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DMRG in the Thermodynamic limitWorkshop and Symposium on DMRG Technique for Strongly Correlated
Systems in Physics and Chemistry
Ian McCulloch
University of QueenslandCentre for Engineered Quantum Systems (EQuS)
23/6/2015
Ian McCulloch (UQ) iDMRG 23/6/2015 1 / 54
Outline
1 Constructing Matrix Product States - alternative viewsFrom classical to quantum statesSequential generation
2 Matrix Product Operators
3 Infinite size DMRG
4 Broken symmetries
5 Scaling relations in the thermodynamic limit
6 Expectation values of iMPO’sHigher momentsBinder cumulant
7 Infinite Boundary Conditions
8 2D
Ian McCulloch (UQ) iDMRG 23/6/2015 2 / 54
Method 1: quantize a classical state
Start from a classical (product) state
|ψ〉 = |s1〉 |s2〉 |s3〉 |s4〉 · · ·
Each |si〉 is a classical vector, with real (or c-number) coefficients in somebasis
|si〉 = axi |x〉+ ay
i |y〉+ azi |z〉
Turn our (commuting) numeric coefficients into a matrix
|si〉jk = Axjk|x〉+ Ay
jk|y〉+ Azjk|z〉
We can recover an amplitude at the end by taking the trace, or arranging thatthe boundary matrices are 1× D and D× 1.
|ψ〉 = Tr∑
si
As1 As2 As3 As4 · · ·|s1〉 |s2〉 |s3〉 |s4〉 · · ·
Ian McCulloch (UQ) iDMRG 23/6/2015 3 / 54
Method 1: quantize a classical state
Start from a classical (product) state
|ψ〉 = |s1〉 |s2〉 |s3〉 |s4〉 · · ·
Each |si〉 is a classical vector, with real (or c-number) coefficients in somebasis
|si〉 = axi |x〉+ ay
i |y〉+ azi |z〉
Turn our (commuting) numeric coefficients into a matrix
|si〉jk = Axjk|x〉+ Ay
jk|y〉+ Azjk|z〉
We can recover an amplitude at the end by taking the trace, or arranging thatthe boundary matrices are 1× D and D× 1.
|ψ〉 = Tr∑
si
As1 As2 As3 As4 · · ·|s1〉 |s2〉 |s3〉 |s4〉 · · ·
Ian McCulloch (UQ) iDMRG 23/6/2015 3 / 54
Method 2: quantum finite-state machines
What is a Matrix Product State?
Another way to visualizing them (from Greg Crosswhite)
A finite-state machineis a model of a systemthat can transitionbetween a finitenumber of states.
Ian McCulloch (UQ) iDMRG 23/6/2015 4 / 54
A classical finite-state machine is always in one discrete state.
In a quantum finite-state machine, we choose every possible transition withsome probability amplitude
(from Crosswhite and Bacon, Phys. Rev. A 78, 012356 (2008))
|ψ〉 =
| ↑〉| ↓〉
Ian McCulloch (UQ) iDMRG 23/6/2015 5 / 54
A classical finite-state machine is always in one discrete state.
In a quantum finite-state machine, we choose every possible transition withsome probability amplitude
(from Crosswhite and Bacon, Phys. Rev. A 78, 012356 (2008))
|ψ〉 =
| ↑↑〉| ↓↑〉+ | ↑↓〉
Ian McCulloch (UQ) iDMRG 23/6/2015 5 / 54
A classical finite-state machine is always in one discrete state.
In a quantum finite-state machine, we choose every possible transition withsome probability amplitude
(from Crosswhite and Bacon, Phys. Rev. A 78, 012356 (2008))
|ψ〉 =
| ↑↑↑〉| ↓↑↑〉+ | ↑↓↑〉+ | ↑↑↓〉
Ian McCulloch (UQ) iDMRG 23/6/2015 5 / 54
A classical finite-state machine is always in one discrete state.
In a quantum finite-state machine, we choose every possible transition withsome probability amplitude
(from Crosswhite and Bacon, Phys. Rev. A 78, 012356 (2008))
|ψ〉 = | ↓↑↑↑〉+ | ↑↓↑↑〉+ | ↑↑↓↑〉+ | ↑↑↑↓〉
Ian McCulloch (UQ) iDMRG 23/6/2015 5 / 54
Matrix Product StatesThis quantum finite-state machine has a transition matrix associated with it
W-state
|ψ〉 =1√N
(| ↓↑↑↑ . . .〉+ | ↑↓↑↑ . . .〉+ | ↑↑↓↑ . . .〉+ . . .)
