Pomeranchuk Instability Research Articles

Superconducting Order Parameter Structure in the Nematic Phase of Iron-based Materials

We consider the effect of the nematic order on the formation of the superconducting state in iron pnictides and chalcogenides. Nematic order with the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{B}_{{2g}}}$$\\end{document} symmetry is modelled as the d-type Pomeranchuk instability and treated within the mean-field approach. Calculated nematic order parameter depends on the nematic interaction coefficient and abruptly changes with the coefficient’s increase. The superconducting solution is obtained within the spin-fluctuation pairing theory. We show that the leading solution in the nematic phase has a \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{s}_{{\\pi \\pm }}}$$\\end{document} structure. From the critical temperature \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{T}_{c}}$$\\end{document} estimations, we conclude that the nematic superconducting state of the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{s}_{{\\pi \\pm }}}$$\\end{document} type is more favorable than the usual \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{s}_{ \\pm }}$$\\end{document} and \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{d}_{{{{x}^{2}} - {{y}^{2}}}}}$$\\end{document} type states appearing in the absence of the nematicity.

Nematicity-enhanced superconductivity in systems with a non-Fermi liquid behavior

We explore the interplay between nematicity (spontaneous breaking of the sixfold rotational symmetry), superconductivity, and non-Fermi liquid behavior in partially flat-band (PFB) models on the triangular lattice. A key result is that the nematicity (Pomeranchuk instability), which is driven by many-body effect and stronger in flat-band systems, enhances superconducting transition temperature in a systematic manner on the T c dome. There, a plausible -wave symmetry, in place of the conventional -wave, governs the nematicity-enhanced pairing with a sharp rise in the T c dome on the filling axis. When the sixfold symmetry is spontaneously broken, the pairing interaction is shown to become stronger with more compact pairs in real space than when the symmetry is enforced. These are accompanied by a non-Fermi character of electrons in the PFBs with many-body interactions.

Asymptotic Pomeranchuk instability of Fermi liquids in half-filled Landau levels

We present a theory of spontaneous Fermi surface deformations for half-filled Landau levels (filling factors of the form nu =2 , n+1/2). We assume the half-filled level to be in a compressible, Fermi liquid state with a circular Fermi surface. The Landau level projection is incorporated via a modified effective electron-electron interaction and the resulting band structure is described within the Hartree-Fock approximation. We regulate the infrared divergences in the theory and probe the intrinsic tendency of the Fermi surface to deform through Pomeranchuk instabilities. We find that the corresponding susceptibility never diverges, though the system is asymptotically unstable in the n rightarrow infty limit.

Collective effects in an incompressible electronic liquid.

Starting from Landau's kinetic equation, we show that an electronic liquid in d = 2, 3 spatial dimensions depicted by a Landau-type effective theory will become incompressible on condition that the Landau parameters satisfy either (i) [Formula: see text] or (ii) [Formula: see text]. Condition (i) is the Pomeranchuk instability in the current channel and suggests a quantum spin liquid (QSL) state with a spinon Fermi surface; while condition (ii) means that the strong repulsion in the charge channel leads to a conventional charge and thermal insulator. In the collisionless regime (ωτ ≫ 1) and the hydrodynamic regime (ωτ ≪ 1), the zero and first sound modes have been studied and classified by symmetries, including the longitudinal and transverse modes in d = 2, 3 and the higher angular momentum modes in d = 3. The sufficient (and/or necessary) conditions of these collective modes have been revealed. It has been demonstrated that some of these collective modes will behave in quite different manners under incompressibility condition (i) or (ii). Possible nematic QSL states and a hierarchy structure for gapless QSL states have been proposed in d = 3.

