Disorder effects at a nematic quantum critical point in d-wave cuprate superconductor
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Disorder effects at a nematic quantum critical point in d -wave cuprate superconductor Jing Wang, Guo-Zhu Liu,
1, 2 and Hagen Kleinert Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China Institut f ¨ u r Theoretische Physik, Freie Universit ¨ a t Berlin, Arnimallee 14, D-14195 Berlin, Germany A d -wave high temperature cuprate superconductor exhibits a nematic ordering transition at zerotemperature. Near the quantum critical point, the coupling between gapless nodal quasiparticlesand nematic order parameter fluctuation can result in unusual behaviors, such as extreme anisotropyof fermion velocities. We study the disorder effects on the nematic quantum critical behavior andespecially on the flow of fermion velocities. The disorders that couple to nodal quasiparticles aredivided into three types: random mass, random gauge field, and random chemical potential. Arenormalization group analysis shows that random mass and random gauge field are both irrelevantand thus do not change the fixed point of extreme velocity anisotropy. However, the marginalinteraction due to random chemical potential destroys this fixed point and makes the nematic phasetransition unstable. PACS numbers: 73.43.Nq, 74.72.-h, 74.25.Dw
I. INTRODUCTION
One important reason that high-temperature cupratesuperconductors are hard to understand is that they havevery complicated phase diagram. The competitions andtransitions between different phases give rise to manyunusual properties, and hence have attracted consider-able theoretical and experimental efforts in the past twodecades. Among the widely studied competing orders,the various phase of the anisotropic electronic liquid areof particular interests. Kivelson, Fradkin, and Emeryproposed that due to the local electronic phase separa-tion, a number of novel electronic liquid crystal phasescan exist in a doped Mott insulator [1]. The simplestof such phases is the electronic nematic phase, in whichthe rotational symmetry is broken but the translationalsymmetry is preserved. In recent years, the nematic or-dering phase transition has been investigated extensively[2, 3]. The resistivity anisotropy observed by Ando etal. in two types of cuprate superconductors provided theearly evidence for the predicted nematic phase [4]. Morerecently, the neutron-scattering experiments performedin YBa Cu O . also pointed to the existence of nematicphase [5]. Further evidences came from the observed in-plane anisotropy of the Nernst effect in the pseudogapregion of YBa Cu O y [6] and from the scanning tunnel-ing microscopy experiments performed in the pseudogapregion of Bi Sr CaCu O δ [7]. Interestingly, there arealso compelling experimental indications for the existenceof nematic phase in Sr Ru O [8] and in newly discoverediron-based superconductor [9].In the language of field theory, the nematic phase hasan Ising-type order parameter that can be represented bya real scalar field φ . However, the dynamics of the systemclose to the critical point can not be fully described byan effective φ theory [10] when there are itinerant elec-trons. The interaction between quantum fluctuation ofthe nematic order parameter and the itinerant electronshas to be treated carefully. Close to the quantum crit-ical point, this interaction becomes singular, which was shown to be able to produce highly unusual, non-Fermiliquid like, behaviors [11–14]. The nematic physics is in-timately related to Pomeranchuk instability and has alsobeen investigated from this point of view [15, 16].Besides the pseudogap phase of underdoped cuprates,it is also interesting to study the nematic transition thatoccurs in the d -wave superconducting phase [17–21]. Thisis a new example of quantum phase transitions happeningin the superconducting dome [22–24], which is a widelystudied topic. In the superconducting phase, the nematicorder parameter interacts strongly with the gapless nodalquasiparticles, which are the low-energy excitations of a d -wave superconductor. This interaction remarkably af-fects the dynamics of both nodal quasiparticles and ne-matic order parameter. An early work of Vojta et al. [17]presented a detailed renormalization group (RG) analy-sis of various types of Yukawa couplings in the d -wavesuperconducting phase, including nematic type coupling.More recently, Kim et al. studied the effect of quantumfluctuations of nematic order parameter on the spectralproperties of nodal quasiparticles [18].In actual d -wave cuprate superconductor, the gaplessnodal quasiparticles have a Fermi velocity v F and a gapvelocity v ∆ , which are not equal. Indeed, the ratio v ∆ /v F may be as small as 1 /
20 [25]. This small ratio plays animportant role because it appears in a number of observ-able quantities [25]. For instance, it was found [26] thatthe dc thermal conductivity contains the large inverse ofthis ratio as κT ∝ k B ~ (cid:18) v ∆ v F + v F v ∆ (cid:19) (1)at nearly zero temperature. This easily accessible ma-terial property is universal — it is independent of theamount of disorder [26]. This universality was confirmedby transport measurements [27]. Since the inverse of v ∆ /v F is so large it completely dominates κ/T .An interesting problem is how the velocity ratio is in-fluenced by the nematic phase transition. Recently, Huhand Sachdev [19] studied this problem by making a care-ful RG analysis within an effective field theory of nematicordering transition. They found that v ∆ /v F flows to afixed point with v ∆ /v F →
