Optical conductivity of a two-dimensional metal at the onset of spin-density-wave order

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by  Andrey V. Chubukov

Optical conductivity of a two-dimensional metal
at the onset of spin-density-wave order
Andrey V. Chubukova), Dmitrii L. Maslovb), and Vladimir I. Yudsonc)
a)Department of Physics, University of Wisconsin-Madison, 1150 Univ. Ave., Madison, WI 53706-1390
b)Department of Physics, University of Florida, P. O. Box 118440, Gainesville, FL 32611-8440
c)Institute for Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow region, 142190, Russia
(Dated: January 8, 2014)
We consider the optical conductivity of a clean two-dimensional metal near a quantum spin-
density-wavetransition. Criticalmagneticfluctuationsareknowntodestroyfermioniccoherenceat
“hotspots”oftheFermisurfacebutcoherentquasiparticlessurviveintherestoftheFermisurface.
4 AlargepartoftheFermisurfaceisnotreally“cold”butrather“lukewarm”inasensethatcoherent
1 quasiparticles in that part survive
0 but are strongly renormalized compared to the non-interacting case. We discuss the self-energy
2 of lukewarm fermions and their contribution to the optical conductivity, σ(Ω), focusing specifically
on scattering off composite bosons made of two critical magnetic fluctuations. Recent study [S.A.
n
Hartnolletal.,Phys. Rev. B84,125115(2011)]foundthatcompositescatteringgivesthestrongest
a
contributiontotheself-energyoflukewarmfermionsandsuggestedthatthismaygiverisetoanon-
J
Fermi liquid behavior of the optical conductivity at the lowest frequencies. We show that the most
7
singular term in the conductivity coming from self-energy insertions into the conductivity bubble,

σ(cid:48)(Ω)∝ln3Ω/Ω1/3,iscanceledoutbythevertex-correctionandAslamazov-Larkindiagrams. How-
]
l ever, the cancelation does not hold beyond logarithmic accuracy, and the remaining conductivity
e stilldivergesas1/Ω1/3. Wefurtherarguethatthe1/Ω1/3 behaviorholdsonlyatasymptoticallylow
-
r frequencies, well inside the frequency range affected by superconductivity. At larger Ω, up to fre-
st quenciesabovetheFermienergy,σ(cid:48)(Ω)scalesas1/Ω,whichisreminiscentofthebehaviorobserved
. in the superconducting cuprates.
t
a
m
I. INTRODUCTION port effective mass” (m is bare electron mass). If the
-
d fermionicself-energyΣ=Σ(cid:48)+iΣ(cid:48)(cid:48) hasastrongerdepen-
n Understanding the behavior of fermions near a dence on the frequency than on the momentum across
co quantum-critical point (QCP) remains one of the most the Fermi sur(cid:16)face (FS(cid:17)),−1m∗tr(Ω)/m is equal to 1/Z(Ω),
[ challengingproblemsinthephysicsofstronglycorrelated where Z = 1+ ∂Σ(cid:48) is the quasiparticle residue.
materials. Asonepossiblemanifestationofquantumcrit- ∂Ω
1 The transport scattering rate 1/τ (Ω) is proportional
icality, the resistivity ρ(T) of optimally-doped cuprates, tr
v to Σ(cid:48)(cid:48)(Ω), but may be much smaller than the latter
Fe-pnictides,heavy-fermioncompounds,andothermate-
1 if the dominant scattering mechanism involves small
6 rials exhibits a linear-in-T behavior over a wide range of
momentum transfers. For an ordinary FL with inter-
4 temperatures1–3 insteadoftheT2 behavior,expectedfor
actions roughly the same at all momentum transfers,
1 a Fermi liquid (FL) with umklapp scattering4. Another
Σ(cid:48)(cid:48)(Ω,T) ∼ 1/τ (Ω,T) ∝ max{Ω2,T2} and Z = const
. type of the non-FL (NFL) behavior, ρ(T) ∝ Tb with tr
1 (Ref.9). Equation(1.1)thenpredictsthatσ(cid:48)(Ω)=const
0 b ≈ 3/2, has been observed near the end point of the
at low frequencies, when ZΣ(cid:48)(cid:48)(Ω) (cid:28) Ω. Instead, the
4 superconducting phase in the hole- and electron-doped
measuredσ(cid:48)(Ω)ofmanystrongly-correlatedmaterialsde-
1 cuprates,5,6 whereas ρ(T) ∝ Tc with c ≈ 5/3 has been
: observed near ferromagnetic criticality in a number of pends strongly on the frequency, often as a power-law
v σ(cid:48)(Ω) ∝ 1/Ωd with positive exponent d, meaning that
i three-dimensional itinerant ferromagnets.7
X σ(cid:48)(Ω) increases as Ω gets smaller. For example, σ(cid:48)(Ω)
In addition to the dc resistivity, the optical conductiv-
of several underdoped and optimally doped cuprates in
r ity provides useful information about the energy depen-
a the(x,Ω)domainoutsidethepseudogapphase(xstands
dencesofthescatteringrateandeffectivemass. Thereal
for doping) was described by a power-law form with ei-
part of of the conductivity σ(cid:48)(Ω), measured at Ω (cid:29) T,
ther d ≈ 1 (Ref. 10) in a wide frequency range, roughly
can be described by a “generalized Drude formula”8
from 100 meV to about 1 eV, or with d ≈ 0.7 (Ref. 11)
Ω2 1/τ (Ω) and d = 0.65 (Ref.12 and 13) in the intermediate fre-
σ(cid:48)(Ω)= p tr , (1.1) quency range Ω∼100−500 meV. Likewise, σ(cid:48)(Ω) of the
4π (cid:104)Ωm∗tmr(Ω)(cid:105)2+(cid:16)τtr1(Ω)(cid:17)2 r1u6t)h,eansawteesllSorfRthueOh3e(liRmeaf.g1n4e)tManndSCi(aaRtuaOm3b(ieRnetfsp.r1es5suarned,
where Ω is the effective plasma frequency, τ (Ω) is Ref. 17), follows a power-law form with d≈1/2.
p tr
the transport scattering time, and m∗(Ω) is the “trans-
tr
2
TABLE I. List of notations
Notation Meaning Relation to other parameters
g¯ coupling constant of the spin-fermion model (in units of energy) −(cid:48)(cid:48)−
γ Landau damping coefficient γ =4g¯/πv2
F
k arbitrary point on the Fermi surface
F
k location of the hot spot
h.s.
