arXiv subjects

Andrey V. Chubukov

Publications and source records attributed to Andrey V. Chubukov.

At least 55 records · Page 3Linked to original sources

arXiv2022

Spin-triplet superconductivity at the onset of isospin order in biased bilayer graphene

The quest for unconventional superconductivity governed by Coulomb repulsion between electrons rather than phonon attraction received new momentum with the advent of moiré graphene. Initially, delineating the phonon and Coulomb-repulsion-based pairing mechanisms has proven to be a challenging task, however the situation has changed after recent discovery of superconductivity in non-twisted graphene bilayers and trilayers. Superconductivity occurring at the phase boundaries of spin and valley polarized orders calls for non-phonon scenarios, yet the specific pairing mechanisms remain to be understood. Here we analyze a striking example -- superconductivity in graphene bilayers occurring at the onset of valley-polarized order induced by a magnetic field. We describe an attraction-from-repulsion mechanism for pairing mediated by a quantum-critical mode, which fully explains the observed phenomenology. While it is usually notoriously difficult to infer the pairing mechanism from the observed superconducting phases, this case presents a rare exception, allowing for a fairly unambiguous identification of the origin of the pairing glue. A combination of factors such as the location of superconducting phase at the onset of isospin-polarized phase, a threshold in a magnetic field, above which superconductivity occurs, and its resilience at high magnetic fields paints a clear picture of a triplet superconductivity driven by quantum-critical fluctuations.

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arXiv2022

Odd frequency pairing in a quantum critical metal

We analyze a possibility for odd-frequency pairing near a quantum critical point(QCP) in a metal. We consider a model with dynamical pairing interaction $V(Ω_n)\sim 1/|Ω_n|^γ$ (the $γ$-model). This interaction gives rise to a non-Fermi liquid in the normal state and is attractive for pairing. The two trends compete with each other. We search for odd-frequency solutions for the pairing gap $Δ(ω_m)= -Δ(ω_m)$. We show that for $γ<1$, odd-frequency superconductivity looses the competition with a non-Fermi liquid and does not develop. We show that the pairing does develop in the extended model in which interaction in the pairing channel is larger than the one in the particle-hole channel. For $γ>1$, we argue that the original model is at the boundary towards odd-frequency pairing and analyze in detail how superconductivity is triggered by a small external perturbation. In addition, we show that for $γ>2$, the system gets frozen at the critical point towards pairing in a finite range in the parameter space. This give rise to highly unconventional phase diagrams with flat regions.

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arXiv2022

Monte Carlo study of the pseudogap and superconductivity emerging from quantum magnetic fluctuations

The origin of the pseudogap behavior, found in many high-$T_c$ superconductors, remains one of the greatest puzzles in condensed matter physics. One possible mechanism is fermionic incoherence, which near a quantum critical point allows pair formation but suppresses superconductivity. Employing quantum Monte Carlo simulations of a model of itinerant fermions coupled to ferromagnetic spin fluctuations, represented by a quantum rotor, we report numerical evidence of pseudogap behavior, emerging from pairing fluctuations in a quantum-critical non-Fermi liquid. Specifically, we observe enhanced pairing fluctuations and a partial gap opening in the fermionic spectrum. However, the system remains non-superconducting until reaching a much lower temperature. In the pseudogap regime the system displays a "gap-filling" rather than "gap-closing" behavior, consistent with experimental observations. Our results provide the first unambiguous lattice model realization of a pseudogap state in a strongly correlated system, driven by superconducting fluctuations.

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arXiv2022

The dynamical exponent of a quantum critical itinerant ferromagnet: a Monte Carlo study

We consider the effect of the coupling between 2D quantum rotors near an XY ferromagnetic quantum critical point and spins of itinerant fermions. We analyze how this coupling affects the dynamics of rotors and the self-energy of fermions.A common belief is that near a $q=0$ ferromagnetic transition, fermions induce an $Ω/q$ Landau damping of rotors (i.e., the dynamical critical exponent is $z=3$) and Landau overdamped rotors give rise to non-Fermi liquid fermionic self-energy $Σ\propto ω^{2/3}$. This behavior has been confirmed in previous quantum Monte Carlo (QMC) studies.Here we show that for the XY case the behavior is different.We report the results of large scale quantum Monte Carlo simulations,which show that at small frequencies $z=2$ and $Σ\propto ω^{1/2}$. We argue that the new behavior is associated with the fact that a fermionic spin is by itself not a conserved quantity due to spin-spin coupling to rotors, and a combination of self-energy and vertex corrections replaces $1/q$ in the Landau damping by a constant. We discuss the implication of these results to experiments.

