arXiv subjects

Grigory Tarnopolsky

Publications and source records attributed to Grigory Tarnopolsky.

At least 37 records · Page 2Linked to original sources

arXiv2019

Prismatic Large $N$ Models for Bosonic Tensors

We study the $O(N)^3$ symmetric quantum field theory of a bosonic tensor $ϕ^{abc}$ with sextic interactions. Its large $N$ limit is dominated by a positive-definite operator, whose index structure has the topology of a prism. We present a large $N$ solution of the model using Schwinger-Dyson equations to sum the leading diagrams, finding that for $2.81 < d < 3$ and for $d<1.68$ the spectrum of bilinear operators has no complex scaling dimensions. We also develop perturbation theory in $3-ε$ dimensions including eight $O(N)^3$ invariant operators necessary for the renormalizability. For sufficiently large $N$, we find a "prismatic" fixed point of the renormalization group, where all eight coupling constants are real. The large $N$ limit of the resulting $ε$ expansions of various operator dimensions agrees with the Schwinger-Dyson equations. Furthermore, the $ε$ expansion allows us to calculate the $1/N$ corrections to operator dimensions. The prismatic fixed point in $3-ε$ dimensions survives down to $N\approx 53.65$, where it merges with another fixed point and becomes complex. We also discuss the $d=1$ model where our approach gives a slightly negative scaling dimension for $ϕ$, while the spectrum of bilinear operators is free of complex dimensions.

hep-th↗

arXiv2019

Magic Angle Hierarchy in Twisted Graphene Multilayers

When two monolayers of graphene are stacked with a small relative twist angle, the resulting band structure exhibits a remarkably flat pair of bands at a sequence of 'magic angles' where correlation effects can induce a host of exotic phases. Here, we study a class of related models of $n$-layered graphene with alternating relative twist angle $\pm θ$ which exhibit magic angle flat bands coexisting with several Dirac dispersing bands at the Moiré K point. Remarkably, we find that the Hamiltonian for the multilayer system can be mapped exactly to a set of decoupled bilayers at {\it different} angles, revealing a remarkable hierarchy mathematically relating all these magic angles to the TBG case. For the trilayer case ($n = 3$), we show that the sequence of magic angle is obtained by multiplying the bilayer magic angles by $\sqrt{2}$, whereas the quadrilayer case ($n = 4$) has two sequences of magic angles obtained by multiplying the bilayer magic angles by the golden ratio $φ= (\sqrt{5} + 1)/2 \approx 1.62$ and its inverse. We also show that for larger $n$, we can tune the angle to obtain several narrow (almost flat) bands simultaneously and that for $n \rightarrow \infty$, there is a continuum of magic angles for $θ\lesssim 2^o$. Furthermore, we show that tuning several perfectly flat bands for a small number of layers is possible if the coupling between different layers is different. The setup proposed here can be readily achieved by repeatedly applying the "tear and stack" method without the need of any extra tuning of the twist angle and has the advantage that the first magic angle is always larger than the bilayer case.

cond-mat.str-el↗

arXiv2019

Symmetry Breaking in Coupled SYK or Tensor Models

We study a large $N$ tensor model with $O(N)^3$ symmetry containing two flavors of Majorana fermions, $ψ_1^{abc}$ and $ψ_2^{abc}$. We also study its random counterpart consisting of two coupled Sachdev-Ye-Kitaev models, each one containing $N_{\rm SYK}$ Majorana fermions. In these models we assume tetrahedral quartic Hamiltonians which depend on a real coupling parameter $α$. We find a duality relation between two Hamiltonians with different values of $α$, which allows us to restrict the model to the range of $-1\leq α\leq 1/3$. The scaling dimension of the fermion number operator $Q=iψ_1^{abc} ψ_2^{abc}$ is complex and of the form $1/2 +i f(α)$ in the range $-1\leq α<0$, indicating an instability of the conformal phase. Using Schwinger-Dyson equations to solve for the Green functions, we show that in the true low-temperature phase this operator acquires an expectation value. This demonstrates the breaking of an anti-unitary particle-hole symmetry and other discrete symmetries. We also calculate spectra of the coupled SYK models for values of $N_{\rm SYK}$ where exact diagonalizations are possible. For negative $α$ we find a gap separating the two lowest energy states from the rest of the spectrum; this leads to exponential decay of the zero-temperature correlation functions. For $N_{\rm SYK}$ divisible by $4$, the two lowest states have a small splitting. They become degenerate in the large $N_{\rm SYK}$ limit, as expected from the spontaneous breaking of a $\mathbb{Z}_2$ symmetry.