0, i.e., the inverse velocityratio v F /v ∆ diverges. Therefore, the nematic orderingtransition in d -wave superconductor is accompanied bythe appearance of an extreme velocity anisotropy. Sincethe diverging velocity ratio v F /v ∆ enters various physicalproperties, the predicted extreme anisotropy should haveobservable effects. In particular, the low-temperature dcthermal conductivity is expected to be significantly en-hanced near the critical point. By using a Boltzmannequation approach, Fritz and Sachdev [21] calculatedthe thermal conductivity enhancement near the nematicquantum critical point due to the divergence of v F /v ∆ .If this enhancement were observed in transport experi-ments at certain doping concentration, this would serveas an important evidence for the existence of nematictransition in d -wave cuprate superconductor.When studying the low-temperature transport proper-ties of an interacting electron system, it is hardly possibleto ignore the disorder effects. First of all, the fermionsare always scattered by certain amount of disorder in anyrealistic physical system. Moreover, although the elasticscattering due to quenched disorder is less important athigh temperature, it dominates over the inelastic scatter-ing due to inter-particle interactions at very low temper-ature. The disorder effects should be taken into accountwhen calculating the low-temperature thermal conduc-tivity. The nematic order parameter fluctuation can leadto significant enhancement of thermal conductivity onlywhen the fixed point of extreme velocity anisotropy isstable against disorder scattering. If the fixed point ischanged or even destroyed by disorder, the thermal con-ductivity enhancement will not occur in practice. It istherefore crucial to examine the disorder effects on theRG flow of fermion velocities, especially on the stabilityof extreme anisotropy of velocities.In general, the disorders coupled to gapless nodalquasiparticles in d -wave superconductor can be dividedinto three types: random chemical potential, randomgauge field, and random mass. The difference comes fromthe different Pauli matrices used to define the fermion-disorder interacting terms. The effects of these disorderson the low-temperature transport properties of nodalquasiparticles have been discussed extensively [28, 29].These disorders will alter the RG flows of fermion ve-locities. On the other hand, the RG flow of strengthparameters of fermion-disorder couplings are determinedby fermion velocities, and thus should be calculated self-consistently with the flow of fermion velocities.In this paper, we present a RG analysis of the inter-play between nematic order parameter fluctuation anddisorder scattering. We derive a series of coupled RGequations of fermion Fermi velocity v F , gap velocity v ∆ ,and disorder strength parameter g . In the cases of ran-dom mass and random gauge field, the correspondingdisorder strength parameters both flow to zero at lowenergy. Therefore, these two kinds of disorders do not change the flow of fermion velocities and hence the fixedpoint of extreme velocity anisotropy is stable. However,the strength parameter of random chemical potential re-mains a constant even down to the lowest energy, andthus is able to modify the fermion velocities significantly.We found that the fermion velocities do not flow to anyfixed points, but indeed oscillate rapidly between posi-tive and unphysical negative values. This implies thatthe extreme anisotropy fixed point is destroyed and thenematic phase transition may become unstable due torandom chemical potential.In Sec. II, we define the model action in the presenceof both nematic order parameter fluctuation and disor-der. In Sec. III, we calculate the fermion self-energycorrections due to nematic order parameter and disorderscattering. The fermion-disorder vertex corrections dueto nematic and disorder interactions are also computedin this section. In Sec. IV, we make the RG analysis andobtain the self-consistent RG equations for fermion veloc-ities and disorder strength parameter. These equationsare solved both analytically and numerically in Sec. V.From the solutions, we found that the fixed point of ex-treme velocity anisotropy is not changed by random massand random gauge field. However, the random chemicalpotential destroys this fixed point and indeed makes thenematic phase transition unstable. In Sec. VI, we brieflysummarize the results obtained in this paper and discussthe possible experimental detection of the predicted ex-treme velocity anisotropy. II. MODEL
We start from the following action S = S ψ + S φ + S ψφ , (2)where the free action for nodal quasiparticles is S ψ = Z d k (2 π ) dω π ψ † a ( − iω + v F k x τ z + v ∆ k y τ x ) ψ a + Z d k (2 π ) dω π ψ † a ( − iω + v F k y τ z + v ∆ k x τ x ) ψ a , (3)where τ ( x,y,z ) denote Pauli matrices. The linear disper-sion of Dirac fermions originates from the d x − y -wavesymmetry of the energy gap of cuprate superconductor.Here, the spinor ψ † represents nodal quasiparticles ex-cited from the ( π , π ) and ( − π , − π ) nodal points, and ψ † the other two nodal points [17]. The repeated spin index a is summed from 1 to N f , the number of fermion spincomponents. The ratio v ∆ /v F ≈ /
20 between Fermivelocity and gap velocity is determined by experiments[25]. The effective action S φ describes the Ising type ne-matic order parameter, which is expanded (for notationalsimplicity) in real space as S φ = Z d x dτ n