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arXiv2022

Quasiparticle scattering in a superconductor near a nematic critical point: resonance mode and multiple attractive channels

We analyze the scattering rate for 2D fermions interacting via soft nematic fluctuations. The ground state is an s-wave superconductor, but other pairing channels are almost equally attractive. This strongly alters the scattering rate: At energies beyond the pairing gap $Δ$, it is renormalized by contributions from all pairing channels. At energies of order $Δ$, it is determined by the competition between scattering into a gapped continuum and dispersing nematic resonance. The outcome is a "peak-peak-dip-hump" spectrum, similar, but not identical, to the "peak-dip-hump" structure in the cuprates.

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arXiv2021

The Effect of Repulsion on Superconductivity at Low Density

We examine the effect of repulsion on superconductivity in a three-dimensional system with a Bardeen-Pines-like interaction in the low-density limit, where the chemical potential $μ$ is much smaller than the phonon frequency $ω_L$. We parameterize the strength of the repulsion by a dimensionless parameter $f$, and find that the superconducting transition temperature $T_c$ approaches a nonzero value in the $μ= 0$ limit as long as $f$ is below a certain threshold $f^*$. In this limit, we find that $T_c$ goes to zero as a power of $f^*-f$, in contrast to the high density limit, where $T_c$ goes to zero exponentially quickly as $f$ approaches $f^*$. For all nonzero $f$, the gap function $Δ(ω_m)$ changes sign along the Matsubara axis, which allows the system to partially overcome the repulsion at high frequencies. We trace the position of the gap node with $f$ and show that it approaches zero frequency as $f$ approaches $f^*$. To investigate the robustness of our conclusions, we then go beyond the Bardeen-Pines model and include full dynamical screening of the interaction, finding that $T_c$ still saturates to a non-zero value at $μ= 0$ when $f < f^*$.

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arXiv2021

SU(4) symmetry in twisted bilayer graphene - an itinerant perspective

We study symmetry-broken phases in twisted bilayer graphene at small filling above charge neutrality and at Van Hove filling. We argue that the Landau functionals for the particle-hole order parameters at these fillings both have an approximate SU(4) symmetry, but differ in the sign of quartic terms. We determine the order parameter manifold of the ground state and analyze its excitations. For small fillings, we find a strong 1st-order transition to an SU(3)$\otimes$U(1) manifold of orders that break spin-valley symmetry and induce a 3-1 splitting of fermionic excitations. For Van Hove filling, we find a weak 1st-order transition to an SO(4)$\otimes$U(1) manifold of orders that preserves the two-fold band degeneracy. We discuss the effect of particle-hole orders on superconductivity and compare with strong-coupling approaches.

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arXiv2021

Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. VI. The $γ$ model and its phase diagram at $2 < γ<3$

In this paper, the sixth in series, we continue our analysis of the interplay between non-Fermi liquid and pairing in the effective low-energy model of fermions with singular dynamical interaction $V(Ω_m) = {\bar g}^γ/|Ω_m|^γ$ (the $γ$ model). The model describes low-energy physics of various quantum-critical metallic systems at the verge of an instability towards density or spin order, pairing of fermions at the half-filled Landau level, color superconductivity, and pairing in SYK-type models. In previous Papers I-V we analyzed the $γ$ model for $γ\leq 2$ and argued that the ground state is an ordinary superconductor for $γ<1$, a peculiar one for $1<γ<2$, when the phase of the gap function winds up along real frequency axis due to emerging dynamical vortices in the upper half-plane of frequency, and that there is a quantum phase transition at $γ=2$, when the number of dynamical vortices becomes infinite. In this paper we consider larger $2< γ<3$ and address the issue what happens on the other side of this quantum transition. We argue that the system moves away from criticality in that the number of dynamical vortices becomes finite and decreases with increasing $γ$. The ground state is again a superconductor, however a highly unconventional one with a non-integrable singularity in the density of states at the lower edge of the continuum. This implies that the spectrum of excited states now contains a level with a macroscopic degeneracy, proportional to the total number of states in the system. We argue that the phase diagram in variables $(T,γ)$ contains two distinct superconducting phases for $γ<2$ and $γ>2$, and an intermediate pseudogap state of preformed pairs.

