Thermoelectric transport near pair breaking quantum phase transition out of d-wave superconductivity

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by  Daniel Podolsky

Thermoelectric transport near the pair breaking quantum phase transition out of a
d-wave superconductor
Daniel Podolsky1, Ashvin Vishwanath1,2, Joel Moore1,2, and Subir Sachdev3
1Department of Physics, University of California, Berkeley, CA 94720
2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
3Department of Physics, Harvard University, Cambridge, MA 02138
7 (Dated: Printed February2, 2008)
0
We study electric, thermal, and thermoelectric conductivities in the vicinity of a z = 2
0
superconductor-diffusive metal transition in two dimensions, both in the high and low frequency
2
limits. Wefind violation of the Wiedemann-Franzlaw and a dcthermoelectric conductivityα that

n doesnotvanishatlowtemperatures,incontrasttoFermiliquids. WeintroduceaLangevinequation
a formalism to study critical dynamics overa broad region surrounding thequantumcritical point.
J

7
1 INTRODUCTION T 0. Hence,thereisastrongviolationofWiedemann-
Fra→nz law (WF) in the bosonic sector, with an apparent
l] Transportofheatandchargeprovideausefulprobeof excess of charge carriers. The total conductivities are
e
- stronglycorrelatedelectronicsystems. Forexample,avi- sums of fermionic and bosonic parts, σ = σf +σΦ (and
r olationoftheWiedemann-Franzlawatlowtemperatures similarly for α, κ)[16]. Thus, the magnitude of the ex-
t
s signals non-Fermi liquid physics. One possible origin of pected WF violation is difficult to estimate. Another
.
at such a violation is proximity to a quantum critical point striking effect is that αΦ has a very weak T dependence
m (QCP), where the presence of low energy critical modes at low temperatures ( lnlnT), unlike αf, which van-
∼
leadstonovelphenomena. Therehasbeenmuchtheoreti- ishes linearly with T. Hence, experimental measurement
-
d calandexperimentalattentiondirectedtowardselectrical ofthe anomalousthermoelectric conductivity wouldgive
n transportatQCP’s,inthecontextofthesuperconductor- directinformationregardingcriticaltransportproperties,
o insulator transition in amorphous films, and the Quan- as well as yielding the dimensionless “thermoelectric fig-
[c tum Hall transitions (see [1, 2] for reviews). Recently, ure of merit” ZT = ασ2κT, a measure of the strength of
experimental investigations of thermal and thermoelec- particle-hole breaking.
2
tric properties at quantum phase transitions have been
v
7 pursuedwithveryinterestingresults. Theseincludemea-
T =0 Ω=0
9 surements of thermal conductivity in highly underdoped
5 cupratesystemsnearcriticaldoping[3,4],andofthermo- σ const (2he)28πb2ηlnTΛ
10 ealehcetarvicytfrearnmsipoonrtQ(CNPer[n5s]t. aHnodwSeveeebr,etchkecroeehffiacsiebnetesn)rneelaar- κ˜/αT ∼∼iΩTΩ2 2ehkB 8cπo12nηsltnlnTΛ
5 T2
0 tively little theoretical work on thermal and thermoelec-
/ tric transport at QCP’s. In this paper we calculate the TABLE I: Asymptotic low T behavior of transport coef-
t
a nontrivial electric, thermal, and thermoelectric conduc- ficients along the line s = sc in the zero temperature
m tivities at a z = 2, d = 2 quantum critical point for the (Ω/T → ∞) and dc (Ω/T → 0) limits. The prefac-
- transition from an unconventional (non s-wave) super- tor of σ(Ω = 0) is given in an extreme low T limit b =
d [tan−1(1/η)+4η/(1+η2)]/2lnln(Λ/T). Improvedvaluesfor
conductor to a diffusive metal in 2 dimensions.
