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DOE OSTI2023

An effective field theory for non-maximal quantum chaos

In non-maximally quantum chaotic systems, the exponential behavior of out-of-time-ordered correlators (OTOCs) results from summing over exchanges of an infinite tower of higher “spin” operators. We construct an effective field theory (EFT) to capture these exchanges in (0 + 1) dimensions. The EFT generalizes the one for maximally chaotic systems, and reduces to it in the limit of maximal chaos. The theory predicts the general structure of OTOCs both at leading order in the 1/N expansion (N is the number of degrees of freedom), and after resuming over an infinite number of higher order 1/N corrections. These general results agree with those previously explicitly obtained in specific models. We also show that the general structure of the EFT can be extracted from the large q SYK model.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

The connection between nonzero density and spontaneous symmetry breaking for interacting scalars

We consider U(1)-symmetric scalar quantum field theories at zero temperature. At nonzero charge densities, the ground state of these systems is usually assumed to be a superfluid phase, in which the global symmetry is spontaneously broken along with Lorentz boosts and time translations. We show that, in d > 2 spacetime dimensions, this expectation is always realized at one loop for arbitrary non-derivative interactions, confirming that the physically distinct phenomena of nonzero charge density and spontaneous symmetry breaking occur simultaneously in these systems. We quantify this result by deriving universal scaling relations for the symmetry breaking scale as a function of the charge density, at low and high density. Moreover, we show that the critical value of μ above which a nonzero density develops coincides with the pole mass in the unbroken, Poincaré invariant vacuum of the theory. The same conclusions hold non-perturbatively for an O(N) theory with quartic interactions in d = 3 and 4, at leading order in the 1/N expansion. We derive these results by computing analytically the zero-temperature, finite-μ one-loop effective potential, paying special attention to subtle points related to the iε terms. We check our results against the one-loop low-energy effective action for the superfluid phonons in λΦ 4 theory in d = 4 previously derived by Joyce and ourselves, which we further generalize to arbitrary potential interactions and arbitrary dimensions. As a byproduct, we find analytically the one-loop scaling dimension of the lightest charge-n operator for the λΦ 6 conformal superfluid in d = 3, at leading order in 1/n, reproducing a numerical result of Badel et al. For a λΦ 4 superfluid in d = 4, we also reproduce the Lee-Huang-Yang relation and compute relativistic corrections to it. Finally, we discuss possible extensions of our results beyond perturbation theory.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

$\mathrm{BPS}$ coherent states and localization

We introduce coherent states averaged over a gauge group action to study correlators of half BPS states in $\mathcal{N}$= 4 SYM theory. The overlaps of these averaged coherent states are a generating function of correlators and can be written in terms of the Harish-Chandra-Itzykzon-Zuber (HCIZ) integral. We show that this formula immediately leads to a computation of the normalization of two point functions in terms of characters obtained originally in the work of Corley, Jevicki and Ramgoolam. We also find various generalizations for A n–1 quivers that follow directly from other solvable integrals over unitary groups. All of these can be computed using localization methods. When we promote the parameters of the generating function to collective coordinates, there is a dominant saddle that controls the effective action of these coherent states in the regime where they describe single AdS giant gravitons. We also discuss how to add open strings to this formulation. These will produce calculations that rely on correlators of matrix components of unitaries in the ensemble that is determined by the HCIZ integral to determine anomalous dimensions. We also discuss how sphere giants arise from Grassman integrals, how one gets a dominant saddle and how open strings are added in that case. The fact that there is a dominant saddle helps to understand how a 1/N expansion arises for open strings. We generalize the coherent state idea to study 1/4 and 1/8 BPS states as more general integrals over unitary groups.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Strong coupling expansion of free energy and BPS Wilson loop in $$ \mathcal{N} $$ = 2 superconformal models with fundamental hypermultiplets