A =
(| ↑〉 0| ↓〉 | ↑〉
)Practically all prototype wavefunctions studied in quantum information have alow-dimensional MPS representation
GHZ state – long-range entangled, S = ln 2
|ψ〉 =1√2
(| ↑↑↑ . . .〉+ | ↓↓↓ . . .〉)
A =
(| ↑〉 00 | ↓〉
)AKLT state
A =
( √1/3|0〉 −sqrt2/3|+〉√2/3|−〉 −
√1/3|0〉
)Ian McCulloch (UQ) iDMRG 23/6/2015 6 / 54
Matrix Product StatesThis quantum finite-state machine has a transition matrix associated with it
W-state
|ψ〉 =1√N
(| ↓↑↑↑ . . .〉+ | ↑↓↑↑ . . .〉+ | ↑↑↓↑ . . .〉+ . . .)
A =
(| ↑〉 0| ↓〉 | ↑〉
)Practically all prototype wavefunctions studied in quantum information have alow-dimensional MPS representation
GHZ state – long-range entangled, S = ln 2
|ψ〉 =1√2
(| ↑↑↑ . . .〉+ | ↓↓↓ . . .〉)
A =
(| ↑〉 00 | ↓〉
)AKLT state
A =
( √1/3|0〉 −sqrt2/3|+〉√2/3|−〉 −
√1/3|0〉
)Ian McCulloch (UQ) iDMRG 23/6/2015 6 / 54
Matrix Product StatesThis quantum finite-state machine has a transition matrix associated with it
W-state
|ψ〉 =1√N
(| ↓↑↑↑ . . .〉+ | ↑↓↑↑ . . .〉+ | ↑↑↓↑ . . .〉+ . . .)
A =
(| ↑〉 0| ↓〉 | ↑〉
)Practically all prototype wavefunctions studied in quantum information have alow-dimensional MPS representation
GHZ state – long-range entangled, S = ln 2
|ψ〉 =1√2
(| ↑↑↑ . . .〉+ | ↓↓↓ . . .〉)
A =
(| ↑〉 00 | ↓〉
)AKLT state
A =
( √1/3|0〉 −sqrt2/3|+〉√2/3|−〉 −
√1/3|0〉
)Ian McCulloch (UQ) iDMRG 23/6/2015 6 / 54
The Matrix Product Ansatz: beyond groundstatesThe key advantage of MPS formulation: arithmetic manipulationsThe sum (superposition) of two matrix product states is also a matrix productstate
|C〉 = |A〉+ |B〉
C =∑si
Tr Cs1 Cs2 . . .CsL |s1s2 . . . sL〉
whereCsi = Asi ⊕ Bsi
The dimension of the matrices increases: dim(C) = dim(A) + dim(B)Action of an operator on a state: if the operator is a product of local terms:
O = O1 ⊗ O2 ⊗ . . .
|C〉 = O|A〉 is a Matrix Product State, with
Csi =∑
s′i
Osi,s′ii As′i
Ian McCulloch (UQ) iDMRG 23/6/2015 7 / 54
The Matrix Product Ansatz: beyond groundstatesThe key advantage of MPS formulation: arithmetic manipulationsThe sum (superposition) of two matrix product states is also a matrix productstate
|C〉 = |A〉+ |B〉
C =∑si
Tr Cs1 Cs2 . . .CsL |s1s2 . . . sL〉
whereCsi = Asi ⊕ Bsi
The dimension of the matrices increases: dim(C) = dim(A) + dim(B)Action of an operator on a state: if the operator is a product of local terms:
O = O1 ⊗ O2 ⊗ . . .
|C〉 = O|A〉 is a Matrix Product State, with
Csi =∑
s′i
Osi,s′ii As′i
Ian McCulloch (UQ) iDMRG 23/6/2015 7 / 54
Matrix Product OperatorsIPM J. Stat. Mech. P10014 (2007), arXiv:0804.2509
At each iteration we have a set of block operators, acting on them-dimensional auxiliary spaceIt is natural to use a Matrix Product approach to constructing the blockoperators used in DMRG
Ising model H =∑<i,j>
Szi S
zj + λ
∑i
Sxi , adding a site to the block:
(identity operator) I → I ⊗ Ilocal(z-spin acting on right-most site) Sz → I ⊗ Sz
local(block Hamiltonian) H → λI ⊗ Sx
local + Sz ⊗ Szlocal + H ⊗ Ilocal
In matrix form:
(H Sz I)′︸ ︷︷ ︸new block operators
= (H Sz I)︸ ︷︷ ︸old block operators
×
ISz
λSx Sz I
︸ ︷︷ ︸
local
Ian McCulloch (UQ) iDMRG 23/6/2015 8 / 54
Matrix Product OperatorsThis form can represent many operators
fermionic c†k=0: Wc†k=0=
(Ic† P
), P = (−1)N , J-W string
finite momentum b†k : Wb†k=
(I
b† eikI
)Advantages of the MPO representation: arithmetic operations!
H1 + H2 direct sum of the MPO representationsH1 × H2 direct product of the MPO representations
also derivatives, etcThis preserves the lower triangular form.Can we evaluate an expectation value of an MPO in the thermodynamic limit?
