[PPT] - Detecting topological orders from Matrix-Product State based PowerPoint Presentation

SLIDE 1 International Workshop on Tensor Networks and Quantum Many-Body Problems 2016 Max Planck Institute for the Physics of Complex Systems Frank Pollmann Detecting topological orders from   Matrix-Product State based simulations

SLIDE 2 (1) Matrix-product states and efficient simulations

Review: Entanglement and matrix-product states (MPS)
MPS for infinite systems
Time evolving block decimation (TEBD)

(2) Extracting fingerprints of topological order

Symmetry protected topological phases
Characterizing intrinsic topological orders
Symmetry enriched topological phases
(Dynamics of topological excitations)

Overview Detecting topological orders from   Matrix-Product State based simulations

SLIDE 3 Matrix-product states ψj1,j2,j3,j4,j5 ≈

SLIDE 4 Entanglement

A generic quantum state has a dimensional Hilbert space

|ψi = X j1,j2,...,jL ψj1,j2,...,jL|j1i|j2i . . . |jLi ... A B ...

Decompose a state into a superposition  
f product states (Schmidt decomposition)

Entanglement entropy as a measure for the

amount of entanglement  S = − P α Λ2 α log Λ2 α , jn = 1 . . . d

Equivalent to with

S = −TrρA log ρA ρA = TrB|ψihψ|

SLIDE 5 L R (a) (b) N (d) (c) Many body Hilbert space Area law states L R (a) (b) N (d) (c) Many body Hilbert space Area law states Area law for ground states of local (gapped) Hamiltonians   in one dimensional systems Entanglement All ground states live in a tiny corner of the Hilbert space! S(L) = const. [Srednicki ’93, Hastings ’07] |ψi = X α Λα|αiA ⌦ |αiB L R (a) (b) N (d) (c) Many body Hilbert space Area law states L

SLIDE 6

Example:

|ψ = NA X i=1 NB X j=1 Cij|iA|jB = X γ λγ|φγA|φγB Compression of quantum states C =    0.23 · · · 0.56 . . . ... . . . 0.22 · · · 0.34    =    0.23 · · · 0.56 . . . ... . . . 0.22 · · · 0.34    χ = 1200

Matrix can represent an image (array of pixel)

≈

Reconstruction of the matrix (image) from a small

number of Schmidt states (SVD):

SLIDE 7 χ = 4 χ = 16 χ = 64 χ = 256 χ = 1200 Important features visible already for states! < 16 Compression of quantum states

SLIDE 8 Compression of quantum states [Mondrian]

SLIDE 9 Compression of quantum states

Coefficients in the many-body wave function:

Rank-L tensor: diagrammatic representation

ψj1,j2,j3,j4,j5 =

L R (a) (b) N (d) (c) Many body Hilbert space Area law states

Successive Schmidt decompositions: matrix-product states
|ψi =

d X j1=1 d X β=1 A[1]j1 β Λ[1] β |j1i|βi[2,...N]

SLIDE 10 Matrix-Product States

Matrix-product states: Reduction of variables:

ψj1,j2,j3,j4,j5 ≈

Aj

αβ = [M. Fannes et al. 92, Schuch et al ‘08]

Matrix-product operators
M

M M M M

M

M M M M (b)

(a)
M

M M M M (c)

*

* * * * Oj0 1,j0 2,j0 3,j0 4,j0 5 j1,j2,j3,j4,j5 = [1] [2] [3] [4] [5] [Verstraete et al ’04] dL → Ldχ2

SLIDE 11 Infinite matrix-product states

∞

SLIDE 12 A A A A A A A [1] [2] [3] [4] [5] [6] [7]

Infinite and translationally invariant systems:

. . . . . . |ψi : Ldχ2 → dχ2 Infinite MPS and the canonical form

MPS is not unique

= ˜ Ain ➡ describes the same state! ˜ Ain = XAinX−1

SLIDE 13

Choose a convenient representation in Canonical Form:

Bond index corresponds to Schmidt decomposition!