12 ( ∂τ φ ) + c ∇ φ ) + r φ + u φ o , (4)where τ is imaginary time and c is velocity. The massparameter r tunes the nematic phase transition with r =0 defining the quantum critical point. The parameter u is the quartic self-interaction strength. The nematicorder parameter couples to nodal quasiparticles via theYukawa term S ψφ = Z d x dτ { λ φ ( ψ † a τ x ψ a + ψ † a τ x ψ a ) } . (5)Following Huh and Sachdev [19], we now perform theRG analysis in the framework of a 1 /N f expansion. Theinverse of the free propagator of the nematic order pa-rameter field behaves as q + r . After taking into accountthe polarization effects, there will be an additional lin-ear q -term. At low energy regime, the q -term dominatesover the q -term, which then can be neglected. Near thequantum critical point, we keep only the mass term andassume that φ −→ φ/λ and r −→ N f rλ , leading to S = S ψ + Z d x dτ n N f r φ + φ [ ψ † a τ x ψ a + ψ † a τ x ψ a ] o . (6)After integrating out fermion degrees of freedom, the ef-fective action for the scalar field becomes S φ N = 12 Z d q (2 π ) [ r + Π( q )] | φ ( q ) | + O ( φ ) . (7)The lowest-order Feynman diagram for the polarizationfunction is shown in Fig. 1 and symbolizes the integralΠ( q , ǫ ) = Z d k (2 π ) dω π Tr[ τ x G ψ ( k , ω ) τ x G ψ ( k + q , ω + ǫ )] , where the free fermion propagator is G ψ ( k , ω ) = 1 − iω + v F k x τ z + v ∆ k y τ x . (8)As shown previously [19], the propagator for the nematicorder parameter is given by G − φ ( q , ǫ ) = Π( q , ǫ )= 116 v F v ∆ ( ǫ + v F q x )( ǫ + v F q x + v q y ) / + 116 v F v ∆ ( ǫ + v F q y )( ǫ + v F q y + v q x ) / (9)in the vicinity of nematic quantum critical point r = 0.Disorders are present in almost all realistic condensedmatter systems and play important roles in determiningthe low-temperature behaviors. In the present problem,the nodal quasiparticles can interact with three typesof random potentials, which represent different disorderscattering processes. According to the coupling betweennodal quasiparticle and disorders, there are three types ofrandom fields in d -wave superconductors: random mass,random chemical potential, and random gauge potential.All these types of disorders have been investigated in the FIG. 1: The polarization function for nematic order param-eter. The solid line represents the fermion propagator andwavy line represents the boson propagator.FIG. 2: One loop fermion self-energy correction due to (a)nematic order parameter fluctuation and (b) disorder. Thedashed line represents disorder scattering. contexts of d -wave cuprate superconductor [28, 29], quan-tum Hall effect [30], and graphene [31, 32]. In the gen-eral analysis to follow, we shall consider the three typesof disorders.The fermion field couples to a random field A ( x ) as Z d x ψ † ( x )Γ ψ ( x ) A ( x ) , (10)For random chemical potential, the matrix Γ is Γ = I.For a random mass it is, Γ = τ y , and for a random gaugefield Γ = ( τ x , τ z ). The random potential A ( x ) is assumedto be a quenched, Gaussian white noise field with thecorrelation functions h A ( x ) i = 0; h A ( x ) A ( x ) i = gv δ ( x − x ) . (11)The dimensionless parameter g represents the concen-tration of impurity, and the parameter v Γ measures thestrength of a single impurity. It will be convenient toredefine the random potential as A ( x ) → v Γ A ( x ), andthen write the fermion-disorder interaction term as [31] S dis = v Γ Z d x ψ † ( x )Γ ψ ( x ) A ( x ) , (12)with the random potential distribution h A ( x ) i = 0; h A ( x ) A ( x ) i = gδ ( x − x ) . (13)Now the RG flow of disorder strength can be calculatedby studying the vertex correction to the fermion-disorderinteraction term. After a Fourier transformation, the cor-responding action has the form S dis = v Γ Z d k d k dωψ † ( k , ω )Γ ψ ( k , ω ) A ( k − k ) . (14)This action will be analyzed together with the actions (3),(6), and (7). In order to perform perturbative expansion,we assume that g and v Γ are both small in magnitude,corresponding to the weak disorder case. III. FERMION SELF-ENERGY ANDFERMION-DISORDER VERTEX CORRECTIONS
According to the Dyson equation, the interactions in-duce a self-energy correction to the free propagator ofDirac fermion, yielding G − ψ ( k , ω ) = − iω + v F k x τ z + v ∆ k y τ x − Σ nm ( k , ω ) − Σ dis ( k , ω ) , (15)where self-energy functions Σ nm and Σ dis come from ne-matic ordering and disorder scattering, respectively. Tothe leading order, the corresponding Feynman diagramsare presented in Fig. 2.The nematic self-energy Σ nm has already been ob-tained by Huh and Sachdev [19], who found that d Σ nm ( k , ω ) d ln Λ = C ( − iω ) + C v F k x τ z + C v ∆ k y τ x , (16)where C = 2( v ∆ /v F ) N f π Z ∞−∞ dx Z π dθ × x − cos θ − ( v ∆ /v F ) sin θ ( x + cos θ + ( v ∆ /v F ) sin θ ) G ( x, θ ) , (17) C = 2( v ∆ /v F ) N f π Z ∞−∞ dx Z π dθ × cos θ − x − ( v ∆ /v F ) sin θ ( x + cos θ + ( v ∆ /v F ) sin θ ) G ( x, θ ) , (18) C = 2( v ∆ /v F ) N f π Z ∞−∞ dx Z π dθ × x + cos θ − ( v ∆ /v F ) sin θ ( x + cos θ + ( v ∆ /v F ) sin θ ) G ( x, θ ) , (19) G − = x + cos θ q x + cos θ + ( v ∆ /v F ) sin θ + x + sin θ q x + sin θ + ( v ∆ /v F ) cos θ . (20)The computational details of C , , are presented in theappendix.The fermion self-energy due to disorder Σ dis ( iω ) canbe computed asΣ dis ( iω ) = gv Z d k (2 π ) Γ G ψ ( k , ω )Γ= gv πv F v ∆ iω ln Λ . (21)From this expression, we know that Σ dis ( iω ) has the sameresult for all possible expressions of Γ. Another impor-tant feature is that Σ dis ( iω ) is independent of momen-tum, which reflects the fact that the quenched disorderis static. It is easy to have d Σ dis ( iω ) d ln Λ = C g iω, (22) FIG. 3: Fermion-disorder vertex correction due to (a) nematicorder parameter and (b) disorder parameter. where C g = gv πv F v ∆ . (23)The fermion-disorder interaction parameter v Γ is alsosubjected to RG flow. To get its flow equation, we needto calculate the fermion-disorder vertex corrections. For-mally, the vertex correction has the form v Γ Γ ′ = v Γ Γ + V nm + V dis , (24)where V nm represents the vertex correction due to ne-matic order parameter fluctuation and V dis represents thevertex correction due to disorder interaction. The corre-sponding diagrams are shown in Fig. 3. They will becalculated explicitly in the following for all three kindsof disorders.
