I. Fixed points with two relevant directions

Supplemenatry Material to “On mul- timatrix models motivated by random Noncommutative Geometry I: the Func- tional Renormalization Group as a flow in the free algebra”

I. Fixed points with two relevant directions

In the main text the fixed points corresponding to the unique solution developing a single relevant direction were reported. Next tables cor- respond to two relevant directions, where uniqueness of the solution is lost. The selection criteria were the same mentioned in Sections 7.6 and 7.7.

I.1. Geometry (0, 2) or ( − , − )

With two relevant directions five solutions are found. The critical exponents are in that case

θ

1

θ

2

1,2

+1.0318 +0.274913

(two-fold multiplicity)

3,4,5

+0.301298 +0.027688

(three-fold multiplicity)

and the fixed-point coupling constant values are given in Table A.

Coupling 1 2 3 4 5

η −0.3625 −0.3625 −0.3418 −0.3418 −0.3418 a4 −0.07972 −0.07972 −0.05812 −0.05812 −0.1194 a6 0 0 −5.897×10⁻⁶ −5.897×10⁻⁶ −0.00002453

c1111 0 0 +0.06126 −0.06126 0

c₂₁₂₁ 0 0 −0.00001863 −0.00001863 −3.053×10⁻⁹ c22 −0.03986 −0.03986 −0.05969 −0.05969 +0.001568

c3111 0 0 −0.00003726 +0.00003726 0

c42 0 0 −0.00003632 −0.00003632 +9.418×10⁻⁷ d_2|02 −0.01337 0.08013 −0.01289 −0.01289 +0.001253 d2|04 0 0 −0.00002598 −0.00002598 +3.476×10⁻⁶

d2|1111 0 0 −0.00005407 +0.00005407 0

d2|2 −0.005156 −0.03632 −0.004297 −0.004297 −0.009011 d_2|22 0 0 −0.000106 −0.000106 +2.107×10⁻⁶ d2|4 0 0 −0.00002598 −0.00002598 −0.0001095 d11|11 −0.3782 −0.004201 −0.05657 −5.008×10⁻⁶ −5.008×10⁻⁶

d_11|31 0 0 −0.000226 −1.14×10⁻⁸ −1.14×10⁻⁸

d12|3 0 0 −0.00005331 +5.71×10⁻⁷ +9.208×10⁻⁷

d21|21 0 0 −0.00008135 +2.855×10⁻⁷ −1.231×10⁻⁸

d3|3 0 0 −8.735×10⁻⁶ +2.855×10⁻⁷ −0.00003585

Table A. All the coupling constant values at those fixed points of the RG-flow for the (0, 2)-geometry characterized by two relevant directions; for solu- tions with single relevant direction, cf. eqs. (7.5)

I.2. Geometry (2, 0) or (+, +)

Solutions with two relevant directions are six:

θ

₁

θ

₂

1

+1.0318 +0.274913

2

+1.0318 +0.274913

3,4,5,6

+0.3013 +0.02779

(four-fold multiplicity)

corresponding to the couplings shown in Table B.

2

Coupling 1 2 3 |4,5,6| η −0.3625 −0.3625 −0.3418 +0.3418 a4 −0.07972 −0.07972 −0.1194 +0.05812

a6 0 0 +0.00002453 +5.897×10⁻⁶

c1111 0 0 0 +0.06126

c₂₁₂₁ 0 0 +3.053×10⁻⁹ +0.00001863

c22 −0.03986 −0.03986 +0.001568 +0.05969

c3111 0 0 0 +0.00003726

c42 0 0 −9.418×10⁻⁷ +0.00003632 d_2|02 +0.08013 −0.01337 +0.001253 +0.01289

d2|04 0 0 +3.476×10⁻⁶ +0.00002598

d2|1111 0 0 0 +0.00005407

d2|2 −0.03632 −0.005156 −0.009011 +0.004297 d_2|22 0 0 −2.107×10⁻⁶ +0.000106

d2|4 0 0 +0.0001095 +0.00002598

d12|3 0 0 −9.208×10⁻⁷ +5.71×10⁻⁷

d_21|21 0 0 +1.231×10⁻⁸ +2.855×10⁻⁷

d3|3 0 0 +0.00003585 +2.855×10⁻⁷

d1|12 −0.00985 −0.00985 +0.0003325 +0.0003721 d_1|14 0 0 −1.925×10⁻⁷ +3.972×10⁻⁷

d1|2111 0 0 +1.275×10⁻⁹ +4.158×10⁻⁷

d1|3 −0.00985 −0.00985 −0.02262 +0.0003721 d_1|32 0 0 −1.269×10⁻⁶ +1.21×10⁻⁶

d1|5 0 0 +0.00005281 +3.972×10⁻⁷

d01|01 −0.2543 −0.2543 −0.3901 +0.009373

d_11|11 −0.004201 −0.3782 −5.008×10⁻⁶ +5.008×10⁻⁶

d11|31 0 0 +1.14×10⁻⁸ +1.14×10⁻⁸

Table B. All the coupling constant values at those fixed points of the RG-flow for the (2, 0)-geometry characterized by two relevant directions. The bars around the fixed points 3, 4, 6 mean that we report only the couplings in absolute value (the three solu- tions differ from sign change in some coupling con- stants).

