arXiv subjects

Andrey Chubukov

Publications and source records attributed to Andrey Chubukov.

At least 19 recordsLinked to original sources

arXiv2026

Superconductivity mediated by nematic fluctuations -- the dispersion of collective modes

We analyze the spectrum of collective modes in a superconductor in which pairing is mediated by long-range nematic fluctuations. Previous experimental and theoretical studies have found that the superconducting gap in such a system is highly anisotropic and, at any finite $T<T_c$, vanishes on four arcs of the Fermi surface, even when the pairing symmetry is $s$ wave ($s^{+-}$ between hole and electron pockets). We derive the expression for the pair susceptibility $χ(\mathbf{q},Ω)$ at finite momentum $\mathbf{q}$ and frequency $Ω$ deep in the superconducting phase. We analyze the spectral function, $\operatorname{Im}χ(\mathbf{q},Ω)$, and its pole structure in the transverse (phase) and longitudinal (amplitude) channels, and compare the results with those of a conventional $s$-wave superconductor. We find that the analytic structure of the pair susceptibility in both channels is qualitatively distinct from that in a BCS superconductor. This gives rise to a highly unconventional dispersion of phase and amplitude collective modes.

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arXiv2026

Limits of validity for Migdal-Eliashberg theory: role of polarons/bi-polarons

It is widely believed that in an adiabatic limit a Fermi liquid state of an electron-phonon system described by Migdal-Eliashberg theory remains stable before a dressed phonon softens. Using Holstein model as a prototypical example and variational/analytic considerations we demonstrate that in a wide range of fillings both in 3D and 2D, a polaronic/bi-polaronic state emerges before phonon softening; at small filling in 3D this happens already at weak coupling. We show that a polaronic/bi-polaronic state emerges, upon increasing coupling, via an intermediate pseudogap-type mixed state, in which some fermions regain Fermi liquid behavior, yet Luttinger theorem is broken. At even larger couplings the density of states gradually approaches its form in the atomic limit.

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arXiv2026

Breakdown of the Migdal-Eliashberg theory for electron-phonon systems. Role of polarons/bi-polarons

The Migdal-Eliashberg theory (MET) describes electrons interacting with phonons in the adiabatic limit when the phonon Debye frequency is much smaller than the Fermi energy. A conventional belief is that MET holds even at strong coupling, when electron self-energy is large, and breaks down only near the point where the dressed phonon spectrum softens to near zero. We analyze numerically and analytically a different option -- collapse to a polaronic/bipolaronic ground state. The last scenario has never been analyzed in precise quantitative terms for a generic electron density. Using variational considerations, we establish rigorous upper bounds on the coupling $λ$, at which a FL state transforms into the bipolaron/polaron state. We show that at small and near-maximum densities, this happens well before a dressed phonon softens. This is true both in 2D and 3D systems; in the latter the upper bound on $λ$ tends to zero in the limit of small or near-full density. We present analytical reasoning for this behavior based on hints extracted from exact diagrammatic treatment of the on-site Holstein model for the spin polarized case and argue that polarons are produced by fermions with energies comparable to the bandwidth; i.e., polaron formation is outside the realm of MET. Closer to half-filling, the leading instability upon increasing $λ$ is towards a charge-density-wave state (CDW), and there exists a strong coupling regime of MET near this instability, while the polaron/bipolaron state develops at larger $λ$ out of a CDW-ordered state and inherits a CDW order over some range of coupling.

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arXiv2024

Unconventional Superconductivity Mediated by Nematic Fluctuations in a Multi-Orbital System -- Application to doped FeSe

We analyze superconductivity in a multi-orbital fermionic system near the onset of a nematic order, using doped FeSe as an example. We associate nematicity with a spontaneous polarization between $d_{\text{xz}}$ and $d_{\text{yz}}$ orbitals (a Pomeranchuk-type order) and analyze the pairing mediated by soft nematic fluctuations. Such a pairing gives rise to a highly anisotropic gap function whose structure strongly varies with temperature, and leads to strongly non-BCS behavior in thermodynamics, spectroscopy and transport. We compute the specific heat and its directional variation with a magnetic field, magnetic susceptibility, density of states, tunneling conductance, Raman intensity, superfluid stiffness and penetration depth without and with impurity scattering and for the latter computed also optical conductivity and $T_c$ variation. We find good agreement with the existing data for FeSe$_{1-x}$S$_x$ and FeSe$_{1-x}$Te$_x$ and suggest new experiments.

