Strange metal from incoherent bosons
Anurag Banerjee, Maxence Grandadam, Hermann Freire, Catherine Pépin
IIncoherent transport in a model for the strange metal phase
A. Banerjee, M. Grandadam, H. Freire, and C. P´epin Institut de Physique Th´eorique, Universit´e Paris-Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France. Instituto de
F´ısica , Universidade Federal de Goi´as, 74.001-970, Goiˆania-GO, Brazil.
We present a scenario for the strange metal phase in the cuprates, where diffusive, charge-two, finite momen-tum bosons are present in a vast region of the phase diagram. The presence of these bosons emerging from pairsof high-energy electrons can account for a regime of linear-in- T resistivity. Diffusive bosons are incoherent,and as such, they do not contribute to the Hall conductivity. Surprisingly, these incoherent bosons contributeto the longitudinal Drude conductivity with the corresponding transport time given by τ ∼ (cid:126) / ( k B T ) , reminis-cent of the Planckian dissipators associated with a putative quantum critical point in the strange metal phase.We also obtain a linear-in- H magnetoresistance when the diffusive bosons originate from electron pairs of atriplet. The presence of such bosons in the strange metal phase of the cuprates can shed light on recent transportmeasurements in overdoped compounds [J. Ayres et al. , unpublished (2020)]. I. INTRODUCTION
Arguably the most enduring enigma of the phase diagramof the cuprate superconductors is the strange metal regimeobserved in optimally-doped and overdoped compounds.
Over a wide temperature and doping region, the Landau Fermiliquid paradigm of transport breaks down, with a resistivityshowing a linear-in- T dependence from low temperature upto the melting point of the material. Interestingly, such abreakdown of the Fermi Liquid paradigm is observed acrossa wide variety of materials.
Correspondingly, the opticalconductivity displays ω/T scaling and exhibits a transportscattering rate given by (cid:126) τ − ∼ k B T , which is the maxi-mal rate allowed by the Heisenberg uncertainty principle (also called the Planckian dissipation limit). Interestingly, forfrequencies lower than the aforementioned transport scatter-ing rate, i.e., ω < τ − , the optical conductivity remarkablyfollows the classic Drude form, in addition to showing alinear-in- T resistivity.A recent groundbreaking experiment reports the appear-ance of incoherent transport in the strange metal (SM) phaseof the cuprates in the optimally-doped and overdoped re-gions. At high magnetic fields, the magnetoresistance alsodisplays a positive linear-in- H dependence, in addition toa linear-in- T dependence at low fields. Furthermore, thereare reports of linear in field magnetoresistance in the electron-doped cuprates, iron chalcogenide, and iron pnictides. Moreover, this component of the conductivity is insensitiveto the magnetic field’s orientation, implying a vanishing Hallconductivity. Such vanishing Hall conductivity is also re-ported in the normal state of the cuprates and in the super-conducting (SC) thin films, among others. Thus, the mys-terious SM phase acquires another ingredient – On the onehand, it shows linear-in- T resistivity with the optical conduc-tivity following the clasic Drude form, and the detection of anincoherent transport component insensitive to the orientationof the magnetic field. On the other hand, the experimental re-sult since the early days of the cuprates exhibits a secondtransport time (cid:126) τ − H ∼ T which controls the cotangent of theHall angle over the whole phase diagram.Early attempts to demystify the strange metal phase in-clude the Marginal Fermi Liquid theory, which can heuris- tically explain the temperature dependence of Hall angle. Subsequent transport theories have put forward two types ofquasiparticles with different scattering rates to capture thedifferent temperature evolution of longitudinal conductivityand Hall angle.
More recently, the Hall transport time, τ H , are satisfactorily described by the presence of quasielec-trons with an anisotropic transport time around the Fermi sur-face. However, the latter theories encounter difficulties inaccounting for the linear-in- T resistivity and the correspond-ing Planckian limit of the scattering rate. Furthermore, theregime for the Drude form of the optical conductivity, alongwith the recent report of incoherent non-orbital contributionto transport remains to be addressed. Given that situation, aregime of very strong coupling, possibly obtained by eitherholographic techniques or other transport methods have been invoked to account for some of these observedproperties.To address this formidable problem, a simple and intuitivephenomenological model would be helpful. In this paper, wepropose a scenario in which the quasielectrons are not the solecharge carriers in the strange metal regime of the cuprates,but charge-two bosons with finite wavevector are also present.These latter diffusive excitations emerge from pairs of high-energy electrons in the system. Within this scenario, the quasi-electrons around the Fermi surface will naturally account forthe observed coherent transport in the material (since they re-act to the magnetic field according to the Hall lifetime τ H ). Incontrast, the diffusive bosons will, in turn, be responsible forthe incoherent transport reported recently. II. THE MODEL
We propose a model consisting of quasielectrons scatter-ing off each other via hydrodynamic fluctuations as wellas charge-two bosons. The bosons originate from pairs ofhigh-energy electrons, which interact with the low-energyquasielectrons, with strength, g I , and with themselves withstrength, g b . With the application of an external mag-netic field, the corresponding gauge-invariant Hamiltonian be- a r X i v : . [ c ond - m a t . s up r- c on ] S e p comes ˆ H = (cid:88) k ,α c † k ,α (cid:34) ( k − e A ) m − (cid:15) F (cid:35) c k ,α + V e − e + 12 (cid:88) q b † q (cid:2) ( q − e A ) + µ (cid:3) b q − (cid:88) k , k (cid:48) α,α (cid:48) c † k + k (cid:48) ,α ( (cid:126)σ αα (cid:48) . H ) c k (cid:48) ,α (cid:48) + g b (cid:88) q , p , k b † k b k + q b † p − q b p + g I (cid:88) k , q ,α (cid:2) b † q c k ,α c − k + q , ± α + h.c. (cid:3) , (1)where c † k ,α is the creation operator for conduction electrons, α is the spin projection of the electrons, b † q is the creation op-erator for charge-two bosons, with implicit assumptions thatthe bosons have a minimum dispersion around a finite wave-vector Q , such that q = Q − Q is the deviation of themomentum Q from this wavevector. This is illustrated inFig. (1b), where the bosonic dispersion has a minimum at Q = Q . The quantities e and m are, respectively, the ele-mentary charge and the quasielectron mass, whereas A is thevector potential associated with the magnetic field given by H = ∇ × A . The quantities (cid:15) F and µ denote, respectively,the chemical potential of the electrons and the bare bosonicmass term. The next term refers to the coupling of the elec-tron spins to the Zeeman field, where (cid:126)σ αα (cid:48) are the Pauli ma-trices. The term V e − e represents the interactions between theelectrons and the environment that can consist of other typesof hydrodynamic modes or impurities. Finally, the last twoterms in Eq. (1) are, respectively, the boson-boson interactionand the fermion-boson interaction. In the interaction term thatcontains g I , we allow for the possibility of the bosons to beeither spin-0 or spin-1. III. RESULTS
We study the electromagnetic response of the system withinthe Kubo formalism, considering not only the electronic, butalso the bosonic response to the electromagnetic field. TheFeynman diagrams contributing to the charge transport prop-erties, as well as to the self-energy corrections are presentedin Fig. (2). Naturally, charged bosons have a markedly differ-ent behavior from fermions. At low temperatures, fermionsscatter around the Fermi surface, and scattering with finitewavevectors affects only small regions of the Fermi surface,creating a transport anisotropy commonly referred to as “hotspots” and “cold spots”. The hot-spots are shown by the cir-cles in the Fig. (1a). Such fermions participate both in thecoherent transport and in the Drude weight.
