(Almost) 25 Years of DMRG - What Is It About? Ulrich Schollwck - - PowerPoint PPT Presentation

(Almost) 25 Years of DMRG - 
 What Is It About?

Ulrich Schollwöck

University of Munich

Outline

Concerning your talk (65min + 10 min for questions) we would prefer if you could provide a broad introduction into DMRG. This may include in particular: 1) Brief historic discussion 2) Why does one use DMRG? 3) Concept of entanglement and correlations and area laws 5) Details on how to implement DMRG/How does it work? 6) benchmark calculations and comparison to results obtained by resorting to different methods 7) comments on scaling behavior (w.r.t. number of electrons, dimension of truncated 1-particle Hilbert space and "bond length \chi") 8) Open problems While working out your presentation please be aware of related talks (for instance the

• ne by Markus Reiher, "DMRG in Quantum Chemistry", building up on your presentation).

fundamental problem of solid state

what do we need DMRG for? problem class:
 
 fundamental Hamiltonian (without lattice vibrations…!):

kinetic
 energy electron-electron
 interaction lattice 
 potential

we don’t know how to solve the Schrödinger equation!
 
 problem: electron-electron interactions

H =

e−

X

j=1

p2

j

2me + 1 2 1 4⇡✏0 q2

e

|ri − rj| +

e−

X

j

Veff(rj)

many-body problem of solid state I

scenario I
 valence electrons well delocalized
 interactions well screened

lattice potential electron cloud energy DOS half-filled conductor

many metals, semiconductors: single-electron picture OK
 
 density functional theory (DFT)

many-body problem of solid state II

scenario II
 valence electrons tightly bound
 strong local interactions

lattice potential energy DOS half-filled insulator
 


• eg. high-Tc

parent compounds

many particle picture: strongly correlated materials
 
 model Hamiltonian methods - here DMRG comes in!

why strong correlations?

0 dimensions

magnetic impurity physics quantum dots

1 dimension

spin chains & ladders Luttinger liquid

doping T Fermi liquid Non-Fermi liquid Néel order superconductivity strange metal pseudo gap

2 dimensions

frustrated magnets high-Tc superconductors

3 dimensions

realistic modelling: transition metal, rare earth compounds

transition metal oxides and rare earths

belated filling of the d- and f-shells
 valence electrons quite tightly bound

ultra-cold bosonic atoms form Bose-Einstein condensate
 
 standing laser waves: optical lattice

Greiner et al (Munich group), Nature ’02
 
 


extension to fermions (spin = hyperfine levels)

cold atoms in optical lattices

U t

e.g. M. Köhl et al (Esslinger group), PRL ‘05 interaction tunable via lattice depth bosonic Hubbard model

tunability

controlled tuning of interaction U/t in time via lattice depth adiabatic change of U/t: quantum phase transition

momentum distribution function

sudden change of U/t to Mott insulator: collapse and revival „state engineering“ for generic quantum many-body systems

adiabatic increase of interaction U time superfluid Mott insulator

compression of information

compression of information necessary and desirable

diverging number of degrees of freedom emergent macroscopic quantities: temperature, pressure, ...

classical spins

thermodynamic limit: degrees of freedom (linear)

quantum spins

superposition of states thermodynamic limit: degrees of freedom (exponential)

N → ∞ N → ∞ 2N

2N

classical simulation of quantum systems

compression of exponentially diverging Hilbert spaces what can we do with classical computers?

exact diagonalizations

limited to small lattice sizes: 40 (spins), 20 (electrons)

stochastic sampling of state space

quantum Monte Carlo techniques negative sign problem for fermionic systems

physically driven selection of subspace: decimation

variational methods renormalization group methods how do we find the good selection? DMRG!

DMRG: a young adult

09.11.1992 S.R. White: Density Matrix Formulation for Quantum Renormalization Groups (PRL 69, 2863 (1992))

„This new formulation appears extremely powerful and versatile, and we believe it will become the leading numerical method for 1D systems; and eventually will become useful for higher dimensions as well.“

~2004 old insight „DMRG is linked to MPS (Matrix Product States)“ goes viral (some) reviews:

• U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005) - „old“ statistical physics perspective, applications
• U. Schollwöck, Ann. Phys. 326, 96 (2011) - „new“ MPS perspective, technical

F. Verstraete,

• V. Murg, J. I. Cirac, Adv. Phys. 57, 143 (2008) - as seen from quantum information