n the Ω = 0 results, valid for a broader T range appear below
o InanappliedelectricfieldE andtemperaturegradient Eq. (19). Explicit expressions for σ(T = 0) and κ/˜T(Ω= 0)
c T,theelectric(σ), thermal(κ),andthermoelectric(α)
v: c∇onductivities are defined by, are given in Eqs. (7) and (8).
i
X
j σ α E
= . (1)
r j αT κ˜ T
a (cid:18) Q (cid:19) (cid:18) (cid:19)(cid:18)−∇ (cid:19)
AQCP with either LorentzorGalileaninvariancewill
where j and j are the electrical and heat currents, re- have infinite thermal conductivity, because a “boosted”
Q
spectively. Thermal conductivity measurements are al- thermal distribution will never decay; the same logic ap-
ways carried in open circuit (j = 0) boundary condi- plies to electric conductivity if there is only one sign of
tions, so that κ = κ˜ α2T/σ. The difference between chargecarrier. We findbelowthatafiniteη atthe z =2
−
κ and κ˜ is negligible in metals, but this need not be QCPregularizesthe thermalconductivity. It is alsocru-
the case in systems with strong particle-hole symmetry cialthat the z =2 theory breaksparticle-hole symmetry
breaking, as considered here. Our results are shown in sothatthethermoelectriccoefficientcanbenonzero;the
Table I. We find that, while the dc thermal conduc- z =2theoryisthusthesimplestcriticaltheoryforwhich
tivity is metallic, the dc electric conductivity divergesas all the transport coefficients in (1) are finite.
2
MODEL AND CURRENTS
Our starting point is an electron model with a pair- quantum critical
ing interaction favoring unconventional superconductiv-
ity,andadisorderpotentialwhosemaineffectistorender diffusive
electronsdiffusive. Then,introducingaHubbard-Strata- metal
KT transition
novichfieldΦ inthe Cooperchannelandintegratingout
fermions yields a Ginsburg-Landau action for Φ,[6]
unconventional
β 1
S[Φ] = dτ d2r Φ∗ ∂ +η ∂ 2+s Φ superconductor
τ τ
| |− 2m∇
Z0 Z (cid:20) (cid:18) (cid:19)
V
+ Φ4 , (2)
2| |
(cid:21)
FIG. 1: Phase diagram in thevicinity of theQCP as a func-
where the dissipative term ∂ , shorthand for the Mat-
| τ| tion of disorder, parameterized by the lifetime τ of electrons
subara expression ω , arises from the decay of Cooper
n in the diffusive metal. At finite T, interactions shift the
| |
pairsintogaplessfermions. Notethat,forunconventional transition line to the right, so that the diffusive metal phase
superconductors, disorder is pair-breaking. This insures lies above the QCP. The crossover into the quantum critical
thatthecriticaltheory(2)islocal[6]. Naivepowercount- regime occurs when R≈T.
ingofmodel(2)yieldsdynamicalcriticalexponentz =2.
Althoughdisorderdominatesextremelyclosetothetran-
sition, one can choose microscopic parameters such that the order of the limits Ω 0 and T 0[8]. Sending
→ →
eq.(2)describesalargecrossoverregionneartheQCP[6]. Ω 0 first yields the dc conductivities, which are more
→
For instance, for a clean sample with an elastic mean readily accessible to experiment, but are typically more
free path of 100 nm, Model (2) is valid provided T is difficult to compute than their T =0 analogs.
larger than a few millikelvin[15]. This paper focuses on For a d = 2, z = 2 theory, the quartic interaction V
transportin this region. Equation(2)also applies to the is dangerously irrelevant. Thus, at the T = 0 QCP, the
onset of antiferromagnetic order in an itinerant electron correctdynamicsisobtainedfromthelimitV 0+. We
→
system[7],withΦanO(3)orderparameter. Manyofour first consider this non-interacting case. Note that, even
results apply to this QCP as well (eg. thermal conduc- without interactions, finite transport is plausible, due to
tivity) but the field Φ carries spin, and not charge. dissipation in action (2). Indeed, σ is metallic in the
The electric and heat currents j = ∂S and j = ∂S zerotemperaturelimit[6],asshowninthemiddlecolumn
are obtained by making the action (∂2A)egauge-cQovar∂iaAnTt of Table I. On the other hand, to compute dc conduc-
through the substitution ieA iA (i∂ ) tivities, we shall consider finite temperatures, for which
e T t
∇→D≡∇− −
(see Ref. [9]), interactions give important logarithmic corrections.
