Interacting fermions in two dimensions: beyond the perturbation theory

0.2 MB·English

by  S. Gangadharaiah

Interacting fermions in two dimensions: beyond the perturbation theory
Suhas Gangadharaiah1, Dmitrii L. Maslov1∗,2, Andrey V. Chubukov3, and Leonid I. Glazman4
1Department of Physics, University of Florida, P. O. Box 118440, Gainesville, FL 32611-8440
2Abdus Salam International Center for Theoretical Physics, 11 Strada Costiera, 34014 Trieste, Italy
3Department of Physics, University of Maryland, College Park, MD 20742-4111
4Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455
(Dated: February 2, 2008)
5
Weconsiderasystemof2Dfermionswithshort-rangeinteraction. Astraightforwardperturbation
0
theory is shown to be ill-defined even for an infinitesimally weak interaction, as the perturbative
0
series for the self-energy diverges near the mass shell. We show that the divergences result from
2
theinteractionoffermionswiththezero-soundcollective mode. Byre-summingthemostdivergent

n diagrams, we obtain a closed form of the self-energy near the mass shell. The spectral function
a exhibitsathresholdfeatureattheonsetoftheemissionofthezero-soundwaves. Wealsoshowthat
J theinteraction with thezero sound does not affect a non-analytic, T2-part of the specific heat.

3
PACSnumbers: 71.10.Ay,71.10.Pm

]
l
e
- The Landau Fermi-liquid (FL) theory states that the ory must be re-summed even for an arbitrarily weak in-
r
t low-energy properties of an interacting fermion system teraction [3, 5].
s
. are similar to those of an ideal Fermi gas [1]. A neces-
t
a sary (but not sufficient) condition for the validity of this In this Letter, we report the results of an asymptot-
m theory is that various characteristic properties (thermo- ically exact re-summation of the divergent perturbation
- dynamicparameters,quasi-particlelifetime, etc.) canbe theory for D = 2. In addition to obtaining a closed and
d
expressed via regular perturbative expansions in the in- divergence-free form of the self-energy to all orders in
n
teraction. Backinthe50s,itwasshowntobethecasefor theinteraction,thisprocedurealsoallowsonetoidentify
o
c a model of 3D fermions with both short- and long-range the nature of the singularities in a perturbation theory
[ (Coulomb) repulsion. Soon thereafter, it was reazlied as originating from the interactionbetween the fermions

that the perturbation theory is singular in 1D and, as a and the zero-sound (ZS) collective mode. At any finite
1
v result, the FL is destroyed. The case of D =2 had been order of the perturbation theory, the collective mode co-
3 thesubjectofanactiveandfairlyrecentdiscussion,with incides with the upper edge of the particle-hole (PH)
1 anumberofproposals,mostnotablybyAnderson[2],for continuum. In 2D, this degeneracy is strong enough to
0
the breakdownof the FL in 2D. Although the prevailing generate the divergences in the self-energy. Once per-
1
opinion currently is that the FL is stable in 2D for suf- turbations are summed up to all orders, the zero-sound
0
5 ficiently short-range (including Coulomb) interactions, a mode splits off from the continuum, and the power-law
0 full description of the FL in 2D is still lacking. divergences disappear. The remaining log-divergences
/ are eliminated by restoring the finite curvature of the
t
a Oneoftheproblemsindescribinga2Dinteractingsys- spectrumneartheFermisurface[5,6]. Thusthe FLsur-
m
tem is that, in a contrastto the 3D case, a naive pertur- vives. However, the resulting self-energy contains sev-
- bative expansion in the interaction is singular. Namely, eral non-perturbative features that signal a deviation
d
for a linearized single-particle spectrum, the imaginary from the standard FL behavior. First, the imaginary
n
o part of the self-energy diverges on the mass shell. The part of a non-perturbative, ZS contribution to the self-
c divergence is logarithmic to second order in the interac- energy, Σ (ω,k), is a non-monotonic function of the
ZS
:
v tion [3, 4, 5, 6] but, as we will show in this paper, it is “distance” to the mass-shell, ∆ ω ǫk, and exhibits
≡ −
i amplified to a power-law, starting from the third order. an anomaly at the threshold for emission of ZS waves.