On the otherhand, bosons do not have a Fermi surface and, consequentlythey scatter uniformly through other species in the sample.Therefore, the bosonic pathway of charge transport is pro-tected against short circuit of hot regions by the cold onesunlike the fermionic counterpart. Doping (x) T e m p e r a t u r e ( T ) Fermi Liquid(a) (b)(c) PseudoGapped - π π - π π k x k y Drude Q ω Q FIG. 1. (a)
Shows the Fermi surface observed in overdoped cuprateswith the hot-spots denoted by brown circles. (b)
Presents the disper-sion of the bosons as a function of the momentum Q . The dispersionhas a minimum at a finite wave-vector Q (for the representative caseshown, there is a gap of µ ). Note that µ varies with the doping andvanishes at the quantum critical point (QCP). (c) Displays a skele-ton phase diagram in the temperature-doping plane. The T ∗ sets thepsuedogap energy scale in the system, which vanishes at the QCP, x c . For larger dopings, the Fermi liquid behavior is established. Thestrange metal phase is expected to reside in the quantum critical fanin between these two regions. The energy scale that separates thetwo distinct regions is µ /γ , where µ is the doping-dependent barebosonic mass term, and the γ is the Landau damping coefficient ofthe diffusive bosons. A momentum relaxation mechanism is necessary to obtaina steady current flow upon the application of external elec-tric field. However, the zero momentum bosons preservethe translational invariance, and hence it is essential to invokeother mechanisms that can break the Galilean symmetry. Onthe other hand, bosons with finite momentum have a naturalscattering mechanism to decay the current. Take for examplethe charge-two bosons b † q = (cid:68) c † k c †− k + Q + q (cid:69) made of elec-trons on the Fermi surface of Fig. (1a). Since Q relates twoportions of the Fermi surface, which are distinct translationsin reciprocal space by factors of π , the boson-boson scatter-ing generates Umklapp processes through the coupling to thelattice. Therefore, finite-momentum charged bosons have thelifetime equal to the transport time. Hence, this study is de-voted to the careful analysis of the finite momentum bosoniccontribution to the transport. Since the bosons have a finitewavevector, Q , and are made of high-energy fermions, suchthat they naturally participate in the decay of the electric cur-rent. Consequently, the two species contribute to the trans-port, both terms must be included to obtain the total opticalsum rule.When the coupling between the bosons is stronger than thedamping coefficient, our key findings are encapsulated in thephase diagram of Fig. (1c). Above a threshold temperature T > µ /γ , we find a linear-in- T resistivity and a vanish-ing Hall conductance (where γ is the Landau-damping coef-ficient of the diffusive bosons to be defined shortly). Here, µ is the doping-dependent bare mass of the boson as illus-trated in Fig. (1b), which vanishes at the quantum criticalpoint or a critical phase. Our phenomenological study can-not distinguish between a quantum critical point and quan-tum critical phase as observed in Ref. 48. Furthermore, when T > µ /γ , the incoherent bosons contribute to the Drude-like conductivity with a scattering rate reminiscent of Planck-ian dissipation. On the other hand, when the temper-ature is below
T < µ /γ , the traditional Fermi-liquid behav-ior is established due to the additional presence of a coherentfermionic pathway. The bare bosonic mass, µ determinesthe crossover from the strange metallic to conventional metalregime, as exhibited in Fig. (1c). (a) (b)(c) (d) FIG. 2. Feynman diagrams corresponding to the transport proper-ties and the interactions of the bosons among themselves and withthe fermions of the model defined in Eq. (1): (a), (b) and (d) rep-resent contributions to the bosonic self-energy in the present the-ory, whereas (c) stands for the diagram associated with the current-current correlation function.
A. Boson scattering via the fermions
In the overdoped and optimally doped region of thecuprates, the electrons are rather coherent. Hence, we con-sider the scattering process of bosons from electrons as thepredominant one. Evaluating the diagram on Fig. (2a) (de-tailed calculations are given in Appendix A), we note thatsuch polarization bubble is proportional to g I and producesa Landau damping term. This distinctive feature is typicalof a charge-two boson with finite momentum, which couplesto electrons in the same way as a pair-density-wave (PDW). After integrating out the electronic degrees of freedom thebosonic propagator reads D − ( q , iω n ) = γ | ω n | + q + µ ( T ) . (2)Here, ω n is the Matsubara frequency, wherethe Landau-damping constant is given by γ = g I N ( (cid:15) F ) / (2 π (cid:112) (2 k F Q ) − Q ) , where N ( (cid:15) F ) isdensity of states at the Fermi energy and k F is the cor-responding Fermi momentum and µ ( T ) is the bosonic“mass-term” at finite temperatures. This form of the bosonicGreen’s function is valid for all the frequencies below ω c ≈ k F Q . From Eq. (2), it becomes clear that the bosonsare diffusive near the critical point (or critical phase) wherethe bare mass of the boson vanishes. Moreover, the formfactor of the electron pairs does not have a qualitativeinfluence on this diffusive behavior of the bosons. We havechecked numerically, e.g., that a d -wave form factor for theelectron pairs also lead to such Landau damping term, albeitwith a different coefficient. We show below that the bosonicpropagator of Eq. (2) can contribute to the incoherent part ofthe resistivity that was recently reported in Ref. 13 and 15. B. Kubo formula for the conductivity
Since the charge-two boson directly couples to the electro-magnetic field, the main bosonic contribution to the longitudi-nal resistivity is given within the Kubo formula by the diagramin Fig. (2c) (see Appendix B, for detailed evaluation of this di-agram). The leading-order contribution to the conductivity isgiven by σ ij ( ω ) = Tω n (cid:88) ε n (cid:90) dx (cid:90) dx (cid:48) {− δ ij δ ( x − x (cid:48) ) D ( ε n , x, x (cid:48) )+ˆ v i D ( ε n , x, x (cid:48) )ˆ v j D ( ε n + ω n , x (cid:48) , x ) } , (3)where the analytical continuation iω n → ω + iδ needs tobe performed, the indices i, j refer to the spatial directional, ˆ v x = (cid:0) − i∂ x − iH∂ k y (cid:1) and ˆ v y = ( − i∂ y + iH∂ k x ) are thevelocity kernels. The longitudinal conductivity (independentof the magnetic field H ) is then given by σ (0) xx ( ω n ) = Tω n (cid:88) ε n L (cid:88) q (cid:2) Q D ( ε n , q ) D ( ε n + ω n , q )+ D ( ε n , q )] . (4)Note that since the bosons have a finite momentum, the ve-locity kernels in Eq. (3) become proportional to Q , whichresult in a prefactor for the above integral. Thus, performingthe corresponding integration, we find in the first regime, i.e., T (cid:28) (cid:112) ω / µ /γ , that the optical conductivity becomes σ xx ( iω → ω + iδ ) = σ b τ (cid:16) − i γω µ (cid:17) , (5)with σ b = Q / (2 π γ ) . Strikingly, Eq. (5) is reminiscentof a Drude conductivity, with the scattering rate given by (cid:126) τ − = (2 µ/γ ) . However, in the second regime, i.e., T (cid:29) (cid:112) ω / µ /γ , the optical conductivity does not exhibit thetraditional Drude form σ ( ω ) = Q µ π γ T (cid:18) − i γω µ (cid:19) . (6)We will show in the next section that this latter regime (non-Drude-like) is never obtained if the coupling strength betweenthe bosons is larger than the Landau-damping parameter. C. Renormalization of the bosonic “mass-term”
In order to figure out the temperature dependence ofthe static resistivity, we evaluate the renormalization of thebosonic “mass-term” due to its scattering with strength g b .This is given by the diagram in Fig. (2b) which is propor-tional to the number of bosons, N b = T (cid:80) ν n (cid:80) q D ( ν n , q ) .The bosonic mass term of the Eq. (2) is renormalized by µ = µ + g b N b , (7)where µ is the temperature independent part of the mass-term, which vanishes at the quantum critical point. Theleading-order correction to the mass-term evaluates to (for de-tails refer to Appendix B) µ = µ + ˜ g b T log (cid:18) γTµ (cid:19) for γT (cid:29) µ ,µ for γT (cid:28) µ , (8)where we have defined, ˜ g b = g b / (4 π ) . Therefore, for an in-termediate to strong coupling regime, i.e., ˜ g b ≥ γ , we willalways have γT (cid:28) µ . For this reason, the second regimedisplaying the non-Drude form of the optical conductivityis not attained if the coupling is stronger than the damping.In the main text, we mainly focus on the ˜ g b ≥ γ regime.The possibility of the other theoretical limits are explored inAppendix C 1. Thus, plugging the temperature-dependenceof the bosonic “mass-term” calculated in Eq. (8) back intoEq. (5), the static ω → becomes ρ xx ( T ) = π µ Q + 4 π ˜ g b Q T log (cid:18) γTµ (cid:19) for γT (cid:29) µ , π µ Q for γT (cid:28) µ . (9)Therefore, up to logarithmic corrections, we obtain a linear-in- T regime for the resistivity when γT ≥ µ . The first termis a T -independent contribution. To further confirm our ana-lytical results, we perform numerical integration to obtain thestatic resistivity as a function of temperature. The Fig. (3a)shows a clear linear-in- T behavior of resistivity for the follow-ing parameter choices, γ = 1 . , Q = π/ , µ = 0 . and ˜ g b = 1 . . As can be seen, there is a very good match between the numerical and the approximate analytical behavior. Sim-ilarly, in Fig. (3b) we show the same for a larger interactionparameter ˜ g b = 1 . , which again displays a linear dependencewith temperature, albeit with a different slope (for details, seeAppendix C). FIG. 3. Displays the linear-in- T evolution of resistivity obtainedfrom the analysis of the model. In all the plots, we set the Landau-damping constant equal to γ = 1 . and the temperature independentmass term µ = 0 . . Besides, we choose also the input parameters (a) ˜ g b = 1 . and (b) ˜ g b = 1 . . Above T > µ /γ the linear-in- T behavior sets in. We also compare both the numerical and the ana-lytical expressions in these plots which are in good agreement witheach other. Our calculations reveal that the incoherent transport due tothe charged bosons contributes to the Drude-like response atfinite frequencies. Furthermore, in this regime the momen-tum relaxation rate, τ − ∼ k B T / (cid:126) scales linearly with tem-perature up to logarithmic corrections. In overdoped cuprates,line-shapes of the optical conductivity as a function of fre-quency remarkably follows such classic Drude form. A closerelationship between the scattering rates of the charge carriersand linear-in- T behavior is established across several fami-lies of overdoped cuprates. In Fig. (3a) we present the fulloptical conductivity as a function of frequency ω for the fol-lowing parameter choices, γ = 1 . , Q = π/ , µ = 0 . and ˜ g b = 1 . . The real part of the optical conductivity showsa sharp peak at low temperature, T = 0 . . The peak broad-ens progressively as the temperature increases to T = 0 . aspresented in Fig. (4b) to Fig. (4d). We obtain from Eq. (9) alongitudinal conductivity that varies as T − (up to logarithmiccorrections), which participates in a Drude-like response at fi-nite frequency. This result is unusual enough to be noticed,since, within the holographic framework, it is currently a mat-ter of intense discussions to decide whether a fixed point canproduce incoherent transport, which contributes to the Drudeconductivity. Our simple model provides an example ofsuch behavior.
