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arXiv:1707.09352 · Journal of High Energy Physics

Notes on melonic O(N)q−1 tensor models

Sayantan Choudhury, Anshuman Dey, Indranil Halder, Lavneet Janagal, Shiraz N. Minwalla, Rohan R. Poojary

Published Jun 1, 2018• 92 citations
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Abstract

Abstract It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group O(N) q−1 agrees with the large N limit of the SYK model. In these notes we investigate aspects of the dynamics of the O(N) q−1 theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has $$ \left(q-1\right)\frac{N^2}{2} $$ q − 1 N 2 2 new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like E ln E (i.e. faster than Hagedorn) up to energies of order N 2. The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order 1/ln N. We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in J/m.

DOI
10.1007/jhep06(2018)094

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Abstract It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group O(N) q−1 agrees with the large N limit of the SYK model. In these notes we investigate aspects of the dynamics of the O(N) q−1 theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has (q−1)N22\left(q-1\right)\frac{N^2}{2}(q−1)2N2​ q − 1 N 2 2 new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like E ln E (i.e. faster than Hagedorn) up to energies of order N 2. The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order 1/ln N. We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in J/m.
We study the operators in the large NNN tensor models, focusing mostly on the fermionic quantum mechanics with O(N)3O(N{)}^{3}O(N)3 symmetry which may be either global or gauged. In the model with global symmetry, we study the spectra of bilinear operators, which are in either the symmetric traceless or the antisymmetric representation of one of the O(N)O(N)O(N) groups. In the symmetric traceless case, the spectrum of scaling dimensions is the same as in the Sachdev-Ye-Kitaev (SYK) model with real fermions; it includes the h=2h=2h=2 zero mode. For the operators antisymmetric in the two indices, the scaling dimensions are the same as in the additional sector found in the complex tensor and SYK models; the lowest h=0h=0h=0 eigenvalue corresponds to the conserved O(N)O(N)O(N) charges. A class of singlet operators may be constructed from contracted combinations of mmm symmetric traceless or antisymmetric two-particle operators. Their two-point functions receive contributions from mmm melonic ladders. Such multiple ladders are a new phenomenon in the tensor model, which does not seem to be present in the SYK model. The more typical 2k2k2k-particle operators do not receive any ladder corrections and have quantized large NNN scaling dimensions k/2k/2k/2. We construct pictorial representations of various singlet operators with low kkk. For larger kkk, we use available techniques to count the operators and show that their number grows as 2kk!{2}^{k}k!2kk!. As a consequence, the theory has a Hagedorn phase transition at the temperature which approaches zero in the large NNN limit. We also study the large NNN spectrum of low-lying operators in the Gurau-Witten model, which has O(N)6O(N{)}^{6}O(N)6 symmetry. We argue that it corresponds to one of the generalized SYK models constructed by Gross and Rosenhaus. Our paper also includes studies of the invariants in large NNN tensor integrals with various symmetries.
Abstract We explore computationally tractable deformations of the SYK model. The deformed theories are described by the sum of two SYK Hamiltonians with differing numbers, q and q∼ \overset{\sim }{q} q∼​ q ~ , of interacting fermions. In the large N limit, employing analytic and numerical tools, we compute finite temperature correlation functions and thermodynamic quantities. We identify a novel analytically solvable model in the large q limit. We find that, under certain circumstances, the thermal RG flow in the strongly coupled infrared phase exhibits two regions of linear-in-temperature entropy, which we interpret in terms of Schwarzian actions. Using conformal perturbation theory we compute the leading relevant correction away from the intermediate near-conformal fixed point. Holographic spacetimes in two spacetime dimensions that reproduce the thermodynamics of the microphysical theory are discussed. These are flow geometries that interpolate between two Euclidean near-AdS2 spacetimes with different radii. The Schwarzian soft mode corresponding to the AdS2 region in the deep interior resides entirely within the geometric regime.
We contrast some aspects of various SYK-like models with large-N melonic behavior. First, we note that ungauged tensor models can exhibit symmetry breaking, even though these are 0+1 dimensional theories. Related to this, we show that when gauged, some of them admit no singlets, and are anomalous. The uncolored Majorana tensor model with even N is a simple case where gauge singlets can exist in the spectrum. We outline a strategy for solving for the singlet spectrum, taking advantage of the results in arXiv: 1706.05364, and reproduce the singlet states expected in N = 2. In the second part of the paper, we contrast the random matrix aspects of some ungauged tensor models, the original SYK model, and a model due to Gross and Rosenhaus. The latter, even though disorder averaged, shows parallels with the Gurau-Witten model. In particular, the two models fall into identical Andreev ensembles as a function of N. In an appendix, we contrast the (expected) spectra of AdS 2 quantum gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta function.
In this thesis, we discuss the Sachdev-Ye-Kitaev (SYK) model and tensor models with similar properties. The SYK model is a quantum field theoretical model describing N interacting fermions, whose coupling constants are drawn from a Gaussian ensemble. Noteworthy properties of the SYK model include that it is analytically solvable in the large N limit, that it exhibits conformal symmetry at low energies and that it is maximally chaotic. These properties are remarkably similar to those of a 1 + 1 dimensional Schwarzschild black hole. It has been conjectured the SYK model is a holographic dual to the black hole. We introduce a set of Feynman rules for the SYK model. Using these rules, we show that in the large N limit the diagrams that contribute to the two-point function are all so-called iterated melonic diagrams. This allows us to derive a Schwinger-Dyson equation for the two-point function, which, in turn, can be solved exactly in the infrared limit. We also consider the four-point function. In the large N limit, the leading-order correction to the four-point function is given by so-called ladder diagrams. This allows us to derive an explicit expression for the four-point function. The SYK model can be generalized in a few different ways. In this thesis, we consider the generalization where the fermions act through q-fold interactions instead of quartic interactions present in the original SYK model. In particular, considerable simplifications can be achieved in the q → ∞ limit or q = 2 case, which we study. While the SYK model has many interesting properties, its random couplings limit its usability especially as a dual to a Schwarzschild black hole. We therefore also consider tensor models which do not have this drawback but manage to preserve the interesting properties of the SYK model. In the last chapter, we briefly inspect the chaotic behaviour of the SYK and tensor models and derive Lyapunov exponent for them. It can be shown that the expression saturates an upper bound for Lyapunov exponents of a large class of quantum systems, including large N systems.
We contrast some aspects of various SYK-like models with large-N melonic behavior. First, we note that ungauged tensor models can exhibit symmetry breaking, even though these are 0+1 dimensional theories. Related to this, we show that when gauged, some of them admit no singlets, and are anomalous. The uncolored Majorana tensor model with even N is a simple case where gauge singlets can exist in the spectrum. We outline a strategy for solving for the singlet spectrum, taking advantage of the results in arXiv:1706.05364, and reproduce the singlet states expected in N = 2. In the second part of the paper, we contrast the random matrix aspects of some ungauged tensor models, the original SYK model, and a model due to Gross and Rosenhaus. The latter, even though disorder averaged, shows parallels with the Gurau-Witten model. In particular, the two models fall into identical Andreev ensembles as a function of N . In an appendix, we contrast the (expected) spectra of AdS2 quantum gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta function.
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.