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Probing the d_{x2-y2}-wave Pomeranchuk instability by ultrasound

Selection rules of ultrasound attenuation and sound velocity renormalization are analyzed in view of their potential application to identify Pomeranchuk instabilities (electronic nematic phase). It is shown that the transverse sound attenuation along [110] direction is enhanced by the Fermi surface fluctuations near a d_{x2-y2}-wave Pomeranchuk instability, while the attenuation along [100] direction remains unaffected. Moreover the fluctuation regime above the instability is analyzed by means of a self-consistent renormalization scheme. The results could be applied directly to Sr3Ru2O7 which is a potential candidate for a Pomeranchuk instability at its metamagnetic transition in strong magnetic fields.


I. INTRODUCTION
In 1958, Pomeranchuk 1 considered stability conditions for normal isotropic Fermi liquids and argued that, besides a uniform (spin-dependent) deformation of the Fermi surface, i.e., an itinerant ferromagnetic instability, residual quasiparticle interactions can induce a nonuniform deformation of the Fermi surface. Since then, an inhomogeneous deformation of a Fermi surface caused by quasiparticle interactions is frequently referred to as "Pomeranchuk instability (PI)" It is sometimes called "nematic" electron liquid 2 because such a Fermi surface deformation can be characterized by a director order parameter which is used to describe a nematic phase in conventional liquid crystals. The nematic order introduces an anisotropy in the otherwise isotropic physical quantities, and the strong anisotropy of the longitudinal resistivity observed in a clean two-dimensional electron system under high magnetic fields was interpreted as a signal of the development of the nematic order. 3 In lattice systems the term PI is commonly used to refer to a Fermi surface deformation which breaks the symmetry of the underlying lattice. In case of twodimensional square lattices, it was shown 4-7 that a PI with d x 2 −y 2 symmetry can be realized near van Hove fillings, which then reduces the C 4 symmetry of the original Fermi surface to C 2 symmetry. These findings have stimulated a number of further theoretical works 8-20 on the nature of a d x 2 −y 2 -wave PI. Recently this phase has attracted a renewed interest in several fields of solid state physics. The occurrence of a d x 2 −y 2 -wave PI is hotly debated as a possible explanation for the curious phase found in the bilayer ruthenate Sr 3 Ru 2 O 7 under strong magnetic field. 21,22 Also a d x 2 −y 2 -wave PI may provide an explanation for the anisotropy in the magnetic excitation spectrum observed in cuprate superconductors, 23,24 and a similar argument is ongoing in the context of new ironpnictide superconductors. 25,26 An analogous state in the spin channel was also proposed [27][28][29] and used to explain the so-called hidden-order phase found in URu 2 Si 2 30 . Despite these extensive discussions, however, an iden-tification of such a d x 2 −y 2 -wave PI is not an easy task. The problem lies in the difficulty in finding an experimental probe which couples to the order parameter of the d x 2 −y 2 -wave PI. Unlike a magnetic transition where we can detect the magnetic susceptibility using neutron scattering experiment, it is hard to detect the corresponding susceptibility for a d x 2 −y 2 -wave PI which occurs in the charge degrees of freedom. In Ref. 31, it was suggested that a spatial pattern of the local density of states provides a direct probe of the d x 2 −y 2 -wave PI. It is however applicable only through surface probes, a scanning tunneling microscopy, which is easily influenced by surface conditions. Therefore, it is desirable to have another way to identify the d x 2 −y 2 -wave PI based on the bulk properties of a sample.
From the point of view that the characteristic anisotropy of a d x 2 −y 2 -wave PI can be easily wiped out in a macroscopic scale by a formation of domains between two degenerate ground states [ FIG. 1 (b) and (c)], 21,31 it had better concentrate on the fluctuation effects above the transition temperature 32,33 rather than concentrate on the phenomena deep inside of the ordered state. The basic idea presented in this paper is that we can use phonons for the detection of the d x 2 −y 2 -wave PI. This idea is based on the fact that phonons not only have a coupling to the order parameter of a d x 2 −y 2 -wave PI through electron-phonon interactions, but also have a propagation-direction-selective coupling. By combining this remarkable property of phonons with a well known fact that damping of phonons (sound attenuation) exhibits a remarkable divergence near a second order phase transitions, 34 we can pick up information on the d x 2 −y 2wave PI. In other words, the fluctuation sound attenuation caused by a d x 2 −y 2 -wave PI can be regarded as a direct probe of the unconventional charge susceptibility characterizing this ordered state.