X
This mass-shellsingularity in 2D is a weakerform of the Thisanomalygivesrisetoanon-Lorentzianshapeofthe
r
a “infraredcatastrophe”in 1D. There, anon-shell fermion spectral function. Second, as the theory is regularized
can emit an infinite number of soft bosons–quanta of by the difference in the ZS and Fermi velocities, which
charge-andspin-densityfluctuations[7],whichgivesrise is small for a weak interaction,Σ is stronglyenhanced
ZS
toapower-lawdivergenceoftheself-energyonthemass- near the mass shell. On the mass shell, ReΣ is of the
ZS
shell. In 2D, this mechanism is weakened but not com- same – U2ω ω –order, as the perturbative contribution.
| |
pletely eliminated (the “memory”about the 1D infrared It has been argued that the non-analytic, perturbative
catastrophe is erased completely only for D > 2). At self-energy ReΣ U2ω ω gives rise to a non-analytic,
∝ | |
a first glance, the breakdown of the perturbation theory T2-term in the specific heat C(T) [6, 8, 9]. Therefore, it
confirms the conjecture that the FL is destroyed in 2D becomes an issue whether the non-perturbative part of
[2]. However, all it really means is that in order to ob- the self-energyalso contributes to the T2 -termin C(T).
tain physically meaningful results the perturbation the- Weshowthatthis isnotthecase,asthe thermodynamic
2
k1 k1 k1 k1 k1 k2
α α α α α β
a)
k
2 β βk2 k2 β βk2 k1α β k2
a) b) c)
FIG.2: Forward(a) andbackscattering(b+c)scatteringpro-
cesses.
b)
Higher-order diagrams contain higher powers of di-
vergent PH bubbles; these bubbles are either explicit,
as in the ring diagrams of Fig. 1, or generated by
integrating out the fermionic energy and momenta.
One can verify that the square-root singularities ac-
cumulate, and the higher-order diagrams for the self-
FIG. 1: Maximally divergent self-energy diagrams to second
(a) and third (b) orders in theinteraction. energy diverge as powers of ∆−1. At the n-th order,
ImΣ unω2(ω/∆)n/2−1θ(ω/∆), where θ(x) is the
F ∝ 0
step-function. By Kramers-Kronigrelation, these singu-
potential is not affected by the mass-shell singularities. larities generate similar power-law divergences in ReΣ .
F
In whatfollows,we presentthe results andoutline the Hence,tofindtheself-energyinD =2onehastore-sum
main steps. Detailed calculations can be found in the the perturbative series even for an infinitesimally small
extended version of this communication [10]. U.
To understand the origin of the mass-shell singularity Thissummationcanbe carriedoutexplicitly forsmall
inarbitrarydimensionD,considertheself-energytosec- U by collecting diagrams with the maximum number of
ondorderintheinteraction(firsttwodiagramsinFig.1), polarization bubbles at each order (to third order, such
and focus on small-angle scattering of fermions with al- diagrams are shown in Fig. 1). The result is
mostparallelmomenta(forwardscattering),asshownin
Fig. 2a. 1
Σ (p)= G(p q) 4U(0) 2U2(0)Π(q)
The imaginary part of the retarded self-energy on the F 2 Zq − " −
mass shell is given by (for ω >0)
U(0) 3U(0)
+ , (2)
0
ImΣF(ω =ǫk) ∝ u20Z−ωdΩZ dQQD−1Z dOD 1−U(0)Π(q) − 1+U(0)Π(q)#
δ(Ω+v Qcosϑ)ImΠ(Ω,Q),(1) where q (Q,Ω) and T d2q/(2π)2....
× F ≡ q··· ≡ Ωm
The last two terms in Eq. (2) correspond to the inter-
R P R
whereu0 U(0)m/2πisthedimensionlesscouplingcon- action in the charge and spin channels, respectively; the
≡
stant, Π(Ω,Q) is the polarization bubble, and dO1 = first two terms contain unaccounted parts of first- and
δ(ϑ)dϑ, dO2 = dϑ, dO3 = sinϑdϑ. ImΠ(Ω,Q) is the second-order diagrams. All terms in Eq. (2) have per-
probabilityamplitudetogenerateaPHpairoffrequency turbative contributions from the PH continuum. In ad-
Ω and momentum Q. For D = 1, the PH continuum dition, the charge part contains a non-perturbative con-
consists of a single line Ω = vFQ and ImΠ(Ω,Q) tribution from the ZS pole in the charge-channel prop-
aδ(Ωno−n-vinFtQeg)r.abAleprsoindguucltaroiftyt,wwohδi-cfhunicstitohnes oinrig(1in) yoifelad∝ns ac2gQat2o)r,.wNheearer cth=e pvol(e1, +[1u−2/U2(0+)Π..(.q))]−is1t∝heuZ20SQv2/el(oΩc2it−y.