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arXiv2024

Non-BCS behavior of the pairing susceptibility near the onset of superconductivity in a quantum-critical metal

We analyze the dynamical pairing susceptibility $χ_{pp} (ω_m)$ at $T=0$ in a quantum-critical metal, where superconductivity emerges out of a non-Fermi liquid ground state once the pairing interaction exceeds a certain threshold. We obtain $χ_{pp} (ω_m)$ as the ratio of the fully dressed dynamical pairing vertex $Φ(ω_m)$ and the bare $Φ_0 (ω_m)$ (both infinitesimally small). For superconductivity out of a Fermi liquid, the pairing susceptibility is positive above $T_c$, diverges at $T_c$, and becomes negative below it. For superconductivity out of a non-Fermi liquid, the behavior of $χ_{pp} (ω_m)$ is different in two aspects: (i) it diverges at the onset of pairing at $T=0$ only for a certain subclass of bare $Φ_0 (ω_m)$ and remains non-singular for other $Φ_0 (ω_m)$, and (ii) below the instability, it becomes a non-unique function of a continuous parameter $ϕ$ for an arbitrary $Φ_0 (ω_m)$. The susceptibility is negative in some range of $ϕ$ and diverges at the boundary of this range. We argue that this behavior of the susceptibility reflects a multi-critical nature of a superconducting transition in a quantum-critical metal when immediately below the instability an infinite number of superconducting states emerges simultaneously with different amplitudes of the order parameter down to an infinitesimally small one.

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arXiv2023

Unconventional Superconductivity near a Nematic Instability in a Multi-Orbital system

We analyze superconductivity in a multi-orbital fermionic system near the onset of a nematic order, using doped FeSe as an example. We associate the nematic order with spontaneous polarization between $d_{xz}$ and $d_{yz}$ orbitals. We derive the pairing interaction, mediated by soft nematic fluctuations, and show that it is attractive, and that its strength depends on the position on the Fermi surface. As the consequence, right at the nematic quantum-critical point (QCP), superconducting gap opens up at $T_c$ only at special points and extends into finite arcs at $T < T_c$. In between the arcs the Fermi surface remains intact. This gives rise to highly unconventional behavior of the specific heat, with no jump at $T_c$ and an apparent finite offset at $T=0$, when extrapolated from a finite $T$. We argue that this behavior is consistent with the specific heat data for FeSe$_{1-x}$S$_x$ near critical $x$ for the onset of a nematic order. We discuss the behavior of the gap away from a QCP and the pairing symmetry, and apply the results to FeSe$_{1-x}$S$_x$ and FeSe$_{1-x}$Te$_x$, which both show superconducting behavior near the QCP distinct from that in a pure FeSe.

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arXiv2021

Superconductivity of incoherent electrons in Yukawa-SYK model

We study a model of $N$ fermions in a quantum dot, coupled to $M$ bosons by a disorder-induced complex Yukawa coupling (Yukawa-SYK model), in order to explore the interplay between non-Fermi liquid and superconductivity in a strongly coupled, (quantum-)critical environment. We analyze the phase diagram of the model for an arbitrary complex interaction and arbitrary ratio of $N/M$, with special focus on the two regimes of non-Fermi-liquid behavior: an SYK-like behavior with a power-law frequency dependence of the fermionic self-energy and an impurity-like behavior with frequency independent self-energy. We show that the crossover between the two. can be reached by varying either the strength of the fermion-boson coupling or the ratio $M/N$. We next argue that in both regimes the system is unstable to superconductivity if the strength of time-reversal-symmetry-breaking disorder is below a certain threshold. We show how the corresponding onset temperatures vary between the two regimes. We argue that the superconducting state is highly unconventional with an infinite set of minima of the condensation energy at $T=0$, corresponding to topologically different gap functions. We discuss in detail similarities and differences between this model and the model of dispersion-full fermions tuned to a metallic quantum-critical point, with an effective singular dynamical interaction $V(Ω) \propto 1/|Ω|^γ$ (the $γ-$model).