hep-th↗

arXiv2019

Gauge theory for the cuprates near optimal doping

We describe the phase diagram of a 2+1 dimensional SU(2) gauge theory of fluctuating incommensurate spin density waves for the hole-doped cuprates. Our primary assumption is that all low energy fermionic excitations are gauge neutral and electron-like, while the spin density wave order is fractionalized into Higgs fields transforming as adjoints of the gauge SU(2). The confining phase of the gauge theory is a conventional Fermi liquid with a large Fermi surface (and its associated $d$-wave superconductor). There is a quantum phase transition to a Higgs phase describing the `pseudogap' at lower doping. Depending upon the quartic terms in the Higgs potential, the Higgs phase exhibits one or more of charge density wave, Ising-nematic, time-reversal odd scalar spin chirality, and $\mathbb{Z}_2$ topological orders. It is notable that the emergent broken symmetries in our theory of fluctuating spin density waves co-incide with those observed in diverse experiments. For the electron-doped cuprates, the spin density wave fluctuations are at wavevector $(π,π)$, and then the corresponding SU(2) gauge theory only has a crossover between the confining and Higgs regimes, with an exponentially large confinement scale deep in the Higgs regime. On the Higgs side, for both the electron- and hole-doped cases, and at scales shorter than the confinement scale (which can be infinite when $\mathbb{Z}_2$ topological order is present), the electron spectral function has a `fractionalized Fermi liquid (FL*)' form with small Fermi surfaces. We also describe the deconfined quantum criticality of the Higgs transition in the limit of a large number of Higgs flavors, and perturbatively discuss its coupling to fermionic excitations.

cond-mat.str-el↗

arXiv2019

Spectrum of Majorana Quantum Mechanics with $O(4)^3$ Symmetry

We study the quantum mechanics of 3-index Majorana fermions $ψ^{abc}$ governed by a quartic Hamiltonian with $O(N)^3$ symmetry. Similarly to the Sachdev-Ye-Kitaev model, this tensor model has a solvable large $N$ limit dominated by the melonic diagrams. For $N=4$ the total number of states is $2^{32}$, but they naturally break up into distinct sectors according to the charges under the $U(1)\times U(1)$ Cartan subgroup of one of the $O(4)$ groups. The biggest sector has vanishing charges and contains over $165$ million states. Using a Lanczos algorithm, we determine the spectrum of the low-lying states in this and other sectors. We find that the absolute ground state is non-degenerate. If the $SO(4)^3$ symmetry is gauged, it is known from earlier work that the model has $36$ states and a residual discrete symmetry. We study the discrete symmetry group in detail; it gives rise to degeneracies of some of the gauge singlet energies. We find all the gauge singlet energies numerically and use the results to propose exact analytic expressions for them.

hep-th↗

arXiv2018

TASI Lectures on Large $N$ Tensor Models

The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large $N$ limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large $N$ limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of $O(N)$, $O(N)^2$, or $O(N)^3$ symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large $N$ limits. In the latter example we review how the dominance of melonic diagrams follows when a special "tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large $N$ limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with $O(N)^3$ symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with $SU(N)^2\times O(N)\times U(1)$ symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large $N$ tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.

hep-th↗

arXiv2018

Origin of Magic Angles in Twisted Bilayer Graphene

Twisted Bilayer graphene (TBG) is known to feature isolated and relatively flat bands near charge neutrality, when tuned to special magic angles. However, different criteria for the magic angle such as the vanishing of Dirac speed, minimal bandwidth or maximal band gap to higher bands typically give different results. Here we study a modified continuum model for TBG which has an infinite sequence of magic angles $θ$ at which, we simultaneously find that (i) the Dirac speed vanishes (ii) absolutely flat bands appear at neutrality and (iii) bandgaps to the excited bands are maximized. When parameterized in terms of $α\sim 1/θ, $ they recur with the simple periodicity of $Δα\simeq 3/2$, which, beyond the first magic angle, differs from earlier calculations. Further, in this model we prove that the vanishing of the Dirac velocity ensures the exact flatness of the band and show that the flat band wave functions are related to doubly-periodic functions composed of ratios of theta functions. Also, using perturbation theory up to $α^8$ we capture important features of the first magic angle $θ\approx1.09^{\circ}$ ($α\approx 0.586$), which precisely explains the numerical results. Finally, based on our model we discuss the prospects for observing the second magic angle in TBG.