Write tensor as product of

Ain αβ : Diagonal matrix with Schmidt values : Tensor relating to Schmidt basis

(d)

(b)

(a)

(c) (e) R L ' ' '

L

L

(d)

(b)

(a)

(c) (e) R L ' ' '

L

L

(d)

(b)

(a)

(c) (e) R L ' ' '

L

L |ψi = χ X α=1 Λα|αiL ⌦ |αiR with hα|α0i = δαα0 Infinite MPS and the canonical form [Vidal ‘03]

SLIDE 14

Schmidt states in terms of the MPS:
(d)

(b)

(a)

(c) (e) R L ' ' '

L

L

(d)

(b)

(a)

(c) (e) R L ' ' '

L

L Infinite MPS and the canonical form

Orthogonality:

R ' ' ' * * *

SLIDE 15

Conditions for the canonical form:
(d)

(b)

(a)

(c) (e) R L ' ' '

L

L

(d)

(b)

(a)

(c) (e) R L ' ' '

L

L

Left and right transfer matrix have dominant eigenvalue
ne and the corresponding eigenvector is the identity

T L αα0;ββ0 = X j ΛαΛα0 Γj αβ

Γj

α0β0 ∗ Infinite MPS and the canonical form

SLIDE 16

Example: Affleck-Kennedy-Lieb-Tasaki (AKLT)
Ground state the spin-1 Hamiltonian

H = X j ~ Sj ~ Sj+1 + 1 3(~ Sj ~ Sj+1)2, Infinite MPS and the canonical form

SLIDE 17

Example: Affleck-Kennedy-Lieb-Tasaki (AKLT)

Γ−1 = r 4 3σ+, Γ0 = − r 2 3σz, Γ1 = − r 4 3σ− Λ = r 1 2 ✓ 1 1 ◆ and d = 3 χ = 2 Infinite MPS and the canonical form

SLIDE 18

Efficient evaluation of expectation values:

Infinite MPS and the canonical form Λ2 Λ2 Γ Γ∗ hψ|On|ψi = Λ2 Λ2 Γ Γ∗ Γ Γ∗ Γ Γ∗ Γ Γ∗ Γ Γ∗ Λ Λ Λ Λ Λ Λ Λ Λ T R hψ|OmOn|ψi =

SLIDE 19

Correlation length of an MPS:

⇠ = − 1 log |✏2|, is the second largest eigenvalue of the transfer matrix ✏2

Degeneracy of largest

eigenvalue (unity) shows  that the MPS is a cat-state   Infinite MPS and the canonical form

(d)

(b)

(a)

(c) (e) R L ' ' '

L

L

SLIDE 20 Efficient numerical simulations

i~ ∂ ∂t|ψi = H|ψi

SLIDE 21

Transverse field Ising model

Ordered Quantum Disordered g/J H = − X j (Jσz j σz j+1 + gσx j ) m = hσzi Efficient numerical simulations

Ground state properties

σz(t) ... ... σ+ 0 |ψi

Dynamics

SLIDE 22 Time evolving block decimation

Assume we have a Hamiltonian of the form

H = X j h[j,j+1] |ψ0i = lim τ→∞ exp(Hτ)|ψii || exp(Hτ)|ψii||

Time evolution in imaginary time

|ψti = exp(iHt)|ψt=0i

Time evolution in real time

Disclaimer: Simple!  Not perfect for all uses!

SLIDE 23

Consider a Hamiltonian

H = X j h[j,j+1] Time evolving block decimation [F [r], F [r0]] = 0 ([G[r], G[r0]] = 0) [G, F] 6= 0

We observe

but F F F G G H=F+G F ≡ X even j F [j] ≡ X even j h[j,j+1] G ≡ X

dd j

G[j] ≡ X

dd j

h[j,j+1]

Decompose the Hamiltonian as

[Vidal ‘03]

SLIDE 24

Apply Suzuki-Trotter decomposition of order p

    with , , etc. exp (−i(F + G)δt) ≈ fp [exp(−Fδt), exp(−Gδt)] f1(x, y) = xy f2(x, y) = x1/2yx1/2 Time evolving block decimation UF = Y even r exp(−iF [r]δt) UG = Y

dd r

exp(−iG[r]δt)

Two chains of two-site gates

SLIDE 25

Time Evolving Block Decimation algorithm (TEBD)

1 1 2 2 3 3 4 4 5 5 6 7 1 1 2 2 3 3 4 4 5 5 6 7 Time evolving block decimation

How do we get the original form back?