q SDW wavevector q =(π,π)
π π
k component of k along the normal to the Fermi surface
⊥
δk distance from the hot spot along the Fermi surface
m∗ effective mass defined by Eq. (2.23)
E∗ effective Fermi energy E =m∗v2/2
F F F
K (2+1) momentum K =(k,Ω)
K (2+1) momentum on the Fermi surface K =(k ,Ω)
F F F
z dynamic scaling exponent
Z =Z quasiparticle residue Eq. (2.12)
kF δk
Γ(P,K;P(cid:48),K(cid:48))≡Γ(P,K;Q) composite vertex Eq. (2.18)
Σ self-energy due to scattering by a single SDW fluctuation Eq. (2.9)
qπ
Σ one-loop self-energy due to scattering by composite bosons Eqs. (2.20) and (2.21)
comp1
two-loop self-energy in the 2D regime Eqs. (2.26) and (2.28)
Σ
comp2 two-loop self-energy in the 1D regime Eqs. (2.35) and (2.36)
Σ three-loop self-energy
comp3
Ω lower boundary of the 1/Ω behavior of the conductivity g¯2/E
min F
Ω upper boundary of the 1/Ω behavior of the conductivity E2/g¯
max F
Ω crossover between FL and NFL forms of the self-energy (v δk)2/g¯
b F
k¯ dimensionless distance from the hot spot along the Fermi surface k¯=v δk/g¯
F
Ω¯ dimensionless frequency Ω¯ =Ω/g¯
Ω¯ dimensionless form of Ωmin Ω¯ /g¯=g¯/E
min min F
Ω¯ dimensionless form of Ωmax Ω¯ /g¯=(g¯/E )2
max max F
Ω¯ dimensionless form of Ωb Ω /g¯=k¯2
b b
σ(cid:48)(Ω) real part of the optical conductivity at T =0
σ(cid:48) (Ω) σ(cid:48)(Ω) obtained by taking into account self-energy insertions only
Σ
Among the various deviations from the FL scenario, the tivity. For a finite-q QCP, only a subset of points on
linear scaling of ρ(T) with T and concomitant 1/Ω scal- the FS around hot spots is affected by criticality, while
ing of σ(cid:48)(Ω) are considered as the most ubiquitous and fermions on the rest of the FS preserve a regular FL be-
universal ones.2,3 As the temperature and frequency de- havior. Becausebothresistivityandopticalconductivity
pendencesoftheconductivityarelikelytooriginatefrom are obtained by averaging over the Fermi surface, a NFL
thesamescatteringmechanism,thecombinationofthese contribution from hot fermions is short-circuited by the
two scalings imposes some important constraints on the contributions from other parts of the FS, i.e., the largest
form of the fermionic self-energy. contributions to ρ(T) and σ(cid:48)(Ω) come from outside the
hot regions (the “Hlubina-Rice conundrum”19).
These constraints form the basis of the phenomeno-
IntheMFLphenomenology, thisproblemisby-passed
logical “marginal FL” (MFL) theory,18 which stipulates
by assuming that the critical bosonic field is purely lo-
that Σ(cid:48)(cid:48)(Ω,T) scales as Ω or T (whichever is larger) at
cal, i.e., that the corresponding susceptibility does not
any point on the FS, and also that Σ(cid:48)(cid:48)(Ω,T) is compa-
depend on q (Ref. 20) and diverges at the QCP for all
rable to 1/τ (Ω,T). However, attempts to derive the
tr momenta. Then,ononehand,typical1/τ isofthesame
MFL form of 1/τ (Ω,T) microscopically, in some model tr
tr order as Σ(cid:48)(cid:48)(Ω), on the other, every point on the FS is
for interacting electrons near a QCP in 2D, have been
hot. However, a scenario in which a bosonic susceptibil-
largely unsuccessful. Problems arise both for Pomer-
ity softens simultaneously at all momenta is very special
anchuk (q = 0) and density-wave (finite q) types of a
and not likely to be applicable to all systems in which
QCP, in either charge or spin channel. For a q =0 QCP,
a linear-in-T resistivity and 1/Ω scaling of the optical
thefactthatcriticalscatteringinvolvessmallmomentum
conductivity have been observed.
transfersimpliesthat1/τ issmallerthanΣ(cid:48)(cid:48)(Ω,T),and
tr
not only differs from it in magnitude but also scales dif- Analternativeroute,whichwewillfollowinthispaper,
ferently with Ω and T, so that a MFL behavior of the is to revisit the “conventional” theory of a density-wave
self-energy does not translate into that of the conduc- instabilitywithsoftfluctuationspeakednearaparticular
3
q, and to analyze in more detail contributions to the lation to those by HHMS.
resistivityandopticalconductivitycomingfromfermions
outside the hot regions.
An important step in this direction has recently been A. Optical conductivity near a QCP: summary of
made by Hartnoll, Hofman, Metlitski, and Sachdev prior results
(HHMS) in Ref. 21. They considered the optical con-
ductivity of a 2D metal near a spin-density-wave (SDW) 1. Pomeranchuk QCP (q=0)
instability with ordering wavevector q = (π,π) and fo-
π
cusedprimarilyonthecontributiontoσ(Ω)comingfrom A Pomeranchuk-like QCP separates two spatially uni-
fermions in “lukewarm” regions, located just outside the formphases,e.g.,aparamagnetandferromagnet. Thisis
hot regions. Lukewarm fermions behave as FL quasipar- acontinuousphasetransition, andthecorrelationlength
ticlesevenrightattheQCP,buttheirresidueissmalland oflong-wavelengthorder-parameterfluctuationsdiverges
dependsonthedistancetothehotspotinasingularway. atthecriticalpoint. Scatteringoffermionsbythesefluc-
TheleadingcontributiontoΣ(cid:48)(cid:48)(Ω)oflukewarmfermions tuationsisstrong, buttypicalmomentumtransfersq˜are
comesnotfromdirectscatteringbyq ,astheinitialand small compared to k . For a generic FS,22 the opti-
π F
finalstatesofthisprocesscannotbesimultaneouslynear cal conductivity is finite even in the absence of umklapp
theFS,butfromacompositescatteringprocesswhichin- scattering and disorder, and is described by Eq. (1.1)
volves two critical bosonic fields. Scattering by one field with 1/τ (Ω) that differs from Σ(cid:48)(cid:48)(Ω) by the “transport
tr
takesafermionoutoftheFS,whilescatteringbyanother factor” (q˜/k )2. Critical scaling implies that q˜∝ Ω2/z,
F
bringsitbacktotheFSand,furthermore,tothevicinity where z = 3 is the dynamical exponent for a Pomer-
ofitsoriginallocation. Theself-energyΣ(cid:48)(cid:48)(Ω)fromcom- anchuktransition.23 Asaresult,scalingoftheconductiv-
positescatteringhasaFLformbutdependsinasingular ity is different from that of Σ(cid:48)(cid:48), both in 3D and 2D.22–25
way on the distance to the nearest hot spot. Substitut- In3D,bothΣ(cid:48)(cid:48)(Ω)∝ΩD/z andm∗/m=1/Z =1/|lnΩ|
tr
ing this self-energy into the conductivity bubble, HHMS fit the MFL scheme, while the transport scattering rate
obtained a NFL form of the optical conductivity at the 1/τ ∼Σ(cid:48)(cid:48)(Ω)(q˜/k )2 ∝Σ(cid:48)(cid:48)(Ω)Ω2/z scalesasΩ5/3. Then
tr F
smallestΩ: σ(cid:48)(Ω)∝1/Ω1/3 totwoloop-order. (Hereand theconductivityσ(cid:48)(Ω)∼Σ(cid:48)(cid:48)(Ω)Z2Ωz2−2 scalesas1/Ω1/3
in the rest of the paper, Ω is assumed to be positive so (modulo a logarithm), which is very different from the
thatallnon-analyticfunctionsofΩaretobeunderstood MFL, 1/Ω form. In 2D, both Σ(cid:48)(Ω) and Σ(cid:48)(cid:48)(Ω) scale
as real.) asΩ2/3 (modulologarithmicrenormalizationsbyhigher-
HHMS further argued that the self-energy from com- order processes26,27). In this situation, the quasiparticle
posite scattering comes predominantly from 2kF pro- Z factor is frequency dependent and scales as Z(Ω) ≈
cesses (two incoming particles have nearly opposite mo- (∂Σ(cid:48)/∂Ω)−1 ∝ Ω1/3 for frequencies below some charac-
menta),inwhichcasevertexcorrectionsdonotcancelthe teristic scale. As a result, σ(cid:48)(Ω) ∼ Σ(cid:48)(cid:48)(Ω)Z2(Ω)Ωz2−2
self-energy contribution, hence the final result for σ(cid:48)(Ω) tends to a constant value in the low-frequency limit,
shouldbethesameasobtainedsimplybyreplacing1/τtr as in an ordinary FL. At higher frequencies, when the
by Σ(cid:48)(cid:48). Z factor is almost equal to unity, σ(cid:48)(Ω) behaves as
Inthispaper,wereporttworesults. First,weanalyzed Σ(cid:48)(cid:48)(Ω)Ω2/z/Ω2 ∝ Ω−2/3. At even higher frequencies,
the interplay between the self-energy, vertex-correction, where q˜∼k , σ(cid:48)(Ω) scales as Ω−4/3.
