Renormalization group analysis of quantum critical points in d-wave superconductors

0.27 MB·English

by  Matthias Vojta

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RENORMALIZATION GROUP ANALYSIS OF QUANTUM
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CRITICAL POINTS IN d-WAVE SUPERCONDUCTORS

]
el MATTHIASVOJTA,* YINGZHANG,ANDSUBIRSACHDEV
-
Department of Physics, Yale University
r
t P.O. Box 208120, NewHaven, CT06520-8120, USA
s
.
t
a
Wedescribeasearchforrenormalizationgroupfixedpointswhichcontrolasecond-order
m
quantum phase transition between a dx2−y2 superconductor and some other supercon-
- ductinggroundstate.Onlyafewcandidatefixedpointsarefound.Inthefinitetempera-
d ture(T)quantum-criticalregionofsomeofthesefixedpoints,thefermionquasiparticle
n lifetimeis very short and the spectral function has an energy width of order kBT near
o theFermipoints.Underthesameconditions,thethermalconductivity isinfiniteinthe
c scaling limit. We thus provide simple, explicit, examples of quantum theories in two
[
dimensions for which a purely fermionic quasiparticle description of transport is badly

violated.
2
v
8 1. Introduction
4
0 Thequasiparticleexcitationsofthedx2−y2-wavehightemperaturesuperconductors
8 have been subjected to intense scrutiny in the past few years.An especially impor-
0
tant test of our understanding of the underlying physics is whether quasiparticle
0
0 relaxation processes measured by different experimental probes can be reconciled
/ with each other. A striking dichotomy appears to have emerged recently in such
t
a a context. Photoemission experiments1 indicate that the nodal quasiparticles have
m
veryshortlifetimes inthe superconductingstate,with their spectralfunctions hav-
-
ing linewidths of order k T. In contrast, transport experiments, especially mea-
d B
n surements of the thermal conductivity, when interpreted in a simple quasiparticle
o model, indicate far larger quasiparticle lifetimes in the superconductor.2
c
Motivated primarily by the photoemission experiments,1 we recently proposed
:
v a scenario3,4 in which the large linewidths in the fermion spectral functions are
i
X explained by fluctuations near a quantum critical point between the dx2−y2 su-
r perconductor and some other superconducting state X (see Fig 1). In this paper
a
we will review and extend earlier arguments which classify various possibilities for
the state X. We will provide the details of a renormalization group (RG) analysis
which shows that only a small number of the candidates for X are associated with
a RG fixed point which describes a second-order phase boundary between X and
the dx2−y2 superconductor that can be generically crossed by varying a single pa-
rameter r; we will identify the subset of these fixed points which lead to fermion
spectrallinewidths of order k T.We will also initiate a discussionof the transport
B
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T Tc
Quantum
critical
T
X
superconducting dx2-y2
state X superconductor
0
rc r
Fig. 1. Finite-temperature phase diagram3 of the d-wave superconductor close to a quantum-
critical point. Superconductivity ispresent forT <Tc. Thelong-range order associated withthe
stateXvanishesforT >TX,butfluctuationsofthisorderprovideanomalousdampingofthenodal
quasiparticles in the quantum-critical region. The tuning parameter r is some coupling constant
in the Hamiltonian, with the suitable choice depending upon the identity of the state X. It is
possible,althoughnotnecessary,thatincreasingrcorrespondstoincreasingdopingconcentration,
δ.Thecriticalpointr=rcisnotrequiredtobeintheexperimentallyaccessibleparameterregime,
justnottoofarfromthedx2−y2 superconductor.
propertiesofthese fixedpoints,andarguethatthey offerattractivepossibilities for
explaining the transport experiments as well.
We begin with a brief review of studies of quantum phase transitions in the
cuprate superconductors. The subject was initiated in the work of Chakravarty et
al.,5 who presented a field-theoretic study of the destruction of N´eel order in insu-
lating antiferromagnets, but focused mainly on thermal fluctuations above a N´eel
ordered state. Subsequent work examined the nature of the paramagnetic ground
stateintheinsulator,6 andofitsquantum-criticalpointtotheN´eelstate.7,8 Inpar-
ticular, it was proposed7 that destroying the N´eel order by adding a finite density
of mobile charge carriers also led a quantum-critical point in the same universality
class as in an insulating antiferromagnet, with dynamic critical exponent z = 1.