〈A〉L = polynomial function of L
Examples:Energy: 〈H〉L = L εHamiltonian block operator matrix elements to restart a calculationSingle-mode approximation: 〈S−k HS+
k 〉L/〈S−k S+
k 〉LIan McCulloch (UQ) iDMRG 23/6/2015 9 / 54
Matrix Product OperatorsThis form can represent many operators
fermionic c†k=0: Wc†k=0=
(Ic† P
), P = (−1)N , J-W string
finite momentum b†k : Wb†k=
(I
b† eikI
)Advantages of the MPO representation: arithmetic operations!
H1 + H2 direct sum of the MPO representationsH1 × H2 direct product of the MPO representations
also derivatives, etcThis preserves the lower triangular form.Can we evaluate an expectation value of an MPO in the thermodynamic limit?
〈A〉L = polynomial function of L
Examples:Energy: 〈H〉L = L εHamiltonian block operator matrix elements to restart a calculationSingle-mode approximation: 〈S−k HS+
k 〉L/〈S−k S+
k 〉LIan McCulloch (UQ) iDMRG 23/6/2015 9 / 54
DMRG in the infinite size limit (arxiv:0804.2509)
Infinite-size translationally invariant MPS
The “infinite size” DMRG algorithm has existed since the start (1992)It doesn’t produce a translationally invariant MPS fixed pointNo prescription for constructing the initial wavefunction at next iterationiTEBD produces a translationally invariant MPS, but for groundstatesimaginary time evolution is not so fast
’
Ian McCulloch (UQ) iDMRG 23/6/2015 10 / 54
DMRG in the infinite size limit (arxiv:0804.2509)
Infinite-size translationally invariant MPS
The “infinite size” DMRG algorithm has existed since the start (1992)It doesn’t produce a translationally invariant MPS fixed pointNo prescription for constructing the initial wavefunction at next iterationiTEBD produces a translationally invariant MPS, but for groundstatesimaginary time evolution is not so fast
? ’
Ian McCulloch (UQ) iDMRG 23/6/2015 10 / 54
A recurrence relation for MPS
Suppose we have an initial state:0
Λ
Suppose we also have the MPS enlarged with an extra unit cell:R
Λ
ΛL
Note: ΛL and ΛR are not necessarily diagonal
Now we can insert one more unit cell:Λ1
Λ1 = ΛR Λ−10 ΛL
Ian McCulloch (UQ) iDMRG 23/6/2015 11 / 54
Variant of the finite system algorithmDifferent treatment of the boundaries
2 1
132
0
0 3 1
42
3
3
2 4
5
4
2
5 3
1
Λ23 = Λ21 Λ−101 Λ03
Λ43 = Λ41 Λ−121 Λ23
Λ45 = Λ43 Λ−123 Λ25
Ian McCulloch (UQ) iDMRG 23/6/2015 12 / 54
Broken symmetries
Finite size MPS: No broken symmetries (to O(truncation error))Infinite size MPS: The Ansatz can break all symmetries
even continuous symmetries in one dimension
How to understand this?
Matrix elements connecting symmetry sectors vanish as∼ exp(−L) → 0Continuous symmetries cannot break in exact 1D because theassociated goldstone modes would destroy the order parametercompletely (percolation threshold!)But if the goldstone modes are gapped due to finite basis size, thesymmetry can breakAlternatively: in order to get a finite correlation length we must perturb theHamiltonian with a relevant perturbation. No reason why that perturbationshould not break any (or all) symmetry.
Ian McCulloch (UQ) iDMRG 23/6/2015 13 / 54
Broken symmetries
Finite size MPS: No broken symmetries (to O(truncation error))Infinite size MPS: The Ansatz can break all symmetries
even continuous symmetries in one dimension
How to understand this?
Matrix elements connecting symmetry sectors vanish as∼ exp(−L) → 0Continuous symmetries cannot break in exact 1D because theassociated goldstone modes would destroy the order parametercompletely (percolation threshold!)But if the goldstone modes are gapped due to finite basis size, thesymmetry can breakAlternatively: in order to get a finite correlation length we must perturb theHamiltonian with a relevant perturbation. No reason why that perturbationshould not break any (or all) symmetry.
Ian McCulloch (UQ) iDMRG 23/6/2015 13 / 54
Prototypical example: Bose-Hubbard model
H =U2
∑i
Ni(Ni − 1)− J∑<i,j>
b†i bj + b†j bi − µN
Mean-field approximation: β = 〈b〉
HMF =U2
∑i
Ni(Ni − 1)− Jβ∑
i
(b†i + bi)− µN
Mean field Hamiltonian breaks U(1) particle number conservationGroundstate is an m = 1 infinite MPS (product state!)
|ψ〉 = (|0〉+ a1|1〉+ a2|2〉 . . .)⊗L
An iMPS with no symmetries reduces to mean-fieldImposing quantum number symmetries reduces the quality of thevariational state (for fixed m)But usually worth the cost in computational efficiency
Ian McCulloch (UQ) iDMRG 23/6/2015 14 / 54
Prototypical example: Bose-Hubbard model
H =U2
∑i
Ni(Ni − 1)− J∑<i,j>
b†i bj + b†j bi − µN
Mean-field approximation: β = 〈b〉
HMF =U2
∑i
Ni(Ni − 1)− Jβ∑
i
(b†i + bi)− µN
Mean field Hamiltonian breaks U(1) particle number conservationGroundstate is an m = 1 infinite MPS (product state!)