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arXiv2021

Dynamical vortices in electron-phonon superconductors

We analyze the structure of an $s-$wave superconducting gap in systems with electron-phonon attraction and electron-electron repulsion. Earlier works have found that superconductivity develops despite strong repulsion, but the gap, $Δ(ω_m)$, necessarily changes sign along the Matsubara axis. We analyze the sign-changing gap function from a topological perspective using the knowledge that a nodal point of $Δ(ω_m)$ is the center of dynamical vortex. We consider two models with different cutoffs for the repulsive interaction and trace the vortex positions along the Matsubara axis and in the upper frequency half plane upon changing the relative strength of the attractive and repulsive parts of the interaction. We discuss how the presence of dynamical vortices affects the gap structure along the real axis, detectable in ARPES experiments.

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arXiv2021

One-dimensional scattering of two-dimensional fermions near quantum criticality

Forward and backscattering play an exceptional role in the physics of two-dimensional interacting fermions. In a Fermi liquid, both give rise to a non-analytic $ω^2 \log(ω)$ form of the fermionic scattering rate at second order in the interaction. Here we argue that higher powers of $\log(ω)$ appearin the backscattering contribution at higher orders. We show that these terms come from "planar" processes, which are effectively one-dimensional. This is explicitly demonstated by extending a Fermi liquid to the limit of $N \gg 1$ fermionic flavors, when only planar processes survive. We sum the leading logarithms for the case of a 2D Fermi liquid near a nematic transition and obtain an expression for the scattering rate at $T=0$ to all orders in the interaction. For a repulsive interaction, the resulting rate is logarithmically suppressed, and the result is valid down to $ω= 0$. For an attractive interaction, the ground state is an $s$-wave superconductor with a gap $Δ_0$. We show that in this case the scattering rate increases as $ω$ is reduced towards $Δ_0$. At $ω\geq Δ_0$, the behavior of the scattering rate is rather unconventional as many pairing channels compete near a nematic critical point, and $s$-wave wins only by a narrow margin. We take superconductivity into consideration and obtain the scattering rate also at smaller $ω\simeq Δ_0$.

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arXiv2021

Specific Heat and the gap structure of a Nematic Superconductor, application to FeSe

We report the results of our in-depth analysis of spectroscopic and thermodynamic properties of a multi-orbital metal, like FeSe, which first develops a nematic order and then undergoes a transition into a superconducting state, which co-exists with nematicity. We analyze the angular dependence of the gap function and specific heat $C_V (T)$ of such nematic superconductor. We specifically address three issues: (i) angular dependence of the gap in light of the competition between nematicity-induced $s$-$d$ mixture and orbital transmutation of low-energy excitations in the nematic state, (ii) the effect of nematicity on the magnitude of the jump of the specific heat $C_V (T)$ at $T_c$ and the temperature dependence of $C_V (T)$ below $T_c$, and (iii) a potential transition at $T_{c1} < T_c$ from an $s+d$ state to an $s + e^{iη} d$ state that breaks time-reversal symmetry. We consider two scenarios for a nematic order: scenario A, in which this order develops between $d_{xz}$ and $d_{yz}$ orbitals on hole and electron pockets and scenario B, in which there is an additional component of the nematic order for $d_{xy}$ fermions on the two electron pockets.

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arXiv2021

Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal: V. The $γ$ model and its phase diagram. The case $γ=2$

This paper is a continuation and a partial summary of our analysis of the pairing at a quantum-critical point (QCP) in a metal for a set of quantum-critical systems, whose low-energy physics is described by an effective model with dynamical electron-electron interaction $V(Ω_m) = ({\bar g}/|Ω_m|)^γ$ (the $γ$-model). Examples include pairing at the onset of various spin and charge density-wave and nematic orders and pairing in SYK-type models. In previous papers, we analyzed the physics for $γ<2$. We have shown that the onset temperature for the pairing $T_p$ is finite, of order ${\bar g}$, yet the gap equation at $T=0$ has an infinite set of solutions within the same spatial symmetry. As the consequence, the condensation energy $E_c$ has an infinite number of minima. The spectrum of $E_c$ is discrete, but becomes more dense as $γ$ increases. Here we consider the case $γ=2$. The $γ=2$ model attracted special interest in the past as it describes the pairing by an Einstein phonon in the limit when the dressed phonon mass $ω_D$ vanishes. We show that for $γ=2$, the spectrum of $E_c$ becomes continuous. We argue that the associated gapless "longitudinal" fluctuations destroy superconducting phase coherence at a finite $T$, such that at $0<T< T_p$ the system displays pseudogap behavior. We show that for each gap function from the continuum spectrum, there is an infinite array of dynamical vortices in the upper half-plane of frequency. For the electron-phonon case, our results show that $T_p =0.1827 {\bar g}$, obtained in earlier studies, marks the onset of the pseudogap behavior of preformed pairs, while the actual superconducting $T_c$vanishes at $ω_D \to 0$.