SLIDE 26 truncation χ3 Time evolving block decimation

Scales with the matrix dimension as
Time Evolving Block Decimation algorithm (TEBD)

SLIDE 27 ΓA Λ A ΓBΛ B Λ B (i) (ii) Λ A (iii) X Y (iv) ΓA Λ A ΓBΛ B Λ B ~ ~ ~ ~ ( )

1

SVD Θ ~ Λ A X Y ~ Λ B Λ B Λ B

1

Λ B ( ) = Θ ~ E0 L R L R ~ ~ ΓA Λ B ~ (Γ ) A * Λ B ~ ΓBΛ B ~ (Γ ) B *Λ B ~ Θ ~ M M Θ (v) truncation χ3

Scales with the matrix dimension as
Density matrix renormalization group (DMRG)

Time evolving block decimation

SLIDE 28

Assume that is translational invariant and :

infinite Time Evolving Block Decimation algorithm (iTEBD) |ψi N = ∞

Time evolution achieved by repeated local application  
f gates (parallel)

Γ[2r] = ΓA, λ[2r] = λA, Γ[2r+1] = ΓB, λ[2r+1] = λB

A

B B B B A A A A A

Partially break translational symmetry to simulate

the action of the gates infinite Time evolving block decimation [Vidal ‘07]

SLIDE 29 Python!

Python + numpy provide useful tools to simply implement

the algorithm as key functions are already implemented Xijk = X m YimZmjk X=tensordot(Y,Z,axes=(1,0)) X=reshape(X,(dim1*dim2,dim3)) Xijk → X(ij)k X=transpose(X,(0,2,1)) Xijk → Xikj

SLIDE 30

Finite entanglement scaling: Entanglement and correlation

length are always finite in an MPS Infinite MPS and the canonical form   FP , S. Mukerjee, A.M. Turner, J. Moore Phys. Rev. Lett. 102, 255701 (2009).

SLIDE 31

Finite entanglement scaling: Entanglement and correlation

length are always finite in an MPS Infinite MPS and the canonical form

SLIDE 32 Infinite MPS and the canonical form S = c 6 log ξ c = 1/2

Finite entanglement scaling: Extract central charge c

➡ Ising critical  point: [Calabrese & Cardy ‘04]

SLIDE 33 Extracting fingerprints of topological order

SLIDE 34

Topological phases cannot be described

by symmetry breaking

Quantum Hall effects
(gapped) spin-liquids 
Topological insulators
Haldane spin chain

[Klitzing ’80, Tsui ’82, Laughlin ’83] [Haldane ‘83] [Kane & Mele ’05] Topological phases of matter

Fascinating features: quantized conductance,

fractionalization, protected edge states, … [Anderson ‘73]

SLIDE 35

Spin-1 Heisenberg chain
Haldane phase: Gapped + no symmetry breaking

[Haldane ‘83] H = P j ~ Sj · ~ Sj+1 E ... ... E ×4 [Affleck et al ‘87]

Spin-1/2 excitations at the edges: Protected by symmetry

Symmetry protected topological phases

Edge spins have been observed in the

NMR profile close to the chain ends of   Mg-doped Y2BaNiO5 [S.H. Glarum, et al., Tedoldi et al. ‘99]

SLIDE 36   FP , A.M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010). 

Hamiltonian and ground state symmetric under

|ψ0i … = |ψ0i : ➡ Boundary: Projective representations  (e.g., spin-1/2) UgUh = eiφ(g,h)Ugh U L g U R g g, h ∈ G ➡ Bulk: Linear on-site representation  (e.g., spin-1) uguh = ugh |ψ0i : Symmetry protected topological phases (…is complete ) [Chen et al. ’11; Schuch et al ’11] ➡ Classification of Symmetry protected topological phases H2[G, U(1)]

Classified by the second cohomology group

[Schur 1911] ug ug ug ug ug ug ug ug

SLIDE 37 Which symmetries can stabilize topological phases?

Example : Rotation about single axis 
Redefining removes any phase

Zn Rn = 1 ⇒ U n R = eiφ1 ˜ UR = e−iφ/nUR   FP , A.M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010). 

Example : Phase for pairs
Phases cannot be gauged

away: Distinct topological phases Z2 × Z2 RxRz = RzRx ⇒ UxUz = eiφxzUzUx φxz = 0, π Symmetry protected topological phases

SLIDE 38 1D symmetry protected topological phases FP and A.M. Turner, Phys. Rev. B 86, 125441 (2012). 