cond-mat.str-el↗

arXiv2018

Spectra of Eigenstates in Fermionic Tensor Quantum Mechanics

We study the $O(N_1)\times O(N_2)\times O(N_3)$ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks $N_i$ are all equal, this model has a large $N$ limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of $SO(N_1)\times SO(N_2)\times SO(N_3)$ invariant states for any set of $N_i$. For equal ranks the number of singlets is non-vanishing only when $N$ is even, and it exhibits rapid growth: it jumps from $36$ in the $O(4)^3$ model to $595354780$ in the $O(6)^3$ model. We derive bounds on the values of energy, which show that they scale at most as $N^3$ in the large $N$ limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order $1/N$. For $N_3=1$ the tensor model reduces to $O(N_1)\times O(N_2)$ fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with $SU(N_1)\times SU(N_2)\times U(1)$ symmetry. Finally, we study the $N_3=2$ case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only $O(N_1)\times O(N_2)\times U(1)$. All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard 't Hooft large $N$ limits where the ground state energies are of order $N^2$, while the energy gaps are of order $1$.

hep-th↗

arXiv2018

$Z_2$ fractionalized phases of a solvable, disordered, $t$-$J$ model

We describe the phases of a solvable $t$-$J$ model of electrons with infinite-range, and random, hopping and exchange interactions, similar to those in the Sachdev-Ye-Kitaev models. The electron fractionalizes, as in an `orthogonal metal', into a fermion $f$ which carries both the electron spin and charge, and a boson $ϕ$. Both $f$ and $ϕ$ carry emergent $\mathbb{Z}_2$ gauge charges. The model has a phase in which the $ϕ$ bosons are gapped, and the $f$ fermions are gapless and critical, and so the electron spectral function is gapped. This phase can be considered as a toy model for the underdoped cuprates. The model also has an extended, critical, `quasi-Higgs' phase where both $ϕ$ and $f$ are gapless, and the electron operator $\sim f ϕ$ has a Fermi liquid-like $1/τ$ propagator in imaginary time, $τ$. So while the electron spectral function has a Fermi liquid form, other properties are controlled by $\mathbb{Z}_2$ fractionalization and the anomalous exponents of the $f$ and $ϕ$ excitations. This `quasi-Higgs' phase is proposed as a toy model of the overdoped cuprates. We also describe the critical state separating these two phases.

cond-mat.str-el↗

arXiv2018

Spectra of Operators in Large $N$ Tensor Models

We study the operators in the large $N$ tensor models, focusing mostly on the fermionic quantum mechanics with $O(N)^3$ 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 $O(N)$ groups. In the symmetric traceless case, the spectrum of scaling dimensions is the same as in the SYK model with real fermions; it includes the $h=2$ zero-mode. For the operators anti-symmetric 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 $h=0$ eigenvalue corresponds to the conserved $O(N)$ charges. A class of singlet operators may be constructed from contracted combinations of $m$ symmetric traceless or antisymmetric two-particle operators. Their two-point functions receive contributions from $m$ 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 $2k$-particle operators do not receive any ladder corrections and have quantized large $N$ scaling dimensions $k/2$. We construct pictorial representations of various singlet operators with low $k$. For larger $k$ we use available techniques to count the operators and show that their number grows as $2^k k!$. As a consequence, the theory has a Hagedorn phase transition at the temperature which approaches zero in the large $N$ limit. We also study the large $N$ spectrum of low-lying operators in the Gurau-Witten model, which has $O(N)^6$ 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 $N$ tensor integrals with various symmetries.

hep-th↗

arXiv2018

On large $q$ expansion in the Sachdev-Ye-Kitaev model

We consider the Sachdev-Ye-Kitaev (SYK) model where interaction involves $q$ fermions at a time. We find the next order correction to the thermal two-point function in the large $q$ expansion. Using this result we find the next order correction to the SYK free energy.

