A test of the g-ology model for one-dimensional interacting Fermi systems

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by  Andrey V. Chubukov

A test of the g ology model for one-dimensional interacting Fermi systems
−
Andrey V. Chubukov1, Dmitrii L. Maslov2 and Fabian H.L. Essler3
1Department of Physics, University of Wisconsin-Madison,
1150 Univ. Ave., Madison, WI 53706-1390, USA
2Department of Physics, University of Florida, P. O. Box 118440, Gainesville, FL 32611-8440, USA
2 The Rudolf Peierls Centre for Theoretical Physics,
University of Oxford, 1 Keble Road Oxford, OX1 3NP, UK
Bosonization predictsthatthespecificheat,C(T),ofaone-dimensionalinteractingFermisystem
is a sum of the specific heats of free collective charge and spin excitations, plus the term with the
8 runningbackscattering amplitudewhich flowsto zerologarithmically with decreasing T. Weverify
0 whether this result is reproduced in the g−ology model. Of specific interest are the anomalous
0 termsinC(T)thatdependonthebarebackscatteringamplitude. Weshowthatthesetermscanbe
2 incorporatedintoarenormalized spinvelocity. Wedothisbyprovingtheequivalenceoftheresults
for C(T) obtained within the g-ology model and by bosonization with velocities obtained by the
n
a numerical solution of theBethe-ansatz equations for theHubbardmodel.
J
PACSnumbers:
1
1

One-dimensionalinteractingfermionic systemsarebe- twoof us haveobtained[11] the specific heatofinteract-
]
l lieved to be described by an effective low-energy theory: ing 1D fermions to order g3 and found that the effect of
e i
- the g-ology model [1, 2, 3]. This model involves a small 2kF scatteringinthespinchannelismoreinvolvedthanit
r
number of interaction vertices describing small momen- had been previously thought – in addition to terms that
t
s
tum scattering of fermions near the same and opposite depend on the running coupling g (T), the free energy
. 1
t Fermi points (g and g , correspondingly) and 2k scat- also contains terms that depend on the bare coupling g
a 4 2 F 1
m teringg [for fermionsonalattice, thereis alsoanUmk- (see Eq. (9) below).
1
lapp vertex (g )]. The effective vertices are, in principle, ¿From the field-theoretical point of view, these terms
- 3
d obtained by integrating out high-energy fermions in mi- should regarded as anomalies, i.e., they can be viewed
n
croscopicmodels,e.g.,theHubbardmodel. Tofirstorder equivalently either as low- or high-energy contribu-
o
in U, g =U/(2πv ); beyondfirst order,allg are differ- tions.Theseanomaloustermscannotbe simply absorbed
c i F i
[ ent, and each of them is represented by a series in U. into the Gaussian part of the bosonic Hamiltionian, and

Apowerfulwaytotreattheg-ologymodelisbosoniza- it is not a priori clear whether such terms can be incor-
1
tion, which transforms interacting fermions into the col- poratedintothe renormalizedchargeandspinvelocities.
v
7 lective bosonic excitations in the charge and spin chan- Thegoalofthispaperistoprovethattheanswertothe
3 nels[3,5]. Forthecaseofarepulsiveinteractionbetween question formulated above is affirmative, at least within
8 originalfermions,consideredinthis paper,the bosoniza- perturbationtheory. Weshowthisbycomparingthespe-
1
tion shows that the charge sector is a free Gaussian the- cific heat obtained from second-order perturbation the-
.
1 ory,while the spin sector becomes asymptotically free in ory in the g ology model, calculated in Ref. 11, with
0 thelow-energylimit,whenthecouplingofthemarginally Eq. (1),wher−ethevelocitiesareobtainedfromtheBethe
8
irrelevant process (2k scattering of fermions of oppo- ansatz solution of the Hubbard model [4].
0 F
site spins) flows to zero [6, 7, 8, 9] As a consequence, Our starting point is the Hamiltonian of the 1D Hub-
:
v bosonizationpredictsthatatthelowesttemperaturesthe bard model
i
X specificheatofinteracting1Dfermionsisthesameasthe
1
r specific heat oftwo systems of acoustic 1D phonons,i.e., H= (ǫk−µ)c†k,σck,σ+UN c†k,↑c†q,↓c↓,k+lc↑,q−l,
a
k,σ k,q,l
X X
πT 1 1 (2)
C(T)= + , (1)
3 (cid:18)vρ vσ(cid:19) where ǫk =−2tcosk and µ=−2tcoskF is the chemical
potential (the lattice spacing is set to unity). For U =0
where vρ and vσ are the charge and spin velocities, cor- the Fermivelocity is relatedto the Fermimomentum kF
respondingly. Eq. (1) was verified by a weak-coupling by v =2tsink .
