arXiv subjects

Andrey V. Chubukov

Publications and source records attributed to Andrey V. Chubukov.

At least 91 records · Page 5Linked to original sources

arXiv2018

Pairing in quantum-critical systems: $T_c$, $Δ$, and their ratio

We compute the ratio of the pairing gap $Δ$ at $T=0$ and $T_c$ for a set of quantum-critical models in which the pairing interaction is mediated by a gapless boson with local susceptibility $χ(Ω) \propto 1/|Ω|^γ$ (the $γ$ model). The limit $γ= 0+$ ($χ(Ω) =\log {|Ω|}$) describes color superconductivity, and models with $γ>0$ describe superconductivity in a metal at the onset of charge or spin order. The ratio $2Δ/T_c$ has been recently computed numerically for $0<γ<2$ within Eliashberg theory and was found to increase with increasing $γ$ [T-H Lee et al, arXiv:1805.10280]. We argue that the origin of the increase is the divergence of $2Δ/T_c$ at $γ=3$. We obtain an approximate analytical formula for $2Δ/T_c$ for $γ\leq 3$ and show that it agrees well with the numerics. We also consider in detail the opposite limit of small $γ$. Here we obtain the explicit expressions for $T_c$ and $Δ$, including numerical prefactors. We show that these prefactors depend on fermionic self-energy in a rather non-trivial way. The ratio $2Δ/T_c$ approaches the BCS value $3.53$ at $γ\to 0$.

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arXiv2018

Superconductivity in FeSe: the role of nematic order

Bulk FeSe is a special iron-based material in which superconductivity emerges inside a well-developed nematic phase. We present a microscopic model for this nematic superconducting state, which takes into account the mixing between $s-$wave and $d-$wave pairing channels and the changes in the orbital spectral weight promoted by the sign-changing nematic order parameter. We show that nematicity gives rise to a $\cos2θ$ variation of the pairing gap on the hole pocket that agrees with ARPES and STM data for experimentally-extracted Fermi surface parameters. We further argue that, in BCS theory, $d_{xz}$ and $d_{yz}$ orbitals give nearly equal contributions to the pairing glue, i.e. nematic order alone accounts for the gap anisotropy, but has little effect on $T_{c}$. This result questions the validity of the concept of orbital-selective pairing. Self-energy corrections, however, make $d_{xz}$ orbital more incoherent and reduce its contribution to pairing.

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arXiv2018

Time-reversal symmetry-breaking nematic superconductivity in FeSe

FeSe is a unique member of the family of iron-based superconductors, not only because of the high values of $T_{c}$ in FeSe monolayer, but also because in bulk FeSe superconductivity emerges inside a nematic phase without competing with long-range magnetic order. Near $T_{c}$, superconducting order necessarily has $s+d$ symmetry, because nematic order couples linearly the $s$-wave and $d$-wave harmonics of the superconducting order parameter. Here we argue that the near-degeneracy between $s$-wave and $d$-wave pairing instabilities in FeSe, combined with the sign-change of the nematic order parameter between hole and electron pockets, allows the superconducting order to break time-reversal symmetry at a temperature $T^{*}<T_{c}$. The transition from an $s+d$ state to an $s+\mathrm{e}^{iα}d$ state should give rise to a peak in the specific heat and to the emergence of a soft collective mode that can be potentially detected by Raman spectroscopy.

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arXiv2018

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.

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arXiv2018

Effects of Lifshitz transitions in ferromagnetic superconductors: the case of URhGe

In ferromagnetic superconductors, like URhGe, superconductivity co-exists with magnetism near zero field, but then re-appears again in a finite field range, where the system also displays mass enhancement in the normal state. We present theoretical understanding of this non-monotonic behavior. We explore the multi-band nature of URhGe and associate re-entrant superconductivity and mass enhancement with the finite field Lifshitz transition in one of the bands. We found good agreement between our theory and a number of experimental results for URhGe, such as weakly first order reentrant transition, the dependence of superconducting $T_c$ on a magnetic field, and the field dependence of the effective mass, the specific heat and the resistivity in the normal state. Our theory can be applied to other ferromagnetic multi-band superconductors.