1. Random chemical potential
We first calculate the vertex correction due to nematicordering. To this end, we employ the method proposedby Huh and Sachdev [19]. At zero external momenta andfrequencies, the vertex correction is expressed as V nm = v Γ Z d Q (2 π ) H ( Q ) K ( q Λ ) . (25)There is an useful formula [19], dV nm d ln Λ = v Γ v F π Z ∞−∞ dx Z π dθH ( ˆ Q ) , (26)where H ( ˆ Q ) = 1 N f τ x − iv F x + v F cos θτ z + v ∆ sin θτ x ) I × − iv F x + v F cos θτ z + v ∆ sin θτ x ) τ x Q ) . (27)Here, the matrix I corresponds to the coupling betweenDirac fermion and random chemical potential. It will bereplaced by τ y in the case of random mass and τ x,z inthe case of random gauge field. After straightforwardcomputation, we have dV nm d ln Λ = C v Γ I , (28)where C = − v ∆ /v F ) N f π Z ∞−∞ dx Z π dθ × ( x − cos θ − ( v ∆ /v F ) sin θ )( x + cos θ + ( v ∆ /v F ) sin θ ) G ( x, θ )= − C . (29)The vertex correction due to averaging over disorder is V dis = gv Z d p (2 π ) I G ψ ( ω, p ) v Γ I G ψ ( ω, p + k )I . (30)Again, the matrix I should be replaced by certain Paulimatrix in the case of random mass or random gauge field.Taking the external momentum k = 0 and keeping onlythe leading divergent term, we have dV dis d ln Λ = C Γ v Γ I , (31)where C Γ = v g πv F v ∆ = C g . (32)
2. Random mass
The calculation of vertex correction in the case of ran-dom mass parallels the process presented above, so wejust state the final result. The nematic ordering inducedvertex correction is dV nm d ln Λ = C v Γ τ y , (33)where C = 2( v ∆ /v F ) N f π Z ∞−∞ dx Z π dθ × ( x + cos θ + ( v ∆ /v F ) sin θ )( x + cos θ + ( v ∆ /v F ) sin θ ) G ( x, θ )= C − C − C . (34)The disorder induced vertex correction is dV dis d ln Λ = − C Γ ( v Γ τ y ) , (35)where C Γ = v g πv F v ∆ = C g . (36)
3. Random gauge potential
The random gauge potential has two components,characterized by τ x and τ z respectively. For the τ x com-ponent, the nematic ordering contribution to vertex cor-rection is dV nm d ln Λ = C A v Γ τ x , (37)where C A = − v ∆ /v F ) N f π Z ∞−∞ dx Z π dθ × ( x + cos θ − ( v ∆ /v F ) sin θ )( x + cos θ + ( v ∆ /v F ) sin θ ) G ( x, θ )= − C . (38)For the τ z component, we have dV nm d ln Λ = C B v Γ τ z , (39)where C B = − v ∆ /v F ) N f π Z ∞−∞ dx Z π dθ × ( x − cos θ + ( v ∆ /v F ) sin θ )( x + cos θ + ( v ∆ /v F ) sin θ ) G ( x, θ )= − C . (40)The disorder contribution can be calculated similarly.To both the τ x and τ z component, we have V dis ( ω ) = finite , (41)so that dV dis d ln Λ = 0 . (42)for both two components. IV. RG EQUATIONS FOR FERMIONVELOCITIES AND DISORDER STRENGTH
In order to perform the RG analysis of the fermionvelocity and disorder strength, it is convenient to makethe following scaling transformations [19, 33] k i = k ′ i e − l , (43) ω = ω ′ e − l , (44) ψ , ( k , ω ) = ψ ′ , ( k ′ , ω ′ ) e R l (4 − η f ) dl , (45) φ ( q , ǫ ) = φ ′ ( q ′ , ǫ ′ ) e R l (5 − η b ) dl , (46)where i = x, y and b = e − l with l >
0. The parameters η f and η b will be determined by the self-energy and nematic-fermion vertex corrections. Note the energy is required toscale in the same way as the momentum, so the fermionvelocities are forced to flow under RG transformations.In the spirit of RG theory [33], to specify how a fieldoperator transforms when the energy and momenta arere-scaled, the standard method is to require that its ki-netic term remains invariant. In the present problem,however, the random potential A ( x ) does not have anown kinetic term. In order to find out its scaling behav-ior, we write the Gaussian white noise distribution in themomentum space as h A ( k ) A ( k ) i = gδ ( k + k ) . (47)When the momentum k becomes b k , the delta functionis rescaled to δ ( k + k ) → δ ( b k + b k ) = b − δ ( k + k ) . (48)If we require that the disorder distribution Eq. (47) isinvariant under scaling transformations, then the randompotential should transform as A ( k ) → b − A ( k ) . (49)Now we have to assume that A ( k ) = A ′ ( k ′ ) e l . (50)According to the RG technique presented in [33], themomentum shell between b Λ and Λ will be integratedout, while keeping the − iω term invariant. From thenematic ordering and disorder contributions to fermionself-energy function, we have Z b Λ d k dωψ † [ − iω − C ( − iω ) ln Λ b Λ + C g ( − iω ) ln Λ b Λ ] ψ = Z b Λ d k dωψ † ( − iω )[1 + ( C g − C ) l ] ψ ≈ Z b Λ d k dωψ † ( − iω ) e ( C g − C ) l ψ. (51)After the scaling transformation, this term should goback to the free form, so that η f = C g − C . (52)The kinetic terms should also be kept invariant underscaling transformation, which leads to dv F dl = ( C − C − C g ) v F , (53) dv ∆ dl = ( C − C − C g ) v ∆ . (54)Based on these expressions, the ratio between gap veloc-ity and Fermi velocity is d ( v ∆ /v F ) dl = ( C − C )( v ∆ /v F ) . (55) The disorder strength g appears in the above expres-sions. Due to the interplay of nematic ordering and disor-der, this parameter also flows under RG transformation.The flow equation depends on the type of disorder, whichwill be studied in the following.We first consider the case of random chemical poten-tial. The bare fermion-disorder action is v Γ Z d k d k dωψ † ( k , ω )Γ ψ ( k , ω ) A ( k − k ) . (56)Including corrections due to nematic and disorder inter-actions yields Z b Λ d k d k dωψ † ( k , ω )[ v Γ I − C v Γ I ln Λ b Λ+ C g v Γ I ln Λ b Λ ] ψ ( k , ω ) A ( k − k )= Z b Λ d k d k dωψ † ( k , ω ) v Γ I[1 + ( C g − C ) l ] × ψ ( k , ω ) A ( k − k ) ≈ Z b Λ d k d k dωψ † ( k , ω ) v Γ I e ( C g − C ) l × ψ ( k , ω ) A ( k − k ) . (57)After redefining energy, momentum, and field operators,we have Z Λ d k ′ d k ′ dω ′ ψ ′† ( k ′ , ω ′ ) v Γ I × e ( C g − C ) l ψ ′ ( k ′ , ω ′ ) e − η f l A ′ ( k ′ − k ′ ) . (58)Since η f = C g − C , it is easy to obtain the following RGflow equation for v Γ , dv Γ dl = 0 . (59)Apparently, the parameter v Γ does not flow and thus canbe simply taken to be a constant.In the case of random mass, the flow equations forfermion velocities have the same expressions as Eq.