F F
renormalization group (RG) treatment of the original Away from half-filling, umklapp scattering requires
fermionicmodel,inwhichallnon-logarithmiccorrections collisions of more than two fermions and is therefore ne-
to the couplings were neglected [1, 3, 5, 10]. However, glected in our treatment. The spin and charge veloc-
it has never been provenexplicitly that the bosonization ities can be obtained by solving certain linear integral
result in Eq. (1) is valid to all orders in the interaction. equationsobtainedfromtheBetheansatzsolutionofthe
Recent results call for a further study of the validity Hubbardmodel[4]. Atstrongcoupling,thespinvelocity
ofEq.(1)beyondthe weak-couplinglimit. Inparticular, is v = (π/2)(4t2/U)(1 sin2k /2k ) [12]. The weak
s F F
−
2
n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 to appear in these expansions. We have determined the
c -0.51 -0.56 -0.64 -0.74 -0.87 -1.01 -1.16 -1.31 coefficients c and c′ for a number of different densities.
′ TheresultsareshowninTableIAswewillseebelow,we
c 0.49 0.44 0.36 0.26 0.13 -0.01 -0.16 -0.31
will only need the difference c′ c to verify the validity
TABLEI:Coefficientscandc′fordifferentdensitiesinthe1D ofEq. (1). Inallcaseswe findt−hatwithinthe numerical
Hubbard model. The density n=2kF/π; n=1 corresponds accuracy of our computation
to half-filling.
c′ c=1. (4)
−
coupling limit has been studied in [13]. It is possible to
deriveasmallU expansionforthespinandchargeveloc- Thedeviationfrom1islessthanonepercentinallcases.
ities analyticallybysolvingthe integralequationsdrived Thespinandchargevelocitiescanalternativelybecalcu-
in Ref. [13] by Wiener-Hopf methods. As we are inter- lated in perturbation theory for the g ology model [11,
−
estedinthe (U2)termsoftheexpansions,theresulting 14]. An explicit calculation of C(T) within this model
O
calculations are somewhat involved. We therefore have is somewhat involved as the low-energy g ology model
−
solvedtheintegralequationsnumericallyforsmallUand contains two momentum cutoffs, Λf and Λb, constrain-
found that the results are well fit by the series ingtheintegrationoverthefermionicdispersionandover
the momentum transfers near 2k , respectively. [The
F
U U 2 g-ology model is only valid when Λf > Λb, i.e., when
v v 1+ +c (3a)
ρ ≃ F 2πvF (cid:18)2πvF(cid:19) ! the interaction vanishes at the cutoff set by the disper-
sion.] Some terms in C(T) are cutoff-independent while
2
U U some depend logarithmically on the ratio Λ /Λ . Fortu-
v v 1 +c′ . (3b) f b
σ ≃ F − 2πvF (cid:18)2πvF(cid:19) ! nately,at leastto secondorderin g, allcutoff-dependent
renormalizations can be absorbed into the renormalized
Basedonthe analyticresultfor the spinvelocity athalf- backscattering amplitude g˜ = g 2g2log(Λ /Λ ), so
1 1 − 1 f b
filling[4]andthefactthattheonlymarginallyirrelevant that the specific heat is expressedin terms ofg , g , and
4 2
operator(the interactionofspin currents)is the same at g˜ without any explicit dependence on the cutoffs [11].
1
andbelowhalf-fillingwedonotexpectlogarithmicterms To second order in g, C(T) is given by
2
2πT 1 3
C(T)= 1+(g˜ g )+(g˜ g )2+g2+ g g˜ + g˜2+O(g3) , (5)
3vF " 1− 4 1− 4 4 (cid:18) 2− 2 1(cid:19) 4 1 #
where all vertices are measured in the units of 2πv . rameterizedby the couplings g and g are mapped onto
F 4 2
To compare Eq. (5) with Eqs. (1) where v and v the free, Gaussian part of the bosonized Hamiltonian.
ρ σ
given by Eqs. (3a,3b), we first note that the g-ology The 2k term, parameterized by g , leads to non-linear,
F 1
model can be bosonized by expressing the operators of cosine terms in the bosonic Hamiltonian, which give rise
right- and left-moving fermions, R and L (α= , ) as to interactions in the spin channel.
