News on tensor network algorithms Romn Ors Donostia International - - PowerPoint PPT Presentation

Román Orús

Donostia International Physics Center (DIPC) December 6th 2018

News on tensor network algorithms

• S. S. Jahromi, RO, M. Kargarian, A. Langari, PRB 97, 115162 (2018)
• S. S. Jahromi, RO, PRB 98, 155108 (2018)
• S. S. Jahromi, RO, arXiv:1808.00680
• P. Schmoll, S. Singh, M. Rizzi, RO, arXiv:1809.08180
• A. Kshetrimayum, M. Rizzi, J. Eisert, RO, arXiv:1809.08258
• P. Schmoll, A. Haller, M. Rizzi, RO, arXiv:1812.01311

Applications welcomed

Entanglement and Tensor Networks Strongly correlated systems Quantum information and computation Quantum simulations Classical statistical mechanics Quantum chemistry Materials science HEP and lattice gauge theories Quantum gravity, string theory and AdS/CFT Numerical tensor calculus Nuclear physics AI, Deep learning & Linguistics

Tens nsor ne networks ks everyw ywhe here

Outli line ne

1) iPEPS for the ruby model 2) iPEPS & VBS phases on the star lattice 3) gPEPS and the structure matrix 4) iDMRG for SU(2)-invariant chiral ladder Overview, omit many technical details

Outli line ne

1) iPEPS for the ruby model 2) iPEPS & VBS phases on the star lattice 3) gPEPS and the structure matrix 4) iDMRG for SU(2)-invariant chiral ladder

• S. S. Jahromi, RO, M. Kargarian, A. Langari, PRB 97, 115162 (2018)

The he rub ruby mo model

Coarse-graining triangles leads to brickwall lattice Study with iPEPS

H = − X

α=x,y,z

Jα X

α−links

σα

i σα j

Reproduces the low-energy sector of the topological color code model in its gapped phase (Z2 x Z2 topological order)

• E. Bombin, M. A. Martin-Delgado, PRL 97, 180501 (2006)

ε0

−0.62 −0.61 −0.6 −0.59 −0.58 −0.57 −0.56 −0.55

1/D

0.1 0.2 0.3 0.4 0.5 0.6 Jc=(0.8, 0.8, 0.4)

iPEPS FU ED (N=24)

ε0

−1.14 −1.13 −1.12 −1.11 −1.1 −1.09 −1.08 −1.07 −1.06 −1.05

1/D

0.1 0.2 0.3 0.4 0.5 0.6

iPEPS FU ED (N=24)

J=(1, 1, 1)

Comparison to exact diag. for 24 sites. Good convergence. Multicritical point challenging.

Ground state energy

Multicritical point

Phase diagram

Jx + Jy + Jz = 2

3 phases separated by 2nd order QPTs. QPTs meet at a multicritical point. A1 phase: gapped, Z2 x Z2 TO. A2 & A3 phases: gapless. Gapped by a perturbation, colored Ising anyons.

• S. S. Jahromi, M. Kargarian, S. F. Masoudi, A. Langari, PRB 94,

125145 (2016)

Ene nergies a and nd pha hase di diag agram ram

d(λ1,λ2)=

C1 C2 C3 C4

a

C1 C2 C3 C4 Tu Td Tr Tl C1 C2 C3 C4 Tu Td C1 C2 C3 C4 Tr Tl

Ground state fidelity per site ln d(λ1, λ2) ≡ lim

N→∞

ln F(λ1, λ2) N

F(λ1, λ2) = |hΨ(λ2)|Ψ(λ1)i| ⇠ d(λ1, λ2)N

H.-Q- Zhou, RO, G. Vidal, PRL 100, 080601 (2008)

With CTMs

Compatible with other observables:

• Entanglement entropy
• 2-point correlators

Many open questions:

• Top. nature of A2 and A3?
• Multicritical point?