D. Higher order terms in the self-energy
We now turn to the next-to-leading order correction regard-ing the “mass-term” renormalization, namely, the rainbow di-agram represented in Fig. (2d). In addition, the imaginary partfrom this diagram renormalizes the Landau-damping constant (a) T=0.07 σ ' ( ω ) σ " ( ω ) σ ( ω ) (b) T=0.2 (c) T=0.4 ω σ ( ω ) (d) T=0.7 ω FIG. 4. Shows real and imaginary parts of the optical conductivity σ ( ω ) = σ (cid:48) ( ω ) + iσ (cid:48)(cid:48) ( ω ) for the following parameter choices: γ =1 . , ˜ g b = 1 . and µ = 0 . . (a) T = 0 . , (b) T = 0 . , (c) T = 0 . and (d) T = 0 . . Here, σ ( ω ) shows the traditional Drudeform with the width of real part increasing with temperature. Thus,the linear-in- T resistivity from the incoherent bosons contributes toa Drude-like response at finite frequencies. γ in Eq. (2). The corresponding polarization bubble reads Π ( q ) = g b T (cid:88) ν n ,ω n (cid:88) p , q D ( ν n − ω n + q , − p + k ) × D ( ν n , k ) D ( ω n , p ) , (10)where q is the incoming frequency that is assumed to be asmall parameter during the evaluation. The renormalizationof the µ and γ to the second order for γT (cid:29) µ is given by(details presented in Appendix D) µ ≈ µ + g b π log (cid:18) γTµ (cid:19) + 2 c γλπ log ( γT /µ ) , (11) ˜ γ ≈ γ + c γπ log ( γT /µ ) , (12)where λ = min [ µ , γT ] . Now we take the limit γT /µ (cid:29) and find that the second-order terms are negligible. Next, eval-uating the same quantities for γT (cid:28) µ , we get µ ≈ µ + c λ ( γT ) π γµ , (13) ˜ γ ≈ γ + c ( γT ) π γµ . (14)If we assume γT /µ (cid:28) , the second order contributions thenbecome negligible. The constants c = 0 . and c = 0 . are evaluated by employing numerical techniques. Therefore,in both regimes, the higher-order terms are small compared tothe first-order ones and, therefore, we can safely ignore theireffects from now on in our analysis. Effect of a magnetic fieldE. Hall conductivity
We begin the discussion on the effect of magnetic fieldon the charged bosons with the Hall conductivity. The termlinear-in- H in Eq. (3) leads to the following expression forthe Hall conductivity, which is given by σ (1) xy = 1 ω n Im (cid:26) T (cid:88) ε n L (cid:88) q iH [ q x D ( ε n , q ) ∂ q y D ( ε n + ω n , q ) − ∂ q y D ( ε n , q ) q x D ( ε n + ω n , q )] (cid:27) . (15)Evaluating this term with Eq. (2), we obtain that it naturallyvanishes (details can be found in Appendix E). This result isnot surprising, since the bosons are incoherent and the the-ory has a particle-hole symmetry. This can be confirmedby noting that the bosonic propagator in Eq. (2) is symmet-ric under ω → − ω transformation. The fact that diffusivebosons do not participate in the Hall number could explainthe recent studies where the number of Hall carriers is seento gradually decrease, as the doping is reduced from the over-doped region to the underdoped regime. Similarly, vanishingHall conductivity is reported in the normal state of the stripe-ordered cuprates, and in two-dimensional superconductingthin-films. The emergence of particle-hole symmetry ofthe charged incoherent bosons in this study also implicates atendency towards the vanishing Hall conductivity.
F. Second-moment of conductivity
The contribution quadratic in H of the conductance inEq. (3) writes σ (2) xx = − H ω n Im (cid:26) T (cid:88) ε n (cid:88) q [ ∂ q x D ( ε n + ω n , q ) ∂ q x D ( ε n , q )] (cid:27) . (16)This orbital contribution from Eq. (16) is a bit more involved(see Appendix F) and it reads in the two possible theoreticalregimes as follows σ (2) xx = γ Q T H π µ for γT (cid:28) µ, T γQ H πµ for γT (cid:29) µ. (17)Again, we emphasize that the second regime is never realizedwhen the interaction between the electrons is stronger than theLandau damping coefficient. Armed with the expression for σ (0) xx , σ (1) xx , and σ (2) xx , we proceed to evaluate the magnetic fielddependence of magnetoresistance. G. Magnetoresistance
We now turn to the evaluation of the magnetoresistance(MR) of the system. For a system with vanishing Hall con-ductivity σ xy , the magnetoresistance is evaluated (details pro-vided in Appendix H) through ∆ ρ xx ρ xx (0) = ρ xx ( H ) − ρ xx (0) ρ xx (0) = σ xx (0) − σ xx ( H ) σ xx ( H ) , (18)where σ xx (0) denotes the conductance measured at zero mag-netic field. The longitudinal conductivity, however, has contri-butions from both σ (0) xx and σ (2) xx . In order to proceed, the massrenormalization due to the Zeeman field needs to be evaluated.Two cases then arise due to the symmetry of the spins of theelectron pairs.
1. Spin-zero case:
First, let us consider that the diffusive bosons have spin-zero, i.e., the spins of the electron pairs have the symmetry ofa singlet. The Zeeman coupling to the spin of the electrons(diagram in Fig. (2a)) renormalizes the bosonic mass term.The resulting renormalization is independent of the magneticfield H and is given by µ = µ + µ T , (19)where µ T = ˜ g b T log( γT /µ ) Hence, the mass-renormalization is insensitive to the magnetic field (detailsprovided in Appendix G 1). On the other hand, since the or-bital contribution Eq. (16) gives a term in H , it leads to a H dependence of the MR (evaluated in detail in Appendix H 1).The regimes are then determined by the maximum among µ and µ T . The MR is given by ∆ ρ xx ρ xx (0) = κβ H , (20)where in the first regime when µ > µ T , the constant is givenby κβ ≡ − γ T µ . By contrast, when µ < µ T , the constantis given by κβ ≡ − γ T µ + µ T ) . In both regimes, the magne-toresistance has a quadratic dependence on the magnetic field.Thus, particle-particle pairs with singlet symmetry contributeto the magnetoresistance identically as the conduction elec-trons would do, typical of the conventional Landau Fermi liq-uid theory.
