Fermi Surface Fluctuations and Single Electron Excitations Near

PHYSICAL REVIEW B 73, 045127 ͑2006͒

Fermi surface ﬂuctuations and single electron excitations near Pomeranchuk instability in two dimensions

Luca Dell’Anna and Walter Metzner Max-Planck-Institut für Festkörperforschung, D-70569 Stuttgart, Germany ͑Received 22 July 2005; revised manuscript received 18 November 2005; published 27 January 2006; corrected 3 March 2006͒

A metallic electron system near an orientational symmetry breaking Pomeranchuk instability is characterized by a “soft” Fermi surface with enhanced collective fluctuations. We analyze fluctuation effects in a two- dimensional electron system on a square lattice in the vicinity of a Pomeranchuk instability with d-wave symmetry, using a phenomenological model which includes interactions with a small momentum transfer only. We compute the dynamical density correlations with a d-wave form factor for small momenta and frequencies, the dynamical effective interaction due to a fluctuation exchange, and the electron self-energy. At the quantum critical point the density correlations and the dynamical forward scattering interaction diverge with a dynamical exponent z=3. The singular forward scattering leads to large self-energy corrections, which destroy Fermi liquid behavior over the whole Fermi surface except near the Brillouin zone diagonal. The decay rate of single-particle excitations, which is related to the width of the peaks in the spectral function, exceeds the excitation energy in the low-energy limit. The dispersion of maxima in the spectra flattens strongly near those portions of the Fermi surface which are remote from the zone diagonal. The contribution from classical fluctuations to the self-energy spoils ͑␻/T͒ scaling in the quantum critical regime.

DOI: 10.1103/PhysRevB.73.045127 PACS number͑s͒: 71.10.Fd, 74.20.Mn, 71.18.ϩy

I. INTRODUCTION will, however, focus on symmetry-breaking Fermi surface deformations in an otherwise normal state. Since a Pomeran- Under suitable circumstances electron-electron interac- chuk instability of electrons on a lattice breaks only a dis- tions can generate a spontaneous breaking of the rotation crete symmetry, no Goldstone mode exists and symmetry- symmetry of an itinerant electron system without breaking broken states are stable also at finite temperature in dജ2 translation invariance. From a Fermi liquid perspective such dimensions. an instability is driven by forward scattering interactions and Electron systems in the vicinity of a Pomeranchuk insta- leads to a symmetry-breaking deformation of the Fermi sur- bility have peculiar properties due to a “soft” Fermi surface, face. Alluding to a stability limit for forward scattering de- which can be easily deformed by anisotropic perturbations. rived long ago by Pomeranchuk,1 it is therefore frequently In particular, it was shown that dynamical fluctuations of referred to as “Pomeranchuk instability.” On a square lattice, such a soft Fermi surface lead to a strongly enhanced decay a Pomeranchuk instability with dx2−y2 symmetry, where the rate for single-particle excitations and thus to non-Fermi liq- 11 Fermi surface expands along the kx axis and shrinks along uid behavior. Close to a dx2−y2 Pomeranchuk instability the ͑ ͒ 2 the ky axis or vice versa , was first considered for the t-J, decay rate is maximal near the kx and ky axes and minimal Hubbard,3,4 and extended Hubbard model.5 Fermi surface near the diagonal of the Brillouin zone. The decay rate was symmetry breaking in fully isotropic ͑not lattice͒ systems has computed in random phase approximation ͑RPA͒ for a phe- also been analyzed.6 A Pomeranchuk instability leads to a nomenological model describing electrons on a square lattice state with the same symmetry reduction as the “nematic” which interact exclusively via scattering processes with electron liquid defined by Kivelson et al.7 in their discussion small momentum transfers.11 Subsequently it was shown that of similarities between doped Mott insulators with charge the putative continuous Pomeranchuk transition in this model stripe correlations and liquid crystal phases. is usually preempted by a first order transition at low tem- While we deal with a particular Pomeranchuk instability peratures; that is, the Fermi surface symmetry changes in the charge channel, Pomeranchuk instabilities in the spin abruptly before the fluctuations become truely critical.12,13 channel can also lead to interesting consequences. For ex- However, for reasonable choices of hopping and interaction ample, a p-wave spin Pomeranchuk instability gives rise to a parameters the system is nevertheless characterized by a dynamical generation of spin-orbit coupling.8 drastically softened Fermi surface on the symmetric side of A Pomeranchuk instability usually has to compete with the first order transition, and hence by strongly enhanced other instabilities, but can also coexist with other symmetry- Fermi surface fluctuations.14 Furthermore, by adding a uni- breaking orders. For example, in the two-dimensional Hub- form repulsion to the forward scattering interaction, the first bard model with a sizable next-to-nearest neighbor hopping order transition can be suppressed, and for a favorable but and an electron density near van Hove filling a superconduct- not unphysical choice of model parameters even a genuine ing state with a d-wave deformed Fermi surface is stabilized quantum critical point can be realized.14 at least at weak coupling.9 Superconducting nematic states In this work we present a detailed analysis of dynamical have also been included in a general classification of possible Fermi surface fluctuations in the vicinity of a Pomeranchuk 10 symmetry-breaking patterns by Vojta et al. In this work we instability with dx2−y2-wave symmetry in two dimensions and

1098-0121/2006/73͑4͒/045127͑18͒/$23.00045127-1 ©2006 The American Physical Society L. DELL’ANNA AND W. METZNER PHYSICAL REVIEW B 73, 045127 ͑2006͒ their effect on single electron excitations. We compute the spectively, the dispersion relation is given by dynamical density correlations with a d-wave form factor for ⑀ ͓ ͑ ͒ Ј small momenta and frequencies, the dynamical effective in- k =−2 t cos kx + cos ky +2t cos kx cos ky teraction due to fluctuation exchange, and the electron self- ͑ ͔͒ ͑ ͒ + tЉ cos 2kx + cos 2ky . 2 energy, from which the spectral function for single-particle excitations is obtained. In the quantum critical regime the For a simplified treatment, which however fully captures the density correlations and the dynamical forward scattering in- crucial physics, we consider an interaction of the form teraction diverge with a singularity familiar from other quan- ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ tum phase transitions in itinerant electron systems with dy- fkkЈ q = u q + g q dkdkЈ 3 namical exponent z=3. The singular forward scattering leads with u͑q͒ജ0 and g͑q͒Ͻ0, and a form factor d with d 2 2 to large self-energy contributions ⌺͑k,␻͒, which are propor- k x −y symmetry, such as d =cos k −cos k . The coupling functions tional to d2, where d is a form factor with d-wave symme- k x y k k u͑q͒ and g͑q͒ vanish if ͉q͉ exceeds a certain small momen- try. At the quantum critical point, Im ⌺͑k ,␻͒ scales to zero F tum cutoff ⌳. This ansatz mimics the effective interaction in as ͉␻͉2/3. Fermi liquid behavior is thus destroyed over the the forward scattering channel as obtained from renormaliza- whole Fermi surface except near the Brillouin zone diagonal. tion group calculations3 and perturbation theory15 for the We also compute the crossover from the non-Fermi to Fermi two-dimensional Hubbard model near van Hove filling. The liquid behavior in the case of a large finite correlation length uniform term originates directly from the repulsion between ␰ in the ground state. In the quantum critical regime at low electrons and suppresses the electronic compressibility of the finite temperatures the self-energy has a part due to quantum system. The d-wave term drives the Pomeranchuk instability. fluctuations, which obeys ͑␻/T͒ scaling, but also a classical ␰͑ ͒ The mean-field solution of the f model has been analyzed part, which is proportional to T T near the Fermi surface. for various choices of parameters in a series of recent ͑␻ ͒ The classical contribution violates /T scaling and was articles.12–14 In the plane spanned by the chemical potential overlooked in Ref. 11. ␮ and temperature T the symmetry-broken phase is formed The paper is structured as follows. In Sec. II we define ͑␮͒ below a dome-shaped transition line Tc with a maximal our phenomenological model and review its mean-field transition temperature near van Hove filling. The phase tran- phase diagram. In Sec. III we analyze the dynamical effec- sition is usually first order near the edges of the transition tive interaction of the model in a regime where the nearby line, that is, where Tc is relatively low, and always second Pomeranchuk instability leads to strong Fermi surface fluc- order near its center.13 The two tricritical points at the ends tuations. The low-energy behavior of the self-energy in the of the second order transition line can be shifted to lower presence of these fluctuations is then computed in Sec. IV, temperatures by a sizable uniform repulsion u included in and results for the corresponding spectral function are dis- 14 fkkЈ. For a favorable choice of hopping and interaction pa- cussed in Sec. V. We finally summarize and discuss the pos- rameters, with a finite tЉ and uϾ0, one of the first order sible role of d-wave Fermi surface fluctuations in cuprate edges is suppressed completely such that a quantum critical superconductors in Sec. VI. point is realized. Although quantum critical points are usually prevented by first order transitions at low temperatures II. F MODEL in the f model, the Fermi surface is nevertheless already very soft near the transition, such that fluctuations can be ex- To analyze fluctuation effects in the vicinity of a Pomer- pected to be important.14 anchuk instability we consider a phenomenological lattice model with an interaction which drives a Fermi surface symmetry breaking, but no other instability. The model Hamil- III. EFFECTIVE INTERACTION tonian, which has been introduced already in Ref. 11, reads In this section we derive and analyze the dynamical effec- 1 tive interaction for the f model, which is closely related to ⑀ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 16 H = ͚ knk␴ + ͚ fkkЈ q nk q nkЈ − q , 1 the dynamical d-wave density correlation function ␴ 2V k, k,kЈ,q ϱ ⑀ ͑ ͒ ͑ ␯͒ ͵ i␯t͓͗ ͑ ͒ ͑ ͔͒͘ ͑ ͒ where k is a single-particle dispersion, nk q Nd q, =−i dt e nd q,t ,nd − q,0 , 4 ͚ † 0 = ␴ck−q/2,␴ck+q/2,␴, and V is the volume of the system. Since the Pomeranchuk instability is driven by interactions with where the d-wave density fluctuation operator is defined as vanishing momentum transfers, that is forward scattering, we ͑ ͒ choose a function fkkЈ q which contributes only for rela- ͑ ͒ ͑ ͒ ͑ ͒ nd q = ͚ dknk q . 5 tively small momenta q, and refer to the above model as the k “f model.” Clearly this model is adequate only if the Pomer- ͑ ͒ anchuk instability dominates over other instabilities and fluc- and nd q,t is the corresponding dynamical operator in the ͑ ␯͒ tuations in the system. Otherwise, it would have to be Heisenberg picture. We first analyze Nd q, and the effec- supplemented by interactions with large momentum trans- tive interaction within RPA, and then discuss higher order fers. corrections. For hopping amplitudes t, tЈ, and tЉ between nearest, next- The RPA result for the d-wave density correlation func- nearest, and third-nearest neighbors on a square lattice, re- tion in the f model is simply