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arXiv2020

Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. IV: The $γ$ model and its phase diagram at $1<γ<2$

In this paper we continue our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical (QC) systems with an effective dynamical electron-electron interaction $V(Ω_m) \propto 1/|Ω_m|^γ$ (the $γ$-model). In previous papers we studied the cases $0<γ<1$ and $γ\approx 1$. We argued that the pairing by a gapless boson is fundamentally different from BCS/Eliashberg pairing by a massive boson as for the former there exists an infinite number of topologically distinct solutions for the gap function $Δ_n (ω_m)$ at $T=0$ ($n=0,1,2...$), each with its own condensation energy $E_{c,n}$. Here we extend the analysis to larger $1< γ<2$. We argue that the discrete set of solutions survives, and the spectrum of $E_{c,n}$ gets progressively denser as $γ$ approaches $2$ and eventually becomes continuous at $γ\to 2$. This increases the strength of "longitudinal" gap fluctuations, which tend to reduce the actual superconducting $T_c$ and give rise to a pseudogap region of preformed pairs. We also detect two features on the real axis for $γ>1$ which become critical at $γ\to 2$. First, the density of states evolves towards a set of discrete $δ-$functions. Second, an array of dynamical vortices emerges in the upper frequency half-plane. These two features come about because on a real axis, the real part of the interaction, $V' (Ω) \propto \cos(πγ/2)/|Ω|^γ$, becomes repulsive for $γ>1$, and the imaginary $V^{''} (Ω) \propto \sin(πγ/2)/|Ω|^γ$, gets progressively smaller at $γ\to 2$. The features on the real axis are consistent with the development of a continuum spectrum of $E_{c,n}$ obtained using $Δ_n (ω_m)$ on the Matsubara axis. We consider the case $γ=2$ separately in the next paper.

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arXiv2020

Valley magnetism, nematicity, and density wave orders in twisted bilayer graphene

We analyze density-wave and Pomeranchuk orders in twisted bilayer graphene. This compliments our earlier analysis of the pairing instabilities. We assume that near half-filling of either conduction or valence band, the Fermi level is close to Van Hove points, where the density of states diverges, and study potential instabilities in the particle-hole channel within a patch model with two valley degrees of freedom. The hexagonal symmetry of twisted bilayer graphene allows for either six or twelve Van Hove points. We consider both cases and find the same two leading candidates for particle-hole order. One is an SU(2)-breaking spin state with ferromagnetism within a valley. A subleading inter-valley hopping induces antiferromagnetism between the valleys. The same state has also been obtained in strong coupling approaches, indicating that this order is robust. The other is a mixed state with $120^\circ$ complex spin order and orthogonal complex charge order. In addition, we find a weaker, but still attractive interaction in nematic channels, and discuss the type of a nematic order.

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arXiv2020

Identification of non-Fermi liquid fermionic self-energy from quantum Monte Carlo data

Quantum Monte Carlo (QMC) simulations of correlated electron systems provide unbiased information about system behavior at a quantum critical point (QCP) and can verify or disprove the existing theories of non-Fermi liquid (NFL) behavior at a QCP. However, simulations are carried out at a finite temperature, where quantum-critical features are masked by finite temperature effects. Here we present a theoretical framework within which it is possible to separate thermal and quantum effects and extract the information about NFL physics at $T=0$. We demonstrate our method for a specific example of 2D fermions near a Ising-ferromagnetic QCP. We show that one can extract from QMC data the zero-temperature form of fermionic self-energy $Σ(ω)$ even though the leading contribution to the self-energy comes from thermal effects. We find that the frequency dependence of $Σ(ω)$ agrees well with the analytic form obtained within the Eliashberg theory of dynamical quantum criticality, and obeys $ω^{2/3}$ scaling at low frequencies. Our results open up an avenue for QMC studies of quantum-critical metals.