II. Noncommutative Hessians and Laplacians

This appendix lists first the (twisted) NC-Laplacians

∇

²_τ

Tr

A2

( O

_I|I⁰

(A, B)) = Tr

2

Hess

^στ

Tr

A2

[ O

_I|I⁰

(A, B)]

(by Claim 2.1) of the double-trace operators O

_I|I0

(A, B) (in the left column). Subsequently, the Hessians also used in the proof of Theo- rem 7.2 are provided. The operators that do not appear in the next list can be obtained from these by the A ↔ B, e

_a

↔ e

_b

exchange (and in the case of the Hessian, with pertinent changes in the matrix structure).

3

O_I|I0(A,B) ∇²_τTrA₂(O_I|I0(A,B)) N⁻¹1N⊗(A·A) 2ea1N⊗1N

A⊗A 2ea1N⊗τ1N

N⁻¹1N⊗(A·A·A·A) 4ea(1N⊗(A·A) + (A·A)⊗1N+A⊗A) N⁻¹1N⊗(A·A·B·B) eb1N⊗(A·A) +eb(A·A)⊗1N

+ea1N⊗(B·B) +ea(B·B)⊗1N

N⁻¹1N⊗(A·B·A·B) 2(ebA⊗A+eaB⊗B) (A·B)⊗(A·B) 2(ebA⊗τA+eaB⊗τB)

(A·A)⊗(B·B) 2·1N⊗_τ1N(ebTrN(A·A) +eaTrN(B·B)) A⊗(A·A·A) 3ea(TrN(A)(A⊗τ1N+ 1N⊗τA)

+1N⊗(A·A) + (A·A)⊗1N) A⊗(A·B·B) ebTrN(A)(A⊗τ1N+ 1N⊗τA)

+ea1N⊗(B·B) +ea(B·B)⊗1N

(A·A)⊗(A·A) 4ea(1N⊗τ1NTrN(A·A) + 2A⊗A) (B·B)⊗(B·B) 4eb(1N⊗τ1NTrN(B·B) + 2B⊗B) N⁻¹1N⊗(A·A·A·A·A·A) 6ea(1N⊗(A·A·A·A) + (A·A·A·A)⊗1N

+A⊗(A·A·A) + (A·A·A)⊗A +(A·A)⊗(A·A))

N⁻¹1N⊗(A·A·A·A·B·B) ea1N⊗(A·A·B·B) +ea1N⊗(A·B·B·A) +ea1N⊗(B·B·A·A) +ea(A·A·B·B)⊗1N

+ea(A·B·B·A)⊗1N+ea(B·B·A·A)⊗1N

+eaA⊗(A·B·B) +eaA⊗(B·B·A) +ea(A·A)⊗(B·B) +ea(B·B)⊗(A·A) +ea(B·B·A)⊗A+eb1N⊗(A·A·A·A) +ea(A·B·B)⊗A+eb(A·A·A·A)⊗1N

N⁻¹1N⊗(A·A·B·A·A·B) 2[ea1N⊗(B·A·A·B) +ea(B·A·A·B)⊗1N

+eaB⊗(A·B·A) +ea(A·B)⊗(A·B) +ea(A·B·A)⊗B+eb(A·A)⊗(A·A))

+ea(B·A)⊗(B·A)]

O_I|I0(A,B) ∇²_τTrA₂(O_I|I0(A,B))

N⁻¹1N⊗(A·A·A·B·A·B) ea1N⊗(A·B·A·B) +ea1N⊗(B·A·B·A) +ea(A·B·A·B)⊗1N+ea(B·A·B·A)⊗1N

+eaA⊗(B·A·B) +eaB⊗(A·A·B) +eaB⊗(B·A·A) +ea(A·B)⊗(B·A)

+ea(A·A·B)⊗B+ea(B·A·A)⊗B +ebA⊗(A·A·A) +eb(A·A·A)⊗A +ea(B·A)⊗(A·B) +ea(B·A·B)⊗

A⊗(A·B·B·B·B) ebTrN(A)

1N⊗τ(A·B·B) + 1N⊗τ(B·A·B) +1N⊗_τ(B·B·A) + (A·B·B)⊗_τ1N

+(B·B·A)⊗τ1N+A⊗τ(B·B) +B⊗τ(A·B) +B⊗τ(B·A) + (A·B)⊗τB+ (B·A)⊗τB

+(B·A·B)⊗τ1N+ (B·B)⊗τA +ea1N⊗(B·B·B·B) +ea(B·B·B·B)⊗1N

A⊗(A·A·A·B·B) TrN(A)[ea1N⊗τ(A·B·B) +ea1N⊗τ(B·B·A) +ea(A·B·B)⊗τ1N+ea(B·B·A)⊗τ1N

+eaA⊗τ(B·B) +ea(B·B)⊗τA +eb1N⊗_τ(A·A·A) +eb(A·A·A)⊗_τ1N] +ea[1N⊗(A·A·B·B) + 1N⊗(A·B·B·A)