045127-2 FERMI SURFACE FLUCTUATIONS AND SINGLE¼ PHYSICAL REVIEW B 73, 045127 ͑2006͒

function ⌸0͑q,␯͒ in three dimensions have been derived already long ago in the context of almost ferromagnetic metals by Moriya.17 ⌸0 At zero frequency d is a real function of q, which can be expanded as ⌸0͑ ͒ ͑ ͒ ͑ ͉͒ ͉2 O͉͑ ͉4͒͑͒ FIG. 1. Series of bubble chains contributing to the effective d q,0 = a T + c T q + q 12 interaction ⌫. for small q. The coefﬁcient a͑T͒ is always negative and can be written as ⌸0͑q,␯͒ N ͑q,␯͒ = d ͑6͒ d ͑ ͒⌸0͑ ␯͒ 1−g q d q, ͑ ͒ ͵ ⑀ Ј͑⑀ ␮͒ 2͑⑀͒͑͒ a T = d f − Nd 13 with the bare d-wave polarization function

2 ͑⑀ ␮͒ ͑⑀ ␮͒ with a weighted density of states N 2͑⑀͒ d p f p+q/2 − − f p−q/2 − d ⌸0͑ ␯͒ ͵ 2 ͑ ͒ 2 2 2 d q, =− dp. 7 =͐d p/͑2␲͒ ␦͑⑀−⑀ ͒d . A low temperature ͑Sommerfeld͒ ͑2␲͒2 ␯ + i0+ − ͑⑀ − ⑀ ͒ p p p+q/2 p−q/2 expansion yields The infinitesimal imaginary part in the denominator specifies ␲2 that we consider retarded functions. Note that the coupling 2 4 a͑T͒ =−N 2͑␮͒ − NЉ ͑␮͒T + O͑T ͒, ͑14͒ u͑q͒ does not enter here because mixed polarization func- d 6 d2 tions with a constant and a d-wave vertex vanish for small q. ͑␮͒ The RPA effective interaction is defined by the series of provided that Nd2 and its second derivative are finite. The chain diagrams sketched in Fig. 1, yielding coefficient c͑T͒ can be expressed as u͑q͒ g͑q͒ ⌫ ͑ ␯͒ 1 1 kkЈ q, = + dkdkЈ, ͑ ͒ ͵ ⑀ Ј͑⑀ ␮͒ Љ ͑⑀͒ Ј ͑⑀͒ ͑ ͒ ͑ ͒⌸0͑ ␯͒ ͑ ͒⌸0͑ ␯͒ c T = d f − ͓ 48 Nd2v2 − 16 Nd2⌬⑀ ͔, 15 1−u q q, 1−g q d q, ͑ ͒ 8 ͑⑀͒ ͓͐ 2 ͑ ␲͒2͔␦͑⑀ ⑀ ͒ 2⌬⑀ where Nd2⌬⑀ = d p/ 2 − p dp p with 0 2 2 ͒⑀and N 2 2͑ , ץ+ ץ=⌬ where ⌸ ͑q,␯͒ is the conventional polarization function, de- the Laplacian px py d v ͉ ⑀ٌ͉ fined as ⌸0͑q,␯͒ but without the form factor d . ͓͐ 2 ͑ ␲͒2͔␦͑⑀ ⑀ ͒ 2 2 d k = d p/ 2 − p dpvp with vp = p . Primes denote Near the Pomeranchuk instability the denominator derivatives with respect to ⑀. The sign of c͑T͒ depends on the ͑ ͒⌸0͑ ␯͒ ͑ ͒ ͑ ͒ 1−g q d q, in Eqs. 6 and 8 becomes very small for dispersion and other model parameters. In the low tempera- q→0 and ␯→0, if ␯ vanishes faster than q, while ture limit it will be sufficient to use 1−u͑q͒⌸0͑q,␯͒ remains of order 1 or even larger. The effec- ͑ ͒ 1 Ј ͑␮͒ 1 Љ ͑␮͒ ͑ ͒ tive interaction is then dominated by the second term in Eq. c = c 0 = 16 Nd2⌬⑀ − 48 Nd2v2 . 16 ͑8͒ and can be written as For finite frequencies ⌸0͑q,␯͒ is generally complex. For ⌫ ͑q,␯͒ = gS ͑q,␯͒d d ͑9͒ d kkЈ d k kЈ small q and ␯ with ␯/͉q͉→0 its imaginary part behaves as ͑ ͒ with g=g 0 and the “dynamical Stoner factor” ␯ Im ⌸0͑q,␯͒ → − ␳͑qˆ ,T͒ , ͑17͒ 1 d ͉ ͉ S ͑q,␯͒ = . ͑10͒ q d 1−g͑q͒⌸0͑q,␯͒ d where ␳͑qˆ ,T͒Ͼ0 depends only on the direction qˆ of q.At ͑ ␯͒ The d-wave density correlation function Nd q, is also pro- low temperatures it is sufficient to use ␳͑qˆ ͒=␳͑qˆ ,0͒, which ͑ ␯͒ portional to Sd q, . In the case of a second order transition can be expressed as the static Stoner factor 1 1 1 ͑ ␯͒͑͒ 2 Sd = lim lim Sd q, 11 ␳͑qˆ ͒ = ͚ d 0 . ͑18͒ ͉͒ ͑ ٌ ͉ ␯→ ␲ k → q 0 0 4 0 F vk0 tk0 · k0 qˆ · vk0 kF F F F F diverges on the transition line. Near the first order transition 0 Here kF are points on the Fermi surface satisfying the con- obtained typically for low temperatures in the mean-field so- 0 dition qˆ ·vk0 =0 and tk0 is a tangential unit vector in k . lution of the f-model, Sd is still drastically enhanced, that is F F F 14 Since the velocity vk0 is perpendicular to the Fermi surface of order 10 and more. F 0 Within RPA, the dynamical d-wave density fluctuations in kF and also perpendicular to qˆ , the Fermi surface has to be 0 near the Pomeranchuk instability and the corresponding sin- parallel to qˆ in kF. For convex reflection symmetric Fermi gularity of the effective interaction are obviously determined surfaces in two dimensions there are two such points for each by the asymptotic behavior of the d-wave polarization func- given qˆ , which are antipodal to each other. ⌸0͑ ␯͒ ␯ tion d q, for small q and , which we describe in the Motivated by the RPA result, but envisaging already cor- following; for derivations, see the Appendix. Analogous for- rections beyond RPA, we parametrize the dynamical Stoner mulas for the expansion of the conventional polarization factor for small q and ␯, with ␯/͉q͉ also small, as follows:

045127-3 L. DELL’ANNA AND W. METZNER PHYSICAL REVIEW B 73, 045127 ͑2006͒

1 Corrections to RPA and corresponding subleading correc- S ͑q,␯͒ = . ͑19͒ d ␯ tions to Fermi liquid behavior due to regular interactions, in ͑␰ ␰͒2 ␰2͉ ͉2 0/ + 0 q − i a generic stable Fermi liquid regime, have been analyzed u͑qˆ ͉͒q͉ thoroughly in the last few years.19 The low-energy behavior of most quantities receives nonanalytic corrections to Fermi Within RPA, the length scales ␰ and ␰ and the velocity u are 0 liquid behavior in dimensions dഛ3. For example, in two related in a simple manner to the expansion coefficients of ͉ ͉ ͉ ͉2 ⌸0 ͑ ͒ dimensions the spin susceptibility varies as q instead of q d and the coupling function g q . Assuming that the latter for small q at T=0. Remarkably, the charge susceptibility for ͑ ͒ ͉ ͉2 can be expanded as g q =g+g2 q +¯ for small q,weget small q remains unaffected. In this case nonanalytic contributions appearing on the level of single Feynman diagrams ␰2 =−gc͑T͒ − g a͑T͒, ͑20͒ 0 2 cancel systematically when all relevant diagrams at a certain order are summed.19 The arguments establishing the cancel- ͑␰ ␰͒2 −1 ͑ ͒ ͑ ͒ 0/ = Sd =1−ga T , 21 lation of nonanalytic corrections for the charge susceptibility can be readily extended to our case of a d-wave density 1 instead of the conventional density operator.20 The crucial u͑qˆ ͒ =− . ͑22͒ point is that a perturbing field coupling to the d-wave density g␳͑qˆ ,T͒ operator does not alter the singularities in the polarization 18 For gϽ0 the velocity u͑qˆ ͒ is always positive. The static function at q=0 and q=2kF. Hence, corrections due to regu- Stoner factor S diverges at the Pomeranchuk transition, if it lar interactions may shift the parameters with respect to the d ͑ ␯͒ ⌫ ͑ ␯͒ is continuous, and the correlation length ␰ diverges accord- RPA result for Nd q, and kkЈ q, , but they do not yield ͱ ␰ any qualitative changes. Within the f model, one such cor- ingly as Sd. The relation for 0 makes sense only if the right-hand side is positive. This is guaranteed if we restrict rection appears already on mean-field level: the u term in ͑ ͒ ͑ ͒ ourselves to systems where the Pomeranchuk transition is the fkkЈ q generates a constant momentum-independent Har- leading instability. For −gc͑T͒−g a͑T͒Ͻ0 a charge density tree self-energy correction, which renormalizes the relation 2 between ␮ and the density.14 wave instability with a wave vector q0 sets in first. The Near a continuous phase transition order parameter corre- parameters ␰ and u͑qˆ ͒ remain finite at the Pomeranchuk 0 lations are strongly renormalized with respect to the mean- transition and do not vary much in its vicinity. The correla- field or RPA result, if the dimensionality of the system is ␰͑␦ ͒ tion length ,T near the transition depends sensitively on below the upper critical dimension.21 These renormalizations control parameters ␦, such as the chemical potential, and on are most naturally treated by a renormalization group analy- the temperature. If the transition is continuous, ␰͑␦,T͒ di- sis of an effective field theory, where the order parameter → ͑␦͒ ␰͑␦ ͒ verges for T Tc . Within RPA, ,T diverges as fluctuations are represented by a bosonic field. The propaga- ͑ ͒−1/2 ͑␦͒Ͼ −1 ␦ T−Tc if Tc 0, and as T if is tuned to a quantum tor of that field corresponds to the order parameter correla- ␦ ͑ ␯͒ critical point c. tion function, that is to Nd q, in our case. The singular For small q and ␯, with ␯/͉q͉ also small, the d-wave den- effective interaction between electrons, which is generated ͑ ␯͒ sity correlation function can be written as Nd q, by large order parameter fluctuations, is then mediated by the ␬0 ͑ ␯͒ ␬0 ⌸0͑ ␯͒ bosonic field. In the bosonic representation, corrections to =− dSd q, , where d =−limq→0lim␯→0 d q, is the static d-wave compressibility.14 Its imaginary part is then given by the RPA result for the order parameter correlations are due to interactions of the Bose field. Close to a continuous Pomer- ␯ anchuk transition at finite Tc these terms are relevant and u͉q͉ lead to the classical non-Gaussian asymptotic behavior of the Im N ͑q,␯͒ =−␬0 . ͑23͒ Ising universality class in two dimensions. In the following d d ␯ 2 ͓͑␰ ␰͒2 ␰2͉ ͉2͔2 ͩ ͪ we focus, however, on the behavior in the quantum critical 0/ + 0 q + ␦ ␦ u͉q͉ regime near the zero temperature critical point at = c.In that regime the upper critical dimension separating Gaussian For ␰ӷ␰ and small finite q this function exhibits a pro- 0 from non-Gaussian behavior is dc =4−z, where z is the dy- nounced peak at low frequencies with a steep slope for namical exponent.22 In our case of a charge instability at ␯→0, in agreement with recent numerical results for q=0 one has z=3, in complete analogy to the ferromagnetic ͑ ␯͒ 22 Nd q, obtained for the t-J model within slave boson quantum critical point in itinerant electron systems. Hence, 16 RPA. we have dc =1, while the dimensionality of our system is We now discuss corrections due to contributions beyond two, such that Gaussian behavior is stable. However, as first ͑ ␯͒ 23 RPA. The exact density correlation function Nd q, can be pointed out by Millis, the irrelevant quartic interaction of written in the form of Eq. ͑6͒, with the full polarization func- the order parameter fluctuations changes the temperature de- ⌸ ͑ ␯͒ pendence of the correlation length near the quantum critical tion d q, , which is dressed by interactions, instead of the bare one. Analogously the full effective interaction is given point completely compared to the RPA result. In particular, by Eq. ͑8͒ with dressed polarization functions ⌸͑q,␯͒ and in a two-dimensional system with z=3, the correlation length ␦ 23 ⌸ ͑ ␯͒ at c behaves as d q, . Close to a continuous phase transition two types of interaction corrections can be distinguished, namely regular 1 ␰͑␦ ͒ ϰ ͑ ͒ interactions which remain finite at the transition and singular c,T 24 ͱT͉log T͉ effective interactions associated with large order parameter fluctuations which diverge. instead of the naive T−1-divergence.