|ψ〉 = (|0〉+ a1|1〉+ a2|2〉 . . .)⊗L
An iMPS with no symmetries reduces to mean-fieldImposing quantum number symmetries reduces the quality of thevariational state (for fixed m)But usually worth the cost in computational efficiency
Ian McCulloch (UQ) iDMRG 23/6/2015 14 / 54
0 0.05 0.1 0.15 0.2 0.25J/U
0
0.1
0.2
0.3
0.4
0.5
0.6
Supe
rflu
id D
ensi
ty
m=10m=20m=30m=40m=50m=60m=70m=80
Bose-Hubbard Model Mott-Superfluid Transitionµ=0.25
0.219 0.2192 0.2194 0.2196 0.21981e-06
0.001
1
Ian McCulloch (UQ) iDMRG 23/6/2015 15 / 54
Correlation Functions
The form of correlation functions are determined by the eigenvalues of thetransfer operator
All eigenvalues ≤ 1One eigenvalue equal to 1,corresponding to the identityoperator
Expansion in terms of eigenspectrum λi:
〈O(x)O(y)〉 =∑
i
ai λ|y−x|i
Ian McCulloch (UQ) iDMRG 23/6/2015 16 / 54
50 100 150 200Number of states kept
0
0.2
0.4
0.6
0.8
1λ
(0,0) Singlet(1,0) Spin triplet(0,1) Holon Triplet(1/2,1/2) Single-particle
Hubbard Model transfer matrix spectrumHalf-filling, U/t = 4
Ian McCulloch (UQ) iDMRG 23/6/2015 17 / 54
64 128Number of states kept
1
10
100
Cor
rela
tion
leng
th
(0,0) Singlet(1,0) Spin triplet(0,1) Holon Triplet(1/2,1/2) Single-particle
Hubbard model transfer matrix spectrumHalf-filling, U/t=4
Ian McCulloch (UQ) iDMRG 23/6/2015 18 / 54
Critical scaling exampleTwo-species bose gas with linear tunneling Ω, from F. Zhan et al, Phys. Rev. A 90, 023630 2014
1
10
100
50 100 200
Ω = 0.2148Ω = 0.215Ω = 0.2152Ω = 0.2154Ω = 0.2156Ω = 0.2158
ξ
mIan McCulloch (UQ) iDMRG 23/6/2015 19 / 54
CFT Parameters
For a critial mode, the correlation length increases with number of states m asa power law,
ξ ∼ mκ
[T. Nishino, K. Okunishi, M. Kikuchi, Phys. Lett. A 213, 69 (1996)M. Andersson, M. Boman, S. Östlund, Phys. Rev. B 59, 10493 (1999)L. Tagliacozzo, Thiago. R. de Oliveira, S. Iblisdir, J. I. Latorre, Phys. Rev. B 78, 024410 (2008)]
This exponent is a function only of the central charge,
κ =6√
12c + c
[Pollmann et al, PRL 2009]
Note: in practice this usually isn’t a good way to determine c – better to useentropy scaling
Ian McCulloch (UQ) iDMRG 23/6/2015 20 / 54
Scaling dimensions
Suppose we have a two-point correlator that has a power-law at largedistances
〈O(x)O(y)〉 = |y− x|−2∆
As we increase the number of states kept m the correlation length increases,so the region of validity of the power law increases.
Take two different calculations with m1 and m2
Correlation lengths ξ1 and ξ2
We expect:O(ξ2)
O(ξ1)=
(ξ2
ξ1
)∆
for x large, we have: O(x) ' a λx (with ξ = −1/ lnλ)Prefactor a is overlap of operator O with next-leading eigenvector oftransfer operator
a ∝ ξ−∆
This gives directly the operator scaling dimensions by direct fit
Ian McCulloch (UQ) iDMRG 23/6/2015 21 / 54
Scaling dimensions
Suppose we have a two-point correlator that has a power-law at largedistances
〈O(x)O(y)〉 = |y− x|−2∆
As we increase the number of states kept m the correlation length increases,so the region of validity of the power law increases.
Take two different calculations with m1 and m2
Correlation lengths ξ1 and ξ2
We expect:O(ξ2)
O(ξ1)=
(ξ2
ξ1
)∆
for x large, we have: O(x) ' a λx (with ξ = −1/ lnλ)Prefactor a is overlap of operator O with next-leading eigenvector oftransfer operator
a ∝ ξ−∆
This gives directly the operator scaling dimensions by direct fit
Ian McCulloch (UQ) iDMRG 23/6/2015 21 / 54
Scaling dimensions
Suppose we have a two-point correlator that has a power-law at largedistances
〈O(x)O(y)〉 = |y− x|−2∆
As we increase the number of states kept m the correlation length increases,so the region of validity of the power law increases.
