Entire spectrum of fully many-body localized systems
using tensor networks
Thorsten B. Wahl
Rudolf Peierls Centre for Theoretical Physics, University of Oxford
10 February 2017
in collaboration with
Arijeet Pal, Steve Simon (Oxford)
T. B. Wahl, A. Pal, and S. H. Simon, arXiv:1609.01552
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Ergodicity in classical systems
H({xi , pi }) =
X p2
i
i
2mi
+
X
V (xi − xj ) = E
i<j
{pi }
H({xi }, {pi })=E
{xi }
equiprobability
all {xi }, {pi } consistent with H({xi }, {pi }) = E
are reached with equal probabiltiy
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
E = EA + EB + EAB
B
A
microcanonical → canonical
p({xAi },{pAi })dEA
EA ({xAi }, {pAi })
= exp −
dEA
T
|A| |B|
ergodicity
the system acts as its own heat bath
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Ergodicity in quantum systems
H|ψn i = En |ψn i
H = HA + HB + HAB
hψn |ÔA |ψn i =
tr ÔA exp (−βn HA )
tr (exp (−βn HA ))
B
A
Eigenstate Thermalization Hypothesis
(ETH)
J. M. Deutsch, Phys. Rev. A. 43, 2046 (1991)
M. Srednicki, Phys. Rev. E 50, 888 (1994)
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Violation of ETH
Anderson impurity model:
H=
X
i
i ai† ai −
X
tij ai† aj + h.c.
i<j
i
tij
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Violation of ETH
Anderson impurity model:
H=
X
i ai† ai −
i
X
tij ai† aj + h.c.
i<j
Simplifications:
1
impurities lie on a lattice
2
thi,ji = t, zero otherwise
3
i ∈ [−W , W ]
ψ
delocalized
W =0
x
localized
x
W > 0 in 1D and 2D,
W > W3D in 3D
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Switching on interactions
X
†
1D chain: H = X a† a − X t a† a
+
a
a
+
ni ni+1
i i i
i+1
i
i
i+1
i
i
i
localization survives for W > Wc :
Many-body localization
D. Basko, I. Aleiner, and B. Altshuler, Ann. Phys. 321, 1126 (2006).
I. Gornyi, A. Mirlin, and D. Polyakov, Phys. Rev. Lett. 95, 206603 (2005).
M. Schreiber, et. al, Science 349, 842 (2015).
no heat or electrical conductivity
system retains memory of initial state
topological protection at all energy scales (quantum memory)
Y. Bahri, R. Vosk, E. Altman, and A. Vishwanath, Nat. Comm. 6, 7341 (2015).
Motivation
Many-body localization
Table of content
1
Motivation
2
Many-body localization
3
Tensor Network ansatz
4
Numerical Results
Tensor Network ansatz
Numerical Results
Motivation
Many-body localization
Table of content
1
Motivation
2
Many-body localization
3
Tensor Network ansatz
4
Numerical Results
Tensor Network ansatz
Numerical Results
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Many-body localization (MBL)
Disordered Heisenberg antiferromagnet: MBL for W > Wc ≈ 3.5
H=
N
X
(JSi · Si+1 + hi Siz ),
i=1
hi ∈ [−W , W ]
ξL
τiz
↓
τiz = Uσiz U †
[H, τiz ] = [τiz , τjz ] = 0
↑
↓
↑
↑
↓
↑
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Many-body localization (MBL)
Disordered Heisenberg antiferromagnet: MBL for W > Wc ≈ 3.5
H=
N
X
(JSi · Si+1 + hi Siz ),
i=1
hi ∈ [−W , W ]
ξL
τiz
↓
↑
↓
τiz = Uσiz U †
[H, τiz ] = [τiz , τjz ] = 0
H|ψi1 ...iN i = Ei1 ...iN |ψi1 ...iN i
τ1z |ψ↑i2 ...iN i = |ψ↑i2 ...iN i
τ1z |ψ↓i2 ...iN i = −|ψ↓i2 ...iN i etc.
↑
↑
↓
↑
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Many-body localization (MBL)
Disordered Heisenberg antiferromagnet: MBL for W > Wc ≈ 3.5
H=
N
X
(JSi · Si+1 + hi Siz ),
i=1
hi ∈ [−W , W ]
ξL
τiz
↓
↑
↓
τiz = Uσiz U †
[H, τiz ] = [τiz , τjz ] = 0
H|ψi1 ...iN i = Ei1 ...iN |ψi1 ...iN i
↑
↑
↓
↑
En
Emax
1
MBL
thermal, ETH
τ1z |ψ↑i2 ...iN i = |ψ↑i2 ...iN i
τ1z |ψ↓i2 ...iN i = −|ψ↓i2 ...iN i etc.
