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arXiv:1807.09099 · Physical review. B./Physical review. B

Low-high voltage duality in tunneling spectroscopy of the Sachdev-Ye-Kitaev model

N. V. Gnezdilov, Jimmy A. Hutasoit, C. W. J. Beenakker

Published Aug 31, 2018• 33 citations
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Abstract

The Sachdev-Ye-Kitaev (SYK) model describes a strongly correlated metal with all-to-all random interactions (average strength $J$) between $N$ fermions (complex Dirac fermions or real Majorana fermions). In the large-$N$ limit a conformal symmetry emerges that renders the model exactly soluble. Here we study how the non-Fermi-liquid behavior of the closed system in equilibrium manifests itself in an open system out of equilibrium. We calculate the current-voltage characteristic of a quantum dot, described by the complex-valued SYK model, coupled to a voltage source via a single-channel metallic lead (coupling strength $\mathrm{\ensuremath{\Gamma}}$). A one-parameter scaling law appears in the large-$N$ conformal regime, where the differential conductance $G=dI/dV$ depends on the applied voltage only through the dimensionless combination $\ensuremath{\xi}=eVJ/{\mathrm{\ensuremath{\Gamma}}}^{2}$. Low and high voltages are related by the duality $G(\ensuremath{\xi})=G(\ensuremath{\pi}/\ensuremath{\xi})$. This provides for an unambiguous signature of the conformal symmetry in tunneling spectroscopy.

DOI
10.1103/physrevb.98.081413

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The Sachdev-Ye-Kitaev (SYK) model describes a strongly correlated metal with all-to-all random interactions (average strength JJJ) between NNN fermions (complex Dirac fermions or real Majorana fermions). In the large-NNN limit a conformal symmetry emerges that renders the model exactly soluble. Here we study how the non-Fermi-liquid behavior of the closed system in equilibrium manifests itself in an open system out of equilibrium. We calculate the current-voltage characteristic of a quantum dot, described by the complex-valued SYK model, coupled to a voltage source via a single-channel metallic lead (coupling strength \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ). A one-parameter scaling law appears in the large-NNN conformal regime, where the differential conductance G=dI/dVG=dI/dVG=dI/dV depends on the applied voltage only through the dimensionless combination \ensuremathξ=eVJ/\ensuremathΓ2\ensuremath{\xi}=eVJ/{\mathrm{\ensuremath{\Gamma}}}^{2}\ensuremathξ=eVJ/\ensuremathΓ2. Low and high voltages are related by the duality G(\ensuremathξ)=G(\ensuremathπ/\ensuremathξ)G(\ensuremath{\xi})=G(\ensuremath{\pi}/\ensuremath{\xi})G(\ensuremathξ)=G(\ensuremathπ/\ensuremathξ). This provides for an unambiguous signature of the conformal symmetry in tunneling spectroscopy.
We consider a recent proposal for a physical realization of the Sachdev-Ye-Kitaev (SYK) model in the zeroth-Landau-level sector of an irregularly shaped graphene flake. We study in detail charge transport signatures of the unique non-Fermi-liquid state of such a quantum dot coupled to noninteracting leads. The properties of this setup depend essentially on the ratio ppp between the number of transverse modes in the lead MMM and the number of the fermion degrees of freedom NNN on the SYK dot. This ratio can be tuned via the magnetic field applied to the dot. Our proposed setup gives access to the nontrivial conformal-invariant regime associated with the SYK model as well as a more conventional Fermi-liquid regime via tuning the field. The dimensionless linear-response conductance acquires distinct p\sqrt{p}p​ and 1/p1/\sqrt{p}1/p​ dependencies for the two phases, respectively, in the low-temperature limit, with a universal jump at the transition. We find that corrections scale linearly and quadratically in either temperature or frequency on the two sides of the transition. In the weak-tunneling regime, we find differential conductance proportional to the inverse square root of the applied voltage bias UUU. This dependence is replaced by a conventional Ohmic behavior with constant conductance proportional to 1/T1/\sqrt{T}1/T​ for bias energy eUeUeU smaller than temperature scale kBT{k}_{B}TkB​T. We also describe the out-of-equilibrium current-bias characteristics and discuss various crossovers between the limiting behaviors mentioned above.
The Sachdev-Ye-Kitaev (SYK) model describes interacting fermionic zero modes in zero spatial dimensions, e.g. quantum dot, with interactions strong enough to completely washout quasiparticle excitations in the infrared. In this note, we consider the complex-valued SYK model at temperature TTT coupled to a zero temperature reservoir by a quench. We find out that the tunneling current dynamics reveals a way to distinguish the SYK non-Fermi liquid (nFL) initial state of the subsystem from the disordered Fermi liquid. Temperature dependent contribution to the current's half-life scales linearly in TTT at low temperatures for the SYK nFl state, while for the Fermi liquid it scales as T2T^2T2. This provides a characteristic signature of the SYK non-Fermi liquid in a non-equilibrium measurement.
The Sachdev-Ye-Kitaev (SYK) model describes interacting fermionic zero modes in zero spatial dimensions, e.g., quantum dot, with interactions strong enough to completely washout quasiparticle excitations in the infrared. In this paper we consider the complex-valued SYK model at initial temperature TTT and chemical potential $\ensuremath{\mu}$ coupled to a large reservoir by a quench at time t=0t=0t=0. The reservoir is kept at zero temperature and charge neutrality. We find that the dynamics of the discharging process of the SYK quantum dot reveals a distinctive characteristic of the SYK non-Fermi liquid (nFl) state. In particular, we focus on the tunneling current induced by the quench. We show that the temperature dependent contribution to the current's half-life scales linearly in TTT at low temperatures for the SYK nFl state, while for the Fermi liquid it scales as T2{T}^{2}T2.
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 SYK model of Sachdev, Ye and Kitaev is a recent model of N Majorana fermions coupled via quartic random interaction. It has the extraordinary properties of being exactly solvable at large N , having emergent conformal symmetry in the IR, and exhibiting maximal quantum chaos. The confluence of these properties make SYK a highly attractive candidate for a solvable AdS/CFT correspondence. What’s more, it was found to capture elements of non-Fermi-liquid behavior in strange metals. This thesis introduces the SYK model and discusses a recent variant, the SYK∗ model of Marcus and Vandoren. In the latter, the effective action is derived and branches of conformal solutions are discussed. The parameter space of SYK∗ is investigated for large and small M . The former reveals a phase separation and a N critical point in the T−M plane is claimed to exist.
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).