Functional renormalization of spinless triangular-lattice fermions: N-patch vs. truncated-unity scheme

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 occurs at Van Hove filling and f-wave and p-wave pairing emerge when the filling is reduced. In the repulsive case, we obtain a charge density wave at Van Hove filling and extended p-wave pairing with reduced filling. The p-wave pairing solution is doubly degenerate and can realize chiral p+ip superconductivity with different Chern numbers in the ground state. We discuss implications for strongly correlated spin-orbit coupled hexagonal electron systems such as moiré heterostructures.Graphic

Phonon renormalization and Pomeranchuk instability in the Holstein model

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 $\lambda_0$ suggests that at zero temperature the model exhibits a Pomeranchuk instability characterized by a divergent uniform compressibility at a critical value of $\lambda_0$ 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 Holstein model indeed exhibits a tricritical fixed point associated with a Pomeranchuk instability. This non-Gaussian fixed point is ultraviolet stable and is closely related to the well-known ultraviolet stable fixed point of $\phi^3$-theory above six dimensions. To realize the Pomeranchuk critical point in the Holstein model at fixed density both the electron-phonon coupling $\lambda_0$ and the adiabatic ratio $\omega_0 / \epsilon_F$ have to be fine-tuned to assume critical values of order unity, where $\omega_0$ is the phonon frequency and $\epsilon_F$ is the Fermi energy. On the other hand, for dimensions $d \leq 3$ we find that the RG flow of the Holstein model does not have any critical fixed points. This rules out a quantum critical point associated with a Pomeranchuk instability in $d \leq 3$.

Theoretical Insights into Electronic Nematic Order, Bond-Charge Orders, and Plasmons in Cuprate Superconductors

In this article, we focus on the charge degree of freedom in cuprate superconductors and review theoretical insights into the electronic nematic order, bond-charge orders, and plasmons. The low-energy charge dynamics is controlled by the spin-spin interaction J, which generates various bond-charge ordering tendencies including the electronic nematic order. The nematic order is driven by a d-wave Pomeranchuk instability and is pronounced in the underdoped region as well as around van Hove filling in the hole-doped case; the nematic tendency is weak in the electron-doped region. Nematicity consistent with the d-wave Pomeranchuk instability was reported for hole-doped cuprates in various experiments. Although the t-J and Hubbard models correctly predicted the proximity to the nematic instability in cuprates far before the experimental indications were obtained, full understanding of the charge ordering tendencies in hole-doped cuprates still requires further theoretical studies. In electron-doped cuprates, on the other hand, the d-wave bond-charge excitations around momentum q=(0.5pi,0) explain the resonant x-ray scattering data very well. Plasmon excitations are also present and the agreement between the large-N theory of the t-J model and resonant inelastic x-ray scattering measurements is nearly quantitative in both hole- and electron-doped cuprates. Theoretically the charge dynamics in cuprates is summarized as a dual structure in energy space: the low-energy region scaled by J, where the nematic and various bond-charge orders are relevant, and the high-energy region typically larger than J, where plasmons are predominant.

Pomeranchuk instabilities in holographic metals

We develop a method to detect instabilities leading to nematic phases in strongly coupled metallic systems. We do so by adapting the well-known Pomeranchuk technique to a weakly coupled system of fermions in a curved asymptotically AdS bulk. The resulting unstable modes are interpreted as corresponding to instabilities on the dual strongly coupled holographic metal. We apply our technique to a relativistic 3 + 1-dimensional bulk with generic quartic fermionic couplings, and explore the phase diagram at zero temperature for finite values of the fermion mass and chemical potential, varying the couplings. We find a wide region of parameters where the system is stable, which is simply connected and localized around the origin of coupling space.

Nematicity and superconductivity: Competition versus cooperation

Electronic nematic behavior has been identified and studied in iron-based superconductors for some time, particularly in the well-known BaFe$_2$As$_2$ system, where it is well-known to compete with superconductivity. On the other hand, it has been shown recently that FeSe displays a negligible effect of nematicity on superconductivity near the superconducting transition, and actual cooperation between the two orders when the system is doped with S. Recently it has also been proposed that LiFeAs undergoes a nematic transition in the superconducting state itself. Generally, we expect superconductivity to be anisotropic when it coexists with nematic order, but it is not clear under what circumstances the two orders compete or cooperate, nor how the anisotropy of the superconducting state correlates with that in the nematic state. To address this, we study a simple mean field model of a $d$-wave Pomeranchuk instability together with a mixed $s,d$ pairing interaction, and identify when nematicity is enhanced or suppressed by superconductivity. We show that the competition or cooperation depends significantly on the distortion of the Fermi surface due to nematicity relative to the anisotropy of the superconducting gap function. Further, we discuss the implications of our results for FeSe and LiFeAs.