Östlund, Rommer, PRL 75, 3537 (1995), Dukelsky, Martin-Delgado, Nishino, Sierra, EPL43, 457 (1998) Vidal, PRL 93, 040502 (2004), Daley, Kollath, Schollwöck, Vidal, J. Stat. Mech. P04005 (2004),
 White, Feiguin, PRL 93, 076401 (2004), Verstraete, Porras, Cirac, PRL 93, 227205 (2004), Verstraete, Garcia-Ripoll, Cirac, PRL 93, 207204 (2004), Verstraete, Cirac, cond-mat/0407066 (2004)

matrix product states: definitions

{σi} i ∈ {1, 2, . . . , L} | "i, | #i H = ⌦L

i=1Hi

Hi = {|1ii, . . . , |dii} |ψi = X

σ1,...,σL

cσ1...σL|σ1 . . . σLi {σ} = σ1 . . . σL c{σ}

quantum system living on L lattice sites d local states per site example: spin 1/2: d=2 Hilbert space: most general state (not necessarily 1D): abbreviations:

(matrix) product states

Hi = {| "ii, | #ii} H = H1 ⊗ H2 |ψi = c↑↑| ""i + c↑↓| "#i + c↓↑| #"i + c↓↓| ##i c↑↓ 6= c↑c↓ |ψi = 1 p 2 | "#i 1 p 2 | #"i cσ1 · cσ2 · . . . · cσL → M σ1 · M σ2 · . . . · M σL cσ1...σL = cσ1 · cσ2 · . . . · cσL

exponentially many coefficients! standard approximation: mean-field approximation

dL → dL coefficients

• ften useful, but misses essential quantum feature: entanglement

consider 2 spin 1/2: singlet state: generalize product state to matrix product state:

matrix product states

|ψi = X

σ1,...,σL

M σ1M σ2 . . . M σL|σ1σ2 . . . σLi (1 × D1), (D1 × D2), . . . , (DL−2 × DL−1), (DL−1 × 1) M σi → M σiX M σi+1 → X−1M σi+1 XX−1 = 1

useful generalization even for matrices of dimension 2: AKLT (Affleck-Kennedy-Lieb-Tasaki) model
 general matrix product state (MPS):
 matrix dimensions:
 non-unique: gauge degree of freedom


matrix product states

Why are matrix product states interesting?


any state can be represented as an MPS 
 (even if numerically inefficiently) MPS are hierarchical: matrix size related to degree of entanglement MPS emerge naturally in renormalization groups MPS can be manipulated easily and efficiently MPS can be searched efficiently: 
 which MPS has lowest energy for a given Hamiltonian?

popular notation: (left) singular vectors
 key workhorse of MPS manipulation and generally very useful! general matrix A of dimension then with dim. (ON col); if :

• dim. diagonal: non-neg.:

singular values, non-vanishing = rank

• dim. (ON row); if : 


singular value decomposition (SVD)

(m × k) U †U = I m = k UU † = I (k × k) si ≥ 0 r ≤ k s1 ≥ s2 ≥ s3 ≥ . . . (k × n) V †V = I k = n V V † = I |uii U = [|u1i|u2i . . .] (m × n) k = min(m, n) A = USV † U S V †

SVD and EVD (eigenvalue decomp.)

si ≥ 0 s1 ≥ s2 ≥ s3 ≥ . . . AU = UΛ λi U = [|u1i|u2i . . .] A†A = V SU †USV † = V S2V † ⇒ (A†A)V = V S2 AA† = USV †V SU † = US2U † ⇒ (AA†)U = US2 A†A AA† A = USV †

singular value decomposition (always possible):
 eigenvalue decomposition (for special square matrices):
 eigenvectors
 connection by „squaring“ A:
 eigenvalues = singular values squared eigenvectors = left, right singular vectors


slice U into d matrices:


any state can be decomposed as MPS

cσ1σ2...σL → Ψσ1,σ2...σL = X

a1

Uσ1,a1Sa1,a1V †

a1,σ2...σL

Uσ1,a1 → {Aσ1} with Aσ1

1,a1 = Uσ1,a1

ca1σ2σ3...σL = Sa1,a1V †

a1,σ2...σL

cσ1σ2...σL = X

a1

Aσ1

1,a1ca1σ2σ3...σL

ca1σ2σ3...σL → Ψa1σ2,σ3...σL = X

a2

Ua1σ2,a2Sa2,a2V †

a2,σ3...σL

Aσ2

a1,a2 = Ua1σ2,a2

cσ1σ2...σL = X

a1,a2

Aσ1

1,a1Aσ2 a1,a2ca2σ3σ3...σL

reshape coefficient vector into matrix of dimension and SVD:
 slice U into d row vectors:
 rearrange SVD result:
 reshape coefficient vector into matrix of dim. and SVD:
 rearrange SVD result: and so on!