(53)and Eq.(54). However, the flow equation for disorderstrength is different from Eq.(58), and has the form dv Γ dl = ( C − C − C g ) v Γ , (60)which couples self-consistently to flow equations offermion velocities.Following the steps presented above, we find the follow-ing RG equations in the case of random gauge potential dv F dl = ( C − C − C gi ) v F , (61) dv ∆ dl = ( C − C − C gi ) v ∆ , (62)which couple to the flow equations of disorder strength dv Γ1 dl = [( C − C g ) − C ] v Γ1 , (63) dv Γ2 dl = [( C − C g ) − C ] v Γ2 , (64) - - - - - g=10 -7 g=10 -5 g=10 -4 g=10 -3 g=10 -2 e - l v / v ( a ) - - - - - v / v ( b ) g=10 -7 g=10 -5 g=10 -4 g=10 -3 g=10 -2 e - l g=10 -7 g=10 -5 g=10 -4 g=10 -3 g=10 -2 - - - - - v / v (c) e - l FIG. 4: (a) v Γ for random mass; (b) v Γ for random gaugepotential τ x component; (c) v Γ for random gauge potential τ z component. where C g i = v i g πv F v ∆ , i = 1 , . (65)Here, the equations denoted by i = 1 , τ x and τ z components of random gauge potential,respectively. V. STABILITY OF EXTREME ANISOTROPYAGAINST DISORDERS
The RG flows of fermion velocities v F and v ∆ withgrowing scale l can be obtained by numerically solvingthe corresponding coupled equations with the initial val-ues v F , v ∆0 , and v Γ0 . First of all, in the clean limit -8 -7 -7 -7 g=0 g=10 -7 g=10 -5 g=10 -4 g=10 -3 g=10 -2 v F / v F0 e - l -10 -10 -10 -10 e - l v / v g=0 g=10 -7 g=10 -5 g=10 -4 g=10 -3 g=10 -2 FIG. 5: The flows of v F (upper one) and v ∆ (lower one) inthe case of random mass. g = 0, the equations reduce to that obtained by Huh andSachdev. In this case, it was already known that an ex-treme anisotropy of fermion velocities, i.e., v ∆ /v F →
0, iscaused by nematic order parameter fluctuation. The ef-fects of various disorders on this fixed point will be trans-parent when the dimensionless parameter g is increasedsmoothly.We first consider the case of random mass. As l grows, v Γ first increases and then decreases, eventuallyapproaching zero, as shown in Fig. 4(a). Although theRG equations for fermion velocities are modified by scat-tering due to random mass, the disorder parameter v Γ flows to zero as l → ∞ . Apparently, the random mass isirrelevant in the present problem. As can be easily seenfrom see Fig. 5, the fermion velocities v F and v ∆ bothdecrease as l grows. More concretely, v ∆ goes down tozero rapidly, but v F decreases much more slowly and ac-tually approaches a finite value. These results imply theexistence of extreme velocity anisotropy with v ∆ /v F → v Γ with growing l areshown in Fig. 4(b) for component τ x and in Fig. 4(c) forcomponent τ z . For both components, the correspond-ing v Γ decrease as l grows and finally approaches zero as l → ∞ . Similar to the case of random mass, the randomgauge potential makes no important contributions to theflow of fermion velocities. Therefore, as in the clean limit,both v F and v ∆ decreases with l until approaching zero -8 -7 -7 -7 e - l v F / v F0 g=0 g=10 -7 g=10 -5 g=10 -4 g=10 -3 g=10 -2 -10 -10 -10 -10 g=0 g=10 -7 g=10 -5 g=10 -4 g=10 -3 g=10 -2 e - l v / v FIG. 6: The running v F and v ∆ for the τ x component ofrandom gauge potential. The running of fermion velocitiesfor component τ z are very similar to this case, and thus arenot shown. for component τ x depicted in Fig. 6; the flows of v F and v ∆ for component τ z with growing l will not be shownsince they are similar to those in the case of component τ x . It is obvious that random gauge potential does notchange the extreme velocity anisotropy.Unlike random mass and random gauge potential, thedisorder strength parameter v Γ in the case of randomchemical potential does not flow with l and thus shouldbe kept as a constant. As such, the influence of scat-tering due to random chemical potential can not be ne-glected and indeed the flows of velocities v F and v ∆ de-pend heavily on the magnitude of v Γ . At first glance,the flow equation of velocity ratio v ∆ /v F is independentof disorder strength v Γ , as shown in Eq. (55), and thusappears to have fixed point at v ∆ /v F = 0 as in the cleanlimit. However, this solution is artificial. In the presentproblem, the flow equation of v ∆ /v F is derived from themore fundamental equations of v ∆ and v F , and thereforeis reliable only when v ∆ and v F have well-defined fixedpoints. If the equations for v ∆ and v F have no fixedpoints, the equation of v ∆ /v F becomes meaningless.To see the effect of random chemical potential, wemake a qualitative analysis based on the RG equationsof fermion velocities v ∆ and v F . The fixed points can beobtained by requiring that dv F dl = ( C − C ) v F − v g πv ∆ = 0 , (66) dv ∆ dl = ( C − C ) v ∆ − v g πv F = 0 . (67)We assume that v ∗ F and v ∗ ∆ correspond to the fixed points.If both v ∗ F and v ∗ ∆ are finite, then the above equationsimply that ( C − C ) v ∗ ∆ v ∗ F = ( C − C ) v ∗ ∆ v ∗ F , which cannot be satisfied since v ∗ ∆ = 0. If v ∗ ∆ = 0, then v ∗ F = v g πv ∗ ∆ ( C − C ) . (68)From the expressions for C and C , this implies that1 ∝ / ( v ∗ ∆ ) , which is clearly inconsistent with the as-sumption of v ∗ ∆ = 0. Before going to v ∗ F = 0 case, wedefine D i = ( v F /v ∆ ) C i , i = (1 , , D − D ) v ∆ = v g πv ∆ , (69)( D − D ) v v F = v g πv F . (70)If v ∗ F = 0, by both analytical and numerical analysis, wefound that these equations have no solution.In summary, as shown by the above analysis, there isno fixed point of the fermion velocities v F and v ∆ whenthe fermions interact with random chemical potential.Therefore, v F and v ∆ do not approach any stable val-ues at the low energy regime. Alternatively, straight-forward numerical calculations show that they oscillaterapidly between positive and unphysical negative valuesas l grows. In this case, the extreme velocity anisotropyfixed point is destroyed. We interpret the occurrence ofunphysical negative velocities as a signature of the in-stability of nematic phase transition in the presence ofrandom chemical potential. VI. SUMMARY AND DISCUSSION
