
because a scaling argument (J J )L1/ν is no longer a
c
− Higgs peak, δm /∆ = 1.5. It has been claimed [11] that
H
constant value. In our preliminary survey, scanning the
the Higgs peak for the O(3)-symmetric model should be
parameter space (J J )L1/ν, we observed the following
c
− broadened significantly. Our result supports this claim.
tendency. For (J J )L1/ν > 2, the scaled Higgs mass
c
−
m /∆ = 2.7 is kept invariant. In closer look, however,
H
3 Summary and Discussions
for (J J )L1/ν >3.5, the Higgs peak drifts to the high-
c
−
energy side gradually possibly because of the finite-size ThecriticalityoftheHiggs-excitationspectrumχ′′(ω)[Eq.
s
artifact (limitation of the tractable system size). The mi- (3)] for the bilayer Heisenberg model (1) was investigated
croscopic origin of the drift is as follows. The spectral by means of the numerical diagonalization method; the
weight transfers from the primary subpeak [see Fig. 4 spectralfunctionχ′′(ω)isaccessibledirectlyviathecontinued-
s
6 Yoshihiro Nishiyama: Universal scaled Higgs-mass gap for thebilayer Heisenberg model in theordered phase
fraction expansion [10]. The spectral function appears to 3D O(3) universality class
Higgs: massive
obeythe scalingformula(4)satisfactorily,indicating that Goldstone: massless J(/J’)
0 Jc=0.435
the simulation data already enter the scaling regime. As Dimer phase Ordered phase
(paramagnetic)
aresult,weestimatedthescaledHiggsmassm /∆=2.7
H
withthepeakwidthδm /∆=1.5.Sofar,withthe(quan- Fig. 1. A schematic phase diagram [3,4] for the bilayer
H
Heisenberg model (1) is presented. A critical point locates
tum)MonteCarlomethod,thescaledHiggsmasshasbeen
at J = 0.435 [4], separating the paramagnetic and ordered
estimated as m /∆ = 2.2(3) [6] and 2.6(4) [7]. Accord- c
H
phases.Inthelatterphase,theHiggs(Goldstone)excitationis
ing to the normalization-group analysis, the scaled Higgs
massive(massless). TheHiggs-excitation peakforthespectral
mass was estimated as m /∆= 2.7 [8] and 1.64 [9]. Our
H
function (3) is our concern.
result agrees with these preceding estimates [6,7,8]; the
errormarginofour estimate should be bounded by half a
peak width, 0.75.
≈
5
TheGinzburg-Landautheory(basedonthewine-bottle- 4.5
4
3.5
bottompotential)yieldsthecriticalamplituderatiom /∆=
H 3
G
m 2.5
√2. Clearly, the Ginzburg-landau theory fails in describ- L 2
1.5
ing the spectral property for the d = 3 O(3) universal- 1
0.5
0
ity class.In other words, such a spectral property reflects 0 0.2 0.4 0.6 0.8 1
J
a character of each universality class rather sensitively.
Fig. 2. The scaled Goldstone mass Lm is plotted for var-
As a matter of fact, as for the “deconfined critical” phe- G
ious J and (+) N(= 2L2) = 30, (×) 32, (∗) 34 and (✷) 36.
nomenon [20], an exotic spectral property was predicted.
The result agrees with the preceding estimate J = 0.435 [4];
c
Aconsiderationtowardthis directionis left forthe future
note that the intersection point of thecurves indicates thelo-
study.
cation of the critical point. In the ordered phase J > J , the
c
Goldstone-excitation gap closes, m [< O(L−1)] → 0, in the
G
thermodynamiclimit,L→∞.Onthecontrary,intheparam-
Acknowledgment agneticphaseJ <Jc,anexcitationgapopens;thegap[seeEq.
(5)]setsafundamentalenergyscaleforthesubsequentscaling
analyses.
This work was supported by a Grant-in-Aid for Scientific
Research (C) from Japan Society for the Promotion of
Science (Grant No. 25400402).