α α
↑ ↓
1
Rα(x),Lα(x)= √2πbexp[±i(φα(x)∓θα(x))], (6) To first order in g˜1, backscattering just renormalizes
the prefactors in the Gaussian part of the bosonized
wherebisashort-distancecutoffrelatedtothefermionic Hamiltonian [15, 16], so that the g ology model can
−
momentum cutoff (Λf) of the g ology model. Under be reduced to a gas of free acoustic bosons with HG =
bosonization,thetermsintheferm−ionicHamiltonianpa- H(ρ)+H(σ), where
G G
1
H(ρ) = dx(1+2g +2g 2g˜ )(∂ φ )2+(1+2g 2g )(∂ θ )2,
G 2 4 2− 1 x ρ 4− 2 x ρ
Z
1
H(σ) = dx(1 2g˜ )(∂ φ )2+(∂ θ )2, (7)
G 2 − 1 x σ x σ
Z
3
in U and to select the contributions which comes from
a) b) c) d)
high energies. By construction, such contributions are
absorbed into the bare couplings of the g ology model.
−
Therearefoursecond-orderdiagramsfortheg ampli-
4
tudetoorderU2(seeFig. 1). ForaconstantU,diagrams
a)andb)cancelleachother. Diagramc)containsthepo-
larization bubble Π(0)=(1/2π)2 dkdω/(iω ǫ +µ)2.
k
−
FIG. 1: One-loop diagrams for the interaction vertices. In Renormalizations due to Π(0) are within the low-energy
R
this paper, we need only renormalizations coming from the theory, as one can evaluate Π(0) in such a way that the
high-energy scales. The high-energy renormalizations of g4 result Π(0) = 1/(πvF) is determined entirely by the
and g1 come only from the Cooper diagram d). states near the−Fermi energy [11]. The remaining dia-
gramd)describesrenormalizationintheCooperchannel.
Up to a prefactor, it is given by
and the charge and spin bosons are defined as φ =
ρ,σ
(φ φ )/√2 and θ =(θ θ )/√2. 1
↑± ↓ ρ,σ ↑± ↓ dkdω . (11)
If this were the only effect of backscattering, the spe- Z (iω−ǫk+kF +µ)(iω+ǫk−kF −µ)
cific heat would be given by Eq. (1) with the effective
The momentum integration in Eq. (11) is not confined
spin and charge velocities v˜ and v˜ read off from Eq.
σ ρ to the Fermi surface, i.e., this diagram does contribute
(7):
to high-energy renormalization of g . In the Hubbard
4
model, the momentum integration is limited from above
v˜2=v2 (1+2g g˜ )2 (2g g˜ )2 ,
ρ F 4− 1 − 2− 1 by the Brillouin zone. The lower limit Λf can be safely
v˜σ2=vF2 (cid:0)(1−g˜1)2−g˜12 (cid:1) (8) set to zero as the integral is infrared-finite, i.e., dk =
π
dk. Integrating over ω and then over k, we obtain
Re-expressing Eq.(cid:0)(5) in terms of(cid:1)v˜ and v˜ , we obtain −π R
σ ρ
R U U
πT 1 1 πT g4 = 1 . (12)
C(T)= 3 v˜ + v˜ + 3v g˜12+O(g˜13). (9) 2πvF (cid:18) − 2πvF(cid:19)
(cid:18) ρ σ(cid:19) F Note that the only depence on the density is through
We see that this expression differs from Eq. (1). This is vF =2tsinkF. We verifiedthat amodel offermionsin a
not surprising because the effect of backscattering can- continuum with dispersion k2/2m gives the same result.
not be simply absorbed into the Gaussian part of the Renormalization of g1 is more involved because g1 is
bosonized Hamiltonian, beyond the first order in g . a running coupling, and the second-orderresult depends
1
Theissuethereforeiswhethertheextrag˜2 terminEq. logarithmicallyontheuppercutoffofthelow-energythe-
1
(9) can be absorbed into renormalization of the veloci- ory, which is the lower limit of the integrationfor “high-
ties v˜ v, so that the specific heat is still given by Eq. energy” renormalization. The diagrams are the same as
→
(1) with the renormalizedvelocities vρ and vσ. A simple inFig. 1. Asforg4,diagramsa)andb)canceleachother
extension of the previous analysis to a non-SU(2) sym- while diagram c) contains Π(0). Therefore, only Cooper
metric case shows that, beyond the firstorder,2k scat- diagram d) contributes to the “high-energy” renormal-
F
tering contributes only to the spin part of the bosonized ization. Evaluating Cooper diagram for backscattering,
Hamiltonian. The real issue then is renormalization of we obtain
the spin velocity v˜σ vσ. The charge velocity given by U
→ g =(U/2πv ) 1 L , (13)
EqA.(8s)trmaiugshttfboerwtahredswamayetaoscthheeckextahcistiosnteo,ci.oem.,pv˜aρr=eEvqρs.. 1 F (cid:18) − 2πvF(cid:19)
where
(1) and (9) with Eq. (3a,3b). Quite generally, one can
write vσ = v˜σ +avFg˜12, where a is a dimensionless con- L= vF π dk ∞ dω 1 (14)
stant. Only if a = −1, the extra g˜12 term in Eq.(9) can 2πPZ−π Z−∞ ω2+(ǫk−µ)2
be absorbed into v , i.e., Eq. (9) can be cast into Eq.