(d × dL−1)

(d2 × dL−2)

bipartition of „universe“ AB into subsystems A and B: read coefficients as matrix entries, carry out SVD:


Schmidt decomposition

1 {|j〉B} {|i〉A} L ℓ+1 ℓ

|ψi =

dim HA

X

i=1 dim HB

X

j=1

ψij|iiA|jiB |ψi =

r

X

α=1

sα|αiA|αiB |αiA =

dim HA

X

i=1

Uiα|iiA |αiB =

dim HB

X

j=1

V ∗

jα|jiB

Schmidt decomposition


• rthonormal

sets!


arbitrary bipartition
 AAAAAAAA AAAAAAAAAAAAAAA reduced density matrix and bipartite entanglement

bipartite entanglement in MPS

S = −

• α

wα log2 wα

ˆ ρS =

• α

wα|αS⟩⟨αS|

|ψ⟩ =

M

• α

√wα|αS⟩|αE⟩

≤ log2 M

codable maximum use Schmidt decomposition

measuring bipartite entanglement S: reduced density matrix

|ψ⟩ =

• ψij|i⟩|j⟩

ˆ ρ = |ψ⟩⟨ψ| → ˆ ρS = TrE ˆ ρ S = −Tr[ ˆ ρS log2 ˆ ρS] = −

• wα log2 wα

system |i> environment universe |j>

entanglement grows with system surface: area law for ground states!
 
 
 
 
 
 
 
 
 
 


entanglement scaling: gapped systems

S(L) ∼ L S(L) ∼ L2

S(L) ∼ cst.

gapped states

M > 2cst.

M > 2L

M > 2L2

S ≤ log2 M ⇒ M ≥ 2S

black hole

Latorre, Rico, Vidal, Kitaev (03) Bekenstein `73 Callan, Wilczek `94 Eisert, Cramer, Plenio, RMP (10)

random state in Hilbert space: entanglement entropy extensive expectation value for entanglement entropy extensive and maximal states with non-extensive entanglement set of measure zero merit of MPS: 
 parametrize 
 this set efficiently!

Hilbert space size: just an illusion?

ground states are here! Hilbert space

work with MPS: diagrammatics

a1 σ1 aL-1 σL aℓ-1 aℓ σℓ σℓ aℓ-1 aℓ

σ1 σL

matrix: vertical lines = physical states, horizontal lines = matrix indices
 left edge bulk right edge complex conjug.
 rule: connected lines are contracted (multiplied and summed)
 matrix product state in graphical representation


block growth, decimation and MPS

|a`i = X

a`−1,`

ha`−1, σ`|a`i|a`−1i|σ`i ⌘ X

a`−1,`

M `

a`−1,a`|a`−1i|σ`i

M `

a`−1,a` = ha`−1, σ`|a`i

|a`i = X

1,...,`

(M 1M 2 . . . M `)1,a`|σ1σ2 . . . σ`i

1 ℓ-1 ℓ 1 ℓ |aℓ-1〉A |aℓ〉A |σℓ〉

σ1 aℓ-1 σ1 aℓ σℓ

RG schemes: grow blocks while decimating basis
 simple rearrangement of expansion coefficients into matrices:
 recursion easily expressed as matrix multiplication:


(left and right) normalization

I = X

σ`

Bσ`Bσ`†

AAAAAMBBBBBBBBB

aℓ a´ℓ

=

aℓ a´ℓ

=

δa0

`,a` =

ha0

`|a`i =

X

a0

`10 `a`1`

M 0

`⇤

a0

`1,a0 `M `

a`1,a0

`ha0

`1σ0 `|a`1σ`i

= X

a`1`

M `⇤

a`1,a0

`M `

a`1,a0

` =

X

`

(M `†M `)a0

`,a`

I = X

σ`

M σ`†M σ` ≡ X

σ`

Aσ`†Aσ`

both state decomposition and block growth scheme give special gauge
 left normalization (called A); more compact representation:
 right normalization (called B): mixed normalization:

matrix product operators (MPO)