+(A·A·B·B)⊗1N+ (A·B·B·A)⊗1N

+1N⊗(B·B·A·A) + (B·B·A·A)⊗1N] A⊗(A·A·B·A·B) TrN(A)[ea1N⊗_τ(B·A·B) +ea(B·A·B)⊗_τ1N

+eaB⊗τ(A·B) +eaB⊗τ(B·A) +ea(A·B)⊗τB +ea(B·A)⊗τB+ebA⊗τ(A·A) +eb(A·A)⊗τA]

+ea[1N⊗(B·A·A·B) + 1N⊗(B·A·B·A) +(B·A·A·B)⊗1N+ (B·A·B·A)⊗1N

+1N⊗(A·B·A·B) + (A·B·A·B)⊗1N] (A·B)⊗(A·A·A·B) eaTrN(A·B)(1N⊗τ(A·B) + 1N⊗τ(B·A)

+(A·B)⊗_τ1N+ (B·A)⊗_τ1N+A⊗_τB+B⊗_τA) +ea[B⊗(A·A·B) +B⊗(A·B·A) +B⊗(B·A·A)] +ea[(A·A·B)⊗B +(A·B·A)⊗B+ (B·A·A)⊗B]

+ebA⊗(A·A·A) +eb(A·A·A)⊗A

5

O_I|I0(A,B) ∇²_τTrA2(OI|I0(A,B)) A⊗(A·A·A·A·A) 5ea

_Tr

N(A)[1N⊗τ(A·A·A) + (A·A·A)⊗τ1N

+A⊗τ(A·A) + (A·A)⊗τA]

+1N⊗(A·A·A·A) + (A·A·A·A)⊗1N

(A·B)⊗(A·B·B·B) ebTrN(A·B)[1N⊗τ(A·B) + 1N⊗τ(B·A) +(A·B)⊗_τ1N+ (B·A)⊗_τ1N+A⊗_τB+B⊗_τA]

+eb[A⊗(A·B·B) +A⊗(B·A·B) +A⊗(B·B·A)] +eb[(A·B·B)⊗A +(B·A·B)⊗A+ (B·B·A)⊗A]

+eaB⊗(B·B·B) +e_a(B·B·B)⊗B

(A·A)⊗(A·A·B·B) TrN(A·A)[eb1N⊗τ(A·A) +eb(A·A)⊗τ1N

+ea1N⊗τ(B·B) +ea(B·B)⊗τ1N] +2ea1N⊗1NTrN(A·A·B·B) +2ea[A⊗(A·B·B) +A⊗(B·B·A)]

+2ea[(A·B·B)⊗A+ (B·B·A)⊗A]

(A·A)⊗(A·B·A·B) 2 TrN(A·A)(ebA⊗τA+eaB⊗τB) +2ea1N⊗1NTrN(A·B·A·B) +4eaA⊗(B·A·B) + 4ea(B·A·B)⊗A

(A·A)⊗(A·A·A·A) 2ea[2 TrN(A·A)(1N⊗τ(A·A) + (A·A)⊗τ1N

+A⊗τA) + 1N⊗τ1NTrN(A·A·A·A) +4A⊗(A·A·A) + 4(A·A·A)⊗A]

(A·A)⊗(B·B·B·B) 4ebTrN(A·A)[1N⊗τ(B·B) + (B·B)t⊗1N

+B⊗τB] + 2ea1N⊗1NTrN(B·B·B·B)

(A·A·A)⊗(A·A·A) 6ea

(A⊗τ1N+ 1N⊗τA) TrN(A·A·A) +3(A·A)⊗(A·A)

(A·B·B)⊗(A·A·A) (A⊗τ1N+ 1N⊗τA)[3eaTrN(A·B·B) +ebTrN(A·A·A)]

+3ea(A·A)⊗(B·B) + 3ea(B·B)⊗(A·A)

(A·A·B)⊗(A·A·B) 2

e_a(B⊗τ1N+ 1N⊗τB) TrN(A·A·B) +ea[(A·B)⊗(A·B) + (A·B)⊗(B·A) +(B·A)⊗(A·B) + (B·A)⊗(B·A)]

+eb(A·A)⊗(A·A)

O_I|I0 HessσTrA₂(O_I|I0) (untwisted)

1

N1⊗A·A·A·A·A·A

6ea(1⊗A⁴+A⁴⊗1 +A⊗A³+A²⊗A²+A³⊗A) 0

0 0

(A·B)⊗(A·B)

2eaB⊗τB 2(1⊗1 Tr(AB) +B⊗τA) 2(1⊗1 Tr(AB) +A⊗τB) 2ebA⊗τA

(A·B·B)⊗(A·A·A)

3ea[(A⊗1 + 1⊗A] Tr(ABB) +A²⊗τB²+B²⊗τA²) TrA³(B⊗1 + 1⊗B) + 3A²⊗τAB+ 3A²⊗τBA TrA³(B⊗1 + 1⊗B) + 3AB⊗τA²+ 3BA⊗τA² eb(A⊗1 + 1⊗A) TrA³

A⊗(A·A·A·A·A)

5ea[TrA(1⊗A³+A³⊗1 +A⊗A²+A²⊗A) + 1⊗τA⁴+A⁴⊗τ1] 0

0 0

(A·A·A)⊗(A·A·A)

