Quantum criticality plays a central role in understanding non-Fermi liquid behavior and unconventional superconductivity in strongly correlated systems. In this review, we explore the quantum critical Eliashberg theory, which extends conventional Eliashberg approaches to non-Fermi liquid regimes governed by critical fluctuations. We discuss the theoretical foundations and recent developments in the field, focusing on the interplay between electronic interactions and bosonic modes near quantum phase transitions as described in the Yukawa-coupled version of the Sachdev–Ye–Kitaev model. Special emphasis is placed on the breakdown of quasiparticle coherence, anomalous scaling behavior, Cooper pairing without quasiparticles, and emergent universality in different physical settings. Starting from a zero-dimensional quantum dot model, we discuss the generalization to higher spatial dimensions and demonstrate the connection between quantum critical Eliashberg theory and holographic superconductivity. Our analysis provides a perspective on how quantum criticality shapes the dynamics of strongly correlated metals and superconductors.Quantum criticality plays a central role in understanding non-Fermi liquid behavior and unconventional superconductivity in strongly correlated systems. In this review, we explore the quantum critical Eliashberg theory, which extends conventional Eliashberg approaches to non-Fermi liquid regimes governed by critical fluctuations. We discuss the theoretical foundations and recent developments in the field, focusing on the interplay between electronic interactions and bosonic modes near quantum phase transitions as described in the Yukawa-coupled version of the Sachdev–Ye–Kitaev model. Special emphasis is placed on the breakdown of quasiparticle coherence, anomalous scaling behavior, Cooper pairing without quasiparticles, and emergent universality in different physical settings. Starting from a zero-dimensional quantum dot model, we discuss the generalization to higher spatial dimensions and demonstrate the connection between quantum critical Eliashberg theory and holographic superconductivity. Our analysis provides a perspective on how quantum criticality shapes the dynamics of strongly correlated metals and superconductors.

Jan 1, 20260 citations

Quantum Critical Eliashberg Theory

Ilya Esterlis, Schmalian, Joerg

Quantum criticality plays a central role in understanding non-Fermi liquid behavior and unconventional superconductivity in strongly correlated systems. In this review, we explore the quantum critical Eliashberg theory, which extends conventional Eliashberg approaches to non-Fermi liquid regimes governed by critical fluctuations. We discuss the theoretical foundations and recent developments in the field, focusing on the interplay between electronic interactions and bosonic modes near quantum phase transitions as described in the Yukawa-coupled version of the Sachdev-Ye-Kitaev model. Special emphasis is placed on the breakdown of quasiparticle coherence, anomalous scaling behaviour, Cooper pairing without quasiparticles, and emergent universality in different physical settings. Starting from a zero-dimensional "quantum-dot" model, we discuss the generalization to higher spatial dimensions and demonstrate the connection between quantum-critical Eliashberg theory and holographic superconductivity. Our analysis provides a perspective on how quantum criticality shapes the dynamics of strongly correlated metals and superconductors.Quantum criticality plays a central role in understanding non-Fermi liquid behavior and unconventional superconductivity in strongly correlated systems. In this review, we explore the quantum critical Eliashberg theory, which extends conventional Eliashberg approaches to non-Fermi liquid regimes governed by critical fluctuations. We discuss the theoretical foundations and recent developments in the field, focusing on the interplay between electronic interactions and bosonic modes near quantum phase transitions as described in the Yukawa-coupled version of the Sachdev-Ye-Kitaev model. Special emphasis is placed on the breakdown of quasiparticle coherence, anomalous scaling behaviour, Cooper pairing without quasiparticles, and emergent universality in different physical settings. Starting from a zero-dimensional "quantum-dot" model, we discuss the generalization to higher spatial dimensions and demonstrate the connection between quantum-critical Eliashberg theory and holographic superconductivity. Our analysis provides a perspective on how quantum criticality shapes the dynamics of strongly correlated metals and superconductors.

arXiv:2506.11952Jun 13, 20250 citations

Quantum critical Eliashberg theory, the Sachdev-Ye-Kitaev superconductor and their holographic duals