Wc
full MBL
W
Motivation
|ψi1 ...iN i:
Many-body localization
↓
↑
A
↓
↑
Tensor Network ansatz
A
↑
↓
↑
↓
↓
Numerical Results
↓
A
↑
↑
ρA = trA (|ψi1 ...iN ihψi1 ...iN |)
entanglement entropy S(ρA ) = −tr(ρA ln(ρA )) < const. as |A| → ∞
M. Friesdorf, A. H. Werner, W. Brown, V. B. Scholz, and J. Eisert, Phys. Rev. Lett. 114, 170505 (2015).
approximation by Matrix Product States
↓
Motivation
Many-body localization
Table of content
1
Motivation
2
Many-body localization
3
Tensor Network ansatz
4
Numerical Results
Tensor Network ansatz
Numerical Results
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Matrix Product States
|ψi1 ...iN i ≈
s1
|ψMPS i =
s2
sN
...
X
|s1 , . . . , sN i
s1 ...sN =↑,↓
s
s =
Aαβ
α
A
s0
s
β
X
β
s0
s
Aαβ
Bβγ = α
A
β
B
γ
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Matrix Product States
|ψi1 ...iN i ≈
s1
|ψ̃i1 ...iN i =
s2
sN
...
X
|s1 , . . . , sN i
s1 ...sN =↑,↓
i1
...
i2
iN
s
Asi
αβ = α
A
i
s0
s
β
X
s0i 0
si
Aαβ
Bβγ = α
β
Matrix Product Operator
D. Pekker, B. K. Clark, Phys. Rev. B 95, 035116 (2017).
How to make |ψ̃i1 ...iN i orthonormal?
A
i
β
B
i0
γ
Motivation
Many-body localization
Tensor Network ansatz
Spectral Tensor Networks
Ũ =
!
we want: Ũ Ũ † = 1 and ŨH Ũ † ≈ diagonal matrix
Numerical Results
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Spectral Tensor Networks
Ũ =
!
we want: Ũ Ũ † = 1 and ŨH Ũ † ≈ diagonal matrix
u1,4
u1,3
Ũ =
u2,4
u2,3
u1,2
u1,1
u2,2
u2,1
...
F. Pollmann, V. Khemani, J. I. Cirac, and S. L. Sondhi, Phys. Rev. B 94, 041116 (2016)
Figure of merit: variance
X 1
∆H 2 = N
hψ̃i1 ...iN |H 2 |ψ̃i1 ...iN i − hψ̃i1 ...iN |H|ψ̃i1 ...iN i2
2 i ...i =↓,↑
1
N
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Scaling
approximate integrals of motion: τ̃iz = Ũσiz Ũ †
ξL
τiz
↓
↑
↓
↑
↑
↓
Ũ =
↑
↓
↓
↑
n
σiz
tCPU ∼ exp (n)
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Scaling
approximate integrals of motion: τ̃iz = Ũσiz Ũ †
ξL
`
error ∼ exp − ξ`L
τiz
↓
↑
↓
↑
↑
↓
↑
↓
↓
↑
Ũ =
#required ∼ 22`
n
#actual ∼ n
σiz
tCPU ∼ exp (n) ∼ exp (exp (`))
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Alternative tensor network
#parameters ∼ 2`
u1,2
u1,1
u2,2
u2,1
sufficient to capture τiz
correctly on length scale `
...
`
tCPU ∼ exp(`) ⇒ error ∼
1
poly(tCPU )
error ∼ exp − ξ`L
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Alternative tensor network
#parameters ∼ 2`
u1,2
u1,1
u2,2
u2,1
sufficient to capture τiz
correctly on length scale `
...