6ea[(A⊗1 + 1⊗A) TrA³+ 3A²⊗τA²] 0

0 0

(A·A)⊗(A·A·A·A)

2ea(2 TrA²(1⊗A²+A²⊗1 +A⊗A) + 1⊗1 TrA⁴+ 4A⊗τA³+ 4A³⊗τA) 0

0 0

(A·A)⊗(B·B·B·B)

2ea1⊗1 TrB⁴ 8A⊗τB³

8B³⊗τA 4ebTrA²(1⊗B²+B²⊗1 +B⊗B)

7

4

O_I|I1pA, Bq HessσTr_N^b2pO_I|I1pA, Bqq

pABq b pAAABq

eapTrABp1bAB 1bBA ABb1 BAb1 AbB Bb Aq BbτpAABq BbτpABAq BbτpBAAq pAABq bτ

B pABAq bτB pBAAq bτBq

TrABp1bA² A²b1 AbAq 1b1TrpA³Bq BbτpA³q pAABq bτA pABAq bτA pBAAq bτA

TrABp1bA² A²b1 AbAq 1b1TrpA³Bq Abτ

pAABq AbτpABAq AbτpBAAq pA³q bτB

ebpAbτpA³q pA³q bτAq

Ab pABBBBq

eap1bτB⁴ B⁴bτ1q TrpAqp1bB³ B³b1 BbB² B²bBq 1bτpABBBq 1bτpBABBq 1bτpBBABq 1bτpBBBAq TrpAqp1bB³ B³b1 BbB² B²bBq pABBBq bτ1

pBABBq bτ1 pBBABq bτ1 pBBBAq bτ1 ebTrpAqp1b pABBq 1b pBABq 1b pBBAq pABBq b1 pBABq b1 pBBAq b1 Ab pBBq BbAB BbBA

ABbB BAbB B²bAq

Ab pAABABq

eapTrpAq1b pBABq TrpAqpBABq b1 1bτpABABq 1bτ

pBAABq 1bτpBABAq pABABq bτ1 pBAABq bτ1 pBABAq bτ1 TrpAqBbAB TrpAqBbBA TrpAqABb

B TrpAqBAbBq

TrpAqp1b pABAq 1b pBAAq pAABq b1 pABAq b1 Ab BA ABbAq 1bτpAABAq 1bτpABAAq

TrpAqp1b pAABq 1b pABAq pABAq b1 pBAAq b1 Ab

AB BAbAq pAABAq bτ1 pABAAq bτ1 ebTrpAqpAbA² A²bAq

Ab pAAABBq

eapTrpAq1b pABBq TrpAq1b pBBAq TrpAqpABBq b1 TrpAqpBBAq b1 1bτpAABBq 1bτpABBAq 1bτ

pBBAAq pAABBq bτ1 pABBAq bτ1 pBBAAq bτ1 TrpAqAbB² TrpAqB²bAq

TrpAqp1b pAABq pBAAq b1 AbAB BbA² A²bB BAbAq 1bτpA³Bq 1bτpBA³q

TrpAqp1b pBAAq pAABq b1 AbBA BbA² A²bB

ABbAq pA³Bq bτ1 pBA³q bτ1 ebTrpAqp1b pA³q pA³q b1q

2

OI|I1pA, Bq HessσTr_N^b2pOI|I1pA, Bqq

IdbpAABAABq N

2eap1b pBAABq pBAABq b1 Bb pABAq ABbAB

BAbBA pABAq bBq 2p1b pABAAq pAABAq b1 Ab pBAAq pAABq bAq 2p1b pAABAq pABAAq b1 Ab pAABq pBAAq bAq 2ebA²bA²

pAABq b pAABq

2eapBb1TrpAABq 1bBTrpAABq ABbτAB ABbτ

BA BAbτAB BAbτBAq 2ppAb1 1bAqTrpAABq ABbτA² BAbτA²q 2ppAb1 1bAqTrpAABq A²bτAB A²bτBAq 2ebA²bτA²

pAAq b pABABq

2eap1b1TrpABABq BbBTrA² 2AbτpBABq 2pBABqbτAq 2TrA²p1bBA ABb1q 4AbτpABAq

2TrA²p1bAB BAb1q 4pABAq bτA 2ebAbATrA²

pAAq b pAABBq

eapTrA²1bB² TrA²B²b1 21b1TrpAABBq 2Abτ

pABBq 2AbτpBBAq 2pABBq bτA 2pBBAq bτAq TrA²p1bAB BAb1 AbB BbAq 2AbτpAABq 2AbτpBAAq

TrA²p1bBA ABb1 AbB BbAq 2pAABq bτA 2pBAAq bτA

ebTrA²p1bA² A²b1q

IdbpAAABABq N

eap1b pABABq 1b pBABAq pABABq b1 pBABAq b1 Ab pBABq Bb pAABq Bb pBAAq pABq bBA BAb