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arXiv2018

Orbital order in FeSe - the case for vertex renormalization

We study the structure of the d-wave orbital order in FeSe in light of recent STM and ARPES data, which detect the shapes of hole and electron pockets in the nematic phase. The geometry of the pockets indicates that the sign of the orbital order $Γ= \langle d^{\dagger}_{xz} d_{xz}- d^{\dagger}_{yz} d_{yz}\rangle$ is different between hole and electron pockets $(Γ_h$ and $Γ_e$). We argue that this sign change cannot be reproduced if one solves for the orbital order within mean-field approximation, as the mean-field analysis yields either no orbital order, or order with the same sign of $Γ_e$ and $Γ_h$. We argue that another solution with the opposite signs of $Γ_e$ and $Γ_h$ emerges if we include the renormalizations of the vertices in $d-$wave orbital channel. We show that the ratio $|Γ_e/Γ_h|$ is of order one, independent on the strength of the interaction. We also compute the temperature variation of the energy of $d_{xz}$ and $d_{yz}$ orbitals at the center of electron pockets and compare the results with ARPES data.

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arXiv2018

Itinerant fermions on a triangular lattice: unconventional magnetism and other ordered states

We consider a system of 2D fermions on a triangular lattice with well separated electron and hole pockets of similar sizes, centered at certain high-symmetry-points in the Brillouin zone. We first analyze Stoner-type spin-density-wave (SDW) magnetism. We show that SDW order is degenerate at the mean-field level. Beyond mean-field, the degeneracy is lifted and is either $120^{\circ}$ "triangular" order (same as for localized spins), or a collinear order with antiferromagnetic spin arrangement on two-thirds of sites, and non-magnetic on the rest of sites. We also study a time-reversal symmetric directional spin bond order, which emerges when some interactions are repulsive and some are attractive. We show that this order is also degenerate at a mean-field level, but beyond mean-field the degeneracy is again lifted. We next consider the evolution of a magnetic order in a magnetic field starting from an SDW state in zero field. We show that a field gives rise to a canting of an SDW spin configuration. In addition, it necessarily triggers the directional bond order, which, we argue, is linearly coupled to the SDW order in a finite field. We derive the corresponding term in the Free energy. Finally, we consider the interplay between an SDW order and superconductivity and charge order. For this, we analyze the flow of the couplings within parquet renormalization group (pRG) scheme. We show that magnetism wins if all interactions are repulsive and there is little energy space for pRG to develop. However, if system parameters are such that pRG runs over a wide range of energies, the system may develop either superconductivity or an unconventional charge order, which breaks time-reversal symmetry.

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arXiv2018

The Conditions for $l=1$ Pomeranchuk Instability in a Fermi Liquid

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 with angular momentum $l$. However, it is known that conservation laws for total spin and charge prevent Pomeranchuk instabilities for $l=1$ spin- and charge- current order parameters. Our study aims to understand whether this holds only for these special forms of $l=1$ order parameters, or is a more generic result. To this end we perform a diagrammatic analysis of spin and charge susceptibilities for charge and spin density order parameters, as well as perturbative calculations to second order in the Hubbard $U$. We argue that for $l=1$ spin-current and charge-current order parameters, certain vertex functions, which are determined by high-energy fermions, vanish at $F^{c(s)}_{l=1}=-1$, preventing a Pomeranchuk instability from taking place. For an order parameter with a generic $l=1$ form-factor, the vertex function is not expressed in terms of $F^{c(s)}_{l=1}$, and a Pomeranchuk instability does occur when $F^{c(s)}_1=-1$. We argue that for other values of $l$, a Pomeranchuk instability occurs at $F^{c(s)}_{l} =-1$ for an order parameter with any form-factor