In this paper, we investigate the behavior of the fluctuation enhancement of the sound attenuation caused by a d x 2 −y 2 -wave PI. For this purpose, we use two different approaches. The first one is a phenomenological argument using a Gaussian Ginzburg-Landau-Wilson ac- tion. The second one is a microscopic analysis starting from a quasi-two-dimensional Hubbard model with on-site and nearest-neighbor repulsions, which is supplemented by the self-consistent renormalization (SCR) theory 35 to treat the fluctuation effects. Both approaches are shown to give the same result that there is a selection rule in the fluctuation enhancement of the sound attenuation and sound velocity softening near a d x 2 −y 2wave PI, depending on the propagation directions and polarizations. This leads to a conclusion that we can utilize these properties to detect the d x 2 −y 2 -wave PI with the aid of ultrasound measurements. The plan of this paper is as follows. In Sec. II, we present a phenomenological argument to see how the Fermi surface fluctuations near a d x 2 −y 2 -wave PI manifest themselves in transverse sound attenuation. The main message of this paper is essentially given in this section. In Sec. III, we present a microscopic analysis of a d x 2 −y 2 -wave PI, and discuss the mean field phase diagram as well as the effects of fluctuations. Besides, we present a microscopic calculation of the fluctuation sound attenuation, which reinforces the result of the phenomenological argument in Sec. II. Finally in Sec. IV, we summarize our result and give a discussion on the application of our result to the bilayer ruthenate Sr 3 Ru 2 O 7 . We use the unit k B =h = 1 throughout this paper.

II. PHENOMENOLOGICAL APPROACH
In this section, we present a phenomenological argument to see how the Fermi surface fluctuations near a d x 2 −y 2 -wave PI manifest themselves in phonon prop-erties. We pay special attention to the propagationdirection resolved sound attenuation. Because we consider the fluctuation effects above the transition point and mainly focus on the transverse sound attenuation, our argument is essentially different from that given in Ref. 10 where the longitudinal sound propagation below the transition point was discussed.
The order parameter of a d x 2 −y 2 -wave PI 4,5 which is considered in this work is defined by [see also Eq. (25)] where η x (η y ) denote the bond strength along x (y) direction ( Fig.1) of the underlying square lattice. The order parameter η is odd under the permutation x ↔ y. In the presence of nonzero η, the bond strength along x and y directions are different, hence it deforms the lattice and affects the phonon properties.
Up to the lowest order in η, the mode-coupling term δF GL between lattice (phonon) and the PI order parameter is given by where Q = (q, Q z ) is a three-dimensional wavevector with the in-plane component q and the out-of-plane component Q z , κ is the coupling constant, u xx − u yy = iq x u x − iq y u y with the lattice displacement field U = (u, U z ). The above form of the coupling is allowed from the symmetry reason because under the permutation x ↔ y both η and u xx −u yy are odd, keeping the coupling invariant.
We now study how phonons couple to the order parameter of a d x 2 −y 2 -wave PI depending on their propagation direction and polarization. For the moment, we consider the case where the wavevector and the polarization vector of a sound wave lie within a conducting layer, i.e., Q = (q, 0) and U = (u x , u y , 0), because interesting results come out in this case. Then, the longitudinal (transverse) phonon u L (u T ) can be written as Consider a sound wave propagating along [100]direction (i.e., q y = 0). In this case, using the fact that u xx − u yy = i|q|u L when q y = 0, we can write δF GL in Eq. (2) as On the other hand, when a sound wave is propagating along [110]-direction (i.e., q x = q y ), the mode-coupling term (2) can be written as Equations (3) and (4) where the unrenormalized kernel has the form K The information on phonon dynamics is extracted by studying the retarded quantity K R ν (q, ω) = K ν (q, iω m → ω + i0 + ). In the presence of itinerant electrons, excitations of particle-hole pairs yield an imaginary part in K R ν of the form 38 ImK R ν (q, ω) = −ω/τ ph,ν (q) for small ω. This results in a finite phonon lifetime τ ph,ν (q). The sound attenuation α ν (q) is related to the phonon lifetime as α ν (q) = 1/(s ν τ ph,ν ), or equivalently, The phonon propagator [K R ν (q, ω)] −1 is modified in the presence of η fluctuations. When we consider the Gaussian fluctuation region for simplicity, the corresponding action is given by is the d-wave density correlation function, µ 0 ∝ T − T c0 measures the distance from the mean field transition temperature T c0 , ξ 0 and Γ d (q) = w 0 |q| are the bare correlation length and the damping rate of fluctuations, respectively [for the microscopic definition, see Eqs. (41) and (42)]. After integrating out the d x 2 −y 2 -wave PI order parameter η by performing the Gaussian integral in S GL (η) + δS GL (u, η) [where δS GL (u, η) = dτ δF GL (u, η)], we can show that the couplings [Eqs. (3) and (4)] give an additional contribution to K ν (q, iω m ). This leads to the divergent sound attenuation α ν (q) ∝ 1 ω Imχ R d (q, ω). Recalling that the inplane longitudinal phonon (transverse phonon) couples to η only when the wavevector is along [100] direction ([110] direction), we obtain while there are no divergent behaviors for sounds along the other directions, α L110 : no divergence, (12) α T 100 : no divergence. (13) Here, L100 means the longitudinal sound wave propagating along [100] direction, etc. In addition to Eq. (2) there is another relevant term, (14) in which the longitudinal sound modes couple to the d x 2 −y 2 -wave PI order parameter. As before, by integrat- and assuming a three-dimensional behavior of the fluctuations, we can show [see also the paragraph containing Eq. (63)] that this coupling gives the longitudinal sound attenuation the same divergent contribution as Eqs. (10) and (11), Note that this latter result is the same as that obtained in Ref. 39 for fluctuation sound attenuation near a magnetic transition, because the coupling (14) and the coupling discussed in Ref. 39 [Eq. (1) therein] have the same form. Before proceeding to the summary of this section, we comment on the behavior of the sound velocity. The anomaly in the sound attenuation is intimately related to the softening of the sound velocity. The renormalization of the sound velocity due to the fluctuation of η is given by (16) This means that the divergence in the sound attenuation is accompanied by the reduction in the sound velocity (sound mode softening).