Take two different calculations with m1 and m2
Correlation lengths ξ1 and ξ2
We expect:O(ξ2)
O(ξ1)=
(ξ2
ξ1
)∆
for x large, we have: O(x) ' a λx (with ξ = −1/ lnλ)Prefactor a is overlap of operator O with next-leading eigenvector oftransfer operator
a ∝ ξ−∆
This gives directly the operator scaling dimensions by direct fit
Ian McCulloch (UQ) iDMRG 23/6/2015 21 / 54
0.0078125 0.015625transfer matrix eigenvalue 1 - λ = 1 / ξ
0.03125
0.0625
pref
acto
r of t
he sp
in o
pera
tor a
t thi
s mod
e
iDMRG data for m=15,20,25,30,35y = 0.45126 * x^0.480
Heisenberg model fit for the scaling dimension
Ian McCulloch (UQ) iDMRG 23/6/2015 22 / 54
Generalized Scaling
Alternative viewpoint (Vid Stojevic et al, Phys. Rev. B 91, 035120 (2015))
Scaling relation for large s:
O(sξ) ' aλsξ = aλ−s/ lnλ = ae−s
So we obtain ∆ by scaling O(sξ) versus sξ
However, this also works for s small, eg s 1,
O(sξ) ∝ (sξ)−∆
because for s 1, the correlation function is already (approximately)power-law.
the scaling relation works for any 0 < s <∞ !
But: O(sξ) ' aλsξ only for s 1
Ian McCulloch (UQ) iDMRG 23/6/2015 23 / 54
Generalized Scaling
Alternative viewpoint (Vid Stojevic et al, Phys. Rev. B 91, 035120 (2015))
Scaling relation for large s:
O(sξ) ' aλsξ = aλ−s/ lnλ = ae−s
So we obtain ∆ by scaling O(sξ) versus sξ
However, this also works for s small, eg s 1,
O(sξ) ∝ (sξ)−∆
because for s 1, the correlation function is already (approximately)power-law.
the scaling relation works for any 0 < s <∞ !
But: O(sξ) ' aλsξ only for s 1
Ian McCulloch (UQ) iDMRG 23/6/2015 23 / 54
Generalized Scaling
Alternative viewpoint (Vid Stojevic et al, Phys. Rev. B 91, 035120 (2015))
Scaling relation for large s:
O(sξ) ' aλsξ = aλ−s/ lnλ = ae−s
So we obtain ∆ by scaling O(sξ) versus sξ
However, this also works for s small, eg s 1,
O(sξ) ∝ (sξ)−∆
because for s 1, the correlation function is already (approximately)power-law.
the scaling relation works for any 0 < s <∞ !
But: O(sξ) ' aλsξ only for s 1
Ian McCulloch (UQ) iDMRG 23/6/2015 23 / 54
Expectation values of MPO’s - arxiv:0804.2509
We have seen that we can write many interesting operators in the form of amatrix product operator
Can we evaluate the expectation value of an arbitrary MPO?
If the MPO has no Jordan structure, this is a simple eigenvalue problem
= λW
For a lower triangular MPO, this doesn’t work.
But we can make use of the triangular structure ISz
λSx Sz I
index by index, each component is a function only of the previouslycalculated terms
Ian McCulloch (UQ) iDMRG 23/6/2015 24 / 54
Expectation values of MPO’s - arxiv:0804.2509
We have seen that we can write many interesting operators in the form of amatrix product operator
Can we evaluate the expectation value of an arbitrary MPO?
If the MPO has no Jordan structure, this is a simple eigenvalue problem
= λW
For a lower triangular MPO, this doesn’t work.