Characteristic fingerprints of SPT’s from DMRG

|ψi ≈ H = P j Sj · Sj+1 + D P j(Sz j )2 Z2 × Z2 S = 1 O ∝ tr(UxUzU † xU † z) |0i|0i|0i|0i|0i

stabilizes the

Haldane phase Ug [Ug]αα0 = hφR α| O j∈L gj|φR α0i ➡ Projective representations can directly be extracted P α λα|φL αi|φR αi

SLIDE 39

Transformation of an MPS under symmetry operations

              ...wave function only changes by a phase 

U

U e i

Σ

Σ , [UΣ, Λ] = 0 [Perez-Garcia ’07]

a
a
a
a
b
b
b
b
a
a
a
a
b
b
b
b
1D symmetry protected topological phases

SLIDE 40 | {z } T ΣX=X | {z } T 1=1

Get from the dominant eigenvector  
f the generalized transfermatrix ( )

Overlap of transformed Schmidt states
U

e iL

⇔

Σ

U

e iL

UΣ

X UΣ = X†

U

e iL

T Σ

(αα0);(ββ0) = X j,j0 Σjj0 ˜ Γj0,αβΓ∗ j,α0β0ΛβΛβ0 UΣ = X†

U

U e i ˜ 1D Symmetry protected topological phases

SLIDE 41 (1) Matrix-product states and efficient simulations

Review: Entanglement and matrix-product states (MPS)
MPS for infinite systems
Time evolving block decimation (TEBD)

(2) Extracting fingerprints of topological order

Symmetry protected topological phases
Characterizing intrinsic topological orders
Symmetry enriched topological phases
(Dynamics of topological excitations)

Overview Detecting topological orders from   Matrix-Product State based simulations

SLIDE 42 Intrinsic topological order

SLIDE 43

Tait’s smoke ring experiment in 1867
Vortex rings are extremely stable and long lived!

Topological Order

SLIDE 44

Lord Kelvin developed a theory about matter being knots in

the aether: Inequivalent knots correspond to   different atoms

Turned out it is not true… Michelson–Morley

experiment: No aether Topological Order

SLIDE 45

Still, knots remained a very interesting topic:

Figure out which knots are equivalent

Equivalent knots can be related to each other using

the Reidemeister moves

Jones Polynomial distinguish inequivalent knots

: t + t2 − t4 [Jones ’84] : 1 Topological Order :?

SLIDE 46

Topological Quantum Field

Theory (TQFT): Field Theory   in which amplitudes depend only  

n the topology of the process

Chern-Simons TQFT: Amplitude of a process given by the

Jones Polynomial of the knot

2D TQFT: Anyonic quasiparticle excitations!

Z = Z DφeiS[φ] [Witten ’80] t ≡ eiφ Topological Order

SLIDE 47 eiφ A B |ψi = P α ppα |φA αi |φB α i S = − P α pα log pα Abelian: γ = log( p #anyons) Topological Order

Intrinsic topological order: Gapped quantum phases

that are robust to any small (local) perturbation

Characterized by quasiparticle excitations

that obey fractional statistics “anyons”

Topological degeneracy (= number of anyons)
Topological entanglement entropy :

γ [Kitaev and Preskill ’06, Levin and Wen ‘06] [Wen ’90] S = αL − γ

SLIDE 48 Fractional Quantum Hall

2D electron gas in magnetic field :

highly degenerate “Landau levels”   

Number of degenerate orbits in each Landau level equal

to number of flux quanta: Filling fraction

Incompressible liquid at integer fillings
Fractional quantum Hall effect (FQHE):

Incompressible liquid due to interactions En = heB m (n + 1 2) B B NB ν = Ne/NB | {z } NB Integer QH Fractional QH [Tsui, Stormer ’82, Laughlin ’83] [Klitzing ’80] n = 1

SLIDE 49 Fractional Quantum Hall

M. P

. Zaletel, R. S. K. Mong, FP , PRL 110, 236801 (2013). 

M. P

. Zaletel, Roger S. K. Mong, FP , and E. H. Rezayi, Phys. Rev. B 91, 045115 (2015).

