Critical Flavor Number in the Three Dimensional Thirring Model

0.16 MB·English

by  Stavros Christofi

CRITICAL FLAVOR NUMBER IN THE THREE DIMENSIONAL THIRRING
MODEL
Stavros Christofia, Simon Handsb and Costas Strouthosc
aFrederick Institute of Technology, CY-1303 Nicosia, Cyprus
bDepartment of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. and
c Institute of Chemical Sciences and Engineering,
E´cole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland.
WepresentresultsofaMonteCarlosimulation ofthethreedimensionalThirringmodelwiththe
numberoffermion flavorsN variedbetween2and18. Byidentifyingthelatticecouplingatwhich
f
the chiral condensate peaks, simulations are be performed at couplings g2(N ) corresponding to
f
the strong coupling limit of the continuum theory. The chiral symmetry restoring phase transition
is studied as N is increased, and the critical number of flavors estimated as N = 6.6(1). The
f fc
critical exponents measured at the transition do not agree with self-consistent solutions of the
7
Schwinger-Dyson equations; in particular there is no evidence for the transition being of infinite
0
order. Implications for thecritical flavornumberin QED3 are briefly discussed.
0
2
PACSnumbers: PACS:11.10.Kk,11.30.Rd,11.15.Ha

n
a
J The study of quantum field theories in which the QED4 [10, 11]. Using different sequences of truncation,

2 groundstateshowsasensitivitytothenumberoffermion however,otherSDEapproacheshavefoundvaluesofN
fc
2 flavors N is intrinsically interesting. According to cer- ranging between 3.24 [4] and ∞ [6].
f

1 tainapproximatesolutionsofSchwinger-Dysonequations SinceforNf <Nfcthechiralsymmetrybreakingtran-
(SDEs), in d = 3 spacetime dimensions both quantum sition corresponds to a UV-stable RG fixed point at a
v
6 electrodynamics(QED3)andthe Thirringmodeldisplay critical gc2, which can be analysed using finite volume
1 thisphenomenon. Bothmodelshavebeenproposedasef- scaling techniques, study using lattice Monte Carlo sim-
0 fectivetheoriesdescribingdifferentregionsofthecuprate ulation has proved possible with relatively modest com-
1 phase diagram, Thirring describing the superconduct- puter resources [12, 13, 14, 15]. It has been shown that
0 ing phase, while QED3 supposedly describes the non- for Nf ≤ 5 the model is in the chirally broken phase at
7 superconducting “pseudogap” behaviour seen in the un- strongenoughcouplingg2 <∞,providingalowerbound
0
derdoped regime [1, 2]. The Thirring model is a theory on N , and the critical exponents found to vary with
/ fc
t of fermions interacting via a current contact interaction: N . This should be contrasted with numerical studies
a f
l of QED3: in this case the phase transition in the con-
p- L=ψ¯(∂/ +m)ψ + g2 (ψ¯γ ψ )2, (1) tinuum limit at Nf = Nfc is an IR-stable fixed point
e i i 2N i µ i (see, eg. [16, 17]), and SDE predictions for the separa-
f
h tionofscalesparametrisedbythedimensionlessquantity
v: where ψi,ψ¯i are four-component spinors, m is a parity- hψ¯ψi/α2 aresosmallthatenormouslatticesarerequired
Xi conserving bare mass, and the index i runs over Nf dis- to establish whether the model with a particular Nf lies
tinct fermion flavors. In the chiral limit m → 0 the La- in the broken or symmetric phase [18]. A determination
ar grangian(1)sharesthesameglobalU(1)chiralsymmetry of Nfc for QED3 (whose value may have profound im-
ψ 7→ eiαγ5ψ, ψ¯ 7→ ψ¯eiαγ5 as QED3. Since the coupling plications for the cuprate phase diagram [2]) by purely
g2hasmassdimension−1,naivepower-countingsuggests numerical means currently appears very difficult.
that the model is non-renormalisable. However [3, 4, 5], As already noted, Thirring and QED3 share the same
an expansion in powers of 1/N , rather than g2, is ex- globalsymmetries;moreoverin the strongcouplinglimit
f
actly renormalisable and the model has a well-defined the 1/Nf expansion predicts the existence of a massless
continuumlimitcorrespondingtoaUV-stablefixedpoint spin-1 ψψ¯ bound state in the Thirring spectrum [5, 7].