The results obtained in this section is summarized in FIG. 2. Eqs. (10)- (13) and (15) mean that longitudinal sound attenuation always show a divergent behavior on approaching a second order phase transition of a d x 2 −y 2wave PI. On the other hand, in-plane transverse sound attenuation are divergent only when their wavevectors are along [110]-direction. Likewise, longitudinal sound velocities always show a critical sound mode softening near a d x 2 −y 2 -wave PI, whereas in-plane transverse sound velocities show the softening only when their wavevectors are along [110]-direction. We therefore propose to detect the d x 2 −y 2 -wave PI by using a measurement of propagation-direction resolved transverse sound attenuation and sound velocities.

III. MICROSCOPIC ANALYSIS
In this section, we present a microscopic analysis of fluctuation sound attenuation near a d x 2 −y 2 -wave PI in order to reinforce the phenomenological argument given in the previous section. Firstly, we perform a mean field analysis of a d x 2 −y 2 -wave PI starting from a single-band Hubbard Hamiltonian with on-site and nearest-neighbor repulsions, and draw the mean field phase diagram. Next, we investigate the fluctuation effect above the transition temperature by employing the self-consistent renormalization formalism. 35 Finally, based on the same microscopic model, we present a diagrammatic calculation of the fluctuation sound attenuation above the transition temperature, which is shown to give the same conclusion derived within the more phenomenological approach of the previous section.

A. Mean field phase diagram
We start from a single-band Hubbard Hamiltonian on a two-dimensional square lattice, where and describe the on-site and nearest-neighbor repulsions.
Here n σ (r i ) = c † σ (r i )c σ (r i ) is the number operator for electrons on a lattice site r i with spin projection σ = ±1, and r i , r j means a bond between a nearest-neighbor lattice site r i and r j .
In this work we consider the effect of an external magnetic field H ext perpendicular to the two-dimensional plane. Hence, the first term in Eq. (17) describes the kinetic energy plus the Zeeman energy, (20) where n p,σ = c † p,σ c p,σ and c p,σ = 1 √ N ri c σ (r i )e −ip·ri with the number of lattice sites N . Here ] is the single-particle dispersion with hopping amplitudes −t, −t ′ , −t ′′ between nearest, next-nearest, and third nearest neighbors, ǫ F is the Fermi energy under the zero magnetic field, and h = µ B H ext with the Bohr magneton µ B . Because we neglect the influence of magnetic field on orbital degrees of freedom (spin-orbit interaction, Landau diamagnetism), the magnetic-field effect is absorbed into the spin-dependent Fermi energy ǫ F,σ (h) = ǫ F + σh. Hereafter, Length and energy are measured in unit of the lattice spacing a and the nearest hopping amplitude t.
The second term H imp in Eq. (17) represents the spin-independent short-range isotropic (s-wave) impurity scattering, where the impurity potential u imp obeys the Gaussian ensemble u imp (r i ) = 0, u imp (r i )u imp (r j ) = n imp |u| 2 δ i,j with n imp and u being the impurity concentration and the strength of the impurity potential. We treat the impurity potential using Born approximation, and the impurityaveraged bare Green's function is given by where ξ k,σ = ǫ p − ǫ F,σ , and ε n = ε n + Γ sign(ε n ) with the quasiparticle scattering rate Γ. As discussed in Ref. 7, Eq. (17) contains a phase with d x 2 −y 2 -wave PI as a mean field solution. This phase appears when the Fermi energy coincides with van Hove energy 5 ǫ VH = 4(t ′ − t ′′ ). In this paper we consider a situation where the zero-magnetic-field Fermi energy ǫ F does not satisfies the "van Hove condition" (ǫ F = ǫ VH ), but a moderately large external magnetic field h tunes one of the spin-dependent Fermi energies ǫ F,σ * (h) to a van Hove condition. In this situation, the spin component σ * satisfying ǫ F,σ * (h) = ǫ VH is relevant to the occurrence of the PI, and we hereafter consider only the "active" spin component σ * . The antiferromagnetic correlations described by the on-site repulsion H U are suppressed because of the polarization under the strong external magnetic field. Although the ferromagnetic state 3: Schematic view of the density of states N (ǫ) considered in this paper. The density of states has a peak at van Hove energy ǫVH = 4(t ′ − t"), and a spin-dependent Fermi energy ǫ F,↑ touches ǫVH under magnetic fields. In the situation shown here, the up-spin is assigned as the "active" spin component.
