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arXiv:1908.09831 · Physical review. B./Physical review. B

Pairing instability on a Luttinger surface: A non-Fermi liquid to superconductor transition and its Sachdev-Ye-Kitaev dual

Chandan Setty

Published May 13, 2020• 19 citations
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Abstract

Superconductivity results from an instability of the Fermi surface---contour of poles of the single-particle propagator---to an infinitesimally small attraction between electrons. Here, we instead discuss the analogous problem on a model Luttinger surface, or contour of zeros of the Green function. At zero temperature $(\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty})$ and a critical interaction strength $({u}_{c\ensuremath{\infty}})$ characterized by the residue of self-energy pole, we find that the pair susceptibility diverges leading to a superconducting instability. We evaluate the pair fluctuation partition function and find that the spectral density in the normal state has an interaction-driven, power-law $\frac{1}{\sqrt{\ensuremath{\omega}}}$ type, van-Hove singularity (vHS), indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit $(\ensuremath{\beta}u\ensuremath{\gg}1)$, the leading-order fluctuation free-energy terms in the normal state of this NFL-SC transition resembles the equivalent $\left[O(1)\right]$ terms of the Sachdev-Ye-Kitaev (SYK) model. This free-energy contribution takes a simple form $\ensuremath{-}\ensuremath{\beta}F=\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\ensuremath{-}\ensuremath{\gamma}\phantom{\rule{0.16em}{0ex}}\text{ln}\left(\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\right),$ where $\ensuremath{\gamma}$ is a constant equal to $\frac{1}{2}$. Weak impurity scattering $(\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\beta}}^{\ensuremath{-}1})$ leaves the low-energy spectral density unaffected but leads to an interaction-driven enhancement of superconductivity. Our results shed light on the role played by order-parameter fluctuations in providing the key missing link between Mott physics and strongly coupled toy models exhibiting gravity duals.