This scenario had strong consequences for magnetic experiments, and for the man-
ner inwhichaspin‘pseudo-gap’woulddevelopasT waslowered,andthese appear
to be consistent with observations: crossovers in NMR relaxation rates,9 uniform
susceptibility,7 and dynamic neutron scattering10 all indicate that z = 1. More-
over, a further consequence7,11 of such a scenario was that the paramagnetic state
should have a stable S = 1 ‘resonant’ spin excitation near the antiferromagnetic
wavevector,and this is also borne out by numerous neutron scattering studies.12
If we accept that the mobile charge carriers have a superconducting ground
state (inparticular,ad-wavesuperconductor),thenthe argumentsforthe common
universality of the magnetic quantum critical point in insulating and doped anti-
ferromagnetscanbe sharpened.Forboth cases,it is clearthatthe orderparameter
is a 3-component real field, N , which measures the amplitude of the local anti-
α
ferromagnetic order. In the paramagnetic state, N will fluctuate about N = 0,
α α
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and indeed, the triplet ‘resonance’ modes just mentioned are the 3 normal mode
oscillations of N . The transition to the state with magnetic long-range order (we
α
assumethat the chargesectoris superconducting onboth sidesof the transition)is
describedby the condensationofN ,andthe theoryforthe quantum-criticalpoint
α
willdependuponwhethertheN coupleefficientlytootherlow-energyexcitations,
α
not directly associated with the magnetic transition. In a d-wave superconductor,
theimportantcandidatesfortheselowenergyexcitationsarethefermionicS =1/2
Bogoliubovquasiparticles(onecanalsoconsiderfluctuationsinthephaseofthesu-
perconducting order parameter, but we will argue below that such a coupling can
be safely neglected). Momentum conservation now plays a key role: fluctuations of
N occur primarily at a finite wavevector Q (in the present situation, this is the
α
wavevector at which the magnetic order appears), and the fermions will be scat-
teredby this the wavevector.IfQ doesnotequalthe separationbetweentwonodal
points of the d-wave superconductor [the nodes are the locations in the Brillouin
zoneofgaplessfermionicexcitations,andtheyareat( K, K)withK =0.39π at
± ±
optimal doping], then it is not difficult to show that the fermion scattering serves
mainly to renormalizethe parametersin the effective low energyactionfor the N ,
α
and does not lead to any disruptive low energy damping.13 In such a situation,
there is no fundamental difference between the magnetic fluctuations in a super-
conductor and an insulating paramagnet, and both cases have the same theory for
the quantum critical point to the onset of long-range magnetic order. Conversely,
if Q = (2K,2K), the coupling to the fermionic quasiparticles is important, and a
new theory obtains: such a theory was discussed by Balents et al.14
In this paper we will focus onthe fermionic quasiparticles rather than the mag-
netic excitations. We are interested in damping mechanisms for the fermions and
associated possibilities for the state X in Fig 1. The above discussion on magnetic
properties suggests a natural possibility for X: a state with co-existing magnetic
and superconducting order. However, it should also be clear from the discussion
above that strong damping of the fermionic quasiparticles requires Q = (2K,2K).
For the current experimental values, this condition is far from being satisfied, and
so the magnetic ordering transition is just as in an insulator. This also means that
the magnetic quantum critical point is not currently a favorable candidate for the
quantum critical point in Fig 1. We are therefore led to a search for other possi-
bilities, and this paper will describe the results of such a search. The new cases we
will consider do not have magnetic order parameters,and so we are envisaging two
distinct quantum critical points near the d-wave superconductor: one involving the
magnetic order parameter which is already known to occur with decreasing doping
(andwhich,asdiscussedabove,is inaccordwithnumerousmagneticexperiments),
andanotheroneassociatedwiththestateX whichmayormaynotbepresentalong
the experimentally accessible parameter regime.For completeness, we will also dis-
cuss the magnetic case with Q = (2K,2K) in Section 2.2: this is the only case
which envisages a single quantum phase transition to explain both the magnetic
experiments and the fermion damping.