As a continuation of the study (in arXiv:2102.07696 and arXiv:2104.12625 ) of strong-coupling expansion of non-planar corrections in \( \mathcal{N} \) = 2 4d superconformal models we consider two special theories with gauge groups SU( N ) and Sp(2 N ). They contain N -independent numbers of hypermultiplets in rank 2 antisymmetric and fundamental representations and are planar-equivalent to the corresponding \( \mathcal{N} \) = 4 SYM theories. These \( \mathcal{N} \) = 2 theories can be realised on a system of N D3-branes with a finite number of D7-branes and O7-plane; the dual string theories should be particular orientifolds of AdS 5 × S 5 superstring. Starting with the localization matrix model representation for the \( \mathcal{N} \) = 2 partition function on S 4 we find exact differential relations between the 1 /N terms in the corresponding free energy F and the \( \frac{1}{2} \) -BPS Wilson loop expectation value \( \left\langle \mathcal{W}\right\rangle \) and also compute their large ’t Hooft coupling ( λ » 1) expansions. The structure of these expansions is different from the previously studied models without fundamental hypermultiplets. In the more tractable Sp(2 N ) case we find an exact resummed expression for the leading strong coupling terms at each order in the 1 /N expansion. We also determine the exponentially suppressed at large λ contributions to the non-planar corrections to F and \( \left\langle \mathcal{W}\right\rangle \) and comment on their resurgence properties. We discuss dual string theory interpretation of these strong coupling expansions.

1/N expansion↗
DOE OSTI2021

BPS Wilson loop in $\mathcal{N}$ = 2 superconformal SU(N) “orientifold” gauge theory and weak-strong coupling interpolation

We consider the expectation value \( \left\langle \mathcal{W}\right\rangle \) of the circular BPS Wilson loop in \( \mathcal{N} \) = 2 superconformal SU( N ) gauge theory containing a vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. This theory admits a regular large N expansion, is planar-equivalent to \( \mathcal{N} \) = 4 SYM theory and is expected to be dual to a certain orbifold/orientifold projection of AdS 5 × S 5 superstring theory. On the string theory side \( \left\langle \mathcal{W}\right\rangle \) is represented by the path integral expanded near the same AdS 2 minimal surface as in the maximally supersymmetric case. Following the string theory argument in [5], we suggest that as in the \( \mathcal{N} \) = 4 SYM case and in the \( \mathcal{N} \) = 2 SU( N ) × SU( N ) superconformal quiver theory discussed in [19], the coefficient of the leading non-planar 1/ N 2 correction in \( \left\langle \mathcal{W}\right\rangle \) should have the universal λ 3/2 scaling at large ’t Hooft coupling. We confirm this prediction by starting with the localization matrix model representation for \( \left\langle \mathcal{W}\right\rangle \) . We complement the analytic derivation of the λ 3/2 scaling by a numerical high-precision resummation and extrapolation of the weak-coupling expansion using conformal mapping improved Padé analysis.

1/N Expansion↗
DOE OSTI2025

Failure of the large-N expansion in a bosonic tensor model

We study the tensor model generalization of the quantum p-spherical model in the large-N limit. While the tensor model has the same large-N expansion as the disordered quantum p-spherical model, its ground state is superextensive, in contradiction with large-N perturbation theory. Therefore, the large-N expansion of this model catastrophically fails at arbitrarily large-N, without any obvious signal in perturbation theory.

1/N Expansion↗
DOE OSTI2022

Notes on resonances and unitarity from celestial amplitudes

We study the celestial description of the O(N) sigma model in the large N limit as introduced by Coleman, Jackiw and Politzer. Focusing on three dimensions, we analyze the implications of a UV complete, all-loop order 4-point amplitude of pions in terms of correlation functions defined on the celestial circle. We find these retain many key features from the previously studied tree-level case, such as their relation to Generalized Free Field theories and crossing-symmetry, but also incorporate new properties such as IR/UV softness and S-matrix metastable states. In particular, to understand unitarity, we propose a form of the optical theorem that controls the imaginary part of the correlator based solely on the presence of these resonances. We also explicitly analyze the conformal block expansions and factorization of four-point functions into three-point functions. We find that summing over resonances is key for these factorization properties to hold. We end with some topics for future study.