A2 - A3 QPT

Fideli lity fr from

• m CTMs

CTMs

Outli line ne

1) iPEPS for the ruby model 2) iPEPS & VBS phases on the star lattice 3) gPEPS and the structure matrix 4) iDMRG for SU(2)-invariant chiral ladder

Outli line ne

1) iPEPS for the ruby model 2) iPEPS & VBS phases on the star lattice 3) gPEPS and the structure matrix 4) iDMRG for SU(2)-invariant chiral ladder

• S. S. Jahromi, RO, PRB 98, 155108 (2018)

(a) (b) (c) (d)

σxσx σyσy σzσz

Coarse-graining leads to brick-wall lattice (different setups) Same iPEPS strategy than the ruby lattice

St Star Heisenb nberg Ant ntiferroma magne net

• J. Richter et al, PRB 70, 1 (2004)
• H. Yao, S. A. Kivelson, PRL 99, 247203 (2007)
• G. Kells et al, PRB 81, 104429 (2010)
• S. Dusuel et al, PRB 78, 125102 (2008)
• G. Misguich, P. Sindzingre, JPCM 19, 145202 (2007)
• B. J. Yang, A. Paramekanti, Y. B. Kim, PRB 81 134418 (2010)
• T. P. Choy, Y. B. Kim, PRB 80, 064404 (2009)
• S. J. Ran et al, PRB 97, 075146 (2018)
• G. Y. Huang, S. D. Liang, D. X. Yao, EPJB 86, 379 (2013)

Frustrated triangles. Ground state not fully clear yet. Chiral QSL, VBS, spin-1 trimerized...

H = Je X

hiji2e

Si · Sj + Jt X

hiji2t

Si · Sj

T=0 phase diagram as a function of Jt/Je?

(spin-1/2)

iPE iPEPS PS pha hase di diag agra ram m

Continuous QPT New phase, breaks C3 symmetry (4x4 unit cell, D=9) Agrees with previous results

Setup A Setup B

ε0

−0.256 −0.255 −0.254 −0.253 −0.252 −0.251 −0.25

1/D

0.1 0.2 0.3 0.4 0.5 0.6 2 3 Setup A Setup B

Je=0.05, Jt=1 ε0

−0.37524 −0.37522 −0.3752 −0.37518 −0.37516 −0.37514 −0.37512

1/D

0.1 0.2 0.3 0.4 0.5 0.6 2 3 Setup A Setup B

Je=1, Jt=0.05

Coarse-g

• graini

ning ng se setups tups

Jt/Je << 1 Jt/Je >> 1

Fast convergence! Better for large-D

Ebond

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1

1/D

0.1 0.15 0.2 0.25 2 3 4 3

Jt-bond1 Jt-bond2 Jt-bond3 Je-bond Je=0.05, Jt=1

(a) (b) (c)

Bond nd ene nergies

Degeneracy of the three configurations. RVB possible. iPEPS favors one of them because it has less entanglement. Jt/Je << 1 Jt/Je >> 1 Jt/Je >> 1

Outli line ne

1) iPEPS for the ruby model 2) iPEPS & VBS phases on the star lattice 3) gPEPS and the structure matrix 4) iDMRG for SU(2)-invariant chiral ladder

Outli line ne

1) iPEPS for the ruby model 2) iPEPS & VBS phases on the star lattice 3) gPEPS and the structure matrix 4) iDMRG for SU(2)-invariant chiral ladder

• S. S. Jahromi, RO, arXiv:1808.00680

Store information about the TN on a structure matrix Example: star lattice

Unit cell Links in unit cell Label of edge for tensor

Looping over the columns of the structure matrix we can authomatize tensor updates in imaginary-time evolution for essentially any lattice. Simple update (no environment): automatic! Full update (with environment): work in progress

Graph-b h-based P PEPS ( (gPEPS gPEPS)

Store information about the TN on a structure matrix Example: star lattice

Unit cell Links in unit cell Label of edge for tensor

Looping over the columns of the structure matrix we can authomatize tensor updates in imaginary-time evolution for essentially any lattice. Simple update (no environment): automatic! Full update (with environment): work in progress

Graph-b h-based P PEPS ( (gPEPS gPEPS)

All ll yo you ne need is is the he ma matrix

1d chain

All ll yo you ne need is is the he ma matrix

2d square

All ll yo you ne need is is the he ma matrix

2d triangular

All ll yo you ne need is is the he ma matrix

2d kagome

All ll yo you ne need is is the he ma matrix

3d pyrochlore

Some me ground nd stat state ene nergies

1d 2d 3d 2d 3d 2d 2d

Very simple, efficient, and fully-atomatized way of getting a first idea about the ground state in many cases Simple update, and mean field environment (only lambdas) for observables. Not variational.