But we can make use of the triangular structure ISz
λSx Sz I
index by index, each component is a function only of the previouslycalculated terms
Ian McCulloch (UQ) iDMRG 23/6/2015 24 / 54
Choose bond indices i, j of Wij, and denote TWij
ji W
Example: ISz
λSx Sz I
Eigentensor is (E1 E2 E3)
Starting from E3:E3 = TI(E3) = I
is equivalent to the orthogonality condition - E3 is just the identityE2:
E2 = TSz(E3) = TSz(I) = Sz
E1: doesn’t reach a fixed point, E1 = E1(L) depends on the number ofiterations L
E1(L + 1) = TI(E1(L)) + TSz(E2) + TλSx (E3)= TI(E1(L)) + C
where C is a constant matrix, C = TSz(Sz) + λSx
Ian McCulloch (UQ) iDMRG 23/6/2015 25 / 54
Fixed point equations for E1:
E1(L + 1) = TI(E1(L)) + C
Eigenmatrix expansion of TI :
TI =
m2∑n=1
λn|λ〉〈λ|
giving
E(n)1 (L + 1) = λnE(n)
1 (L) + C(n)
Since λ1 = 1 by construction, this motivates decomposing intocomponents parallel and perpendicular to the identity:
E1(L) = E′1(L) + e1(L) I
where Tr E′1(L)ρ = 0
Ian McCulloch (UQ) iDMRG 23/6/2015 26 / 54
Component in the direction of the identity:
e1(L + 1) = e1(L) + Tr Cρ
Has the solutione1(L) = L Tr Cρ
is the energyComponent perpendicular to the identity:
E′1(L + 1) = TI(E′1(L)) + C′
where C′ = C − (Tr Cρ) I
E′1(L + 1)n = λnE′1(L)n + C′n
Since all |λn| < 1 here, this is a geometric series that converges to a fixedpoint (independent of L),
(1− TI)(E′1) = C′
Linear solver for the unknown matrix E′1
Ian McCulloch (UQ) iDMRG 23/6/2015 27 / 54
Component in the direction of the identity:
e1(L + 1) = e1(L) + Tr Cρ
Has the solutione1(L) = L Tr Cρ
is the energyComponent perpendicular to the identity:
E′1(L + 1) = TI(E′1(L)) + C′
where C′ = C − (Tr Cρ) I
E′1(L + 1)n = λnE′1(L)n + C′n
Since all |λn| < 1 here, this is a geometric series that converges to a fixedpoint (independent of L),
(1− TI)(E′1) = C′
Linear solver for the unknown matrix E′1
Ian McCulloch (UQ) iDMRG 23/6/2015 27 / 54
Summary:
Decompose eigentensor into components parallel and perpendicular tothe identityThe component parallel to the identity is the energy per siteThe perpendicular components reach a fixed point and give theHamiltonian matrix elements
E3 = I Identity operatorE2 = Sz Sz block operatorE1(L) = E′ + Le1 Hamiltonian operator + energy per site
Ian McCulloch (UQ) iDMRG 23/6/2015 28 / 54
Generalization to arbitrary triangular MPO’sarXiv:1008.4667
At the ith iteration, we have
Ei(L + 1) = TWii(Ei(L)) +∑j>i
TWji(Ej(L))︸ ︷︷ ︸= C(L)
Basic idea:if Wii = 0, then Ei = C
if Wii 6= 0, then solve (1− TWii)(Ei) = C
The result will be a polynomial function of L
solve separately for the coefficient of the k-th power of n
If the diagonal element is unitary, then obtain the eigenvalues ofmagnitude 1If any eigenvalues are complex, then expand also in fourier modes
Ian McCulloch (UQ) iDMRG 23/6/2015 29 / 54
Generalization to arbitrary triangular MPO’sarXiv:1008.4667
At the ith iteration, we have
Ei(L + 1) = TWii(Ei(L)) +∑j>i
TWji(Ej(L))︸ ︷︷ ︸= C(L)
Basic idea:if Wii = 0, then Ei = C
if Wii 6= 0, then solve (1− TWii)(Ei) = C
The result will be a polynomial function of L
solve separately for the coefficient of the k-th power of n
If the diagonal element is unitary, then obtain the eigenvalues ofmagnitude 1If any eigenvalues are complex, then expand also in fourier modes
Ian McCulloch (UQ) iDMRG 23/6/2015 29 / 54
Examples 1 - Variance
How close is a variational state to an eigenstate of the Hamiltonian?
Sometimes there is an algorithmic measure, often not.
The square of the Hamiltonian operator determines the energy variance
〈H2〉L − 〈H〉2L = 〈(H − E)2〉L = Lσ2
A universal measure for the quality of a variational wavefunctionlower bound for the energy: there is always an eigenstate within σ of E
We can easily construct an MPO representation of H2
Ian McCulloch (UQ) iDMRG 23/6/2015 30 / 54
0 1×10-6
2×10-6
3×10-6
4×10-6
5×10-6
6×10-6
σ2
-0.443148
-0.443147
-0.443146
-0.443145
-0.443144
-0.443143
-0.443142
ESpin 1/2 Heisenberg Model
Energy per site scaling with variance (exact energy = -ln 2 + 0.25 = -0.44314718056)
0 1×10-7
2×10-7
3×10-7
-0.44314720-0.44314715-0.44314710-0.44314705-0.44314700-0.44314695-0.44314690
Ian McCulloch (UQ) iDMRG 23/6/2015 31 / 54
Momentum distribution
Momentum-dependent operators have a simple form,
b†k =∑
x
eikxb†x
Wb†k=
(I
b† eikI
)
Momentum occupation:
n(k) =1L
b†kbk
Broken U(1) symmetry: 〈b†〉 6= 0 hence n(k = 0) ∝ L (extensive)With U(1) symmetry: n(k = 0) is finite, but diverges with m in superfluidphase
Ian McCulloch (UQ) iDMRG 23/6/2015 32 / 54
-1 -0.5 0 0.5 1k/pi
1
10
100
1000N
(k)
U/J=3.2U/J=3.6U/J=4.0U/J=4.8U/J=6.4
Bose-Hubbard Model N(k)Infinite 1D, one particle per site
Ian McCulloch (UQ) iDMRG 23/6/2015 33 / 54
Higher moments
It is straight forward to evaluate a local order parameter, eg
M =∑
i
Mi
The first moment of this operator gives the order parameter,
〈M〉 = m1(L)
It is also useful to calculate higher moments, eg
〈M2〉 = m2(L)
or generally
〈Mk〉 = mk(L)
These are polynomial functions in the system size L.