F
andAslamazov-Larkindiagramsandfoundthatthelead-
ing contribution to σ(cid:48)(Ω) is canceled between different
diagrams. In this respect our result differs from that 2. Density-wave QCP (finite q)
byHHMS.Wefound,however,thatthecancelationdoes
notholdbeyondthelogarithmicaccuracy,andevenafter A density-wave QCP separates a uniform disordered
cancelations σ(cid:48)(Ω) still diverges at Ω→0 as 1/Ω1/3. phase and a spatially modulated ordered phase, e.g.,
Second, we found that, if the ratio of the spin-fermion a paramagnet and spin-density wave (SDW). For defi-
coupling to the Fermi energy is treated as a small pa- niteness, we consider an SDW with ordering momentum
rameter of the theory, the 1/Ω1/3 behavior holds only at q = (π,π,π) in 3D and q = (π,π) in 2D (the lattice
π π
asymptotically low frequencies, below some scale Ωmin constantissettounity). Becauseonlyasubsetofpoints
which is parametrically smaller than the scale of the on the FS is connected by q , critical fluctuations affect
π
d−wave superconducting transition temperature Tc. At mostly the fermions near such “hot lines” (in 3D) or hot
higher frequencies, the optical conductivity behaves in a spots (in 2D); see Fig. 1 for the 2D case. On the rest
MFL way: σ(cid:48)(Ω) ∝ 1/Ω. This last behavior holds up to of the FS, the interaction mediated by critical fluctua-
frequencies on the order of the fermionic bandwidth. tionstransfersafermionfromaFSpointk tok +q ,
F F π
In the next subsection, we present a brief summary of which is away from the FS. The energy of the final state
the theoretical results for the optical conductivity near (measured from the Fermi level) |ε | is finite, hence
kF+qπ
both q = 0 and finite q critical points, obtained with- atfrequenciessmallerthanthisscalequantumcriticality
out taking into account composite scattering. Then, in doesnotplayarole,andtheself-energyretainsthesame
Sec. IB, we summarize our results and describe their re- FL form as away from the QCP.
4
In 3D, the effective interaction between hot fermions,
1
mediated by critical fluctuations, yields Σ(cid:48)(cid:48)(Ω) ∝ Ω and
m∗/m = 1/Z = 1/|lnΩ|, same as for a q = 0 QCP.
tr
Because q is a large momentum transfer, 1/τ (Ω) is
π tr
the same as Σ(cid:48)(cid:48)(Ω). However the width of the hot re-
gion (the distance from the hot line where Σ(cid:48)(cid:48)(Ω) ∝ Ω) qπ
by itself scales as Ω1/2, hence the conductivity σ(cid:48)(Ω) ∝ 2
2
Σ(cid:48)(cid:48)(Ω)Z2(Ω)Ω1/2/Ω2 scales as 1/Ω1/2, up to logarithmic
corrections.
In 2D, the effective interaction mediated by critical
fluctuations destroys FL quasiparticles in hot regions,
whichismanifestedbyaNFLformoftheself-energy. At
√ √
1
one-loop level, Σ(cid:48)(cid:48)(Ω) ∼ Σ(cid:48)(Ω) ∝ Ω, and Z(Ω) ∝ Ω.
ThewidthofthehotregionscalesasΩ1/2, asin3D,and
the contribution to the conductivity from hot fermions
FIG. 1. A two-dimensional Fermi surface with hot spots de-
tends to a constant value at vanishing Ω. Beyond one-
noted by 1, ¯1, 2, and ¯2. Hot spots 1 and 2 are connected
loop level, this contribution is further reduced by vertex
by the spin-density-waver ordering vector q = (π,π). Hot
corrections21 andscalesasΩa witha>0,i.e.,itvanishes spots ¯1 and ¯2 are the mirror images of hotπspots 1 and 2,
at Ω=0.
correspondingly.
For Ω larger than the maximum value of |ε | the
kF+qπ
whole FS is hot, i.e., Σ(cid:48)(cid:48)(Ω) is approximately the same
at all points on the FS. Parametrically, this holds only
Hotfermionshavethelargestself-energybutthesmall-
at rather high energies, larger than the bandwidth (see
estZ ,andalsothewidthofthehotregionshrinksasΩ
Sec.IIIAbelow),butthescalemaybereducedbyasmall kF
decreases. To two-loop order, the contribution from hot
numerical prefactor. If the self-energy still obeys the fermions to the conductivity σ(cid:48)(Ω → 0) is a frequency-
quantum-criticalforminthisregime,σ(cid:48)(Ω)scalesas1/Ω
√ independent constant, which simply adds up to FL-like,
in3D,andeitheras1/ Ωor1/Ω3/2 in√2D,dependingon constant contributions from the cold regions. Beyond
whetherZ inthisregimestillscalesas Ωorhasalready two-loop order this contribution is further reduced by
saturated at Z =1.28 vertex corrections.21
The issue we considered, following HHMS, is whether
fermions, located away from the hot regions, can give
rise to a NFL behavior of the optical conductivity at
an SDW instability. Phenomenological models that take
into account the interplay between the hot and cold re-
gions in various transport phenomena have been consid-
ered by several many authors.29 We considered this in-
B. Summary of the results of this paper
terplay within a microscopic theory.
