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arXiv:1604.07818 · Physical review. D/Physical review. D.

Remarks on the Sachdev-Ye-Kitaev model

JUAN M. MALDACENA, Douglas Stanford

Published Nov 4, 2016• 1,845 citations
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Abstract

The authors study in detail the quantum mechanical model of $N$ Majorana fermions with random interactions of a few fermions at a time (Sachdev-Ye-Kitaev model) in the large $N$ limit. At low energies, the system is strongly interacting and an emergent conformal symmetry develops. Performing technical calculations, the authors elucidate a number of properties of the model near the conformal point.

DOI
10.1103/physrevd.94.106002

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The authors study in detail the quantum mechanical model of NNN Majorana fermions with random interactions of a few fermions at a time (Sachdev-Ye-Kitaev model) in the large NNN limit. At low energies, the system is strongly interacting and an emergent conformal symmetry develops. Performing technical calculations, the authors elucidate a number of properties of the model near the conformal point.
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 discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) model. These are quantum mechanical models involving NNN Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The N=1\mathcal{N}=1N=1 model with a single supercharge has unbroken supersymmetry at large NNN, but nonperturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large NNN equation, and also by performing numerical computations for small values of NNN. We also compute the large NNN spectrum of ``singlet'' operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an N=2\mathcal{N}=2N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large NNN computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev model. These are quantum mechanical models involving NNN Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The N=1{\cal N}=1N=1 model with a single supercharge has unbroken supersymmetry at large NNN, but non-perturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large NNN equation, and also by performing numerical computations for small values of NNN. We also compute the large NNN spectrum of "singlet" operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an N=2{\cal N}=2N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large NNN computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
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 $\mathrm{\ensuremath{\Gamma}}$). 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 $\ensuremath{\xi}=eVJ/{\mathrm{\ensuremath{\Gamma}}}^{2}$. 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.
The Sachdev-Ye-Kitaev (SYK) model is a quantum-mechanical model of NNN Majorana fermions which displays a number of appealing features---solvability in the strong coupling regime, near-conformal invariance and maximal chaos---which make it a suitable model for black holes in the context of the AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT holography. In this paper, we show for the colored SYK model and several of its tensor model cousins that the next-to-leading order in the large-NNN expansion preserves the conformal invariance of the two-point function in the strong-coupling regime, up to the contribution of the pseudo-Goldstone bosons due to the explicit breaking of the symmetry which are already seen in the leading-order four-point function. We also comment on the composite field approach for computing correlation functions in colored tensor models.
We construct a sign-problem-free variant of the complex Sachdev-Ye-Kitaev (SYK) model which keeps all the essential properties of the SYK model, including the analytic solvability in the large-NNN limit and being maximally chaotic. In addition to the number of complex fermions NNN, our model has an additional parameter MMM controlling the number of terms in the Hamiltonian which we take $M\ensuremath{\rightarrow}\ensuremath{\infty}$ with keeping M/NM/NM/N constant in the large-NNN limit. While our model respects global U(1)\text{U}(1)U(1) symmetry associated with the fermion number conservation, both the large-NNN limit and the sign-problem-free nature become explicit in the Majorana representation. We present a detailed analysis on our model, i.e., the random matrix classification based on the symmetry analysis, analytic approach, and the quantum Monte Carlo simulations. All these analyses show that our model exhibits a non-Fermi-liquid physics, a gapless fermionic system lying beyond the conventional Fermi-liquid picture.