Consider the FQHE on an infinitely long cylinder
Orbitals are localized along the cylinder: Quasi 1D model

using an occupation number basis          x y B [Haldane & Rezayi ’94; Bergholtz et al. ’05, Seidel et al. ’05] | . . . , j0, j1, . . .i ˆ H = X n X k≥|m| Vkmc† n+mc† n+kcn+m+kcn

Coulomb interactions yield quantum many-body problem

Coulomb |0 1 1 0 i |ψ0i : . . . . . . B B B B B B B Bin αβ

SLIDE 50 Fractional Quantum Hall

M. P

. Zaletel, R. S. K. Mong, FP , PRL 110, 236801 (2013). 

M. P

. Zaletel, Roger S. K. Mong, FP , and E. H. Rezayi, Phys. Rev. B 91, 045115 (2015).

Topological entanglement entropy of the FQHE with Coulomb

interactions (“minimally entangled states”) S = αL − γ ν = 1/3 ν = 7/3

SLIDE 51 Fractional Quantum Hall

M. P

. Zaletel, R. S. K. Mong, FP , J. Stat. Mech. P10007 (2014). 

M. P

. Zaletel, R. S. K. Mong, FP , PRL 110, 236801 (2013). (see also Zhang et al. ’12, Tu et al. ’13, Cincio & Vidal ‘13)

Extracting topological content by adding a “twist”

Momentum polarization: topological spin,

central charge, Hall viscosity x y L ξ ν = 2/5 … …

SLIDE 52

M. P

. Zaletel, R. S. K. Mong, FP , Z. Papic arXiv:1505.02843. Fractional Quantum Hall [Xia et al ’04] six primary fields of   the Z3 parafermion  CFT:1, ψ, ψ†, ε, σ, σ†. Momentum Energy 6 3 1 1 x [Read Rezayi ’98, Kitaev ’03]

Charge e/5 quasiparticles: Fibonacci anyons

“Universal topological quantum computing” eiφ

Finding the ground state with filling factor

(Coulomb + finite well thickness at ) ν = 12/5 L = 28`B

SLIDE 53

M. P

. Zaletel, R. S. K. Mong, FP , Z. Papic arXiv:1505.02843. Density profile for anyons with charge e/5 y/`B x/`B Fractional Quantum Hall

SLIDE 54

UΣ

Symmetry enriched topological order

SLIDE 55 No symmetry Trivial Topological  

rder X

With symmetry Topological  

rder Xb

Topological  

rder Xa

Topological  

rder

Y Symmetry enriched topological order

Intrinsic topological order is characterized by its

anyonic quasi particles (QP) (no symmetry required)

Topologically ordered systems have a richer structure

when symmetries are present [Wang and Wen ’12, Wen ’13] [Wen ’91]

SLIDE 56 g ∈ Hψ

Hamiltonian and ground state

have the same symmetry   ( ) 

Fractionalization of symmetry  
perations in terms of the anyonic

quasiparticles (QP) ➡ QP induce projective representations:

Co-homology : Inequivalent projective

representations classify different SET’s (not completely!) Symmetry enriched topological order GH = Gψ H2[G, U(1)] Ug U † g gh = k : UgUh = eiφghUk

SLIDE 57

Topological degeneracy: Ground state degeneracy on

the torus is equal to the number of QP

Minimally entangled ground states (MES) of a topological

system on torus are eigenstates of Wilson loop operators   parallel to the bipartite cut 

QP of type a and are localized at the edges of the cut

Symmetry enriched topological order Wy (a) (b) A B W y [Zhang et al. ’12; Grover et al. ’12; Cincio & Vidal ’12, Wen ’90] (b) A B

a

a x y

SLIDE 58

MES can also be obtained for an infinite cylinder

(locally equivalent to a large torus)

A liquid (e.g., toric code) has

four quasiparticle types

Map cylinder to an effective one-dimensional system:

extract projective representations of QPs like in 1D SPTP Z2 1 e f m [Kitaev ’01]

Symmetry enriched topological order

[Cincio & Vidal ’12, Zaletel et al ‘13] [Cirac et al ’11]

SLIDE 59

bosons model state:

loop gas of AKLT chains

and -particles in projective

( ) representation

Represented by tensor product state

Z2 e f S = 1/2 S = 1 χ = 3 QP 1 e m f UxUzU −1 x U −1 z 1

1

1

1

UΣ Symmetry enriched topological order C.-Y. Huang, X. Chen, and FP , Phys. Rev. B 90, 045142 (2014)

