Lattice effects on nematic quantum criticality in metals

# Lattice effects on nematic quantum criticality in metals

I. Paul and M. Garst Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot-Paris 7 & CNRS, UMR 7162, 75205 Paris, France
Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
July 14, 2019
###### Abstract

Metals near a nematic quantum critical point, where systems are poised to undergo a zero temperature continuous phase transition that breaks rotational symmetry, are of great interest for studying the iron superconductors, cuprates, ruthanates, and quantum Hall systems. Theoretically, it is commonly held that in the quantum critical regime the electronic excitations become incoherent on the entire “hot” Fermi surface, triggering non Fermi liquid behavior. However, such conclusions are based on electron-only theories that ignore a symmetry-allowed coupling between the electronic nematic variable and a suitable crystalline lattice strain. Here we show that including this coupling leads to entirely different conclusions because the critical fluctuations are mostly cutoff by the non-critical lattice shear modes. The thermodynamics remain Fermi liquid type, while, depending on the Fermi surface geometry, either the entire Fermi surface stays cold, or at most there are hot spots. Our results emphasize the importance of this coupling in the study of nematic quantum criticality.

At an Ising nematic quantum critical point (QCP) in solids the ground state transforms from one having discrete rotational symmetry to another in which this symmetry is broken (see Figure 1) Fradkin2010 (); Nie2013 (); Achkar2016 (); Borzi2007 (); Lilly1999 (); Chu2010 (); Fernandes2014 (); Gallais2016 (). An ideal example is the tetragonal to orthorhombic structural transition at temperature in the iron superconductors (FeSC), which is driven by electronic correlations, and where with doping Chu2010 (); Fernandes2014 (); Gallais2016 (); Johnston2010 (). Besides the FeSC, a nematic QCP is often invoked in the context of several other correlated metals, notably the cuprates Fradkin2010 (); Nie2013 (); Achkar2016 (). Consequently, a topic of immediate relevance for a wide variety of complex metals is how the quantum fluctuations associated with this QCP affect the low temperature properties of a metal.

At present it is widely believed that the effective electron-electron interaction becomes long-ranged near the nematic QCP Lohneysen2007 (); Oganesyan2001 (); Metzner2003 (); Garst2010 (). As a result the electrons become unusually massive and short-lived, leading to non Fermi liquid (NFL) behavior both in thermodynamics and in single electron properties almost everywhere on the Fermi surface. Thus, the specific heat coefficient , where is the free energy, diverges as in space dimension , and as in . Simultaneously, almost the entire Fermi surface gets “hot”, and is characterized by a frequency dependent self-energy in , and by in .

These results are based on the simplest treatment of the typical model describing itinerant electrons interacting with the critical nematic collective mode of the electrons themselves Lohneysen2007 (). The latter is characterized by a susceptibility

 χ−10(q,iΩn)=ν−10[r+q2+D(q,iΩn)], (1)

where is a constant with dimension of density of states, and and are dimensionless momentum and Matsubara frequency, respectively. The dynamics of the collective mode is damped due to the excitation of particle-hole pairs close to the Fermi surface, . At the QCP the tuning parameter vanishes, .

More recently, a lot of work has been done to improve the theory in  Lee2009 (); Metlitski2010 (); Mross2010 (); Schattner2015 (); Drukier2012 (); Holder2015 (). However, these works do not question the belief that the electronic properties are NFL type. In fact, it is widely accepted that quantum criticality involving a non-modulating order parameter invariably leads to NFL physics.

Nemato-elastic coupling. Note, the above conclusions are based on an electron-only theory. In practice, in a solid the electronic environment is sensitive to the lattice strains, and this gives rise to a symmetry-allowed nemato-elastic coupling between the electron-nematic variable and a suitable component of the strain tensor of the type

 Hnem−latt=λ∫drϕ(r)ε(r), (2)

where is the coupling constant with dimension of energy. For the sake of concreteness we assume to transform as under the point group operations. Then, is the local orthorhombic strain, is the atomic displacement associated with strain fluctuation, and the uniform macroscopic strain in the symmetry-broken nematic/orthorhombic phase, see Fig. 1(c). The problem is well-posed if we assume that the undistorted lattice is tetragonal, whose elastic energy is given by , , where are the bare elastic constants (for an explicit expression in the more convenient Voigt notation used henceforth, see Supplementary Information (SI)) Landau-Lifshitz ().