ie
j = Φ† Φ ( Φ)†Φ
m D − D
NON-INTERACTING CASE
1(cid:0) (cid:1)
j = (∂ is)Φ† Φ+( Φ)†(∂ +is)Φ
Q t t
2m − D D
The conductivities are obtained from Kubo formulas,
We compute con(cid:0)ductivities from Kubo formulas in(cid:1)volv- G (Ω)= (2e)2−γkBγ Π (Ω+iδ). ForV =0,Π isgivenby
γ iΩTγ γ γ
ing the dynamical correlations of these currents.
a single-loop integral,
iγ d2k k2
ORDER OF T →0 AND Ω→0 LIMITS Πγ(iΩn) = β (2π)2mx2 −ωnγG(k,ωn)2 (4)
Xωn Z (cid:8)
Conductivities near the 2d QCP depend on ratios of + (ωn+Ωn/2)γ (k,ωn) (k,ωn+Ωn) ,
G G }
small energy scales (and possibly a UV cutoff scale Λ),
and on the dimensionless parameters mV and η, where is the Matsubara Green’s functions, (k,ωn)=
G G
(iω η ω ǫ s)−1, where ǫ = k2/2m. Rewrit-
n n k k
Gγ(Ω)=gγ |s−Tsc|νz,TΩ,TΛ,mV,η . (3) ingdωt′−hAis(k|,iωn′)|t,−erm(ks,−ωof)=the spec2tηrωal func,tiwoen,obGt(aki,nω,n) =
(cid:18) (cid:19) 2π iωn−ω′ A (ω−ǫk−s)2+η2ω2
Here,G σ,G α,andG κ˜/T,Ωisthefrequency R
0 ≡ 1 ≡ 2 ≡ 1 ω +ω γ
of the external field, and νz = 1 for the d = 2, z = 2 Π (iΩ ) = 1 2 (k,ω ) (k,ω )
γ n 1 2
model in question. At the QCP, s = sc, Gγ depend on β Xωn Zω1,ω2,k(cid:18) 2 (cid:19) A A
3
kx2 1 This is purely imaginary for real frequencies (z =
| |
m2 (iω ω )(i(ω +Ω ) ω ) zsign(Rez)). Hence, while the energy of a quasi parti-
(cid:26) n− 1 n n − 2
1 cle is not renormalized, quasi particles are given a finite
, (5)
lifetime,
−(iω ω )(iω ω )
n− 1 n− 2 (cid:27)
1
TInh(is5)awcceohuanvtesmfoardethtehefascutbstthiatuttitohneitωimn →e d(eωr1iv+atωiv2e)/i2n. τk = i[Σ(ω+iδ)−Σ(ω−iδ)]ω=ǫk =2ηǫk. (9)
jQ does not commute with the time-ordering symbol, Thisisthe bosonlifetime, atbubble level,inthefermion
andisnecessarytoobtainthecorrectthermalconductiv- model. Here we neglect corrections to the transport life-
ity [10]. Performing the ωn sum and using the identity, time from higher order diagrams.