In summary, we have examined the disorder effect nearthe critical point of nematic phase transition in d -wavecuprate superconductor. We considered three types ofquenched disorders that couple directly to the gaplessnodal quasiparticles: random mass, random gauge field,and random chemical potential. By means of a RG anal-ysis, we have derived a series of self-consistent flow equa-tions for fermion velocities and disorder strength. It wasfound that the fixed point of extreme velocity anisotropydue to critical fluctuation of nematic order parameter isnot changed by random mass and random gauge field,which are both irrelevant at low energy. Therefore, itseems reasonable to expect an enhancement of dc ther-mal conductivity at low temperature if there are onlythese two kinds of disorders. However, when there ismoderately strong random chemical potential, which ismarginal, the extreme anisotropy fixed point is destroyed.Moreover, the nematic phase transition may become un-stable in the presence of such random chemical potential.The extreme velocity anisotropy produced by the crit-ical nematic fluctuations may be probed by the heattransport measurements, since it leads to a remarkableenhancement of the low-temperature thermal conductiv-ity. Apart from transport measurements, such extremeanisotropy can also show its existence in angle-resolvedphoto-emission spectroscopy (ARPES) experiments. In-deed, the currently known value of the velocity ratio v ∆ /v F in d -wave cuprate superconductors was extractedfrom both heat transport [34] and ARPES measurements[35]. Unlike heat transport experiments that can only es-timate the ratio v ∆ /v F , the ARPES measurements areable to determine the Fermi velocity v F and the gapvelocity v ∆ separately [35]. From the solutions of RGequations, we know that, the extreme velocity anisotropyemerges because v ∆ is driven by the critical nematic fluc-tuations to drop rapidly down to zero at large l but v F is driven to decrease very slowly. It should be possible todetect the extreme anisotropy by measuring v F and v ∆ separately by means of the ARPES experiments.In addition, the effects of the nematic order parameterfluctuations can also be reflected in the single-particlespectral function of nodal quasiparticles. Kim et al. [18]investigated this issue and found two important features:strong angle-dependence of quasipaticle scattering, andan enhancement of velocity anisotropy. These predictedfeatures of the fermion spectral function are expected tobe tested by ARPES experiments.We next would like to remark on the disorder effects onthe polarization function and the final results. In our cal-culations, the polarization function is obtained from thebubble diagram shown in Fig. 1, since including inter-nal disorder scattering line will introduce an additionalsuppressing factor gv . To justify this approximation,we now make a qualitative analysis on the disorder ef-fects. It is well known that disorder scattering generatesa fermion damping rate γ , which shifts the energy ofDirac fermions from ω to ω + iγ . In the presence offinite γ , it seems possible to get a full analytical expres-sion for the polarization function Π( q x , q y , ǫ ) only in thestatic ( ǫ = 0) limit. Following the computational proce-dures given in [36, 37], we haveΠ( q x , q y , γ ) = 12 π v F v ∆ Z dx p x (1 − x ) v F q x q v F q x + v q y × arctan (cid:16) γ − q x (1 − x )( v F q x + v q y ) (cid:17) +( q x ←→ q y ) . (71) In order to simplify this expression, we now con-sider the low-energy regime, | q | ≤ γ , whichleads to arctan (cid:16) γ − q x (1 − x )( v F q x + v q y ) (cid:17) ≈ γ − q x (1 − x )( v F q x + v q y ). Using this approximation,the polarization function is found to beΠ( q x , q y , γ ) = v F π v ∆ γ ( q x + q y ) . (72)Substituting this new polarization to the nematic prop-agator and then performing the same RG calculations,we found that the qualitative results presented in Sec. Vdo not change. We thus conclude that it is justified toneglect disorder effects in the polarization function forweak disorders. Admittedly, when disorders are strongenough to cause Anderson localization, the RG approachutilized in our manuscript is no longer applicable and anew RG scheme is needed to deal with the vertex correc-tions generated by disorder scattering.In this paper, we have considered only the coupling be-tween disorders and fermionic nodal quasiparticles sincewe are mainly interested in the disorder effects on theRG flow of fermion velocities. In practice, the effectsof disorders on the nematic transition are more compli-cated. For instance, there might be quenched disordersthat couple directly to the nematic order parameter. Thisissue was briefly discussed by Kim et al. , who argued[18] that quenched disorder may smear the symmetry-breaking type quantum phase transition thereby produc-ing a glassy state. It is currently unclear how the nodalquasiparticles are influenced by such kind of disorders. VII. ACKNOWLEDGMENTS
G.Z.L. and H.K. would like to thank Flavio S. Nogueirafor valuable discussions. J.W. is grateful to Wei Li andJing-Rong Wang for their helps. G.Z.L. acknowledgesthe financial support from the National Science Founda-tion of China under Grant No.11074234 and the projectsponsored by the Overseas Academic Training Funds ofUniversity of Science and Technology of China.