AB pAABq bB pBAAq bB pBABq bAq

1b pAABAq 1b pBA³q pA³Bq b1 pABAAq b1 Ab pABAq A²bBA ABbA² pABAq bA

1b pA³Bq 1b pABAAq pAABAq b1 pBA³q b1 Ab pABAq A²bAB BAbA² pABAq bA

ebpAb pA³q pA³q bAq

IdbpAAAABBq N

eap1b pAABBq 1b pABBAq 1b pBBAAq pAABBq b1 pABBAq b1 pBBAAq b1 Ab pABBq Ab pBBAq A²b

B² B²bA² pABBq bA pBBAq bAq

1b pA³Bq pBA³q b1 Ab pAABq Bb pA³q A²bAB BAbA² pA³q bB pBAAq bA

1b pBA³q pA³Bq b1 Ab pBAAq Bb pA³q A²bBA ABbA² pA³q bB pAABq bA

ebp1b pA⁴q pA⁴q b1q

III. Beta-functions for disconnected couplings

2h2(a4(12a6+ 5d_1|5) + 18a6d_1|3e_a+e_b(c42(d_1|12−c1111e_a) +c22e_a(c3111+d_1|32))) +d_1|5(3η+ 3) =β(d_1|5)

2h2(b4(12b6+ 5d01|05) +eb(18b6d01|03+c22ea(c1311+d01|23)) +c24ea(d01|21−c1111eb)) +d01|05(3η+ 3) =β(d01|05)

h2 −a4c1111eaeb−b4c1111eaeb+ 2c²₁₁₁₁−4c1111c22−8c1111d11|11eaeb+ 4c²₂₂+ 8c22d11|11eaeb+ 4d²_11|11

+h1(−2c1212eb−c1311eb−2c2121ea−c3111ea−2d11|13eb−2d11|31ea) +d11|11(2η+ 2) =β(d11|11)

h2 a4c22+ 4a4d2|02+b4c22+ 4b4d2|02+c²₂₂eaeb+ 12c22d02|02+ 12c22d2|2+ 24d02|02d2|02+ 24d2|02d2|2

+h1 −1 2^c24e_a−1

2^c42e_b−1

2^d02|22e_a−2d_02|4e_b−2d_2|04e_a−1

2^d2|22e_b

₊

d_2|02(2η+ 2) =β(d_2|02)

h₂ 2a²₄e_a+ 6a₄d_1|3−2c₁₁₁₁c₂₂e_a+ 2c₂₂d_1|12

+h1 −6a6−c3111e_ae_b−d12|3e_ae_b−d1|32e_ae_b−5d1|5−6d3|3

₊

d1|3(2η+ 2) =β(d1|3)

h2 2a4c22ea−2b4c1111ea+ 2b4d1|12−4c1111c22eb−4c1111d1|12eaeb+ 4c²₂₂eb+ 4c22d1|12eaeb+ 6c22d1|3

+h1 −2c1212−2c1311−c3111eaeb−2c42eaeb−2d12|12−3d12|3eaeb−3d1|14−d1|2111eaeb−2d1|32eaeb

+d1|12(2η+ 2) =β(d1|12)

h2 −2a4c1111eb+ 2a4d01|21+ 2b4c22eb−4c1111c22ea−4c1111d01|21eaeb+ 4c²₂₂ea+ 6c22d01|03+ 4c22d01|21eaeb

+h1 −c1311eaeb−2c2121−2c24eaeb−2c3111−d01|1211eaeb−2d01|23eaeb−3d01|41−3d21|03eaeb−2d21|21

+d01|21(2η+ 2) =β(d01|21)

h₂ 2b²₄e_b+ 6b₄d_01|03−2c₁₁₁₁c₂₂e_b+ 2c₂₂d_01|21

+h1 −6b6−c1311eaeb−5d01|05−d01|23eaeb−6d03|03−d21|03eaeb

₊

d01|03(2η+ 2) =β(d01|03)

10

2h2(a₄c₂₄+b₄(2c₁₂₁₂+ 2c₁₃₁₁+ 3d_1|14)−6b₆c₁₁₁₁e_ae_b+ 6b₆d_1|12e_b−2c₁₁₁₁c₂₄e_ae_b−2c₁₁₁₁d_1|14e_ae_b +4c1212c22eaeb+ 4c1212d1|12eb+ 4c1311c22eaeb+ 4c1311d1|12eb+ 4c22c24eaeb+c22c3111eaeb+ 2c22c42eaeb

+2c22d1|14eaeb+c22d1|2111eaeb+ 2c22d1|32eaeb+ 2c24d1|12eb+ 3c24d1|3ea) +d1|14(3η+ 3) =β(d1|14)

2h2(5a4c42+ 2a4d1|32+ 6a6c22eaeb+b4(c3111+d1|32)−2c1111c1212eaeb−4c1111c2121eaeb−2c1111c24eaeb

−2c1111d1|2111eaeb+ 2c1212d1|12eb+ 2c1311c22eaeb+ 2c22c3111eaeb+ 4c22c42eaeb+ 2c22d1|14eaeb+ 2c22d1|32eaeb

+5c22d1|5eaeb+ 2c24d1|12eb+ 2c3111d1|12eb+ 2c42d1|12eb+ 9c42d1|3ea) +d1|32(3η+ 3) =β(d1|32)