Electronic multipoles and multiplet pairs induced by Pomeranchuk and Cooper instabilities of Bogoliubov Fermi surfaces

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 moments (diagonal order parameter) and multiplet pair amplitudes (offdiagonal order parameter) of original electrons, which are induced by the internal fields arising from bogolons' ordering. While the bogolons' order parameters partially inherit the characters of the original electrons, many order parameter components mix with similar magnitude. Hence there is no clear-cut distinction whether the phase transition is diagonal or offdiagonal ordering in terms of the original electrons. These ordering instabilities inside the superconducting states provide insights into the superconductors which have the second phase transition below the first transition temperature.

Possibility of Pomeranchuk instability in staggered flux state

In connection with the symmetry-breaking phenomena found in cuprate superconductors, the instability toward x−y and (x + y)−(x − y) anisotropy (Pomeranchuk instability) is studied for a strongly correlated Hubbard model (U/t = 12) on a square lattice with the next-nearest-neighbor transfer t′ using a variational Monte Carlo method. As a variational function, a staggered flux state ΨSF is considered, which is a candidate for the pseudogap state in cuprates. ΨSF is stabilized in a underdoped regime and reduced to the normal state (projected Fermi sea) for larger doping area. By analyzing ΨSF for t′/t = 0, ±0.3, we argue that the appearance of Pomeranchuk instability is limited to the normal state with t′/t = 0 and (σ: doping rate) in the x−y anisotropy. Band-renormalization effects (Fermi-surface modification) owing to strong correlations play a crucial role for the argument.

Atypical behavior of collective modes in two-dimensional Fermi liquids

Using the Landau kinetic equation to study the non-equilibrium behavior of interacting Fermi systems is one of the crowning achievements of Landau’s Fermi liquid theory. While thorough study of transport modes has been done for standard three-dimensional Fermi liquids, an equally in-depth analysis for two dimensional Fermi liquids is lacking. In applying the Landau kinetic equation (LKE) to a two-dimensional Fermi liquid, we obtain unconventional behavior of the zero sound mode c0. As a function of the usual dimensionless parameter s = ω/qvF, we find two peculiar results: first, for |s| > 1 we see the propagation of an undamped mode for weakly interacting systems. This differs from the three dimensional case where an undamped mode only propagates for repulsive interactions and the mode experiences Landau damping for any arbitrary attractive interaction. Second, we find that regardless of interaction strength, a propagating mode is forbidden for |s| < 1. This is profoundly different from the three-dimensional case where a mode can propagate, albeit damped. In addition, we present a revised Pomeranchuk instability condition for a two-dimensional Fermi liquid as well as equations of motion for the fluid that follow directly from the LKE. In two dimensions, we find a constant minimum for all Landau parameters for ℓ ⩾ 1 which differs from the three dimensional case. Finally we discuss the effect of a Coulomb interaction on the system resulting in the plasmon frequency ωp exhibiting a crossover to the zero sound mode.