of the renormalizationgroup(RG). The 1/N expansion The RG fixed point of both models is therefore a theory
f
may not, however, describe the true behaviour of the of massless fermions interacting via massless vector ex-
model at smallN . Chiralsymmetry breaking,signalled change between conserved currents. It is therefore plau-
f
by a condensate hψ¯ψi =6 0, is forbidden at all orders in sible that the two models have the same Nfc, and that
1/N , and yet is predicted by a self-consistent SDE ap- simulation of the Thirring model is the most effective
f
proach [4, 6, 7, 8]. SDEs solved in the limit g2 →∞ [7] means to determine its value. Here, we present results
show that chiral symmetry is spontaneously broken for based on numerical simulations with Nf = 2,...,18 in
Nf <Nfc ≃4.32, close to certain predictions of Nfc for anefforttodetermineNfc withunprecedentedprecision,
non-trivial IR behaviour in QED3 [9]. Based on these exploitingastrategyofidentifyingthevalueofthelattice
results, at N = N the model is expected to undergo couplingg2 correspondingtothe strongcouplinglimitof
f fc
aninfiniteorderorconformalphasetransition,originally the continuum theory.
discussed by Miranskii et al in the context of quenched The lattice actionwe haveused is based on[12] andis
2
as follows: where J(m) is the value of the integral contributed by
1 the second diagram in Fig. 1. The physics described by
S = 2 χ¯i(x)ηµ(x)(1+iAµ(x))χi(x+µˆ)−h.c. continuum1/Nf perturbationtheoryoccursfortherange
Xxµi of couplings g2 ∈ [0,∞), ie. for g2 ∈ [0,g2 ) where to
R lim
N leading order in 1/N g2 = 3 for m = 0. The strong
+m χ¯ (x)χ (x)+ A (x)2 (2) lim 2
Xxi i i 4g2 Xxµ µ cgo2u>plgin2g ltihmeitauisxiltihaerryepforroeparegcaotvoerrbeedcoamt egs2n=egagtl2iimve.,aFnodr
lim
where χ,χ¯ are staggered fermion fields and the flavor the lattice model no longer describes a unitary theory.
indexirunsoverN species. Wehaveintroducedanaux- As already discussed, chiral symmetry breaking is ab-
iliary real-valuedlink field A , superficially resembling a sent in large-N calculations. It may well be that the
µ f
photon field, so that the lattice action can be expressed value of the second diagram in Fig. 1 is considerably al-
as a bilinear form. The formulationis not unique; Gaus- tered in the chirally broken vacuum expected for N <
f
sian integration over the A in (2) results in a lattice N . In this study we use lattice simulations to estimate
µ fc
action resembling the continuum form (1) in that all in- the value of g2 as a function of N . Fig. 2 shows hψ¯ψi
lim f
teractionsremainof the formχ¯χχ¯χ for arbitraryN, but
some ofwhich whenre-expressedinterms of continuum- 0.25
like Dirac spinors are of non-covariant form. As argued EoS fit
in [13], in the 1/Nf expansion these unwanted contri- 0.2 123
butions are probably irrelevant, but more care may be 163
needed when discussing the UV fixed-point theory. m=0.02
0.15 m=0.03
For d = 3 N staggered fermion species describe Nf = <ψψ> N=4
f
2N continuum flavors. We employed the Hybrid Monte
N=8
Carlo(HMC)algorithmtosimulateevenN [13]andthe 0.1 f
f
HybridMolecularDynamics(HMD)algorithmforoddor
non-integerN [15]. FortheHMDsimulationsweuseda 0.05
f
small enough fictitious time step ∆τ ≤ 0.0025 to ensure
that the O(N2∆τ2) systematic errors in the molecular 0
0 0.2 0.4 0.6 0.8
dynamicsstepsweresmallerthanthe statisticalerrorsof -2
g
the various observables.
FIG. 2: Chiral condensate hψ¯ψi vs. g−2. N =6, m=0.01
f
on a 163 lattice unless otherwise stated. The dashed line
denotes a fit of the163, N =6, m=0.01 data to an EoS of
f
theform (5) taking finitevolume scaling into account.
data as a function of g−2 for various N , m and lat-
f
tice volumes. The condensate is non-monotonic, show-
ing a clear maximum for N = 6 flavors at g−2 ≃ 0.3.
FIG. 1: Leading order vacuum polarisation in lattice QED. f
The peak position at g2 = g2 , unlike the value of the
max
Next we review the discussion of [13] regarding the condensate itself, is independent of both lattice volume
non-conservation of the vector current in the lattice andfermion mass,indicating that its originis at the UV
Thirring model. The leading order 1/N quantum cor- scale. Since in an orthodox description of chiral sym-
rections to the photon propagator in lattice QED3 are metry breaking |hψ¯ψi| is expected to increase with the
sketchedinFig.1. The seconddiagram,arisingfromthe strength of the interaction, we interpret the peak as the
gauge-invariantformχ¯xeiAµxχx+µˆ,ispeculiartothe lat- point where unitarity violation sets in, ie. gm2ax ≈ gl2im,
tice regularisation,but is requiredto ensure transversity and hence identify the peak with the location of the
of the vacuum polarisation tensor, ie: strong coupling limit.