2h2(a4(2c2121+ 3c3111+d1|2111)−2c1111c1311eaeb−c1111c24eaeb−2c1111c3111eaeb−2c1111c42eaeb−2c1111d1|2111eaeb

−2c1111d1|32eaeb+ 2c1212c22eaeb+ 2c1311d1|12eb+c2121(8c22eaeb+ 4d1|12eb+ 6d1|3ea) + 4c22c3111eaeb

+c₂₂d_1|14e_ae_b+ 4c₂₂d_1|2111e_ae_b+c₂₄d_1|12e_b+ 2c₃₁₁₁d_1|12e_b+ 6c₃₁₁₁d_1|3e_a) +d_1|2111(3η+ 3) =β(d_1|2111)

2h2(a4(2c2121+ 2c3111+ 3d01|41)−6a6c1111eaeb+ 6a6d01|21ea+b4c42−2c1111c42eaeb−2c1111d01|41eaeb+c1311c22eaeb

+4c2121c22e_ae_b+ 4c2121d01|21e_a+ 2c22c24e_ae_b+ 4c22c3111e_ae_b+ 4c22c42e_ae_b+c22d01|1211e_ae_b+ 2c22d01|23e_aε

b+ 2c22d01|41e_ae_b+ 4c3111d01|21e_a+ 3c42d01|03e_b+ 2c42d01|21e_a) +d01|41(3η+ 3) =β(d01|41)

h2

_a2

24 + 8a4d2|2+^c21111 6 +^c222

3 +8

3^c22d2|02+8d²_2|02 3 + 24d²_2|2

+h1

−a6ea−^c2121eb

3 −^d2|22eb

3 −4d2|4ea

3

+d2|2(2η+ 2) =β(d2|2)

h₂

_b2 4

2 + 8b₄d_02|02+^c21111 6 +^c222

3 +8

3^c22d2|02+ 24d²_02|02+8d²_2|02 3

+h₁

−b₆e_b−^c1212ea

3 −4d02|04eb

3 −^d02|22ea

3

+d_02|02(2η+ 2) =β(d_02|02)

11

2h2(a4(c1311+d01|23) + 5b4c24+ 2b4d01|23+ 6b6c22eaeb−4c1111c1212eaeb−2c1111c2121eaeb−2c1111c42eaeb

−2c1111d01|1211eaeb+ 2c1311c22eaeb+ 2c1311d01|21ea+ 2c2121d01|21ea+ 4c22c24eaeb+ 2c22c3111eaeb+ 5c22d01|05eaeb+ 2c22d01|23eaeb

+2c22d01|41eaeb+ 9c24d01|03eb+ 2c24d01|21ea+ 2c42d01|21ea) +d01|23(3η+ 3) =β(d01|23)

2h2(b4(2c1212+ 3c1311+d01|1211)−2c1111c1311eaeb−2c1111c24eaeb−2c1111c3111eaeb−c1111c42eaeb−2c1111d01|1211eaeb

−2c1111d01|23eaeb+c1212(8c22eaeb+ 6d01|03eb+ 4d01|21ea) + 4c1311c22eaeb+ 6c1311d01|03eb+ 2c1311d01|21ea+ 2c2121c22eaeb

+4c₂₂d_01|1211e_ae_b+c₂₂d_01|41e_ae_b+ 2c₃₁₁₁d_01|21e_a+c₄₂d_01|21e_a) +d_01|1211(3η+ 3) =β(d_01|1211)

2h2(2a4(3c2121+ 2c3111+d11|31)−6a6c1111eaeb+b4c3111−c1111c1311eaeb−4c1111c2121eaeb−2c1111c24eaeb

−6c₁₁₁₁c₃₁₁₁e_ae_b−4c₁₁₁₁c₄₂e_ae_b−4c₁₁₁₁d_11|31e_ae_b+ 4c₁₂₁₂c₂₂e_ae_b+ 2c₁₃₁₁c₂₂e_ae_b+ 8c₂₁₂₁c₂₂e_ae_b+ 8c₂₁₂₁d_11|11

+8c22c3111eaeb+ 8c22c42eaeb+ 2c22d11|13eaeb+ 4c22d11|31eaeb+ 8c3111d11|11+ 4c42d11|11+ 2d11|11d11|31) +d11|31(3η+ 3) =β(d11|31)

2h2(a₄c₁₃₁₁+ 2b₄(3c₁₂₁₂+ 2c₁₃₁₁+d_11|13)−6b₆c₁₁₁₁e_ae_b−4c₁₁₁₁c₁₂₁₂e_ae_b−6c₁₁₁₁c₁₃₁₁e_ae_b−4c₁₁₁₁c₂₄e_ae_b

−c1111c3111eaeb−2c1111c42eaeb−4c1111d11|13eaeb+ 8c1212c22eaeb+ 8c1212d11|11+ 8c1311c22eaeb+ 8c1311d11|11

+4c2121c22eaeb+ 8c22c24eaeb+ 2c22c3111eaeb+ 4c22d11|13eaeb+ 2c22d11|31eaeb+ 4c24d11|11+ 2d11|11d11|13) +d11|13(3η+ 3) =β(d11|13)