`
tCPU ∼ exp(`) ⇒ error ∼
1
poly(tCPU )
Variance as a figure of merit
7`
tCPU ∼ N 2`
error ∼ exp − ξ`L
Motivation
Many-body localization
Our figure of merit
Recap
τiz = Uσiz U †
[H, τiz ] = [τiz , τjz ] = 0
τ̃iz = Ũσiz Ũ †
Tensor Network ansatz
Numerical Results
Motivation
Many-body localization
Tensor Network ansatz
Our figure of merit
Recap
τiz = Uσiz U †
[H, τiz ] = [τiz , τjz ] = 0
τ̃iz = Ũσiz Ũ †
Define figure of merit as:
f ({ux ,y }) =
N
1 X
z
z †
[H,
τ̃
][H,
τ̃
]
tr
i
i
2N i=1
= const. −
N/`
X
x =1
scaling: tCPU ∼ N23` `2
fx (ux ,1 , ux −1,2 , ux ,2 )
Numerical Results
Motivation
Many-body localization
Table of content
1
Motivation
2
Many-body localization
3
Tensor Network ansatz
4
Numerical Results
Tensor Network ansatz
Numerical Results
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Comparison to exact diagonalization
`=2
antiferromagnetic
Heisenberg chain:
H is real and
S z -symmetric
hi ∈ [−W , W ],
Wc ≈ 3.5
N = 16, W = 6
4 layers
`=4
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Comparison to exact diagonalization
`=2
`=4
antiferromagnetic
Heisenberg chain:
H is real and
S z -symmetric
hi ∈ [−W , W ],
Wc ≈ 3.5
N = 16, W = 6
dimHilbert = 216
= 65, 536
`=8
overlap for ` = 8
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Approximation of local observables
plotting:
|hψ̃i1 ...iN |σjz |ψ̃i1 ...iN i|
N = 12, use ` = 6
W =2
W =3
W =4
W =6
Motivation
Many-body localization
Tensor Network ansatz
Scaling with system size
W =6
remember:
f ({ux ,y }) = const.−
X
x
fx
Numerical Results
Summary benchmark results
very high precisions for ` = 6, 8
local observables approximated
accurately at tCPU ∼ N
⇒ simulation of large MBL systems with
high accuracies
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Approaching the phase transition for N = 72
#param (` = 8) = 6307
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Detection of the phase transition
|ψi1 ...iN i
# 1:
↓
↑
N/2
↑
↓
↑
|ψi1 ...iN i → ρi1 ...iN
↓
↑
↓
→ S(ρi1 ...iN ) →
↓
↓
↑
↑
average S(1)
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Detection of the phase transition
|ψi1 ...iN i
# 1:
# 2:
...
# 100:
↓
↑
N/2
↑
↓
↑
|ψi1 ...iN i → ρi1 ...iN
↓
↑
↓
→ S(ρi1 ...iN ) →
↓
↓
↑
↑
average S(1)
average S(2)
...
average S(100)
S, σS
Motivation
Many-body localization
Tensor Network ansatz
Comparison to exact diagonalization
N = 12
Numerical Results
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Now again for a large system of N = 72
|ψ̃i1 ...iN i
# 1:
# 2:
...
# 100:
↓
↑
N/2
↑
↓
↑
|ψ̃i1 ...iN i → ρ̃i1 ...iN
↓
↑
↓
→ S(ρ̃i1 ...iN ) →
↓
↓
↑
↑
average S(1)
average S(2)
...
average S(100)
S, σS
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Now again for a large system of N = 72
|ψ̃i1 ...iN i
# 1:
# 2:
...
# 100:
↓
A
↑
↓
↑
↑
|ψ̃i1 ...iN i → ρ̃i1 ...iN
↓
↑
↓
→ S(ρ̃i1 ...iN ) →
↓
↓
↑
↑
average S(1)
average S(2)
...
average S(100)
S, σS
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Now again for a large system of N = 72
|ψ̃i1 ...iN i
# 1:
# 2:
...
# 100:
↓
A
↑
↓
↑
↑
|ψ̃i1 ...iN i → ρ̃i1 ...iN
↓
↑
↓
→ S(ρ̃i1 ...iN ) →
↓
↓
↑
↑
average S(1)
average S(2)
...
average S(100)
S, σS
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Now again for a large system of N = 72
|ψ̃i1 ...iN i
# 1:
# 2:
...
# 100:
↓
A
↑
↓
↑
↑
|ψ̃i1 ...iN i → ρ̃i1 ...iN
↓
↑
↓
→ S(ρ̃i1 ...iN ) →
↓
↓
↑
↑
average S(1)
average S(2)
...
average S(100)
S, σS
hSicuts , hσS icuts
Motivation
Many-body localization
Tensor Network ansatz
N = 72
Numerical Results
Motivation
Many-body localization
Tensor Network ansatz
Conclusions
tensor network ansatz for fully MBL systems
computational complexity tCPU ∝ N
scalable: error ∼ 1/poly (tCPU ) for given N
figure of merit decomposes into local parts
(improved scaling 27` → 23` )
very high accracies, even in vicinity of
MBL-to-thermal transition
Numerical Results
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results
Comparison to previous scheme for N = 12
`=2
n = 4 layers
`=4
same without imposing symmetries
Motivation
Many-body localization
Tensor Network ansatz
Numerical Results