The figure also shows that g2 depends on N (Cf.
[Π (x)−Π (x−µˆ)]=0. (3) max f
µν µν Fig. 3 of Ref. [13]). For N < N , it is possible to fit
f fc
Xµ condensate data taken at g2 < g2 to an equation of
max
For the A propagator in the lattice Thirring model (2) state (EoS) of the form [13]
µ
however,the seconddiagramisabsent,andthe transver-
sitycondition(3)violatedbyatermofO(a−1). Transver- m=A(g−2−gc−2)hψ¯ψip+Bhψ¯ψiδ, (5)
sityofΠ iscrucialtotherenormalisabilityofthe1/N
µν f
whichdescribesacontinuoustransitioninthe limitm→
expansion; fortunately, the impact of the extra diver-
0atg2 =g2 toachirallysymmetricphase,characterised
gencecanbeabsorbedbyawavefunctionrenormalisation c
by critical exponents δ and a “magnetic” exponent β =
of A and a coupling constant renormalisation
µ (δ−p)−1. Onadatasettakenon123,...,323takingfinite
g2 volume scaling into account we have found the best fit
gR2 = 1−g2J(m), (4) for Nf = 6 given by gc−2 = 0.316(1), δ = 5.75(13), p =
3
1.18(2), to be compared with corresponding quantities shown in Fig. 3 and the predictions of the 1/N expan-
f
EtaobSulfiattefdorfoNrfs=ma6l,lesrhNowfninfor[1t5h].e 1S6in3cdeagtac2 b<∼ygam2daxa,shthede msioaninisnathdeispreagriimtyebNetfw>eenNgfm2c.ax(INnfp)aratnicdultahre, tlahregree-Nref-
line in Fig. 2, is of borderline credibility. prediction gl2im = 23, which may be due to subleading
Fig. 2 also shows that for Nf = 8 the peak moves to correctionsin 1/Nf, or to large finite-volume corrections
the right, and is considerably less pronounced. In Fig. 3 described by the conformal field theory expected in the
we plot gm−a2x vs. Nf for Nf ∈ [2,18]. For Nf <∼ 6, gm−a2x limit Nf →Nfc+.
Fig.4isinterestingbecauseforthefirsttimeitpresents
lattice data in a form suitable for direct comparison
with SDE predictions. One such study of chiral sym-
0.8 metry breaking in the d-dimensional Thirring model
with d ∈ (2,4) using this approach was by Itoh et al
[7]. They calculated the dressed fermion propagator
0.6 S(p) = [A(p2)/p + B(p2)]−1, by first exploiting a hid-
-2
g den local symmetry to fix a gauge in which A ≡ 1,
max
and then approximating the auxiliary propagator and
0.4 fermion-auxiliary vertex by their forms to leading or-
der in 1/N . This enabled a solution for the self-energy
f
function Σ(x) = B(x)/Λ, with Λ a UV cutoff and
x = (p/Λ)d−2, in the strong coupling limit g2 → ∞,
0.2
0 5 10 15 20
yielding a dynamically-generated fermion mass M:
N
f
d−2
FIG. 3: gm2ax vs. Nf. M ∝exp −2π , (7)
(cid:18) Λ (cid:19) Nfc −1
decreasessmoothlyandmonotonically,butintheinterval qNf 
between 6.5 and6.6 there is a sudden sharpincrease;for
6.6 ≤ Nf ≤ 12, g−2 ≃ 0.43(2) is roughly constant, and where Nfc(d = 3) = 128/3π2 ≃ 4.32. The chiral order
then increasesfor N stilllarger. With the identification parameter follows via the relation hψ¯ψi ∝ Σ′(x = 1),
f
of g2 with the strong coupling limit of the continuum leading to the strong coupling prediction
max
theory, as argued above,it is natural to ask whether the
transitionatNf ≃6.5hasanycorrelationwiththechiral hψ¯ψi∝Λd−22Md2. (8)
order parameter. Fig. 4 shows hψ¯ψ(g2 )i vs. N . This
max f
Spontaneous chiral symmetry breaking occurs for N <
f
N in qualitative, but not quantitative agreement
fc
with Fig. 4. Eqn. (7) predicts an infinite-order phase
3
0.3 16 transition. Its nature is further elucidated using the
3
32 anomalous scaling dimension of the ψ¯ψ bilinear γψ¯ψ =
dlnhψ¯ψi/dlnΛ and the relations for the critical expo-
0.2 nents η and δ [19]:
<ψψ>
d+2−η
0.1 η =d−2γψ¯ψ ; δ = d−2+η. (9)
From (8) we obtain γψ¯ψ = (d−2)/2 and hence η = 2,
δ =1. Thisscenariohasbeentermeda“conformalphase
0
0 5 10 15 20
transition” [11], and has also been exposed by SDE ap-
N
f proachesinQED3[16]andquenchedQED4[10],thecon-
trol parameter being respectively N and the fine struc-
FIG. 4: hψ¯ψ(gm2ax)i vs. Nf for m=0.01. Lines denote EoS ture constant α. f
fitsdiscussed in thetext.