DOI
10.1103/physrevb.101.184506

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Physical review. B./Physical review. B•Dec 10, 2021•14 citations
Superconductivity results from an instability of the Fermi surface---contour of poles of the single-particle propagator---to an infinitesimally small attraction between electrons. Here, we instead discuss the analogous problem on a model Luttinger surface, or contour of zeros of the Green function. At zero temperature (\ensuremathβ\ensuremath→\ensuremath∞)(\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty})(\ensuremathβ\ensuremath→\ensuremath∞) and a critical interaction strength (uc\ensuremath∞)({u}_{c\ensuremath{\infty}})(uc\ensuremath∞​) characterized by the residue of self-energy pole, we find that the pair susceptibility diverges leading to a superconducting instability. We evaluate the pair fluctuation partition function and find that the spectral density in the normal state has an interaction-driven, power-law 1\ensuremathω\frac{1}{\sqrt{\ensuremath{\omega}}}\ensuremathω​1​ type, van-Hove singularity (vHS), indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit (\ensuremathβu\ensuremath≫1)(\ensuremath{\beta}u\ensuremath{\gg}1)(\ensuremathβu\ensuremath≫1), the leading-order fluctuation free-energy terms in the normal state of this NFL-SC transition resembles the equivalent [O(1)]\left[O(1)\right][O(1)] terms of the Sachdev-Ye-Kitaev (SYK) model. This free-energy contribution takes a simple form \ensuremath−\ensuremathβF=\ensuremathβuc\ensuremath∞\ensuremath−\ensuremathγln(\ensuremathβuc\ensuremath∞),\ensuremath{-}\ensuremath{\beta}F=\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\ensuremath{-}\ensuremath{\gamma}\phantom{\rule{0.16em}{0ex}}\text{ln}\left(\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\right),\ensuremath−\ensuremathβF=\ensuremathβuc\ensuremath∞​\ensuremath−\ensuremathγln(\ensuremathβuc\ensuremath∞​), where \ensuremathγ\ensuremath{\gamma}\ensuremathγ is a constant equal to 12\frac{1}{2}21​. Weak impurity scattering (\ensuremathτ\ensuremath≫\ensuremathβ\ensuremath−1)(\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\beta}}^{\ensuremath{-}1})(\ensuremathτ\ensuremath≫\ensuremathβ\ensuremath−1) leaves the low-energy spectral density unaffected but leads to an interaction-driven enhancement of superconductivity. Our results shed light on the role played by order-parameter fluctuations in providing the key missing link between Mott physics and strongly coupled toy models exhibiting gravity duals.
We show that fluctuation thermodynamics on a model Luttinger surface -- a contour of \textit{zeros} of the many-body Green function -- mimics black hole thermodynamics in the strong coupling limit. At zero temperature (β→∞\beta \rightarrow \inftyβ→∞) and a critical interaction strength (uc∞u_{c\infty}uc∞​) characterized by the self-energy pole, we find that the pair susceptibility diverges leading to a superconducting instability. We evaluate the pair fluctuation partition function and find that the spectral density in the normal state has an interaction-driven, power-law 1ω\frac{1}{\sqrt{\omega}}ω​1​ type, van-Hove singularity (vHS) indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit (βu≫1\beta u \gg 1βu≫1), the free energy in the normal state of this NFL-SC transition resembles well studied class of models with gravity duals and takes the form −βF=βuc∞−γ ln(βuc∞)-\beta F = \beta u_{c\infty} - \gamma~\text{ln}\left(\beta u_{c \infty}\right)−βF=βuc∞​−γ ln(βuc∞​) where γ\gammaγ is a constant equal to 12\frac{1}{2}21​. Weak impurity scattering (τ≫β−1\tau \gg \beta^{-1}τ≫β−1) leaves the low-energy spectral density unaffected, but leads to an interaction-driven enhancement of superconductivity. Our results shed light on the role played by order-parameter fluctuations in providing the key missing link between Mott physics and strongly coupled toy-models exhibiting gravity duals such as Sachdev-Ye-Kitaev (SYK)-type models.
The pairing of two electrons on a Fermi surface due to an infinitesimal attraction between them always results in a superconducting instability at zero temperature (T=0T=0T=0). The equivalent question of pairing instability on a Luttinger surface (LS)---190a contour of zeros of the propagator---instead leads to a quantum critical point (QCP) that separates a non-Fermi liquid and superconductor. A surprising and little understood aspect of pair fluctuations at this QCP is that their thermodynamics maps to that of the Sachdev-Ye-Kitaev (SYK) model in the strong coupling limit. Here, we offer a simple justification for this mapping by demonstrating that (i) LS models share the reparametrization symmetry of the $q\ensuremath{\rightarrow}\ensuremath{\infty}$ SYK model with qqq-body interactions close to the LS, and (ii) the enforcement of gauge invariance results in a $\frac{1}{\sqrt{\ensuremath{\tau}}}$ ($\ensuremath{\tau}\ensuremath{\sim}{T}^{\ensuremath{-}1}$) behavior of the fluctuation propagator near the QCP, as is the feature of SYK conformal Green functions, but leaves the overall form of the free energy map robust.
We investigate models of electrons in the Sachdev-Ye-Kitaev class with random and all-to-all electron hopping, electron spin exchange, and Cooper-pair hopping. An attractive on-site interaction between electrons leads to superconductivity at low temperatures. Depending on the relative strengths of the hopping and spin exchange, the normal state at the critical temperature is either a Fermi-liquid or a non-Fermi liquid. We present a large-MMM [where spin symmetry is enlarged to SU(M)]\mathrm{SU}(M)]SU(M)] study of the normal state to superconductor phase transition. We describe the transition temperature, the superconducting order parameter, and the electron spectral functions. We contrast between Fermi liquid and non-Fermi liquid normal states: we find that for weaker attractive on-site interaction there is a relative enhancement of Tc{T}_{c}Tc​ when the normal state is a non-Fermi liquid, and correspondingly a strong deviation from BCS limit. Also, the phase transition in this case becomes a first-order transition for strong non-Fermi liquids. On the other hand, for stronger on-site interaction, there is no appreciable difference in Tc{T}_{c}Tc​ between whether the superconductivity emerges from a Fermi liquid or a non-Fermi liquid. Notable features of superconductivity emerging from a non-Fermi liquid are that the superconducting electron spectral function is different from the Fermi-liquid case, with additional peaks at higher energies, and there is no Hebel-Slichter peak in the NMR relaxation rate in the non-Fermi liquid case.
We study a model of NNN fermions in a quantum dot, coupled to MMM bosons by a disorder-induced complex Yukawa coupling [Yukawa Sachdev-Ye-Kitaev (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/MN/MN/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 impuritylike 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/NM/NM/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=0T=0T=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 $V(\mathrm{\ensuremath{\Omega}})\ensuremath{\propto}1/{|\mathrm{\ensuremath{\Omega}}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model).
Characterizing the interplay between non-Fermi-liquid physics and superconductivity is an important open problem in condensed matter physics. From a theoretical standpoint, the challenge stems from the difficulty in identifying controlled models where these phenomena can be simultaneously treated on equal footing. Here, the authors propose a new model of strongly interacting electrons and phonons, in close analogy with the well-known Sachdev-Ye-Kitaev model, and, via exact solution, explore how superconductivity can emerge from quantum critical and highly incoherent non-Fermi-liquid normal states. Many of the results are in qualitative agreement with observations made in iron-based, heavy-fermion, and cuprate superconductors.
One of the fundamental questions about high-temperature cuprate superconductors is the size of the Fermi surface underlying the superconducting state. By analyzing the single-particle spectral function for the Fermi-Hubbard model as a function of repulsion UUU and chemical potential $\ensuremath{\mu}$, we find that the Fermi surface in the normal state undergoes a transition from a large Fermi surface matching the Luttinger volume as expected in a Fermi liquid, to a Fermi surface that encloses fewer electrons that we dub the ``Luttinger breaking'' phase, as the Mott insulator is approached. This transition into a non-Fermi-liquid phase that violates the Luttinger count occurs at a critical density in the absence of any other broken symmetry. We obtain the Fermi-surface contour from the spectral weight ${A}_{\mathbf{k}}(\ensuremath{\omega}=0)$ and from an analysis of the singularities of the Green's function ReGk(E=0)\mathrm{Re}\phantom{\rule{0.16em}{0ex}}{G}_{\mathbf{k}}(E=0)ReGk​(E=0), calculated using determinantal quantum Monte Carlo and analytic continuation methods. We discuss our numerical results in connection with experiments on Hall measurements, scanning tunneling spectroscopy, and angle-resolved photoemission spectroscopy.