Importantly, the above coupling shifts the nematic QCP, and it occurs already at a finite value of given by

 r=r0≡λ2ν0/C0, (3)

where is the bare orthorhombic elastic constant. At this point the renormalized orthorhombic elastic constant vanishes, triggering a simultaneous orthorhombic instability. We take to be a small parameter, i.e., the effective energy scale generated by the coupling is small compared to Fermi energy. Technically, this allows to track how the properties of the familiar electron-only theory are recovered at a sufficiently high temperature.

Direction selective criticality. This is an inherent property of acoustic instabilities of a solid whereby criticality, or the vanishing of the acoustic phonon velocity, is restricted to certain high-symmetry directions. In order to identify the soft directions of a tetragonal-orthorhombic transition, we consider the orthorhombic projection of the displacement vector of a transverse vibration given by , where , and . This can be interpreted as the distortion of a stack of embedded membranes aligned along the direction of momentum within the plane, see yellow lines in Fig. 2(a) and (b). For general orientations of , the membranes are clamped within the atomic crystal. Consequently, fluctuations of the membrane trigger the non-critical strain fields. However, for along , the membranes become “unclamped” from the rest of the crystal at the orthorhombic instability, Fig. 2(c). The atoms then locally fluctuate between the two possible orthorhombic states so that is an eigenmode whose phonon velocity vanishes Cowley1976 ().

In the presence of the nemato-elastic coupling the strain and the electron-nematic degree of freedom hybridize, and the resulting mode inherits the above property. The hybridization can be incorporated by integrating out the strain fluctuations giving rise to a renormalization of the nematic susceptibility of Eq. (1), , with Here is the density, is polarization index, , and is the polarization vector for the bare acoustic phonons with angle-dependent velocity and dispersion . To lowest order in , the frequency dependence of can be dropped. Then both the numerator and the denominator of are . This implies that the effect of the nemato-elastic coupling is to soften the mass of the nematic fluctuations albeit with an angular dependence, . Note, possesses the four-fold symmetry of the crystal lattice in the non-nematic phase. As shown in the SI, an immediate consequence of this angular dependent mass is that criticality is restricted to the two high-symmetry directions only, for which at the QCP, see Fig. 2(d). The remaining directions stay non-critical since even at the QCP.

In the following we assume that all the bare elastic constants are of order , such that the entire lattice effect can be modeled by the single parameter . With this simplification, that does not change the results qualitatively, the critical static nematic susceptibility is given by Note, the criticality around can be deduced by . This leads to two important conclusions. First, even if the electronic sub-system has two-dimensional dispersion, as in the cuprates and the FeSC, the dependence of is generated by the lattice. Second, the direction selective criticality leads to anisotropic scaling with around . Since each non-critical direction scales as twice the critical one, this is equivalent to a theory with isotropic scaling in an enhanced effective space dimension  Cowley1976 (); Folk1976 (); Zacharias2015 (). Thus, the effect of fluctuations are weaker compared to the electron-only theory.

Note, the physics of the lattice-imposed direction selective criticality is well-known from studies of structural transitions Larkin1969 (); Levanyuk1970 (); Villain1970 (), and its relevance for the finite- structural/nematic transition in FeSC has also been pointed out Cano2010 (); Karahasanovic2016 (). Our goal here is to study how this physics affects the metal’s quantum critical properties.

Fermi surface dependent dynamics. The effect of the lattice is indirect. Since is essentially static at small , the critical dynamics is generated by the excitation of particle-hole pairs in the Fermi sea, and is given by of Eq. (1). In electron-only theories this invariably leads to Landau-damping along generic directions , and a dynamical exponent . However, with finite the lattice imposes that is determined by , and the question is whether there is Landau-damping along these directions. As we argue below, this depends on the Fermi surface, leading to two different universality classes.

The important point is that the interaction between the nematic collective mode and the electrons, given by in usual notations, is invariably accompanied by a form factor that transforms as . Note, Landau damping requires electrons to scatter along the Fermi surface. This implies that the damping of a collective mode with momentum along depends on the form factor at those particular points on the Fermi surface where is tangential to the surface.

Ballistic nematicity. Consider the Fermi surface of the cuprates, shown in Figure 3(a). The possibility of Landau damping with bosonic momentum involve points on the Fermi surface which intersect with the dashed line, and along this line the form factor . Thus, there is no Landau damping, and we get , leading to ballistic critical dynamics at the lowest temperatures and frequencies, with dynamical exponent  Zacharias2009 ().