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arXiv2020

Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. I: The $γ$-model and its phase diagram at $T=0$. The case $0 < γ<1$

We analyze a class of quantum-critical models, in which momentum integration and the selection of a particular pairing symmetry can be done explicitly, and the competition between non-Fermi liquid and pairing can be analyzed within an effective model with dynamical electron-electron interaction $V(Ω_m)\sim 1/|Ω_m|^γ$ (the $γ$-model). In this paper, the first in the series, we consider the case $T=0$ and $0<γ<1$. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that superconducting state is highly non-trivial as there exists a discrete set of topologically distinct solutions for the pairing gap $Δ_n (ω_m)$ ($n = 0, 1, 2..., \infty$). All solutions have the same spatial pairing symmetry, but differ in the time domain: $Δ_n (ω_m)$ changes sign $n$ times as a function of Matsubara frequency $ω_m$. The $n =0$ solution $Δ_0 (ω_m)$ is sign-preserving and tends to a finite value at $ω_m =0$, like in BCS theory. The $n = \infty$ solution corresponds to an infinitesimally small $Δ(ω_m)$. As a proof, we obtain the exact solution of the linearized gap equation at $T=0$ on the entire frequency axis for all $0<γ<1$, and an approximate solution of the non-linear gap equation.We argue that the presence of an infinite set of solutions opens up a new channel of gap fluctuations. We extend the analysis to the case where the pairing component of the interaction has additional factor $1/N$ and show that there exists a critical $N_{cr} >1$, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid.We show that all solutions develop simultaneously once $N$ gets smaller than $N_{cr}$.

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arXiv2020

Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. II. The $γ$-model at a finite $T$ for $0<γ<1$

In this paper we continue the analysis of the interplay between non-Fermi liquid and superconductivity for quantum-critical systems, the low-energy physics of which is described by an effective model with dynamical electron-electron interaction $V(Ω_m) \propto 1/|Ω_m|^γ$ (the $γ$ model). In paper I [A. Abanov and A. V. Chubukov, Phys Rev B. 102, 024524 (2020)] two of us analyzed the $γ$ model at $T=0$ for $0<γ<1$ and argued that there exist a discrete, infinite set of topologically distinct solutions for the superconducting gap, all with the same spatial symmetry. The gap function $Δ_n (ω_m)$ for the $n$th solution changes sign $n$ times as the function of Matsubara frequency. In this paper we analyze the linearized gap equation at a finite $T$. We show that there exist an infinite set of pairing instability temperatures, $T_{p,n}$, and the eigenfunction $Δ_n (ω_{m})$ changes sign $n$ times as a function of a Matsubara number $m$. We argue that $Δ_n (ω_{m})$ retains its functional form below $T_{p,n}$ and at $T=0$ coincides with the $n$th solution of the nonlinear gap equation. Like in paper I, we extend the model to the case when the interaction in the pairing channel has an additional factor $1/N$ compared to that in the particle-hole channel. We show that $T_{p,0}$ remains finite at large $N$ due to special properties of fermions with Matsubara frequencies $\pm πT$, but all other $T_{p,n}$ terminate at $N_{cr} = O(1)$. The gap function vanishes at $T \to 0$ for $N > N_{cr}$ and remains finite for $N < N_{cr}$. This is consistent with the $T =0$ analysis.

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arXiv2020

Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. III: The $γ$ model and its phase diagram across $γ= 1$

In this paper we continue our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical models with an effective dynamical electron-electron interaction $V(Ω_m) \propto 1/|Ω_m|^γ$ (the $γ$-model). We analyze both the original model and its extension, in which we introduce an extra parameter $N$ to account for non-equal interactions in the particle-hole and particle-particle channel. In two previous papers(arXiv:2004.13220 and arXiv:2006.02968), we considered the case $0 < γ<1$ and argued that (i) at $T=0$, there exists an infinite discrete set of topologically different gap functions, $Δ_n (ω_m)$, all with the same spatial symmetry, and (ii) each $Δ_n$ evolves with temperature and terminates at a particular $T_{p,n}$. In this paper, we analyze how the system behavior changes between $γ<1$ and $γ>1$, both at $T=0$ and a finite $T$. The limit $γ\to 1$ is singular due to infra-red divergence of $\int d ω_m V(Ω_m)$, and the system behavior is highly sensitive to how this limit is taken. We show that for $N =1$, the divergencies in the gap equation cancel out, and $Δ_n (ω_m)$ gradually evolve through $γ=1$ both at $T=0$ and a finite $T$. For $N \neq 1$, divergent terms do not cancel, and a qualitatively new behavior emerges for $γ>1$. Namely, the form of $Δ_n (ω_m)$ changes qualitatively, and the spectrum of condensation energies, $E_{c,n}$ becomes continuous at $T=0$. We introduce different extension of the model, which is free from singularities for $γ>1$.

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