Tensor Models at N Large and Small
Igor Klebanov

Talk at Strings 2018,
Okinawa Institute of Science and Technology
June 27, 2018

Three Large N Limits• O(N) Vector: solvable because the bubble
diagrams can be summed.
• Matrix (‘t Hooft) Limit: planar diagrams.
Solvable only in special cases.
• Tensor of rank three and higher. Wheninteractions are specially chosen, dominatedby the melonic (ladder) diagrams. Bonzom, Gurau,Riello, Rivasseau; Carrozza, Tanasa; Witten; IK, Tarnopolsky

O(N) x O(N) Matrix Model• Theory of real matrices fab with distinguishable
indices, i.e. in the bi-fundamental
representation of O(N)axO(N)b symmetry.
• The interaction is at least quartic: g tr ffTffT• Propagators are represented by colored doublelines, and the interaction vertex is• In d=0 or 1 special limits describe twodimensional quantum gravity.

• In the large N limit
where gN is held fixed
we find planar Feynman
graphs, and each index
loop may be red orgreen.
• The dual graphs shownin black may be thoughtof as random surfaces
tiled with squares whosevertices have alternatingcolors (red, green, red,green).

From Bi- to Tri-Fundamentals• For a 3-tensor with distinguishable indices thepropagator has index structure• It may be represented graphically by 3 coloredaabbwires cc• Tetrahedral interaction witha bc
O(N)axO(N)bxO(N)c symmetry
1 1 1

Carrozza, Tanasa; IK, Tarnopolsky

a1b2
c2

c1
b2
a2

c2b1a2

• Leading correction to the propagator has 3index loops• Requiring that this “melon” insertion is oforder 1 means thatmust be held
fixed in the large N limit.
• Melonic graphs obtained by iterating

Cables and Wires• The Feynman graphs of the quartic field
theory may be resolved in terms of thecolored wires (triple lines)Non-Melonic Graphs• Most Feynman graphs in the quartic field theory
are not melonic are therefore subdominant in thenew large N limit, e.g.

• Scales as• None of the graphs with an odd number ofvertices are melonic.

The Sachdev-Ye-Kitaev Model• Quantum mechanics of a large number NSYK ofanti-commuting variables with action• Random couplings j have a Gaussian
distribution with zero mean.
• The model flows to strong coupling andbecomes nearly conformal. Georges, Parcollet, Sachdev;
Kitaev; Polchinski, Rosenhaus; Maldacena, Stanford; Jevicki, Suzuki, Yoon;…

• Spectrum for a single realization of NSYK=32
model with q=4. Maldacena, Stanford• No exact degeneracies, but the gaps areexponentially small. Large low T entropy.

SYK-Like Tensor Quantum Mechanics• E. Witten, “An SYK-Like Model WithoutDisorder,” arXiv: 1610.09758. Has 4N3 fermions.
• Appeared on the evening of Halloween:
October 31, 2016.

• It is sometimes tempting to change the term
“melon diagrams” to “pumpkin diagrams.”

The O(N)3 Model• A pruned version: there are N3 Majorana
fermions IK, Tarnopolsky• Has O(N)axO(N)bxO(N)c symmetry under• The SO(N) symmetry charges areψa1 b1 c1• The 3-tensors may be
associated withindistinguishable verticesof a tetrahedron.

c1a1b1
ψa2 b2 c1b2

ψa1 b2 c2c2

a2
ψa2 b1 c2

a1b1c1• This is equivalent toa1b2c2• The 3-line Feynman
graphs are produced
using the propagator

c1b2a2c2b1a2

abcabcSchwinger-Dyson Equations• Some are the same as in the SYK model Kitaev,;
Polchinski, Rosenhaus; Maldacena, Stanford; Jevicki, Suzuki, Yoon...

• Neglecting the left-hand side in IR we find...

• Four point function
t1t3
...

t2t4

t1t3

...

...
t2t4

• If we denote by

the ladder with n rungs

Spectrum of two-particle operators• S-D equation for the three-point function Gross,Rosenhaus• Scaling dimensions of operators

• The first solution is h=2; dual to gravity.

• The higher scaling dimensions areapproaching

Gauge Invariant Operators• Bilinear operators related by the EOM to someof the higher particle “single-sum” operators.

• All the 6-particle
operators vanish by
the Fermi statistics inthe theory of one
Majorana tensor

• The bubbles come from O(N) charges andvanish in the gauged model:• The 17 single-sum 8-particle operators which
do not include bubble insertions areFactorial Growth• There are 24 bubble-free 10-particle; 617 12particle; 4887 14-particle; 82466 16-particle
operators; etc.
• The number of (2k)-particle operators grows
asymptotically as k! 2k. Bulycheva, IK, Milekhin, Tarnopolsky• The Hagedorn temperature of the large N
theory vanishes as 1/log N.
• The tensor models seem to lie “beyond stringtheory.”• Are they related to M-theory?