f0(ω +iΩn) = f0(ω), before analytically continuing to The Boltzmann equation for the occupation function
real frequencies, iΩn Ω+iδ, we find f(k,r,t) in the scattering lifetime approximation yields
→
1 k2 Ω γ 1
Im Πγ(Ω+iδ)= 2Zω2,k mx2 (cid:18)ω2− 2(cid:19) × (6) ∂tf +v·∇rf +k˙ ·∇kf =−τk(f −f0). (10)
(k,ω2 Ω) (k,ω2) f0(ω2 Ω) f0(ω2) , The electric and energy currents are expressed in terms
A − A − −
of f,
where f0(ω) = (exp(βω) 1)(cid:0)−1 is the Bose occupa(cid:1)tion
−
factor. TheintegralsareUVfinite,andhencethedepen- d2k
dence on the cutoff Λ drops out, ja(r,t) = (2e)vkaf(k,r,t) (11)
(2π)2
Z
s s νz Ω d2k
Gsγcaling(Ω)=gγscaling | −Tc| ,T,η . jQa(r,t) = (2π)2 (ǫk−s)vkaf(k,r,t). (12)
(cid:18) (cid:19) Z
Expression(6) canreadily be evaluatedfor generic Ω/T. The conductivities are then
(2e)2−γkγ d2k τk ǫ1+γ ∂f0
G = B k .(13)
Zero temperature (Ω/T →∞) limit γ Tγ¯h (2π)2 1 iΩτk m −∂ǫk
Z − (cid:18) (cid:19)
For η 0, this gives perfect agreement with the pre-
The results in the zero temperature limit at the QCP →
vious results for G in the dc limit. Here, we see that
γ
(s = s = 0) are summarized in the middle column of
c the reasonfor the divergences in σ and α is that low en-
Table I. While σ is metallic, α and κ˜/T are divergent
ergyquasiparticleshavearbitrarilylargelifetimesτk. On
(notethatαispurelyreactive). Theelectricconductivity
the otherhand,these verylong-livedquasiparticles,hav-
depends on the dissipation η,
ing low energies, contribute little to the energy current.
Ω (2e)2/h Hence κ˜/T is finite.
σ = [2η+π(1+η2)]arctanη
T →∞ 4πη2
(cid:18) (cid:19)
πη η(cid:8)2 (1+η2)arctan2η , (7) INTERACTING CASE
− − −
since η is marginal. (cid:9)
For T > 0, the dangerously irrelevant interaction V
must be taken into account. The most important effect
DC (Ω/T →0) limit of interactions is to shift the phase transition, such that
the QCP is approached at finite temperatures from the
diffusive metal phase, as shown in Fig. 1. This is cap-
Ontheotherhand,thedctransportpropertiesatV =
0 are drastically different. Whereas σ = (2e)2 T and tured by a renormalizedmass R, discussed below, which
h 4|Ω| ispositiveabovethetransition. Ractsasaneffectivegap
α= 2ehkB 8π12η ln|TΩ| are divergent, Tκ˜ is metallic, forlowenergyquasiparticles,thusrenderingalldctrans-
port coefficients finite. The situation is in contrast with
κ˜ Ω k2 η+arctan(1/η)
0 = B (8) Ref.[8],whereinteractionsregularizetransportbyintro-
T T → h 12πη ducing quasiparticle scattering. Here, the leading low T
(cid:18) (cid:19) (cid:18) (cid:19)
dcconductivitiesareobtainedfromaHartree-Fock(HF)
In order to elucidate the dc results, we employ a simple
analysis, where the only effect of interactions is to shift
Boltzmann equation approach, which is exact for weak
the quasiparticle mass R. The HF results are shown in
dissipation η 0. The dissipating term in the action
→ the last column of Table I. These results are valid for
(2)canbeinterpretedasthe self-energyofbosonsdue to
extremely low temperatures, such that lnlnΛ/T 1.
their interactions with the fermion bath,
≫
To study transport on a much broader region surround-
Σ(iω )=η ω . ing the QCP, we introduce a classical treatment of the
n n
| |
4
orderparameterwhich,whensupplementedbyLangevin values of s s . This is due to the absence of long-
c
−
dynamics, will be shown to capture the correct quan- range order (LRO) in 2d at finite T. For an O(2) order
tumcriticaltransportbehaviorovertheregionwherethe parameter,however,quasiLROisestablishedataT >0
much weaker condition, lnΛ/T 1, is satisfied. Kosterlitz-Thouless transition.