F 0
infrared catastrophe [7]. For D > 1, the PH contin- Notice that the quanta of ZS are not free bosons: the
uum is a broad band specified by Ω vFQ, and the residueofthe pole vanishesatQ=0,asitis requiredby
| | ≤
fermion spectral function is softened by angle-averaging. the translational invariance of the system. Substituting
For D = 3, the resulting self-energy is finite on the the ZS propagator into Eq. (2), we obtain for the corre-
mass shell. However, for D = 2 the angle-averaged sponding contribution to the self-energy
fermion spectral function still has a square- root sin-
gvFulQar>ity|:Ω|R, wdθhδe(rΩea−s ImvFΠQ(Ωco,sQϑ))∝∝Ω/(v(FvFQQ−−|Ω|Ω|)|−)11//22hfaosr ReΣZS = u820EωF2 FR(cid:18)∆∆∗(cid:19); (3a)
anothersquare-rootsingularityatthecontinuumbound- u2ω2 ∆
ImΣ = 0 F , (3b)
ary. Merging of the two square-root singularities results ZS 4πE I ∆∗
F (cid:18) (cid:19)
in a logarithmic divergence of ImΣ on the mass shell.
Near the mass shell (∆ ω ), ImΣ ln ∆. where ∆∗ u2ω/2, and functions F are the real and
| |≪| | ∝ | | ≡ 0 R,I
3
4 within the perturbation theory the interaction with the
continuum is indistinguishable from the interaction with
F(x)
3 I the zero sound. The logarithmic divergence in the self-
energyis cut off by a finite curvature ofthe spectrumon
a scale ∆ ω2/W, where W E is the bandwidth.
2 ≃ ≃ F
The power-law divergences in the PH contribution are
∗
cutatthesamescale∆ asintheΣ ,thistimebecause
ZS
1
F (x) an increase of U(0)ImΠ near the boundary of the PH
R continuum reduces the effective interaction for ∆ < ∆∗.
0 However, contrary to the zero-sound contribution, Σ
−1 0 1x 2 3 ∗ PH
is smooth at ∆ ∆ as no Cherenkov-type condition is
≃
involved. Near the mass shell, Σ reduces to
PH
u2 ω ∆
FIG. 3: Scaling functions FI(x) and FR(x) ReΣPH = 0| |
− 4E
F
u2ω2 lnu2 for ω ω ;
imaginary parts of ImΣPH = 4π0EF (cid:26)||ln(|0ω||/EF)| for ||ω||≫≪ωcc,
3 3 1+√1 x
F(x)=(1+ 2x)√1−x+ 2x2ln √ x− , (4) where ωc ≡u20EF .