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arXiv2017

Chiral liquid phase of simple quantum magnets

We study a $T=0$ quantum phase transition between a quantum paramagnetic state and a magnetically ordered state for a spin $S=1$ XXZ Heisenberg antiferromagnet on a two-dimensional triangular lattice. The transition is induced by an easy plane single-ion anisotropy $D$. At the mean-field level, the system undergoes a direct transition at a critical $D = D_c$ between a paramagnetic state at $D > D_c$ and an ordered state with broken U(1) symmetry at $D < D_c$. We show that beyond mean field the phase diagram is very different and includes an intermediate, partially ordered chiral liquid phase. Specifically, we find that inside the paramagnetic phase the Ising ($J_z$) component of the Heisenberg exchange binds magnons into a two-particle bound state with zero total momentum and spin. This bound state condenses at $D > D_c$, before single-particle excitations become unstable, and gives rise to a chiral liquid phase, which spontaneously breaks spatial inversion symmetry, but leaves the spin-rotational U(1) and time-reversal symmetries intact. This chiral liquid phase is characterized by a finite vector chirality without long range dipolar magnetic order. In our analytical treatment, the chiral phase appears for arbitrarily small $J_z$ because the magnon-magnon attraction becomes singular near the single-magnon condensation transition. This phase exists in a finite range of $D$ and transforms into the magnetically ordered state at some $D<D_c$. We corroborate our analytic treatment with numerical density matrix renormalization group calculations.

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arXiv2017

Optical conductivity of a two-dimensional metal near a quantum-critical point: the status of the "extended Drude formula"

The optical conductivity of a metal near a quantum critical point (QCP) is expected to depend on frequency not only via the scattering time but also via the effective mass, which acquires a singular frequency dependence near a QCP. We check this assertion by computing diagrammatically the optical conductivity, $σ' (Ω)$, near both nematic and spin-density wave (SDW) quantum critical points (QCPs) in 2D. If renormalization of current vertices is not taken into account, $σ' (Ω)$ is expressed via the quasiparticle residue $Z$ (equal to the ratio of bare and renormalized masses in our approximation) and transport scattering rate $γ_{\text{tr}}$ as $σ' (Ω)\propto Z^2 γ_{\text{tr}}/Ω^2$. For a nematic QCP ($γ_{\text{tr}}\proptoΩ^{4/3}$ and $Z\proptoΩ^{1/3}$), this formula suggests that $σ'(Ω)$ would tend to a constant at $Ω\to 0$. We explicitly demonstrate that the actual behavior of $σ' (Ω)$ is different due to strong renormalization of the current vertices, which cancels out a factor of $Z^2$. As a result, $σ' (Ω)$ diverges as $1/Ω^{2/3}$, as earlier works conjectured. In the SDW case, we consider two contributions to the conductivity: from hot spots and from"lukewarm" regions of the Fermi surface. The hot-spot contribution is not affected by vertex renormalization, but it is subleading to the lukewarm one. For the latter, we argue that a factor of $Z^2$ is again cancelled by vertex corrections. As a result, $σ' (Ω)$ at a SDW QCP scales as $1/Ω$ down to the lowest frequencies.

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arXiv2017

Optical response of correlated electron systems

Recent progress in experimental techniques has made it possible to extract detailed information on dynamics of carriers in a correlated electron material from its optical conductivity, $σ(Ω, T)$. This review consists of three parts, addressing the following three aspects of optical response: 1) the role of momentum relaxation, 2) $Ω/T$ scaling of the optical conductivity of a Fermi-liquid metal, and 3) optical conductivity of non-Fermi-liquid metals. In the first part (Sec. II), we analyze the interplay between the contributions to the conductivity from normal and umklapp scattering. In the second part (Secs. III and IV), we re-visit the Gurzhi formula for the optical scattering rate, $1/τ(Ω,T)\proptoΩ^2+4π^2 T^2$, and show that a factor of $4π^2$ is the manifestation of the "first-Matsubara-frequency rule", which states that $1/τ(Ω,T)$ must vanish upon analytic continuation to the first boson Matsubara frequency. However, recent experiments show that the coefficient $b$ in the Gurzhi-like form, $1/τ(Ω,T)\proptoΩ^2+bπ^2 T^2$, differs significantly from $b=4$ in most of the cases. We suggest that the discrepancy may be due to the presence of elastic scattering, which decreases the value of $b$ below $4$, with $b=1$ corresponding to purely elastic scattering. In the third part (Sec. V), we consider the optical conductivity of metals near quantum phase transitions to nematic and spin-density-wave (SDW) states. In the last case, we focus on "composite" scattering processes, which give rise to a non-Fermi--liquid behavior of the optical conductivity: $σ'(Ω,T)\propto Ω^{-1/3}$ at low frequencies and $σ'(Ω,T)\propto Ω^{-1}$ at higher frequencies. We also discuss $Ω/T$ scaling and show that $σ'(Ω,T)$ in the same model scales in a non-Fermi-liquid way, as $T^{4/3}Ω^{-5/3}$.