≫
A description in terms of a classical action [12] is ap-
propriate whenever logΛ /T 1. In this limit, U T,
ω
Classical action for order parameter so that modes with ω =0 ar≫e significantly gapped≪.
n
6
With interactions, the critical point is shifted away
from s =0. To linear order in V, Dynamics of order parameter
c
1
s =2V We approximate the low frequency dynamics of the
c
Zω,k iωn−η|ωn|−ǫk classicalorderparameterbyaLangevinequation(model
A dynamics of Ref. [13]),
In the vicinity of the QCP, we renormalize V by a one
loopRGequation,uptothescalewherethesystemeither ∂ψ δS
c
= (i+η) +f (17)
developsagap,orwhentherescaledtemperaturereaches ∂t − δψ∗ η
an upper frequency cutoff Λω[7, 11], f∗(x,t)f (x′,t′) = 2ηδ(x x′)δ(t t′)
h η η i − −
2π2 1
V Equal time correlators computed with these dynamics
R ≈ m tan−1 1 + 4η ln Λω areequaltothoseoftheclassicalactionS ,asnecessary.
η 1+η2 Max(T,|s−sc|) c
The appearance of the “bare” value of η in eq. (17) is
(cid:16) (cid:17)
The static properties of the finite T model can be stud- due to the fact that dispersion in the quantum action is
ied by integrating out all non-zero Matsubara frequency non-local in time and therefore is not renormalized.
modes. After rescaling, Φ = √2mTψ, r = x/√2m, the Consider the HF approximation, in which (17) be-
ω =0 mode has the following classical action, comes a linear equation with mass R. Solving for σ,
n
U (2e)2 T Ω
S = d2x ψ 2+R˜ ψ 2+ ψ 4 Reσ(Ω) = tan−1
c |∇x | | | 2| | h 2πΩ 4ηR
Z (cid:20) (cid:21)
(2e)2 T forR=0
whereU =2mTV . Thistheoryissuper-renormalizable, 4|Ω|
R
and is rendered UV finite by introducing a renormalized → h ( T forR=0, Ω 0
8πηR 6 →
mass R,
For R=0, this reproduces the non-interacting result, as
Λ d2k 1 expected. On the other hand, for R = 0, we obtain a
R˜ =R 2U . (14) 6
− (2π)2k2+R finite dc conductivity,
Z0
Λ
R hasauniversalexpressionintermsofs sc,reflecting σ ln ω (18)
the contribution of the ω =0 modes − ∼ T
n
6
We note that Eq. (18) disagrees with Ref. [14], which
U ∞ T T
R = (s s )+ dy predicts σ lnln(Λ /T). This is due to an erroneous
c ω
− 2πT y+R − y+s s ∼
Z0 (cid:20) − c computation of T∗ in Eq. (12) of that reference.
∞ dΩ ηΩ 1 Naive use of Eq. (17) yields divergent values of α and
+
Z0 π eΩ/T −1(cid:18)(Ω−y−(s−sc))2+η2Ω2 κ˜/T. This is not surprising: the Langevin equation as-
1 sumes classical modes, whose occupation factors satisfy
+
(Ω+y+s s )2+η2Ω2 equipartition, feq(ω)=T/ω. However, inspection of the
− c (cid:19)(cid:21) Boltzmannapproach,eq.(13),showsthatforsuchdistri-
U T
= s s + ln +F(s s ,η) , (15) bution, α and κ˜/T have UV catastrophes. In this sense,
c c
− 2π R −
(cid:26) (cid:27) the Langevin equation does not capture the correct dy-
namics of high energy quantum modes. However, high
where F(s s ,η) ln 1+η2 as s s . Solving this
− c → → c energy modes are very weakly perturbed by the quar-
self-consistent equation at s=s yields,
p c tic interaction. Thus, the correct result is given by the
T Boltzmann equation with a full Bose distribution, as in
R (16)
Eq.(13),butwithachemicalpotentialsetbyR. Equiva-
∼ ln(Λ /T)
ω
lently, this corresponds to evaluating the one-loop quan-
up to a prefactor of order lnln(Λ /T). Note that, for tumexpression(5)withchemicalpotentialR. Thisyields
ω
T >0, R is always positive, even for arbitrarily negative the last column of Table I.