ˆ O = X

{σ}

X

{σ0}

cσ1...σL,σ0

1...σ0 L|σ1 . . . σLihσ0

1 . . . σ0 L|

cσ1...σL,σ0

1...σ0 L → cσ1σ0 1σ2σ0 2...σLσ0 L

cσ1σ0

1σ2σ0 2...σLσ0 L → cσ1σ0 1 · cσ2σ0 2 · . . . · cσLσ0 L

ˆ Sz

i → ˆ

I1 ⊗ ˆ I2 ⊗ . . . ⊗ ˆ Sz

i ⊗ . . . ⊗ ˆ

IL

cσ1σ0

1σ2σ0 2...σLσ0 L = δσ1,σ0 1 · δσ2,σ0 2 · . . . · ( ˆ

Sz)σi,σ0

i · . . . · δσL,σ0 L

ˆ O = X

{σ}

X

{σ0}

M σ1σ0

1M σ2σ0 2 . . . M σLσ0 L|σ1 . . . σLihσ0

1 . . . σ0 L|

general operator: rearrange indices: „mean-field“ very useful: matrix product operator:

applying an MPO to an MPS

˜ M σi

(ab),(a0b0) =

X

σ0

i

N σiσ0

i

aa0 M σ0

i

bb0

σℓ σ´ℓ σ1 σL σ´1 σ´L σ1 σL σ1 σL

graphical representation with ingoing and outgoing physical states: applying an MPO to an MPS: new MPS with matrix dims multiplied

normalization and compression I

|ψi = X

{σ}

Aσ1Aσ2 . . . Aσ`M σ`+1Bσ`+2 . . . BσL|σ1 . . . σLi Ma`,σ`+1a`+1 = M σ`+1

a`,a`+1

Bσ`+1

a`,a`+1 = V † a`,σ`+1a`+1

problem: matrix dimensions of MPS grow under MPO application solution: compression of matrices with minimal state distance assume state is given in mixed normalized form: stack M matrices into one: carry out SVD, and use results:

Aσ` ← Aσ`U M = USV †

• rthonormality of U !

normalization and compression II

|ψi = X

{σ}

Aσ1Aσ2 . . . Aσ`−1M σ`Bσ`+1 . . . BσL|σ1 . . . σLi |a`iA := X

1,...,`

(A1 . . . A`)1,a`|σ1 . . . σ`i |a`iB := X

`+1,...,L

(B`+1 . . . BL)a`,1|σ`+1 . . . σLi |ψi = X

a`

sa`|a`iA|a`iB

now introduce orthonormal states: read off Schmidt decomposition: compress matrices by keeping D largest singular values

Aσ`, Bσ`+1 Aσ`S → M σ`

mixed rep shifted by 1 site: sweep through chain; also normalization

Heisenberg model:

time-evolution

|ψ(t)i = e−i ˆ

Ht|ψ(0)i

N → ∞ τ → 0 Nτ = T ˆ H =

L−1

X

i=1

ˆ hi ˆ hi = Si · Si+1 e−i ˆ

HT = N

Y

i=1

e−i ˆ

Hτ = N

Y

k=1

e−i PL−1

i=1 ˆ

hiτ !

=

N

Y

k=1 L−1

Y

i=1

e−iˆ

hiτ

assume initial state in MPS representation; time evolution: how to express the evolution operator as an MPO?

• ne solution: Trotterization of evolution operator into small time steps

τ ∼ 0.01

first-order Trotter decomposition

calculation of as matrix:

Trotter decomposition

e

ˆ A+ ˆ B = e ˆ Ae ˆ Be

1 2 [ ˆ

A, ˆ B]

ˆ H = ˆ Hodd + ˆ Heven; ˆ Hodd = X

i

ˆ h2i−1, ˆ Heven = X

i

ˆ h2i e−i ˆ

HT = e−i ˆ Hevenτe−i ˆ Hoddτ;

e−i ˆ

Hevenτ =

Y

i

e−iˆ

h2iτ,

e−i ˆ

Hoddτ =

Y

i

e−iˆ

h2i−1τ

problem: exponential does not factorize if operators do not commute but error is negligible as

τ → 0 [ˆ hiτ, ˆ hi+1τ] ∝ τ 2

convenient rearrangement:

e−iˆ

hiτ

(d2 × d2)