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FIG. 2. Fock diagram relating the self-energy ⌺ to the effective interaction ⌫.

In summary, the d-wave density correlations and the singular part of the effective interaction in the quantum critical regime can be parametrized by the dynamical Stoner factor ͑ ␯͒ ͑ ͒ ␰ ͑ ͒ Sd q, as in Eq. 19 , where the parameters 0 and u qˆ do not vary much, while the correlation length ␰͑␦,T͒ depends ␦ FIG. 3. Decomposition of momentum transfers q in normal and sensitively on control parameters and temperature. The tangential components relative to the Fermi surface in k . asymptotic behavior of ␰͑␦,T͒ in the critical region is F strongly inﬂuenced by interactions of order parameter ﬂuc- tuations. express the effective interaction by the dynamical Stoner factor, Im ⌺ can be written as

gd2 d2q IV. SELF-ENERGY Im ⌺͑k,␻͒ =− k ͵ d␯ ͵ ͓b͑␯͒ + f͑␯ + ␻͔͒ ␲ ͑2␲͒2 We now analyze the low-energy behavior of the self- ϫ ͑ ␯͒ ͑ ␻ ␯͒ ͑ ͒ energy in the presence of strong d-wave Fermi surface ﬂuc- Im Sd q, Im G k + q, + . 27 tuations. We compute the self-energy to ﬁrst order in the Here and in the following G, ⌺, and S are retarded func- ⌫ ͑ ␯͒ d singular interaction kkЈ q, , non-self-consistently as well tions, that is the real frequency axis is approached from as self-consistently, and then discuss the role of vertex cor- ͓ above. The imaginary part of Sd can be written as see Eq. rections. ͑19͔͒

u͑qˆ ͉͒q͉␯ A. Random phase approximation Im S ͑q,␯͒ = . ͑28͒ d ␯2 + ͓͑␰ /␰͒2 + ͑␰ ͉q͉͒2͔2͓u͑qˆ ͉͒q͉͔2 To ﬁrst order in ⌫, the self-energy is given by the Fock 0 0 diagram in Fig. 2. The Hartree term vanishes because the In the non-self-consistent calculation one can use ͑ ␻͒ ␲␦͑␻ ␰ ͒ ␰ ⑀ ␮ ␯ expectation value of the d-wave density operator vanishes in Im G0 p, =− − p , where p = p − , to perform the the symmetric phase. The Fock diagram yields integral analytically, which yields

d2q d2q ⌺͑k,i␻ ͒ =−T͚ ͵ ⌫ ͑q,i␯ ͒G͑k + q,i␻ + i␯ ͒ ⌺͑ ␻͒ 2 ͵ ͓ ͑␰ ␻͒ ͑␰ ͔͒ n 2 kk n n n Im k, = gdk b k+q − + f k+q ␯ ͑2␲͒ ͑ ␲͒2 n 2 i0+͑␻ +␯ ͒ ϫ ͑ ␰ ␻͒ ͑ ͒ ϫe n n , ͑25͒ Im Sd q, k+q − . 29 where G is the propagator of the interacting system in a Since we are interested in the renormalization of low- self-consistent perturbation expansion, which is replaced by energy excitations, with k close to the Fermi surface, and q the bare propagator G0 in the non-self-consistent version. since only small momentum transfers contribute to the ⌫ ⌫ Note that we have approximated kkЈ with kЈ=k+q by kk, self-energy, it is convenient to introduce a local coordinate which makes almost no difference since only small q con- system in momentum space, centered around the Fermi point ͑ tribute and the effective interaction does not vary rapidly as a kF which is reached from k by a normal projection see Fig. ͒ function of k and kЈ. Analytic continuation to the real fre- 3 , such that the vector k−kF is perpendicular to the Fermi quency axis yields surface in kF. We can then parametrize k by the variable ͉ ͉ kr =± k−kF , with a positive sign for k on the exterior side 1 d2q ⌺͑ ␻ +͒ ͵ ␯ ͵ ͓ ͑␯͒ ⌫ ͑ ␯ +͒ of the Fermi surface, and minus inside. The momentum k, + i0 =− d 2 b Im kk q, + i0 ␲ ͑2␲͒ transfer q can be parametrized by a radial variable qr and a tangential variable q , as shown in Fig. 3. ϫ ͑ ␯ ␻ +͒ ͑␯͒⌫ ͑ ␯ ␻ t G k + q, + + i0 − f kk q, − ␰ The excitation energy k+q appearing in the above + ⌺ ␰ − i0 ͒ expressions for Im can be expanded as k+q =v k +v q +͑1/2m ͒q2 for small q and k near k . The ϫ ͑ ␯ +͔͒ ͑ ͒ kF r kF r t t F Im G k + q, + i0 , 26 ␰ parameter mt is given by the second derivative of k in tan- ␰ ͉ . The term of order q2 has 2ץ͉=where b͑␯͒=͓e␤␯ −1͔−1 and f͑␯͒=͓e␤␯ +1͔−1 are the Bose and gential direction, m−1 t kt k k=kF t Fermi functions, respectively. It is convenient to focus on the been included since some asymptotic results are dominated ⌺ ͉ ͉ӷ͉ ͉ imaginary part of , from which the real part can be easily by contributions with qt qr . It is convenient to use recovered via the Kramers-Kronig relation. Using Eq. ͑9͒ to qЈ=q +q2 ͑2m v ͒ instead of q as integration variable ͑in r r t ր t kF r

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͒ ␰ ͑ Ј͒ ͑ ␯͒ ␰ ϱ addition to qt , since the excitation energy k+q =vk kr +qr structure as our Stoner factor Sd q, for = . Different is, F ⌺͑ ␻͒ is linear in that variable; the corresponding Jacobi determi- however, the d-wave form factor making Im kF , nant is one. strongly anisotropic. It is strongest near the van Hove points, while the leading terms vanish on the diagonal of the Bril- 1. Ground state louin zone. Subleading terms and contributions from interac- At T=0 the combination of Bose and Fermi functions tions with large momentum transfers generate at least con- contributing to the RPA self-energy reduces to ventional Fermi liquid decay rates ͑of order ␻2 log͉␻͉͒ on the diagonal, but faster decay may be obtained due to higher − ⌰͑− ␻ Ͻ ␯ Ͻ 0͒ for ␻ Ͼ 0 b͑␯͒ + f͑␯ + ␻͒ = ͭ ͮ, order processes which couple different parts of the Fermi ⌰͑0 Ͻ ␯ Ͻ − ␻͒ for ␻ Ͻ 0 surface. ͑30͒ A strongly anisotropic decay rate for single-particle excitations following a power-law with exponent 2 has also been ⌰͑͒ 3 where =1, if the inequalities in the argument are satis- found for an isotropic continuum ͑not lattice͒ version of our fied, and ⌰͑͒=0 otherwise. In the following we restrict our- model.6 However, that result was obtained for the symmetry- selves to the case ␻Ͼ0 in derivations for definiteness, but broken nematic phase, and the large anisotropic decay rate is state final results also for ␻Ͻ0. due to the anisotropy of the nematic state and its Goldstone At the quantum critical point ͑T=0,␰=ϱ͒, the non-self- modes. At the quantum critical point the decay rate of the consistent RPA self-energy Eq. ͑29͒ can be written as 2 isotropic system also obeys a power law with exponent 3 , ͑␻/v ͒−k but now with a constant prefactor over the whole Fermi sur- kF r dqЈ Im ⌺͑k,␻͒ = gd2 ͵ r face. kF 2␲ −kr For k kF, that is finite kr, the asymptotic behavior of Im ⌺͑k,␻͒, Eq. ͑31͒, is obtained by rewriting the integral u͑qˆ ͉͒q͉͓␻ − v ͑k + qЈ͔͒ dqt kF r r Ј ϫ ͵ with a dimensionless variable ˜qr defined by qr +kr 2␲ ͓␻ − v ͑k + qЈ͔͒2 + ␰4͓u͑qˆ ͔͒2͉q͉6 ͑␻ ͒˜ ˜ ͑␰2 ͒−1/3␻1/3˜ kF r r 0 = /vk qr, and qt defined by qt = 0u qt. Here we F ͑ ͒ ͑31͒ approximate u qˆ by a constant u for simplicity. In the limit ␻→0 one can replace ͉q͉ by ͱq2 +k2. Carrying out the ˜q ␻Ͼ t r r for 0. One may impose a cutoff on the qt integral, which integral one then obtains ͑including the case ␻Ͻ0͒ however does not affect the asymptotic behavior for small ␻. ␻ 2 1/3 2/3 ϱ For k=kF, that is kr =0, the asymptotic dependence of gdk u ͉␻͉ dq˜ ⌺ Im ⌺͑k,␻͒ = F ͵ t ͱ˜q2 + ␬2 Im can be extracted by introducing dimensionless vari- 2␲v ␰4/3 2␲ t ables ˜q and ˜q , which are defined by qЈ=͑␻/v ͒˜q and kF 0 0 r t r kF r q =͑␰2u ͒−1/3␻1/3˜q , respectively, where u =u͑t ͒. For 1 t 0 kF t kF kF ϫ ͫ ͬ ͑ ͒ ␻ ln 1+ 2 2 3 , 34 small and kr =0 the above q integral is dominated by al- ͑˜q + ␬ ͒ ͉ ͉ӷ͉ ͉ t most tangential q vectors, that is qt qr ; more precisely ͉ Ј͉ ͉ ͉3 ͉ ͉ ͉ ͉2 with ␬=͑u␰2 /͉␻͉͒1/3k . Two different types of asymptotic be- qr scales as qt , and qr consequently as qt . Hence, we 0 r ͉ ͉ ͉ ͉ ͑ ͒ can replace q by qt and u qˆ by uk . This yields, for havior are separated by the frequency scale ␻→ F 0, ␻ = u␰2͉k ͉3. ͑35͒ kr 0 r gd2 u1/3␻2/3 1 ϱ ͉ ͉͑ ͒ kF kF dq˜r dq˜t ˜qt 1−˜qr For ͉␻͉ӷ␻ , corresponding to ␬Ӷ1, one recovers the result ⌺͑ ␻͒ → ͵ ͵ kr Im kF, 4/3 2 6 . v ␰ 2␲ ϱ 2␲ ͑1−˜q ͒ + ˜q Eq. ͑33͒, while for ͉␻͉Ӷ␻ one can expand the integrand in kF 0 0 − r t kr ␬−1 ͑32͒ and obtains 2 This asymptotic result does not depend on any cutoff. The gdk 1 Im ⌺͑k,␻͒ = F ␻2. ͑36͒ definite integral can be done analytically; the result is ␲2 4 4 6 vk u␰ k ͑4ͱ3␲͒−1. For ␻Ͻ0 one obtains the same with ͑−␻͒2/3. F 0 r Hence we have shown that In the latter limit momentum transfers normal to the Fermi