At first glance, the interaction between fermions lo-
Following earlier work,21,27,28 we adopt the spin-
cated away from the hot regions is unable to give rise to
fermion model as a microscopic low-energy theory for aNFLbehaviorofσ(cid:48)(Ω). Indeed, theinteractionpeaked
a system of interacting fermions at an SDW instabil-
at q moves a fermion away from the FS, into a region
π
ity. This model has two characteristic energy scales –
where its energy (measured from the Fermi level) is fi-
the Fermi energy E ∼ v k and the effective spin-
F F F nite. One could then expect quantum criticality to be
mediated four-fermion interaction g¯. To decouple the irrelevant, and the corresponding contribution to σ(cid:48)(Ω)
low-andhigh-energysectorsofthetheory,wechoosethe
to approach a constant value at T = 0, as in an ordi-
ratio g¯/E to be small. We found that in this case the
F nary FL. This reasoning is, however, oversimplified be-
whole FS becomes hot only at Ω>Ω ≡E2/g¯>E .
max F F cause, besides processes with momentum transfer qπ,
At such high energies, results of the low-energy theory
there also exist composite processes, which involve an
can hardly be valid. To obtain a true low-energy be-
even number of critical bosonic fields. These composite
havior, one then has to consider the situation when only
processeshavebeenintroducedinRef.27andconsidered
some parts of the FS are hot while the rest of it is cold.
in detail by HHMS. (For more recent work, see Ref. 30.)
In this case, σ(cid:48)(Ω) is given by an average over the FS:
HHMS introduced a composite boson, with a propaga-
(cid:73) 1/τ (k ,Ω) tor made from two critical propagators of the primary
σ(cid:48)(Ω)∝ dkF(Ω/Z )2+tr[1/Fτ (k ,Ω)]2, (1.2) bosonic fields and two Green’s functions of intermediate
kF tr F fermions (see Fig. 4). They found (and we confirmed
where k indicates a point on the FS and Z = their result) that the imaginary part of the fermionic
F kF
m/m∗(k ,Ω) depends on the position of k relative to self-energy from “one-loop” composite scattering (dia-
tr F F
the nearest hot spot. gram a in Fig. 5) scales as Σ(cid:48)(cid:48) (Ω) ∝ Ω3/2 for any
comp
1
5
point on the FS. This singular self-energy, however, does behavior actually extends to even higher frequencies, up
not crucially affect σ(cid:48)(Ω) because the Ω3/2 term comes to Ω ≡ E2/g¯ > E , at which scale the whole FS
max F F
fromsmall-momentumscattering,and1/τ (Ω)issmaller becomes hot. In the range g¯ < Ω < Ω , the domi-
tr max
thanΣ(cid:48)(cid:48)(Ω)bypowerofΩ,whichmakesthiscontribution nant contribution to conductivity comes from direct q
π
smaller than a regular FL term. scattering.
A more interesting contribution to the self-energy Whether σ(cid:48) (Ω) gives a good approximation for the
Σ
comesfrom“two-loop”compositescatteringoflukewarm actual optical conductivity depends on the interplay be-
fermions (Fig. 6). To be specific, a fermion located away tween self-energy and vertex-correction insertions into
from a hot spot by distance δk along the FS is classified the conductivity bubble. HHMS argued that the self-
as “lukewarm”, if δk is large enough for the self-energy energy and vertex-correction diagrams for 2kF scatter-
toassumeaFLformwithΣ∼O(Ω)+iO(Ω2), yetsmall ing add up rather than cancel each other because the
enough such that ∂Σ(cid:48)(δk,Ω)/∂Ω > 1. The characteris- current vertices in the self-energy diagram are near the
tic frequency separating the hot and lukewarm regimes same hot spot, while the current vertices in the vertex-
is Ω ≡(v δk)2/g¯, and the boundary between lukewarm correction diagram are near the opposite hot spots. We
b F
and cold regimes is |δk| ∼ g¯/v ; the lukewarm behavior obtained a somewhat different result. Namely, we found
F
holds for k (Ωg¯/E2)1/2 <|δk|<g¯/v . that that the ln3Ω/Ω1/3 contributions to σ(cid:48)(Ω) are can-
F F F
celed within each of the two groups of diagrams. The
HHMS put a special emphasis on a particular two-
firstgroupcontainstheself-energyandvertex-correction
loop composite process (Fig. 6) in which intermediate
insertions (diagrams A and B in Fig. 12), while the sec-
fermionsbelongto“lukewarm”regionsnearoppositehot
spots located at k and −k , e.g., spots 1 and ¯1 in ond one contains two Aslamazov-Larkin–type diagrams
hs hs
(diagrams C and D in Fig. 13). HHMS considered only
Fig. 1. For a system with a circular FS, such a process
one diagram in each group, and, consequently, did not
is often called a “2k process” and we will use this ter-
F
minology here.31 find the cancelation.
We found, however, that the cancelation does not
Our result for the self-energy of a lukewarm fermion
holdbeyondthelogarithmicaccuracy: aftercancelations,
frombothtwo-loopforwardand2k compositeprocesses
F σ(cid:48)(Ω) still diverges at vanishing frequency as σ(cid:48)(Ω) ∝
is FL-like at the smallest Ω, i.e., Σ(cid:48)(cid:48) scales as Ω2; how-
1/Ω1/3. We also found that at higher frequencies, Ω >
ever,theprefactoroftheΩ2termdependsstronglyonδk:
Σ(cid:48)c(cid:48)omp2(Ω) ∝ (cid:2)g¯2Ω2EF/(vFδk)4(cid:3)ln3 ΩΩb. This is para- Ωacmtoinr,btuhtenvoetrttehxecfuonrrcetciotinoanlsfochrmanogfetthhee1n/uΩmsecrailcianlg,pir.eef.-,
metrically (by a factor of k /|δk|) larger than Σ(cid:48)(cid:48) ∝
F qπ the the final result for the conductivity in this range is
g¯2Ω2/(vFδk)3 from direct scattering by qπ.27,28,32 The σ(cid:48)(Ω)∼σ(cid:48) (Ω)∝1/Ω.
HHMSresultforΣ(cid:48)(cid:48) (Ω)differsfromours–theyfound Σ
comp Theoutcomeofouranalysisisthatcompositescatter-
2
that Σ(cid:48)(cid:48) (Ω) is the same as Σ(cid:48)(cid:48) , up to a logarithmic ingoflukewarmfermionsdoesgiverisetoaNFLbehavior
comp2 qπ
factor. The difference is due to the fact that we consid- oftheopticalconductivityatanSDWinstability,namely
ered a 2D FS with finite curvature, of order 1/k , while
F (cid:26)
HHMS assumed that the curvature is zero at the bare σ(cid:48)(Ω)∝ Ω−1/3, forΩ<Ωmin; (1.3)
level but generated dynamically by the interaction. The Ω−1, forΩmin <Ω<g¯.