Gian Andrea Inkof, Koenraad Schalm, Jörg Schmalian

Abstract Superconductivity is abundant near quantum critical points, where fluctuations suppress the formation of Fermi liquid quasiparticles and the BCS theory no longer applies. Two very distinct approaches have been developed to address this issue: quantum-critical Eliashberg theory and holographic superconductivity. The former includes a strongly retarded pairing interaction of ill-defined fermions, the latter is rooted in the duality of quantum field theory and gravity theory. We demonstrate that both are different perspectives of the same theory. We derive holographic superconductivity in form of a gravity theory with emergent space-time from a quantum many-body Hamiltonian—the Yukawa Sachdev-Ye-Kitaev model—where the Eliashberg formalism is exact. Exploiting the power of holography, we then determine the dynamic pairing susceptibility of the model. Our holographic map comes with the potential to use quantum gravity corrections to go beyond the Eliashberg regime.Abstract Superconductivity is abundant near quantum critical points, where fluctuations suppress the formation of Fermi liquid quasiparticles and the BCS theory no longer applies. Two very distinct approaches have been developed to address this issue: quantum-critical Eliashberg theory and holographic superconductivity. The former includes a strongly retarded pairing interaction of ill-defined fermions, the latter is rooted in the duality of quantum field theory and gravity theory. We demonstrate that both are different perspectives of the same theory. We derive holographic superconductivity in form of a gravity theory with emergent space-time from a quantum many-body Hamiltonian—the Yukawa Sachdev-Ye-Kitaev model—where the Eliashberg formalism is exact. Exploiting the power of holography, we then determine the dynamic pairing susceptibility of the model. Our holographic map comes with the potential to use quantum gravity corrections to go beyond the Eliashberg regime.

arXiv:2108.11392npj Quantum MaterialsJun 1, 202235 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.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, 20240 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).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:2106.12078Physical review. B./Physical review. BSep 14, 202123 citations

Sachdev-Ye-Kitaev Models and Beyond: A Window into Non-Fermi Liquids

Debanjan Chowdhury, Antoine Georges, Olivier Parcollet +1 more

We present a review of the Sachdev-Ye-Kitaev (SYK) model of compressible quantum many-body systems without quasiparticle excitations, and its connections to various theoretical studies of non-Fermi liquids in condensed matter physics. The review is placed in the context of numerous experimental observations on correlated electron materials. Strong correlations in metals are often associated with their proximity to a Mott transition to an insulator created by the local Coulomb repulsion between the electrons. We explore the phase diagrams of a number of models of such local electronic correlation, employing a dynamical mean field theory in the presence of random spin exchange interactions. Numerical analyses and analytical solutions, using renormalization group methods and expansions in large spin degeneracy, lead to critical regions which display SYK physics. The models studied include the single-band Hubbard model, the $t$ - $J$ model and the two-band Kondo-Heisenberg model in the presence of random spin exchange interactions. We also examine non-Fermi liquids obtained by considering each SYK model with random four-fermion interactions to be a multi-orbital atom, with the SYK-atoms arranged in an infinite lattice. We connect to theories of sharp Fermi surfaces without any low-energy quasiparticles in the absence of spatial disorder, obtained by coupling a Fermi liquid to a gapless boson; a systematic large $N$ theory of such a critical Fermi surface, with SYK characteristics, is obtained by averaging over an ensemble of theories with random boson-fermion couplings. Finally, we present an overview of the links between the SYK model and quantum gravity and end with an outlook on open questions.We present a review of the Sachdev-Ye-Kitaev (SYK) model of compressible quantum many-body systems without quasiparticle excitations, and its connections to various theoretical studies of non-Fermi liquids in condensed matter physics. The review is placed in the context of numerous experimental observations on correlated electron materials. Strong correlations in metals are often associated with their proximity to a Mott transition to an insulator created by the local Coulomb repulsion between the electrons. We explore the phase diagrams of a number of models of such local electronic correlation, employing a dynamical mean field theory in the presence of random spin exchange interactions. Numerical analyses and analytical solutions, using renormalization group methods and expansions in large spin degeneracy, lead to critical regions which display SYK physics. The models studied include the single-band Hubbard model, the $t$-$J$ model and the two-band Kondo-Heisenberg model in the presence of random spin exchange interactions. We also examine non-Fermi liquids obtained by considering each SYK model with random four-fermion interactions to be a multi-orbital atom, with the SYK-atoms arranged in an infinite lattice. We connect to theories of sharp Fermi surfaces without any low-energy quasiparticles in the absence of spatial disorder, obtained by coupling a Fermi liquid to a gapless boson; a systematic large $N$ theory of such a critical Fermi surface, with SYK characteristics, is obtained by averaging over an ensemble of theories with random boson-fermion couplings. Finally, we present an overview of the links between the SYK model and quantum gravity and end with an outlook on open questions.

arXiv:2109.05037Sep 10, 202114 citations

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).

t

J

model and the two-band Kondo-Heisenberg model in the presence of random spin exchange interactions. We also examine non-Fermi liquids obtained by considering each SYK model with random four-fermion interactions to be a multi-orbital atom, with the SYK-atoms arranged in an infinite lattice. We connect to theories of sharp Fermi surfaces without any low-energy quasiparticles in the absence of spatial disorder, obtained by coupling a Fermi liquid to a gapless boson; a systematic large

N

theory of such a critical Fermi surface, with SYK characteristics, is obtained by averaging over an ensemble of theories with random boson-fermion couplings. Finally, we present an overview of the links between the SYK model and quantum gravity and end with an outlook on open questions.