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In Section 2, we present the details of a renormalization group (RG) analysis
which searches for candidate fixed points describing the quantum phase transition
between X and the d-wave superconductor: among the many a priori possibilities,
only a few stable fixed points are found. The structure of the T 0 single fermion
≥
Green’s functions near these fixedpoints is discussedinSection3:these results can
be comparedwith photoemissionexperiments.Spin, thermal, andchargetransport
properties of all the fixed points are subsequently consideredin Section 4; a signifi-
cantresultisthatthethermalconductivityofthequantumfieldtheoriesdescribing
the vicinities of these fixed points is infinite in the scaling limit.
2. Renormalization group analysis
Letusassumethatthe orderparameterassociatedwithX carriestotalmomentum
Q. We have argued3,4 above that its coupling to the fermions can be relevant only
if a nesting conditionis satisfied:orderparameterfluctuations willscatter fermions
by a momentum Q, and such scattering events are important only if they occur
between low energy fermions, i.e., the wavevector Q (or an integer multiple of Q)
must equal the separation between two nodal points. If such a condition is not
satisfied, then, as noted above, fermion scattering events can be treated as virtual
processes which modify the coupling constants in the effective action, but do not
lead to a fundamental change in the form of the low energy theory.
Three natural possibilities can satisfy the nesting condition:15 Q =0, Q =
(2K,2K), and Q = (2K,0),(0,2K). We will consider these possibilities in turn
in the following subsections. Of these, the first condition can be naturally satisfied
for a range of parameter values, while the last two require fine-tuning unless, for
some reason, there is a mode-locking between the values of Q and K.
2.1. Order parameters with Q = 0
We will assume that the Q = 0 order parameter is a spin-singlet fermion bilinear
(spin triplet condensation at Q = 0 would imply ferromagnetic correlations which
are unlikely to be present, while order parameters involving higher-order fermion
correlationsarenotexpectedtohavearelevantcouplingtothe fermions15).Simple
group theoretic arguments4 permit a complete classification of such order parame-
ters. The order parameter for X must be built out of the following correlators(c
qa
annihilates an electron with momentum q and spin a= , )
↑ ↓
c† c = A
h qa qai q
c c = [∆ (cosq cosq )+B ]eiϕ, (1)
q↑ −q↓ 0 x y q
h i −
where ∆0 is the background dx2−y2 pairing which is assumed to be non-zero on
both sides of the transition, ϕ is the overall phase of the superconducting order,
andA andB containthepossibleorderparametersforthestateX corresponding
q q
to condensation in the particle-hole (or excitonic) channel or additional particle-
particlepairingrespectively.Itisclearthatϕhastheusualcharge2transformation
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d-wave superconductor
(D)
(E)
(F)
(A), (B)
fully gapped spectrum (G)
Fig.2. Evolutionofthe nodal points whenpassingfromr>rc tor<rc atT =0for6ofthe
Q= 0 states X discussed in Section 2.1. There is no change in the position of the nodal points
forcaseC,onlyachangeinvelocity.
undertheelectromagneticgaugetransformation,andsogradientsofϕmeasureflow
of physical electrical current; because of the non-zero superfluid density associated
with ∆ , ϕ fluctuations remain non-critical and we will show that they can be
0
neglected. It follows that the order parameter B , which is in general a complex
q
number, carries no electrical charge. Similarly, A is also neutral but must be real.
q
To classify the order parameters for X, we expand A and B in terms of
q q
the basis functions of the irreducible representation of the tetragonal point group
C . This group has 4 one-dimensional representations, which we label as (basis
4v
functions in parentheses) s (1), dx2−y2 (cosqx cosqy), dxy (sinqxsinqy), and g
−
(sinq sinq (cosq cosq )),andone2-dimensionalrepresentationp(sinq ,sinq ).