2h2(a4(2c2121+ 6c42+ 3d_2|22) + 2(3a6c22e_ae_b−c1111c1311e_ae_b−c1111c3111e_ae_b−2c1111d_2|1111e_ae_b+c1212c22e_ae_b +4c1212d2|02eaeb+ 2c2121c22eaeb+ 12c2121d2|2+ 2c22c24eaeb+ 2c22c42eaeb+c22d02|22eaeb+ 2c22d2|04eaeb

+c22d2|22eaeb+ 2c22d2|4eaeb+ 6c24d2|02eaeb+ 18c42d2|2+ 2d02|22d2|02eaeb+ 6d2|2d2|22) +b4(2c2121+d2|22)) +d2|22(3η+ 3) =β(d2|22)

2h2(3a4c3111+ 2a4d2|1111−2c1111c1212eaeb−2c1111c42eaeb−2c1111d2|22eaeb+ 2c1311c22eaeb+ 8c1311d2|02eaeb

+4c22c3111eaeb+ 2c22d02|1111eaeb+ 4c22d2|1111eaeb+ 24c3111d2|2+ 4d02|1111d2|02eaeb+ 12d2|1111d2|2) +d2|1111(3η+ 3) =β(d2|1111)

12

2h2(6a4(3a6+d_2|4) + 72a6d_2|2−c1111c3111e_ae_b+ 2c2121c22e_ae_b+c22c42e_ae_b+ 2c22d_02|4e_ae_b+c22d_2|22e_ae_b

+4c42d2|02eaeb+ 4d02|4d2|02eaeb+ 12d2|2d2|4) +d2|4(3η+ 3) =β(d2|4)

2h2(a₄(c₂₄+ 2d_2|04) +b₄(c₂₄+ 4d_2|04) + 6b₆c₂₂e_ae_b+ 24b₆d_2|02e_ae_b+ 2c₂₂c₂₄e_ae_b+c₂₂c₄₂e_ae_b+ 2c₂₂d_02|04e_ae_b+

c22d2|22eaeb+ 12c24d2|2+ 4d02|04d2|02eaeb+ 12d2|04d2|2) +d2|04(3η+ 3) =β(d2|04)

2h2(a₄(2c₁₂₁₂+d_02|22) +b₄(2c₁₂₁₂+ 6c₂₄+ 3d_02|22) + 2(3b₆c₂₂e_ae_b−c₁₁₁₁c₁₃₁₁e_ae_b−c₁₁₁₁c₃₁₁₁e_ae_b−2c₁₁₁₁d_02|1111e_ae_b +2c1212c22eaeb+ 12c1212d02|02+c2121c22eaeb+ 4c2121d2|02eaeb+ 2c22c24eaeb+ 2c22c42eaeb+ 2c22d02|04eaeb+c22d02|22eaeb

+2c22d02|4eaeb+c22d2|22eaeb+ 18c24d02|02+ 6c42d2|02eaeb+ 6d02|02d02|22+ 2d2|02d2|22eaeb)) +d02|22(3η+ 3) =β(d02|22)

2h2(3b4c1311+ 2b4d02|1111−2c1111c2121eaeb−2c1111c24eaeb−2c1111d02|22eaeb+ 4c1311c22eaeb+ 24c1311d02|02

+2c22c3111eaeb+ 4c22d02|1111eaeb+ 2c22d2|1111eaeb+ 8c3111d2|02eaeb+ 12d02|02d02|1111+ 4d2|02d2|1111eaeb) +d02|1111(3η+ 3) =β(d02|1111)

2h2(6b4(3b6+d02|04) + 72b6d02|02−c1111c1311eaeb+ 2c1212c22eaeb+c22c24eaeb+c22d02|22eaeb+ 2c22d2|04eaeb

+4c24d2|02eaeb+ 12d02|02d02|04+ 4d2|02d2|04eaeb) +d02|04(3η+ 3) =β(d02|04)

2h2(a4(c42+ 4d02|4) + 6a6c22eaeb+ 24a6d2|02eaeb+b4(c42+ 2d02|4) +c22c24eaeb+ 2c22c42eaeb+c22d02|22eaeb

+2c22d2|4eaeb+ 12c42d02|02+ 12d02|02d02|4+ 4d2|02d2|4eaeb) +d02|4(3η+ 3) =β(d02|4)

13

2h2(a4(9a6+ 6d3|3) +eaeb(c22(c3111+d12|3)−c1111c2121)) +d3|3(3η+ 3) =β(d3|3)

2h2(a₄(c₃₁₁₁+ 3(c₄₂+d_12|3)) +e_ae_b(6a₆c₂₂−c₁₁₁₁c₂₄

−2c1111c42−2c1111d12|3+ 2c1212c22+ 2c1311c22+ 2c22c3111

+4c22c42+ 2c22d12|12+ 2c22d12|3+ 6c22d3|3) +b4(c3111+d12|3)) +d12|3(3η+ 3) =β(d12|3)

2h2(a4(4c2121+ 3c3111+ 2d21|21) +eaeb(−3a6c1111−c1111(2c1212+c1311

+2(2c2121+c3111+c42+ 2d21|21)) +c22(c1311+ 4c2121+ 2c24

+4c3111+ 4c42+ 3d21|03+ 4d21|21)) +b4c2121) +d21|21(3η+ 3) =β(d21|21)