Appendix
In order to maintain a self-consistency of this paper,here we provide a detailed calculation of the fermion self-energy (16). Following Ref. [19], we can defineΣ nm ( K ) = Z d Q (2 π ) F ( Q + K ) G ( Q ) K (cid:18) ( q + k ) Λ (cid:19) K (cid:18) q Λ (cid:19) , (73)where K ≡ ( k , ω ) and Q ≡ ( q , ǫ ) are 3-momenta. Here K ( y ) is an arbitrary function with K (0) = 1, and it falls offrapidly with y , e.g. K ( y ) = e − y . However, the results are independent of the particular choices of K ( y ). It is easy to0identify that, G ( Q ) = 1Π( q , ǫ ) , (74) F ( Q + K ) = 1 N f i ( ǫ + ω ) − v F ( q x + k x ) τ z + v ∆ ( q y + k y ) τ x ( ǫ + ω ) + v F ( q x + k x ) + v ( q y + k y ) . (75)Expanding F ( Q + K ) K (cid:16) ( q + k ) Λ (cid:17) at Q + K = Q , and retaining the first order, we have F ( Q + K ) K (cid:18) ( q + k ) Λ (cid:19) ≈ K µ (cid:20) ∂F ( Q + K ) ∂Q µ K (cid:18) ( q + k ) Λ (cid:19) + F ( Q + K ) 2( q + k ) µ Λ K ′ (cid:18) ( q + k ) Λ (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) Q µ + K µ = Q µ = K µ (cid:20) ∂F ( Q ) ∂Q µ K (cid:18) q Λ (cid:19) + F ( Q ) 2 q µ Λ K ′ (cid:18) q Λ (cid:19)(cid:21) . (76)Then the self-energy becomesΣ nm ( K ) ≈ K µ Z d Q (2 π ) (cid:20) ∂F ( Q ) ∂Q µ G ( Q ) K (cid:18) q Λ (cid:19) + F ( Q ) G ( Q ) 2 q µ Λ K (cid:18) q Λ (cid:19) K ′ (cid:18) q Λ (cid:19)(cid:21) , (77)which leads to d Σ nm ( K ) d Λ = K µ Z d Q (2 π ) (cid:26)h − q Λ ∂F ( Q ) ∂Q µ − F ( Q ) q µ Λ i G ( Q ) K (cid:16) q Λ (cid:17) K ′ (cid:16) q Λ (cid:17) − q q µ Λ F ( Q ) G ( Q ) h K (cid:16) q Λ (cid:17) K ′′ (cid:16) q Λ (cid:17) + K ′ (cid:16) q Λ (cid:17)i(cid:27) . (78)It is convenient to introduce the following cylindrical coordinates, Q µ = y Λ( v F x, cos θ, sin θ ) , ˆ Q µ = ( v F x, cos θ, sin θ ) ,q µ = y Λ(0 , cos θ, sin θ ) , ˆ q µ = (0 , cos θ, sin θ ) ,d Q = y Λ v F dxdydθ. It is straightforward to obtain F ( ˆ Q ) = 1 N f v F (cid:18) ix − cos θτ z + ( v ∆ /v F ) sin θτ x x + cos θ + ( v ∆ /v F ) sin θ (cid:19) ,G ( ˆ Q ) = 1Π( ˆ Q ) = 16 v ∆ G ( x, θ ) , (79)where G − = x + cos θ q x + cos θ + ( v ∆ /v F ) sin θ + x + sin θ q x + sin θ + ( v ∆ /v F ) cos θ . (80)Since F and G are homogenous functions, F ( Q ) = 1 y Λ F ( ˆ Q ) , G ( Q ) = 1 y Λ G ( ˆ Q ) . (81)1We thus haveΛ d Σ nm ( K ) d Λ ≈ v F K µ (2 π ) Z ∞−∞ dx Z π dθ Z ∞ dy nh − y ∂F ( ˆ Q ) ∂ ˆ Q µ − y ˆ q µ F ( ˆ Q ) i G ( ˆ Q ) K ( y ) K ′ ( y ) − y ˆ q µ F ( ˆ Q ) G ( ˆ Q ) h K ( y ) K ′′ ( y ) + K ′ ( y ) io = v F K µ π Z ∞−∞ dx Z π dθ nh − ∂F ( ˆ Q ) ∂ ˆ Q µ − q µ F ( ˆ Q ) i G ( ˆ Q ) Z ∞ ydy K ( y ) K ′ ( y ) − q µ F ( ˆ Q ) G ( ˆ Q ) Z ∞ y dy h K ( y ) K ′′ ( y ) + K ′ ( y ) io . (82)After integrating y out, we find Z ∞ y dy h K ( y ) K ′′ ( y ) + K ′ ( y ) i = − Z ∞ ydy K ( y ) K ′ ( y ) = 14 . (83)Therefore, Λ d Σ nm ( K ) d Λ = v F K µ π Z ∞−∞ dx Z π dθ nh ∂F ( ˆ Q ) ∂ ˆ Q µ + ˆ q µ F ( ˆ Q ) i G ( ˆ Q ) − ˆ q µ F ( ˆ Q ) G ( ˆ Q ) o = v F K µ π Z ∞−∞ dx Z π dθ ∂F ( ˆ Q ) ∂ ˆ Q µ G ( ˆ Q ) . (84)Formally, the fermion self-energy function can be expanded as d Σ nm ( K ) d ln Λ = C ( − iω ) + C v F k x τ z + C v ∆ k y τ x . (85)When K = ω and ˆ Q = v F x , we finally have C ( − iω ) = v F ω π Z ∞−∞ dx Z π dθ ∂F ( ˆ Q ) ∂v F x G ( ˆ Q )= 2 v ∆ ωN f π v F Z ∞−∞ dx Z π dθ i ( x + cos θ + ( v ∆ /v F ) sin θ ) − x ( ix − cos θτ z + ( v ∆ /v F ) sin θτ x )( x + cos θ + ( v ∆ /v F ) sin θ ) G ( x, θ )= 2( v ∆ /v F ) N f π Z ∞−∞ dx Z π dθ x − cos θ − ( v ∆ /v F ) sin θ ( x + cos θ + ( v ∆ /v F ) sin θ ) G ( x, θ )( − iω ) , (86)which directly leads to C = 2( v ∆ /v F ) N f π Z ∞−∞ dx Z π dθ x − cos θ − ( v ∆ /v F ) sin θ ( x + cos θ + ( v ∆ /v F ) sin θ ) G ( x, θ ) . (87) C and C can be obtained similarly. [1] S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature(London), , 550 (1998).