Ω2 behavior of Σ(cid:48)(cid:48) holds up to a characteristic scale
comp2 The 1/Ω behavior furthermore extends to even higher
Ω ≡ (v δk)3/(g¯E ) ∼ Ω (δk/k ) < Ω (modulo loga-
b1 F F b F b frequencies,uptoΩ . Atg¯<Ω<Ω itcomesfrom
rithms). InthefrequencyintervalΩ <Ω<Ω ,thecur- max max
b1 b hot fermions. These are the key results of this paper.
vatureoftheFSbecomesirrelevant, andcompositescat-
The rest of the paper is organized as follows. Sec-
tering becomes effectively a 1D process. In this regime,
tionIIisdevotedtothefermionicself-energy. Webriefly
we confirmed the HHMS result that the self-energy ac-
review the spin-fermion model near an SDW transition
quires a MFL-like form with Σ(cid:48)(cid:48) (Ω)∝g¯Ω/(v δk).
comp F in Sec. IIA and discuss the fermionic self-energy for hot,
2
WeextendedtheanalysistotheregionoflargerΩ>g¯
lukewarm,andcoldfermionsduetolarge-q scatteringby
and v |δk|>g¯, not considered by HHMS, and obtained
F a primary bosonic field in Sec. IIB. In Sec. IIC, we con-
thefullformsoftheself-energyduetoone-loopandtwo-
sider small-q scattering by a composite field made from
loop composite scattering (see Figs. 7 and 8).
twoprimaryfields. InSec.IID,wesummarizetheresults
Asubstitutionofthesefullformsintothecurrentbub- for the self-energy to two-loop order and present the hi-
ble gives the “self-energy” contribution to the optical erarchies of crossovers in Σ as a function of Ω and δk.
conductivity, σ(cid:48) (Ω), which does not take into account In Sec. IIF, we analyze the effect of higher loop correc-
Σ
vertex corrections. We found that σ(cid:48) (Ω) ∝ ln3Ω/Ω1/3 tions. Section III is devoted to the optical conductivity.
Σ
atfrequenciesbelowthelowestenergyscaleofthemodel, InSec.IIIA,weconsiderthecontributiontotheconduc-
i.e., for Ω < Ω ≡ g¯2/E < g¯. The exponent 1/3 tivityobtainedbyinsertingthefermionicself-energyinto
min F
coincides with the HHMS result to two-loop order. At theconductivitybubble. InSec.IIIC2,weshowtheself-
higher frequencies, Ω < Ω < g¯, we found that two- energy and vertex-correction diagrams mutually cancel
min
loop composite scattering of lukewarm fermions give rise eachotherifoneneglectsthevariationsofthequasiparti-
to a MFL-like conductivity: σ(cid:48) (Ω) ∝ 1/Ω. The 1/Ω cleresidueovertheFS.InSec.IIIC3,weshow,however,
Σ
6
that if this variation is taken account, the NFL power- of the self-energy on θ is not crucial, as long as θ is not
law singularities in the conductivity [Eq. (1.3)] survive too small and, to shorten the formulas below, we assume
after cancelations between the self-energy and vertex- thatθ =π/2(i.e.,v =v ). Thisassumptionholdswhen
x y
correction diagrams. In Sec. IIIB, we explain how this the hot spots are located close to (0,π) and symmetry-
result can be understood in the framework of the semi- related points.
classical Boltzmann equation. Our conclusions are pre- For a FS of the type shown in Fig. 1, the fermionic
sented Sec. IV. For the readers convenience the list of bandwidthisofthesameorderastheFermienergyE ∼
F
notations is given in Table I. v k , where k is the Fermi momentum averaged over
F F F
the FS. At the QCP, where ξ−1 = 0, we then have only
two relevant energy scales: E and g¯ (we remind that
F
II. SPIN-FERMION MODEL AND FERMIONIC g¯ is chosen to be smaller than E ). We will see that
F
SELF-ENERGY the frequency dependence of the conductivity exhibits
crossovers at two energies:
A. Spin-fermion model
g¯2 E2
Ω ≡ and Ω ≡ F. (2.4)
min E max g¯
The spin-fermion model has been discussed several F
times in recent literature,21,27,28 so we will be brief. The
The hierarchy of energy scales in the model is then
model assumes that the low-energy physics near a SDW
instability in a 2D metal can be described via approx- Ω <g¯<E <Ω . (2.5)
min F max
imating the fully renormalized fermion-fermion interac-
tion by an effective interaction in the spin channel. This Here and in the rest of the paper, we use weak inequali-
interaction is mediated by nearly-gapless antiferromag- ties(<and>)insteadofstrongones((cid:28)and(cid:29))because
netic spin fluctuations: the actual crossovers are determined not only by param-
eters but also by numbers, which we do not attempt to
(cid:88)
H = V(k−p)c† c† c c (cid:126)σ ·(cid:126)σ , compute in this paper. Also, ∼ means “equal in order of
int k,α k(cid:48)β k+k(cid:48)−p,γ p,δ αδ βγ
magnitude” and ≈ means “approximately equal”.
k...p(cid:48)
(2.1)
with
B. Self-energy due to q scattering
π
V(k−p)=g¯χ(k−p), (2.2)
1. One-loop order
whereg¯istheeffectivecoupling(inunitsofenergy),and
1 First,weconsiderthefermionicself-energyduetoscat-
χ(q)=χ(q,Ω=0)= , (2.3)
tering mediated by a single spin fluctuation peaked at
ξ−2+|q−q |2
π
q =(π,π). A self-consistent treatment of the fermionic
π
is proportional to the static spin susceptibility peaked and bosonic self-energies shows27,28,32 that close to criti-
near q . cality,i.e.,whenξg¯/v islargerthanunity,thefermionic
π F
Theinputparametersofthemodelareg¯,thespincor- self-energydependsmuchstrongeronthefrequencythan
relation length ξ, and the Fermi velocity v which, in on the momentum transverse to the FS. The self-energy
F
general, depends on the location along the FS. The cou- is also the largest at the hot spots, because a fermion
plingg¯isassumedtobesmallerthanthefermionicband- scattered from one of the hot spots lands almost exactly
width, otherwise the low- and high-energy sectors of the onanotherhotspot. Toone-looporder, thebosonicself-
theorydonotdecouple. Landaudampingofspinfluctua- energy(theLandaudampingterm)isequaltoγΩ,where
tionsisgenerateddynamically,asthebosonicself-energy,
4g¯
and is due to the same spin-fermion coupling (2.1) that γ = (2.6)
gives rise to the fermionic self-energy. πvF2
Asinpreviouswork, weconsideraFSthatcrossesthe
is the Landau damping coefficient. The fermionic self-
magnetic Brillouin zone boundary at eight points – the
energy right at the hot spot is given by
hot spots (see Fig. 1). There are two hot spots in each
quadrant of the Brillouin zone, and four out of the eight 3g¯ (cid:16)(cid:112) (cid:17)
Σ (k ,Ω)=i −iγΩ+ξ−2−ξ−1 . (2.7)
hot spots are the mirror images of the other four. qπ hs 2πv γ
F
The Fermi velocities at the two hot spots connected
by q are given by v (k ) = (v ,v ) and v (k + The one-loop bosonic self-energy can be absorbed into
π F hs x y F hs
q )=(−v ,v ),wherethelocalxaxisisalongthe(π,π) the staggered spin susceptibility. Correspondingly, the
π x y
vector connecting the two hot spots and y is orthogonal effective interaction becomes dynamic: V(q)→V(q,Ω),
to it. Instead of v and v , it is more convenient to use where
x y
v = (v2 +v2)1/2 and the angle θ between v (k ) and g¯
F x y F hs V(q,Ω)= . (2.8)
vF(khs+qπ): θ =arccos(vx2−vy2)/vF2. The dependence ξ−2+(q−qπ)2−iγΩ
7
As long as ξ is finite, Σqπ(khs,Ω) at the lowest Ω lukewarm k <1 k >1
hΣa(cid:48)(cid:48)s(akca,nΩo)ni∝calΩF2L. fRorigmh,twaitththΣe(cid:48)qQπ(CkPhs,,Ωξ)=∝∞Ω, aanndd Σ'' hot Z Σ'' cold hot Z 1
qπ hs √ 1
Σ (k ,Ω) has a NFL form: Σ (k ,Ω) ∝ iΩ. In
qπ hs qπ hs
thiscase,Σ(cid:48) (k ,Ω)andΣ(cid:48)(cid:48) (k ,Ω)areofcomparable
qπ hs qπ hs
magnitudes,andbotharelargerthanthebareΩtermin k
the fermionic propagator.