x y x y x y
−
Ananalysisofallexcitonicorderandadditionalpairings(bothrealandimaginary)
inthese functions hasbeen carriedout,4 andit wasfoundthat 7inequivalentorder
parametersare allowedfor X. Of these, the first 6 (A-F) involve a one-dimensional
representationofC ,andsothe orderparameterisIsing-likeandrepresentedbya
4v
real field φ, while the 7th (G) involves the 2-dimensional representation and 2 real
fields φ ,φ . The 7 possibilities for X are
x y
(A) is pairing: A =0, B =iφ
q q
(B) id pairing: A =0, B =iφsinq sinq
xy q q x y
(C) ig pairing: A =0, B =iφsinq sinq (cosq cosq )
q q x y x y
−
(D) s pairing: A =0, B =φ
q q
(E) d excitons: A =φsinq sinq , B =0
xy q x y q
(F) d pairing: A =0, B =φsinq sinq
xy q q x y
(G) p excitons: A =φ (sinq +sinq )+φ (sinq sinq ), B =0 (2)
q x x y y x y q
−
Apart from case C, these possible orderings for X change the nature of the nodal
excitations, and these are sketched in Fig 2. Cases A, B open up a gap in the
fermionspectrum overthe entire Brillouinzone (suggesting anenergetic preference
for these cases), case C leaves the positions of the nodal points unchanged but
changesavelocityinthedispersionrelation,whilecasesD–GbreakC symmetries
4v
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by moving the nodal points as shown.
Knowledge of the order parameters in (2) and simple symmetry considerations
allow us to write down the quantum field theories for the transition between the
dx2−y2 superconductor and X.
The first term in the action, S , is simply that for the low energy fermionic
Ψ
excitations in the dx2−y2 superconductor. We denote the components of cqa in the
vicinity of the four nodal points ( K, K) by (anti-clockwise) f , f , f , f ,
1a 2a 3a 4a
± ±
and introduce the 4-component Nambu spinors Ψ = (f ,ε f† ) and Ψ =
1a 1a ab 3b 2a
(f ,ε f† ) where ε = ε and ε =1 [we willfollow the conventionof writing
2a ab 4b ab − ba ↑↓
out spin indices (a,b) explicitly, while indices in Nambu space will be implicit].
Expanding to linear order in gradients from the nodal points, we obtain
d2k
S = T Ψ† ( iω +v k τz +v k τx)Ψ
Ψ Z (2π)2 1a − n F x ∆ y 1a
Xωn
d2k
+ T Ψ† ( iω +v k τz +v k τx)Ψ . (3)
Z (2π)2 2a − n F y ∆ x 2a
Xωn
Hereω isa Matsubarafrequency,τα arePaulimatriceswhichactinthe fermionic
n
particle-hole space, k measure the wavevector from the nodal points and have
x,y
been rotated by 45 degrees from q co-ordinates,and v , v are velocities.
x,y F ∆
Thesecondterm,S describestheeffectiveactionfortheorderparameteralone,
φ
generated by integrating out high energy fermionic degrees of freedom. For cases
A–F this is the familiar field theory of an Ising model in 2+1 dimensions
1 c2 r u
S = d2xdτ (∂ φ)2+ ( φ)2+ φ2+ 0φ4 ; (4)
φ τ
Z h2 2 ∇ 2 24 i
here τ is imaginary time, c is a velocity, r tunes the system across the quantum
critical point, and u is a quartic self-interaction. For case G, the generalization of
0
S is
φ
1
S = d2xdτ (∂ φ )2+(∂ φ )2+c2(∂ φ )2+c2(∂ φ )2+c2(∂ φ )2 (5)
φ Z (cid:20)2 τ x τ y 1 x x 2 y x 2 x y
(cid:8)
e 1
+ c2(∂ φ )2+e(∂ φ )(∂ φ )+r(φ2 +φ2) + u (φ4 +φ4)+2v φ2φ2 .
