Introduction to iPEPS (second lecture) Philippe Corboz, Institute - - PowerPoint PPT Presentation

Introduction to iPEPS

Philippe Corboz, Institute for Theoretical Physics, University of Amsterdam

(second lecture)

• Part I: iPEPS ansatz
• Part II: Contraction of PEPS / iPEPS
• Interlude: Example application: the Shastry-Sutherland model
• Part III: Optimization

✦ Imaginary time evolution ✦ Variational optimization (iterative energy minimization)

• Part IV: Computational cost & benchmarks

✦ How large does D need to be? ✦ Comparison with 2D DMRG ✦ Comments on extrapolations

• Part

V: Advanced tensor network applications

✦ SU(N) Heisenberg models

• Outlook & summary

Outline

PART III: Optimization

MPS

Structure

Variational ansatz

iterative optimization

• f individual tensors

(energy minimization) imaginary time evolution Contraction of the tensor network exact / approximate

Find the best (ground) state

|˜ Ψ

Compute

• bservables

˜ Ψ|O|˜ Ψ⇥

PEPS 2D MERA 1D MERA

✓ ✓

Summary: Tensor network algorithm for ground state

Optimization via imaginary time evolution

• Idea:

exp(−τ ˆ Hb)

• At each step: apply a two-site operator to a bond and truncate bond back to D

Keep D largest singular values

U √ ˜ s √ ˜ sV

SVD s

U V †

Time Evolving Block Decimation (TEBD) algorithm

Note: MPS needs to be in canonical form

... ...

exp(β ˆ H)|Ψii

β → ∞

|ΨGSi

τ = β/n exp(−β ˆ H) = exp(−β X

b

ˆ Hb) = exp(−τ X

b

ˆ Hb) !n ≈ Y

b

exp(−τ ˆ Hb) !n

Trotter-Suzuki decomposition:

• 1D:

Optimization via imaginary time evolution

Cluster update Wang, Verstraete, arXiv:1110.4362 (2011)

• 2D: same idea: apply

to a bond and truncate bond back to D

exp(−τ ˆ Hb)

• However, SVD update is not optimal (because of loops in PEPS)!

simple update (SVD)

★ “local” update like in TEBD ★ Cheap, but not optimal (e.g. overestimates magnetization in S=1/2 Heisenberg model)

Jiang et al, PRL 101 (2008)

full update

★ Take the full wave function into account for truncation ★ optimal, but computationally more expensive ★ Fast-full update [Phien et al, PRB 92 (2015)]

Jordan et al, PRL 101 (2008)

Optimization: simple update

• iPEPS with “weights” on the bonds (takes environment effectively into account)

A B C D

..." ..." ..." ..."

λ1 λ2 λ5 λ4 λ3 λ3 λ6 λ6 λ8 λ8 λ7 λ7

..." ..." ..." ..."

ΓA ΓB Γ

ΓD

λ6

λ1

λ2

λ3

=

A ΓA

keep only D largest singular values

˜ Γ

˜ λ

=

Θ

SVD$

˜ Γ

λ6

λ3

λ2

=

Γ'A ˜ Γ

λ8

λ3

λ4

=

Γ'B ˜ Γ

A

Jiang, et al., PRL 101, 090603 (2008)

• Update works like in 1D with iTEBD (infinite time-evolving block decimation)

g λ1 λ6 λ3 λ2 λ8 λ4 λ3 Θ

=

ΓA ΓB

• G. Vidal, PRL 91, 147902 (2003)

Trick to make it cheaper

• Idea: Split off the part of the tensor which is updated

g λ1 λ6 λ3 λ2 λ8 λ4 λ3 ΓA ΓB

=

g

=

U sV T

=

g

SVD$

=

λ6

−1

λ3

−1

λ2

−1

=

Γ'A ˜ Γ

A

λ8

−1

λ3

−1

λ4

−1

=

Γ'B ˜ Γ

B

g

=

˜ Γ

A

˜ λ

1

˜ Γ

B

keep only D largest singular values

Optimization: full update

• Approximate old PEPS + gate with a new PEPS with bond dimension D

g A B

Environment

|˜ Ψi = g|Ψi

Environment

A' B'

≈

|Ψ0i

≈

• The full wave function is taken into account for the truncation!
• Environment has to be computed: expensive... but optimal!