Damped nematicity. Now consider the typical Fermi surface of the FeSC with hole and electron pockets around the zone center, and around and , respectively, as shown in Figure 3(b). For the same reason as above, the hole pocket does not give rise to Landau damping of the critical mode. But, since the centers of the electron pockets are shifted, is finite everywhere on the electron Fermi surface, and the critical mode gets damped. This leads to the standard and exponent . The damping only involves certain hot spots of the electron pockets, on which we comment further below.

Critical thermodynamics. For the sake of concreteness henceforth we assume that the electronic dispersion is two-dimensional. The free energy of the nematic fluctuations is , and the critical phase diagram is summarized in Fig. (4). There are two important regions in -space: and . For , the entire nemato-elastic coupling can be neglected, and we get the susceptibility of the electron-only theory with , where . Since it covers a larger volume in -space, the contribution from gives the leading term. Thus, above the temperature scale we recover the usual electron-only theory with isotropic two-dimensional criticality and . However, for this mode becomes massive giving Fermi-liquid type (FL) contribution . In this low -regime the nemato-elastic coupling sets in, and direction selective criticality is restricted to region . The associated thermodynamics is as follows.

Ballistic nematicity (Cuprates). In this case where the last term indicates that Landau damping requires a finite -component. The competition between these two terms yields an additional crossover scale . For the dynamics is ballistic, giving the scaling . But, above the dynamics is damped, with the scaling . In both these two regimes momentum scaling is anisotropic, and the anisotropy extends up to the temperature . The critical free energy can be estimated from the above scaling (for detailed calculation see SI). For , including the non-critical contribution from we get

 γ(T)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩c1r1/20EF+c2T3/2r0E5/2F,T≪T∗,c1r1/20EF+c3T4r60E5F,T∗≪T≪TFL. (4)

Note, the contribution appears also in the context of elastic quantum criticality (EQC), even in the absence of itinerant electrons Zacharias2015 ().

Damped nematicity (FeSC). In this case there is finite Landau damping even for so that . There is no physics related to the crossover . For this leads to

 γ(T)=c1r1/20EF+c4T2/3r0E5/3F,T≪TFL. (5)

In the above are numerical prefactors. For both cases, once the nemato-elastic coupling sets in below , the leading thermodynamics is Fermi liquid type, while the critical contribution is subleading, in stark contrast to what the electron-only theory predicts.

Electron Self-energy. We calculate at zero temperature the frequency dependence of the electron self-energy on the Fermi surface, i.e., , where , and is the electron Green’s function. As in the free energy calculation, the regions and of the -space are important. At sufficiently high frequency the contribution from gives , and the entire Fermi surface is hot (barring the points where ). Thus, at high frequency we recover the properties of the electron-only critical theory. For low frequency this contribution turns into a non-critical Fermi liquid correction with , which guarantees that the real part of the self energy stays Fermi liquid type everywhere on the Fermi surface.

The contribution from region can lead to singular self-energy provided it involves electrons scattering parallel to the Fermi surface. This implies that at most we expect “hot spots” where electronic lifetimes are short, see Figure (3). However, for the Fermi surface of the cuprates, as well as for the hole Fermi pockets of the FeSC, the vanishing form factor at these points imply that the hot spots do not survive. On the other hand, the hot spots do survive on the electron pockets of the FeSC for which . As shown in the SI, the region gives a subleading critical contribution to self-energy , which leads to a reduced lifetime for electrons at these hot spots. The arc lengths of the spots scale as , and thus their contribution to , which is consistent with Eq. (5).

In conclusion, the presence of a crystalline lattice has profound consequences on nematic quantum criticality in metals. In particular, below a certain temperature the system is a Fermi liquid even at the QCP. Non canonical Fermi liquid behavior is restricted to “hot spots” at most, predicted for the electron pockets of the iron superconductors. This can be tested by photoemission, and by quasiparticle interference effects in tunneling spectroscopy.

We thank C. Max and A. Rosch for helpful discussions. I.P. acknowledges financial support from ANR grant “IRONIC” (ANR-15-CE30-0025-01). M.G. acknowledges support from SFB 1143 “Correlated Magnetism: From Frustration To Topology”.

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Supplementary Material for “Lattice effects on nematic quantum criticality in metals”

## Appendix A Nemato-elastic coupling and direction selective criticality

In this section we provide the mathematical details of how nemato-elastic coupling leads to direction selective criticality. We start with an explicit expression for the elastic energy, and then we discuss how the electronic nematic susceptibility , see equation (1) of the main text, is renormalized to by the equation

 χ−1(q,iΩn)=χ−10(q,iΩn)−Π(q,iΩn) (S1)

in the presence of the nemato-elastic coupling given by equation (2) of the main text. We derive the expression for , and we show how this leads to the concept of a four-fold symmetric mass, and, thus, to direction selective criticality.