ThesumΣ +Σ isthetotalcontributiontotheself-
− ZS PH
correspondingly. Plots of F are presented in Fig. 3. energyfromforwardscattering. Another contributionto
R,I
Notice that F (x<0)=0 and F (x>1)=0. the self-energy comes from the processes in which the
I R
Forlargex,i.e., farawayfromthemassshell,F (x) fermionsmoveinitiallyinalmostoppositedirections,and
I
∝
1/√x, and ImΣ u3ω5/2/√∆. This behavior repro- theneitherscatterbysmallangle(Fig.2b)orbackscatter
ZS ∝ 0
duces the result of the perturbation theory. Expanding (Fig. 2c). We refer to both these processesas “backscat-
further in 1/x, one obtains higher powers of 1/∆. How- tering”. The backscattering part of the self-energy, ΣB,
ever, we see from (4) that the emerging singularity at is regular on the mass shell and, for weak interaction,
∆ = 0 is actually cut off at x 1, i.e., at ∆ ∆∗. needs to be evaluated only to second order [6]. Taking
≃ ≃
For small x, FI(x) = 3π2x2/8, and ImΣZS ∆2 for the result for ΣB from Ref. [6] and adding it up with
∆ ∆∗. At ∆ = 0, ImΣZS vanishes. Van∝ishing of the results for ΣZS andΣPH, we obtainfor the totalself-
≪
ImΣ on the mass shell is a Cherenkov-type effect: be- energy on the mass shell
ZS
causethezero-soundvelocityc>v ,anon-shellfermion
F
ω ω
cannot emit a ZS boson. Th∗e ∆2-behavior of ImΣZS in ReΣ(ω) = 8E| | u20− u20+u22kF −u0u2kF
between ∆ = 0 and ∆ = ∆ tracks an increase of the F
phase space available for the emission of ZS bosons. We ω ω (cid:0) (cid:2) (cid:3)(cid:1)
emphasize that as ∆∗ ∝ u20, the crossover at ∆ ≃ ∆∗ = 8E|F| u2kF (u0−u2kF); (5a)
could have not been obtained within the perturbation u2 ω2 E ω
∗ ImΣ(ω) = 0 ln FΦ | | , (5b)
theory. Near∆=∆ ,thederivativeofImΣ(∆)diverges: 2π E ω u2 E
ImΣ (x) 1/√∆ ∆∗. The physical meaning of scale F | | (cid:18) max F(cid:19)
I
∆∗ canbe∝understo−odbynoticingthataCherenkov-type whereu mU(2k )/2πandu max u , u .
2kF ≡ F max ≡ { 0 2kF}
constraint restricts the frequencies of emitted bosons to ThescalingfunctionΦ(x)approachestheconstantvalues
the range 0 < Ω < (∆/∆∗)ω (for ω > 0). This con- of1(1/2)+u (u u )/(2u ),forx andx 0,
straint is relevant for ∆ < ∆∗; for larger ∆, emission of correspondin2gklFy. 2kF− 0 0 →∞ →
bosonswithfrequencies inthe entireinterval,allowedby The first term in the first line of Eq. (5a) is the
the Pauli principle (0<Ω<ω) is possible. non-perturbative contribution from forward scattering,
The real part of the self-energy varies slowly near the whereas the term in square brackets is the backscatter-
mass shell as F (x 1) = 1+x. Right on the mass ing contribution. We see that near the mass shell both
R
≪
shell, ReΣ u2ω2. Notice that ReΣ is of order contributions are of the same order, i.e., the interaction
ZS ∝ 0 ZS
u2, although it is obtained by summing up terms of or- with the zero-sound collective mode modifies significantly
0
der u3 and higher. This enhancement is due to a non- the perturbative result for the self-energy. Forcontactin-
0
perturbative cut-off of the mass-shell divergences in the teraction (u = u ), the zero-sound contribution can-
0 2kF
perturbation theory. celsoutwiththebackscatteringone,sothatthenetReΣ
The remaining contribution to the forward-scattering vanishes at the mass shell. At the same time, the effect
part of the self-energy comes from the PH continuum. of the interaction with ZS on the on-shell ImΣ is rather
It contains a logarithmic divergence to second order and benign: all one has is a smooth crossoverfunction inter-
alsopower-lawdivergencestothirdandhigherorders,as polating between two limiting values of the prefactor in
4
103 formalism, in which the polarization operator Πm =
(m/2π)[1 Ω / (v Q)2+Ω2 ]isregularatanyΩ
− −| m| F m m
andQ. Asaresult,the MatsubaraseriesforΞconverges
102 p
x) for a weak, short-rangeinteraction, and there is no need
A( forre-summationoftheperturbationtheoryforΞ. Eval-
E F 101 uating Ξ to second-order in U yields
2 u
π 2 100 δC(T)= u2+u2 u u 3mζ(3) T2. (8)
− 0 2kF − 0 2kF π E
F
(cid:0) (cid:1)
10−1 This result coincides with C(T) obtained in Ref. [6] by
−2 −1 0 1 2
x finding the perturbative, backscattering part of the self-
energyfirstandthenusingtherelationbetweenC(T)and
Σ[1]. Theagreementbetweenthetworesultsshowsthat
non-parturbative,forward-scatteringself-energydoesnot
FIG.4: Alog-plot of thespectral function A(ω,k) as afunc-
tion of x = 2(ω−ǫk)/u2ω for fixed ω. Lω = 2 and γ = 0.1 affect C(T).