SLIDE 60

RVB state on the kagome lattice:

spin liquid states (spin 1/2 singlets)

and -particles in projective

( ) representation

Represented by tensor product state

Z2 e f S = 1/2 χ = 3 (a) (b) (c) 1.5 (b) (d) Symmetry enriched topological order QP 1 e m f UxUzU −1 x U −1 z 1

1

1

1

UΣ C.-Y. Huang, X. Chen, and FP , Phys. Rev. B 90, 045142 (2014)

SLIDE 61

“String order (SO) parameter”: SO detects the projective

representations of anyons

Selection rule forces SO to vanish if edge spins are

fractionalized  5 10 15 20 0.0 0.1 0.2 0.3 0.4 4 L-1 SO(n) n m-MES L=2 L=3 L=4 L=5 L=6 L=7 4 8 12 16 20 0.0 0.1 0.2 0.3 0.4 n f-MES L=2 L=3 L=4 L=5 L=6 L=7 5 10 15 20 0.0 0.1 0.2 0.3 0.4 n e - MES L=2 L=3 L=4 L=5 L=6 L=7 5 10 15 20 0.0 0.1 0.2 0.3 0.4 n I - MES L=2 L=3 L=4 L=5 L=6 L=7 SO = hψ0| n Y j=1 exp(iSz j )|ψ0i 1 e f m n ... ... XL XR ... (a) (b) a Symmetry enriched topological order C.-Y. Huang, X. Chen, and FP , Phys. Rev. B 90, 045142 (2014)

SLIDE 62 Dynamics of topological excitations

SLIDE 63 Kitaev-Heisenberg model

Density-Matrix Renormalization Group
Infinite cylinders with circumference up to
Extract both static and dynamic properties

L = 12 [White ‘92]

SLIDE 64 Kitaev-Heisenberg model

Kitaev model on the honeycomb lattice is

an exactly solvable quantum spin liquid   [Kitaev ’06]

Fractionalized excitations:

gapless majoranas and static gapped fluxes

Kitaev-Heisenberg model

[Jackeli and Khaliullin ‘09]             Experimental relevance: Iridates, alpha-RuCl3,… hi, ji

SLIDE 65 Kitaev-Heisenberg model

Phase diagram of the Kitaev-Heisenberg model

[Jackeli and Khaliullin ‘09] [Matthias Gohlke, Ruben Verresen, FP , Roderich Moessner]

SLIDE 66

Dynamical structure factor:

S(~ q, !) = X n

h n|S+

~ q | 0i

2

(! + !0 !n) [Mourigal et al, ’13] |f i

Magnetic field - Spin waves

No magnetic field - Fractionalized spinons

Fractional spinon excitations in the quantum

Heisenberg antiferromagnetic chain (CuSO4·5D2O) Dynamical Response Dynamical Response

SLIDE 67 Spin-1 Heisenberg S(k, ω) C(x, t) = hψ0|S− x (t)S+ 0 (0)|ψ0i S(k, ω) = X x Z ∞ −∞ dte−i(kx+ωt)C(x, t)

Dynamical structure factor

Dynamical Response

SLIDE 68 1 + t X x Hx | {z } ✏∼L2t2 ≈ 1 + t X x Hx + t2 X x<y HxHy + t3 X x<y<z HxHyHz + . . . Neglect overlapping   terms in expansion Compact matrix product 

perator representation

α β jn W [n]jnj0 n αβ = j0 n

Hamiltonian expressed as a sum of terms

Expand for : H = P x Hx t ⌧ 1 U = exp(−itH) | {z } ✏∼Lt2 1 + t X x Hx → Y x (1 + tHx) [Zaletel et al ‘15] MPO based time evolution

SLIDE 69 D D − 1 dimensional  Hamiltonian MPO dimensional  time evolution MPO

Matrix product operators….

MPO based time evolution

SLIDE 70

Dynamical correlation functions in the

Haldane Shastry model [Haldane & Zirnbauer ’93] HHS = X x,r>0 Sx · Sx+r r2 . MPO based time evolution

SLIDE 71 Dynamical correlations of the Kitaev-HB model anisotropy + Heisenberg [Matthias Gohlke, Ruben Verresen, FP , Roderich Moessner]

SLIDE 72 Dynamical correlations of the Kitaev-HB model [Matthias Gohlke, Ruben Verresen, FP , Roderich Moessner]