Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2101.01171 · Physical review. B./Physical review. B

Superconductivity from Luttinger surfaces: Emergent Sachdev-Ye-Kitaev physics with infinite-body interactions

Chandan Setty

Published Jan 4, 2021• 17 citations
arXivHTMLPDF

Abstract

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 ($T=0$). 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 $q$-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.

DOI
10.1103/physrevb.103.014501

Related papers

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

Chandan Setty

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.

arXiv:1908.09831•Physical review. B./Physical review. B•May 13, 2020•19 citations

Pairing Instability on a Luttinger Surface: A Non-Fermi Liquid to Superconductor Transition and its Gravity Dual

Chandan Setty

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 ($\beta \rightarrow \infty$) and a critical interaction strength ($u_{c\infty}$) 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 $\frac{1}{\sqrt{\omega}}$ type, van-Hove singularity (vHS) indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit ($\beta u \gg 1$), 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 $-\beta F = \beta u_{c\infty} - \gamma~\text{ln}\left(\beta u_{c \infty}\right)$ where $\gamma$ is a constant equal to $\frac{1}{2}$. Weak impurity scattering ($\tau \gg \beta^{-1}$) 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.

•Aug 26, 2019•0 citations

Superconducting critical temperature in the extended diffusive Sachdev-Ye-Kitaev model

F. Salvati, A. Tagliacozzo

Models for strongly interacting fermions in disordered clusters forming an array, with electron hopping between sites, reproduce the linear dependence on temperature of the resistivity, typical of the strange metal phase of high temperature superconducting materials [extended Sachdev-Ye-Kitaev (SYK) models]. We identify the low energy collective excitations as neutral energy excitations, diffusing in the lattice of the thermalized non-Fermi liquid phase. However, the diffusion is heavily hindered by coupling to the pseudo-Goldstone modes of the conformal broken symmetry SYK phase, which are local in space. The imaginary time evolution of the extended model in the strong interaction and $1/N$ expansion limit is presented in the incoherent nonchaotic regime. On the other hand, a Fermi electronic liquid at low energy becomes marginal when perturbed by the SYK dots. A critical temperature for superconductivity is derived, which is not BCS-like, in case the collective excitations are assumed to mediate an attractive Cooper pairing.

arXiv:2101.05627•Physical Review Research•Aug 5, 2021•13 citations

Superconductivity of incoherent electrons in the Yukawa Sachdev-Ye-Kitaev model

Laura Classen, Andrey V. Chubukov

We study a model of $N$ fermions in a quantum dot, coupled to $M$ 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 $N/M$, 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 $M/N$. 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 $V(\mathrm{\ensuremath{\Omega}})\ensuremath{\propto}1/{|\mathrm{\ensuremath{\Omega}}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model).

arXiv:2106.12078•Physical review. B./Physical review. B•Sep 14, 2021•23 citations

Superconducting critical temperature in the extended SYK model

F. Salvati, A. Tagliacozzo

Models for strongly interacting fermions in disordered clusters forming an array, with electron hopping between sites, reproduce the linear dependence on temperature of the resistivity, typical of the strange metal phase of High Temperature Superconducting materials (Extended Sachdev-Ye-Kitaev (SYK) models). Our hydrodynamical approach to the marginal Fermi liquid emerging out of the interaction, identifies the low energy collective excitations of the system in its coherent phase. These neutral excitations diffuse in the lattice, but the diffusion is heavily hindered by coupling to the pseudo Goldstone modes of the conformal broken symmetry SYK phase, which are local in space. A critical temperature for superconductivity arises in the electron liquid,in case these excitations are assumed to mediate an attractive Cooper-pairing, in the electron liquid, which is not BCS-like.

•Jan 14, 2021•0 citations

Quantum Criticality and Superconductivity in Systems Without Quasiparticles

Li, Chenyuan

The discovery of high-temperature superconductors has raises questions that extend beyond the scope of quasiparticles, a foundational concept in condensed matter theory. This dissertation explores new theoretical approaches to quantum matter without quasiparticle excitations, with a particular focus on quantum criticality and the associated superconductivity in correlated electron compounds. The soluble Sachdev-Ye-Kitaev (SYK) model has recently emerged as a fascinating platform to address the unusual metallic states with T-linear resistivity. We first consider differ- ent variants of random Hubbard models, which share many similarities to key aspects of the original SYK model. Using a large M analysis and a renormalization group method, we propose the existence of quantum critical points with fractionalized excitations separating two distinct phases, which provides insights into the quantum phase transitions and non-Fermi liquid behaviors observed in cuprate superconductors. In the second part of this dissertation, we examine the interplay between non-Fermi liquids and superconductivity with two different pairing mechanisms. Under the framework of random Hubbard model, we demonstrate that superconductivity emerging from non-Fermi liquids strongly deviates from the BCS theory by tuning the relative strength of hopping and exchange interactions. Furthermore, we investigate a two-dimensional Yukawa-SYK model with spatially randomn interactions coupling a Fermi surface to a scalar field associated with a quantum phase transition. Our theory agrees well with transport properties analysed in cuprates, such as a T-linear resistivity and Planckian dissipation.