Spectra of Energy Eigenstates• Generalize the Majorana tensor model to havesymmetry• The traceless Hamiltonian is• The Hilbert space has dimension• Eigenstates of H form irreducible
representations of the symmetry.

Complete Diagonalizations• Generally possible only for small ranks. Krishnan,Pavan Kumar, Sanyal, Bala Subramanian, Rosa; Chaudhuri et al.; IK, Roberts,Stanford, Tarnopolsky• For example IK, Milekhin, Popov, Tarnopolsky

• Spectra for N3=2
• For the O(2)3 model
only two singlets atenergies -2g and 2g.

Energy Bounds• The bound on the singlet ground state energy
IK, Milekhin, Popov, Tarnopolsky• In the melonic limit, this correctly scales as N3.
• The gap to the lowest non-singlet state scales
as 1/N.
• For unequal ranks the bound is

A Fermionic Matrix Model• For N3=2 the bound simplifies to• Saturated by the ground state.
• This is a fermionic matrix model with symmetry

Gauge Singlets• To eliminate large degeneracies, focus on thestates invariant under• Their number can be found by gauging thefree theory

Singlets in the Matrix Model• Their number grows slowly. For N1=N2=10 only
24 singlets out of 2100 states.

Gauge Singlets in the O(N)3 Model• Their number vanishes for odd N due to a QM
anomaly for odd numbers of flavors.
• Grows very rapidly for even N• The large low-temperature entropy suggests
tiny gaps for singlet excitations ~Spectrum of the Gauged N=4 Model• Work in progress on this system of 32 qubits withK. Pakrouski, F. Popov and G. Tarnopolsky.
• Need to isolate the 36 states invariant underSO(4)3 out of the 601080390 “half-filled” states(those with 16 ones and 16 zeros).
• Diagonalize 4H/g + 100 C where C is the sum of
three Casimir operators.
• A Lanczos type algorithm is well suited for thissparse operator.
• Find 15 distinct SO(4)3 invariant energy levels:
E=0 and 7 “mirror pairs” (E, -E).

Discrete Symmetries• Act within the SO(N)3 invariant sector and canlead to small degeneracies.
• Z2 parity transformation within each group like• Interchanges of the groups flip the energy• Z3 symmetry generated by,Preliminary Numerical Results• The maximum degeneracy at non-zero energy is 3.
• The lowest singlet state is non-degenerate and has
E0=- 40.035 g.
• This is likely the ground state of H.
• It is not far from our lower bound -41.569 g• The next SO(4)3 invariant states are at -24.255 g;they have degeneracy 3.
• The highest degeneracy is at E=0.

Unstable Bosonic Tensor Model• Action with a potential that is not positive
definite IK, Tarnopolsky; Giombi, IK, Tarnopolsky• Schwinger-Dyson equation for 2pt function
Patashinsky, Pokrovsky• Has solution

Spectrum of two-particle spin zero
operators• Schwinger-Dyson equation• In d<4 the first solution is complex

Complex Fixed Point in 4-e Dimensions• The tetrahedron operator
mixes with the pillow and double-sum operators• The renormalizable action is• The large N scaling is• The 2-loop beta functions and fixed points:• The scaling dimension ofis

Stable Bosonic Model in 2.9 Dimensions• Work in progress with S. Giombi, F. Popov, S.
Prakash and G. Tarnopolsky on the theory
dominated by the positive “prism” interaction• To obtain the large N solution
it is convenient to rewrite• Tensor counterpart of a bosonic SYK-like model.
Murugan, Stanford, Witten• The IR solution in general dimension:• In• For d=2.9 find numerically• Graphical solution for dimensions of bilinear
operators in d=2.9• The first root is• For d<2.8056,

becomes complex.

Renormalized Perturbation Theory• For 2.8056 < d <3 the large N theory is stable.
• To make the theory renormalizable in d=3
need to add 7 more O(N)3 invariant terms.
• The 8 coupled beta functions have a nontrivial real fixed point.
• The resulting epsilon expansions agree in thelarge N limit with the solutions of theSchwinger-Dyson equations.

Conclusions• The vector and matrix large N limits have beenused extensively for many years in various
theoretical physics problems.
• The tensor large N limits for rank 3 and higher
are relatively new.
• The O(N)3 fermionic tensor quantum
mechanics seems to be the closest
counterpart of the basic SYK model for
Majorana fermions. Yet, there are somedifferences between the two.

• Gauging the SO(N)3 symmetry leaves
interesting spectra of operators andeigenstates.
• Energy gaps should become very small alreadyfor N=6.
• Higher dimensional generalizations arepossible, e.g. a stable sextic scalar theory in
2.8056 < d < 3, which is solvable in the large Nlimit.
• In 3-e dimensions it may be studied for finite
N using standard perturbation theory.