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arXiv2019

Superconductivity near a nematic quantum critical point -- the interplay between hot and lukewarm regions

We present a strong coupling dynamical theory of the superconducting transition in a metal near a QCP towards $Q = 0$ nematic order. We use a fermion-boson model, in which we treat the ratio of effective boson-fermion coupling and the Fermi energy as a small parameter $λ$. We solve, both analytically and numerically, the linearized Eliashberg equation. Our solution takes into account both strong fluctuations at small momentum transfers $\sim λk_F$ and weaker fluctuations at large momentum transfers. The strong fluctuations determine $T_c$, which is of order $λ^2 E_F$ for both s- and d- wave pairing. The weaker fluctuations determine the angular structure of the superconducting order parameter $F(θ_k)$ along the Fermi surface, separating between hot and lukewarm regions. In the hot regions $F(θ_k)$ is largest and approximately constant. Beyond the hot region, whose width is $θ_h\simλ^{1/3}$, $F(θ_k)$ drops by a factor $λ^{4/3}$. The s- and d- wave states are not degenerate but the relative difference $(T_c^s-T_c^d)/T_c^s\simλ^2$ is small.

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arXiv2019

Implicit renormalization approach to the problem of Cooper instability

In the vast majority of cases, superconducting transition takes place at exponentially low temperature $T_c$ out of the Fermi liquid regime. We discuss the problem of determining $T_c$ from known system properties at temperatures $T \gg T_c$, and stress that this cannot be done reliably by following the standard protocol of solving for the largest eigenvalue of the original gap-function equation. However, within the implicit renormalization approach, the gap-function equation can be used to formulate an alternative eigenvalue problem, solving which leads to an accurate prediction for both $T_c$ and the gap function immediately below $T_c$. With the diagrammatic Monte Carlo techniques, this eigenvalue problem can be solved without invoking the matrix inversion or even explicitly calculating the four-point vertex function.

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arXiv2018

Multiple pairing states and temperature-dependent gap anisotropy for superconductivity near a nematic quantum-critical point

Superconductivity in many strongly correlated materials appears in proximity to a density-wave or nematic order and is believed to be mediated by quantum-critical (QC) fluctuations of the corresponding order parameter. We argue that fingerprints of QC pairing can be extracted from the angular dependence of the gap $Δ(θ)$. We consider pairing by QC nematic fluctuations and show that there exist multiple pairing instabilities within the same symmetry ($s-$wave in our case), with closely spaced transition temperatures $T_{c,n}$. The corresponding $Δ_n (θ)$ change sign $8n$ times along the FS. Only the solution with the highest $T_{c,0} =T_c$ develops, but other gap components are induced below $T_c$ and get resonantly enhanced below $T_{c,n}$. This gives rise to strong variation of the angular dependence of the gap below $T_c$. The effect gets much weaker away from a quantum-critical point.

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arXiv2018

Pairing Mechanism in Hunds Metal Superconductors and the Universality of the Superconducting Gap to Critical Temperature Ratio

We analyze a simple model containing the physical ingredients of a Hund's metal, the local spin fluctuations with power-law correlators, $(Ω_0/|Ω|)^γ$, with $γ$ greater than one, interacting with electronic quasiparticles. While the critical temperature and the gap change significantly with varying parameters, the $2Δ_{max}/k_BT_c$ remains close to twice the BCS value in agreement with experimental observations in the iron-based superconductors (FeSC).