The SDE analysis of [7] was later extended to cover
plot necessarilyputs anupper bound onthe condensate. g2 <∞ by Sugiura [8]. In this case the chiral transition
It is difficult to draw any conclusion other than there takes place for N <N , and the solution for the order
f fc
being a chiral symmetry restoring phase transition at parameter is of the form
Nfc =6.6(1). (6) hψ¯ψi∝Λd−2M. (10)
Eqn. (6) is the main result of this Letter. The same chain of arguments leads to critical exponents
Inordertorefinethis picture,furthertheoreticalanal- η = 4−d, δ = d−1, coincident with those of the d-
ysis is needed to establish contact between the results dimensional Gross-Neveu model in the large-N limit
f
4
[19]. The nature of the transition predicted by SDEs ted values for δ and ν together with those from EoS fits
thus appearssensitiveto the orderofthe limits g2 →∞, to data from N <N compiled from Refs. [13, 14, 15],
f fc
N →N . as shown in Fig. 5, we see that the estimate for δ(N )
f fc fc
We have been motivated by these considerations to lies within a general trend of δ increasing with N , with
f
attempt an EoS fit to the data of Fig. 4 of the form no evidence for non-commutativity ofthe limits g2 →∞
andN →N (there issomeindicationforajump from
f fc
m=A[(Nf −Nfc)+CL−ν1]hψ¯ψip+Bhψ¯ψiδ, (11) ν ≈ 0.5 to ν ≈ 1 as Nf increases from 6 to Nfc, which
needs to be confirmed in a more refined study). In any
where L is the linear extent of the system and the case, the fitted values of δ lie well above those predicted
exponent ν is given by the hyperscaling relation ν = in the SDE approach.
(δ+1)/d(δ−p). ThetermproportionaltoC accountsfor
Insummary,forthe firsttime we havebeenable using
finitevolumecorrections. OurbestfittodatawithN ∈
f lattice Monte Carlo simulation to study a chiral phase
[4,8]yieldsN =6.89(2),δ =6.90(3),p=4.23(2),with
fc transition as the number of fermion flavors N is var-
χ2/dof=129/15. f
ied continuously. We find a continuous phase transition
Besides the relativelypoor quality,andsignificantdis-
in the strong coupling limit at a critical flavor number
agreement with Eq. (6), the fit should not be taken too N = 6.6(1), in qualitative, but not quantitative agree-
fc
seriouslyfortwofurtherreasons,namelythecombination
mentwiththepredictionsofananalysisusingSchwinger-
of“exact”HMCdatawithHMDcontainingasystematic
Dysonequations. Inparticular,thecriticalexponentsare
error due to ∆τ 6= 0, and the as yet unquantified errors
notthoseeitherofaconformalphasetransitionorthe3d
due to the identification of g2 with g2 . Nonetheless
lim max Gross-Neveu model. It is plausible that our result may
the disparity of the value of the exponent δ with the
inform estimates of the corresponding critical number of
SDE prediction is striking. Moreover, if we plot the fit-
flavorsfor chiral symmetry breaking in QED3, where di-
rect lattice simulations are hampered by a large separa-
8
tion of scales. Note that a recent perturbative analysis
δ
ν of RG flow equations in the large-Nf limit of QED3 pre-
dicts N = 6 [20]. In future it will be interesting to
6 fc
check whether the same universal features of the strong
couplinglimitemergeusingthealternativelatticeformu-
4 lations of the Thirring model reviewed in [13].
2 Acknowledgments
0 Thesimulationswereperformedonaclusterof2.4GHz
2 3 4 5 6 7
N OpteronprocessorsattheFrederickInstituteofTechnol-
f
ogy, Cyprus.
FIG. 5: Critical exponentsδ and ν vs. N .
f
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