### a.1 Elastic free energy & normal modes

The most general elastic free energy for a tetragonal system, to lowest order in the strains, is given by Landau-LifshitzSI (); Cowley1976SI ()

 FE =∫d3r[c112(εxx(r)2+εyy(r)2)+c332εzz(r)2+2c44(εxz(r)2+εyz(r)2)+2c66εxy(r)2 +c12εxx(r)εyy(r)+c13(εxx(r)+εyy(r))εzz(r)]. (S2)

Here , with are the local strains, which has an uniform component and a fluctuating part that describe the acoustic phonons. The latter is defined in terms of the atomic displacement , and etc. denote elastic constants in Voigt notation. In particular, the local orthorhombic strain, that enters equation (2) of the main text, is defined by , and the associated elastic constant is . Note, in the theory the bare elastic medium is stable, and the bare elastic constants are finite and temperature independent.

The dynamical matrix is defined by the relation , where summation over repeated indices is implied. We get , , , , , . We write

 u(q)=∑μUq,μ^uq,μ, (S3)

where are the polarization vectors, are the associated displacements at , and is the polarization index. The eigenvalue equation , with the mass density , defines the bare phonon dispersions .

### a.2 Nemato-elastic coupling & renormalization of nematic susceptibility

The electronic nematic variable is characterized by the susceptibility in the electron-only theory, and is given in equation (1) of the main text. Here denotes Landau damped dynamics, and a dynamical exponent , which is standard in an electron-only theory for a quantum critical point (QCP) where the instability is at , such as a nematic one. In particular, the above susceptibility implies a mean field Landau free energy density for the transition, where is the electron nematic order parameter with dimension of energy. In other words, the QCP is at . In the nematic phase the electronic dispersions along and directions are inequivalent, and this is manifested by a Fermi surface which is symmetric rather than symmetric, see Fig. (1a) in the main text. Our goal is to study how the critical theory is modified by the symmetry-allowed nemato-elastic coupling where is the coupling constant with dimension of energy.

At the mean field level the effect of is to couple with the uniform orthorhombic strain , such that the Landau free energy density for the transition is modified to

 fL=rν02ϕ20+C02ε2+λν0ϕ0ε. (S4)

This implies that the QCP is shifted to a positive value of . This is the content of equation (3) in the main text. is the ratio of a lattice generated energy scale to Fermi energy, and it can be taken as a small parameter of the theory. Simultaneously, the renormalized orthorhombic elastic constant

 ¯C0≡C0−λ2ν0/r, (S5)

softens to zero at the QCP, and in the nematic phase the lattice is orthorhombic with , see Fig. (1c) in the main text. Note, since the remaining strains, defined in equation (A.1), do not couple to at the mean field level, they remain non-critical variables in the theory. That is, their corresponding elastic constants are not renormalized by from their bare values.

At the level of fluctuations the nemato-elastic coupling can be written as

 Hnem−latt=iλ∑q≠0aq⋅uqϕ−q, (S6)

where . Expressing the displacements in terms of the normal modes given by equation (S3), and noting that the phonon Green’s function is , it is simple to infer that the result of integrating out the lattice variables is to obtain equation (S1) with

 Π(q,iΩn)=λ2ρ∑μ(aq⋅^uq,μ)2ω2q,μ+Ω2n. (S7)

### a.3 Direction selective criticality

The acoustic phonon dispersion is linear in momentum with a direction dependent bare velocity , i.e., . This implies that is only a function of the two angles . Thus, the main effect of the nemato-elastic coupling is to renormalize the mass of the nematic fluctuations, which is isotropic in the electron-only theory, and which becomes a four-fold symmetric function of . In other words,

 r→r(^q)≡r−ν0Π(q,Ωn=0). (S8)

Note, collective modes with direction dependent masses are well known from studies of certain structural phase transitions, such as an uniaxial ferroelectric transition that involves long range dipolar interaction Larkin1969SI (), and acoustic instabilities where the long range force is mediated by the shear modes of the solid Levanyuk1970SI ().