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arXiv2017

Dynamical susceptibility of a near-critical non-conserved order parameter and B2g Raman response in Fe-based superconductors

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 polarization $Π(0, Ω)$ has similar behavior in the two cases, but the relations between $Π(0, Ω)$ and the bosonic susceptibility $χ(0, Ω)$ are different, leading to different forms of $χ^{\prime \prime} (0, Ω)$, as measured by Raman scattering. We compare our results with polarization-resolved Raman data for the Fe-based superconductors FeSe$_{1-x}$S$_x$, NaFe$_{1-x}$Co$_x$As and BaFe$_2$As$_2$. We argue that the data for FeSe$_{1-x}$S$_x$ are well described within Pomeranchuk scenario, while the data for NaFe$_{1-x}$Co$_x$As and BaFe$_2$As$_2$ are better described within the "independent" scenario involving a composite spin order.

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arXiv2017

Dynamical susceptibility near a long-wavelength critical point with a nonconserved order parameter

We study the dynamic response of a two-dimensional system of itinerant fermions in the vicinity of a uniform ($\mathbf{Q}=0$) Ising nematic quantum critical point of $d-$wave symmetry. The nematic order parameter is not a conserved quantity, and this permits a nonzero value of the fermionic polarization in the $d-$wave channel even for vanishing momentum and finite frequency: $Π(\mathbf{q} = 0,Ω_m) \neq 0$. For weak coupling between the fermions and the nematic order parameter (i.e. the coupling is small compared to the Fermi energy), we perturbatively compute $Π(\mathbf{q} = 0,Ω_m) \neq 0$ over a parametrically broad range of frequencies where the fermionic self-energy $Σ(ω)$ is irrelevant, and use Eliashberg theory to compute $Π(\mathbf{q} = 0,Ω_m)$ in the non-Fermi liquid regime at smaller frequencies, where $Σ(ω) > ω$. We find that $Π(\mathbf{q}=0,Ω)$ is a constant, plus a frequency dependent correction that goes as $|Ω|$ at high frequencies, crossing over to $|Ω|^{1/3}$ at lower frequencies. The $|Ω|^{1/3}$ scaling holds also in a non-Fermi liquid regime. The non-vanishing of $Π(\mathbf{q}=0, Ω)$ gives rise to additional structure in the imaginary part of the nematic susceptibility $χ^{''} (\mathbf{q}, Ω)$ at $Ω> v_F q$, in marked contrast to the behavior of the susceptibility for a conserved order parameter. This additional structure may be detected in Raman scattering experiments in the $d-$wave geometry.

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arXiv2017

Conservation laws, vertex corrections, and screening in Raman spectroscopy

We present a microscopic theory for the Raman response of a clean multiband superconductor accounting for the effects of vertex corrections and long-range Coulomb interaction. The measured Raman intensity, $R(Ω)$, is proportional to the imaginary part of the fully renormalized particle-hole correlator with Raman form-factors $γ(\vec k)$. In a BCS superconductor, a bare Raman bubble is non-zero for any $γ(\vec k)$ and diverges at $Ω= 2Δ+0$, where $Δ$ is the largest gap along the Fermi surface. However, for $γ(\vec k) =$ const, the full $R(Ω)$ is expected to vanish due to particle number conservation. It was long thought that this vanishing is due to the singular screening by long-range Coulomb interaction. We argue that this vanishing actually holds due to vertex corrections from the same short-range interaction that gives rise to superconductivity. We further argue that long-range Coulomb interaction does not affect the Raman signal for $any$ $γ(\vec k)$. We argue that vertex corrections eliminate the divergence at $2Δ$ and replace it with a maximum at a somewhat larger frequency. We also argue that vertex corrections give rise to sharp peaks in $R(Ω)$ at $Ω< 2Δ$, when $Ω$ coincides with the frequency of one of collective modes in a superconductor, e.g, Leggett mode, Bardasis-Schrieffer mode, or an excitonic mode.