2 1/3 surface dominate, and the above result can thus be easily gdk uk generalized to a direction dependent u͑qˆ ͒ replacing u by Im ⌺͑k ,␻͒ → F F ͉␻͉2/3 ͑33͒ F ͱ ␲ ␰4/3 u͑n ͒. Note that for small k , the low frequency behavior of 4 3 vk 0 kF r F Im ⌺͑k,␻͒ deviates from the ͉␻͉2/3 behavior only below a for small ͉␻͉. Note that ␯=v qЈ−␻=͑˜q −1͒␻ vanishes ͉ ͉3 kF r r very small scale of order kr . The same crossover behavior faster than ͉q͉ for ␻→0, which justifies our expansion of has already been obtained previously for fermions coupled to ͑ ␯͒ ␯ ͉ ͉ 27 Sd q, for small ratios / q . a gauge field. ⌺͑ ␻͒ ⌺͑ ␻͒ Not unexpectedly, Im kF , has the same energy de- We now analyze the behavior of Im kF , at T=0 in pendence as for the quantum critical points near phase the symmetric phase at some small distance from the quan- separation24 and ferromagnetism25 in two dimensions, and tum critical point, where the correlation length ␰ is large, but also for fermions coupled to a U͑1͒-gauge field.26,27 In both not infinite. Of interest is, in particular, at which scale the ͉␻͉2/3 ⌺͑ ␻͒ ␰ cases the fluctuation propagator has the same singularity behavior of Im kF , is affected by a large finite .

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tively weak dependence of the results on u . kF 2. Low finite temperature At TϾ0 the self-energy in the non-self-consistent RPA approximation, Eq. ͑29͒, can be written as Im ⌺͑k,␻͒ dqЈ dq 2 ͵ r ͵ t ͕ ͓ ͑ Ј͒ ␻͔ = gd b vk kr + q − kF 2␲ 2␲ F r + f͓v ͑k + qЈ͔͖͒ kF r r ͑ ͉͒ ͉͓ ͑ Ј͒ ␻͔ u qˆ q vk kr + qr − ϫ F . ͓v ͑k + qЈ͒ − ␻͔2 + ͓͑␰ /␰͒2 + ͑␰ ͉q͉͒2͔2͓u͑qˆ ͉͒q͉͔2 kF r r 0 0 ͑40͒ FIG. 4. Non-self-consistent RPA result for the imaginary part of the self-energy Im ⌺͑k ,␻͒, divided by gd2 , as a function of ␻ in F kF To tackle the asymptotic behavior of this integral for low T, ͑T ͒ ␻ the ground state =0 for various choices of the correlation length small , and small kr, it is instructive to consider first the ␰; the other relevant parameters are ␰ =v =u =1. ␻ 0 kF kF special case kr = =0, that is Ј 2 dqr dqt Introducing the same dimensionless variables ˜qr and ˜qt as for ⌺͑ ͒ ͵ ͵ ͓ ͑ Ј͒ ͑ Ј͔͒ Im kF,0 = gdk b vk qr + f vk qr ␰=ϱ, one finds that the q integral is still dominated by tan- F 2␲ 2␲ F F ␻ gential momentum transfers for small , such that we can u͑qˆ ͉͒q͉v qЈ ͉ ͉ ͉ ͉ ͑ ͒ kF r approximate q by qt and u qˆ by uk . The ˜qr integral can ϫ . F ͑v qЈ͒2 + ͓͑␰ /␰͒2 + ͑␰ ͉q͉͒2͔2͓u͑qˆ ͉͒q͉͔2 then be carried out analytically, which yields kF r 0 0 2 1/3͉␻͉2/3 ϱ ͑41͒ gdk uk dq˜ Im ⌺͑k ,␻͒ = F F ͵ t˜q F 2␲v ␰4/3 2␲ t We introduce dimensionless variables ˜qr and ˜qt via the rela- kF 0 0 Ј ␰2 ␰3 ␰ tions qr = 0˜qr / and qt =˜qt / , respectively. We now assume 1 that ␰ diverges faster than T−1/3 for T→0, as is indeed the ϫlnͫ1+ ͬ, ͑37͒ ˜q2͑˜q2 + ␨2͒2 case when we approach the quantum critical point from the t t quantum critical region.23 Then the above integral is domi- ␨ ͑ ␰2 ͉␻͉͒1/3␰−1 where = uk / . The ˜qt integral can be done ana- nated by momenta with a small ratio v qЈ/T, such that the F 0 kF r lytically, leading however to a rather lengthy expression. Bose function can be expanded as b͑v qЈ͒→T/͑v qЈ͒, and kF r kF r There are two asymptotic regimes, separated by the charac- the Fermi function can be neglected. Furthermore we can teristic frequency scale exploit that the integral is dominated by momenta q which ␰ ␻ ␰2 ␰3 ␰−1 −3/2 ͑ ͒ are almost tangential to the Fermi surface for large , since ␰ = uk 0/ = uk 0 Sd . 38 ͉ Ј͉ ͉ ͉3 ͉ ͉ ͉ ͉2 F F qr scales as qt , and hence qr as qt . We can thus replace ͉q͉ by ͉q ͉ and u͑qˆ ͒ by u . We then obtain the simple result For ͉␻͉ӷ␻␰ one has ␨Ӷ1, which leads back to the result Eq. t kF ͑33͒. For ͉␻͉Ӷ␻␰, an expansion of the integral yields the 2 ϱ ϱ gdk dq˜ dq˜ asymptotic behavior Im ⌺͑k ,0͒ → F T␰͵ r ͵ t F ␰2 ␲ ␲ 0 −ϱ 2 −ϱ 2 2 4 gdk ␰ ␻ ⌺͑ ␻͒ F ␻2 ␰ ͑ ͒ Im kF, = ln . 39 u ͉˜q ͉ 6␲2v u ␰4 ͉␻͉ kF t kF k 0 ϫ F ͑v ˜q ͒2 + ͑1+˜q2͒2͑u ˜q ͒2 kF r t kF t Hence, below the scale ␻␰ one recovers Fermi liquid behav- ␰ gd2 ior. Close to the quantum critical point, that is for large and kF ͉␻͉2/3 = T␰ ͑42͒ Sd, the crossover from scaling to Fermi liquid behavior 4v ␰2 sets in only for very small frequencies, and the coefficient in kF 0 2 front of the asymptotic ␻ log ␻␰ /͉␻͉ law is anomalously for T→0. A similar contribution of order T␰ has been found large. A similar non-Fermi to Fermi liquid crossover occurs already earlier for almost antiferromagnetic28,30 and almost 28 25 31 ⌺͑ ͒ near antiferromagnetic and ferromagnetic quantum criti- ferromagnetic metals. Note that Im kF ,0 does not obey cal points. In Fig. 4 we show the non-self-consistent RPA the same power law as a function of T as Im ⌺͑k ,␻͒ as a ⌺͑ ␻͒ F result for Im kF , in the ground state for various choices function of ␻ at T=0. The T2/3 law proposed in Ref. 11, of ␰, as obtained from a numerical integration of the finite ␰ which one might expect by identifying T and ␻ scaling, does analog of Eq. ͑31͒. The choice of parameters ␰ =v =1 cor- 0 kF not describe the leading asymptotic behavior at low T. responds to fixing a length and energy scale. The somewhat We now generalize the preceding analysis to finite ␻, arbitrary choice of u is not very critical due to the rela- kF which shall however be sufficiently small that we can still