2. Spin-one case:
Next, we consider the situation where the spins of theparticle-particle pairs have a triplet symmetry. In this sce-nario, the boson scattering off conduction electrons generatesa mass-correction due to the Zeeman field H given by µ = µ + µ T + µ H , (21) where µ H = αH and α ≡ γπ coth − (cid:18) k F + Q √ k F − Q (cid:19) . For acomprehensive evaluation of this mass renormalization, referto Appendix G 2. Again, the regimes will be determined bythe maximum among µ , µ T , and µ H . As a result, we have aregime where the mass-term couples linearly to the magneticfield. As mentioned before, we focus on the situation whenthe interaction between bosons is stronger than the Landaudamping coefficient, i.e., ˜ g b ≥ γ . Taking the limit γT /µ (cid:28) in Eq. (17) it is clear that the orbital contribution becomesnegligible in this regime. Hence, the spin-one contribution tothe magnetoresistance becomes ∆ ρ xx ρ xx (0) = αµ + µ T H for max ( µ , µ T , µ H ) = µ H ,κβ H otherwise , (22)where κβ ≡ − γ T µ , when max ( µ , µ T , µ H ) = µ .If max ( µ , µ T , µ H ) = µ T , we obtain instead that κβ ≡ − γ T µ + µ T ) . For a detailed evaluation of all these quan-tities, refer to Appendix H 2.Note that µ H (cid:29) µ T can be recast in the form H (cid:29) ηT .Here, up to a logarithmic corrections, η is just a constant .Consequently, in this high field regime, we have a linear-in- H magnetoresistance. However, in the low field regime, H (cid:28) ηT , we have a quadratic H dependence of the mag-netoresistance. Notice that similar field evolution of magne-toresistance is recently reported in overdoped cuprates. As aresult, our calculations unveil that the incoherent bosons canexplain such a behavior of the MR.Lastly, we comment on the scaling of the in-plane magne-toresistance with that observed experimentally. The in-planeMR is given by ∆ ρ xx = ρ xx ( H, T ) − ρ xx (0 , . (23)Near the QCP, ∆ ρ xx follows a quadrature dependence , i.e., ∆ ρ xx = √ a T + b H , where a and b are constants.In the low-field and high-field limits, this quantity, therefore,scales as ∆ ρ xx ∝ H for H (cid:29) T,H T for H (cid:28) T. (24)Our phenomenological model cannot obtain the exact depen-dence of the in-plane MR. However, as will become clearshortly, we can obtain the correct scaling forms in both low-field and high-field limits for this quantity.We concentrate on the physical regime when the interac-tion between the bosons is stronger than the Landau dampingparameter, i.e., ˜ g b ≥ γ , i.e., µ (cid:29) γT . We also restrict ourattention to the case when the bosons emerge from pairs ofhigh-energy electrons that have spin-triplet symmetry. Con-sequently, the mass-term is given by Eq. (21). Again, as wementioned before, the maximum among µ , µ T , and µ H de-termines the different regimes in the present theory. There-fore, the leading order contribution to this quantity becomes ∆ ρ xx ∝ H for H (cid:29) ηT,H T for H (cid:28) ηT, (25)where, up to logarithmic corrections, η is just a constant. Thedetailed evaluation is presented in Appendix H 3. As a re-sult, although our calculations cannot determine exactly thequadrature dependence for ∆ ρ xx , a similar scaling behavioris found in the low-field and high-field limits. Therefore, ∆ ρ xx calculated within the present theory of incoherent finite-momentum bosons suggests a similar quadrature dependence. IV. DISCUSSION
In this paper, we suggest a phenomenological model to ac-count for the recent puzzling observations in optimally-dopedand overdoped cuprates. The presence of diffusive, charge-two bosons in this part of the phase diagram contributes to thelinear-in- T resistivity and leads to a broad Drude componentin the optical conductivity with the dissipation of “Planckian”-type. Since the bosons are incoherent, they do not con-tribute to the Hall conductivity, thereby explaining the miss-ing number of carriers reported in the most recent study ofthe Hall conductivity. If bosons are spin-one, they also pro-duce a linear-in- H , positive magnetoresistance. Of course,our model also contains conduction electrons, which providethe coherent part of the transport. The scattering around theFermi surface has to show a form of anisotropy in the trans-port lifetime, according to which both the c-axis magneto-resistance and the electron lifetime (extracted from thecotangent of the Hall angle cot θ H ∼ τ − H ∼ T ) can besuccessfully reproduced.A few microscopic theories can lead to diffusive, charge-two bosons in the optimally doped and overdoped regions ofthe cuprates phase diagram. Firstly it is suggested that thepseudogap is a transition towards a “fluctuating” pair densitywave (PDW) phase. This scenario could naturally lead tothe presence of charge-two finite momentum bosons in thestrange metal phase. Another recent proposal suggests that thepseudogap can result from fractionalizing a PDW state.
In this proposal, the gap opening at T ∗ results from a de-confining transition of a PDW order parameter into an SCand charge density wave fields. The fluctuations of the gaugefield associated to the fractionalization produce the pseudo-gap. At T = 0 , this involves a coherent superposition ofparticle-particle and particle-hole orders. Here again, pre-formed PDW pairs can exist above T ∗ . Several microscopicmodels are also proposed to examine the possibility ofsuch PDW sates. In the presence of either time-reversal orparity symmetry, the strong correlation between electrons be-comes an essential ingredient for the generation of the PDWstates. Nevertheless, these PDW pairs are typically ex-pected to have a singlet spin symmetry and cannot lead to alinear-in- H magnetoresistance. A recent study explores the possibility of the PDW states in the triplet channel, andsome proposals have suggested to fractionalize a stripe ora spin density wave order. As a final remark, we note that since incoherent charge-twobosons contribute to the Drude peak observed in the opticalconductivity, these pairs could also, be a good candidate forexplaining the missing spectral weight in the superfluid den-sity that was ubiquitously reported to be present in this regionof the phase diagram.
V. ACKNOWLEDGMENTS
The authors thank Saheli Sarkar, Nigel Hussey, Yvan Sidis,Dmitrii Maslov and Debmalya Chakraborty for valuable dis-cussions. This work has received financial support fromthe ERC, under grant agreement AdG694651-CHAMPAGNE.H.F. acknowledges funding from CNPq under Grants No.405584/2016-4 and No. 310710/2018-9.
Appendix A: Scattering through fermions
In this section, we formally show that the scattering throughfermions leads to a diffusive imaginary part of the self-energyof finite momentum bosons. Fig. (5a) shows the relevantFeynman diagram. The bosons emerge from the pairs offermions with finite-momentum Q . The wavy-lines repre-sent the bosons, and the solid lines denote the fermions. (a) (b)T=0.07 ω I m Π ( ω ) (c)T=0.35 ω FIG. 5. (a)
The leading order boson propagator correction, given byEq. (A1). The solid line is the bare electronic Green’s function, G .The wavy lines are the finite momentum bosons with ordering wave-vector Q . (b) Comparison of imaginary part of Π( ω ) for numericaland approximate analytical evaluations for low temperature, T =0 . and g I = 1 , Q = π/ (b) Same for the higher temperature T = 0 . . The expression for the diagram in Fig. (5a) reads as Π( ω n , Q ) = g I L (cid:80) k T (cid:80) ε n G ( − ε n , − k ) G ( ε n + ω n , k + Q )+ G ( − ε n , − k ) G ( ε n − ω n , k − Q ) . (A1)Here L is the volume of of the system, T is the temperatureand g I is the interaction strength between the finite momen-tum bosons and fermions. The frequencies, ε n and ω n arefermionic and bosonic Matsubara frequencies, respectively.The Green’s functions, G , denote the free fermionic propa-gators given by G − ( k , ω n ) = i(cid:15) n − ξ k , (A2)where ξ k = (cid:126) k / m e . For simplicity of notations, we set (cid:126) / m e = 1 , from now on. In order to perform the Matsubarasummation, we go to the complex plane by performing thesubstitution, i(cid:15) n → z . The first term of the RHS of Eq. (A1)becomes Π( ω n , Q ) = − g I L (cid:80) k (cid:72) C dz πi n F ( z )( z + ξ − k )( z + iω n − ξ k + Q0 ) . (A3) The integral is evaluated using the residue theorem and obtain Π( ω n , Q ) = − g I L (cid:80) k − n F ( ξ − k ) − n F ( ξ k + Q0 ) iω n − ξ − k − ξ k + Q0 . (A4)We perform the analytic continuation by letting iω n → ω + i + and then taking the imaginary partIm Π( ω, Q ) = πg I L (cid:80) k [1 − n F ( ξ − k ) − n F ( ξ k + Q )] × δ ( ω − ξ − k − ξ k + Q ) . (A5)The k -summation is performed by converting it to an integral.We can approximate ξ k + Q ≈ k + Q + 2 k F Q cos( θ ) ,where θ is the angle between Fermi-momentum k F and theordering wave-vector, Q . Furthermore, we use the flat-bandapproximation with the density of states at the Fermi energygiven by N ( (cid:15) F ) , the integral in two dimensions becomesIm Π( ω, Q ) = g I N ( (cid:15) F )16 π (cid:90) π dθ (cid:20) tanh (cid:18) ω + Q + 2 k F Q cos( θ )4 T (cid:19) + tanh (cid:18) ω − Q − k F Q cos( θ )4 T (cid:19)(cid:21) . (A6)In the limit, T → , we can approximate tanh( x/T ) ≈ sgn ( x ) . In this low-temperature regime, the integrand in the squarebrackets in Eq. (A6), which we simply denote as I ( θ ) from now on, is approximately given by I ( θ ) = if θ ∈ (cid:20) cos − (cid:18) ω − Q k F Q (cid:19) , cos − (cid:18) − ω − Q k F Q (cid:19)(cid:21) , if θ ∈ (cid:20) π − cos − (cid:18) − ω − Q k F Q (cid:19) , π − cos − (cid:18) ω − Q k F Q (cid:19)(cid:21) , otherwise . (A7)The form of I ( θ ) is used to evaluate the integral in Eq. (A6)and it reads asIm Π( ω, Q ) = g I N ( (cid:15) F )4 π (cid:20) cos − (cid:18) − ω − Q k F Q (cid:19) − cos − (cid:18) ω − Q k F Q (cid:19)(cid:21) . (A8)Finally, expanding the function for ω (cid:28) k F Q , we arrive atthe resultIm Π( ω, Q ) = g I N ( (cid:15) F )2 π ω (cid:112) (2 k F Q ) − Q . (A9)This shows there is a linear dependence on ω . Performingsimilar calculations for the second term in Eq. (A1) and theimaginary part of the self-energy readsIm Π( ω, Q ) = γ | ω | , (A10)with γ = g I N ( (cid:15) F )2 π √ (2 k F Q ) − Q . We have checked our approxi-mate expression against numerical evaluation of Eq. (A6). A good agreement between them is observed in Fig. (5b) at lowtemperature, and in Fig. (5c) at high temperature. Appendix B: Renormalization of the “mass” term – Number ofbosons
In this section, we present the detailed evaluation of theleading order term in the self-energy, which renormalizes themass of the bosons. Fig. (6a) shows the relevant diagram,where the wavy-lines represent the bosons, which interactwith other bosons with the interaction strength being repre-sented by g b . The mass term renormalization is given by thereal part of this diagram, i.e. N b = 1 L (cid:88) q T (cid:88) ν n γ | ν n | + q + µ . (B1)The Matsubara summation over ε n is carried out by using thespectral decomposition of the bosonic Green’s function. Thespectral function A ( E, q ) is given by A ( q , E ) = − Im [ D R ( q , E )] = − γT ( γT ) + ( q + µ ) . (B2)Noting that D ( q , ν n ) = (cid:82) ∞−∞ dE π A ( q ,E ) iν n − E , the summation istaken to the complex plane by promoting iν n → z and T (cid:80) ν n → (cid:72) C dz πi n B ( z ) , where C covers the whole of thecomplex plane. Therefore the expression becomes N b = 1 L (cid:88) q (cid:73) C dz πi (cid:90) ∞−∞ dE π A ( q , E ) n B ( z ) z − E ,N b = − π (cid:90) ∞ dq (cid:90) ∞−∞ dE π n B ( E ) γT ( γT ) + ( q + µ ) . (B3)After performing the integral over q exactly, N b becomes N b = − π (cid:90) ∞−∞ dE (cid:20) π sgn ( E ) − tan − (cid:18) µγT (cid:19)(cid:21) n B ( E ) . (B4) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● (b) γ =0.75 Analytics ● Numerics N b (a) FIG. 6. (a)
The first-order diagram of the bosonic self-energy. Thewavy lines denote the bosons, which interact with the strength g b . (b) The temperature dependence of the number of bosons, evaluated bysolving the Eq. (B4) numerically and compared with the expressionarrived at analytically in Eq. (B8). The perfect match between thetwo evaluations gives us confidence in our analytical results.