1/N expansion↗
DOE OSTI2025

Generalized entropy of gravitational fluctuations

The corrections to holographic entanglement entropy from bulk quantum fields in a classical gravitational background are now well understood. They lead, in particular, to unitary Page curves for evaporating black holes. However, the correct treatment of quantum fluctuations of the metric, including graviton excitations, is a longstanding problem. We provide a gauge-invariant prescription for the generalized entropy of gravitons in anti-de Sitter space in terms of areas and bulk entanglement entropy, generalizing the quantum extremal surface prescription to accommodate fluctuations in the semiclassical spacetime geometry. This task requires a careful treatment of the area operator on the graviton Hilbert space and the definition of a “quantum extremal gauge” in which the extremal surface is unperturbed. It also requires us to determine the correct vacuum modular Hamiltonian for the graviton field, which we fix by requiring that it doesn’t contain a boundary term in extremal gauge. We check our prescription with an explicit computation of the vacuum-subtracted generalized entropy of states containing a graviton in an AdS-Rindler background. Our results exactly match vacuum-subtracted von Neumann entropies for stress-tensor excited states in holographic conformal field theory with d > 2 dimensions. We also use covariant phase space techniques to give a partial proof of our prescription when the entanglement wedge for the background spacetime has a bifurcate Killing horizon. Along the way, we identify a class of perturbative graviton states that have parametrically larger generalized entropy, in the small G N expansion, than any low-energy excitations of an ordinary quantum field.

1/N expansion↗
DOE OSTI2023

Soft gravitons in the BFSS matrix model

BFSS proposed that asymptotically flat M-theory is dual to a large N limit of the matrix quantum mechanics describing N nonrelativistic D0-branes. Recent insights on the soft symmetries of any quantum theory of gravity in asymptotically flat space are applied to the BFSS matrix model. It is shown that soft gravitons are realized by submatrices whose rank is held fixed in the large N M-theory limit, rather than the usual linear scaling with N for hard gravitons. The soft expansion is identified with the large N expansion and the soft theorem becomes a universal formula for the quantum mechanical scattering of such submatrix excitations. This formula is shown to be the Ward identity of large type IIA U(1) RR asymptotic gauge symmetry in the matrix model, whose asymptotic boundaries are at future and past timelike infinity.

1/N Expansion↗
DOE OSTI2021

Heptagon functions and seven-gluon amplitudes in multi-Regge kinematics

We compute all 2 → 5 gluon scattering amplitudes in planar $ \mathcal{N} $ N = 4 super-Yang-Mills theory in the multi-Regge limit that is sensitive to the non-trivial (“long”) Regge cut. We provide the amplitudes through four loops and to all logarithmic accuracy at leading power, in terms of single-valued multiple polylogarithms of two variables. To obtain these results, we leverage the function-level results for the amplitudes in the Steinmann cluster bootstrap. To high powers in the series expansion in the two variables, our results agree with the recently conjectured all-order central emission vertex used in the Fourier-Mellin representation of amplitudes in multi-Regge kinematics. Our results therefore provide a resummation of the Fourier-Mellin residues into single-valued polylogarithms, and constitute an important cross-check between the bootstrap approach and the all-orders multi-Regge proposal.

1/N expansion↗
DOE OSTI2025

All loop scattering as a counting problem

Abstract This is the first in a series of papers presenting a new understanding of scattering amplitudes based on fundamentally combinatorial ideas in the kinematic space of the scattering data. We study the simplest theory of colored scalar particles with cubic interactions, at all loop orders and to all orders in the topological ’t Hooft expansion. We find novel integral formulas for the amplitudes of this theory, with no trace of the conventional sum over Feynman diagrams, but instead determined by a beautifully simple counting problem attached to any order of the topological expansion. These results represent a significant step forward in the decade-long quest to formulate the fundamental physics of the real world in a radically new language, where the rules of spacetime and quantum mechanics, as reflected in the principles of locality and unitarity, are seen to emerge from deeper mathematical structures.

1/N Expansion↗
DOE OSTI2022

Non-planar BCFW Grassmannian geometries

In this paper, we study non-adjacent BCFW recursion relations and their connection to positive geometry. For an adjacent BCFW shift, the n-point N k MHV tree-level amplitude in N = 4 SYM theory is expressed as a sum over planar on-shell diagrams, corresponding to canonical “dlog” forms on the cells in the positive Grassmannian G+(k, n). Non-adjacent BCFW shifts naturally lead to an expansion of the amplitude in terms of a different set of objects, which do not manifest the cyclic ordering and the hidden Yangian symmetry of the amplitude. We show that these terms can be interpreted as dlog forms on the non-planar Grassmannian geometries, generalizing the cells of the positive Grassmannian G+(k, n) to a larger class of objects which live in G(k, n). We focus mainly on the case of NMHV amplitudes and discuss in detail the Grassmannian geometries. We also propose an alternative way to calculate the associated on-shell functions and dlog forms using an intriguing connection between Grassmannian configurations and the geometry in the kinematical space.