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use the expansion of the Bose function and neglect the Fermi function. To this end we set ␻=v x/␰, where x is a dimen- kF sionless scaling variable which is kept fixed in the low temperature limit. Once again we introduce dimensionless integration variables ˜q and ˜q , now defined by qЈ−␻/v r t r kF ␰2 ␰3 ␰ ␰→ϱ = 0˜qr / and qt =˜qt / , respectively. For we can then replace q in ͉q͉=ͱq2 +q2 by ␻/v , which yields ͉q͉=˜q/␰ r t r kF ͱ 2 2 with ˜q= ˜qt +x . The Bose function can again be expanded and the Fermi function neglected for T→0, if ␰ diverges faster than T−1/3. We then obtain gd2 ⌺͑ ␻͒ → kF ␰ ͑ ͒͑͒ Im kF, T l x 43 4v ␰2 kF 0 with a dimensionless scaling function ͑ ͒ ϱ ϱ ͑ ͒ FIG. 5. Scaling function s x describing the asymptotic behavior dq˜ dq˜ 4vk u qˆ ˜q ⌺͑ ␻͒ ␻ ͑ ͒ ͵ r ͵ t F of the quantum contribution to Im kF , at low and T; the l x = 2 2 2 2 . ϱ 2␲ ϱ 2␲ ͑v ˜q ͒ + ͑1+˜q ͒ ͓u͑qˆ ͒˜q͔ horizontal line indicates the asymptotic value of s͑x͒ for large x. − − kF r ͑44͒ ␯ ͑ ͒ ara frequencies n 0 in Eq. 25 . After analytical continua- c The unit vector qˆ can be parametrized by x and ˜qt, it does not tion to real frequencies, Im ⌺ was obtained from the Bose q depend on ˜qr. Doing the elementary ˜qr integral, the velocities function singularity T/␯. To analyze Im ⌺ for real frequen- v and u͑qˆ ͒ in the above expression for l͑x͒ drop out com- ␯ kF cies we thus subtract T/ from the Bose function, that is we pletely. Carrying out the remaining ˜qt integral, we obtain the replace b͑␯͒ by the regular function ¯b͑␯͒=b͑␯͒−T/␯ in Eq. simple universal result ͑40͒. The asymptotic behavior of Im ⌺q at low-energy scales ͑ ͒ 1 low frequency and temperature can be extracted by using l͑x͒ = . ͑45͒ the same dimensionless variables ˜q and ˜q as already in the ͱ 2 t r 1+x case T=0, and scaling ␻ as T by keeping ␻˜ =␻/T fixed in the → ␻→ ͉ ͉ ͉ ͉ For ␻=0 one has l͑0͒=1 and the above special result for limit T 0, 0. Asymptotically one can replace q by qt and u͑qˆ ͒ by u as for T=0. Furthermore, one can neglect Im ⌺͑k ,0͒ is recovered. For large x the scaling function kF F ␰−2 ␰−2 decays as x−1. Hence, the contribution from the Bose func- in the denominator of Im Sd since scales to zero faster ⌺͑ ␻͒ than ͉q͉2. The ˜q integral can then be carried out analytically, tion singularity to Im kF , leads to a peak with a height t scaling as T␰͑T͒ and a width of order v /␰͑T͒. The product and we obtain kF of height and width is thus proportional to T. Since the con- 2 1/3͉␻͉2/3 gdk uk tribution proportional to the Bose function to Im ⌺͑k,␻͒, Eq. Im ⌺q͑k ,␻͒ → F F s͑␻˜ ͒, ͑48͒ F ␰4/3 ͑ ͒ ␻ ␻ vk 0 40 , depends only via the linear combination −vk kr on F ͑ ͒ F and kr, the right-hand side of Eq. 43 is applicable also for with the universal dimensionless scaling function kk , where it yields the contribution from the expanded F ͑␻͒ ϱ Bose term for T→0, ␻→v k , with fixed x=͑␻/v −k ͒␰. sgn ˜ dq˜r 1 1 kF r kF r ͑␻˜ ͒ ͵ ͫ s = ␻͑ ͒ − ͱ ␲ ˜ ˜qr−1 ␻͑ ͒ The above asymptotic result is entirely due to “classical” 3 3 −ϱ 2 e −1 ˜ ˜qr −1 ␯ fluctuations, corresponding to the contribution with n =0 to 1 ˜q −1 the Matsubara frequency sum in Eq. ͑25͒. The analytic con- ͬ r ͑ ͒ + ␻ . 49 ˜˜qr ͉ ͉4/3 tinuation of that contribution to real frequencies reads e +1 1−˜qr d2q Note that the above integral converges since the Bose func- ⌺c͑ ␻͒ ͵ ⌫ ͑ ͒ ͑ ␻͒ ͑ ͒ k, =−T kk q,0 G k + q, . 46 ͑␻͒ ͑2␲͒2 tion pole has been subtracted. A plot of s ˜ is shown in Fig. 5. For ͉␻˜ ͉→ϱ the scaling function tends to 1/4ͱ3␲, and one ⌫ ͑ ͒ Note that kk q,0 is real and does not depend on the pa- recovers the zero temperature result, Eq. ͑33͒. The conver- ͑ ͒ rameter u qˆ . For G=G0 one can easily show that indeed gence to the zero temperature limit is however rather slow. −2/3 2 For small ͉␻˜ ͉, s͑␻˜ ͒ is negative and proportional to ͉␻˜ ͉ , gdk ⌺c͑ ␻͒ → F ␰ ͓͑␻ ͒␰͔͑͒ such that Im k, T l /vk − kr 47 4v ␰2 F kF 0 2 1/3 gdk uk Im ⌺q͑k ,0͒ → ␣ F F T2/3, ͑50͒ with l͑x͒ from Eq. ͑45͒. A similar scaling behavior of the F ␰4/3 vk 0 self-energy in almost antiferromagnetic metals has been de- F rived in Ref. 30. where ␣Ϸ−0.15 is a numerical constant. Note that c q ⌺q͑ ͒ We now split the total self-energy as ⌺=⌺ +⌺ , where Im kF ,0 is positive but smaller than the absolute value the “quantum” contribution is obtained by summing Matsub- of the classical contribution for low T, since the latter is

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FIG. 6. Non-self-consistent RPA result for the imaginary FIG. 7. Non-self-consistent RPA result for Im ⌺͑k,␻͒ as in 2 part of the self-energy Im ⌺͑k ,␻͒, divided by gd ,asa Fig. 6, but now for kr =0.1, that is k kF. F kF function of ␻ for several temperatures T with a correlation length ␰͑T͒=3/͉T ln T͉1/2; the other relevant parameters are 2 ␻ Ј gdk dqr dqt ␰ =v =u=1. Im ⌺͑k,␻͒ =− ͵ d⑀ ͵ ͵ Im S ͑q,⑀ − ␻͒ 0 kF ␲ ␲ ␲ d 0 2 2 proportional to T␰, such that the imaginary part of ⌺c +⌺q ϫIm G͑k + q,⑀͒. ͑51͒ remains negative, as it should. For k kF, that is for finite kr, ⌺q͑ ␻͒ the momentum dependence of Im k, is negligible for We focus on the quantum critical point, ␰=ϱ, such that ͉␻͉ӷ␻ , with ␻ =u␰2͉k ͉3, as in the zero temperature case. kr kr 0 r To summarize, at low finite T the RPA self-energy is u͑qˆ ͉͒q͉␯ ⌺c ␰ ͑ ͒ ͑ ␯͒ ͑ ͒ given by a classical contribution of order T , see Eq. 47 , Im Sd q, = . 52 q 2/3 2/3 ␯2 ␰4͓ ͑ ͔͒2͉ ͉6 and a quantum contribution ⌺ of order T and ͉␻͉ , which + 0 u qˆ q obeys ␻/T scaling. For ͉␻͉ӷ␻ the latter is described by kr Eq. ͑48͒. A similar structure of the self-energy, with a clas- Motivated by the perturbative results in Sec. IV A, we as- ⌺͑ ␻͒ sical part and a quantum part obeying ͑␻/T͒ scaling, has sume that the low-frequency behavior of Im kF , is ͉␻͉2/3 been obtained already earlier for electrons coupled to strong given by C2 , where C2 is a negative constant, and 31 28 ⌺͑ ␻͒ ferromagnetic or antiferromagnetic fluctuations in the Im k, with a small distance kr from the Fermi surface is quantum critical regime. In the latter case the self-energy is given by the same behavior for frequencies above the small ⌺q ␻ ␰2 3 ⌺ singular at special hot spots on the Fermi surface, and scale k =u 0kr . The analytical properties of in the com- 1 2 r scales with an exponent 2 instead of 3 . In Fig. 6 we show plex plane then dictate ⌺͑ ␻͒ results for Im kF , as obtained from a numerical integration of Eq. ͑40͒ at various temperatures. The correlation ϱ ͉␻ ͉2/3 ␻ C2 Ј length has been chosen as ␰͑T͒ϰ͑T͉log T͉͒−1/2, that is the Re ⌺͑k ,␻͒ =− P͵ d␻Ј = C sgn͑␻͉͒␻͉2/3, F ␲ ͑␻ − ␻Ј͒␻Ј 1 temperature dependence derived by Millis.23 Analogous re- −ϱ sults for k kF are plotted in Fig. 7. In agreement with the ͑53͒ above asymptotic analysis the graphs show a dispersing finite T structure sitting on top of an almost momentum inde- P ͱ where denotes the principal value and C1 = 3C2. Antici- pendent background which is proportional to ͉␻͉2/3 for ␻ ͑ ͒ pating that for small and k=kF the integral in Eq. 51 is TӶ͉␻͉. ⑀ӷ ␰2 3 dominated by contributions with u 0qr , we can approximate B. Self-consistency The self-energy obtained above modifies the propagator G C ͉⑀͉2/3 ͑ ⑀͒ 2 Im G kF + q, = strongly at low-energy scales. We now check the self- ͓⑀ − v qЈ − C sgn͑⑀͉͒⑀͉2/3͔2 + ͑C ͉⑀͉2/3͒2 consistency of the results obtained in the preceding subsec- kF r 1 2 tion, that is we analyze to what extent the self-energy ob- ͑54͒ tained from the self-consistent RPA, Eq. ͑25͒ with a dressed propagator G, differs from the one computed with G0. under the integral. Scaling out the ␻ dependence from the integral in Eq. ͑51͒ one finds that the integration variable ⑀ 1. Ground state ␻ ␻1/3 Ј ␻2/3 ␻ scales as , qt as , and qr as . Hence, for small one ␻Ͼ ⌺͑ ␻͒ can replace ͉q͉ by ͉q ͉ and u͑qˆ ͒ by u . This implies that the At T=0 and for 0, Im k, in self-consistent RPA, t kF ͑ ͒ Ј Eq. 27 , can be written as qr integration acts only on Im G, yielding simply

045127-9 L. DELL’ANNA AND W. METZNER PHYSICAL REVIEW B 73, 045127 ͑2006͒

Ј 2 dqr 1 gdk ˜ ͵ ͑ ⑀͒ ͑ ͒ c F dqr 1 Im G kF + q, =− . 55 ⌺ ͑ ͒ ␰ ͵ ␲ Im kF,0 = 2 T 2 2vk ␰ 2␲ ͱ 2 F 2 0 1+˜qr