The Bose-Einstein distribution is approximated as n B ( x ) = if x > T,Tx if | x | < T, − if x < − T. (B5)Performing the integral in the regime where | E | < T , therenormalization of the mass term reads N (1) b = T π log (cid:18) γTµ (cid:19) for γT (cid:29) µ,µ π γ for γT (cid:28) µ. (B6) Similarly, performing the integral for E < − T , we obtain N (2) b = π (Λ − T ) for γT (cid:29) µ, for γT (cid:28) µ, (B7)where Λ is the ultraviolet energy cutoff of the system. There-fore, N b will be independent of temperature in this regime, as Λ will be the dominant energy scale. This gives the numberof bosons that condenses to the ground state. The mass term µ to the first order is given by setting µ = µ , where µ isthe bare mass of the bosons, which is naturally temperatureindependent. Therefore, to first order in g b , we obtain µ = µ + g b (cid:18) T π log (cid:18) γTµ (cid:19)(cid:19) for γT (cid:29) µ ,µ for γT (cid:28) µ . (B8)The constant terms are absorbed in the µ , which becomesclose to zero near the quantum critical point. The temperaturedependence of N b calculated numerically from Eq. (B4) andanalytical form displayed in Eq. (B8) matches over a widerange of temperature, as can be seen in Fig. (6b) Appendix C: Longitudinal conductivity: Kubo formula
The longitudinal conductivity is given in terms of correla-tion functions K by K ( ω n ) = − T (cid:88) ν n L (cid:88) q [ D ( ν n , q )+ q D ( ν n , q ) D ( ν n + ω n , q ) (cid:3) . (C1)The first term is the diamagnetic term and the second term isthe paramagnetic current-current correlation. The momentum, q = Q − Q is the deviation of the momentum Q from theordering wavector Q , where the dispersion has a minimum.The Eq. (C1) can be approximated by K ( ω n ) ≈ − T (cid:88) ν n L (cid:88) q [ D ( ν n , q )+ Q D ( ν n , q ) D ( ν n + ω n , q ) (cid:3) . (C2)The optical conductivity is then evaluated by σ ( ω ) = − K ( ω n ) ω n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) iω n → ω + i + . (C3)The evaluation of the K is carried out in the following way:The integral is evaluated in the contour shown in Fig. (7b).There are two branch cuts – at z (cid:48) = 0 and z (cid:48) = iω n . Theintegrals over the Γ and Γ contours cancel the diamagneticterm. Therefore, only the Γ − contour contributes to the opti-cal conductivity. The integral becomes0 (a) (b) FIG. 7. (a)
The leading order diagram to evaluate the conductivity. (b)
The contours used to evaluate the Kubo formula for finite-momentumbosons. The two dashed lines are the branch cuts. K ( ω n ) = − Q L (cid:88) q πi (cid:73) Γ dz n B ( z )( iγz + q + µ ) (( − iz + ω n ) γ + q + µ ) . (C4)The poles of z lie outside the Γ -contour and hence the full integrals collapse to the real line integrals along the branch cuts. Theresulting expression becomes K ( ω ) = Q L (cid:88) q πi (cid:90) ∞−∞ dx n B ( x − ω/ − n B ( x + ω/ (cid:0) iγx − iγ ω + q + µ (cid:1) (cid:0) − iγx − iγ ω + q + µ (cid:1) . (C5)The summation over q is converted to an integral and it isperformed by usual means, i.e. K ( ω ) = − Q ω π γ (cid:90) ∞−∞ dx (cid:18) ∂n B ∂x (cid:19) x log (cid:18) − iγx − iγ ω + µiγx − iγ ω + µ (cid:19) . (C6)From the approximate form of the n B given in Eq. (B5), weobtain ∂n B ∂x = if | x | > T, − Tx if | x | < T. (C7)Using Eq. (C7), the optical conductivity becomes σ ( ω ) = − iQ T π γ (cid:90) T − T dx x log (cid:32) − x − ω − iµγ x − ω − iµγ (cid:33) , (C8)where we defined ˜ µ = ω + i µγ . As a result, performing theintegral, we obtain, σ ( ω ) = − iQ T π γ (cid:20) − µT + 12˜ µ log (cid:18) ˜ µ + T ˜ µ − T (cid:19) − µ log (cid:18) ˜ µ − T ˜ µ + T (cid:19) + 12 T log (cid:18) ˜ µ − T ˜ µ + T (cid:19) − T log (cid:18) ˜ µ + T ˜ µ − T (cid:19)(cid:21) . (C9) We expand the above expression in two regimes: For thefirst regime, T (cid:28) (cid:112) ω / µ /γ we find that the opticalconductivity displays the Drude form σ ( ω ) = Q π µ (cid:16) − i γω µ (cid:17) . (C10)We have compared the static conductivity given in the aboveequation against the numerical evaluation for the same usingEq. (C6). This comparison is displayed in Fig. (3a) andFig. (3b). A remarkable match between the two computationsover a wide range of temperatures is observed. The Drudeconductivity is naturally given by: σ ( ω ) = σ τ − iωτ . Fromthat expression, one can easily read off the σ = Q π γ whilethe scattering time of the bosons is given by τ = γ µ .On the other hand, for the second regime T (cid:29) (cid:112) ω / µ /γ the optical conductivity does notexhibit the traditional Drude form σ ( ω ) = Q µ π γ T (cid:18) − i γω µ (cid:19) . (C11)In the next section, we discuss the temperature dependence ofthe dc conductivity.1
1. Static Conductivity – The regimes
Next, we elaborate on the regimes of the static conductivity.Taking a ω → limit, we obtain the static conductivity in thetwo theoretical regimes as ρ xx ( T ) = π µQ for γT (cid:28) µ ( T ) , π γ T Q µ for γT (cid:29) µ ( T ) . (C12)The bosonic mass-renormalization is evaluated in Eq. (B8).Let us define ˜ g b = g b / (4 π ) . Next, we find the temperaturescale T (cid:48) , where γT (cid:48) = µ + ˜ g b T (cid:48) log( γT (cid:48) /µ ) . Solving for T (cid:48) , we get T (cid:48) = − µ ˜ g b W [ − γ/ ˜ g b exp( − γ/ ˜ g b )] , (C13)where W [ x ] is the Lambert W function. For different cou-pling strength ˜ g b , the form for this function is given by W (cid:20) − γ ˜ g b exp (cid:18) − γ ˜ g b (cid:19)(cid:21) = − γ ˜ g b for ˜ g b ≥ γ, − γ ˜ g b exp (cid:18) − γ ˜ g b (cid:19) for ˜ g b < γ. (C14)Putting this in Eq. (C13), we get the temperature scale T (cid:48) = µ γ for ˜ g b ≥ γ,µ γ exp (cid:18) γ ˜ g b (cid:19) for ˜ g b < γ. (C15)It can be seen that if ˜ g b ≥ γ , the temperatures scale T (cid:48) col-lapses on µ /γ . Consequently, we are always in the γT <µ ( T ) regime. In other words, γT > µ ( T ) regime is never at-tained if the coupling between the bosons is stronger than thatof the Landau damping coefficient. In Fig. (1c), we have al-ready presented the phase diagram for this scenario. In thisregime, the static conductivity is given by ρ xx ( T ) = π µ Q + 4 π ˜ g b Q T log (cid:18) γTµ (cid:19) for γT (cid:29) µ , π µ Q for γT (cid:28) µ . (C16)So, the incoherent charged bosons have linear-in- T resistiv-ity when γT ≥ µ . This contribution also leads to the Drudeform of optical conductivity, as shown in Eq. (C10). Thebosonic contribution becomes independent of temperature be-low this temperature. However, the presence of conductionelectrons will lead to a quadratic T -dependence of resistivity,just like in the Fermi liquid.Next, we focus on the situation when the interaction be-tween the bosons is lower than the Landau damping constant, Doping (x) T e m p e r a t u r e ( T ) PseudoGapped
Drude
Non-DrudeNon-Drude
Fermi Liquid
FIG. 8. The phase diagram for the scenario when bosonic interac-tion strength is weaker than the Landau damping parameter. Here,we have an intermediate regime bounded by the dotted black curve,where the optical conductivity does not conform to the conventionalDrude form. i.e., ˜ g b < γ . In this situation, there will be an intermediatetemperature regime, µ /γ < T < T (cid:48) , where γT > µ ( T ) .The resulting phase diagram is presented in Fig. (8). The re-gion bounded by the dotted line can harbor a non-Drude likeoptical conductivity as evaluated in Eq. (C11). The static con-ductivity in this limit is given by ρ xx ( T ) ≈ π γ Q ˜ g b log( γT /µ ) T for γT (cid:29) µ , π γ Q µ T for γT (cid:28) µ . (C17)Consequently, up to logarithmic corrections, we still havea linear-in- T resistivity even when the bosonic interactionstrength is weaker than the damping and γT > µ . However,such linear-in- T resistivity does not subscribe to the Drudeform of the optical conductivity. Below this temperature,the incoherent bosons also contribute to the T -resistivity ex-pected in the Fermi-liquid regime. Thus, for weak coupling,the crossover from the strange metallic to Fermi-liquid behav-ior occurs through this intermediary region.In the pseudogap phase, the opening of a gap at the temper-ature T ∗ results from a deconfining transition of a PDW orderparameter into a SC and CDW fields. Above T ∗ , the inco-herent bosons have a bare mass of µ . This is illustrated inmore details in Appendix (C 2). Using the bare mass for thebosons, a temperature region T ∗ < T < T (cid:48) exists where thenon-Drude form of the optical condutivity survives for weaklycoupled bosons, i.e., ˜ g b < γ .2 q q g b g b p , ω n k − p , ν n − ω n + q k , ν n FIG. 9. The second-order term for the self-energy of the incoherentbosons.