Already at this point it is clear that at the QCP, defined by , the renormalized mass cannot vanish along all the directions. This leads to the concept of direction selective criticality, see Figure (2) of the main text. As we argue below, only the two high symmetry directions become critical, while the remaining stay non-critical, i.e., at the QCP

 r(^q≠±^q1,2)>0. (S9)

The evaluation of the renormalized mass and the identification of the critical directions can be performed simply by diagonalizing the 3 3 dynamical matrix . However, this is cumbersome and less insightful. Instead, we will restrict the evaluation of to only along the high symmetry directions, and this is sufficient to identify the critical directions.

Thus, along the direction the only lattice eigenmode that contributes to mass renormalization has eigenvalue and eigenvector . This leads to . Similarly, along the direction the contributing lattice eigenmode has eigenvalue and eigenvector , and this leads to . Using tetragonal symmetry we also infer that , and . On the other hand, along we get . The stability of the bare lattice ensures that (one of the Born stability criteria) Born1954SI (). Thus, among the high symmetry directions, the mass is the softest along . Furthermore, at the QCP , and we get , while the mass along the remaining high symmetry directions stay positive. The physics of this phenomena is the following. Among the various independent strains only the orthorhombic strain is critical, since the associated renormalized elastic constant softens to zero at the QCP, c.f., equation (S5). All the remaining strains, with positive elastic constants, are non-critical. Now, for , the electronic nematic variable couples to one or more non-critical strain along , and this leads to a finite mass . This includes the generic non high symmetry directions as well. While along the special directions , triggers only the critical strain, and, therefore, at the QCP . This completes the argument that only are critical directions, while the rest are non-critical.

From the above discussion one can infer that small deviations from a critical direction, say , can be expressed as , where are pre-factors that depend on the bare elastic constants. Now, in this work we are interested only in the leading temperature and frequency dependencies and the associated exponents of various quantities, and not their numerical prefactors. On the other hand, in a typical metal, the bare elastic constants are all of the order of 10 GPa, and their ratios are simply numbers of order one, that we are not interested to track in this work. Consequently, the calculation simplifies immensely if we assume that the bare elastic constants are all of the order of , such that the entire lattice effect can be modeled by the single parameter . It is easy to show that . With this simplification the asymptotic form of the renormalized static nematic susceptibility can be written as

 χ−1(q≈q1)=ν−10[r0(q22+q2z)/q21+q21/k2F]. (S10)

Thus, direction selective criticality leads to anisotropic scaling with , see Figure (2d) in the main text. This is standard for critical elasticity Cowley1976SI (); Folk1976SI (); Zacharias2015SI ().

## Appendix B Fermi surface dependent dynamics

At small the dynamics generated by the phonons can be ignored, and that of the nematic boson is entirely generated by its interaction with the electrons. This interaction has the structure,

 Hnem−el∝∑q,khkc†k+q/2ck−q/2ϕq, (S11)

where is a form factor that transforms as . This leads to a nematic polarization of the form

 D(q,iΩn)∝−T∑k,ωnh2kGk(iωn)Gk+q(iωn+iΩn), (S12)

where is the electron Green’s function. For the sake of concreteness, from now on we assume that the electronic sub-system is two-dimensional, as in the case of the iron based superconductors and the cuprates. We simplify the evaluation of the above by assuming a two-dimensional circular Fermi surface where and . Note, the qualitative conclusions will not change for Fermi surfaces with crystalline anisotropy. The low energy contribution to the dynamics of the collective mode can be turned into a Fermi surface integral of the type

 D(q,iΩn)∝−iΩn∫2π0dψk2πh2kiΩn−vFq⋅^k.

The evaluation of the above is standard, and we get Zacharias2009SI ()

 D(q,iΩn)∝|Ωn|vFqsin22θq+2Ω2n(vFq)2cos4θq, (S13)

where .

Due to direction selective criticality, we need to evaluate along the critical directions , and this depends crucially whether itself vanishes along these directions. As shown in Figure (3) of the main text, this is indeed the case for a typical cuprate Fermi surface and for that of the hole pockets in the iron superconductors. Thus, for the cuprate Fermi surface the Landau damping vanishes along the critical directions for which , and we get

 D(q≈q1,iΩn)cuprate∝Ω2n(vFq1)2, (S14)

which gives ballistic critical dynamics, and dynamical exponent . On the other hand, for the iron superconductors (FeSC), the typical Fermi surface is comprised of hole pockets centred at , and electron pockets centred at and . For the same reason as above, the hole pockets cannot provide Landau damping of the critical nematic fluctuations. But, on the electron pockets the form factor , and in this case we get

 D(q≈q1,iΩn)FeSC∝|Ωn|vFq1. (S15)

In other words, we get back standard Landau damping and exponent . Thus, depending on the underlying Fermi surfaces, we get two different classes of nematic quantum criticality.