could compete with the PI, we discuss the case where the PI is stabilized by choosing the parameters V ≈ U/2.
We first make a mean field decoupling by taking the two contractions, one by one. This yields the following mean field Hamiltonian, where η δ (r i ) = V c † σ * (r i +δ)c σ * (r i ) , and we have picked up only the "active" spin component σ * satisfying the van Hove condition. We further assume that the parity of η δ with respect to δ is even (i.e., η δ = η −δ ). Then, we have where η( Because η ′ possesses the four-fold symmetry of the underlying square lattice and can be regarded as a mass renormalization of the quasiparticles, we hereafter neglect it. On the other hand, a nonzero value of η corresponds to the deformation of the Fermi surface which expands along the p x axis and shrinks along p y axis (or vice versa) and Eq.(25) describes a d x 2 −y 2 -wave PI. Going to the momentum representation, we have where d p = cos p x − cos p y , and η(r i ) = q η(q)e iqri . Several comments are in order. In a situation where the PI could be stabilized under the zero magnetic field, it would compete 40 with the so-called d-density wave state, a state with η(q = ±(π, π)). Under a moderately large magnetic field, however, the d-density wave state is not expected to be competitive anymore since this state shows diamagnetic properties 41 suggesting that such a state is suppressed under sizable magnetic fields. Solving these problems is beyond our scope because it would require to include Landau diamagnetism such that the resulting analysis becomes considerably more complicated. Therefore in the following we assume that a uniform η(q = 0) gives the ground state, and the momentum q appearing in Eq. (26) is small (i.e., q ≪ 1).
The equation determining the impurity-averaged Green's function for the "active" spin component σ * is diagrammatically shown in FIG. 4 where the single solid line represents the impurity-averaged bare Green's function [Eq. (22)]. Hereafter, we neglect the last diagram in FIG. 4 assuming that the impurity potential is sufficiently weak. Due to the symmetry difference between η and the impurity potential, this approximation is exact up to the linear order in η. Physically this means that the impurity effect on the second order transition line is rigorously treated while the first order transition line may slightly deviate from the exact result. We note here that even with this simplification, the required full numerical calculation is rather involved. With this approximation, the equation for the impurity-averaged Green's function is solved as where the order parameter η is determined by the following self-consistent equation Here we have introduced a shorthand notation p = 1 N p , and the quantity may be considered as the impurity-averaged Fermi distribution function. The corresponding mean field free energy is given by In the pure limit (Γ → 0), the function f Γ becomes equal to the Fermi distribution function f 0 (x) = (1 + e (x)/T ) −1 due to the identity T εn G (0) σ * (p, iε n ) = f 0 (ξ p,σ * ). This would simplify the self-consistent equation (28), and the free energy (30) would be reduced to a simpler form FMF N = η 2 2V − T p ln |(1 + e −(ξp−dpη)/T )/(1 + e −ξp/T )|. However because we consider the effect of non-magnetic impurities in this work, we need to use Eqs. (28) and (30).
The mean field phase diagram can be drawn by making a Landau expansion 17 of the free energy (30) as where the coefficients a GL and b GL are given by with a convention f ′ Γ (ξ p,σ ) = ∂ ∂ξp,σ f Γ (ξ p,σ ), etc. A condition a GL = 0 defines a second order transition line T c0 (h) in the mean field approximation, if the coefficient of the quartic term of the GL free energy (b GL ) is positive. In case of a negative quartic term the transition becomes of the first order, and the first order transition line T * c (h) is determined by the conditions F MF (η) = 0 and ∂F MF (η)/∂η = 0 for a nonzero η where F MF is defined in Eq. (30).
We calculate the mean field phase diagram by using a square mesh of 500 × 500 in the Brillouin zone. In FIG. 5  (a), we show the calculated mean field phase diagram for a moderately clean sample (Γ = 0.01). Not surprisingly, the phase diagram in this clean case is quite similar to that already obtained in the pure limit (Γ = 0) in the previous work, 42 since as will be discussed below Eq. (37) in the next section, our model is similar to that used in Ref. 42 except for the difference in a spin mixing. The important feature in the phase diagram is that the transition is of second order at higher temperatures while at lower temperatures a first order transition occurs. In the figure, we used a non-zero third-nearest neighbor hopping t ′′ = 0.2. This is because when we use smaller values of t ′′ , a finite momentum state η(q = 0) is stabilized due to a nesting condition.