Ian McCulloch (UQ) iDMRG 23/6/2015 34 / 54
For finite systems, the Binder cumulant of the order parameter cancels theleading-order finite size effects
UL = 1− 〈m4〉3〈m2〉2
0
0.25
0.5
0.75
1
0.18 0.2 0.22 0.24
L = 50L = 100L = 150L = 200iDMRGfinite size scaling
|〈∆NL/2〉|
Ω
0
0.2
0.4
0.6
0.8
0.18 0.2 0.22 0.24
L = 50L = 100L = 150L = 200
UL
Ω
0.4
0.425
0.215 0.216
The 2-component Bose-Hubbard model, with a linear coupling betweencomponents, has an Ising-like transition from immiscible (small Ω) to miscible(large Ω).
H =∑
<i,j>,σ
b†i,σbj,σ +H.c.+ U∑i,σ
nσ(nσ−1)+ U12
∑i
n↑n↓+Ω∑<i,j>
b†i,↑bj,↓+H.c.
Ian McCulloch (UQ) iDMRG 23/6/2015 35 / 54
Cumulant expansions
Express the moments mi in terms of the cumulants per site κj,
m1(L) = κ1Lm2(L) = κ2
1L2 + κ2Lm3(L) = κ3
1L3 + 3κ1κ2L2 + κ3Lm4(L) = κ4
1L4 + 6κ21κ2L3 + (3κ2
2 + 4κ1κ3)L2 + κ4L
κ1 is the order parameter itselfκ2 is the variance (related to the susceptibility)κ3 is the skewnessκ4 is the kurtosis
The cumulants per site κk are well-defined for an iMPS
Note: the cumulants are normally written such that they are extensivequantities→ Lκk.
Ian McCulloch (UQ) iDMRG 23/6/2015 36 / 54
iMPS for two-component bose gas
The cumulant expansion already gives a lot of information
κ1 is the order parameter itself
0.1 0.12 0.14 0.16 0.18 0.2Ω
0
0.2
0.4
0.6
0.8
κ1
Ising transition in 2-component Bose gasOrder parameter |N_a - N_b|
Ian McCulloch (UQ) iDMRG 23/6/2015 37 / 54
The second cumulant gives the susceptibility
0.1 0.12 0.14 0.16 0.18 0.2Ω
0
10
20
30
40
50
κ2
2-component Bose gasSecond cumulant (susceptibility)
Different to a finite-size scaling, the susceptibility exactly diverges at thecritical point.Sufficiently close to the critical point, it looks mean-field-like (so will generallygive the wrong exponent!)
Ian McCulloch (UQ) iDMRG 23/6/2015 38 / 54
The fourth cumulant changes sign at the transition.
0.1 0.12 0.14 0.16 0.18 0.2Ω
-8e+05
-6e+05
-4e+05
-2e+05
0
κ4
2-component Bose gas
fourth cumulant
Ian McCulloch (UQ) iDMRG 23/6/2015 39 / 54
Binder Cumulant for iMPS
Naively taking the limit L→∞ for the Binder cumulant doesn’t produceanything useful:
if the order parameter κ1 6= 0,
UL = 1− 〈m4〉L
3〈m2〉2L→ 2
3
if κ1 = 0, then m4(L) = 3k22L2 + k4L
Hence
UL = 1− 3k22L2 + k4L3k2
2L2 → 0
Finally, a step function that detects whether the order parameter isnon-zero
Better approach, in the spirit of finite-entanglement scaling: Evaluate themoment polynomial using L ∝ correlation length
Ian McCulloch (UQ) iDMRG 23/6/2015 40 / 54
0.98 1 1.02λ
0.3
0.4
0.5
0.6
0.7
Um
m=4m=5m=6m=7m=8m=9m=10m=11m=12
Transverse field Ising modelBinder cumulant, scale factor s=5
0.98 0.99 1 1.01 1.02λ
00.10.20.30.40.50.60.7
Um
m=4m=8m=12
Order parameter
Ian McCulloch (UQ) iDMRG 23/6/2015 41 / 54
String parametersOrder parameters do not have to be local
Mott insulator string order parameter
O2P = lim
|j−i|→∞〈Πj
k=i(−1)nk〉
We can write this as a correlation function of ‘kink operators’,
pi = Πk<i (−1)nk
This turns the string order into a 2-point correlation function:
O2P = lim
|j−i|→∞〈 pi pj 〉
Or as an order parameter:P =
∑i
pi
Then O2p = 1
L2 〈P2〉
Ian McCulloch (UQ) iDMRG 23/6/2015 42 / 54
Real example: 3-leg Bose-Hubbard model with flux phase (F. Kolley, M. Piraud,IPM, U. Schollwoeck, F. Heidrich-Meisner, in preparation)
For density n = 1/3 (one particle per rung), near flux φ ∼ π, there is atransition from a Mott to critical as a function of J⊥
P has a simple MPO representation
P =
(II (−1)n
)Hence we can calculate higher moments of P.
