We derive renormalization group equations which allow us to treat order parameter fluctuations near quantum phase transitions in cases where an expansion in powers of the order parameter is not possible. As a prototypical application, we analyze the nematic transition driven by a d-wave Pomeranchuk instability in a two-dimensional electron system. We find that order parameter fluctuations suppress the first order character of the nematic transition obtained at low temperatures in mean-field theory, so that a continuous transition leading to quantum criticality can emerge.
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 …
Nematicity in quantum Hall systems has been experimentally well established at excited Landau levels. The mechanism of the symmetry breaking, however, is still unknown. Pomeranchuk instability of Fermi liquid parameter $F_{\ell} \le -1$ in the angular momentum $\ell=2$ channel has been argued to be the relevant mechanism, yet there are no definitive theoretical proofs. Here we calculate, using the variational Monte Carlo technique, Fermi liquid parameters $F_\ell$ of the composite fermion Fermi liquid with a finite layer width. We consider $F_{\ell}$ in different Landau levels $n=0,1,2$ as a f…
We perform a microscropic analysis of how the constraints imposed by conservation laws affect $q=0$ Pomeranchuk instabilities in a Fermi liquid. The conventional view is that these instabilities are determined by the static interaction between low-energy quasiparticles near the Fermi surface, in the limit of vanishing momentum transfer $q$. The condition for a Pomeranchuk instability is set by $F^{c(s)}_l =-1$, where $F^{c(s)}_l$ (a Landau parameter) is a properly normalized partial component of the anti-symmetrized static interaction $F(k,k+q; p,p-q)$ in a charge (c) or spin (s) sub-channel w…
We study magnetic susceptibilities of two-dimensional itinerant electron systems exhibiting symmetry-breaking Fermi surface distortions, the so-called d-wave Pomeranchuk instability, in a magnetic field. In a pure forward scattering model, the longitudinal susceptibility chi^{zz} is found to exhibit a jump at a critical point. The magnitude of this jump diverges at a tricritical point. When scattering processes involving finite momentum transfers are allowed for, chi^{zz} is expected to diverge also at a critical point. The system displays multiple critical fluctuations. We argue that the feat…
The Holstein model with dispersionless Einstein phonons is one of the simplest models describing electron-phonon interactions in condensed matter. A naive extrapolation of perturbation theory in powers of the relevant dimensionless electron-phonon coupling $λ_0$ suggests that at zero temperature the model exhibits a Pomeranchuk instability characterized by a divergent uniform compressibility at a critical value of $λ_0$ of order unity. In this work, we re-examine this problem using modern functional renormalization group (RG) methods. For dimensions $d > 3$ we find that the RG flow of the Hols…
Interplay of Pomeranchuk instability (spontaneous symmetry breaking of the Fermi surface) and d-wave superconductivity is studied for the repulsive Hubbard model on the square lattice with the dynamical mean field theory combined with the fluctuation exchange approximation (FLEX+DMFT). We show that the four-fold symmetric Fermi surface becomes unstable against a spontaneous distortion into two-fold near the van Hove filling, where the symmetry of superconductivity coexisting with the Pomeranchuk distorted Fermi surface is modified from the d-wave pairing to (d+s)-wave. By systematically shifti…
We explore features of a 3D Fermi liquid near generalized Pomeranchuk instabilities using a tractable crossing symmetric equation method. We approach the instabilities from the ordered ferromagnetic phase. We find quantum multi-criticality as approach to the ferromagnetic instability drives instability in other channel(s). It is found that a charge nematic instability precedes and is driven by Pomeranchuk instabilities in both the l = 0 spin and density channels.