For a fermion located away from a hot spot, a FL be-
havior holds even at criticality (ξ = ∞), but the pref-
actors of of the FL forms of Σ(cid:48) (k ,Ω) and Σ(cid:48)(cid:48) (k ,Ω)
qπ F qπ F
depend crucially on the distance from a hot spot along
the FS, δk. At ξ =∞, Ω =k2 1 Ω 1 Ω =k2 Ω
b b
3g¯ (cid:16)(cid:112) (cid:17)
Σ (k ,Ω)=i −iγΩ+(δk)2−|δk|
qπ F 2πv γ FIG.2. Imaginarypartofthefermionicself-energyfromq
F π
≡Ω4πv3Fg¯|δk|S(cid:18)(vFg¯|δΩk|)2(cid:19), (2.9) psZcak¯an(tΩe¯tle):r(ik¯nrigg>,hΣt1(cid:48)qa.(cid:48)πxD(isk¯i),m,Ω¯ae)sn(salieoffuntnlaecsxtsiisov)n,aaroinfadΩb¯l.tehsLeeaqfrtuepadsaienpfieanlr:etdki¯cl<aecrc1eo;srirddigiunhegt,
to Eqs. (2.14) and (2.15).
where
(cid:32)(cid:114) (cid:33)
iπ 4ix
S(x)= 1− −1 (2.10) We also introduce dimensionless quantities
2x π
Ω (cid:18)E (cid:19)2 Ω g¯
Ω¯ = max = F , Ω¯ = min = ,
withS(0)=1andS(x(cid:29)1)≈(iπ/x)1/2. Finiteδkplays max g¯ g¯ min g¯ E
F
thesameroleasfiniteξ−1: bothweakenaNFLbehavior
Ω
of the fermionic self-energy. Expanding Eq. (2.9) in Ω, Ω¯b = g¯b =k¯2. (2.15)
we obtain
In these variables, a crossover between the FL and NFL
wΣhqerπe(kF,Ω)=Ω(cid:18)Z1kF −1(cid:19)+ 4π32(vFg¯|δ2k|)3iΩ2+(2..1.1.) rZaetk¯gT(fiiΩm¯hxe)eeds≡bok¯ec,ZhcaukanFvrsdioaarirtsneΩ¯Fsohi∼fgo.wΩΣ¯3nb(cid:48)q,(cid:48)π=ai(nsk¯k¯,aF2Ω¯.ifg)u.n2c≡,tioansΣao(cid:48)q(cid:48)fπfuk¯(knaFctt,iΩofinx)eodafnΩ¯Ω¯d.
1 3g¯ The distinction between cold, lukewarm, and hot behav-
=1+ . (2.12) iors depends on the energy, and is best seen in Fig. 3,
Z 4πv |δk|
A crossover betweenkFthe FL andFNFL regimes occurs at kw¯.here Σ(cid:48)q(cid:48)π(k¯,Ω¯) and Zk¯(Ω¯) are plotted as a function of
the characteristic energy Wedefineafermionas“cold”if,atgivenΩ¯,Σ(cid:48)(cid:48) (k¯,Ω¯)
has a FL, Ω¯2, form and Zk¯(Ω¯) ≈ 1. The coldqrπegions
(v |δk|)2 (cid:18)|δk|(cid:19)2 are indicated in the right panels of Figs. 2 and 3. With
Ω ≡ F ∼Ω . (2.13)
b g¯ max k this definition, cold fermions are described by a weak-
F
coupling FL theory, and thus contribute to the FL-like
At Ω<Ω , the self-energy has a FL form, Eq. (2.11), at part of of the conductivity. Next, we define a fermion as
ΩI>nΩthbe, Σrbeqsπt(okfFt,hΩe)psacpaelers, wase√wiΩll.be focusing on scaling “isluskmeawlalerrmt”haifnΣu(cid:48)qn(cid:48)πit(yk¯,aΩ¯n)dsstcialllehsaassak¯FfoLrfk¯or<m1b.uTthZek¯c(oΩ¯r)-
dependences while discarding numerical prefactors. respondingregionsareshownintheleftpanelsofFigs.2
and 3. Finally, we define a fermion as “hot” if Σ(cid:48)(cid:48) (k¯,Ω¯)
√ qπ
hasaNFLform,i.e.,Σ(cid:48)(cid:48) ∝ Ω¯ toone-looporder. With
qπ
thislastdefinition,thehotregiongraduallyextendswith
2. Classification of fermions as “cold”, “lukewarm”, and
“hot” in the presence of q -scattering increasing frequency and, for Ω¯ > 1, includes the range
π
where the quasiparticle residue is close to unity. One
could,inprinciple,separatethisrangefromatruly“hot”,
It is convenient to measure energies in units of g¯ and
NFL behavior at Ω¯ <1, where not only Σ(cid:48)(cid:48) (k¯,Ω¯) scales
momenta in units of g¯/vF. Accordingly, we define the √ qπ
dimensionless energy and momentum as as Ω¯ but also the quasiparticle residue is smaller than
unity. We will not do this, however, because our main
Ω v δk goalistodistinguishbetweentheFL-andNFL-likeforms
Ω¯ ≡ , k¯ ≡ F . (2.14)
g¯ g¯ oftheopticalconductivity,whichisdeterminedprimarily
8
One can try to control the logarithmic series by ex-
tending the model to N fermionic flavors and taking the
limit N → ∞. In this case, the Landau damping pa-
rameter becomes of order N and the expansion param-
eter becomes small as 1/N. However, it has recently
been found that this procedure brings the theory only
under partial control because some perturbative terms
from n≥4-loop orders do not contain 1/N.26,27,32 Hav-
ing this in mind, we will keep N = 1 in our analysis
andcheckwhetherhigher-ordertermsintheloopexpan-
sion introduce a qualitatively new behavior, not seen at
lower orders. To be more specific, in the next section we
discuss how higher-loop terms affect the structure of the
imaginarypartoftheself-energyforalukewarmfermion.
Atone-looporder,Σ(cid:48)(cid:48)(k ,Ω)∝g¯2Ω2/(v |δk|)3. Itturns
F F
FIG. 3. Imaginary part of the fermionic self-energy from q out that, beyond the one-loop level, there are contribu-
π
sZck¯a(tΩ¯te)r(inrigg,hΣt(cid:48)qa(cid:48)πx(isk¯),,Ω¯a)s(aleffutnacxtiiso)n,aonfdk¯.thLeefqtupaasnipeal:rtΩ¯icl<er1e;sirdiguhet, tsitornonsgtehratdegpivenedpeanrcaemeeittrhiecralolynlΩargoerroΣn(cid:48)(cid:48)δ(kkF. ,TΩo),awnaitlhyzae
panel: Ω¯ > 1. Dimensionless variables are defined according these terms, we follow Ref. 21 and introduce the notion
to Eqs. (2.14) and (2.15). of composite scattering.
by Σ(cid:48)(cid:48). Besides, as we discuss in Sec. IIE, the distinc- C. Self-energy due to composite scattering
tion between hot and lukewarm fermions becomes more
subtle once composite scattering is taken into account.