This is independent of C1 and C2, which means that the 1 self-energy drops out completely. Carrying out the integral ϫIm . ͑60͒ v ˜q + ␰⌺ ͑˜q /␰,0͒ ⑀ ⌺͑ ␻͒ kF r kF r over and qt one then recovers the result for Im kF , obtained already within the non-self-consistent RPA. Hence, The perturbative result for the self-energy indicates that the replacement of G by G does not affect the asymptotic 0 ⌺ ͑˜q /␰,0͒ is of the order T␰. For ␰͑T͒ϰ͑T͉log T͉͒−1/2 the ⌺͑ ␻͒ kF r low-frequency behavior of Im kF , at all, it does not ␰⌺ ͑ ␰ ͒ self-energy correction k ˜qr / ,0 in the above integral is even modify the prefactor. The same result has already been ␰ F → ͑␰→ϱ͒ obtained earlier in the formally similar problem of fermions then suppressed as 1/log for T 0 and can thus be ͑ ͒ 29 neglected compared to ˜qr, albeit only with logarithmic accu- coupled to a U 1 -gauge field, and also for antiferromag- −1 −1 28 racy, such that Im͓v ˜q +␰⌺ ͑˜q /␰,0͔͒ →␲v ␦͑˜q ͒ and netic quantum critical points. kF r kF r kF r ͑ ͒ ⌺͑ ͒ The above arguments can be easily extended to the case we recover the perturbative result Eq. 42 for Im kF ,0 . ͑ k kF with a small finite kr, since the latter affects More generally, the perturbative result is not modified as- Im ⌺͑k,␻͒ only for frequencies below the small scale ymptotically͒ in a self-consistent treatment as long as ␰͑T͒ −1/2 ␻ ϰk3. increases slower than T for T→0. kr r ͑ ͒ ␻ For general k at or away from kF and finite we can still neglect ␯ in G, such that G−1͑k+q,␻+␯͒ 2. Low finite temperature ␻ ͑ Ј͒ ⌺ ͑ Ј ␻͒ = −vk kr +qr − k kr +qr , . Then Im Sd can again be F F ␯ We first analyze to what extent the peak-shaped classical integrated independently over and qt, which yields contribution to Im ⌺͑k,␻͒ is modified by self-energy feed- G 2 back into . As before, we compute this contribution by gdk dq˜ 1 expanding the Bose function, b͑␯͒ϷT/␯, which yields Im ⌺c͑k,␻͒ = F T␰ ͵ r ␰2 2␲ ͱ 2 2 0 1+˜qr gd2 d␯ dqЈ dq Im ⌺c͑k,␻͒ =− k T ͵ ͵ r ͵ t Im S ͑q,␯͒ 1 ␲ ␯ 2␲ 2␲ d ϫIm . v ˜q + ␰͓v k − ␻ + ⌺ ͑k + ˜q /␰,␻͔͒ kF r kF r kF r r ϫIm G͑k + q,␻ + ␯͒, ͑56͒ ͑61͒ where G is the full propagator. To scale out ␰ from ͑ ␯͒ ␯ Im Sd q, , we use dimensionless variables ˜ and ˜q defined The classical contribution to the self-energy feedback in the ␯ ␰2␯ ␰3 ͉ ͉ ␰ ␰ by =vk 0˜ / and q =˜q/ , respectively, such that above integral is again suppressed at least as 1/log , such F that only the quantum contribution ⌺q remains on the right- ␰2 v u͑qˆ ͒˜q˜␯ hand side. The latter is not negligible at ␻0, it rather domi- ͑ ␯͒ kF ͑ ͒ Im Sd q, = . 57 ␰2 v2 ˜␯2 + ͑1+˜q2͒2͓u͑qˆ ͒˜q͔2 nates over the bare frequency dependence of G for small 0 kF finite ␻. Let us assume that the RPA result for ⌺q is not The inverse propagator can be written as G−1͑k+q,␻+␯͒ modified qualitatively by self-consistency, as will be indeed =␻+␯−v ͑k +qЈ͒−⌺͑k+q,␻+␯͒. verified below. For kr =0 or kr scaling to zero more rapidly kF r r ͉␻͉1/3 ⌺ ͑ ␰ ␻͒→⌺q ͑␻͒ ␻ than one thus has simply k kr +˜qr / , , Consider first the case k=kF and =0. Then F kF ⌺͑ ␻ ␯͒ ⌺͑ ͒ → where ⌺q ͑␻͒ is of order ͉␻͉2/3 for small ␻,T with TӶ͉␻͉ k+q, + can be replaced by kF +q,0 for T 0, kF since ␯ scales to zero as ␰−3. Motivated by the perturbative and of order T2/3 for small ␻,T with ͉␻͉ӶT. In the latter case result we assume that Im ⌺͑k,0͒ is of the order T␰ for ⌺q is smaller than ⌺c, such that the self-energy feedback can k=k and decreases for momenta away from k . Note that be neglected completely. For ͉␻−v k ͉Ӷ͉⌺q ͑␻͉͒, the term F F kF r kF T␰ϰ͑␰ log ␰͒−1, since ␰͑T͒ϰ͑T log T͒−1/2. Then ␯ can be ne- ␻−v k can be neglected in Eq. ͑61͒, such that Im ⌺c͑k,␻͒ kF r glected completely in the expression for the propagator, such becomes independent of kr and the frequency dependence is that G−1͑k +q,␯͒=−v qЈ−⌺ ͑qЈ,0͒, where ⌺ ͑qЈ,␻͒ F kF r kF r kF r given by a scaling function which decreases monotonically ⌺͑ ␻͒ ␯ ␰͉␻͉2/3 = kF +q, . Now the integral over acts only on Im Sd, with a dimensionless scaling variable proportional to . ⌺c͑ ␰ ͒ yielding Hence, in the self-consistent calculation Im k, k is not k independent but rather decreases with increasing ͉k ͉ 2 r r d␯ ␰ 1 −3/2 ͵ ͑ ␯͒ ␲ ␲ ͑ ͒ ͑ ͒ on a scale of order ␰ , that is quite rapidly. For Im Sd q, = = Sd q,0 . 58 ␯ ␰2 1+˜q2 ͉␻−v k ͉ӷ͉⌺q ͑␻͉͒ the self-energy feedback is negligible 0 kF r kF Writing ˜q2 =˜q2 +˜q2, we can also carry out the integral over q and one recovers the non-self-consistent RPA result. In gen- r t t eral, one will find a crossover between the limiting cases. In and obtain ⌺c͑ ␻͒ particular, the decrease of Im kF , as a function of in- ␯ ␲ ␰ ͉␻͉ ␰−3/2 dqt d 1 creasing occurs on a scale of order for the smallest ͵ ͵ Im S ͑q,␯͒ = . ͑59͒ ␰−1 2␲ ␯ d 2 ␰2 ͱ 2 frequencies and then, more slowly, on the larger scale . 0 1+˜qr ⌺c͑ ␻͒ For k kF the self-consistent result for Im k, depends ЈϷ ␰ ␻ Using qr qr =˜qr / and collecting all terms we get on kr and in a more complicated way than just via the

045127-10 FERMI SURFACE FLUCTUATIONS AND SINGLE¼ PHYSICAL REVIEW B 73, 045127 ͑2006͒ difference ␻−v k . Asymptotically it depends only on ␻ if kF r ͉␻͉2/3 kr scales to zero faster than . We now check possible modiﬁcations of ⌺q due to the self-energy feedback at low ﬁnite temperatures. Subtracting the pole from the Bose function, the quantum contribution to Eq. ͑27͒ can be written as

gd2 dqЈ Im ⌺q͑k,␻͒ =− k ͵ d␯ ͵ r ␲ 2␲ dq T ϫ ͵ t ͫb͑␯͒ − + f͑␯ + ␻͒ͬ 2␲ ␯ ϫ ͑ ␯͒ ͑ ␻ ␯͒ ͑ ͒ Im Sd q, Im G k + q, + . 62

We consider the case k=kF. Frequencies are scaled with temperature, that is ␻=T␻˜ and ␯=T˜␯. At low T and ␻ ͉ ͉ ͉ ͉ ͑ ͒ one can, once again, replace q by qt and u qˆ by u in Im S ͑q,␯͒, and neglect ␰−2. Then the qЈ kF d r integration acts only on Im G. The inverse propagator can be written as G−1͑k +q,␻+␯͒=␻+␯−v qЈ−⌺c ͑qЈ,␻+␯͒ F kF r kF r ⌺q ͑ Ј ␻ ␯͒ ⌺c ͑ Ј ␻ ␯͒ − k qr , + . The question now is whether k qr , + F Ј F leads to a significant qr dependence, which could spoil the ͑ ͒ Ј simple result Eq. 55 for the qr integral of Im G. Since ␻+␯ scales to zero as T, and thus faster than ␰−3/2, one can replace ⌺c ͑qЈ,␻+␯͒ by ⌺c ͑qЈ,0͒. The latter tends to the kF r kF r ⌺c ͑ ͒ ͉ Ј͉Ӷ␰−1 ͉ Ј͉ constant k 0,0 for qr . The largest qr contributing ͐ Ј F ␰ to dqr Im G are proportional to T , which indeed scales to zero faster than ␰−1 for T→0, albeit only by a factor of order log10 T. Hence, to logarithmic accuracy, we can indeed ne- Ј ͐ Ј glect the qr dependence of the self-energy in dqr Im G, such that the simple formula Eq. ͑55͒ is still valid, and the ⌺q͑ ␻͒ non-self-consistent result for Im kF , is confirmed. The FIG. 8. Self-consistent RPA results for the imaginary part same result is obtained by the same arguments for kk . 2 F of the self-energy Im ⌺͑kF ,␻͒, divided by gd , as a function ⌺c kF A comparison of the asymptotic result for with a nu- of ␻ for several temperatures T, with a correlation length merical evaluation of the self-consistency equations reveals ␰͑ ͒ ͉ ͉1/2 ␰ ͉ ͉ 2 T =3/ T ln T and 0 =vk =u=1. Upper panel, g d =1; lower ⌺c ͑ ͒ F kF that the feedback of into Eq. 61 becomes negligible only panel, ͉g͉d2 =4. on extremely small energy scales. This is not surprising as kF we have shown that this feedback vanishes only logarithmi- ͑ ͒ cally. By contrast, the effects of the self-energy feedback into 40 numerically and subtracting the classical contribution. ⌺͑ ␻͒ ⌺q are indeed relatively small also for finite temperatures and In Figs. 8 and 9 we show results for Im k, as obtained frequencies. In the following, we, therefore, use the non-self- by the above procedure. One can see that the self-energy consistent RPA result for ⌺q, but compute ⌺c from a self- feedback suppresses the peak generated by classical fluctuations, the effect being of course stronger for larger gd2 . consistent solution of the complex version of Eq. ͑61͒, that is kF In summary, the self-energy feedback in a self-consistent ⌺c͑k,␻͒ calculation affects only the contribution from classical fluc- 2 tuations to ⌺͑k,␻͒ at TϾ0. gdk dq 1 = F T ͵ r ␰2 ␲ ͱ −2 2 2 0 2 ␰ + q r C. Vertex corrections 1 ϫ . At zero temperature, vertex corrections and their feedback v ͑q + k ͒ − ␻ + ⌺q ͑k + q ,␻͒ + ⌺c ͑k + q ,␻͒ kF r r kF r r kF r r on the self-energy have been analyzed in detail by Altshuler 29 ͑ ͒ ͑63͒ et al. for fermions coupled to a U 1 -gauge field, which share many features with our model. They showed that ver- This equation can be easily solved numerically by iteration. tex corrections may lead to moderate finite renormalizations, Due to the weak radial momentum dependence of ⌺q one can but the qualitative behavior of the self-energy, in particular, 2 neglect the qr dependence in its argument. Instead of using the power law with the exponent 3 , remains unchanged. This the asymptotic scaling form of ⌺q, which is valid only at was confirmed by a renormalization group analysis of the sufficiently low ␻ and T, we compute ⌺q by integrating Eq. gauge theory, and also for the formally similar case of a

045127-11 L. DELL’ANNA AND W. METZNER PHYSICAL REVIEW B 73, 045127 ͑2006͒

FIG. 10. Feynman diagram representing the ﬁrst order vertex correction.

dq˜ 1 ␥c ͑ ͒ ϰ ␰2 ͵ r k 0,0 T . F ͱ ˜2 ͓v ˜q + ␰⌺ ͑˜q /␰,0͔͒2 1+qr kF r kF r ͑65͒

For T→0, the self-energy term ␰⌺ ͑˜q /␰,0͒ is of the order kF r ␰2 T for small ˜qr, and vanishes thus logarithmically. The above integral diverges thus logarithmically as ͑T␰2͒−1, which cancels precisely the prefactor. Hence, the classical vertex correction remains ﬁnite. We have not analyzed any higher order vertex corrections. However, by virtue of the above results and arguments we do not expect that vertex corrections will modify the RPA results for the self-energy drastically in the quantum critical regime.