We analyze a two-dimensional Pomeranchuk-Nematic instability, trigger by the Landau parameter $F_2<0$, in the presence of a small magnetic field. Using Landau Fermi liquid theory in the isotropic phase, we analyze the collective modes near the quantum critical point $F_2=-1,ω_c=0$ (where $ω_c$ is the cyclotron frequency). We focus on the effects of parity symmetry breaking on the Fermi surface deformation. We show that, for studying the critical regime, the linear response approximation of the Landau-Silin equation is not sufficient and it is necessary to compute corrections at least of order …
We propose that cuprate superconductors are in the vicinity of a spontaneous d-wave type Fermi surface symmetry breaking, often called a d-wave Pomeranchuk instability. This idea is explored by means of a comprehensive study of magnetic excitations within the slave-boson mean-field theory of the t-J model. We can naturally understand the pronounced xy anisotropy of magnetic excitations in untwinned YBa_{2}Cu_{3}O_{y} and the sizable change of incommensurability of magnetic excitations at the transition temperature to the low-temperature tetragonal lattice structure in La_{2-x}Ba_{x}CuO_{4}. In…
We present a computational study of antiferromagnetic transition in RuO$_2$. The rutile structure with the magnetic sublattices coupled by $π/2$-rotation leads to a spin-polarized band structure in the antiferromagnetic state, which gives rise to a $d$-wave modulation of the Fermi surface in the spin-triplet channel. We argue a finite spin conductivity that changes sign in the $ab$ plane is expected RuO$_2$ because of this band structure. We analyze the origin of the antiferromagnetic instability and link it to presence of a nodal line close to the Fermi level.
Motivated by recent transport and neutron scattering experiments suggesting an orientational symmetry breaking in underdoped cuprates we present a theoretical study of Raman scattering near a d-wave Pomeranchuk instability (PI). The d-wave component of Raman scattering from electrons and phonons allows to study directly order parameter fluctuations associated with the PI. Approaching the PI from the normal state by lowering the temperature a central peak emerges both in electronic and, as an additional low-frequency feature, in phononic scattering. Approaching the PI in the superconducting sta…
The Pomeranchuk instability, in which an isotropic Fermi surface distorts and becomes anisotropic due to strong interactions, is a possible mechanism for the growing number of experimental systems which display transport properties that differ along the $x$ and $y$ axes. We show here that the gauge-gravity duality can be used to describe such an instability in fermionic systems. Our holographic model consists of fermions in a background which describes the causal propagation of a massive neutral spin-two field in an asymptotically AdS spacetime. The Fermi surfaces in the boundary theory distor…
We consider collective excitations of a Fermi liquid. For each value of the angular momentum $l$, we study the evolution of longitudinal and transverse collective modes in the charge (c) and spin (s) channels with the Landau parameter $F_l^{c(s)}$, starting from positive $F_l^{c(s)}$ and all the way to the Pomeranchuk transition at $F_l^{c(s)} = -1$. In each case, we identify a critical zero-sound mode, whose velocity vanishes at the Pomeranchuk instability. For $F_l^{c(s)} < -1$, this mode is located in the upper frequency half-plane, which signals an instability of the ground state. In a cle…
We study the competition between a Pomeranchuk instability in the spin channel with angular momentum $\ell=1$ and an attractive interaction, favoring Cooper-pair formation. We found that the superconducting gap strongly suppresses the phase space for the Pomeranchuk instability. We computed a mean-field phase diagram displaying a first order transition between two superconductor phases with different symmetries: p-wave (with spontaneously generated spin-orbit interaction) and s-wave for greater values of the coupling constant. Moreover, we have looked for a possible modulated superconducting p…
As a paradigmatic example of multi-scale quantum criticality, we consider the Pomeranchuk instability of an isotropic Fermi liquid in two spatial dimensions, d=2. The corresponding Ginzburg-Landau theory for the quadrupolar fluctuations of the Fermi surface consists of two coupled modes, critical at the same point, and characterized by different dynamical exponents: one being ballistic with dynamical exponent z=2 and the other one is Landau-damped with z=3, thus giving rise to multiple dynamical scales. We find that at temperature T=0, the ballistic mode governs the low-energy structure of the…
We perform a two-loop renormalization group (RG) analysis of a 2D effective multiband model, which is relevant for describing the low-energy properties of some iron-chalcogenide superconducting materials. Crucial ingredients in this analysis are the calculation of higher-order contributions in the RG scheme that go beyond the widely-used parquet approximation and the consequent inclusion of nontrivial self-energy effects of the model that yield an anisotropic renormalization of the quasiparticle weight in the system. The motivation of our work is the experimental discovery by Sprau et al. (201…