Next let us see how a slight increase in the impurity scattering modifies the mean field phase diagram. 43 In FIG. 5 (b), the mean field phase diagram for a dirtier case (Γ = 0.06) is shown. Firstly, we see that the PI is easily suppressed by the increased impurity scattering. This is understandable because the order parameter of the PI and the impurity potential have different symmetry. Hence, the underlying mechanism is analogous to what happens to a disordered d-wave superconductor where Anderson's theorem is violated due to the unconventional nature of the order parameter. Secondly, the first order transition line at low temperatures disappears while the second order transition still survives. These two features concerning impurity effects are important when we discuss the relevance of the PI-scenario to the metamagnetic transition found in Sr 3 Ru 2 O 7 in Sec. IV.

B. Self-consistent Renormalization (SCR) treatment of the Fermi surface fluctuation
In this subsection, we study how the mean field result obtained in the previous subsection is modified by fluctuation effects. Following Refs. 14 and 18, we define the (thermal) d-wave density correlation function where n d (q) = p d p c † p+q/2,σ * c p−q/2,σ * is the d-wave density operator in the "active" spin component σ * . In general, χ d has the following structure where χ irr d is the irreducible part of χ d . To proceed further to the detailed calculation of χ d , it is convenient to rewrite the interaction Hamiltonian H V in the momentum space, where V p = 2V (cos p x +cos p y ). Now we decouple V p1−p2 p2 ], where d p = cos k x − cos k y , s p = cos k x + cos k y , and p (±) p = sin k x ± sin k y . Because the four van Hove points in the Brillouin zone enhance the interaction with d x 2 −y 2 form factor 5 d p , we hereafter consider only this channel. After changing the ordering of fermion operators and setting p 1 = p + q/2, p 2 = p ′ + q/2, p 3 = p ′ − q/2, we have This is essentially the same model as used in Refs.12 and 42 except for the fact that the interaction H ′ V acts only among the active spin component σ * .
The so-called random phase approximation (RPA) for χ d is obtained when we adopt the simplest building block as the irreducible χ irr d . Hereafter, we write the d-wave density correlation function in a dimensionless form by introducing χ d = χ d /χ irr d,0 . Then, the RPA d-wave density correlation function is given by Near the second order PI, the retarded function χ R d,RPA (q, ω) = χ d,RPA (q, iω m → ω + i0 + ) takes the same form as Eq. (8), where µ 0 = V a GL ∝ T − T c0 with a GL and T c0 being the quadratic coefficient in Eq. (32) and the mean field transition temperature. Also, ξ 2 0 and w 0 are given by (see Appendix A) where M j = (1/2)∂ 2 ξ p /∂k 2 j for j = x, y and q c is the wavevector cutoff. We note that, as was already mentioned above Eq. (27), neglecting the impurity vertex corrections, 44 which would otherwise transform the dynamical behavior into a diffusive one, remains exact in the calculation of these quantities.
In the presence of interlayer coupling, the critical behavior is expected to be governed by three-dimensional fluctuations, 45 and we assume in this work an anisotropic three-dimensional (3D) behavior of the fluctuations. Microscopically, the three dimensionality is introduced by adding a z-axis dispersion δǫ Pz = −δt cos(P z ) into Eq. (20). Instead of doing so, however, we here introduce the three dimensionality in a more phenomeno-logical way, by replacing the two-dimensional wavevector q appearing in χ d (q, iω m ) with a three-dimensional wavevector Q ′ = (q, Q z /γ), namely, where γ = ξ 0 /ξ c0 is the anisotropy parameter given by the ratio of the in-plane correlation length to the out-ofplane correlation length. The above procedure is justified at least for a system with anisotropic ellipsoidal 3D Fermi surface. Here we would like to mention the recent proposal in the context of high-T c cuprates 46 that the interlayer-configuration of the nematic order becomes alternate pattern of η > 0 [ FIG. 1 (b)] and η < 0 [FIG. 1  (c)]. We do not take into account such a possibility here, since this configuration costs the elastic energy of the crystal lattice. We would like to point out that, while a minor change in the form of the propagator in Eq. (40) is needed if the d x 2 −y 2 -wave PI has such a configuration, the main result of this paper is not changed, at least qualitatively. Now we consider the effect of mode coupling between the fluctuations. In the bosonic languages, 47,48 this is given by the self-energy renormalization for the fluctuation propagator coming from the quartic term of the action. In the fermionic languages which we adopt in this work, the corresponding mode coupling is given by the three diagrams 49 shown in Fig. 6, which give (45) where b GL is the coefficient of the quartic term in the Ginzburg-Landau free energy given by Eq. (31). Substituting Eq. (44) into Eq. (35) and expanding it in terms of δχ irr d , we obtain where δ χ irr d also depends on χ d