Yoshihiro Nishiyama: Universal scaled Higgs-mass gap for thebilayer Heisenberg model in theordered phase 7
6 0.04
5 0.035
0.03
4
LmG 3 ν2/χ’’s 0 0.0.0225
2 3-L 0.015
1 0.01
0.005
0
-4 -2 0 2 4 0
(J-J)L1/ν 0 1 2 3 4 5 6 7 8
c ω/∆
Fig.3. ThescalingplotfortheGoldstonemass,(J−J )L1/ν- Fig. 5. The scaling plot (4) for the spectral function, ω/∆-
c
Lm , is plotted for (+) N = 30, (×) 32, (∗) 34 and (✷) 36. L3−2/νχ′′(ω), is shown with fixed (J−J )L1/ν =2.5 and δ =
G s c
Here, the scaling parameters, J = 0.435 and ν = 0.7112, are 1.7∆ for various N =32 (dotted),34 (solid) and 36 (dashed);
c
takenfromtheexistingliteratures,Refs.[4]and[16,17],respec- here, the scaling parameters, J and ν, are the same as those
c
tively.Namely,therearenoadjustable parametersinvolvedin of Fig. 3. The scaled Higgs peak locates at m /∆=2.7.
H
thescaling analysis.
0.04
0.035
0.1 0.03
0.09
0.08 χ’’s 0.025
0.07 ν2/ 0.02
0.06 3-L 0.015
χ’’s 0.05 0.01
0.04
0.005
0.03
0.02 0
0 1 2 3 4 5 6 7 8
0.01
ω/∆
0
0 1 2 3 4 5 6 7
ω
Fig. 6. The scaling plot (4) for the spectral function, ω/∆-
Fig. 4. Thespectralfunctionχ′′(ω)(3)isplotted forvarious L3−2/νχ′′(ω), is shown with fixed N = 36 and δ = 1.7∆ for
s s
ω with fixedJ =0.8and N =36. Theω-resolutionparameter variousscalingarguments(J−Jc)L1/ν =2(dotted),2.5(solid)
δ is set to δ = 1.4 (solid) and 0.3 (dotted). The main peak and 3 (dashed); here, the scaling parameters, Jc and ν, are
at ω = 2.2 (solid) corresponds to the Higgs excitation. The the same as those of Fig. 3. The curves do not necessarily
main peak consists of primary (ω = 1.9) and secondary (ω = overlap,becausethescalingargument(J−Jc)L1/ν isranging.
3.5) subpeaks; the Higgs-excitation peak acquires an intrinsic The scaled Higgs-peak position mH/∆ = 2.7 seems to be a
width. universalconstant.
A Numerical algorithm: Screw-boundary bergmodel(1).Weimplementedthescrew-boundarycon-
dition[15],withwhichoneisabletotreatavarietyofsys-
condition [15]
tem sizes N = 30,32,... (N: the number of constituent
In this Appendix, we explain the simulation algorithm to spins) systematically. According to Ref. [15], an align-
diagonalizetheHamiltonianmatrixforthebilayerHeisen- ment of spins σ (i = 1,2,...,M) with both nearest-
i
8 Yoshihiro Nishiyama: Universal scaled Higgs-mass gap for thebilayer Heisenberg model in theordered phase
and √Mth-neighbor interactions is equivalent to a two- 7. M.Loh¨ofer,T.Coletta,D.G.Joshi,F.F.Assaad,M.Vojta,
dimensional cluster under the screw-boundary condition; S. Wessel and F. Mila, arXiv:1508.07816.
here, the periodical boundary condition as to the spin 8. F.Rose,F.L´eonardandN.Dupuis,Phys.Rev.B91(2015)
alignment, namely, σM+i =σi, is imposed. Based on this 224501.
9. Y. T. Katan and D. Podolsky, Phys. Rev. B 91 (2015)
idea, we express the Hamiltonian matrix
2 N/2 2 N/2 075132.
= J (P−√N/2SaiP√N/2) Sai J Sa,i+1 Sai
H − · − 1·0. E. R. Gagliano and C. A. Balseiro: Phys. Rev. Lett. 59
Xa=1Xi=1 Xa=1Xi=1
N/2
(1987) 2999.
+J′ S1i S2i, (7)
·
Xi=1 11. S. Gazit, D. Podolsky, A. Auerbach and D. P. Arovas,
with the translation operator (by one lattice spacing) P
Phys. Rev.B 88 (2013) 235108.
[15]; namely, a relation P−δSaiPδ =Sa,i+δ holds. We di- 12. K. Chen, L. Liu, Y. Deng, L. Pollet and N. Prokof’ev,
agonalized the above Hamiltonian matrix (7) with the Phys. Rev.Lett. 110 (2013) 170403.
Lanczos method so as to evaluate the ground-state vec- 13. A.Ran¸conandN.Dupuis,Phys.Rev.B89(2014)180501.
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g
| i
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