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arXiv2017

Gradient terms in quantum-critical theories of itinerant fermions

We investigate the origin and renormalization of the gradient ($Q^2$) term in the propagator of soft bosonic fluctuations in theories of itinerant fermions near a quantum critical point (QCP) with $Q =0$. A common belief is that (i) the $Q^2$ term comes from fermions with high energies (roughly of order of the bandwidth) and, as such, should be included into the bare bosonic propagator of the effective low-energy model, and (ii) fluctuations within the low-energy model generate Landau damping of soft bosons, but affect the $Q^2$ term only weakly. We argue that the situation is in fact more complex. First, we found that the high- and low-energy contributions to the $Q^2$ term are of the same order. Second, we computed the high-energy contributions to the $Q^2$ term in two microscopic models (a Fermi gas with Coulomb interaction and the Hubbard model) and found that in all cases these contributions are numerically much smaller than the low-energy ones, blue especially in 2D. This last result is relevant for the behavior of observables at low energies, because the low-energy part of the $Q^2$ term is expected to flow when the effective mass diverges near QCP. If this term is the dominant one, its flow has to be computed self-consistently, which gives rise to a novel quantum-critical behavior. Following up on these results, we discuss two possible ways of formulating the theory of a QCP with $Q=0$.

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arXiv2017

Quantum phase transitions in Heisenberg $J_1-J_2$ triangular antiferromagnet in a magnetic field

We present the zero temperature phase diagram of a Heisenberg antiferromagnet on a frustrated triangular lattice with nearest neighbor ($J_1$) and next nearest neighbor ($J_2$) interactions, in a magnetic field. We show that the classical model has an accidental degeneracy for all $J_2/J_1$ and all fields, but the degeneracy is lifted by quantum fluctuations. We show that at large $S$, for $J_2/J_1 <1/8$, quantum fluctuations select the same sequence of three sublattice co-planar states in a field as for $J_2 =0$, and for $1/8 1$, the transition remains first order, with a finite hysteresis width, but for $S=1/2$ and, possibly, $S=1$, there appears a new intermediate phase, likely without a spontaneous long-range order.

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arXiv2017

Half-magnetization plateau in a Heisenberg antiferromagnet on a triangular lattice

We present the phase diagram in a magnetic field of a 2D isotropic Heisenberg antiferromagnet on a triangular lattice. We consider spin-$S$ model with nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) interactions. We focus on the range of $1/8<J_2/J_1<1$, where the ordered states are different from those in the model with only nearest neighbor exchange. A classical ground state in this range has four sublattices and is infinitely degenerate in any field. The actual order is then determined by quantum fluctuations via "order from disorder" phenomenon. We argue that the phase diagram is rich due to competition between competing four-sublattice quantum states which break either $\mathbb{Z}_3$ orientational symmetry or $\mathbb{Z}_4$ sublattice symmetry. At small and high fields, the ground state is a $\mathbb{Z}_3$-breaking canted stripe state, but at intermediate fields the ordered states break $\mathbb{Z}_4$ sublattice symmetry. The most noticeable of such states is "three up, one down" state in which spins in three sublattices are directed along the field and in one sublattice opposite to the field. Such a state breaks no continuous symmetry and has gapped excitations. As the consequence, magnetization has a plateau at exactly one half of the saturation value. We identify gapless states, which border the "three up, one down" state and discuss the transitions between these states and the canted stripe state.