## Appendix C Critical thermodynamics

In this section we give details of the calculation of the free energy of the nematic fluctuations. The results are summarized in the phase diagram of Figure (4) in the main text. The free energy is given by

 F=(T/2)∑q,Ωnlogχ−1(q,iΩn). (S16)

As mentioned in the main text, there are two regions in the momentum space that are important. (i) The region , where critical fluctuations survive once the nemato-elastic coupling is significant. (ii) The region for which the entire nemato-elastic coupling can be neglected, and where we get the susceptibility of the electron-only theory with

 χ−1∝r0+q22d/k2F+|Ωn|/(vFq2d), (S17)

and . Note, since (ii) spans a larger volume in momentum space than (i), the leading contribution to thermodynamics is from (ii) at all temperatures.

The contribution from (ii) is straightforward to evaluate. Above the temperature scale we recover the usual electron-only theory with isotropic two-dimensional criticality and . However, for it behaves as a massive mode, giving Fermi liquid type contribution with . In this low -regime the nemato-elastic coupling sets in, and criticality is direction selective which is restricted to region (i) of -space. The associated thermodynamics now depends on the type of the dynamics of the fluctuations, and, therefore, on the Fermi surface of the electrons.

### c.1 Ballistic nematicity (Cuprates)

Combining equations (S10) and (S13), the critical nematic susceptibility for is

 χ−1(q≈q1,iΩn)=ν−10[r0(q22+q2z)/q21+q21/k2F+Ω2n/(vFq1)2+(q2/q1)2|Ωn|/(vFq1)]. (S18)

This leads to the following two regimes.

Low temperature regime . Here the dynamics is ballistic and the last term in the above equation can be neglected. This leads to the scaling , and . The critical free energy is given by

 F(T)=T∫dq1dq2dqz(2π)3ln(1−e−Eq/T),

where . Note, the above momentum integral is restricted to where scaling holds. To leading order we get

 F(T)=−a1(2π)2T3.5r0E2.5F, (S19)

where . Note, the numerical prefactor depends on the ultraviolet cutoff of the scaling , i.e., on the upper cutoff of the -integral. As such, it is a non-universal quantity whose order of magnitude is significant, rather than its precise value. It translates into a specific heat coefficient . The contribution to also appears in the context of quantum critical elasticity even in the absence of itinerant electrons Zacharias2015SI (). Note, this critical contribution is only a subleading term, the leading one being the Fermi liquid type contribution of the region (ii).

Intermediate temperature regime . Here the last term in equation (S18) wins over the ballistic term, and the dynamics is damped. This leads to the scaling and , and a critical free energy

 F(T) =−1π4∫ω/(EFr0)0dq1∫q21/r1/200dq2dqz∫∞0dωnB(ω)tan−1(ωq22/(q1EF)q41+r0q22+r0q2z) =−2a2a3T6r60E5F, (S20)

where is the Bose function, , and . Note, depends on the ultraviolet cutoff of the momentum integrals, and is non-universal. This gives a , which is a rather weak temperature dependence that is indistinguishable from higher order analytic Fermi liquid corrections in powers of . As before, the leading dependence in this regime is the Fermi liquid type contribution of the region (ii).

### c.2 Damped nematicity (FeSC)

In this case the critical fluctuations stay damped down to the lowest temperatures, and can be set to zero. Combining equations (S10) and (S15), the critical nematic susceptibility for is

 χ−1(q≈q1,iΩn)=ν−10[r0(q22+q2z)/q21+q21/k2F+|Ωn|/(vFq1)]. (S21)

This gives the scaling and , and a critical free energy

 F(T) =−1π4∫(ω/(EF)1/30dq1∫q21/r1/200dq2dqz∫∞0dωnB(ω)tan−1((ωq1)/EF)q41+r0q22+r0q2z) =−a4a5T8/3r0E5/3F, (S22)

, and . As in the earlier cases, the pre-factor is non-universal. This leads to a critical which is subleading to the Fermi liquid contribution from region (ii).

This completes the demonstration that, for both the universality classes, below the scale the leading thermodynamics is Fermi liquid type.

## Appendix D Electron self-energy

In this section we give the details of the calculation of the electron self-energy due to scattering with the critical nematic fluctuations. This can be written as

 Σk(iωn)∝h2kSk(iω