Jordan, Orus, Vidal, Verstraete, Cirac, PRL (2008) Corboz, Orus, Bauer, Vidal, PRB 81, 165104 (2010)

• Minimize
• Iteratively / CG / Newton / ...

|| |˜ Ψi |Ψ0i ||2 = h˜ Ψ|˜ Ψi + hΨ0|Ψ0i h˜ Ψ|Ψ0i hΨ0|˜ Ψi

Full-update: details

• Split off the part of the tensor

which is updated

=

X p A

=

Y q B

=

Environment

• f p and q

tensors U

(sV)

d(p0, q0) = h˜ Ψ|˜ Ψi + hΨ0|Ψ0i h˜ Ψ|Ψ0i hΨ0|˜ Ψi

“Cost-function”

+

• q

p q p p0 q0 p0 q0 g g p† q† g† q p p† q†

≈

find new p’, and q’ to minimize: || |˜

Ψi |Ψ0i || |˜ Ψi = g|Ψ(p, q)i |Ψ0(p0, q0)i

2

Finding p’ and q’ through sweeping

• Initial guess with SVD:

= p0

0 q0

q p

• Keep q’ fixed and optimize with respect to p’

∂ ∂p0⇤ d(p0, q0) = 0

Mp0

b

=

• Solve linear system:

new p’

SVD

= 0

[ [

∂ ∂p0⇤

+

• p0†

p0† p0

=

p0 g

Finding p’ and q’ through sweeping

• Initial guess with SVD:

= p0

0 q0

q p

• Keep q’ fixed and optimize with respect to p’:

∂ ∂p0⇤ d(p0, q0) = 0

Mp0

b

=

• Solve linear system:

new p’

• Keep p’ fixed and optimize with respect to q’:

∂ ∂q0⇤ d(p0, q0) = 0

=

• Solve linear system:

new q’

˜ Mq0

˜ b

• Repeat above until convergence in

d(p0, q0)

SVD

g

• Retrieve full tensors again:

=

X p' A'

=

Y q' B'

Optimization: full update

• Approximate old PEPS + gate with a new PEPS with bond dimension D

g A B

Environment

|˜ Ψi = g|Ψi

Environment

A' B'

≈

|Ψ0i

≈

• The full wave function is taken into account for the truncation!
• At each step the environment has to be computed! expensive... but optimal!

Jordan, Orus, Vidal, Verstraete, Cirac, PRL (2008) Corboz, Orus, Bauer, Vidal, PRB 81, 165104 (2010)

• Minimize
• Iteratively / CG / Newton / ...

|| |˜ Ψi |Ψ0i ||2 = h˜ Ψ|˜ Ψi + hΨ0|Ψ0i h˜ Ψ|Ψ0i hΨ0|˜ Ψi

Optimization: simple vs full update

simple update

★ “local” update like in TEBD ★ Cheap, but not optimal (e.g. overestimates magnetization in S=1/2 Heisenberg model)

full update

★ Take the full wave function into account for truncation ★ optimal, but computationally more expensive

0.1 0.2 0.3 0.4 0.5 0.3 0.32 0.34 0.36 0.38 0.4 0.42 1/D m [%] Simple update Full update

exact result

Example: 2D Heisenberg model

• Combine the two: Use simple update to get an initial state for the full update
• Don’t compute environment from scratch but recycle previous one

fast full update

Phien, Bengua, Tuan, PC, Orus, PRB 92 (2015)

• 1. Select one of the PEPS tensors A

Variational optimization for PEPS

E = hΨ|H|Ψi hΨ|Ψi

tensor A reshaped as a vector tensor network including all Hamiltonian terms tensor network from norm term

• 2. Optimize tensor A (keeping all the others fixed) by minimizing the energy:

H x = E N x

solve generalized eigenvalue problem

minimize

Verstraete, Murg, Cirac, Adv. Phys. 57 (2008)

in 1D:

ˆ H

H = N =

• 1. Select one of the PEPS tensors A

Variational optimization for PEPS

E = hΨ|H|Ψi hΨ|Ψi

tensor A reshaped as a vector tensor network including all Hamiltonian terms tensor network from norm term

• 2. Optimize tensor A (keeping all the others fixed) by minimizing the energy:

H x = E N x

solve generalized eigenvalue problem

minimize

Verstraete, Murg, Cirac, Adv. Phys. 57 (2008)

• 3. Take the next tensor and optimize (keeping other tensors fixed)
• 4. Repeat 2-3 iteratively until convergence is reached

Variational optimization for iPEPS

E = hΨ|H|Ψi hΨ|Ψi

tensor A reshaped as a vector tensor network including all Hamiltonian terms tensor network from norm term

H x = E N x

minimize

B C F G D A H D A H E B C F G D A H E D A H E B C F G D A H E D A H E E

) iPEPS

Main challenges:

• 1. Need to take into account infinitely many Hamiltonian contributions

✦ Solution: use corner-transfer matrix method [PC, arXiv:1605.03006] ✦ Alternative: use “channel-environments” [Vanderstraeten et al, PRB 92, arxiv:1606.09170] ✦ Or: Use PEPO (similar to 3D classical) [cf. Nishino et al. Prog. Theor. Phys 105 (2001)]

• 2. Tensor A appears infinitely many times! (Min. problem highly non-linear)

✦ Take adaptive linear combination of old and new tensor [PC, arXiv:1605.03006]

[see also Nishino et al. Prog. Theor. Phys 105 (2001), Gendiar et al. PTR 110 (2003)]

✦ Alternative: use CG approach [Vanderstraeten, Haegeman, PC, Verstraete, arXiv:1606.09170]

H-environment

E = hΨ|H|Ψi hΨ|Ψi

tensor network including all Hamiltonian terms tensor network from norm

H x = E N x

minimize

✓

C1 C2 C3 C4 T1

T2 T3 T4 a

• Need additional H-environment tensors:

C2 C3 C4 T1 T2 T3 T4 a

˜ C1

... ... ... ...

+ . . .

... ... ... ... ... ... ... ... ... ... ... ...

+ + +

=

˜ C1 ➡ taking into account all Hamiltonian contributions in the infinite upper left corner

But how about H ??

H-environment

+ + + + + + + + + + + + + + + + + + +

=

hΨ| ˆ H|Ψi =

=

+

+ +

=

+ +

=

+ . . ˜ = =

=

=

˜

= =

= =

Local terms

Terms between a corner and an edge tensor Corner terms

H-environment: bookkeeping

CTM left move:

=

+

=

= = = + + = = +

= = = ˜ C0

˜ T 04 ˜ C0

˜ C0

T 0o

=

... and similarly for right-, top-, bottom-move

• We can sum up all Hamiltonian contributions in an iterative way

= +

= + = = + + = = +

= = = = = = ˜ C0

˜ C0

˜ C0

˜ TT 04

=

Practical scheme

E = hΨ|H|Ψi hΨ|Ψi

tensor A reshaped as a vector

H x = E N x

minimize

• However, the solution A of the GEVP is NOT the optimum

~

λ ∈ [0.5, 1.5]

• Find which minimizes (using only a few steps)

A0(λ)[x,y] = ˜ A[x,y] sin λπ − A[x,y] cos λπ.

• Make ansatz for solution A’

see also [Nishino et al. PTR 105 (2001)]

E(λ)

• Repeat iteratively for all tensors in the unit cell

Practical scheme

E = hΨ|H|Ψi hΨ|Ψi

tensor A reshaped as a vector

H x = E N x

minimize

• However, the solution A of the GEVP is NOT the optimum

~

λ ∈ [0.5, 1.5]

• Find which minimizes (using only a few steps)

A0(λ)[x,y] = ˜ A[x,y] sin λπ − A[x,y] cos λπ.