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arXiv2016

Interplay between magnetism, superconductivity, and orbital order in a 5-pocket model for iron-based superconductors - a parquet renormalization group study

We report the results of the parquet renormalization group (RG) analysis of the phase diagram of the most general 5-pocket model for Fe-based superconductors. We use as an input the orbital structure of excitations near the five pockets made out of $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals and argue that there are 40 different interactions between low-energy fermions in the orbital basis. All interactions flow under RG, as one progressively integrates out fermions with higher energies. We find that the low-energy behavior is amazingly simple, despite the large number of interactions. Namely, at low-energies the full 5-pocket model effectively reduces either to a 3-pocket model made of one $d_{xy}$ hole pocket and two electron pockets, or a 4-pocket model made of two $d_{xz}/d_{yz}$ hole pockets and two electron pockets. The leading instability in the effective 4-pocket model is a spontaneous orbital (nematic) order, followed by $s^{+-}$ superconductivity. In the effective 3-pocket model orbital fluctuations are weaker, and the system develops either $s^{+-}$ superconductivity or stripe SDW. In the latter case, nematicity is induced by composite spin fluctuations.

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arXiv2016

Hund interaction, spin-orbit coupling and the mechanism of superconductivity in strongly hole-doped iron pnictides

We present a novel mechanism of s-wave pairing in Fe-based superconductors. The mechanism involves holes near dxz/dyz pockets only and is applicable primarily to strongly hole doped materials. We argue that as long as the renormalized Hund's coupling J exceeds the renormalized inter-orbital Hubbard repulsion U', any finite spin-orbit coupling gives rise to s-wave superconductivity. This holds even at weak coupling and regardless of the strength of the intra-orbital Hubbard repulsion U. The transition temperature grows as the hole density decreases. The pairing gaps are four-fold symmetric, but anisotropic, with the possibility of eight accidental nodes along the larger pocket. The resulting state is consistent with the experiments on KFe2As2.

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arXiv2016

Competing instabilities, orbital ordering and splitting of band degeneracies from a parquet renormalization group analysis of a 4-pocket model for iron-based superconductors: application to FeSe

We report the results of a parquet renormalization group (RG) study of competing instabilities in the full 2D four pocket, three orbital low-energy model for iron-based superconductors. We derive and analyze the RG flow of the couplings, which describe all symmetry-allowed interactions between low-energy fermions. Despite that the number of the couplings is large, we argue that there are only two stable fixed trajectories of the RG flow and one weakly unstable fixed trajectory with a single unstable direction. Each fixed trajectory has a finite basin of attraction in the space of initial system parameters. On the stable trajectories, either interactions involving only $d_{xz}$ and $d_{yz}$ or only $d_{xy}$ orbital components on the electron pockets dominate, while on the weakly unstable trajectory interactions involving $d_{xz}$ ($d_{yz}$) and $d_{xy}$ orbital states on the electron pockets remain comparable. The behavior along the two stable fixed trajectories has been analyzed earlier [A.V. Chubukov, M. Khodas, and R.M. Fernandes, arXiv:1602.05503]. Here we focus on the system behavior along the weakly unstable trajectory and apply the results to FeSe. We find, based on the analysis of susceptibilities along this trajectory, that the leading instability upon lowering the temperature is towards a {\it three-component} d-wave orbital nematic order. Two components are the differences between fermionic densities on $d_{xz}$ and $d_{yz}$ orbitals on hole pockets and on electron pockets, and the third one is the difference between the densities of $d_{xy}$ orbitals on the two electron pockets. We argue that this order is consistent with the splitting of band degeneracies, observed in recent photoemission data on FeSe by A. Fedorov et al [arXiv:1606.03022].

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arXiv2016

Superconductivity near a quantum-critical point: The special role of the first Matsubara frequency

Near a quantum-critical point in a metal strong fermion-fermion interaction mediated by a soft collective boson gives rise to incoherent, non-Fermi liquid behavior. It also often gives rise to superconductivity which masks the non-Fermi liquid behavior. We analyze the interplay between superconductivity and fermionic incoherence for a set of quantum-critical models with effective dynamical interaction between low-energy fermions. We argue that superconductivity always develops above a quantum-critical point, no matter how strong the fermionic self-energy is. We argue that superconductivity should not be viewed as a pairing of incoherent fermions, as previously thought. Rather, $T_c$ is non-zero due to the fact that the self-energy is suppressed at the two lowest fermionic Matsubara frequencies $ω_m = \pm πT$. We obtain the analytic formula for $T_c$ which reproduces well numerical results for the electron-phonon model at vanishing Debye frequency.

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