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arXiv:2407.20733 · Physical review. B./Physical review. B

Current correlations and conductivity in SYK-like systems: An analytical study

Rishabh Jha, Stefan K. Kehrein, Jan C. Louw

Published Jan 6, 2025• 2 citations
arXivHTMLPDF

Abstract

We present a functional-based approach to compute thermal expectation values for actions expressed in the $G\text{\ensuremath{-}}\mathrm{\ensuremath{\Sigma}}$ formalism, applicable to any time sequence ordering. Utilizing this framework, we analyze the linear response to an electric field in various Sachdev-Ye-Kitaev (SYK) chains. We consider the SYK chain where each dot is a complex $q/2$-body interacting SYK model, and we allow for $r/2$-body nearest-neighbor hopping where $r=\ensuremath{\kappa}q$. We find exact analytical expressions in the large-$q$ limit for conductivities across all temperatures at leading order in $1/q$ for three cases, namely, $\ensuremath{\kappa}={1/2,1,2}$. When $\ensuremath{\kappa}={1/2,1}$, we observe linear-in-temperature $T$ resistivities at low temperatures, indicative of strange metal behavior. Conversely, when $\ensuremath{\kappa}=2$, the resistivity diverges as a power law at low temperatures, namely, as $1/{T}^{2}$, resembling insulating behavior. As $T$ increases, there is a crossover to Fermi liquid behavior ($\ensuremath{\sim}{T}^{2}$) at the minimum resistivity. Beyond this, another crossover occurs to strange metal behavior ($\ensuremath{\sim}T$). In comparison to previous linear-in-$T$ results in the literature, we also show that the resistivity behavior exists even below the MIR bound, indicating a true strange metal instead of a bad metal. In particular, we find for the $\ensuremath{\kappa}=2$ case a smooth crossover from an insulating phase to a Fermi liquid behavior to a true strange metal and eventually becoming a bad metal as temperature increases. We extend and generalize previously known results on resistivities to all temperatures, do a comparative analysis across the three models where we highlight the universal features and invoke scaling arguments to create a physical picture out of our analyses. Remarkably, we find a universal maximum DC conductivity across all three models when the hopping coupling strength becomes large.