• Make ansatz for solution A’

see also [Nishino et al. PTR 105 (2001)]

E(λ)

• Repeat iteratively for all tensors in the unit cell

Minimization scheme

• If E(λ = 0.5) < E(λ = 1), A0 = ˜

A, exit

• Define initial step size ∆0 = 0.1
• If E(1 + h) < E(1), then ∆ = ∆0, else ∆ = −∆0
• Do

– If E(1 + ∆) < E(1) → λ = 1 + ∆, exit – else ∆ = ∆/2

Comparison: Heisenberg model

2 4 6 0.3 0.32 0.34 0.36 0.38 0.4 0.42 D m E /J

• Energy and order parameter are substantially improved

with the variational optimization

• Highest accuracy (D=6): -0.66941
• Extrapolated QMC result: -0.66944 [Sandvik&Evertz 2010]

2 4 6 10

−5

10

−4

10

−3

10

−2

D ∆ Es

simple update full update variational update

Benchmark: Heisenberg model (D=4)

− −0.6685 − −0.6675 0.05 0.1 0.15 −0.669 −0.6688 −0.6686 −0.6684 τ full update variatonal update

s

Es/J

D=4

• Full-update: lower accuracy

even in the limit

τ → 0

Es/J

• Faster convergence,

similar computational cost

200 400 −0.669 −0.6685 −0.668 −0.6675 seconds 0.15

Run-time on laptop

Trotter step τ

Benchmark: t-J model

4 5 6 7 8 9 10 11 −1.55 −1.5 −1.45 −1.4 −1.35 D Ehole/t full update variational update

Better results by performing a few (3-5) additional steps with the variational optimization

PC, arXiv:1605.03006

Benchmark: Shastry-Sutherland model (SrCu2(BO3)2)

10 20 30 40 −0.3865 −0.386 −0.3855 −0.385 −0.3845 −0.384 −0.3835 −0.383 −0.3825 −0.382 β or iteration Es

full update variational update

Initial state: Néel state Full update: gets stuck! Variational optimization: converges to correct plaquette state (plaquette state)

J0/J = 0.7, D = 4

ˆ H = J0 X

hi,ji

Si · Sj + J X

hhi,jiidimer

Si · Sj + h X

i

Sz

i

J0 J

Benchmark: Shastry-Sutherland model + external field

ˆ H = J0 X

hi,ji

Si · Sj + J X

hhi,jiidimer

Si · Sj + h X

i

Sz

i

Oninzuka, et al.

1/8 plateau 1/4 plateau 1/3 plateau

Onizuka, et al., JPSJ 69 (2000)

★ iPEPS: crystals of bound states (for m<1/4)

[PC, F. Mila, PRL 112 (2014)]

★ Previous theories: crystals of triplets

simple update: not symmetric state! Bound state of 2 triplet excitations

2 4 6 0.02 0.04 0.06 0.08 0.1 Symmetry error D simple update full update variational

Better results also for large unit cells!

Overview: optimization in iPEPS

• Imaginary time evolution

✦ Simple update:

Jiang et al, PRL 101 (2008)

✦ Cluster update:

Wang et al, arXiv:1110.4362

✦ Full update:

Jordan et al, PRL 101 (2008)

✦ Fast-full update:

Phien et al, PRB 92 (2015)

cheap and simple, but not accurate improved accuracy high accuracy, more expensive

• Energy minimization / variational optimization

✦ DMRG-like sweeping:

PC, arXiv:1605.03006

✦ CG-approach:

Vanderstraeten, Haegeman, PC, Verstraete, arXiv:1606.09170

✦ ... more to explore...! ✦ See also variational optimization in the context of 3D classical models

Nishino et al. Prog. Theor. Phys 105 (2001), Gendiar et al. Prog. Theor. Phys 110 (2003)

higher accuracy, similar cost as FFU higher accuracy, similar cost as FFU high accuracy, cheaper than FU

+ COMBINATIONS!

MPS

Structure

Variational ansatz

iterative optimization

• f individual tensors

(energy minimization) imaginary time evolution Contraction of the tensor network exact / approximate

Find the best (ground) state

|˜ Ψ

Compute

• bservables

˜ Ψ|O|˜ Ψ⇥

PEPS 2D MERA 1D MERA

✓ ✓ ✓

Summary: Tensor network algorithms (ground state)

PART IV: Computational cost & benchmarks

Computational cost

It depends on the amount of entanglement in the system!