hep-th↗

arXiv2017

On Large $N$ Limit of Symmetric Traceless Tensor Models

For some theories where the degrees of freedom are tensors of rank $3$ or higher, there exist solvable large $N$ limits dominated by the melonic diagrams. Simple examples are provided by models containing one rank-$3$ tensor in the tri-fundamental representation of the $O(N)^3$ symmetry group. When the quartic interaction is assumed to have a special tetrahedral index structure, the coupling constant $g$ must be scaled as $N^{-3/2}$ in the melonic large $N$ limit. In this paper we consider the combinatorics of a large $N$ theory of one fully symmetric and traceless rank-$3$ tensor with the tetrahedral quartic interaction; this model has a single $O(N)$ symmetry group. We explicitly calculate all the vacuum diagrams up to order $g^8$, as well as some diagrams of higher order, and find that in the large $N$ limit where $g^2 N^3$ is held fixed only the melonic diagrams survive. While some non-melonic diagrams are enhanced in the $O(N)$ symmetric theory compared to the $O(N)^3$ one, we have not found any diagrams where this enhancement is strong enough to make them comparable with the melonic ones. Motivated by these results, we conjecture that the model of a real rank-$3$ symmetric traceless tensor possesses a smooth large $N$ limit where $g^2 N^3$ is held fixed and all the contributing diagrams are melonic. A feature of the symmetric traceless tensor models is that some vacuum diagrams containing odd numbers of vertices are suppressed only by $N^{-1/2}$ relative to the melonic graphs.

hep-th↗

arXiv2017

Bosonic Tensor Models at Large $N$ and Small $ε$

We study the spectrum of the large $N$ quantum field theory of bosonic rank-$3$ tensors, whose quartic interactions are such that the perturbative expansion is dominated by the melonic diagrams. We use the Schwinger-Dyson equations to determine the scaling dimensions of the bilinear operators of arbitrary spin. Using the fact that the theory is renormalizable in $d=4$, we compare some of these results with the $4-ε$ expansion, finding perfect agreement. This helps elucidate why the dimension of operator $ϕ^{abc}ϕ^{abc}$ is complex for $d<4$: the large $N$ fixed point in $d=4-ε$ has complex values of the couplings for some of the $O(N)^3$ invariant operators. We show that a similar phenomenon holds in the $O(N)^2$ symmetric theory of a matrix field $ϕ^{ab}$, where the double-trace operator has a complex coupling in $4-ε$ dimensions. We also study the spectra of bosonic theories of rank $q-1$ tensors with $ϕ^q$ interactions. In dimensions $d>1.93$ there is a critical value of $q$, above which we have not found any complex scaling dimensions. The critical value is a decreasing function of $d$, and it becomes $6$ in $d\approx 2.97$. This raises a possibility that the large $N$ theory of rank-$5$ tensors with sextic potential has an IR fixed point which is free of perturbative instabilities for $2.97<d<3$. This theory may be studied using renormalized perturbation theory in $d=3-ε$.

hep-th↗

arXiv2017

On Large N Expansion of the Sphere Free Energy

We propose formulas for the $1/N$ correction to the sphere free energy of theories with 4-fermion interactions, which are conformal for $d>2$. We also propose a formula for the scalar $O(N)$ model. Expanding these formulas in small $ε$ near various integer dimensions we find a perfect agreement with results obtained using $ε$-expansion technique. In $d=3$, the large $N$ results with the $1/N$ correction included are in good agreement with the Pade resummed $ε$-expansion.

hep-th↗

arXiv2017

Yukawa CFTs and Emergent Supersymmetry

We study conformal field theories with Yukawa interactions in dimensions between 2 and 4; they provide UV completions of the Nambu-Jona-Lasinio and Gross-Neveu models which have four-fermion interactions. We compute the sphere free energy and certain operator scaling dimensions using dimensional continuation. In the Gross-Neveu CFT with $N$ fermion degrees of freedom we obtain the first few terms in the $4-ε$ expansion using the Gross-Neveu-Yukawa model, and the first few terms in the $2+ε$ expansion using the four-fermion interaction. We then apply Pade approximants to produce estimates in $d=3$. For $N=1$, which corresponds to one 2-component Majorana fermion, it has been suggested that the Yukawa theory flows to a ${\cal N}=1$ supersymmetric CFT. We provide new evidence that the $4-ε$ expansion of the $N=1$ Gross-Neveu-Yukawa model respects the supersymmetry. Our extrapolations to $d=3$ appear to be in good agreement with the available results obtained using the numerical conformal bootstrap. Continuation of this CFT to $d=2$ provides evidence that the Yukawa theory flows to the tri-critical Ising model. We apply a similar approach to calculate the sphere free energy and operator scaling dimensions in the Nambu-Jona-Lasinio-Yukawa model, which has an additional $U(1)$ global symmetry. For $N=2$, which corresponds to one 2-component Dirac fermion, this theory has an emergent supersymmetry with 4 supercharges, and we provide new evidence for this.