Collective modes near a Pomeranchuk instability in two dimensions

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 clean Fermi liquid the critical mode may be either purely relaxational or almost propagating, depending on the parity of $l$ and on whether the response function is longitudinal or transverse. These differences lead to qualitatively different types of time evolution of the order parameter following an initial perturbation. A special situation occurs for the $l = 1$ order parameter that coincides with the spin or charge current. In this case the residue of the critical mode vanishes at the Pomeranchuk transition. However, the critical mode can be identified at any distance from the transition, and is still located in the upper frequency half-plane for $F_1^{c(s)} < -1$. The only peculiarity of the charge/spin current order parameter is that its time evolution occurs on longer scales than for other order parameters. We also analyze collective modes away from the critical point, and find that the modes evolve with $F_l^{c(s)}$ on a multi-sheet Riemann surface. For certain intervals of $F_l^{c(s)}$, the modes either move to an unphysical Riemann sheet or stay on the physical sheet but away from the real frequency axis. In that case, the modes do not give rise to peaks in the imaginary parts of the corresponding susceptiblities.

Universal collective modes in two-dimensional chiral superfluids

In this work, we utilize semi-classical kinetic equations to investigate the order parameter collective modes of a class of 2 dimensional superfluids. Extending the known results for $p$-wave superfluids, we show for any chiral ground state of angular momentum $L\geq 1$, there exists a sub-gap mode with mass $\sqrt{2}\, \Delta$ in the BCS limit, where $\Delta$ is the magnitude of the ground state gap. We determine the most significant Landau parameter which contributes to the mass renormalization and show explicitly the renormalized modes become massless at the Pomeranchuk instability of the fermion vacuum. Particularly for $L=1$, we propose a continuous field theory to include the Fermi liquid effect in quadrupolar channel and produce the same result under consistent approximations. They provide potential diagnostics for distinguishing 2 dimensional chiral ground states of different angular momenta with order parameter collective modes and reveal another low energy degrees of freedom near nematic transition.

Orbital selectivity versus Pomeranchuk instability in the iron–chalcogenide superconductors: A two-loop renormalization group study

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. (2017) that orbitally-selective renormalization of the quasiparticle weight in the Hund’s metal phase at moderate temperatures underpins the highly unusual gap in the superconducting phase of the FeSe compound at lower temperatures. One prediction we arrive here is that the underlying origin of nematicity in these systems may indeed come from orbital-selectivity, instead of a Pomeranchuk instability in the d± channel. This orbital selectivity is driven by the presence of stripe-type antiferromagnetic fluctuations in the model. Therefore, we argue that the present RG results may provide a scenario from a weak-to-moderate coupling perspective, in which the role of orbital selectivity to describe the physical properties of some iron–chalcogenide superconductors is emphasized.

Fermi surface instabilities of symmetry-breaking and topological types on the surface of a three-dimensional topological insulator

The emergence of the Pomeranchuk instability (PI) in a Helical Fermi liquid (HFL) residing on the surface of a three-dimensional topological insulator (3D TI) is addressed at the mean-field level. An expression for the PI condition is derived in terms of a few microscopic parameters in each angular momentum channel corresponding to a central interaction between the helical electrons. It is found that because of the presence of strong spin-orbit coupling (SOC) the Landau parameter, $\bar{F}_{l}$ corresponding to a particular angular momentum channel $l$ depends not only on the electron-electron interaction in the same channel but also interactions in $(l+1)$ and $(l-1)$ channels. The formalism automatically excludes the $l=1$ PI in the HFL where the Galilean invariance is broken because of the presence of strong SOC. It is also found that the competing PIs can only be avoided until the appearance of $l=2$ PI. In this case, the corresponding nematic instability can even be achieved in the $l=1$ angular momentum channel of interaction between the electrons. The range of interaction between the electrons plays a pivotal role in bringing out the PIs. This is established by analysing a few realistic profiles of the interaction. Another class of instability, involving a change in the topology of the Fermi surface without breaking the rotational symmetry, is found which competes with the PIs. Quantum phase transition originating from this instability is quite similar to the Lifshitz transition but is driven by electron-electron interaction. Possible connections of this instability with experiments are also described briefly.