•Jan 1, 2024•0 citations
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→\ensuremath∞q\ensuremath{\rightarrow}\ensuremath{\infty}q\ensuremath→\ensuremath∞ SYK model with qqq-body interactions close to the LS, and (ii) the enforcement of gauge invariance results in a 1\ensuremathτ\frac{1}{\sqrt{\ensuremath{\tau}}}\ensuremathτ​1​ (\ensuremathτ\ensuremath∼T\ensuremath−1\ensuremath{\tau}\ensuremath{\sim}{T}^{\ensuremath{-}1}\ensuremathτ\ensuremath∼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.
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 [O(1)]\left[O(1)\right][O(1)] terms of the Sachdev-Ye-Kitaev (SYK) model. This free-energy contribution takes a simple form $\ensuremath{-}\ensuremath{\beta}F=\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\ensuremath{-}\ensuremath{\gamma}\phantom{\rule{0.16em}{0ex}}\text{ln}\left(\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\right),$ where $\ensuremath{\gamma}$ is a constant equal to 12\frac{1}{2}21​. 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.
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.
Models for strongly interacting fermions in disordered clusters forming an array, with electron hopping between sites, reproduce the linear dependence on temperature of the resistivity, typical of the strange metal phase of high temperature superconducting materials [extended Sachdev-Ye-Kitaev (SYK) models]. We identify the low energy collective excitations as neutral energy excitations, diffusing in the lattice of the thermalized non-Fermi liquid phase. However, the diffusion is heavily hindered by coupling to the pseudo-Goldstone modes of the conformal broken symmetry SYK phase, which are local in space. The imaginary time evolution of the extended model in the strong interaction and 1/N1/N1/N expansion limit is presented in the incoherent nonchaotic regime. On the other hand, a Fermi electronic liquid at low energy becomes marginal when perturbed by the SYK dots. A critical temperature for superconductivity is derived, which is not BCS-like, in case the collective excitations are assumed to mediate an attractive Cooper pairing.
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).
Models for strongly interacting fermions in disordered clusters forming an array, with electron hopping between sites, reproduce the linear dependence on temperature of the resistivity, typical of the strange metal phase of High Temperature Superconducting materials (Extended Sachdev-Ye-Kitaev (SYK) models). Our hydrodynamical approach to the marginal Fermi liquid emerging out of the interaction, identifies the low energy collective excitations of the system in its coherent phase. These neutral excitations diffuse in the lattice, but the diffusion is heavily hindered by coupling to the pseudo Goldstone modes of the conformal broken symmetry SYK phase, which are local in space. A critical temperature for superconductivity arises in the electron liquid,in case these excitations are assumed to mediate an attractive Cooper-pairing, in the electron liquid, which is not BCS-like.
The discovery of high-temperature superconductors has raises questions that extend beyond the scope of quasiparticles, a foundational concept in condensed matter theory. This dissertation explores new theoretical approaches to quantum matter without quasiparticle excitations, with a particular focus on quantum criticality and the associated superconductivity in correlated electron compounds. The soluble Sachdev-Ye-Kitaev (SYK) model has recently emerged as a fascinating platform to address the unusual metallic states with T-linear resistivity. We first consider differ- ent variants of random Hubbard models, which share many similarities to key aspects of the original SYK model. Using a large M analysis and a renormalization group method, we propose the existence of quantum critical points with fractionalized excitations separating two distinct phases, which provides insights into the quantum phase transitions and non-Fermi liquid behaviors observed in cuprate superconductors. In the second part of this dissertation, we examine the interplay between non-Fermi liquids and superconductivity with two different pairing mechanisms. Under the framework of random Hubbard model, we demonstrate that superconductivity emerging from non-Fermi liquids strongly deviates from the BCS theory by tuning the relative strength of hopping and exchange interactions. Furthermore, we investigate a two-dimensional Yukawa-SYK model with spatially randomn interactions coupling a Fermi surface to a scalar field associated with a quantum phase transition. Our theory agrees well with transport properties analysed in cuprates, such as a T-linear resistivity and Planckian dissipation.