through Eq. (45). As discussed in Ref. 48, the SCR formalism is a selfconsistent one-loop approximation for the mass renormalization Substituting Eq. (47) into Eq. (46) and making the contour deformation for the Matsubara summation, we obtain the following SCR equation to determine µ, where Q ′ c is the wavevector cutoff. The frequency integral in Eq. (49) can be performed using the relation coth(x/2) = 1 + 2/(e z − 1) and Then, introducing the normalized wavevector Q ′ = ξ 0 Q ′ , the final result can be written as where Ψ(Z) is the digamma function, µ Q ′ = µ + Q ′2 , ] → 1 and the above expression reproduces the result of the Hartree approximation for the mass renormalization due to classical fluctuations, 37 which means that Eq. (50) can describe both quantum and classical fluctuation regions. In FIG. 7, we plot the temperature dependences of several parameters characterizing the bare d-wave density correlation function, the mass term µ 0 , correlation length ξ 0 , and damping coefficient w 0 , as well as the mode-coupling parameter b GL . Strictly speaking, w 0 ( q) in Eq. (42) depends on q but we average them using two values at q = x and q = ( x + y)/2, i.e., w 0 (q) → w av 0 = (1/2){w 0 ( x) + w 0 [( x + y)/2]}. From the figure, we can see the following three points. First, as was already discussed in the pure limit 42 (Γ = 0), the bare mass term µ 0 is as small as ∼ 0.1 in a wide temperature region above the mean field transition temperature T c0 . Second, the correlation length is shorter than the lattice spacing a = 1, which can be understood from Eqs. (41) because the quasiparticle velocity |v p | is quite small near van Hove filling. Third, the mode coupling parameter b GL is one order of magnitude larger than other param- eters (µ 0 , ξ 0 , 1/w 0 ) characterizing Gaussian fluctuations. All these features result in strong fluctuations near the d x 2 −y 2 PI.
In FIG. 8, we plot the temperature dependence of the renormalized mass µ = 1/ χ d (Q ′ = 0, iω m = 0) as well as the bare mass µ 0 = 1/ χ d,RPA (Q ′ = 0, iω m = 0). As expected, µ(T ) coincides with µ 0 (T ) well above the mean field transition temperature T c0 but it deviates from the mean field result near and below T c0 . From the figure we determine the fluctuation-renormalized transition temperature T c by a condition µ(T c ) < 10 −3 . In FIG. 5 we have also plotted T c (h) thus determined. We can see that the fluctuation region becomes smaller on approaching the first order transition line.

C. Fluctuation sound attenuation
We consider the electron-phonon interaction following the argument given by Walker, Smith and Samokhin, 50 and begin with the tight-binding Hamiltonian where u(r i ) is the displacement of the ion at r i , δ is the lattice vector, g(δ) = ∇t(δ) with the hopping amplitude t(δ) between site r i and r i +δ. Following the procedure in Ref. 50 and neglecting umklapp processes, the interaction between electrons and a sound wave with wavevector K and polarization ν is given by where B K,ν = b K,ν − b † −K,ν with the phonon annihilation operator b K,ν , e ν is the polarization vector, and the electron-phonon vertex function F { K,ν} (P ) is given by Here, all momenta denoted by capital letters are 3D vectors. However, we consider below the case where the wavevector and the polarization vector of a sound wave lie within a conducting layer because interesting results come out in this case, and we set in the following P = (p, 0), K = (k, 0), etc. In this work we consider the electron-phonon coupling through at most the next nearest neighbor interactions. Assuming that g(δ) is proportional to δ, we make an expansion g(δ) = g n.n. δ n.n. + g n.n.n. δ n.n.n. where g n.n. (g n.n.n. ) is the coupling constant through the nearest (next nearest) neighbor interactions. Then, for transverse phonons propagating along the [100] direction, because the coupling via the nearest neighbor interaction disappears in this case, the dominant coupling is given by the next nearest neighbor interactions as F T 100 (p) = Υ T 100 (2 sin p x sin p y ), while for transverse phonons propagating along the [110] direction, where Υ T 100 = −g n.n.n. L z /ρ ion (s T 100 ) 2 [Υ T 110 = −g n.n L z /ρ ion (s T 110 ) 2 ] with system's c-axis dimension L z . The electron-phonon vertex function for the longitudinal phonons is given by with Υ L = −g n.n. L z /ρ ion (s L ) 2 . The important point here is that in each case F { K,ν} (p) can be expressed as F { K,ν} (p) = Υ W p with a form factor W p and the corresponding electron-phonon coupling constants Υ. For T 100 phonons, W p is equal to the d xy -symmetry form factor d ′ p = 2 sin p x sin p y , while it is equal to the d x 2 −y 2wave form factor d p for T 110 phonons. In case of longitudinal sounds, W p = s p + d p for L100 phonons while W p = s p for L110 phonons where s p is defined below Eq. (36). We note that the inclusion of an isotropic vertex function into Eq. (56), which ensures the charge neutral condition 50 for longitudinal phonons, does not change the main result discussed below.