• How large does have to be?

D

• Leading cost: O(Dk)

(i)PEPS

D

Bond dimension 1 2 3 4 5 6 7 8

MPS

Bond dimension D

MPS for 2D system: D~exp(W) accurate for cylinders up to a certain width

Nvar ∼ D4 cost ∼ (Nvar)2.5

polynomial scaling but large exponent! MPS: k = 3 PEPS: k ≈ 10

Comparison 2D DMRG & iPEPS: 2D Heisenberg model

iPEPS D=6

(variational optimization)

iPEPS D=6 in the thermodynamic limit ~ 2’600 variational pars. MPS D=3000 on finite Ly=10 cylinder ~ 18’000’000

∼

similar accuracy

4 orders of magnitude fewer parameters (per tensor)

2D DMRG and iPEPS provide

complementary

results!!!

Stoudenmire & White, Ann. Rev. CMP 3 (2012)

0.05 0.1 0.15 0.2

1/c

• 0.44
• 0.435
• 0.43

E/site

2D (est.) Torus

DMRG

MERA

Upper Bound Cylinder

Series (HVBC) DMRG, Cyl, Odd DMRG, Cyl, Even DMRG, Torus (Jiang...) Lanczos, Torus

Heisenberg model on the Kagome lattice

iPEPS

accuracy comparable to 2D DMRG YC10-4, m=6000

Classification by entanglement (2D)

high Entanglement low gapped systems band insulators, valence-bond crystals, s-wave superconductors, ... gapless systems with area law Heisenberg model, d-wave / p-wave SC, Dirac Fermions, ... systems with “1D fermi surface” free Fermions, Fermi-liquid type phases, bose-metals?

S(L) ∼ L log L

It depends on the amount of entanglement in the system!

• How large does have to be?

D

Non-interacting spinless fermions (old iPEPS results)

fast convergence with D in gapped phases slow convergence in phase with 1D Fermi surface

Hfree =

• rs⇥

[c†

rcs + c† scr − γ(c† rc† s + cscr)] − 2λ

• r

c†

rcr

1 2 3 4 1 2 3 4 λ γ

critical gapped

1D Fermi surface

1 1.5 2 2.5 3 10

10

10

10

10

10

λ Relative error of energy

γ=0 D=2 γ=0 D=4 γ=1 D=2 γ=1 D=4 γ=2 D=2 γ=2 D=4

D: bond dimension

Li et al., PRB 74, 073103 (2006)

QMC from C(L/2,L/2) 0.2 0.4 0.6 1/D iPEPS QMC extrap. linear fits quadratic fits

Benchmark S=1/2 Heisenberg model: Order parameter

strong finite D effects how to extrapolate? m ≈ 0.295 − 0.314

• rel. error ~ 2%

0.05 0.1 0.3 0.32 0.34 0.36 0.38 0.4 0.42 1/L staggered magnetization QMC from Ms2 QMC from C(L/2,L/2) QMC extrap.

strong finite size effects accurate extrapolation m = 0.30743(1)

Sandvik & Evertz, PRB 82 (2010)

H = J

• i,j⇥A

SiSj +

• i,j⇥B

SiSj

. . . . . . . . . . . .

J

Wenzel, Janke, PRB 79 (2009)

1 Phase diagram

Distinguish between ordered / disordered phase?

2nd order phase transition

0.3969 Dimer phase

S S S S S S S S S S

m=0 Néel phase m>0

0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 1/D m J=0.39 0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 1/D m J=0.3969 0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 1/D m J=0.4

m=0 m~0 m>0

Distinguish between ordered / disordered phase?

Dimer phase

S S S S S S S S S S

J

1

2nd order phase transition

Phase diagram Néel phase 0.3969 m=0 m>0

0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 1/D m J=0.39 0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 1/D m J=0.3969 0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 1/D m J=0.4

m=0 m~0 m>0

Distinguish between ordered / disordered phase?