2h2(b4(9b6+ 6d03|03) +eaeb(c22(c1311+d21|03)−c1111c1212)) +d03|03(3η+ 3) =β(d03|03)

2h2(a4(c1311+d21|03) +b4(c1311+ 3(c24+d21|03)) +eaeb(6b6c22

−2c1111c24−c1111c42−2c1111d21|03+ 2c1311c22+ 2c2121c22

+4c22c24+ 2c22c3111+ 6c22d03|03+ 2c22d21|03+ 2c22d21|21)) +d21|03(3η+ 3) =β(d21|03)

2h2(a4c1212+b4(4c1212+ 3c1311+ 2d12|12) +eaeb(−3b6c1111−c1111(4c1212

+2c1311+ 2c2121+ 2c24+c3111+ 4d12|12) +c22(4c1212+ 4c1311

+4c24+c3111+ 2c42+ 4d12|12+ 3d12|3))) +d12|12(3η+ 3) =β(d12|12)

IV. Intermediate steps in the proof of Theorem 7.2

In order to compute the quantum fluctuations reported in Theorem 7.2, one computed the F P

⁻¹

expansion. Computing to first oder and taking its supertrace one gets the long expression below; from a selected operator, its the large-N limit is taken according to the procedures and scalings described in the main text.

Tr

N²

(A) × (e

a

¯ a

4

− e

a

¯ c

1111

+ 24e

a

¯ d

_2|2

+ 2e

b

¯ d

_11|11

+ 2N ¯ d

_1|12

+ 6N ¯ d

_1|3

) + Tr

_N²

(B) × (2e

a

¯ d

_11|11

+ e

b

¯ b

4

− e

b

¯ c

1111

+ 24e

b

¯ d

_02|02

+ 6N ¯ d

_01|03

+ 2N ¯ d

_01|21

) + Tr

N

(A · A) × 2e

a

e

b

N ¯ d

_01|21

+ 4N

²

e

a

¯ d

_2|02

+ 12N

²

e

a

¯ d

_2|2

+2e

a

N¯ a

4

+ 2e

a

N¯ c

22

+ 6N ¯ d

_1|3

+ Tr

N

(B · B) × 2e

a

e

b

N ¯ d

_1|12

+ 12N

²

e

b

¯ d

_02|02

+ 4N

²

e

b

¯ d

_2|02

+2e

b

N ¯ b

4

+ 2e

b

N¯ c

22

+ 6N ¯ d

_01|03

+ Tr

N

(A · A · A · A) × 2N

²

e

a

¯ d

_2|4

+ 12e

a

N¯ a

6

+ 10e

a

N ¯ d

_1|5

+2N

²

e

_b

¯ d

02|4

+ 2e

b

N¯ c

42

+ 2e

b

N ¯ d

01|41

+ Tr

N

(B · B · B · B) × 2N

²

e

_a

¯ d

2|04

+ 2e

a

N¯ c

24

+ 2e

a

N ¯ d

1|14

+2N

²

e

_b

¯ d

02|04

+ 12e

b

N ¯ b

6

+ 10e

b

N ¯ d

01|05

+ Tr

N

(A · A · B · B) × 2N

²

e

_a

¯ d

2|22

+ 2e

a

N¯ c

42

+ 2e

a

N ¯ d

1|32

+2N

²

e

_b

¯ d

02|22

+ 2e

b

N¯ c

24

+ 2e

b

N ¯ d

01|23

+ Tr

N

(A · B · A · B) × 2N

²

e

_a

¯ d

2|1111

+ 2e

a

N¯ c

3111

+ 2e

a

N ¯ d

1|2111

+2e

b

N

²

¯ d

02|1111

+ 2e

b

N¯ c

1311

+ 2e

b

N ¯ d

01|1211

+ Tr

N

(A · A) Tr

N

(B · B) × (8e

a

N ¯ d

02|4

+ 2e

a

N ¯ d

2|22

+ 2e

a

¯ c

42

+ 6e

a

¯ d

12|3

+2e

b

N ¯ d

_02|22

+ 8e

b

N ¯ d

_2|04

+ 2e

b

¯ c

24

+ 6e

b

¯ d

_21|03

) + Tr

N

(A) Tr

N

(A · A · A) × (10e

a

N ¯ d

_1|5

+ 12e

a

N ¯ d

_3|3

+ 12e

a

¯ a

6

+ 16e

a

¯ d

_2|4

+2e

b

N ¯ d

_12|3

+ 2e

b

N ¯ d

_1|32

+ 2e

b

¯ c

3111

+ 2e

b

¯ d

_11|31

) + Tr

N

(A) Tr

N

(A · B · B) × (6e

a

N ¯ d

_12|3

+ 2e

a

N ¯ d

_1|32

+ 2e

a

¯ c

42

+ 4e

a

¯ d

_2|22

+4e

b

N ¯ d

_12|12

+ 2e

b

N ¯ d

_1|14

+ 2e

b

¯ c

1311

+ 2e

b

¯ d

_11|13

)

15