Using the electron-phonon interaction (52), the sound attenuation is calculated from the phonon Green's function where is the bare phonon Green's function, and Π ν (k, iΩ m ) is the phonon self-energy. The retarded phonon self-energy, Π R ν (k, iΩ m ) = Π ν (k, iΩ m → Ω + i0 + ), determines the phonon lifetime as 1/τ ph,ν = −(ω (0) k,ν ) 2 1 Ω ImΠ R ν (k, Ω), and the sound attenuation α ν (k) for the phonon with wave vector k and polarization ν is given by Strictly speaking, the sound velocity in Eq. (58) should be understood as a renormalized one, with the bare velocity s  (58). Note however that the unrenormalized sound velocity should be used only in Eq. (58). As mentioned at the end of Sec. II, the measurement of the softening of the sound velocity is also a key experiment to detect the d x 2 −y 2 -wave PI.
Consider first the sound attenuation caused by itinerant electrons in the absence of the d x 2 −y 2 -wave PI. In this case the irreducible phonon self-energy is given by 38 [ FIG. 9(a)] This gives the sound attenuation without the Fermi surface fluctuations, (k, Ω). Now we consider the effect of the d x 2 −y 2 -wave PI. The relevant diagrams are shown in FIG. 9(b)-(f) where we have neglected impurity vertex corrections due to the reason mentioned above Eq. (27). The simplest diagram is given by FIG. 9(b) which corresponds to the sound attenuation derived from the mode-coupling term (2). Diagram (b) gives the sound attenuation derived from the mode-coupling term (14), and is analogous to the Aslamazov-Larkin diagram 51 for the fluctuation conductivity in a superconductor. Here we have neglected the For the moment, we focus on the transverse sound attenuation because it is this case where the fluctuation sound attenuation is propagation-direction selective. We first consider diagram (b). For the transverse sound along [100] direction, because the electron-phonon vertex function F T 100 (p) [Eq. (54)] has the d xy -symmetry, the phonon self-energy does not couple to the collective fluctuation of the d x 2 −y 2 -wave PI. Hence the phonon selfenergy has no divergent contribution, and is given by where Π Next we consider diagram (c). In this case, it is easy to show that the triangle blocks in this diagram vanish for both T 100 and T 110 phonons, hence there is no contribution from this diagram. This is consistent with the argument in Sec.II that Eq. (14) couples only to the longitudinal sound, because diagram (b) gives the sound attenuation derived from this mode-coupling term (14).
We then consider diagrams (d)-(f) which were not taken into account in the phenomenological argument in Sec. II. Note that if we pick up only the contribution just at the Fermi surface assuming a constant density of states, then we can show that diagram (d) and diagrams (e) and (f) cancel. However, once we include the energy dependence of the density of states, there is a nonzero contribution from these diagrams as in the case of the quartic term of the GL functional. By power counting, we can show that these diagrams give a logarithmically divergent sound attenuation α T ∼ ln µ where µ is the renormalized mass term (47). We think that in the actual experiment this logarithmic divergence is negligible compared with the stronger divergence caused by Eq. (62). . Here, we have used a moderately large electron-phonon coupling constant Υ 2 = 0.001 to enlarge the signal, whereas the actual softening would be much smaller. 56 Now we come to the case with longitudinal sound modes. Concerning the fluctuation contribution to the longitudinal sound attenuation, it is sufficient to consider diagram (c) in FIG. 9 irrespective of the propagation direction because as you can see in the following this diagram always gives the divergent behavior with the same exponent. After analytical continuation, diagram (c) gives comes from the triangle block in the diagram and Q ′ = (q, Q z /γ) is the rescaled three-dimensional wavevector introduced in Eq. (43). Roughly speaking, B is proportional to the degree of particle-hole asymmetry, and in case of a perfect particle-hole symmetry this contribution vanishes. By power counting, we can show that this gives a divergent sound attenuation α L ∼ µ −2 where µ is the renormalized mass term (47). Therefore, as was already discussed in Sec. II, the longitudinal sound attenuation always show a divergent behavior on approaching a second order PI.
To summarize, a microscopic calculation of the fluctuation sound attenuation in this subsection gives the same result as obtained by a phenomenological argument in Sec. II. Results of this subsection coincide with the main results in Sec. II, i.e., Eqs. (11), (13) and (15), if we replace the bare mass µ 0 with the renormalized mass µ.
In FIG.10 .10(a)] to that for a dirtier system [ FIG.10(b)], we see that the dirtier system shows broader fluctuation behavior in the reduced temperature (T − T c )/T c .