1 y y x x y y x y 24 0 x y 0 x y (cid:21)
(cid:9) (cid:8) (cid:9)
The final term in the action, S couples the bosonic and fermionic degrees of
Ψφ
freedom. From (2) we deduce for A–F that
S = d2xdτ λ φ Ψ† M Ψ +Ψ† M Ψ , (6)
Ψφ Z h 0 (cid:16) 1a 1 1a 2a 2 2a(cid:17)i
where λ is the requiredlinear coupling constant between the order parameterand
0
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a fermion bilinear. The matrices M , M are given by
1 2
(A) M =τy, M =τy
1 2
(B) M =τy, M = τy
1 2
−
(C) λ =0
0
(D) M =τx, M =τx
1 2
(E) M =τz, M = τz
1 2
−
(F) M =τx, M = τx (7)
1 2
−
Note that there is no non-derivative coupling between φ and Ψ for case C: the
fermions are essentially spectators of the transition for this case, which will not be
considered further. Finally, for case G, S generalizes to
Ψφ
S = d2xdτ λ φ Ψ†Ψ +φ Ψ†Ψ . (8)
Ψφ Z h 0(cid:16) x 1 1 y 2 2(cid:17)i
We canalsoconsiederthecouplingbetweentheorderparameterφandthephase
ofthesuperconductingorder,ϕ,in(1).Bysymmetry,thesimplestallowedcoupling
is φ2( ϕ)2. It is easy to show that such a coupling is irrelevant. So the power-law
∇
correlations generated by the superflow are not important.
We now describe our RG analysis of the 6 distinct field theories represented by
S +S +S andS +S +S .The familiarmomentum-shellmethod,inwhich
Ψ φ Ψφ Ψ φ Ψφ
degreesoffreedomwithmomentabetweenΛandΛ dΛaresuccessivelyintegrated
e e −
out, fails: the combination of momentum dependent renormalizations at one loop,
the direction-dependent velocities (v , v , c ...), and the hard momentum cut-off
F ∆
generate unphysicalnon-analytic terms in the effective action. So we will construct
RGequationsby16usingasoftcut-offatscaleΛ,andbytakingaΛ(d/dΛ)derivative
of the renormalized vertices and self energies. We write the bare propagators as
ip +v p τz +v p τx
G = 1 F 2 ∆ 3 K(p2/Λ2) (9)
Ψ1 p2+v2p2+v2p2
1 F 2 ∆ 3
and
1
G = K(p2/Λ2) (10)
φ p2
where K(y) is some decaying cuf-off function with K(0) = 1; e.g., K(y) = e−y
is a convenient choice. [The momentum-shell method would correspond to a step-
function cut-off K(y)=Θ(y 1).] To handle possible anisotropies we use a hybrid
−
approach; the space-time integrals over p = (ω ,p ,p ) are written down in D =
n x y
2+1 dimensions, they can be split via p = pn (where n is a unit vector) into
an angular integral dΩ containing all direction dependent information and an
n
integral over p= p Rwhich is then evaluated in D =4 ǫ dimensions.
| | −
Wedemonstratethismethodbycalculatingthelinear-orderfermionicself-energy
Σ for the cases A,B,D–F. The diagram shown in Fig 3b evaluates to
Ψ1
∞p2dp p2 (p q)2 ip +s v p τz +s v p τx
Σ (q)=λ2 dΩ K K − 1 2 F 2 3 ∆ 3 .
Ψ1 0Z 8π3 Z n (cid:18)Λ2(cid:19) (cid:18) Λ2 (cid:19)(p2+v2p2+v2p2)(p q)2
0 1 F 2 ∆ 3 −
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Ψ
a) φ 1
q −λ2 φ ( ωn2 A1 + c2qx2 A2 + c2qy2 A3) φ
φ
b) Ψ −λ2 Ψ+ ( −iω B + v pτz B + v pτxB ) Ψ
1 1 n 1 F x 2 ∆ y 3 1
p
c1) c2) c3)
−λ3 C1 φ Ψ1+ Ψ1 −u2D1 φ4 −λ4D2 φ4
Fig.3. Diagramsenteringtheone-loopRGequationsforcasesA,B,D–F.Fulllines(dashed)lines
denotefermionic(bosonic)propagators,filled(open)circlesrepresenttheλ0(u0)interaction.The
constants Ai, ... are obtained by evaluating the diagrams, expanding to lowest non-trivial order
intheexternal momentum andtaking theΛ(d/dΛ)derivative asdescribedinthe text. a)andb)
are the velocity renormalizations whereas c) contains the renormalized couplings. (Diagrams for
Ψ2 aresimilar.)