Forcompleteness, we alsoverifiedexplicitly thatthere
A kink at x = 1 is a signature of the interaction with the
zero-sound mode. is no contribution to C(T) from forward scattering.
To this end, one can use the relation between C(T)
and the Green’s function [1] which, to first order in
a familiar ω2ln ω -dependence. This is the consequence (ω ǫ )−1Σ(ω,k), reads
| | k
of the factthat the non-perturbative,ZS contributionto −
ImΣ vanishes on the mass shell. ∞ ∞
mT ∂ 1 ∂n
0
At the same time, away from the mass shell, the δC(T)= dǫk dωω (9)
π ∂T"T −∞ −∞ ∂ω
full self-energy and the the spectral function A(ω,k) = Z Z
( 1/π)ImG(ω,k) are affected by the non-monotonic 1 1
− δ(ω ǫ )ReΣR(ω,k) ImΣR(ω,k) .
variation of ImΣ with ∆ near a threshold for emis- k
ZS ×( − − πPω ǫk )#
sion of ZS bosons. We consider a setup, when ω is fixed −
and the spectral function is measured as a function of Here n (ω) is the Fermi function. It is crucial that both
∗ 0
k. This is equivalent to varying x ∆/∆ at fixed ω. ReΣR(ω,k) and ImΣR(ω,k) are present in Eq. (9). The
≡
Nearthe massshell,the spectralfunctionconsistsoftwo forwardscatteringselfenergy,Σ =Σ +Σ ,depends
F ZS PH
terms A(x)=A (x)+A (x), where ∗
qp ZS on k due to the presence of a non-perturbative scale ∆ ,
and therefore ImΣ yields a finite contribution to C(T).
1 L F
ω
Aqp(x)= π2u2E x2+γ2 (6) Substituting the results for ΣZS and ΣPH into Eq. (9),
F we find that the forward scattering contributions to the
is a quasi-particle contribution and specific heat from ReΣF and ImΣF cancel each other;
hence, there is no contribution to C(T) from forward
FI(x) x2 γ2 scattering.
AZS = π2u2EF (x2+−γ2)2 (7) We acknowledge stimulating discussions with I.
Aleiner,B.Altshuler,A.Andreev,W.Metzner,A.Millis,
isa ZScontribution. We introducedL ln E /u2 ω
ω ≡ F | | and C. Pepin. The researchhas been supported by NSF
and γ = ω L /(2πE ), and assumed for simplicity that
| | ω F (cid:0) (cid:1) DMR 0240238 and Condensed Matter Theory Center at
u u u. The ZS contribution A (x) has a sharp
0 ≡ 2kF ≡ ZS UMD (A. V. Ch.), NSF DMR-0308377 (D. L. M.), and
maximum at x = 1, where function F (x) in Fig. 3 has
I NSF DMR-0237296 (L. I. G.).
a peak. This maximum gives rise to a kink in total A(x)
(see Fig.4). A similar considerationshows that the kink
is present also for the Coulomb interaction. This kink
should be detectable in photoemission experiments on
[*] Permanent address.
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[3] C.Castellani,C.DiCastro,andW.Metzner,Phys.Rev.
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Lett. 72, 316 (1994).
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[4] H.FukuyamaandM.Ogata,J.Phys.Soc.Jpn.63,3923
inition, C(T) = T∂2Ξ/∂T2, and expand the ther-
(1995).
−
modynamic potential Ξ in powers of U. The thermo- [5] C. Halboth and W. Metzner, Phys. Rev. B 57, 8873
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5
[6] a) A. V. Chubukov and D. L. Maslov, Phys. Rev. B 68, L. I. Glazman, cond-mat/0412283.
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[7] Yu.A.Bychkov,L.P.Gor’kov,andI.E.Dzyaloshisnkii, fornon-lineartermsinC(T).However,itcan beverified
Zh. Eksp. Teor. Fiz. 50, 738 (1966) [Sov. Phys. JETP by adirect comparison with thegeneral Luttinger-Ward
23, 489]. formula that the T2-term is correctly described by (9)
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(1993). zero-temperature one.
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