Competition between spin-induced charge instabilities in underdoped cuprates

We study the static charge correlation function in an one-band model on a square lattice. The Hamiltonian consist of effective hoppings of the electrons between the lattice sites and the Heisenberg Hamiltonian. Approximating the irreducible charge correlation function by a single bubble yields the ladder approximation for the charge correlation function. In this approximation one finds in general three charge instabilities, two of them are due to nesting, the third one is the flux phase instability. Since these instabilities cannot explain the experiments in hole-doped cuprates we have included in the irreducible charge correlation function also Aslamasov-Larkin (AL) diagrams where charge fluctuations interact with products of spin fluctuations. We then find at high temperatures a nematic or $d$-wave Pomeranchuk instability with a very small momentum. Its transition temperature decreases roughly linearly with doping in the underdoped region and vanishes near optimal doping. Decreasing the temperature further a secondary axial charge-density wave (CDW) instability appears with mainly $d$-wave symmetry and a wave vector somewhat larger than the distance between nearest neighbor hot spots. At still lower temperatures the diagonal flux phase instability emerges. A closer look shows that the AL diagrams enhance mainly axial and not diagonal charge fluctuations in our one-band model. This is the main reason why axial and not diagonal instabilities are the leading ones in agreement with experiment. The two instabilities due to nesting vanish already at very low temperatures and do not play any major role in the phase diagram. Remarkable is that the nematic and the axial CDW instabilities show a large reentrant behavior.

Non-Fermi-liquid fixed points and anomalous Landau damping in a quantum critical metal

We present a functional renormalization-group calculation of the properties of a quantum critical metal in $d=2$ spatial dimensions. Our theory describes a general class of Pomeranchuk instabilities with ${N}_{b}$ flavors of boson. At small ${N}_{b}$ we find a family of fixed points characterized by weakly non-Fermi-liquid behavior of the conduction electrons and $z\ensuremath{\approx}2$ critical dynamics for the order-parameter fluctuations, in agreement with the scaling observed by Schattner et al. [Phys. Rev. X 6, 031028 (2016)] for the Ising-nematic transition. Contrary to recent suggestions that this represents an intermediate regime en route to the scaling limit, our calculation suggests that this behavior may persist all the way to the critical point. As the number of bosons ${N}_{b}$ is increased, the model's fixed-point properties cross over to $z\ensuremath{\approx}1$ scaling and non-Fermi-liquid behavior similar to that obtained by Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].

Fermi-Liquid Theory and Pomeranchuk Instabilities: Fundamentals and New Developments

This paper is a short review on the foundations and recent advances in the microscopic Fermi-liquid (FL) theory. We demonstrate that this theory is built on five identities, which follow from conservation of total charge (particle number), spin, and momentum in a translationally and $SU(2)$-invariant FL. These identities allows one to express the effective mass and quasiparticle residue in terms of an exact vertex function and also impose constraints on the "quasiparticle" and "incoherent" (or "low-energ" and "high-energy") contributions to the observable quantities. Such constraints forbid certain Pomeranchuk instabilities of a FL, e.g., towards phases with order parameters that coincide with charge and spin currents. We provide diagrammatic derivations of these constraints and of the general (Leggett) formula for the susceptibility in arbitrary angular momentum channel, and illustrate the general relations through simple examples treated in the perturbation theory.

Pomeranchuk Instability of Composite Fermi Liquids.

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_{ℓ}≤-1 in the angular momentum ℓ=2 channel has been argued to be the relevant mechanism, yet there are no definitive theoretical proofs. Here we calculate, using the variational MonteCarlo technique, Fermi liquid parameters F_{ℓ} of the composite fermion Fermi liquid with a finite layer width. We consider F_{ℓ} in different Landau levels n=0, 1, 2 as a function of layer width parameter η. We find that unlike the lowest Landau level, which shows no sign of Pomeranchuk instability, higher Landau levels show nematic instability below critical values of η. Furthermore, the critical value η_{c} is higher for the n=2 Landau level, which is consistent with observation of nematic order in ambient conditions only in the n=2 Landau levels. The picture emerging from our work is that approaching the true 2D limit brings half-filled higher Landau-level systems to the brink of nematic Pomeranchuk instability.