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arXiv:2207.13983 · Physical review. D/Physical review. D.

Spectrum of a Gross-Neveu Yukawa model with flavor disorder in three dimensions

Shiroman Prakash

Published Mar 28, 2023• 12 citations
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Abstract

We show that a variant of the Gross-Neveu Yukawa model with disorder provides a real, nonsupersymmetric generalization of the Sachdev--Ye Kitaev (SYK) model to three dimensions. The model contains $M$ real scalar fields and $N$ Dirac (or Majorana) fermions, interacting via a Yukawa interaction with a local Gaussian random coupling in three dimensions. In the limit where $M$ and $N$ are both large, and the ratio $M/N$ is held fixed, the model defines a line of infrared fixed points parametrized by $M/N$, reducing to the Gross-Neveu vector model when $M/N=0$. When $M/N$ is nonzero, the model is dominated by melonic diagrams and gives rise to SYK-like physics. We compute the spectrum of single-trace operators in the theory, and find that it is real for all values of $M/N$.

DOI
10.1103/physrevd.107.066025

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We show that a variant of the Gross-Neveu Yukawa model with disorder provides a real, nonsupersymmetric generalization of the Sachdev--Ye Kitaev (SYK) model to three dimensions. The model contains MMM real scalar fields and NNN Dirac (or Majorana) fermions, interacting via a Yukawa interaction with a local Gaussian random coupling in three dimensions. In the limit where MMM and NNN are both large, and the ratio M/NM/NM/N is held fixed, the model defines a line of infrared fixed points parametrized by M/NM/NM/N, reducing to the Gross-Neveu vector model when M/N=0M/N=0M/N=0. When M/NM/NM/N is nonzero, the model is dominated by melonic diagrams and gives rise to SYK-like physics. We compute the spectrum of single-trace operators in the theory, and find that it is real for all values of M/NM/NM/N.
The SYK model: fermions with a q-body, Gaussian-random, all-to-all interaction, is the first of a fascinating new class of solvable large N models. We generalize SYK to include f flavors of fermions, each occupying N a sites and appearing with a q a order in the interaction. Like SYK, this entire class of models generically has an infrared fixed point. We compute the infrared dimensions of the fermions, and the spectrum of singlet bilinear operators. We show that there is always a dimension-two operator in the spectrum, which implies that, like in SYK, there is breaking of conformal invariance and maximal chaos in the infrared four-point function of the generalized model. After a disorder average, the generalized model has a global O(N 1) × O(N 2) × … × O(N f ) symmetry: a subgroup of the O(N) symmetry of SYK; thereby giving a richer spectrum. We also elucidate aspects of the large q limit and the OPE, and solve q = 2 SYK at finite N.
The Sachdev-Ye-Kitaev (SYK) model is a model of qqq interacting fermions whose large N limit is dominated by melonic graphs. In this review we first present a diagrammatic proof of that result by direct, combinatorial analysis of its Feynman graphs. Gross and Rosenhaus have then proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, these flavors can be seen as reminiscent of the colors used in random tensor theory. Applying modern tools from random tensors to such a colored SYK model, all leading and next-to-leading orders diagrams of the 2-point and 4-point functions in the large NNN expansion can be identified. We then study the effect of non-Gaussian average over the random couplings in a complex, colored version of the SYK model. Using a Polchinski-like equation and random tensor Gaussian universality, we show that the effect of this non-Gaussian averaging leads to a modification of the variance of the Gaussian distribution of couplings at leading order in NNN. We then derive the form of the effective action to all orders.
Abstract We investigate a model closely related to both the original Sachdev-Ye-Kitaev (SYK) model and the N \mathcal{N} N N = 1 supersymmetric SYK model. It consists of N real Majorana fermions and M auxiliary bosons with Yukawa interactions. We consider the large N and M limit and keep the ratio M/N fixed. The model has two branches characterized by the conformal dimensions of fields, which we compute as a function of the ratio M/N. One of the branches contains the supersymmetric saddle for M = N. As we take the limit M/N → ∞ both branches coincide and we obtain the same conformal dimensions as SYK. Furthermore, we determine the Lyapunov exponent of the model and find maximal chaos independent of M/N.
We investigate a model closely related to both the original Sachdev-Ye-Kitaev (SYK) model and the mathcalN=1\\mathcal{N}=1mathcalN=1 supersymmetric SYK model. It consists of NNN real Majorana fermions and MMM auxiliary bosons with Yukawa interactions. We consider the large NNN and MMM limit and keep the ratio M/NM/NM/N fixed. The model has two branches characterized by the conformal dimensions of fields, which we compute as a function of the ratio M/NM/NM/N. One of the branches contains the supersymmetric saddle for M=NM=NM=N. Furthermore, we determine the Lyapunov exponent of the model and find maximal chaos independent of M/NM/NM/N.
The SYK model proposed by Sachdev, Ye, and Kitaev consists of Majorana fermions that interact randomly four at a time. The model develops a dense spectrum above the ground state, due to which the model becomes nearly conformal. This suggests that a holographic dual may exist, which makes the SYK model interesting in the study of quantum gravity. It has been found that the SYK model is similar to large NNN tensor models: in both models, only the melonic diagrams survive in the large NNN limit. In this thesis, we explore the large NNN tensor model with O(N)3O(N)^3O(N)3 symmetry containing two flavors of Majorana fermions in the fundamental representation. Its quartic Hamiltonian depends on a real parameter βββ. We derive the kernels of the four point functions. With the spectra that we find from the kernels, we calculate the scaling dimensions of several types of conformal primaries. We also find a duality relation between two Hamiltonians of different values of βββ. This is not a perfect duality, because the normalization of energy scales with the transformation. Nevertheless, the ratios of the energies are the same, and the operator dimensions are preserved. In addition, we discover that for β>1β> 1β>1 or β<0β< 0β<0 the scaling dimensions of one of the conformal primaries become complex, rendering the model unstable.
We construct a helical generalization of the Sachdev-Ye-Kitaev (SYK) model in 1+11+11+1 dimensions, built from left- and right-moving Majorana fermions with local quartic interactions and random couplings in flavor-chirality space. These interactions organize into a symmetry-controlled hierarchy of quartic chirality sectors. At the most restrictive end of this hierarchy, symmetry forces the quartic structure into a density-density form, which admits an exact solution using bosonization, rendering the theory integrable. Once the full quartic helical interaction space is allowed, including purely chiral, chirality-balanced, and chirality-imbalanced sectors, this symmetry-protected integrable structure is lost. Nevertheless, the large-NNN infrared limit remains analytically tractable through short-distance selection rules and disorder averaging. Using conformal perturbation theory about the free fixed point, we show that the entire interaction space is marginally irrelevant, and the theory thus becomes free and integrable in the IR.