Antiferromagnetic Heisenberg model on the cubic lattice

An An example le

H = X

ij

Si · Sj

(spin-1/2)

Outli line ne

1) iPEPS for the ruby model 2) iPEPS & VBS phases on the star lattice 3) gPEPS and the structure matrix 4) iDMRG for SU(2)-invariant chiral ladder

Outli line ne

1) iPEPS for the ruby model 2) iPEPS & VBS phases on the star lattice 3) gPEPS and the structure matrix 4) iDMRG for SU(2)-invariant chiral ladder

• P. Schmoll, S. Singh, M. Rizzi, RO, arXiv:1809.08180
• P. Schmoll, A. Haller, M. Rizzi, RO, arXiv:1812.01311

H = X

i

Ji Si · (Si+1 × Si+2)

Chi hiral int nteractions ns on

• n a

a la ladder

(spin-1/2)

Multiple motivations: wire constructionism, simpler case than 2d, etc

H1 H1 = −~ S1 · ⇣ ~ S2 × ~ S3 ⌘ + ~ S2 · ⇣ ~ S3 × ~ S4 ⌘

1 2 3 4

H2 H2 = +~ S1 · ⇣ ~ S2 × ~ S3 ⌘ − ~ S2 · ⇣ ~ S3 × ~ S4 ⌘

1 2 3 4

H3 H3 = −~ S1 · ⇣ ~ S2 × ~ S3 ⌘ − ~ S2 · ⇣ ~ S3 × ~ S4 ⌘

1 2 3 4

H4 H4 = +~ S1 · ⇣ ~ S2 × ~ S3 ⌘ + ~ S2 · ⇣ ~ S3 × ~ S4 ⌘

1 2 3 4

H = X

i

Ji Si · (Si+1 × Si+2)

Singlet ground state S2 = 0 Non-singlet ground state

Chi hiral int nteractions ns on

• n a

a la ladder

(spin-1/2)

Study with SU(2) iDMRG

(omit tech. details)

1) Kadanoff blocking on triangles

Heff = −sgn(J1J2)|J1| + |J2| 3 √ 3 X

n

˜ Sn · ˜ Sn+1

2) Currents by exact diagonalization

J z

ij = i

2(S+

i S− j − S− i S+ j )

H1 pbc 1st excited state (triplet) H3 obc g.s. <S2> = 12, Sz = 3

Fi First st int ntuition

H = X

i

Ji Si · (Si+1 × Si+2)

sgn(J1J2) = -1 Spin-1/2 AFH chain: gapless, c=1 CFT sgn(J1J2) = +1 Spin-1/2 FH chain: non-critical

(1,2) = nn triangles

Compatible with original intuition S

˜ S

101 102 103 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 100 101 102 103 1.5 2 2.5 3 3.5 4

Entanglement entropy: c=1 CFT

• 0.57898
• 0.578975
• 0.57897
• 0.578965
• 0.57896
• 0.578955
• 0.57895

• 0.57898
• 0.578975
• 0.57897
• 0.578965
• 0.57896
• 0.578955
• 0.57895

H1 : : ene nergy a and nd ent ntropy

Algebraic energy convergence: gapless

100 101 102 103 10-4 10-3 10-2 10-1 100 100 101 102 103 10-7 10-6 10-5 10-4 10-3 10-2 10-1

Critical correlation functions Spin-spin Vertical dimer-dimer

C(r) ≈ 1 r

Matches continuum limit

C(r) ≈ 1 r

5 4

Similar algebraic decays for other correlators

H1 : : correla lation func nctions ns

H ≈ g Z dx (JL,1 · JR,2 − JR,1 · JL,2)

E.g., P.-Huang et al, PRB 95, 144413 (2017)

100 101 102 103 5 10 15 20 25 30

Same entang. spectrum Same boundary CFT Same (1+1)d-CFT SU(2)1 WZW theory Compatible with bosonization, Kadanoff blocking, etc Many open questions: H3? More legs? WZW limit?

H1 : : ent ntang ngle leme ment nt spectrum spectrum

Entanglement spectrum of half an infinite chain comparison with spin-1/2 AFH chain

OUTLOOK

1) Phase diagram and phase transitions of the topological ruby model with iPEPS 2) AFH model on the star lattice has a competition between VBS phases. iPEPS shows that one of them is a new phase with a 6-site unit cell 3) Structure matrix can be used to authomatize (at least some) PEPS methods to any lattice 4) SU(2) iDMRG simulations of a chiral ladder suggest that it corresponds to a SU(2)1 WZW theory