DOI
10.1103/physrevb.111.045111

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We present a functional-based approach to compute thermal expectation values for actions expressed in the G\ensuremath-\ensuremathΣG\text{\ensuremath{-}}\mathrm{\ensuremath{\Sigma}}G\ensuremath-\ensuremathΣ formalism, applicable to any time sequence ordering. Utilizing this framework, we analyze the linear response to an electric field in various Sachdev-Ye-Kitaev (SYK) chains. We consider the SYK chain where each dot is a complex q/2q/2q/2-body interacting SYK model, and we allow for r/2r/2r/2-body nearest-neighbor hopping where r=\ensuremathκqr=\ensuremath{\kappa}qr=\ensuremathκq. We find exact analytical expressions in the large-qqq limit for conductivities across all temperatures at leading order in 1/q1/q1/q for three cases, namely, \ensuremathκ=1/2,1,2\ensuremath{\kappa}={1/2,1,2}\ensuremathκ=1/2,1,2. When \ensuremathκ=1/2,1\ensuremath{\kappa}={1/2,1}\ensuremathκ=1/2,1, we observe linear-in-temperature TTT resistivities at low temperatures, indicative of strange metal behavior. Conversely, when \ensuremathκ=2\ensuremath{\kappa}=2\ensuremathκ=2, the resistivity diverges as a power law at low temperatures, namely, as 1/T21/{T}^{2}1/T2, resembling insulating behavior. As TTT increases, there is a crossover to Fermi liquid behavior (\ensuremath∼T2\ensuremath{\sim}{T}^{2}\ensuremath∼T2) at the minimum resistivity. Beyond this, another crossover occurs to strange metal behavior (\ensuremath∼T\ensuremath{\sim}T\ensuremath∼T). In comparison to previous linear-in-TTT results in the literature, we also show that the resistivity behavior exists even below the MIR bound, indicating a true strange metal instead of a bad metal. In particular, we find for the \ensuremathκ=2\ensuremath{\kappa}=2\ensuremathκ=2 case a smooth crossover from an insulating phase to a Fermi liquid behavior to a true strange metal and eventually becoming a bad metal as temperature increases. We extend and generalize previously known results on resistivities to all temperatures, do a comparative analysis across the three models where we highlight the universal features and invoke scaling arguments to create a physical picture out of our analyses. Remarkably, we find a universal maximum DC conductivity across all three models when the hopping coupling strength becomes large.
Prominent systems like the high-T_{c} cuprates and heavy fermions display intriguing features going beyond the quasiparticle description. The Sachdev-Ye-Kitaev (SYK) model describes a (0+1)D quantum cluster with random all-to-all four-fermion interactions among N fermion modes which becomes exactly solvable as N→∞, exhibiting a zero-dimensional non-Fermi-liquid with emergent conformal symmetry and complete absence of quasiparticles. Here we study a lattice of complex-fermion SYK dots with random intersite quadratic hopping. Combining the imaginary time path integral with real time path integral formulation, we obtain a heavy Fermi liquid to incoherent metal crossover in full detail, including thermodynamics, low temperature Landau quasiparticle interactions, and both electrical and thermal conductivity at all scales. We find linear in temperature resistivity in the incoherent regime, and a Lorentz ratio L≡(κρ/T) varies between two universal values as a function of temperature. Our work exemplifies an analytically controlled study of a strongly correlated metal.
The Sachdev-Ye-Kitaev (SYK) model describes a strange metal that shows peculiar non-Fermi-liquid properties without quasiparticles. It exhibits a maximally chaotic behavior characterized by out-of-time-ordered correlators (OTOCs), and is expected to be a holographic dual to black holes. While a faithful realization of the SYK model in condensed-matter systems may be involved, a striking similarity between the SYK model and the Hund-coupling-induced spin-freezing crossover in multiorbital Hubbard models has recently been pointed out. To further explore this connection, we study OTOCs for fermionic single-orbital and multiorbital Hubbard models, which are prototypical models for strongly correlated electrons in solids. We introduce an imaginary-time four-point correlation function with an appropriate time ordering, which by means of the spectral representation and the out-of-time-order fluctuation-dissipation theorem can be analytically continued to real-time OTOCs. Based on this approach, we numerically evaluate real-time OTOCs for Hubbard models in the thermodynamic limit, using the dynamical mean-field theory in combination with a numerically exact continuous-time Monte Carlo impurity solver. The results for the single-orbital model show that a certain spin-related OTOC captures local moment formation in the vicinity of the metal-insulator transition, while the self-energy does not show SYK-type non-Fermi-liquid behavior. On the other hand, for the two- and three-orbital models with nonzero Hund coupling we find that the OTOC exhibits a rapid damping at short times and an approximate power-law decay at longer times in the spin-freezing crossover regime characterized by fluctuating local moments and a non-Fermi-liquid self-energy $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})\ensuremath{\sim}\sqrt{\ensuremath{\omega}}$. These results are in a good agreement with the behavior of the SYK model, providing firm evidence for the close relation between the spin-freezing crossover physics of multiorbital Hubbard models and the SYK strange metal.
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.
"Strange metals" with resistivity depending linearly on temperature T down to low T have been a long-standing puzzle in condensed matter physics. Here, we consider a lattice model of itinerant spin-[Formula: see text] fermions interacting via onsite Hubbard interaction and random infinite-ranged spin-spin interaction. We show that the quantum critical point associated with the melting of the spin-glass phase by charge fluctuations displays non-Fermi liquid behavior, with local spin dynamics identical to that of the Sachdev-Ye-Kitaev family of models. This extends the quantum spin liquid dynamics previously established in the large-M limit of [Formula: see text] symmetric models to models with physical [Formula: see text] spin-[Formula: see text] electrons. Remarkably, the quantum critical regime also features a Planckian linear-T resistivity associated with a T-linear scattering rate and a frequency dependence of the electronic self-energy consistent with the marginal Fermi liquid phenomenology.
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.
This dissertation presents a comprehensive investigation of fermionic quantum matter into the intricate relationship between quantum chaos, thermalization, and their connection to gravitational duals, with particular emphasis on the Sachdev-Ye-Kitaev (SYK) model and its extensions. Through an integrated approach combining analytical solutions in the large-N limit with advanced numerical techniques, we explore fundamental questions in quantum many-body physics, including the interplay of chaos and ergodicity, universal transport in non-Fermi liquids, and the role of gravitational analogs in strongly correlated systems. We first demonstrate that coupled SYK lattices evade instantaneous thermalization despite weak charge fluctuations, governed by a discrete wave equation dependent solely on hopping strength, independent of on-site interactions. Thermodynamically, coupled SYK models exhibit critical exponents universal to van der Waals fluids and AdS black holes, yet diverge from the Hawking-Page transition at ultralow temperatures, revealing a continuum of first-order phase transitions. They simultaneously retain the quantum chaotic properties of both a single-dot SYK model and black holes. In extended SYK chains, we identify a universal DC conductivity bound across interaction regimes, identifying crossovers between insulating, Fermi liquid, strange metal, and bad metal phases with signatures of holographic insulator behavior. Quench dynamics in mixed SYK systems reveal rapid thermalization without prethermal plateaus, enabling direct observation of thermalization in closed quantum systems in the thermodynamic limit. To diagnose ergodicity-breaking, we introduce the Krylov variance, a probe of operator localization in Krylov space, validated through mass-deformed SYK and quantum East models. Finally, we uncover Page curve entanglement dynamics and temporal quantum phase transitions in entanglement Hamiltonians of interacting fermionic chains, where symmetry-sector crossings highlight limitations of perturbative calculations performed around initial times, such as in Hawking’s analysis of black hole evaporation. We provide a condensed matter analog of the holographic “island formula” — a mechanism invoking quantum extremal surfaces to resolve the black hole information paradox. Methodologically, this work integrates large-N-type expansions, Keldysh contour techniques, and numerical methods to bridge condensed matter physics, quantum information, and gravitational duality. The results establish SYK-type systems as a universal framework for studying non-Fermi liquids, quantum criticality, and black hole analogs, while paving the way for exploring bosonic extensions and generalizations to open quantum systems.