★ Extrapolations in D are important to distinguish between

• rdered phase and a disordered one!

★ A better understanding how to accurately extrapolate in D would be very useful [work in progress...] ★ Goal: systematic finite bond-dimension scaling analysis (similar to finite-size scaling analysis)

Improving energy extrapolations

Motivation:

PC, PRB 93 (2016) 0.1 0.2 0.3 0.4 0.5 0.6 −1.37 −1.36 −1.35 −1.34 −1.33 −1.32 −1.31 −1.3 1/D E E(1/D) exact result

Bad estimate based

• n 1/D data!

Extrapolate in truncation error instead!

0.1 0.2 0.3 0.4 0.5 0.6 −1.37 −1.36 −1.35 −1.34 −1.33 −1.32 −1.31 −1.3 1/D or w E E(1/D) E(w) exact result

ˆ H = −t P

⇣ ˆ c†

cjσ + H.c. ⌘ + P

⇣ ˆ c†

c†

c†

c†

⌘

• 1/D extrapolations are typically not accurate
• Extrapolations important to identify true ground state among

several competing states (e.g. Hubbard model)

Truncation error in the full update algorithm

g A B

Environment

|˜ Ψi = g|Ψi

Environment

A' B'

≈

|Ψ0i

≈

w(D) = C(D, β → ∞)/τ

C = || |˜ Ψi |Ψ0i ||

• Cost function:
• Truncation error:

Benchmark: Hubbard model U/t=8, half filling

0.05 0.1 0.15 −0.526 −0.524 −0.522 −0.52 −0.518 −0.516 −0.514 E 1/D or w

E(1/D) E(w) DMET extrap. DMRG extrap. FNMC extrap. AFQMC extrap. AFQMC error

Example: vertical vs diagonal stripe, U/t=8, δ=1/8

0.05 0.1 0.15 0.2 0.25 0.3 0.35 −0.77 −0.76 −0.75 −0.74 −0.73 −0.72 −0.71 −0.7 −0.69 E 1/D or w

E(1/D): vertical stripe E(w): vertical stripe E(1/D): diagonal stripe E(w): diagonal stripe FNMC 20x20 FNMC 16x16 DMRG W=6 extr. AFQMC extr. DMET 5x2 cell

PC, PRB 93 (2016)

[data from LeBlanc, et al., PRX 5 (2015)]

Exploiting global abelian symmetries

Singh, Pfeifer, Vidal, PRB 83 (2011) Bauer, Corboz, Orus, Troyer, PRB 83 (2011)

Hamiltonian with symmetry, e.g. particle number conservation (U(1))

H =

) (

N=0

N=1

N=2 larger bond dimension D can be afforded

Tensors can also be written in block form

reduces computational cost

Part V: iPEPS applications

Examples of iPEPS simulations

• interacting spinless fermions
• honeycomb & square lattice
• t-J model & Hubbard model
• square lattice / honeycomb lattice
• SU(N) Heisenberg models
• N=3 square, triangular, kagome & honeycomb lattice
• N=4 square, honeycomb & checkerboard lattice
• N=5 square lattice
• N=6 honeycomb lattice
• frustrated spin systems
• Shastry-Sutherland model
• Heisenberg model on kagome lattice
• Bilinear-biquadratic S=1 Heisenberg model
• Heisenberg-Kitaev model
• and many more...

iPEPS is a very competitive variational method! Find new physics thanks to (largely) unbiased simulations

SU(N) Heisenberg models (square lattice)

• N=2:

local basis states: | "i, | #i

H = X

hi,ji

SiSj Néel order

| o i, | o i

local basis states:

H = X

hi,ji

Pij

SU(N) Heisenberg models (square lattice)

i j i j

Pij

• N=3

Ground state??

• N=2:
• N=4

Néel order

• N=3
• N=4

87Sr

I = 9/2 :

Nuclear spin

Nmax = 2I + 1 = 10

Identify nuclear spin states with colors:

| o i | o i | o i | o i

|Iz = 1/2i |Iz = 3/2i |Iz = 1/2i |Iz = 3/2i

SU(N) Heisenberg models (square lattice)

| o i, | o i

local basis states:

H = X

hi,ji

Pij

i j i j

Pij

• N=3

Ground state??