2. Bosonic bare mass in the ordered side
Near the ordered phase, i.e., just above T ∗ in Fig. (8),the bosonic propagator attains the bare mass due to the or-dered parameter fluctuations. The Ginzburg-Landau free en-ergy functional is given by F [ ψ ] = (cid:90) d d x (cid:20) µ | ψ ( x ) | + b | ψ ( x ) | (cid:21) . (C18)If ψ ( x ) minimizes F [ ψ ] , we obtain ψ = (cid:114) − µ b . (C19)Expanding around the minima ψ ( x ) = ψ ( x ) + δψ ( x ) ,where δψ ( x ) is the fluctuations. Putting this in Eq. (C18) and noting that the terms linear in δψ ( x ) vanishes, we obtain F [ δψ ] = (cid:90) d d x (cid:2) − µ | δψ ( x ) | + ... (cid:3) . (C20)Therefore, the bare mass of the diffusive bosons just above the T ∗ is given by µ . Appendix D: Mode-Mode Coupling: Higher order terms inself-energy
The second-order bosonic self-energy diagram – whichrenormalizes both the mass-term µ , and the imaginary termof the bosonic propagator, γ – is denoted by Π ( q ) where q is the external frequency. We emphasize that the bosonicdispersion has a minimum at finite momentum Q , which isdifferent from the external frequency in this diagram, q . Thediagram evaluated is depicted in Fig. (9). The integral is givenby Π ( q ) = g b L (cid:88) k , p T (cid:88) ω n ,ν n D ( ν n − ω n + q , k − p ) × D ( ν n , k ) D ( ω n , p ) . (D1)Performing the summation over ν n and ω n and using the spec-tral decomposition, one readily obtains Π ( q ) = g b L (cid:88) k , p (cid:90) ∞−∞ dE π (cid:90) ∞−∞ dE π (cid:90) ∞−∞ dE π [ A ( E , a ) A ( E , b ) A ( E , d ) ( n B ( E ) − n B ( E ))] × (cid:18) n B ( E ) − n B ( E − E ) iq − E + E − E (cid:19) , (D2)where we have defined a = k + µ. (D3) b = ( k − p ) + µ. (D4) d = p + µ. (D5)Analytically continuing iq → q + i + , the imaginary part of the Π becomesIm Π ( q ) = − g b q π L (cid:88) k , p (cid:90) ∞−∞ dE (cid:90) ∞−∞ dE [ A ( E , a ) A ( E , b ) A ( E − E + q , d ) ( n B ( E ) − n B ( E ))] ∂n B ∂ ( E − E ) . (D6)In the regime where | E − E | < T and expanding the spectral function in the q → limit, we obtainIm Π ( q ) = γg b T q π L (cid:88) k , p (cid:90) ∞−∞ dE (cid:90) ∞−∞ dE (cid:20) γE γE (( γE ) + a )(( γE ) + b )(( γ ( E − E )) + d ) (cid:18) n B ( E ) − n B ( E ) E − E (cid:19)(cid:21) . (D7)3Next, approximating n B ( E ) by using Eq. (B5), the integrand will only contribute only both | E | < T and | E | < T . Making achange of variables from ˜ E = γE , we obtainIm Π ( q ) = γg b T q π L (cid:88) k , p (cid:90) γT − γT d ˜ E (cid:90) γT − γT d ˜ E
1( ˜ E + a )( ˜ E + b )(( ˜ E − ˜ E ) + d ) . (D8)Evaluating the integral in the familiar regimes γT (cid:29) µ and γT (cid:28) µ , we obtain the formsIm Π ( q ) = γg b T q L (cid:88) k , p abd ( a + b + d ) for γT (cid:29) µ, γ g b T q π L (cid:88) k , p a b d for γT (cid:28) µ. (D9)Performing the momentum summation and arrive at expressions for the imaginary part Π Im Π ( q ) = c γg b T π µ q for γT (cid:29) µ,c γ g b T π µ q for γT (cid:28) µ, (D10)where c = 0 . and c = 0 . , which are evaluated nu-merically. On the other hand, the real part of Π can be evalu- ated by utilizing Kramers-Kronig relations. The external fre-quency is taken to be small in the above calculations. Thus,a frequency cut-off λ = min [ µ, γT ] is used in the Kramers-Kronig relation. The Kramers-Kronig relation is given byRe Π ( q ) = 2 π P (cid:90) λ ω Im Π ( ω ) ω − q dω. (D11)Therefore, the real-part of the Π becomesRe Π ( q ) = c γg b T π µ (cid:18) λ − q tanh − (cid:18) λq (cid:19)(cid:19) for γT (cid:29) µ,c γ g b T π µ (cid:18) λ − q tanh − (cid:18) λq (cid:19)(cid:19) for γT (cid:28) µ, (D12)where λ is the cut-off energy scale. Now evaluating the renor-malization of the µ and γ up to the second order for γT (cid:29) µ ,we get µ ≈ µ + g b π log (cid:18) γTµ (cid:19) + 2 c γλπ log ( γT /µ ) (D13) ˜ γ ≈ γ + c γπ log ( γT /µ ) . (D14)Taking the limit γT /µ (cid:29) , it is clear that the second-orderterms are negligible. Next, evaluating the same for γT (cid:28) µ ,we get µ ≈ µ + c λ ( γT ) π γµ , (D15) ˜ γ ≈ γ + c ( γT ) π γµ . (D16)Again, taking the limit γT /µ (cid:28) , it becomes clear that thehigher-order terms are negligible compared to the first orderones. Appendix E: Hall conductivity
To discuss the effect of magnetic field, in the first order inmagnetic field, we calculate the Hall conductivity which isgiven by σ (1) xy = iHω n T (cid:88) ε n L (cid:88) q [ q x D ( ε n , q ) ∂ q x D ( ε n + ω n , q ) − q y D ( ε n , q ) ∂ q y D ( ε n + ω n , q )] . (E1)For a particle-hole symmetric theory, the Hall conductivity isnaturally expected to vanish. This means that the incoherentbosons at finite- Q do not contribute to the Hall conductivity.Using the fact that ∂ q x D ( x ) = q x D ( x ) , only the wave-vectornear q x = Q will contribute. As a result, we obtain σ (1) xy ( ω n ) = iHQ ω n T (cid:88) ε n L (cid:88) q (cid:2) D ( ε n , q ) D ( ε n + ω n , q ) −D ( ε n , q ) D ( ε n + ω n , q ) (cid:3) . (E2)4Performing the Matsubara summation by using spectral func-tions, we arrive at σ (1) xy ( ω ) = iHQ L (cid:88) q (cid:90) ∞−∞ dE π (cid:90) ∞−∞ dE π n B ( E ) − n B ( E ) ω ( E − E + ω ) × (cid:16) A ( E , q ) ˜ A ( E , q ) − ˜ A ( E , q ) A ( E , q ) (cid:17) , (E3)where A ( E , q ) is given in Eq. (B2) and the ˜ A ( E , q ) isgiven by ˜ A ( q, E ) = − Im [ D R ( E, q )] = − γE ( q + µ )( γE ) + ( q + µ ) . (E4)Therefore, taking the ω → , the expression for the Hall con-ductivity becomes σ (1) xy (0) = iHQ L (cid:88) q (cid:90) ∞−∞ dE π (cid:90) ∞−∞ dE π A ( E , q ) ˜ A ( E , q ) × (cid:34) coth( E T ) − coth( E T )( E − E ) (cid:35) . (E5)This can be trivially shown to be exactly zero by noting thatthe A ( E, q ) , ˜ A ( E, q ) and coth( E ) are all anti-symmetricfunctions with respect to E . Since I ( − E , − E ) = − I ( E , E ) , as a consequence, the incoherent bosons will in-deed have a vanishing Hall conductivity. Appendix F: Second Moment of Conductivity