V. SINGLE-PARTICLE EXCITATIONS FIG. 9. Self-consistent RPA result for Im ⌺͑k,␻͒ as in Fig. 8, The momentum resolved spectral function for single- but now for kr =0.1. particle excitations is given by quantum critical point near phase separation.32 The argu- 1 1 ments used in the above works can be directly transferred to A͑k,␻͒ =− Im ␲ ␻ ␰ ⌺͑ ␻͒ our system and will not be repeated here. − k − k, We have not performed a complete analysis of vertex cor- ␲−1͉Im ⌺͑k,␻͉͒ rections at finite temperatures. By virtue of ␻/T scaling, one = , ͑66͒ ͓␻ ␰ ⌺͑ ␻͔͒2 ͓ ⌺͑ ␻͔͒2 may expect that quantum contributions ͑from finite Matsub- − k −Re k, + Im k, ara frequencies͒ to vertex corrections behave similarly at ␰ ⑀ ␮ ⌺͑ ␻͒ zero and low finite temperatures, and lead to finite renormal- where k = k − and k, is the self-energy computed izations only. By contrast, contributions from classical fluc- in the preceding section. Close to the Fermi surface tuations at TϾ0 have no counterpart at T=0 and may thus the self-energy for our model obeys the symmetry relations Re⌺ ͑−k ,−␻͒=−Re⌺ ͑k ,␻͒ and Im ⌺ ͑−k ,−␻͒ behave differently. We have therefore analyzed the first order kF r kF r kF r ͑ ͒ =Im⌺ ͑k ,␻͒, which implies A ͑−k ,−␻͒=A ͑k ,␻͒, vertex correction see Fig. 10 due to classical fluctuations in kF r kF r kF r the quantum critical regime. In the static limit, ␯=0 and since ␰ =v k for small k . k kF r r q→0, the classical contribution to the first order vertex correction is given by A. Ground state In Fig. 11 we show results for the spectral function ͑ ␻͒ ␻ A k, as a function of for several choices of kr. The ␥c ͑ ͒ ϰ ͵ 2 Ј͓ ͑ Ј ͔͒2 ͑ Ј ͒ ͑ ͒ underlying self-energy has been computed at the quantum 0,0 T d q G kF + q ,0 Sd q ,0 . 64 kF critical point ͑T=0 and ␰=ϱ͒ by integrating Eq. ͑31͒ numerically. In the following we discuss the most important features by using the analytical results from Sec. IV. At the quantum critical point the asymptotic low-energy ͑ Ј ͒ Ј Ј ␰ Integrating Sd q ,0 over qt and using qr =˜qr / , one obtains result for the self-energy can be summarized as

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d6 ␻ 3 ϰ kF ͑ ͒ c = C , 68 kF v3 kF

which obviously varies very strongly with kF. ͉␰ ͉ӷ␻ ͑ ␻͒ For ﬁxed k with k c the spectral function A k, has almost the Lorentzian shape as a function of ␻, with a maximum near ␻=␰ and a width of order C ͉␰ ͉2/3. The life k kF k time broadening thus decreases more slowly than the excita- ␰ tion energy k,ask approaches the Fermi surface, such that ͉␰ ͉Ϸ␻ no well-deﬁned quasiparticles exist. For k c the maxi- ͑ ␻͒ ␰ mum of A k, is shifted strongly away from k and the ͉␰ ͉Ӷ␻ width is of the order of the peak energy. For k c one can neglect ␻ compared to Re⌺͑k,␻͒ in Eq. ͑66͒, and A͑k,␻͒ is ␻ ¯␰ now peaked at = k, with the renormalized energy scale

¯␰ = sgn͑␰ ͒͑C−1͉␰ ͉͒3/2 ϰ k3/2. ͑69͒ k k kF k r Extracting a dispersion relation from the momentum dependence of the peak position in A͑k,␻͒ one thus obtains a ﬂat band with a vanishing slope near the Fermi surface. The ¯␰ width of the peak centered around k is of the order C ͉¯␰ ͉2/3=͉␰ ͉ and thus linear in k . For k=k the spectral kF k k r F function diverges as ͉␻͉−2/3 for ␻→0. Momentum scans of A͑k,␻͒ perpendicular to the Fermi surface at ﬁxed ␻ lead to Lorentzian peaks centered around k =͑1/v ͓͒␻+C sgn͑␻͉͒␻͉2/3͔. Some such scans are r kF kF shown in Fig. 12 for various choices of ␻. The integral dk 1 ͵ r A͑k,␻͒ = ͑70͒ 2␲ 2␲v FIG. 11. Spectral function A͑k,␻͒ at the quantum critical point kF ␻ as a function of for kr =−0.0405n with n=0,1,2,...,6. Fixed k ␰ ͉ ͉ 2 does not depend on the self-energy. The -integrated density parameters are 0 =vk =u=1. Upper panel, g d =1; lower panel, F kF of states at the Fermi level remains, therefore, unrenormal- ͉g͉d2 =4. kF ized, that is, it is determined by the bare Fermi velocity.

i ⌺͑k ␻͒ → ⌺ ͑␻͒ C ͫ ͑␻͒ ͉ͬ␻͉2/3 B. Low finite temperature , k =− k sgn + , F F ͱ3 At low finite T the results for the self-energy can be summarized as follows. The total self-energy is a sum of two distinct contributions, ⌺=⌺c +⌺q, where ⌺c is due to classi- where cal, and ⌺q due to quantum fluctuations. The quantum contribution ⌺q can be computed from non-self-consistent RPA and obeys ͑␻/T͒ scaling at very low ␻ and T, see Eq. ͑48͒. ͉g͉d2 u1/3 kF kF Its dependence on kr is very weak. The classical contribution C = . ͑67͒ c kF ␲ ␰4/3 ⌺ is affected more strongly by the self-energy feedback, 4 vk 0 F especially by feedback of ⌺q. We compute ⌺c by solving Eq. ͑63͒ self-consistently. The classical part violates ͑␻/T͒ scal- k Strictly speaking this simple k -independent behavior is valid ing and depends significantly on r. r The most significant temperature effect is that Im ⌺͑k ,0͒ only for ͉␻͉ӷ␻ , but the scale ␻ is proportional to k3 and F kr kr r increases quickly from zero to sizable finite values upon in- thus tiny for k near the Fermi surface. The prefactor C kF creasing T. For small T near the quantum critical point we depends strongly on the position of kF. It decreases rapidly obtained for k near the Brillouin zone diagonal, where d vanishes, F kF while it is enhanced near the van Hove points, where v gd2 d2 kF k k T 1/3 Im ⌺͑k ,0͒ → F T␰ ϰ F ͱ ͑71͒ becomes small. The kF dependence of uk is comparatively F ␰2 ͉ ͉ F 4vk vk log T weak. The competition between ␻ and the self-energy in the F 0 F denominator of A͑k,␻͒, Eq. ͑66͒, leads to the characteristic both in the non-self-consistent and self-consistent calcula- ͑ ␻͒ ␻→ energy scale tion. This cuts off the divergence of A kF , for 0 oc-

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FIG. 13. Spectral function A͑k,␻͒ at T=0.003 as a function of FIG. 12. Momentum scans of A͑k,␻͒ at the quantum critical ␻ for k =−0.0405n with n=0,1,2,...,6. Parameters and ␰͑T͒ as in point for ␻=−0.0015–0.0405n with n=0,1,2,...,6. The param- r Fig. 8. Upper panel, ͉g͉d2 =1; lower panel, ͉g͉d2 =4. eters are the same as in Fig. 11. kF kF curring at zero temperature and replaces it by a maximum of d-wave form factor and also as d-wave shaped fluctuations ͱ͉ ͉ ⌺͑ ͒ order log T /T. Note that Im kF ,0 vanishes much of the Fermi surface. They lead to a strong singularity in the slower with T than in conventional Fermi liquids, where one dynamical d-wave density correlation function at small mo- 2 ͑ ͒ 2͉ ͉͑ ͒ has T in 3D or T log T in 2D behavior. menta and frequencies, and to singular forward scattering. ͑ ␻͒ ␻ In Fig. 13 we show results for A k, as a function of The momentum and energy dependence of the singularity is for various choices of kr at a fixed low temperature. The captured essentially correctly by a Gaussian theory ͑RPA͒; underlying self-energy has been computed by the procedure interactions of fluctuations modify only the temperature de- described at the end of Sec. IV B. The most striking differ- ␰͑T͒ ͑ ͒ pendence of the correlation length . ences compared to the ground state results Fig. 11 are, of Single-particle excitations are strongly affected by the course, seen for k near kF, where the peaks are now much ␻ fluctuations. We have analyzed the electron self-energy broader. Note also the steep shoulder near =0 for the spec- ⌺͑k,␻͒ within plain and self-consistent RPA, focusing espe- tra with k near k at strong coupling ͑lower panel͒. The F cially on the low-energy behavior in the quantum critical complementary view A͑k,␻͒ as a function of k for various r regime. The dominant contributions due to singular forward fixed ␻, is shown in Fig. 14. Here the line shape resembles a scattering are proportional to d2, where d is a form factor Lorentzian function with a relatively large width. k k with d-wave symmetry, such as cos kx −cos ky. For k near the Fermi surface this leads to a strong tangential momentum dependence of ⌺͑k,␻͒. The singular contributions vanish on VI. CONCLUSION the diagonal of the Brillouin zone, and have the largest am- In summary, we have presented a detailed analysis of plitude near the van Hove points. By contrast, the momen- quantum critical fluctuations and their effect on single- tum dependence of ⌺͑k,␻͒ perpendicular to the Fermi sur- particle excitations in a two-dimensional electron system face is much weaker at low temperatures. Hence, momentum close to a d-wave Pomeranchuk instability. The fluctuations scans of the spectral function perpendicular to the Fermi sur- can be viewed as long-wavelength density fluctuations with a face have almost a Lorentzian line shape.