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arXiv2017

Critical behavior of itinerant fermions - role of finite size effects

We study the role of finite size effects on a metallic critical behavior near a q = 0 critical point and compare the results with the recent extensive quantum Monte-Carlo (QMC) study [Y. Schattner et al, PRX 6, 0231028]. This study found several features in both bosonic and fermionic responses, in disagreement with the expected critical behavior with dynamical exponent z = 3. We show that finite size effects are particularly strong for z = 3 criticality and give rise to a behavior different from that of an infinite system, over a wide range of momenta and frequencies. We argue that by taking finite size effects into account, the QMC results can be explained within z = 3 theory. Our results also have implications for small interacting fermionic systems, such as magnetic nanoparticles.

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arXiv2016

Distinguishing between $s+id$ and $s+is$ pairing symmetries in multiband superconductors through spontaneous magnetization pattern induced by a defect

The symmetry of the pairing state in iron pnictide superconductor $\mathrm{Ba_{1-x}K_xFe_2As_2}$ is still controversial. At optimal doping ($x \approx 0.4$), it is very likely $s$-wave, but for $x=1$ there are experimental and theoretical arguments for both $s$-wave and $d$-wave. Depending on the choice for $x=1$, intermediate $s+is$ and $s+id$ states have been proposed for intermediate doping $ 0.4 < x < 1$. In both states, the time reversal symmetry is broken and a spontaneous magnetization is allowed. In this work we study a spontaneous magnetization induced by a nonmagnetic defect in the $s+is$ and $s+id$ states by using a perturbation theory and numerical calculations for the Ginzburg-Landau free energy functional. We show that the angular dependence of the magnetization is distinct in these two states due to the difference in symmetry properties of the order parameters. Our results indicate a possible way to distinguish between the $s+is$ and $s+id$ pairing symmetries in multi-band superconductors.

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arXiv2015

Emergent non-Fermi liquid at the quantum critical point of a topological phase transition in two dimensions

We study the effects of Coulomb interaction between 2D Weyl fermions with anisotropic dispersion which displays relativistic dynamics along one direction and Newtonian dynamics along the other. Such a dispersion can be realized in phosphorene under electric field or strain, in TiO$_2$/VO$_2$ superlattices, and, more generally, at the quantum critical point between a nodal semimetal and an insulator in systems with a chiral symmetry. Using the one-loop renormalization group approach in combination with the large-$N$ expansion, we find that the system displays interaction-driven non-Fermi liquid behavior in a wide range of intermediate frequencies and marginal Fermi liquid behavior at the smallest frequencies. In the non-Fermi liquid regime, the quasiparticle residue $Z$ at energy $E$ scales as $Z \propto E^a$ with $a >0$, and the parameters of the fermionic dispersion acquire anomalous dimensions. In the marginal Fermi-liquid regime, $Z \propto (|\log E|)^{-b}$ with universal $b = 3/2$.

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arXiv2015

Superconducting and charge-density-wave orders in the spin-fermion model: a comparative analysis

We present comparative analysis of superconducting and charge-density-wave orders in the spin-fluctuation scenario for the cuprates. That spin-fluctuation exchange gives rise to d-wave superconductivity is well known. Several groups recently argued that the same spin-mediated interaction may also account for charge-density-wave order with momenta $(Q,0)$ or $(0,Q)$, detected in underdoped cuprates. This has been questioned on the basis that charge-density-wave channel mixes fermions from both nested and anti-nested regions on the Fermi surface, and fermions in the anti-nested region do not have a natural tendency to form a bound state, even if the interaction is attractive. We show that anti-nesting is not an obstacle for charge order, but to see this one needs to go beyond the conventional Eliashberg approximation. We show that in the prefect nesting/antinesting case, when the velocities of hot fermions are either parallel or antiparallel, the onset temperatures in superconducting and charge-density-wave channels are of comparable strength for any magnetic correlation length $ξ$. The superconducting $T_{\rm sc}$ is larger than $T_{\rm cdw}$, but only numerically. When the velocities of hot fermions are not strictly parallel/antiparallel, $T_{\rm cdw}$ progressively decreases as $ξ$ decreases and vanishes at some critical $ξ$.

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