We analyze the behavior of an itinerant Fermi system near a charge nematic(n=2) Pomeranchuk instability in terms of the Landau Fermi liquid (FL) theory. The main object of our study is the fully renormalized vertex function $ΓΩ$, related to the Landau interaction function. We derive $Γ^Ω$ for a model case of the long-range interaction in the nematic channel. Already within the Random Phase Approximation (RPA), the vertex is singular near the instability. The full vertex, obtained by resumming the ladder series composed of the RPA vertices, differs from the RPA result by a multiplicative renorm…
It has recently been pointed out that Fermi surfaces can remain even in the superconductors under the symmetric spin-orbit interaction and broken time-reversal symmetry. Using the linear response theory, we study the instability of such systems toward ordering, which is an intrinsic property of the Fermi surfaces. The ordered states are classified into diagonal and offdiagonal ones, each of which respectively indicates the Pomeranchuk instability and Cooper pairing not of original electron but of Bogoliubov particles (bogolons). The corresponding order parameters are expanded by multipole mome…
We study the effects of spin-antisymmetric interactions on the stability of a Landau-Fermi liquid on the square lattice, using the generalized Pomeranchuk method for two-dimensional lattice systems. In particular, we analyze interactions that could induce instabilities of the so called spin-split type, that is when spin-up and spin-down Fermi surfaces are displaced with respect to each other. The phase space is studied as a function of the strength of the interaction $V$, the electron chemical potential $μ$ and an external magnetic field $h$. We find that such interactions produce in general a…
We perform a comprehensive numerical study of d-wave Fermi surface deformations (dFSD) on a square lattice, the so-called d-wave Pomeranchuk instability, including bilayer coupling. Since the order parameter corresponding to the dFSD has Ising symmetry, there are two stacking patterns between the layeres, (+,+) and (+,-). This additional degree of freedom gives rise to a rich variety of phase diagrams. The phase diagrams are classified by means of the energy scale Lambda_{z}, which is defined as the bilayer splitting at the saddle points of the in-plane band dispersion. As long as Lambda_{z} n…
Motivated by the growing evidence of the importance of charge fluctuations in the pseudogap phase in high-temperature cuprate superconductors, we apply a large-N expansion formulated in a path integral representation of the two-dimensional t-J model on a square lattice. We study all possible charge instabilities of the paramagnetic state in leading order of the 1/N expansion. While the d-wave charge density wave (flux phase) becomes the leading instability for various choices of model parameters, we find that a d-wave Pomeranchuk (electronic nematic phase) instability occurs as a next leading …
We analyze the dynamical response of a two-dimensional system of itinerant fermions coupled to a scalar boson $ϕ$, which undergoes a continuous transition towards nematic order with $d-$wave form-factor. We consider two cases: (a) when $ϕ$ is a soft collective mode of fermions near a Pomeranchuk instability, and (b) when it is an independent critical degree of freedom, such as a composite spin order parameter near an Ising-nematic transition. In both cases, the order-parameter is not a conserved quantity and the $d-$wave fermionic polarization $Π(q, Ω)$ remains finite even at $q=0$. The polari…
We analyze an effective model on a square lattice with two types of forward scattering interactions, which, respectively, drive ferromagnetism (FM) and electronic nematic order via a d-wave Pomeranchuk instability (dPI). The FM and dPI in general compete with each other and they are typically separated by a first order phase boundary in the plane of the chemical potential and temperature. Nevertheless there is a parameter region where the dPI occurs inside the FM phase, leading to their coexistence. We also study the effect of a magnetic field by choosing a chemical potential where the ground …
We study possible charge instabilities in doped Mott insulators by employing the two-dimensional t-J model with a positive value of the next nearest-neighbor hopping integral t' on a square lattice, which is applicable to electron-doped cuprates. Although the d-wave charge density wave (flux phase) and d-wave Pomeranchuk instability (nematic order) are dominant instabilities for a negative t' that corresponds to hole-doped cuprates, we find that those instabilities are strongly suppressed and become relevant only rather close to half filling. Instead, various types of bond orders with modulati…
We extend the study of the Fermi surface instability of the Pomeranchuk type into systems with orbital band structures, which are common features in transition metal oxides. Band hybridization significantly shifts the spectra weight of the Landau interactions from the conventional s-wave channel to unconventional non-s-wave channels, which results in anisotropic (nematic) Fermi surface distortions even with ordinary interactions in solids. The Ginzburg-Landau free energy is constructed by coupling the charge-nematic, spin-nematic and ferromagnetic order parameters together, which shows that ne…