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vice versa. Structural transitions which reduce the lattice symmetry of the cuprate planes are quite frequent in cuprates. Close to a Pomeranchuk instability of the electronic system, electronic properties can be expected to react unusu- ally strongly to slight lattice distortions which break the symmetry of the electronic system explicitly. Such “overreac- tions” of electronic properties have indeed been observed early on in several cuprate compounds.33,34 In particular, a slight orthorhombicity of the lattice structure would lead to a relatively strong orthorhombic distortion of the Fermi surface. Yamase and Kohno35 invoked this idea already a few years ago, to explain peculiarities of magnetic excitations in cuprates. Recent experiments on YBCO have established a remarkably strong in-plane anisotropy of electronic and magnetic properties,36–38 although the structural anisotropy of the CuO2 planes, which is induced indirectly by the CuO-chains between the planes in that material, is relatively modest.39 Fermi surface softening with d-wave symmetry due to forward scattering interactions can naturally amplify the effect of a weak or moderate structural anisotropy. Since the Pomeranchuk instability breaks the orientational symmetry of the lattice, it is natural to compare with the mechanisms and consequences of stripe formation, which have been extensively discussed in the context of cuprate superconductors.40 Static stripes also break the translation invariance in addition to the orientational symmetry, and their formation requires interactions with large momentum transfers, such as antiferromagnetic interactions. Many experimental observations, that have been attributed to static or fluctuating stripes,41 actually provide evidence only for a ten- dency to orientational, not translational, symmetry breaking, and could, therefore, be described equally well by a ͑incipi- ent͒ Pomeranchuk instability. FIG. 14. Momentum scans of A͑k,␻͒ at T=0.003 for Strong Fermi surface fluctuations could be at least par- ␻=−0.0405n with n=0,1,2,...,6. Parameters and ␰͑T͒ as in tially responsible for the non-Fermi liquid behavior observed Fig. 8. in the “strange metal” regime of cuprate superconductors near optimal doping. In our model calculation we have ob- At the quantum critical point, the real and imaginary parts tained a strongly anisotropic anomalously large decay rate 2/3 of the self-energy scale as ͉␻͉ with energy. This leads to a for single-particle excitations and a flattening of the disper- complete destruction of quasiparticles near the Fermi surface sion relation near the Fermi surface away from the nodal except on the Brillouin zone diagonal, due to the prefactor direction. The properties of single-particle excitations in 2 dk. The dispersion of the maxima of the spectral function various cuprate compounds have been investigated in consid- A͑k,␻͒ flattens strongly for momenta k near the Fermi sur- erable detail by numerous angular resolved photoemission face away from the zone diagonal. On the other hand, the experiments.42 Extended flat bands in the van Hove region momentum integrated density of states at the Fermi level is have been observed by various groups already in the early not renormalized significantly by the Fermi surface fluctua- 1990s.43–45 Large anisotropic decay rates have been extracted tions. from the linewidth of low-energy peaks in the photoemission In the quantum critical regime at TϾ0 the self-energy spectra observed in optimally doped cuprates, using, in par- consists of a “classical” and a “quantum” part with very dif- ticular, momentum scans perpendicular to the Fermi surface ferent dependences on T and ␻. The classical part, which is at various fixed energies.46,47 The line shape of these scans is due to classical fluctuations, dominates at ␻=0, where it ͱ almost Lorentzian, which is consistent with our results. yields a contribution proportional to T␰͑T͒ϰ T/log T to However, the frequency and temperature dependence of the ⌺͑ ␻͒ Im kF , . The quantum part is generated by quantum self-energy extracted from the experimental raw data differs fluctuations and obeys ͑␻/T͒ scaling in the quantum critical from what we obtained from d-wave Fermi surface fluctua- regime. tions. On the other hand, the functional form of the self- We finally discuss whether soft Fermi surfaces and critical energy chosen by the experimentalists may not be the only Fermi surface fluctuations could play a role in cuprate super- way to achieve consistency with the photoemission data. In conductors. Due to the coupling of electron and lattice de- spite of the impressive progress in this experimental tech- grees of freedom a symmetry-breaking Fermi surface defor- nique, the accuracy of such subtle properties as the energy mation is generally accompanied by a lattice distortion, and and temperature dependence of the electron self-energy is

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͑␰ ͒ ͑␰ ͒ still limited by resolution and background problems. f p+q/2 − f p−q/2 Concerning transport, an anisotropic scattering rate with ͒ 3f͑␰ץ 1 .f͑␰ ͒ + ͚ p q q q + O͉͑q͉5͒ ١ · nodes on the Brillouin zone diagonal can very naturally ac- = q p j1 j2 j3ץ pץ pץ p p 24 count for the pronounced anisotropy between the intra- and j1,j2,j3 j1 j2 j3 inter-plane mobility of charge carriers, as pointed out by ͑A3͒ Ioffe and Millis48 in their phenomenological “cold spot” sce- ͒ ␰ ͒ ͑␰͑ ١ nario. According to their idea, the intra-plane transport is Using pf p = fЈ p vp and dominated by quasiparticles with a long life-time near the ͒ 3f͑␰ץ diagonal of the Brillouin zone, while these carriers are not ͚ p q q q p j1 j2 j3ץ pץ pץ available for inter-plane transport, since transverse hopping amplitudes vanish on the diagonal. j1,j2,j3 j1 j2 j3 ⑀2ץ -Clearly, the possible relevance of d-wave Fermi fluctua fٞ͑␰ ͒͑q · v ͒3 +3fЉ͑␰ ͒͑q · v ͒ ͚ p q q = p j1 j2ץ pץ tions does not reduce the importance of other more fre- p p p p quently discussed fluctuation channels, such as antiferromag- j1,j2 j1 j2 ⑀3ץ netism, pairing, and charge order in the cuprates. The + fЈ͑␰ ͒ ͚ p q q q ͑A4͒ j1 j2 j3 ץ ץ ץ interplay of the former with the latter is an interesting topic p pj pj pj for further investigations. j1,j2,j3 1 2 3 A spin dependent Pomeranchuk instability was recently one obtains invoked to explain a new phase observed in ultrapure crys- ⑀2ץ ͒ ␰ ͒ ͑␰͑ 49 tals of the layered ruthenate metal Sr3Ru2O7, and also to f p+q/2 − f p−q/2 1 p 50 = fЈ͑␰ ͒ + fЉ͑␰ ͒ ͚ q q p j1 j2ץ pץ account for a puzzling phase transition in URu2Si2. For a ⑀ − ⑀ p 8 p broader comparison with experimental data for cuprate su- p+q/2 p−q/2 j1,j2 j1 j2 perconductors and other layered materials, which might un- 1 fٞ͑␰ ͒͑q · v ͒2 + O͉͑q͉4͒͑A5͒ + dergo or be close to a symmetry-breaking Fermi surface de- 24 p p formation, it will be useful to compute experimentally ⑀ accessible response functions in the presence of strong Fermi Note that terms involving third order derivates of p have surface fluctuations. been cancelled. Inserting this expansion in Eq. ͑A1͒ and using the point group symmetries of the square lattice one obtains ACKNOWLEDGMENT We thank S. Andergassen, A. Chubukov, C. Di Castro, D. d2p ⌸0͑q,0͒ = ͵ ͓fЈ͑␰ ͒ + 1 fЉ͑␰ ͒⌬⑀ ͉q͉2 Rohe, M. Vojta, H. Yamase, and especially C. Castellani and d ͑2␲͒2 p 16 p p A. Katanin for very valuable discussions. ͒ ͑ ٞ͑␰ ͒ 2͉ ͉2͔ 2 O͉͑ ͉4͒ 1 + 48 f p vp q dp + q , A6

APPENDIX: BARE POLARIZATION FUNCTION which establishes Eq. ͑12͒ and the formulas for the expansion coefficients a͑T͒ and c͑T͒ presented in Sec. III. In the In this Appendix we derive the asymptotic expressions for special case d =1 the term involving fЉ in the above equa- the bare polarization function ⌸0͑q,␯͒ defined in Eq. ͑7͒ for k d tion can be rewritten in the form of the term proportional to small q and ␯. The derivation is valid for arbitrary form ٞ ͑ ͒ f by a partial integration, yielding the simplified formula factors dp with any symmetry not only d wave , in particu- ⌸0 for the conventional polarization function lar, also for the special case dp =1, for which d reduces to ⌸0 the conventional polarization function . d2p .At zero frequency, ⌸0 can be written as ⌸0͑q,0͒ = ͵ ͓fЈ͑␰ ͒ − 1 fٞ͑␰ ͒v2͉q͉2͔ + O͉͑q͉4͒ d ͑2␲͒2 p 24 p p d2p f͑␰ ͒ − f͑␰ ͒ ͑ ͒ ⌸0͑q,0͒ = ͵ p+q/2 p−q/2 d2 . ͑A1͒ A7 d ͑2␲͒2 ⑀ − ⑀ p p+q/2 p−q/2 We now derive the frequency and momentum dependence ␰ ⑀ ␮ ⌸0͑ ␯͒ ␯ ␯ with p = p − . The numerator and the denominator of the of d q, for small q and to leading order in q and .In fraction under the above integral are odd functions of q, and the limit q→0, ␯→0 the polarization function depends only the integrand is thus an even function. The denominator can via the ratio s=␯/͉q͉ and the unit vector qˆ =q/͉q͉ on q and ␯: be expanded as d2p v · qˆ ⌸0͑q,␯͒ → − ͵ fЈ͑␰ ͒ p d2 . ͑A8͒ ⑀3ץ 1 ⑀ − ⑀ = q · v + ͚ p q q q d ͑2␲͒2 p s + i0+ − v · qˆ p p j1 j2 j3 pץ pץ pץ p+q/2 p−q/2 p 24 j1,j2,j3 j1 j2 j3 In the low temperature limit only momenta on the Fermi + O͉͑q͉5͒͑A2͒ surface contribute to the above integral, since then ͒ Ј͑␰ ͒→ ␦͑␰ ⑀ ١ with vp = p p and j1 , j2 , j3 each running over the two pos- f p − p . The real part is an even function of s which sible directions x and y ͑in two dimensions͒. The numerator tends to a͑T͒ in the limit s→0, with corrections of order s2. is expanded as The imaginary part has the simple form

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d2p s 1 1 ⌸0͑ ␯͒ → ͵ Ј͑␰ ͒ 2 ␦͑ ͒ ͑ ͒ ⌸0͑ ␯͒ → ͚ 2 Im q, f d s s − v · qˆ . A9 Im q, − d 0 , ͉͒ ͑ ١ ͉ d p p p d ␲ k 4␲ 4 0 F vk0 tk0 · k0 qˆ · vk0 kF F F F F For q,␯→0 and small s this simpliﬁes further to ͑A11͒ ⌸0͑ ␯͒→ ␳͑ ͒ Im d q, − qˆ ,T s with d2p ␳͑qˆ ,T͒ =−͵ fЈ͑␰ ͒d2␦͑v · qˆ ͒. ͑A10͒ ␲ p p p 4 where tk0 is a unit vector tangential to the Fermi surface in F 0 At T=0 the integrand in Eq. ͑49͒ contains two ␦ functions. kF, and the sum runs over momenta on the Fermi surface ͑ ͒ which satisfy the equation vk0 ·qˆ =s. The formula 18 for The two-dimensional momentum integral can then be carried F out, yielding ␳͑qˆ ,T͒ at T=0 follows directly.

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