The second moment of the conductivity, the term propor-tional to the square of the field H , is given in terms of thebosonic Green’s function by σ (2) xx ( ω n ) = − H ω n Im (cid:26) T (cid:88) ε n L (cid:88) q ∂ q y D ( ε n , q ) ∂ q y D ( ε n + ω n , q ) (cid:27) . (F1)Using the form of bosonic propagator D ( ω n , q ) , we obtain σ (2) xx ( ω n ) = − Q H ω n Im (cid:26) T (cid:88) ε n L (cid:88) q D ( ε n , q ) D ( ε n + ω n , q ) (cid:27) . (F2)The spectral function in Eq. (E4) is used to perform the Mat-subara summation over ε n . After analytical continuation, thereal part of the second moment of conductivity becomes σ (2) xx ( ω n ) = − Q H ω n L (cid:88) q (cid:90) ∞−∞ dE (cid:104) ˜ A ( E , q ) × ˜ A ( E + ω, q ) ∂n B ∂E (cid:21) . (F3) The Bose function is approximated by Eq. (B5) and the mo-mentum summation is carried out by replacing ( q + µ ) = t ,i.e., σ (2) xx ( ω →
0) = − T Q H π (cid:90) ∞ µ t dt (cid:90) ∞−∞ dE γ { ( γE ) + t } . (F4)Finally, performing the integral over E and t , and then byexpanding in the two familiar limits, we obtain the expressionfor the real part of static second moment of conductivity σ (2) xx = γ Q T H π µ for γT (cid:28) µ, T γQ H πµ for γT (cid:29) µ. (F5) Appendix G: Polarization bubble due to the Zeeman field1. Singlet Case
For particle-particle pairs of singlets, the contribution to theself-energy due to the Zeeman term is evaluated here. Thecorrection to the mass term is given by Π( H, Q ) = g I L (cid:88) k T (cid:88) ε n [ G ( − ε n , ξ − k , ↑ ) G ( ε n , ξ k + Q , ↓ ) −G ( − ε n , ξ − k , ↓ ) G ( ε n , ξ k + Q , ↑ )] , (G1)where ξ k,σ = k − σH where σ = ± . Next, performingthe Matsubara summation over ε n , we arrive at the expressionwhich is independent of magnetic field. The mass term thusbecomes µ = µ + µ T , (G2)where µ T = ˜ g b T log( γT /µ ) . So the mass-term has no con-tribution from the Zeeman field.
2. Triplet Case
Here, we calculate the self-energy correction due to thebosons formed with paired electrons of triplet spin-symmetry.The corresponding expression is given by Π( H, Q ) = g I L (cid:88) k T (cid:88) ε n G ( − ε n , ξ − k , ↑ ) G ( ε n , ξ k + Q , ↑ ) , (G3)where ξ k,σ = k − σH where σ = ± , in our units (cid:126) / (2 m e ) = 1 . Performing the ε n -summation, we get Π( H, Q ) = g I L (cid:88) k (cid:26) − n F ( ξ k − H ) − n F ( ξ k + Q − H ) ξ k + Q + ξ k − H (cid:27) . (G4)Next, using a flat band approximation, we can write the mo-mentum summation in the following form5 Π( H, Q ) = N ( (cid:15) F ) g I π (cid:34)(cid:90) π dθ (cid:90) Λ0 dξ tanh( ξ + ζ − H T ) + tanh( ξ − H T )2 ξ + ζ − H (cid:35) , (G5)where Λ is the largest energy scale of the system. Addition-ally, we have substituted ζ ≡ Q + 2 k F Q cos( θ ) . Now at T → , we will use that tanh( x/T ) → sgn ( x ) and thenperforming the ξ -integral we arrive at Π( H, Q ) = N ( (cid:15) F ) g I π (cid:90) π dθ log (cid:18) − Hζ (cid:19) + log (cid:18) − ζ + 2Λ − Hζ (cid:19) for ζ ≤ , log (cid:18) ζ + 2Λ − Hζ (cid:19) for ζ > and H − ζ ≤ . (G6)Recall that the Λ is the ultraviolet energy cutoff, and hence,expanding in H (cid:28) k F Q (cid:28) Λ , we get Π( H, Q ) = C − N ( (cid:15) F ) g I π (cid:90) π − p p dθ HQ + 2 k F cos( θ ) , (G7)where p ≡ cos − ( Q / (2 k F )) and C is the H -independentconstant. Now integrating over θ , one obtains Π( H, Q ) = C + 2 γπ coth − (cid:32) k F + Q (cid:112) k F − Q (cid:33) H, (G8)where we have used the definition of γ from Appendix (A).The constants can be absorbed in the bare bosonic mass, µ . Therefore, the total bosonic mass renormalization due tothe Zeeman field H , becomes µ = µ + µ T + αH, (G9)where α ≡ γπ coth − (cid:18) k F + Q √ k F − Q (cid:19) and we also define µ H ≡ αH . Thus, we obtain the mass-renormalization dueto the Zeeman field, which is used to evaluate magnetoresis-tance in the next section. Appendix H: Magnetoresistance
In this section, we explicitly show the calculations to ar-rive at the magnetoresistance for diffusive bosons. The mag-netoresistance quantifies the change of resistance due to theapplication of the magnetic field and is given by ∆ ρ xx ( H ) ρ xx ( H = 0) = ρ xx ( H ) − ρ xx (0) ρ xx (0) . (H1)The complete resistivity tensor in terms of the conductivityis written as ρ xx = σ xx σ xx + σ xy . (H2) Notice that, for incoherent transport, we have shown in Ap-pendix E that σ xy = 0 and hence the expression for the mag-netoresistance in terms of conductivity simply reads ∆ ρ xx ( H ) ρ xx ( H = 0) = σ xx (0) − σ xx ( H ) σ xx ( H ) . (H3)Next, the expression for σ xx = σ (0) xx + σ (2) xx where σ (0) xx is al-ready calculated in Eq. (C10) and Eq. (C11) and σ (2) xx is eval-uated in Eq. (F5).
1. Singlet Case
Here, the renormalization of the mass term is independentof the magnetic field and is given by µ = µ + µ T . Theregimes are given by the maximum of µ and µ T . So theexpression for the magnetoresistance becomes ∆ ρ xx ( H ) ρ xx (0) = κH β + κH , (H4)where β and κ are the coefficients of H in σ (0) xx and σ (2) xx ,respectively. We consider that the interaction between thebosons is larger than the Landau damping coefficient, i.e., ˜ g b > γ . In this scenario, if we take the limit γT /µ (cid:28) inEq. (F5) and Eq. (C10), it becomes clear that σ (2) xx is negligiblecompared to the σ (0) xx . Hence, we have the leading contributionto the MR by taking the limit κ/β (cid:28) ρ xx ( H ) ρ xx (0) ≈ κβ H , (H5)where in the first regime when µ > µ T the constant is givenby κβ ≡ − γ T µ . By contrast, when µ < µ T , the constantis given by κβ ≡ − γ T µ + µ T ) . Thus, the bosons arising fromthe singlet pairing of electrons have the same dependence on6 FIG. 10. The figure illustrates the different regimes in the tempera-ture, doping and magnetic field plane. The mass term renormaliza-tion for the particle-particle pairs is given by µ = µ + µ T + µ H .The maximum of the three mass scales determines the regime: Inregime 1, the mass is dominated by µ ; similarly, in regime 2 and 3,it is dominated by µ T and µ H , respectively. H as the conduction electrons would do in the typical Fermiliquid.When the interaction between the bosons is weaker than theLandau damping expanding in κ/β (cid:29) , the magnetoresis-tance becomes independent of H for singlet particle-particlepairs.