• N=2:
• N=4

Néel order

• N=3
• N=4

SU(N) Heisenberg models (square lattice)

Néel order | o i, | o i

local basis states:

H = X

hi,ji

Pij

i j i j

Pij

• N=2:

Cannot use QMC because

• f the sign problem!!!

• N=3
• N=4

SU(N) Heisenberg models (square lattice)

Néel order | o i, | o i

local basis states:

H = X

hi,ji

Pij

i j i j

Pij

• N=2:

3 sub-lattice Néel order

iPEPS & DMRG (Bauer, Corboz, et al. PRB 85, 2012)

ED & flavor-wave theory (Toth et al., PRL 105, 2010)

• N=3
• N=4

SU(N) Heisenberg models (square lattice)

Néel order | o i, | o i

local basis states:

H = X

hi,ji

Pij

i j i j

Pij

• N=2:

New ground state found with iPEPS

Corboz, Läuchli, Mila, Penc, Troyer, PRL ’11

spin-orbital liquid

Wang&Vishwanath ‘09

0.1 0.2 0.3 0.4 0.5 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7 −0.65 −0.6 Es 1/D

2x2 unit cell 4x2 u.c. 4x4 u.c. VMC

V V

1 2 3 4 1 2

V V V V A B B A H H

Corboz, Läuchli, Penc, Troyer, Mila, PRL 107, 215301 (2011)

“Dimer-Néel” order

rotation (SU(2))

SU(4) Heisenberg model: iPEPS results

✓ .

supported by exact diagonalization results

V V

1 2 3 4 1 2

V V V V A B B A H H

“Dimer-Néel” order

Dimerization and SU(4) symmetry breaking

0.1 0.2 0.2 0.4 0.6 0.8 ∆ Eb 1/D

translational symmetry broken

0.1 0.2 0.1 0.2 0.3 0.4 m 1/D

SU(4) symmetry broken

Square lattice: iPEPS results

Corboz, Läuchli, Penc, Troyer, Mila, PRL 107, 215301 (2011)

SU(4) Heisenberg model: iPEPS results

spin-orbital liquid

Wang&Vishwanath ‘09

0.1 0.2 0.3 0.4 0.5 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7 −0.65 −0.6 Es 1/D

2x2 unit cell 4x2 u.c. 4x4 u.c. VMC

Linear flavor-wave theory:

(a)

“Dimer-Néel” order ?

✓

Corboz, Läuchli, Mila, Penc, Troyer, PRL 107, 215301 (2011)

SU(4): Study as a function of bond dimension

D

D=1

product state (mean-field)

infinite ground state degeneracy studying extrapolations

• f the data for

D=5-16 suggests that state is stable D=16

largest D tested in this study

D

increase quantum fluctuations / increase entanglement

D=2

add quantum fluctuations

• n top of product state

same solution as linear flavor-wave theory (a) plaquette state

D=5

Dimer-Néel ordered state

Adding quantum fluctuations systematically...

D

1

control quantum fluctuations (entanglement)

∞

D

Adding quantum fluctuations systematically...

1 ∞

control quantum fluctuations (entanglement) product state

Adding quantum fluctuations systematically...

D

1 ∞

control quantum fluctuations (entanglement) slightly entangled state

Adding quantum fluctuations systematically...

D

1 ∞

control quantum fluctuations (entanglement)

Adding quantum fluctuations systematically...

D

1 ∞

control quantum fluctuations (entanglement)

Adding quantum fluctuations systematically...

D

1 ∞

control quantum fluctuations (entanglement)

D

Adding quantum fluctuations systematically...

1 ∞

control quantum fluctuations (entanglement) strongly entangled state

Summary: SU(4) Heisenberg model

square lattice: Dimer-Néel order

Corboz, Läuchli, Penc, Troyer, Mila, PRL 107, 215301 (2011)

Checkerboard lattice: Quadrumerized!