Ian McCulloch (UQ) iDMRG 23/6/2015 43 / 54
1 1.1 1.2 1.3Jperp
0
0.2
0.4
0.6
0.8
Op
2
m=100m=150m=200m=300m=400m=500m=800
Bose-Hubbard LadderString order parameter
1 1.1 1.2 1.30.001
0.01
0.1
1
Ian McCulloch (UQ) iDMRG 23/6/2015 44 / 54
100 1000m
1
10
100
1000ξ
Jperp = 0.99
Jperp = 1.00
Jperp = 1.01
Jperp = 1.02
Jperp = 1.03
Jperp = 1.04
Jperp = 1.05
Jperp = 1.06
Jperp = 1.10
Jperp = 1.14
Jperp = 1.25
Bose=Hubbard LadderScaling of correlation length
Ian McCulloch (UQ) iDMRG 23/6/2015 45 / 54
0.98 1 1.02 1.04 1.06 1.08Jperp
0.1
0.15
0.2
0.25
0.3
0.35
Uξ
m=200m=250m=300m=350m=400m=450m=500m=550m=600
Bose-Hubbard LadderString parameter Binder cumulant
Ian McCulloch (UQ) iDMRG 23/6/2015 46 / 54
Infinite boundary conditionsH.N. Phien, G. Vidal, IPM, Phys. Rev. B 86, 245107 (2012), Phys. Rev. B 88, 035103 (2013)(see also Zauner et al 1207.0862, Milsted et al Phys. Rev. B 155116 (2013))
Local perturbation to a translationally invariant state
Window (N sites)Left Right
Map infinite system onto a finite MPS, with an effective boundary
Ian McCulloch (UQ) iDMRG 23/6/2015 47 / 54
Key point: Even if the perturbation is correlated at long range, only thetensors at the perturbation are modifiedDecompose the Hamiltonian
H = HL + HLW + HW + HWR + HR
We can calculate HL and HR by summing the infinite series of terms fromthe left and rightAway from the perturbation the wavefunction is approximately aneigenstate, so
exp itHL ∼ I
and we don’t leave the Hilbert space of the semi-infinite strip
Ian McCulloch (UQ) iDMRG 23/6/2015 48 / 54
Spin-1 Heisenberg chain, S+ initial perturbation
60 80 100 120 140
window size = 60
Infinite boundaries
Ian McCulloch (UQ) iDMRG 23/6/2015 49 / 54
Resize the window
We can do better - why keep the size of the window fixed?Window expansion - incorporate sites from the translationally-invariant sectioninto the window
Criteria for expanding: is the wavefront near the boundary?(Calculate from the fidelity of the wavefunction at the boundary)
Ian McCulloch (UQ) iDMRG 23/6/2015 50 / 54
−80 −60 −40 −20 0 20 40 60 80
t = 0
t = 0.7
t = 2.25
t = 4
t = 5.8
t = 7.55
t = 9.35
t = 11.15
t = 12.95
t = 14.75
t = 16.6
t = 18.45
t = 20.3
t = 22.15
t = 24
x
〈Sz(x,t)〉
Expanding windowExpanding windowFixed window
Ian McCulloch (UQ) iDMRG 23/6/2015 51 / 54
Window contraction
Window contraction - incorporate tensors from the window into the boundaryContract the MPS and Hamiltonian MPO
=WWWW
WWWW =
Ian McCulloch (UQ) iDMRG 23/6/2015 52 / 54
Follow the wavefront
60 40 20 0
t = 0
t = 1.75
t = 3.95
t = 6.15
t = 8.35
t = 10.5
t = 12.7
t = 15
t = 17.35
t = 19.8
t = 22.2
t = 24.45
x
〈Sz(x,t)〉
Moving window
Moving window
Fixed window
Ian McCulloch (UQ) iDMRG 23/6/2015 53 / 54
Summary
iDMRG – efficient algorithm for obtaining translationally invariant iMPSMany quantities are natural for iMPS but difficult to calculate for finitesystems (eg correlation length)Finite-entanglement scaling – often easier than finite-size scalingBinder cumulant for detecting phase transitionslocal perturbations – Infinite Boundary Conditions
Future:Equations for the moment expansion have the same structure as theequations for perturbations and excitations (see Frank’s talk!)
Thanks:Fei Zhan, Greg Crosswhite, Phien Ho, Guifre Vidal, lots more...
Ian McCulloch (UQ) iDMRG 23/6/2015 54 / 54
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