HiU = UΛ Hi = UΛU † ⇒ e−iHiτ = Ue−iΛτU † = U · diag(e−iλ1τ, e−iλ2τ, . . .) · U †

tDMRG, tMPS, TEBD

U σ1σ2,σ0

1σ0 2 = hσ1σ2|eiˆ

h1τ|σ0 1σ0 2i

U σ1σ2,σ0

1σ0 2 =

U σ1σ0

1,σ2σ0 2

SV D

= X

b

Wσ1σ0

1,bSb,bWb,σ2σ0 2

= X

b

M σ1σ0

1

1,b

M σ2σ0

2

b,1

bring local evolution operator into MPO form:

initial state

• dd bonds

even bonds

• ne time step: dimension grows as d2

apply one infinitesimal time step in MPO form compress resulting MPS

single-particle excitation

quarter-filled Hubbard chain: U/t=4 add spin-up electron at chain center at time=0 measure charge and spin density separation of charge and spin charge spin

time-dependent DMRG

Kollath, US, Zwerger, PRL 95, 176401 (‘05)

some comments ...

|ψi = X

n

cn|ni ˆ H|ni = En|ni E0 ≤ E1 ≤ E2 ≤ . . .

lim

β→∞ e−β ˆ H|ψi =

lim

β→∞

X

n

e−βEncn|ni = lim

β→∞ e−βE0(c0|0i +

X

n>0

e−β(En−E0)cn|ni = lim

β→∞ e−βE0c0|0i

ground states can be obtained by imaginary time evolution (SLOW!): real time evolution limited by entanglement growth:

S(t) ≤ S(0) + νt S ∼ ln D

in the worst case, matrix dimensions grow exponentially!

• verlaps

hψ(t)|ψ(0)i hSz

i (t)i = hψ(t)| ˆ

Sz

i |ψ(t)i

|ψ〉 〈φ|

hφ|ψi = X

{σ}

X

{σ0}

h{σ0}| ˜ M σ0

1⇤ . . . ˜

M σ0

L⇤M σ1 . . . M σL|{σ}i =

X

{σ}

˜ M σ1⇤ . . . ˜ M σL⇤M σ1 . . . M σL

hφ|ψi = X

{σ}

˜ M σ1∗ . . . ˜ M σL∗M σ1 . . . M σL = X

{σ}

˜ M σL† . . . ˜ M σ1†M σ1 . . . M σL = X

σL

˜ M σL† . . . X

σ2

˜ M σ2† X

σ1

˜ M σ1†M σ1 ! M σ2 ! . . . ! M σL

• verlap contractions:
• rder of contractions: 


zip through the ladder; 
 cost O(dLD3)

O O |ψ〉 〈ψ |

E

E(a`1a0

`1),(a`,a0 `) :=

X

σ`

Aσ`∗

a`1,a`Aσ` a0

`1,a0 `

two-point correlators: long-range or superposition of exponentials

hence: power laws only „by approximation“

dynamical quantum simulator

coherent dynamics! controlled preparation? local measurements? first experiments: period-2 superlattice

• double-well formation
• staggered potential bias
• pattern loading
• odd/even resolved

measurement (Fölling et al. (2007))

first theory proposals:

• prepare
• switch off superlattice
• observe Bose-Hubbard dynamics

|ψ = |1, 0, 1, 0, 1, 0, . . .

Cramer et al., PRL 101, 063001 (2008) Flesch et al., PRA 78, 033608 (2008)

dynamical quantum simulator

45,000 atoms, U=5.2 momentum distribution

Trotzky et al., Nat. Phys. 8, 325(2012)


densities II

fully controlled relaxation in closed quantum system! no free fit parameters! validation of dynamical quantum simulator time range of experiment > 10 x time range of theory real „analog computer“ that goes beyond theory

nearest-neighbour correlators

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 0.25 0.3 1 2 3 4 5 6 7 8 U=1 U=2 U=3 U=4 U=5 U=8 U=12 U=20 U=!

• 0.02

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 1 2 3 4 5 6 7 8 U=1 U=2 U=3 U=4 U=5 U=8 U=12 U=20 U=!