2. Triplet Case
Next, we perform the calculation of the renormalizationof the mass term when the bosons emerge from pairs ofhigh-energy electrons that have spin-triplet symmetry. Thebosonic mass correction due to the Zeeman field is evaluatedin Eq. (G9). Similarly, the expressions for σ (0) xx in terms of µ are evaluated in Eq. (C12) and the same for σ (2) xx are evalu-ated in Eq. (F5). Notice we have different regimes dependingon the renormalization of the mass term from bosonic inter-actions and the Zeeman field. These regimes are illustrated inFig. (10) in the magnetic field, hole doping and temperatureplane. The different scenarios arise because the mass term iseither dominated µ , µ T or µ H . We elaborate on the differentpossibilities one by one in the following. a. ˜ g b ≥ γ and µ T (cid:28) µ H First, if the interaction between the bosons is larger thanthe Landau damping coefficient, i.e., ˜ g b > γ , we are alwaysin γT (cid:28) µ . Additionally, if we are in a regime dominatedby the magnetic field scale , i.e., µ T (cid:28) µ H (see regime 3 inFig. (10)), the mass correction coming from the Zeeman fieldis given by µ = µ + µ T + αH in Eq. (G9). Therefore, the magnetoresistance evaluates to ∆ ρ xx ( H ) ρ xx (0) = Q π ( µ + µ T ) − Q π ( µ + µ T + αH ) − σ (2) xx ( H ) Q π ( µ + µ T + αH ) + σ (2) xx ( H ) . (H6)If we take the limit γT /µ (cid:28) in Eq. (F5), it is clear thatthe σ (2) xx becomes negligible. Therefore, the equation for MRbecomes ∆ ρ xx ( H ) ρ xx (0) ≈ µ + µ T − µ + µ T + αH )1 µ + µ T + αH , ∆ ρ xx ( H ) ρ xx (0) = αµ + µ T H. (H7)Therefore, we obtain a linear-in- H magnetoresistance in theregime 3 of Fig. (10). Note that µ H (cid:29) µ T can be interpretedas H (cid:29) ηT , where η = µ +˜ g b log( γT/µ ) α . Thus up to log-arithmic corrections η is just a constant. We emphasize thatthis a similar high-field regime where linear-in- H magnetore-sistance is observed. b. ˜ g b ≥ γ and µ T (cid:29) µ H Second, we still keep the interaction between the bosonsstronger than the Landau damping coefficient, i.e., ˜ g b > γ .However, if the temperature-correction is larger than the mag-netic field scale, i.e., µ T (cid:29) µ H , the mass correction comingfrom the Zeeman field is independent of the field and is givenby µ = µ + µ T (see regime 2 in Fig. (10)). Consequently,the evaluation of magnetoresistance becomes similar to theone performed for the singlet in Appendix (H 1) ∆ ρ xx ( H ) ρ xx (0) ≈ κβ H , (H8)where κβ ≡ − γ T µ + µ T ) . Again for µ H (cid:28) µ T can be writ-ten as H (cid:28) ηT . Therefore, in the low-field regime shows aquadratic H -dependence of magnetoresistance. c. ˜ g b ≥ γ for µ T (cid:28) µ and µ H (cid:28) µ Similarly, if the temperature or field correction of thebosonic mass term is smaller than the bare bosonic mass, i.e., µ T (cid:28) µ and µ H (cid:28) µ , the mass correction coming fromthe Zeeman field is independent of the field and is given by µ = µ (see regime 1 in Fig. (10)). Again, the magnetoresis-tance becomes ∆ ρ xx ( H ) ρ xx (0) ≈ κβ H , (H9)here we get κβ ≡ − γ T µ . So again we have a H -dependence of magnetoresistance in the regime 1 of Fig. (10).7In this regime we have already established the conventionalFermi liquid behavior.Therefore, when the interaction between the bosons isstronger than the Landau damping coefficient the MR is givenby ∆ ρ xx ( H ) ρ xx (0) = κβ H in regimes 1 and 2 ,αµ + µ T H in regime 3 , (H10)where the coefficient κ/β is different in regimes 1 and 2. No-tice that such an H -evolution of magnetoresistance is recentlyobserved in overdoped cuprates. d. ˜ g b < γ When the coupling is weaker than the Landau damping, atemperature regime survives where µ (cid:28) γT (for details, re-fer to Appendix C 1). We demand the limit µ/ ( γT ) (cid:28) and recognize that σ (0) xx is negligible. Consequently, using theexpression of σ (2) xx from Eq. (F5) in the expression of MR inEq. (H3). We notice that the MR becomes independent of H in all the temperature regimes for ˜ g b < γ .
3. On the quadrature form of the magnetoresistance
This section provides more details in order to compare thescaling of the in-plane magnetoresistance with that observedexperimentally. The in-plane MR is given by ∆ ρ xx = ρ xx ( H, T ) − ρ xx (0 ,
0) = 1 σ xx ( H, T ) − σ xx (0 , , (H11)where in the second equality we have used the fact that theHall conductivity vanishes. Near the QCP, ∆ ρ xx experimen-tally displays a quadrature dependence as follows ∆ ρ xx = (cid:112) a T + b H , (H12)where a and b are constants. As we explained in the main text,in the low-field and high-field limits, this quantity scales as ∆ ρ xx ∝ H for H (cid:29) T,H T for H (cid:28) T. (H13)Although our phenomenological model cannot determine ex-actly the quadrature dependence of Eq. (H12), our results forthe scaling behavior in both low-field and high-field limits cansuggest a similar quadrature ansatz. We concentrate on thephysical regime when the interaction between the bosons isstronger than the Landau damping parameter, i.e., ˜ g b ≥ γ ,i.e., µ (cid:29) γT . We also restrict our attention to the case whenthe bosons emerge from pairs of high-energy electrons thathave spin-triplet symmetry. Consequently, the mass-term is given by Eq. (G9). The maximum among µ , µ T , and µ H de-termines the regime, as shown in Fig. (10). Let us first focuson regime 3 of Fig. (10), where the mass term is dominated by µ H . Mathematically, we are in the regime µ H (cid:29) µ T (cid:29) µ ,or H (cid:29) ηT , where up to logarithmic corrections, η is onlya constant. Using the form of σ (0) xx from Eq. (C12) and σ (2) xx from Eq. (F5) in Eq. (H11), we get ∆ ρ xx = 1 Q π ( µ + µ T + µ H ) + σ (2) xx − π µ Q . (H14)Since the interaction between the bosons is stronger than theLandau damping parameter, i.e., ˜ g b ≥ γ , by taking ( γT ) /µ (cid:28) in Eq. (17), σ (2) xx → . Consequently, we get ∆ ρ xx ≈ π Q ( µ T + µ H ) . (H15)Therefore, in the high field regime H (cid:29) ηT (i.e., the regime3 of Fig. (10)), the leading order H -dependence is given by ∆ ρ xx ∝ H. (H16)The next regime is when the mass-term is dominated by µ T (i.e., the regime 2 in Fig. (10)). Notice that this is the low-fieldregime, H (cid:28) ηT . Here, we have ∆ ρ xx = 1 Q π ( µ + µ T ) + γ Q T H π ( µ + µ T ) − π µ Q . (H17)However, we cannot ignore σ (2) xx to get the leading order H -dependence, since σ (0) xx is independent of the field. Expandingin powers of H , we obtain ∆ ρ xx = 4 π µ T Q − π Q ˜ g b log(( γT ) /µ ) H T . (H18)Therefore, in the low-field regime H (cid:28) ηT , the leading orderscaling is given by ∆ ρ xx ∝ H T . (H19)Next, in the regime 1 of Fig. (10), the mass-term is dominatedby µ . In the latter regime, the in-plane magnetoresistance isgiven by ∆ ρ xx = 4 π µ T Q − π Q ˜ g b log(( γT ) /µ ) H T . (H20)Therefore, in the Fermi liquid regime, the leading order scal-ing is given by ∆ ρ xx ∝ H T µ . (H21)8Finally, combining Eq. (H16), Eq. (H19), and Eq. (H21), wehave the scaling of ∆ ρ xx , to leading order in H , as ∆ ρ xx ∝ H for regime 3 ,H T for regime 2 ,H T µ for regime 1 , (H22) which is identical to the scaling observed from the quadraturedependence in Eq. (H13). We conclude that, although our cal-culations cannot determine exactly the quadrature dependenceof ∆ ρ xx presented in Eq. (H12), we can find a similar scalingbehavior n the low-field and high-field limits. M. Gurvitch and A. T. Fiory, Phys. Rev. Lett. , 1337 (1987). I S. Martin, A. T. Fiory, R. M. Fleming, L. F. Schneemeyer, andJ. V. Waszczak, Phys. Rev. B , 846 (1990). I R. Cooper et al., Science , 603 (2009). I N. E. Hussey, Journal of Physics: Condensed Matter , 123201(2008). I Y. Cao et al., Physical Review Letters , 076801 (2020). I G. R. Stewart, Rev. Mod. Phys. , 797 (2001). A. Rost et al., Proceedings of the National Academy of Sciences , 16549 (2011). I D. Van Der Marel et al., Nature , 271 (2003). I, III, III C R. Lobo, The optical conductivity of high-temperature supercon-ductors, in
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