1/N expansion↗
DOE OSTI2022

Three-point functions in $\mathrm{ABJM}$ and Bethe Ansatz

We develop an integrability-based framework to compute structure constants of two sub-determinant operators and a single-trace non-BPS operator in ABJM theory in the planar limit. In this first paper, we study them at weak coupling using a relation to an integrable spin chain. We first develop a nested Bethe ansatz for an alternating SU(4) spin chain that describes single-trace operators made out of scalar fields. We then apply it to the computation of the structure constants and show that they are given by overlaps between a Bethe eigenstate and a matrix product state. We conjecture that the determinant operator corresponds to an integrable matrix product state and present a closed-form expression for the overlap, which resembles the so-called Gaudin determinant. We also provide evidence for the integrability of general sub-determinant operators. The techniques developed in this paper can be applied to other quantities in ABJM theory including three-point functions of single-trace operators.

1/N Expansion↗
DOE OSTI2022

Long-range fermions and critical dualities

We construct long-range fermionic models with the Gross-Neveu and Gross­Neveu-Yukawa interaction, and argue that their critical regimes are equivalent. To this end, we calculate various CFT data in ϵ- and 1/N-expansion, and demonstrate their agreement in the overlapping regimes of validity.

1/N Expansion↗
DOE OSTI2025

Thermal bootstrap of matrix quantum mechanics

We implement a bootstrap method that combines stationary state conditions, thermal inequalities, and semidefinite relaxations of matrix logarithm in the ungauged one-matrix quantum mechanics, at finite rank N as well as in the large N limit, and determine finite temperature observables that interpolate between available analytic results in the low and high temperature limits respectively. We also obtain bootstrap bounds on thermal phase transition as well as preliminary results in the ungauged two-matrix quantum mechanics.

1/N Expansion↗
DOE OSTI2023

Emergent area laws from entangled matrices

We consider a wavefunction of large N matrices supported close to an emergent classical fuzzy sphere geometry. The SU( N ) Gauss law of the theory enforces correlations between the matrix degrees of freedom associated to a geometric subregion and their complement. We call this ‘Gauss law entanglement’. We show that the subregion degrees of freedom transform under a single dominant, low rank representation of SU( N ). The corresponding Gauss law entanglement entropy is given by the logarithm of the dimension of this dominant representation. It is found that, after coarse-graining in momentum space, the SU( N ) Gauss law entanglement entropy is proportional to the geometric area bounding the subregion. The constant of proportionality goes like the inverse of an emergent Maxwell coupling constant, reminiscent of gravitational entropy.

1/N Expansion↗
DOE OSTI2023

Orbit averaging coherent states: holographic three-point functions of AdS giant gravitons

We study correlation functions of two AdS giant gravitons in AdS 5 × S 5 and a BPS supergravity mode using holography. In the gauge theory these are described by BPS correlators of Schur polynomials of fully-symmetric representations and a single trace operator. We find full agreement between the semiclassical gravity and gauge theory computations at large N, for both diagonal and off-diagonal structure constants. Our analysis in $\mathcal{N}$ = 4 SYM provides a simpler derivation to the results in the literature, and it can be readily generalized to operators describing bound states of AdS giant gravitons as well as bubbling geometries.

1/N Expansion↗
DOE OSTI2021

Cascade of phase transitions in a planar Dirac material

Here we investigate a model of interacting Dirac fermions in 2 + 1 dimensions with M flavors and N colors having the U(M)×SU(N ) symmetry. In the large-N limit, we find that the U(M) symmetry is spontaneously broken in a variety of ways. In the vacuum, when the parity-breaking flavor-singlet mass is varied, the ground state undergoes a sequence of M first-order phase transitions, experiencing M + 1 phases characterized by symmetry breaking U(M)→U(M – k)×U(k) with k ϵ {0, 1, 2, · · · , M}, bearing a close resemblance to the vacuum structure of three-dimensional QCD. At finite temperature and chemical potential, a rich phase diagram with first and second-order phase transitions and tricritical points is observed. Also exotic phases with spontaneous symmetry breaking of the form as U(3)→U(1) 3 , U(4)→U(2)×U(1) 2 , and U(5)→U(2)2×U(1) exist. For a large flavor-singlet mass, the increase of the chemical potential μ brings about M consecutive first-order transitions that separate the low-μ phase diagram with vanishing fermion density from the high-μ region with a high fermion density.

1/N expansion↗