IV. DISCUSSION AND CONCLUSION
We have studied the effect on sound properties of the Fermi surface fluctuations near a d x 2 −y 2 -wave PI, and discussed that there is a propagation-direction-dependent selection rule in the fluctuation transverse sound attenuation and sound velocity softening. As is shown in FIG. 2,  FIG. 10, and FIG. 11, the transverse sound attenuation and sound velocity softening along [110] direction are enhanced by the Fermi surface fluctuations while those along [100] direction are not affected. Also it was argued that there are always fluctuation contributions to the longitudinal sound attenuation and sound velocity softening. We note that a qualitatively similar conclusion can be reached for a second order structural transition which breaks the lattice symmetry in the same way, but is not caused by a Fermi surface instability. Such a transition would not be identified as a genuine Pomeranchuk instability.
As was already mentioned in Sec. I, the possibility of the d x 2 −y 2 -wave PI discussed in this paper has been debated 16,21,22,31,42,57,58 as a possible explanation for the anomalous phase found in Sr 3 Ru 2 O 7 under strong magnetic fields, and it provides us with a good opportunity to apply our results. In earlier experiments for this material, a metamagnetic transition was found 59,60 at a magnetic field B m where the magnetization shows a steep jump. Since several non-Fermi-liquid properties are observed around B m , this system was originally discussed in terms of the metamagnetic quantum critical point, 61 and the importance of van Hove singularity for the metamagnetic transition was discussed. 62 However, the subsequent experiments using an ultra-pure sample found at least two transitions, 63,64 and later on it was revealed 21 that the two transitions reflect the boundary of a new ordered phase which is accompanied by metamagnetic transitions. Based on a consideration on resistivity data, it was proposed 21 and demonstrated 16 that the observed new phase can be an ordered state with d x 2 −y 2 -wave PI.
On this background, it is tempting to compare our results with the experiments for bilayer ruthenate Sr 3 Ru 2 O 7 . Firstly, as was already discussed in Ref. 42, the calculated phase diagram [FIG. 5 (a)] looks quite similar to what was observed in experiments (Fig. 3 in Ref. 21). Secondly, the earlier experiments can be interpreted as a result of the impurity effects which narrow the area of the ordered phase [FIG. 5 (b)], because two closely located transition lines (beyond the experimental resolution) would look like a single transition line in experiments. Thirdly, the non-Fermi-liquid behaviors observed in resistivity, 59,60,65 specific heat, 60,65,66 and thermal expansion 67 can be interpreted as a result of the Fermi surface fluctuations near the d x 2 −y 2 -wave PI discussed in Sec. III B. This argument can also be applied to the non-Fermi-liquid behavior observed in nuclear relaxation rate, 68 if we take into account the spin-orbit scattering. These facts suggest that the d x 2 −y 2 -wave PI is a promising scenario for the curious phase observed in this material. Hence, it is interesting to apply our main result to Sr 3 Ru 2 O 7 , i.e., Eqs. (11), (13) and (15) [note that Eq. (12) is replaced by Eq. (15)], and we propose to measure the propagation-direction resolved transverse sound attenuation and sound velocity softening in Sr 3 Ru 2 O 7 , since it can determine experimentally the presence or absence of the PI in this material.
Finally we comment on the recent papers 69,70 on the microscopic mechanism of the d x 2 −y 2 -wave PI for Sr 3 Ru 2 O 7 . In these articles, the authors point out the importance of the quasi-one-dimensional ruthenium orbitals (d xz and d yz ) for the occurrence of the d x 2 −y 2wave PI. On the other hand, our microscopic analysis in Sec. III essentially uses quasi-two-dimensional ruthenium orbital d xy . We point out that while the microscopic analysis in Sec. III needs small modifications 71 if we start from d xz and d yz orbitals, the main conclusion remains unchanged that the transverse sound attenuation along [110] direction and the longitudinal sound attenuation in all directions are enhanced by the Fermi surface fluctuations while the transverse attenuation along [100] direction is not affected. This can be inferred from the phenomenological nature of the argument given in Sec. II to arrive at Eqs. (11), (13) and (15). ship of the Japan Society for the Promotion of Science and the NCCR MaNEP of the Swiss Nationalfonds. A part of the numerical calculations was carried out on Al-tix3700 BX2 at YITP in Kyoto University. To evaluate χ R d,RPA (q, ω), we need to calculate χ irr,R d,0 (q, ω). On Matsubara (imaginary) axis, χ irr d,0 (q, iω m ) is given by Eq. (38). We transform the Matsubara sum into an integral over the contour as shown in FIG. 12 using T εn F (iε n ) = dε 4πi tanh( ε 2T )F (ε), and perform analytical continuation iω m → ω + i0 + . Contour C 1 and C 3 give the real part of χ R d,0 (q, ω) for small ω, and in this case we can evaluate this quantity in the limit ω → 0,