0.05 0.1 0.15 0.2 0.25 0.3 0.35 5 10 15 20 25 30 35 40

time time

ˆ b†

n(t)ˆ

bn+1(t)⇥

interaction real part imaginary part relaxed correlator

• again three regimes
• U≈3: crossover regime

• at large U, 1/U fit of relaxed correlator


can be understood as perturbation
 to locally relaxed subsystems correlator current

currents

current decay as power law? measurement: split in double wells, measure well oscillations phase and amplitude sloshing; no c.m. motion

nearest neighbour correlations

correlation between neighbours interaction strength

theory experiment

visibility proportional to nearest neighbour correlations momentum distribution general trend, 1/U correct! build-up of quantum coherence

build-up of quantum coherence

discrepancy because original theory ignored trap: trap allows particle migration to the „edges“ energy gained in kinetic energy:

measurement at „long time“

• ld theory prediction for long times

without trap theory prediction in trap

• : measured in trap

Ekin = Jhb†

ibi+1 + b† i+1bii

external potential

liquid

long-time limit of nearest-neighbor correlations (here: visibility of momentum distribution)

theory experiment

neutron scattering at T>0

structure function by neutron scattering (Broholm group) high flux precise lineshapes problem: experiment usually T=4.2K, energy scales at J=O(10K)
 definitely not at T=0! desired feature because of achievable field strengths:
 H should be of order J --- rule of thumb 1K=1T

finite-temperature dynamics

purification
 
 density matrix of physical system:
 pure state of physical system plus auxiliary system finite-temperature dynamics
 
 evolution of pure state in enlarged state space

Verstraete, Garcia-Ripoll, Cirac, PRL ‘04

ˆ ρphys = Traux|ψ⇥ψ|

purification and finite-T evolution

ˆ ρP = X

n

ρn|niP P hn| |ψiP Q = X

n

pρn|niP |niQ ˆ ρP = trQ|ψiP Q P Qhψ|

h ˆ OP iˆ

ρP = trP ˆ

OP ˆ ρP = trP ˆ OP trQ|ψiP Q P Qhψ| = trP Q ˆ OP |ψiP Q P Qhψ| =

P Qhψ| ˆ

OP |ψiP Q

ˆ ρP (t) = e−i ˆ

Htˆ

ρP e+i ˆ

Ht = e−i ˆ HttrQ|ψiP Q P Qhψ|e+i ˆ Ht = trQ|ψ(t)iP Q P Qhψ(t)|

|ψ(t)iP Q = e−i ˆ

Ht|ψiP Q

purification: any mixed state can be expressed by a pure state on a larger system (P: physical, Q: auxiliary state space)

simplest way: Q copy of P

expectation values as before: time evolution as before:

time-evolution of thermal states

e−β ˆ

H = e−β ˆ H/2 · ˆ

IP · e−β ˆ

H/2 = trQe−β ˆ H/2|ρ0iP Q P Qhρ0|e−β ˆ H/2

h ˆ B(2t) ˆ Aiβ = Z(β)−1tr ⇣ [ei ˆ

Hte−β ˆ H/2 ˆ

Be−i ˆ

Ht][e−i ˆ Ht ˆ

Ae−β ˆ

H/2ei ˆ Ht]

⌘

. 2 4 6 8 10 12 0.1 1 10

inverse temperature β scheme A

2 4 6 8 10 12

scheme B

2 4 6 8 10 12 14 16 18 20 22

time t

0.1 1 10 106 107 108 109 1010 1011

• ptimized

scheme

Σi Mi

2 4 6 8 10 12 0.1 1 10

scheme A

2 4 6 8 10 12

scheme B

2 4 6 8 10 12 14 16 18 20 22

time t

0.1 1 10 64 128 256 512 1024 2048

• ptimized

scheme

maxiMi

problem: usually we do not have mixed state in eigenrepresentation thermal states: easy way out by imaginary t-evolution purification of infinite-T state: product of local totally mixed states gauge degree of freedom: arbitrary unitary evolution on Q

lots of room for improvement: build MPOs and compress them:

linear prediction

ansatz: data is linear combination of p previous data points

˜ xn = −

p

• i=1

aixn−i

find prediction coefficients by minimising error for available data

E =

• n

|˜ xn − xn|2 wn

calculation prediction index labels time: time series error estimate

(Barthel, Schollwöck, White, PRB 79, 245101 (2009))

iteratively continue time series from data using ansatz

some results of linear prediction

spinons in spin-1/2 chain:
 experiment vs. numerics

Re S(k,t) time t k=π/4, exact k=π/2, exact k=3π/4, exact DMRG linear prediction

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 5 10 15 20 25 30 tobs-tfit tobs 10-12 10-10 10-8 10-6 10-4 25 50 75 100 125 150

transverse Ising model:
 prediction of S(k,t)

error

extends time domain 10x

50 100

Energy (meV) a) Data 150K

50 100

Wavevector k Energy (meV) b) tDMRG 150K

π 2π 20 40 60 80 100

T=200K f) Energy (meV)