Corboz, Penc, Mila, Läuchli, PRB 86 (2012)

consistent with ED results up to N=20 (by Andreas Läuchli)

also called “simplex solid state”, see Arovas, PRB 77 (2008)

• r “valence cluster state”, see Hermele&Gurarie, PRB 84 (2011)

square lattice: Dimer-Néel order

Checkerboard lattice: Quadrumerized!

Corboz, Läuchli, Penc, Troyer, Mila, PRL 107, 215301 (2011) Corboz, Penc, Mila, Läuchli, PRB 86 (2012)

Summary: SU(4) Heisenberg model

Honeycomb lattice: spin-orbital (4-color) liquid

Corboz, Lajkó, Läuchli, Penc, Mila, PRX 2 (2012)

SU(4) Honeycomb results

Low-D iPEPS solutions

Dimer-Néel ordered? 2x2 unit cell: Color ordered? (same as LFWT) 4x4 unit cell: pictures only reflect short range physics!

NO! Order parameter vanishes in the large D limit!

0.1 0.2 0.1 0.2 0.3 0.4 0.5 1/D m iPEPS 2x2 unit cell iPEPS 4x4 unit cell

NO! Order parameter vanishes in the large D limit!

0.1 0.2 0.1 0.2 0.3 0.4 0.5 ∆ Eb 1/D iPEPS 2x2 unit cell

The ground state seems to be disordered, i.e. a spin-orbital liquid!

Corboz, Lajko, Läuchli, Penc, Mila, PRX 2 (2012)

SU(N) Heisenberg models

)

SU(3) honeycomb: Plaquette state

Corboz, Läuchli, Penc, Mila, PRB 87 (2013)

SU(3) kagome: Simplex solid state

Corboz, Penc, Mila, Läuchli, PRB 86 (2012)

SU(5) square: Color order SU(3) square/triangular: 3-sublattice Néel order

Generalizations

• ther lattices
• ther values of N
• ther representations
• XXZ-type models

3-color quantum Potts: superfluid phases

Bauer, Corboz, et al., PRB 85 (2012) Messio, Corboz, Mila, arXiv:1304.7676

SU(3) honeycomb lattice: competing states!

) )

0.1 0.2 0.3 0.4 0.5 −0.75 −0.7 −0.65 −0.6 −0.55 1/D or 1/N Es 2x2 unit cell dimerized state plaquette state ED

Study of the competing states as a function of D is crucial! Plaquette state Dimerized & color-

• rdered state

The plaquette state wins!

Corboz, Läuchli, Penc, Mila, PRB 87 (2013) Zhao et. al, PRB 85 (2012) Lee&Yang, PRB 85 (2012)

0.02 0.04 0.06 0.08 0.1 −1.1 −1.05 −1 −0.95 −0.9 Energy per site 1/D or 1/Ns iPEPS color−ordered state iPEPS plaquette state VMC chiral state VMC plaquette state ED

a)

SU(6) honeycomb lattice: competing states!

Color-ordered state

Nataf, Lajkó, PC, Läuchli, Penc, Mila, PRB 93 (2016)

Plaquette state

The plaquette state wins... but only for D>30 (full update)!

Outlook & summary

Improvements of 2D TN methods

2D Tensor networks

Monte Carlo sampling Improve extrapolations: finite-D scaling analysis Combinations, e.g.:

• variational wavefunctions
• fixed node MC

Symmetries Parallelization Better optimization / contraction algorithms

Extensions of 2D tensor networks methods

Finite temperature Real-time evolution Ground states Excitation spectra 3D Continuous space

2D Tensor networks

Conclusion

✓ 1D tensor networks: State-of-the-art (MPS, DMRG) ✓ 2D tensor networks: A lot of progress in recent years!

★ iPEPS can outperform state-of-the-art variational methods! ★ Novel phases in SU(N) Heisenberg models ★ New insights into the competing states in the t-J & Hubbard model ★ New understanding of the magnetization process in SrCu2(BO3)2

✓ Big room for improvement. Many possible extensions.

It’s an exciting time to work on tensor networks!

Thank you for your attention!