For Σ each vertex contains the coupling matrix M , the signs s , s = 1 are
Ψ1 1 2 3 ±
thereforegivenbyM τzM =s τz,M τxM =s τx.Expandingthe aboveexpres-
1 1 2 1 1 3
sion to linear order in q gives:
∞ dp p2K(p2/Λ2)K′(p2/Λ2)+Λ2K2(p2/Λ2)
Σ (q) =2λ2 −
Ψ1 0Z 8π3 (cid:20) p2Λ2 (cid:21)
µ
in2q +s v n2q τz +s v n2q τx
dΩ 1 1 2 F 2 2 3 ∆ 3 3 . (11)
×Z n n2+v2n2+v2n2
1 F 2 ∆ 3
We have inserted a lower limit µ in the p integralrepresenting an external momen-
tumtoregularizedtheinfrareddivergence.Notethattheonlyquantityenteringthe
RGequationsistheΛ(d/dΛ)derivativeoftheself-energy.Thisremovestheinfrared
divergence,i.e., wecantakethelimitΛ/µ inthefinalexpression.Writingthe
→∞
cut-off integral involving K in general dimension D we obtain
d 2λ2K in2q +s v n2q τz +s v n2q τx
Λ Σ = 0 1 dΩ 1 1 2 F 2 2 3 ∆ 3 3 , (12)
dΛ Ψ1 (2π)DΛ4−D Z n n2+v2n2+v2n2
1 F 2 ∆ 3
∞
K = dyy(D−4)/2 K(y)K′(y)+y[K′(y)]2+yK(y)K′′(y) .
1
Z −
0 (cid:2) (cid:3)
A simple integration by parts for D = 4 shows that K equals unity, independent
1
of K(y).
At this point it is useful to introduce dimensionless coupling constants λ and u
byλ =λΛǫ/2/S1/2andu =uΛǫ/S withS =Ω /(2π)D =2/[Γ(D/2)(4π)D/2]
0 D 0 D D D
and Ω being the areaof a D-dimensionalunit sphere.FromEq (12) we caneasily
D
read off the values of the B :
i
dΩ s n2
B = 2K n i i (13)
i − 1Z Ω n2+v2n2+v2n2
D 1 F 2 ∆ 3
with s = 1. The other diagrams in Fig 3 are evaluated similarly.
1
−
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The derivation of the RG equations is straightforward and essentially identical
to the momentum shell method. We construct flow equations for all velocities and
couplings to one-loop order in the non-linearities λ, u. It is convenient to choose
the time scale such that c=1 at eachstep (this introduces a non-trivialdynamical
critical exponent z). The procedure results in the following RG equations for cases
A,B,D–F:
A¯ A¯
β(v )= v λ2 B B + 1− 2 ,
F F 2 1
− (cid:18) − 2 (cid:19)
A¯ A¯
β(v )= v λ2 B B + 1− 2 ,
∆ ∆ 3 1
− (cid:18) − 2 (cid:19)
ǫ 3A¯ A¯
β(λ) = λ+λ3 2− 1 +B C ,
1 1
−2 (cid:18) 4 − (cid:19)
λ2u u2
β(u)= ǫu+ (A¯ +3A¯ ) D 24λ4D (14)
1 2 1 2
− 2 − 24 −
with the renormalization constants as defined in Fig 3 and A¯ = 2A , A¯ = A +
1 1 2 2
A . At a fixed point the dynamical critical exponent z and the anomalous field
3
dimensions are given by
z = 1+λ∗2(A¯ A¯ )/2,
1 2
−
η = λ∗2A¯ ,
b 2
η = λ∗2 B (A¯ A¯ )/2 . (15)
f 1 1 2
− −
(cid:2) (cid:3)
From Eq (14) we see that β(v )=β(v )=0 requires B =B at a non-trivial
F ∆ 2 3
fixedpoint.ThesignsoftheB (13)aredeterminedbys ,andthispreemptsafixed
i i
point for the cases D–F: the structure of M leads to s = s , so that B and
1,2 2 3 2
−
B have always different signs.
3
The cases A and B have s = s = 1, and it is easy to show that the fixed
2 3
−
point equations for v and v are simultaneously satisfied only for v∗ = v∗ = 1.