[2] E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein,and A. P. Mackenzie, Annu. Rev. Condens. Matter Phys. , 153 (2010).[3] M. Vojta, Adv. Phys. , 699 (2009).[4] Y. Ando, K. Segawa, S. Komiya, and A. N. Lavrov, Phys.Rev. Lett. , 137005 (2002). [5] V. Hinkov, D. Haug, B. Fauque, P. Bourges, Y. Sidis, A.Ivanov, C. Bernhard, C. T. Lin, and B. Keimer, Science , 597 (2008).[6] R. Daou, J. Chang, D. LeBoeuf, O. Cyr-Choiniere, F.Laliberte, N. Doiron-Leyraud, B. J. Ramshaw, R. Liang,D. A. Bonn, W. N. Hardy, and L. Taillefer, Nature (Lon-don) , 519 (2010).[7] M. J. Lawler, K. Fujita, Jhinhwan Lee, A. R. Schmidt, Y. Kohsaka, Ch. K. Kim, H. Eisaki, S. Uchida, J. C. Davis,J. P. Sethna, and E.-A. Kim, Nature , 347 (2010).[8] R. A. Borzi, S. A. Grigera, J. Farrell, R. S. Perry, S. J.S. Lister, S. L. Lee, D. A. Tennant, Y. Maeno, and A. P.Mackenzie, Science
214 (2007).[9] T.-M. Chuang, M. P. Allan, J. Lee, Y. Xie, N. Ni, S. L.Budko, G. S. Boebinger, P. C. Canfield, and J. C. Davis,Science
181 (2010).[10] J. A. Hertz, Phys. Rev. B , 1165 (1976); A. J. Millis,Phys. Rev. B , 7183 (1993).[11] J. Rech, C. Pepin, and A. V. Chubukov, Phys. Rev. B , 195126 (2006).[12] K. Sun, B. M. Fregoso, M. J. Lawler, and E. Fradkin,Phys. Rev. B , 085124 (2008).[13] M. Garst and A. V. Chubukov, Phys. Rev. B , 235105(2010).[14] M. A. Metlitski and S. Sachdev, Phys. Rev. B , 075127(2010).[15] C. J. Halboth and W. Metzner, Phys. Rev. Lett. , 5162(2000); W. Metzner, D. Rohe, and S. Andergassen, Phys.Rev. Lett. , 066402 (2003).[16] V. Oganesyan, S. A. Kivelson, and E. Fradkin, Phys.Rev. B , 195109 (2001).[17] M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. B ,6721 (2000); Int. J. Mod. Phys. B , 3719 (2000).[18] E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Frad-kin, and S. A. Kivelson, Phys. Rev. B , 184514 (2008).[19] Y. Huh and S. Sachdev, Phys. Rev. B , 064512 (2008).[20] C. Xu, Y. Qi, and S. Sachdev, Phys. Rev. B , 134507(2008).[21] L. Fritz and S. Sachdev, Phys. Rev. B , 144503 (2009).[22] C. Castellani, C. Di Castro, and M. Grilli, Z. f¨ u r Physik , 137 (1997).[23] S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Phys. Rev. B , 155129 (2009).[24] J.-H. She and J. Zaanen, Phys. Rev. B , 184518(2009); S.-X. Yang, H. Fotso, S.-Q. Su, D. Galanakis,E. Khatami, J.-H. She, J. Moreno, J. Zaanen, and M.Jarrell, Phys. Rev. Lett. , 047004 (2011).[25] J. Orenstein and A. J. Millis, Science , 468 (2000).[26] P. A. Lee, Phys. Rev. Lett. , 1887 (1993); A. Durstand P. A. Lee, Phys. Rev. B , 1270 (2000).[27] L. Taillefer, B. Lussier, R. Gagnon, K. Behnia, and H.Aubin, Phys. Rev. Lett. , 483 (1997).[28] A. A. Nersesyan, A. M. Tsvelik, and F. Wenger, Nucl.Phys. B , 561 (1994).[29] A. Altland, B. D. Simons, and M. R. Zirnbauer, Phys.Rep. , 283 (2002).[30] A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G.Grinstein, Phys. Rev. B , 7526 (1994).[31] T. Stauber, F. Guinea, and M. A. H. Vozmediano, Phys.Rev. B , 041406 (2005).[32] M. S. Foster and I. L. Aleiner, Phys. Rev. Lett. ,195413 (2006).[33] R. Shankar, Rev. Mod. Phys. , 129 (1994).[34] M. Chiao, R. W. Hill, Ch. Lupien, L. Taillefer, P. Lam-bert, R. Gagnon, and P. Fournier, Phys. Rev. B , 3554(2000).[35] J. Mesot, M. R. Norman, H. Ding, M. Randeria, J. C.Campuzano, A. Paramekanti, H. M. Fretwell, A. Kamin-ski, T. Takeuchi, T. Yokoya, T. Sato, T. Takahashi, T.Mochiku, and K. Kadowaki, Phys. Rev. Lett. , 840(1999).[36] G.-Z. Liu, W. Li, and G. Cheng, Phys. Rev. B , 205429(2009).[37] G.-Z. Liu and J.-R. Wang, New J. Phys.13