We consider two-dimensional metals near a Pomeranchuk instability which breaks 90$^\circ$ lattice rotation symmetry. Such metals realize strongly-coupled non-Fermi liquids with critical fluctuations of an Ising-nematic order. At low temperatures, impurity scattering provides the dominant source of momentum relaxation, and hence a non-zero electrical resistivity. We use the memory matrix method to compute the resistivity of this non-Fermi liquid to second order in the impurity potential, without assuming the existence of quasiparticles. Impurity scattering in the $d$-wave channel acts as a rand…
We analyze a model of itinerant electrons interacting through a quadrupole density-density repulsion in three dimensions. At the mean field level, the interaction drives a continuous Pomeranchuk instability towards $d$-wave, spin-triplet nematic order, which simultaneously breaks the SU(2) spin-rotation and spatial rotational symmetries. This order results in spin antisymmetric, elliptical deformations of the Fermi surfaces of up and down spins. We show that the effects of quantum fluctuations are similar to those in metallic ferromagnets, rendering the nematic transition first-order at low te…
Crystalline FeSe is known to display strong nematic order below a weak tetragonal-orthorhombic structural transition around $T_s\sim 90$K, and a superconducting transition at $T_c\sim 9K$. Recently, it was shown that electron irradiation, which creates pointlike potential scattering defects, has the surprising effect of enhancing $T_c$ while suppressing $T_s$. Here we discuss a possible scenario for such an effect, which postulates a competition between $s_\pm$ superconductivity and nematic order. The transition to the nematic state is modeled by a mean field theory of a $d$-wave Pomeranchuk i…
We study competing orders of spinless fermions in the triangular-lattice Hubbard model with nearest-neighbor interaction. We calculate the effective, momentum-resolved two-particle vertex in an unbiased way in terms of the functional renormalization group method and compare two different schemes for the momentum discretization, one based on dividing the Fermi surface into patches and one based on a channel decomposition. We study attractive and repulsive nearest-neighbor interaction and find a competition of pairing and charge instabilities. In the attractive case, a Pomeranchuk instability oc…
We report the evolution of the electronic nematic susceptibility in FeSe via Raman scattering as a function of hydrostatic pressure up to 5.8 GPa where the superconducting transition temperature $T_{c}$ reaches its maximum. The critical nematic fluctuations observed at low pressure vanish above 1.6 GPa, indicating they play a marginal role in the four-fold enhancement of $T_{c}$ at higher pressures. The collapse of nematic fluctuations appears to be linked to a suppression of low energy electronic excitations which manifests itself by optical phonon anomalies at around 2 GPa, in agreement with…
The multiband nature of iron pnictides gives rise to a rich temperature-doping phase diagram of competing orders and a plethora of collective phenomena. At low dopings, the tetragonal-to-orthorhombic structural transition is closely followed by a spin density wave transition both being in close proximity to the superconducting phase. A key question is the nature of high-$T_c$ superconductivity and its relation to orbital ordering and magnetism. Here we study the NaFe$_{1-x}$Co$_{x}$As superconductor using polarization-resolved Raman spectroscopy. The Raman susceptibility displays critical enha…
We study the thermal transport in a two-dimensional system with coexisting superconducting (SC) and nematic orders. We analyze the nature of the coexistence phase in a tight-binding square lattice where the nematic state is modelled as a $d$-wave Pomeranchuk type instability and the feedback of the symmetry breaking nematic state on the SC order is accounted for by mixing of the $s$, $d$ paring interaction. The electronic thermal conductivity is computed within the framework of Boltzmann kinetic theory where the impurity scattering collision is treated in the both the Born and Unitary limits. …
Experiments carried over the last years on the underdoped cuprates have revealed a variety of symmetry-breaking phenomena in the pseudogap state. Charge-density waves, breaking of $C_{4}$ rotational symmetry as well as time-reversal symmetry breaking have all been observed in several cuprate families. In this regard, theoretical models where multiple non-superconducting orders emerge are of particular interest. We consider the recently introduced (Phys. Rev. B 93, 085131 (2016)) spin-fermion model with overlapping 'hot spots' on the Fermi surface. Focusing on the particle-hole instabilities we…
Hybrid improper ferroelectricity (HIF) denotes a new class of polar instability by the mixture of two octahedral-distortion modes and can feature the coexistence of abundant head-to-head and tail-to-tail polar domains, of which the domain walls tend to be charged due to the respective screening charges with an opposite sign. However, no such coexisting carriers are available in the materials. Using group-theoretical, microscopic, and spectroscopic analyses, we established the existence of hidden antipolar order parameter in model HIF (Ca,Sr)3Ti2O7 by the condensation of a weak, previously unno…