1. Composite scattering vertex
Inacompositescatteringprocess,afermionlocatedon
With these definitions, at fixed |k¯| < 1 a crossover the FS undergoes an even number of scatterings by the
between the lukewarm and hot regimes occurs at Ω¯ ∼ the primary bosonic field with a propagator peaked at
Ω¯ ∼ k¯2 < 1. There is no range for the cold behavior in q = q [Eq. (2.8)]. At intermediate stages, the fermion
b π
this case. At |k¯| > 1, on the contrary, there is no range can move far away from the FS but it eventually comes
for the lukewarm behavior: as the frequency increases, back to the vicinity of the point of origin.
the crossover between the cold and hot regimes occurs Composite scattering processes can be viewed as 2n-
at Ω¯ ∼ Ω¯ ∼ k¯2 > 1. Again, we will see in Sec. IIE, loop processes in terms of the original spin-fluctuation
b
that the structure of crossovers changes once composite propagator. However,itismoreconvenienttoviewthem
scattering is included. as separate a subclass of processes, which involve small
momentum scattering governed by new composite ver-
tices. The lowest-order composite vertex involves two
3. Higher-order terms and the accuracy of the perturbation scatterings by momenta q +q and q +q , in which
π 1 π 2
theory bothq andq aresmall. Onecanconstructtwovertices
1 2
of this kind, with intermediate processes in the particle-
A peculiar feature of the spin-fermion model near holeandparticle-particlechannels. Suchtwoverticesare
criticality is the absence of a natural small parameter, depicted in panels A and B of Fig. 4, correspondingly.
even if the coupling g¯ is chosen to be small (compared Eachvertexisaconvolutionoftwospin-fluctuationprop-
with the Fermi energy). Although the loop expansion agators with two propagators of intermediate fermions.
goes formally in powers of g¯, a dimensionless parame- As an example, we analyze the particle-hole vertex
ter of the perturbative expansion is not g¯/E but rather (panel A in Fig. 4) for composite scattering between
F
δ ≡ g¯vF2/γ, where γ is the Landau damping coefficient fermions with the initial (2+1) momenta P = (p,Ωp)
[Eq. (2.6)]. Because γ by itself scales as g¯/v2, the spin- and K = (k,Ω), and final momenta P(cid:48) = P −Q and
F
fermion coupling drops out, and δ ∼1, i.e., higher-order K(cid:48) = K + Q, with Q = (q,Ωq). To simplify calcula-
terms inthe loopexpansion for theself-energy are ofthe tions, we will first compute the composite vertex and
same order as the one-loop expression. The functional self-energy in Matsubara frequencies and then perform
forms of the leading terms in the higher-loop fermionic analyticalcontinuation. Neglectingspinindicesforamo-
and bosonic self-energies are then the same as the one- ment, we have
loop result. On a more careful look, however, the two-
(cid:90) d3Q
loop terms contain additional logarithmic factors (lnΩ Γ(P,K;Q)=g¯2 1G(P +Q +Q ) (2.16)
or ln|δk|, depending on the regime), and the powers of (2π)3 π 1
logarithms increase with the loop order.27,28,32 ×G(K+Q+Q +Q )χ(Q +Q )χ(Q +Q +Q),
π 1 π 1 π 1
9
The integral diverges logarithmically at the
lower limit and, to logarithmic accuracy,
P,1 2 P',1 P P' yields (1/4π2γ)ln(cid:2)Λ2/(q2+γ|Ω |)(cid:3), where
q
Λ ∼ min{|δk|,|δp|}. Using Eq. (2.6) for γ, one
A)
  obtains21
g¯ 1 Λ2
K',1(1) 2(2) K,1(1) K' K Γ(PF,KF;Q)= 16πδkδplnq2+γ|Ω |. (2.18)
q
P,1 2 P',1 P P' Notice that the vertex in Eq. (2.18) depends only on Ω ,
q
althoughtheoriginalvertexinEq.(2.16)dependsingen-
eral on all the three frequencies: Ω , Ω, and Ω . The de-
B)
  p q
pendence on Ω and Ω was eliminated by replacing the
p
intermediatestates’Green’sfunctionsbytheirstaticval-
K',1(1) 2(2) K,1(1) K' K ues. This circumstance will be crucial for cancelations
between diagrams for the conductivity in Sec. IIIC.
P P' P P' P P'
The particle-particle vertex (Fig. 4, panel B) differs
C)
  =
  +
  from the particle-hole one only in that the (2+1) mo-
K' K K' K K' K mentum on the double line is replaced by K−Q −Q .
1 π
However, sincetheintermediatefermionsareagainaway
FIG.4. Compositevertexwithintermediateprocessesinthe fromtheFS,theirGreen’sfunctionscanalsobereplaced
particle-hole (A) and particle-particle channel (B). Labels 1, by their static values, 1/v δp and 1/v δk, upon which
¯1, 2, and ¯2 correspond to hot spots in Fig. 1. The notation F F
theparticle-particlevertexbecomesequaltotheparticle-
“P,1” means that the 2D component of P = (p,Ω ) is near
p hole one. The total vertex (Fig. 4, panel C) is equal to
hot spot 1, and similarly for other 2 + 1 momenta. In a
the sum of the particle-hole and particle-particle ones.
forward-scatteringevent,theinitialandfinalstatesbelongto
It is instructive to compare the composite vertex with
thesamehotspot,e.g.,spot1. A2k eventinvolvesfermions
F
from opposite hot spots, 1 and ¯1. Double solid lines denote the bare interaction V(k−p) in Eq. (2.2). First, we ob-
off-shell fermions at hot spots 2 (forward scattering) and ¯2 serve that the composite vertex scales as g¯ rather than
(2k scattering). The total composite vertex (C) is a sum of g¯2 despite the fact that it is formally of second order in
F
verticesAandB. Thewavylinedenotestheeffectivedynamic the spin-fluctuation propagator. One factor of g¯ is can-
interaction carrying a momentum close to qπ [Eq. (2.8)]. celed out by the Landau damping coefficient γ in the
denominator. Next,forfermionsinthelukewarmregion,
V(k−p) ∼ g¯/(cid:2)(δk)2+(δp)2(cid:3). For comparable δk and
where Q = (q ,0). We choose the initial states to be δp, the original and composite vertices are then both of
π π
on the FS with (2+1) momenta P =P ≡(p ,Ω ) and orderg¯/(δk)2,butthecompositevertexhasanadditional
F F p
K =KF ≡(kF,Ω)withsmallΩandΩp,andatdistances logarithmicfactorln Λ2 . Thisextralogarithmgives
δp and δk from the corresponding hot spots. Later, we q2+γ|Ωq|
rise to an additional factor of ln|δk| in the O(Ω) term in
will choose p and k to be either near the same hot
F F therealpartoftheself-energy. Inaddition,thesamelog-
spot, e.g., hot spot 1, or near diametrically opposite hot
arithm leads to two effects in the imaginary part of the
spots, e.g, hot spots 1 and ¯1 in Fig. 1.
self-energy. Thefirstoneisanonanalytic,Ω3/2 termdue
One can make sure that the largest contribution to
toone-loopcompositescattering,discussedinSec.IIC2.