20 40 60 80 100 1 2

T=150K e) Energy (meV) T=75K d)

LL field theory DMRG

1 2 3 4

S(k,ω) (mbarn meV−1 sr−1 per Cu2+)

0.96π<k<1.04π

T=50K c)

data

perfect agreement with high-precision neutron scattering

Lake, … Barthel, US, … PRL 111, 137 (2013) Barthel, US, White (2009)

S(k,ω) frequency ω β=50 1 2 3 4 0.5 1 1.5 2 2.5 S(k,ω) β=10 DMRG & linear pred. DMRG & Parzen filter exact 1 2 3

when does it work?

why do we predict S(k,t) in time and not e.g. G(x,t) 
 (and Fourier transform to momentum space later)?
 
 linear prediction works best for special time series

G(k, ⇥) = 1 ⇥ − k − Σ(k, ⇥)

superposition of exponential decays

• cf. pole structure of momentum-space of Green‘s functions

G(k, t) = a1e−iω1t−η1t xn+m =

p

X

ν=1

cνei(ων−ην)mxn

variational ground state search: DMRG

min hψ| ˆ H|ψi hψ|ψi , min ⇣ hψ| ˆ H|ψi λhψ|ψi ⌘

• λ ×

= 0

• λ ×
• λ

= 0

problem: find MPS (of a given dimension) that minimizes energy graphical representation of expression to be minimized: variational minimization with respect to one matrix:

unnormalized MPS: generalized EV problem mixed normalization MPS: eigenvalue problem multilinear :-(

start with random or guess initial MPS maintaining mixed normalization, sweep „hot site“ forth and back at each step, optimize local matrices by solving eigenvalue problem 


convergence: monitor

ground state DMRG

∂ ∂M σi∗ ⇣ hψ| ˆ H|ψi λhψ|ψi ⌘ ! = 0

X

σ0

ia0 i1a0 i

Hσiai1ai,σ0

ia0 i1a0 iMσ0 ia0 i1a0 i =

X

σ0

ia0 i1a0 i

Nai1ai,a0

i1a0 iδσi,σ0 iMσ0 ia0 i1a0 i ≡

X

σ0

ia0 i1a0 i

Nσiai1ai,σ0

ia0 i1a0 iMσ0 ia0 i1a0 i

Hm = λNm hψ| ˆ H2|ψi (hψ| ˆ H|ψi)2

analytical representation of variational problem: DMRG algorithm:

Hamiltonians in MPO form

start end 1 2 3 4 5 I Sz S+ S- I JSz (J/2)S+ (J/2)S- hSz

ˆ H = J

L−1

X

i=1

1 2( ˆ S+

i ˆ

S−

i+1 + ˆ

S−

i ˆ

S+

i+1) + ˆ

Sz

i ˆ

Sz

i+1 + h L

X

i=1

ˆ Sz

i

ˆ M [i] = X

σi,σ0

i

M σi,σ0

i|σiihσ0

i|

ˆ H = ˆ M [1] ˆ M [2] . . . ˆ M [L]

construct Hamiltonian as automaton that moves through chain (e.g. from right to left) building Hamiltonian

Hamiltonians in MPO form II

ˆ M [i] =        ˆ I ˆ S+ ˆ Sz ˆ S− h ˆ Sz (J/2) ˆ S− Jz ˆ Sz (J/2) ˆ S+ ˆ I       

ˆ M [1] = ⇥ h ˆ Sz (J/2) ˆ S− Jz ˆ Sz (J/2) ˆ S+ ˆ I ⇤ ˆ M [L] = 2 6 6 6 6 6 4 ˆ I ˆ S+ ˆ Sz ˆ S− h ˆ Sz 3 7 7 7 7 7 5

short ranged Hamiltonians find very compact, exact representation!

frustrated magnetism in 2D

J1 J2

„classic“ candidates (spin length 1/2): J1-J2 model on
 a square lattice kagome lattice classical model

• rder only locally coplanar

extensive T=0 entropy agreement: no magnetic order for S=1/2

herbertsmithite ZnCu3(OH)6Cl2 Yan et al, Science (2011) Depenbrock et al, PRL (2012)