σ
and The symbol here implies that momentum inte-
(1). Using Eq. (8), we obtain P
gral does not include the regions near the Fermi points
R
v =v 1+2g g˜ 1(2g g˜ )2 , (10a) of width 2Λf. Integrating over frequency and then over
ρ F 4− 1− 2 2− 1 momentum, we obtain
(cid:18) (cid:19)
v =v 1 g˜ 1−2ag˜2 (10b) L=2log2sinkF (15)
σ F − 1− 2 1 Λf
(cid:18) (cid:19)
Substituting Eq. (15) into Eq. (13), we then obtain
ThesetwoexpressionsshouldbethesameasEqs.(3a,3b).
To compare Eqs. (3a,3b) and (10a,10b), we need to U U 2sink
F
g = 1 log (16)
evaluate renormalizations of g4 and g1 to second order 1 2πvF (cid:18) − πvF Λf (cid:19)
4
In addition, g is renormalized within the low-energy g- Using these results, we have shown that all terms in the
1
ology model. This renormalization depends logarithmi- specificheatintheg ologymodelthatdonotflowunder
−
cally on the ratio of the bosonic and fermionic cutoff RG are absorbed into the specific heat of two free gases
of the g ology model: g˜ = g 2g2log(Λ /Λ ) [11]. of massless bosons. As a result, the full specific heat of
− 1 1 − 1 f b
AddingupthisresultwithEq.(17)wefindthatthecom- a system of interacting fermions in 1D is a sum of the
bination of the high-energy and low-energy renormaliza- specific heats of two free massless Bose gases of charge
tions just replaces Λ by Λ under the logarithm, i.e., andspinexcitations,andthetrueinteractionterm,which
f b
containsthe running backscatteringamplitude andloga-
U U 2sink
F rithmically flows to zero with decreasing T.
g˜ = 1 log . (17)
1
2πv − πv Λ
F (cid:18) F b (cid:19)
Alternatively, one can obtain the full renormalization We acknowledgehelpful discussions with I. L.Aleiner,
of g by excluding the regions of width Λ near 2k H. Frahm, T. Giamarchi, L. I. Glazman, K. B. Efetov,
1 b F
±
fromtheintegrationoverthemomentumtransferk. This A. W. W. Ludwig, K. A. Matveev, A. Melikyan, A.M.
gives the same result as in Eq. (17). Tsvelik, and support from nsf-dmr 0604406 (A.V. Ch),
The value of Λ is unknown: as we said earlier, the nsf-dmr 0308377(D.L.M.)andthe EPSRCundergrant
b
g ology model assumes that the low-energy properties GR/R83712/01(FHLE).
−
of the original system of 1D fermions with a short-range
interaction are the same as in the model where interac-
tions are artificially restricted to narrow regions of mo-
mentum transferseither near zero(for g andg ) or2k
4 2 F
[1] J. S´olyom, Adv.Phys. 28 201 (1979).
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1 F b [2] V. J. Emery, in Highly Conducting One-Dimensional
substantially smaller than a half of the fermionic band-
Solids, eds.J. T.Devreese, R.E.Evrand,abd V.E. van
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b
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(1983)
Eq. (1). This, we believe, is a “numerical proof” of the
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F b
[10] D.L.Maslov,inNanophysics: Coherence andTransport,
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b F
editedbyH.Bouchiat,Y.Gefen,G.Montambaux,andJ.
the g ology model cannot be justified. In particular, at
− Dalibard, Les Houches, Session LXXXI, 2004 (Elsevier,
quarter-filling, Λ sink 0.44. Near half-filling, how-
b F ∼ 2005).
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−
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b
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[14] A. Melikyan and K. A.Matveev (unpublished).
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