5
TheHFresultscanbeobtainedindependentlyfroman
1.8
exact solution of the quantum model (2) in the large N
limit[15], where N is the number of components of the F 1.6
s
order parameter (N = 2 for superconductivity). This
is an important check that the Langevin equation (17) 1.4
captures the correct universal dynamics. We see from
1.2
Eq. (15) that, for s=s and at low T,
c
U 1 1 2 4 6 8 10
R ∝ lnT/R ∼ lnlnΛω/T 0.8 U/R
which justifies HF provided that lnln(Λ /T) is a large
ω
number. Moregenerally,to gobeyondHF,wemustcon- FIG. 2: Estimated scaling function Φσ(U/R,η = 1) from
sider higher order corrections in U. From the Kubo for- numerical integration of the Langevin equation (17). Con-
mulaandthefluctuation-dissipationtheorem,wededuce vergence for static quantities of the Langevin algorithm was
tested by comparison with results from a Wolff cluster algo-
that the dc electric conductivity obeys
rithm. Each datapoint representsat least 2×107 timesteps
on a lattice of spacing a=0.1 and linear size L=64, which
(2e)2 T U
σ = Φ ,η , (19) forstatic quantitieswith η=0isknowntoapproximatewell
h 8πηR σ R the continuum limit a → 0. Results were normalized by the
(cid:18) (cid:19)
U/R=0resultandfitbyaquadraticpolynomial(solidline).
for some scaling function Φ satisfying Φ (0,η) = 1. A
σ σ
similaranalysisappliestoαandκ˜/T,withtheimportant
differencethatsubstractionsarenecessarytocancellead-
CONCLUSIONS
ing UV divergences, as discussed above. When working
with the renormalized R, the only UV divergence comes
ACKNOWLEDGMENTS
from the diagrams already computed. Thus,
2e U We benefited from useful discussions with B. Binz,
α=α + Φ ,η ,
quantum,1loop α A. Paramekanti, T. Senthil, and members of the 2005
2 R
(cid:18) (cid:19) Aspen Center for Physics Workshop on Competing Or-
κ˜ κ˜ R U
= quantum,1loop + Φ ,η , ders, where part of this work was completed. This work
κ
T T 2T R
(cid:18) (cid:19) was supported by NSF grants DMR-0238760 (J.M.) and
DMR-0537077(S.S.), the Hellman Fund (J.M.), and the
where the small R limit of the (quantum, 1 loop) results
LDRD program of LBNL under DOE grant DE-AC02-
is 2ekB lnT for α, and Eq. (8) for κ˜/T. The functions
8π2ηh R 05CH11231(D.P. and A.V.).
Φ canbeevaluatednumericallybyintroducingalattice,
γ
Ua2
S = ψ ψ 2+ R˜ a2 ψ 2+ ψ 4
cL i j L i i
| − | | | 2 | |
Xhiji Xi (cid:20) (cid:21) [1] S. Sachdev, Quantum Phase Transitions (Cambridge
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shouldbeapplicableatmanyothertransitions. Here,we
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6
[15] D.Podolsky,A.Vishwanath,J.E.Moore,andS.Sachdev fluctuating conductivities are less divergent than the
(toappear). bosonic(Aslamazov-Larkin)conductivitiesinTableI[15]
[16] Maki-Thompson and density of states corrections to the