hep-th↗

arXiv2017

Uncolored Random Tensors, Melon Diagrams, and the SYK Models

Certain models with rank-$3$ tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large $N$ limit, where $g^2 N^3$ is held fixed. In this limit the perturbative expansion in the quartic coupling constant, $g$, is dominated by a special class of "melon" diagrams. We study "uncolored" models of this type, which contain a single copy of real rank-$3$ tensor. Its three indexes are distinguishable; therefore, the models possess $O(N)^3$ symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large $N$ limit dominated by the melon diagrams. The quantum mechanics of a real anti-commuting tensor therefore has a similar large $N$ limit to the model recently introduced by Witten as an implementation of the Sachdev-Ye-Kitaev (SYK) model which does not require disorder. Gauging the $O(N)^3$ symmetry in our quantum mechanical model removes the non-singlet states; therefore, one can search for its well-defined gravity dual. We point out, however, that the model possesses a vast number of gauge-invariant operators involving higher powers of the tensor field, suggesting that the complete gravity dual will be intricate. We also discuss the quantum mechanics of a complex 3-index anti-commuting tensor, which has $U(N)^2\times O(N)$ symmetry and argue that it is equivalent in the large $N$ limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension $d$. While the quartic interaction is not positive definite, we construct the large $N$ Schwinger-Dyson equation for the two-point function and show that its solution is consistent with conformal invariance. We carry out a perturbative check of this result using the $4-ε$ expansion.

hep-th↗

arXiv2016

On $C_{J}$ and $C_{T}$ in Conformal QED

QED with a large number $N$ of massless fermionic degrees of freedom has a conformal phase in a range of space-time dimensions. We use a large $N$ diagrammatic approach to calculate the leading corrections to $C_T$, the coefficient of the two-point function of the stress-energy tensor, and $C_J$, the coefficient of the two-point function of the global symmetry current. We present explicit formulae as a function of $d$ and check them versus the expectations in 2 and $4-ε$ dimensions. Using our results in higher even dimensions we find a concise formula for $C_T$ of the conformal Maxwell theory with higher derivative action $F_{μν} (-\nabla^2)^{\frac{d}{2}-2} F^{μν}$. In $d=3$, QED has a topological symmetry current, and we calculate the correction to its two-point function coefficient, $C^{\textrm{top}}_{J}$. We also show that some RG flows involving QED in $d=3$ obey $C_T^{\rm UV} > C_T^{\rm IR}$ and discuss possible implications of this inequality for the symmetry breaking at small values of $N$.

hep-th↗

arXiv2016

On $C_J$ and $C_T$ in the Gross-Neveu and $O(N)$ Models

We apply large $N$ diagrammatic techniques for theories with double-trace interactions to the leading corrections to $C_J$, the coefficient of a conserved current two-point function, and $C_T$, the coefficient of the stress-energy tensor two-point function. We study in detail two famous conformal field theories in continuous dimensions, the scalar $O(N)$ model and the Gross-Neveu model. For the $O(N)$ model, where the answers for the leading large $N$ corrections to $C_J$ and $C_T$ were derived long ago using analytic bootstrap, we show that the diagrammatic approach reproduces them correctly. We also carry out a new perturbative test of these results using the $O(N)$ symmetric cubic scalar theory in $6-ε$ dimensions. We go on to apply the diagrammatic method to the Gross-Neveu model, finding explicit formulae for the leading corrections to $C_J$ and $C_T$ as a function of dimension. We check these large $N$ results using regular perturbation theory for the Gross-Neveu model in $2+ε$ dimensions and the Gross-Neveu-Yukawa model in $4-ε$ dimensions. For small values of $N$, we use Pade approximants based on the $4-ε$ and $2+ε$ expansions to estimate the values of $C_J$ and $C_T$ in $d=3$. For the $O(N)$ model our estimates are close to those found using the conformal bootstrap. For the GN model, our estimates suggest that, even when $N$ is small, $C_T$ differs by no more than $2\%$ from that in the theory of free fermions. We find that the inequality $C_T^{\textrm{UV}} > C_T^{\textrm{IR}}$ applies both to the GN and the scalar $O(N)$ models in $d=3$.

hep-th↗