DMRG in two dimensions

map 2D lattice to 1D (vertical) „snake“ with long-ranged interactions horizontally: ansatz obeys area law: easy axis, long at linear cost vertically: ansatz violates area law: hard axis, long at exponential cost consider long cylinders of small circumference c: mixed BC

vertically OBC vertically PBC: extra cost! circumference c length L

S ∼ log2 M → M ∼ 2L

S ∼ log2 M L = L log2 M S ∼ log2(M 2)L = 2L log2 M

ground state energies

fully SU(2) invariant DMRG code up to 3,800 representatives (16,000 U(1) DMRG states) cylinders up to circumference c=17.3, N=726 tori up to N=(6x6)x3=108 sites

100% increase 50% increase ED: 48 sites

TD limit energy estimate: -0.4386(5) iDMRG (infinite cylinder) upper bounds below HVBC; YC8: -0.4379

iDMRG: I.P . McCulloch, arXiv:0804.2509

• 0.45
• 0.445
• 0.44
• 0.435
• 0.43

0.05 0.1 0.15 0.2 0.25 0.3 energy per site inverse circumference DMRG cylinders Yan et al HVBC DMRG upper bound MERA upper bound 2D estimate, Yan 2D estimate, this work

triplet gap

fully SU(2) invariant DMRG code eliminates need for special edge manipulations of U(1) DMRG:
 ground state of S=1 sector bulk excitation much smoother gap curve triplet gap estimate: 0.13(1)

bond energy deviations from mean singlet gap estimate: approx 0.05 (Yan et al. (2011))
 triplet gap for infinitely long cylinders

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 0.3 triplet gap inverse circumference SU(2) DMRG Cylinders Yan et al linear fit to SU(2) data

topological entanglement entropy (TEE)

ground state in 2D obeying area law

S(L) = aL − γ γ

for smooth (circular) surface of bipartition topological entanglement entropy (TEE) reduces conventional entanglement entropy global property can be used to detect topological phases (complete classification)

Preskill, Kitaev, PRL 2006 Levin, Wen, PRL 2006


γ = log2 D

D = √ 2 D = √ 3 D = 2

chiral QSL
 Z2 QSL double semion phase
 quantum dimension
 torus degeneracy: D2


topological entanglement and DMRG

X Y PBC OBC Y X PBC OBC

smooth surface minimizes subleading corrections

• ne optimal cut for XC,

YC cylinders each MPS structure of DMRG gives direct access to reduced density operator entanglement entropy for free in 1D mapping: cut of arbitrary link of path XCn YCn 2D view:
 2 cylinders A, B A B

cut cut cut

TEE with DMRG: Renyi entropies

S1(ρ) = −tr ρ log2 ρ

• 10
• 9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

500 1000 1500 2000 2500 ’./spectrum.log’

MPS: M non-zero eigenvalues of only tail of spectrum not captured well? von Neumann entropy correct?

index of eigenvalue
 log10 of eigenvalue


ρA

Renyi entropy S0(ρ) = rank ρ

S∞(ρ) = − log2 w1

Sα(ρ) = 1 1 − α log2 tr ρα

(0 ≤ α < ∞)

low-α Renyi: focus on tail 
 high-α Renyi: focus on head of spectrum - measured accurately! theorems:
 
 topological entanglement entropy independent of alpha


head
 tail


TEE in the kagome lattice

extrapolate Renyi entropies 
 to circumference c=0 negative intercept is TEE find topological order! TEE extracted from random state in GS manifold lower bound true value for so-called minimum entropy state DMRG seems to systematically pick those

γ ≈ 0.94 D ≈ 2

• 1

1 2 3 4 5 6 7 2 4 6 8 10 12 14 Renyi entropy Sα circumference c S2 S3 S4 S5 fit to S2 fit to S3 fit to S4 fit to S5

Zhang, Grover, Turner, Oshikawa, Vishvanath, PRB (2012) 


conclusions

1D: DMRG/MPS currently most powerful method ground states time-evolution, also at non-zero temperature limitation: exponential growth of resources; entanglement growth 2D: DMRG/MPS starts making very interesting forays long cylinders suboptimal ansatz, but numerically extremely stable barring new ideas, key challenges for powerful codes: parallelization (non-)Abelian quantum numbers non-trivial geometries (impurity solvers, quantum chemistry) convergence of ground states