Γ(P ,K ;Q) at small Q comes from the range of inte-
F F The second one is the enhancement of the prefactor in
grationwhenallthreecomponentsofQ aresmall. Such
1 theself-energyduetotwo-loopcompositescattering,dis-
a scattering event transfers fermions from the points p
F cussed in Sec. IIC3. We will consider the one- and two-
and k on the FS to the intermediate states with 2-
F loop composite processes separately.
momenta about p +q and k +q , while changing
F π F π
the frequencies only by a small amount (of order Ω ).
q
Since the energies of the intermediate fermions, ε
pF+qπ 2. “One-loop” self-energy due to composite scattering
and ε , are large compared with their frequencies,
kF+qπ
Ω +Ω and Ω+Ω +Ω , the corresponding fermionic
p q1 q q1 Thelowest-ordercontributiontotheself-energydueto
Green’s functions can be approximated by their static
composite scattering is given by diagram a in Fig. 5. In
limits, 1/v δp and 1/v δk, and taken outside the inte-
F F termsoftheoriginalinteraction(wavyline),thisdiagram
gral. The remainder of Γ(P ,K ;Q) contains a product
F F is equivalent to diagram b. Explicitly,
oftwospinpropagatorsintegratedoverthe2+1momen-
tum Q1 =(q1,Ωq1) Σ (k ,Ω)=(cid:90) d3Q G(K +Q)Γ(K ,K +Q;Q).
comp1 F (2π)3 F F F
(cid:90) d2q1dΩq1 1 1 . (2.19)
(2π)3 q 2+γ|Ω |(q +q)2+γ|Ω +Ω | The intermediate fermion’s momentum is P =K +Q,
1 q1 1 q1 q F F
(2.17) i.e., if Q is small, P should be close to K . Integrating
F F
10
3. Two-loop self-energy due to composite scattering
K
a. Main features of two-loop composite scat-
a)
K+Q tering. Anotherroutetoobtainalargeimaginarypart
K of the self-energy is to make use of the singularities in
thedynamicpartoftheparticle-holepolarizationbubble
b) bothatsmalland2k momentumtransfers.33–39Thepo-
F
+ larizationbubbl√ebehavesasΩ/qforΩ/vF <q <kF,and
asΩθ(2k −q)/ 2k −q forq near2k . Inageneric2D
K K K K F F F
FL liquid, both types of singularities give rise to a non-
K+Q K+Q
analytic form of the self-energy, Σ(cid:48)(cid:48)(k ,Ω) ∝ Ω2lnΩ,
F
whichdiffersfromthecanonicalFLform,Ω2,bya“kine-
FIG. 5. a): One-loop self-energy due to composite scat- matic” logarithmic factor.
tering. The hatched box is the composite vertex in Fig. 4C.
b): Equivalent representations of diagram a) in terms of the
original interaction in Eq. (2.8). As in Fig. 4, a double line P −Q
denotes an off-shell fermion.
P
over the component of q transverse to the FS and then
over Ω , using an explicit form of Γ from Eq. (2.18), and
q
continuing to real frequencies, we obtain
g¯ (cid:20) (cid:90) ∞ (cid:16) g¯ (cid:17)(cid:21)
Σ(cid:48)(cid:48) (k ,Ω)∼ Im iΩ dxln 1−iΩ
comp1 F (vFδk)2 0 x2
(g¯)3/2Ω3/2 K K +Q K
∼ , (2.20)
(v δk)2
F
where x = vFδq and δq is a component of q tangential FIG. 6. Two-loop self-energy due to composite scatter-
to the FS. ing.The hatched box is the composite vertex in Fig. 4C.
A non-analytic, Ω3/2 dependence of Σ(cid:48)(cid:48) (k ,Ω)
comp F
1
from one-loop composite scattering was obtained by
Similar singularities occur also in the two-loop self-
HHMSalongwithanontriviallogarithmicprefactor. Ob-
energy from composite scattering, shown in Fig. 6. In
serve that, for the lowest Ω, this form of Σ(cid:48)(cid:48) (k ,Ω)
comp F caseofforwardscattering, allthethreeinternalfermions
1
holdsforallδkoutsidethehotregion. Theonlycondition
(with 2-momenta p−q, p, and k+q) are near the same
ofvalidityofEq.(2.20)isthesmallnessofq2 ∼γΩ ∼γΩ
q pointontheFSastheinitialone(with2-momentumk).
compared with (δk)2, i.e., Ω must be smaller than Ω ,
b In case of 2k scattering, one of the internal momenta is
F
where Ω is defined by Eq. (2.13). This is the same con-
b nearkwhiletheremainingtwoarenearthediametrically
dition which separates lukewarm (or cold) fermions from
opposite point, −k. In terms of the composite vertex,
hot fermions for q scattering.
π Γ(P,K;Q) with 2-momentum transfer q, both processes
Comparing Eq. (2.20) to the Ω2 term in Σ(cid:48)(cid:48) (k ,Ω)
qπ F correspond to small q, with typical vFq ∼Ωq ∼Ω, while
due to q scattering [Eq. (2.11)], we see that one-loop
π k is either near p (forward scattering) or near −p (2k
F
composite scattering gives a larger contribution to the
scattering).
imaginary part of the self-energy at frequencies Ω<Ω ,
b A special feature of composite scattering is an addi-
i.e.,fermionsoutsidethehotregionsaredampedstronger
tionallogarithmicsingularityofthecompositevertex[cf.
by composite scattering than by q scattering. At Ω >
π √ Eq. (2.18).] For both forward and 2k scattering, the
F
Ω ,Eq. (2.20)isnolongervalidbecausetypicalq ∼ γΩ
b vertex can be approximated by
becomecomparabletoδk,andthelogarithmicsingularity
inthecompositevertexdisappears. Atthesefrequencies,
√ g¯ Z v |δk|
qπ scatteringyieldsΣ(cid:48)(cid:48)(Ω)∝ Ωandone-loopcomposite Γ≡Γ(PF,KF;Q)∼ (δk)2 ln kF ΩF . (2.22)
scatteringaddsonlyadditionallogarithmicfactorstothis
dependence.27,28
Although the iΩ2lnΩ term in the self-energy of a 2D
For comparison with other contributions, it is conve-
FLcomesfromprocessesinwhichallfermionicmomenta
nienttore-writeEq.(2.20)intermsofthedimensionless
are either parallel or antiparallel to each other, it would
variables from Eqs. (2.14) and (2.15), which yields
be incorrect to think that these processes occur as if the
Ω¯3/2 system were one-dimensional (1D). Indeed, the informa-
Σ(cid:48)c(cid:48)omp1(k¯,Ω¯)∼g¯ k¯2 (2.21) tionabout2DgeometryoftheFSinencodedinthepref-
actoroftheΩ2lnΩterm,whichcontainsthelocalcurva-
valid for Ω¯ <Ω¯ ∼k¯2. ture of the FS. Namely, if the single-particle dispersion
b