Tensor renormalization group andthe volume independence
in 2D U(N ) and SU(N ) gauge theories
Atis Yosprakob1, Akira Matsumoto1, Mitsuaki Hirasawa3, Jun Nishimura1,2
Based on J. High Energy. Phys. 2021, 11 (2021)
1: The Graduate University for Advanced Studies (SOKENDAI)
2: High Energy Accelerator Research Organization (KEK)
3: INFN (Milano-Bicocca section)@ Kyoto (15/12/2021)

Tensor renormalization group (TRG)• Rewriting the partition function as a network of tensors [Levin & Nave,’07]• Non-stochastic = no sign problem!

• Can access large volumes with logarithmic cost

• Can handle Grassmann fields directly2Application of TRG in gauge theories• There were works on U(1), SU(2), and SU(3) gauge theories in 2d[Bazavov et al.,’19; Kuramashi & Yoshimura,’20; Fukuma et al.,‘21]• Our interest: higher rank gauge group in 2d (character expansion)• Questions to be answered:1.

How to impose the cutoff on representations?

2.

Interesting large-N behaviors?

3Outlineo Introductiono TRG for 2d gauge theories• Efficient cut-off condition for irreps
o Singular value analysis• SV vs couplings• Large-N expressions of SVs
o Numerical results• GWW transition and parity SSB at θ = π• Eguchi-Kawai reduction• New volume reduction at strong couplingo Summary and discussion

TRG for 2d gauge theories

Brief review of 2d TRG
1.

Write the partition function as a network of tensor

2.

Decompose the tensor with SVD

3.

Recombine the tensors into the coarse-grained network(keep the tensor rank fixed to Dcut)

4.

Repeat at step 2

(1)

(2)

(3)
6

2d gauge theories with a theta term

Action:Topological charge:

The model can be exactly solved via character expansion.

7

2d gauge theories with a theta term

original lattice
integrate over group manifold

dual lattice
sum over group representations

8

Representations of U(N ) and SU(N )for SU(N )SU(N ):

U(N ):

Dimensionality(matrix size):9Tensor construction
U(N ) :

Because the tensor is diagonal, singular values have a simple scaling behavior:

The partition function can be exactly evaluated10

Tensor construction
SU(N ) :Because the tensor is diagonal, singular values have a simple scaling behavior:

The partition function can be exactly evaluated11Representation cut-off
There are infinitely many irreps! → need a cut-off
Examples• U(1): cut-off on the charge• SU(2): cut-off on the spin

• SU(3) ?

Cut-off becomes nontrivial!

1?

??

12

Representation cut-off
Our general strategy :1.

Calculate SVs of all irreps that is within the cut-off condition
Larger numberThis number is usually larger than Dcut

= less efficiency

2.

Keep only Dcut irreps in the calculation

3.

Extend the cut-off until the calculation is unchanged (Dcut kept fixed)

4.

The `U(1) charge’ for U(N ) can be cut off independently and straightforwardly

Question: what is the most efficient cut-off condition?
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Representation cut-off
Example 1: usingas the cut-off condition

SU(3):(2,0,0)(1,0,0)(3,1,0)(2,1,0)Problem: the number of irreps grows like

(3,2,0)1(0,0,0)(for fixed N )Too large too quickly for large N !

(1,1,0)(2,2,0)(3,3,0)14

Representation cut-off
Example 2: using dimensionality as the cut-off condition= smallest dim. of irreps withFor each, consider only
those with

Representation cut-off
Example 2: using dimensionality as the cut-off condition

Irrelevant representations!

Representation cut-off
Question: what is the most efficient cut-off condition?

number of irreps
within the cut-off

cutting by dimensionality is the most efficient condition so far

17Singular value analysis

Singular values vs ’t Hooft couplings
SV profile
SU(3)free energy vs Dcut
SU(3)SV decays faster for larger ’t Hooft couplings
This is true for both U(N) and SU(N)19Singular value profile at large N
Questions:• Does large N pose any problem for TRG calculation?
• Any dominant representation?
• Is there a definite profile?

SU(N )
λ=320

Singular value profile at large NSU(N )λ=3• U(N ) and SU(N ) have the sameprofile for λ > 2 (strong coupling)• But they have different profiles
for λ < 2 (weak coupling)21

Large -N expansions
SU(N):
U(N) λ < 2:
U(N) λ > 2:
this term suppresses contributions
from most of the charges(only one q survives at large N)
U(1) d.o.f. is ‘trivialized’U(N ) ~ SU(N )22Singular values with nonzero θ• U(1) charge is related to the theta term!
q0: U(1) charge thatgives the largest SV• Large -N expansion
Only q = q0 survives at large N

weakstrong23

Numerical resultsBasic results
Gross-Witten-Wadia 3rd-order transition

SU(10)
weakcoupling
phase

strongcoupling
phase

The transition starts to appear at N = 1025

Basic results
1st-order transition at θ = odd x πU(3)parity spontaneously broken at large volume*very difficult in Monte Carlo simulations
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Alternative interpretation for Eguchi-Kawai reduction
EK-reduction: certain properties of large-N theories are independent of volume*if center sym. is unbroken

reduced model

original model
Original proof: demonstrating the equivalence of S-D equations for Wilson loops[Eguchi&Kawai,’92]27

Alternative interpretation for Eguchi-Kawai reduction

volume-independent at large N
TRG’s perspective: the theory is volume-independentwhen the SVs have a scaling behavior.

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Large N and nonzero θRecall: parity spontaneously broken at θ = odd x π at large volume
calculation at small volume (V=2x2)

Large NLarge Nweak•

parity remains unbroken at large Nstrong••

parity becomes broken even at small volsmall vol ~ large vol at large N ?? 29

New volume reduction
Recall: only q = q0 survives at large N and strong coupling
can be pulled out

sub-leading volume independence;
can be observed via
topological charge density

A new kind of volume
independence different from EK
reduction30

Summary and Discussions• We consider using TRG based on character expansion to study 2d non-abelian gauge theories.
• For that, we propose an efficient way to cut-off the irreps:that is based on dimensionality.
• Various known results are reproduced: GWW transition and parity SSB at θ = π• By looking at the behavior of singular values, Eguchi-Kawai reduction can be explained.
A new kind of volume independent at strong coupling is also observed.

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Summary and Discussions• We expect that our method to cut-off the irreps can be useful for more general non-Abeliantheories (d>2, or with matter fields, etc.)• Since Large-N reduction at θ ≠ 0 persists even at volume as small as 2x2 where the notionof topology is ambiguous, how is the topological information stored in the large-N matrix?

32Thank you!

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