F ∆ F ∆
This implies that the resulting fixed point is Lorentz invariant;3 for D = 4 the
renormalization constants are given by A¯ = 4, B = 1/2, C = 1, D = 36,
1,2 i 1 1
− −
D =2. The RG equations have the infrared stable fixed point3 with z =1 and
2
ǫ 16ǫ
λ∗2 = , u∗ = . (16)
7 21
Let us briefly discuss the remaining case G. By choosing the time scale we keep
the velocity c =1 fixed. The flow equations for v , v , and λ have the same form
1 F ∆
as above, but with A¯ replaced by A , since each fermion field couples to only
1,2 1,2
one species of bosons. The other RG equations read:
β(e)= eλ2A , β(c2)=c2λ2(A A ),
2 2 2 3− 2
λ2u u2+v2
β(u)= ǫu+ (A +3A ) D 24λ4D ,
1 2 1 2
− 2 − 24 −
λ2v uv v2
β(v) = ǫv+ (A +3A ) E E (17)
1 2 1 2
− 2 − 24 − 24
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where E and E renormalizing v arise from diagrams similar to Fig 3c2. Analysis
1 2
oftheseequationsshowsthatβ(v )=β(v )=β(c2)=0is onlyfulfilled forc =1
F ∆ 2 2
and v = v . Furthermore we must have e = 0 at a fixed point, stability requires
F ∆
A > 0. Explicit evaluation in D = 4 shows that the zeros of β(v ) have A < 0
2 F 2
which proves the non-existence of a fixed point for case G.
We summarize this subsection by restating the main conclusions. Among the
Q = 0 order parameters considered here, only cases A and B possess stable RG
fixed points which can describe a second-order quantum phase transition to the
state X, and with strong damping of quasiparticle spectral functions. The fixed
points for both cases are Lorentz invariant, and for future convenience, we display
this explicitly. We perform the unitary transformation
(τx+τz)
Ψ = Ψ (18)
2a √2 2˙a
(this ensures that the structure of the τ matrices in (3) is the same for Ψ and
1
Ψ ), define Ψ = iτyΨ† , and introduce the Lorentz index µ = τ,x,y. Then
2˙ 1,2˙a − 1,2˙a
S +S +S can be written as
Ψ φ Ψφ
1 r u
S = d3x iΨ ∂ γµΨ +iΨ ∂ γµΨ + (∂ φ)2+ φ2+ 0φ4
1 Z (cid:16) 1a µ 1a 2˙a µ 2˙a 2 µ 2 24
iλ φ(Ψ Ψ Ψ Ψ ) , (19)
− 0 1a 1a∓ 2˙a 2˙a (cid:17)
where γµ = ( τy,τx,τz), we have set v = v = c = 1 and the (+) sign in the
F ∆
− −
last term is for case A (B).
2.2. Order parameters with Q = (2K,2K)
Oneplausiblecandidatefororderingatsuchawavevectorisantiferromagneticorder
at Q = (π,π), K = π/2; this case was discussed in Section 1, and has been ana-
lyzed by Balents et al.14 (The generalizationto magnetic order at incommensurate
wavevectors is not difficult, but we will not consider it because incommensuration
along the diagonal direction in the Brillouin zone has not been observed in the su-
perconducting cuprates.) We represent the strength of the antiferromagnetic order
by a real, three-component field N (α = x,y,z; for the incommensurate case N
α α
is complex).The transitionfroma d-wavesuperconductorwith N =0 to astate
α
h i
X which is a d-wave superconductor with N = 0 is described by the following
α
h i 6
continuum action near the critical point14
1 r u
S = d3x iΨ ∂ γµΨ +iΨ ∂ γµΨ + (∂ N )2+ N2+ 0(N2)2
2 Z (cid:16) 1a µ 1a 2˙a µ 2˙a 2 µ α 2 α 24 α
+iλ N ε σα [Ψ Ψ Ψ Ψ +H.c.] , (20)
0 α ac cb 1aC 1b− 2˙aC 2˙b (cid:17)
whereσα arePaulimatricesinspinspace,and =iτy isamatrixinNambuspace.
C
Like S , S has the property of Lorentz invariance (Ψ Ψ and ΨΨ are Lorentz
1 2
C
scalars). Also like S , the couplings λ and u approach14 fixed point values under
1