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Rigorous free fermion entanglement renormalization from
wavelets
arXiv: 1707.06243
Jutho Haegeman
Ghent University
in collaboration with:
Brian Swingle, Michael Walter,
Jordan Cotler, Glen Evenbly, Volkher Scholz
Computational Complexity and High Energy Physics
August 1st,
2017
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OVERVIEW
Tensor networks and quantum field theories
MERA: quantum circuits, renormalization, wavelets
One-dimensional Dirac fermions
Fermi surface: Non-relativistic two-dimensional fermions
Rigorous approximation result
Outlook and extensions
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TENSOR NETWORKS …
Diagrammatic notationvector:matrix:matrix product:Yang-Baxter
equation:
=
H = (Cd)�NQuantum system of N spins:
|�� =d�
sn=1
�s1,s2,...,sN |s1� � |s2� � · · · � |sN �
dimensional vectordN
rank N tensor
�
s1s2 sN
‘approximate’ with
a tensor network
decomposition
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TENSOR NETWORKS …
Variational families of states for quantum many body systems,
motivated by the structure of entanglement in low energy states
(area law) ……
� =(MPS)
(PEPS)(MERA)
|0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i
⇒ classical simulation⇒ Complexity scaling (MPS): Hastings;
Arad, Kitaev, Landau, Vazirani, Vidick, …
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… AND QUANTUM FIELD THEORY (1)
Variational approach to lattice gauge theory (Hamiltonians)
Very successful for (1+1)d QFT, e.g. Schwinger model
TMR Byrnes
et al, B Buyens et al, MC Bañuls et al, S Montangero et al, …
(Partial) string breaking for heavy probe charges:
L · g0 3 6 9 12 15
0
5
10
15
20
25
VQ(L)/g (m/g = 1)
0.75
1
1.75
2.5
3.25
4.5
5
Q
L · g0 3 6 9 12 15
0
5
10
15
20
25VQ(L)/g (m/g = 0.5)
L · g0 3 6 9 12 15
0
5
10
15
20
25VQ(L)/g (m/g = 0.25)
Boye Buyens, Jutho Haegeman, Henri Verschelde, Frank Verstraete,
Karel Van Acoleyen, PRX 6, 041040 (2016)
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… AND QUANTUM FIELD THEORY (1)
Variational approach to lattice gauge theory (Hamiltonians)
Very successful for (1+1)d QFT, e.g. Schwinger model
TMR Byrnes
et al, B Buyens et al, MC Bañuls et al, S Montangero et al, …
(Partial) string breaking for heavy probe charges:
Boye Buyens, Jutho Haegeman, Henri Verschelde, Frank Verstraete,
Karel Van Acoleyen, PRX 6, 041040 (2016)
z · g-15 -10 -5 0 5 10 15
-1.5
-1
-0.5
0
0.5
1
1.5⟨Ψ̄(z)γ0Ψ(z)⟩ (Q = 4.5)
z · g-15 -10 -5 0 5 10 15
-5
-4
-3
-2
-1
0
1
2⟨E(z)⟩/g (Q = 4.5)
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… AND QUANTUM FIELD THEORY (1)
Variational approach to lattice gauge theory (Hamiltonians)
Very successful for (1+1)d QFT, e.g. Schwinger model
TMR Byrnes
et al, B Buyens et al, MC Bañuls et al, S Montangero et al, …
(Partial) string breaking for heavy probe charges:
Boye Buyens, Jutho Haegeman, Henri Verschelde, Frank Verstraete,
Karel Van Acoleyen, PRX 6, 041040 (2016)
z · g-15 -10 -5 0 5 10 150
0.05
0.1
0.15
0.2
0.25
∆SQ(z) (L · g = 0.55)
100200300400
x
z · g-15 -10 -5 0 5 10 150
0.05
0.1
0.15
0.2
0.25
∆SQ(z) (L · g = 2.55)
z · g-15 -10 -5 0 5 10 150
0.05
0.1
0.15
0.2
0.25
∆SQ(z) (L · g = 7.35)
z · g-15 -10 -5 0 5 10 150
0.05
0.1
0.15
0.2
0.25
∆SQ(z) (L · g = 13.15)
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… AND QUANTUM FIELD THEORY (1)
Variational approach to lattice gauge theory (Hamiltonians)
Very successful for (1+1)d QFT, e.g. Schwinger model
TMR Byrnes
et al, B Buyens et al, MC Bañuls et al, S Montangero et al, …
Real time evolution: quenches, onset of thermalization?
Boye Buyens, Jutho Haegeman, Florian Hebenstreit, Frank
Verstraete, Karel Van Acoleyen, arXiv:1612.00739
t · g0 3 6 9 12 15 18
0
1
2
3
4
5
∆S(t)
t · g0 3 6 9 12 15 18
0
0.1
0.2
0.3
0.4
N(t)
N(t)Nβ0±0.05
t · g0 3 6 9 12 15 18
-1.5
-1
-0.5
0
0.5
1
1.5E(t)/g
E(t)/gEβ0±0.05/g
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… AND QUANTUM FIELD THEORY (1)
Variational approach to lattice gauge theory (Hamiltonians)
Very successful for (1+1)d QFT, e.g. Schwinger model
TMR Byrnes
et al, B Buyens et al, MC Bañuls et al, S Montangero et al, …
Can be extended to (2+1)d (and (3+1)d?):
first explorations by
L Tagliacozzo, E Zohar, …
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Tensor networks for continuous systems:
Continuous MPS: (Verstraete & Cirac, 2010)Chiral condensate
in the Gross-Neveu model (N→∞)
Continuous MERA:So far: only GaussiansInteresting analytical
tool to investigate relation with holography
⇒ Advertisement: PhD & PostDoc positions available
@UGent
… AND QUANTUM FIELD THEORY (2)
λ(Λ)σ/
Λ0.01
0.1
1
λ-1(Λ) = [(N - 1)g(Λ)2]-10.6 0.8 1.0 1.2 1.4 1.6
exactafitbD = 6
D = 8D = 10D = 16
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Multiscale entanglement renormalization ansatz (G Vidal)
Captures power law decay of correlations, logarithmic violation
of area law in (1+1)d, …Possible relation with holography (AdS/CFT
correspondence)
MERA, RENORMALIZATION & WAVELETS
|0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i
Coarse-grain quantum state
Disentangle high energy
degrees of freedom
Variational
ansatz
Quantum
circuit that
prepares state
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Tensor network renormalization interpretation:
(G Evenbly &
G Vidal; S Yang, ZC Gu, XG Wen)
MERA, RENORMALIZATION & WAVELETS
imag
inar
y tim
e
ev
olut
ion
…
d⌧
-
Tensor network renormalization interpretation:
(G Evenbly &
G Vidal; S Yang, ZC Gu, XG Wen)
MERA, RENORMALIZATION & WAVELETS
2d⌧
imag
inar
y tim
e
ev
olut
ion
…
-
Tensor network renormalization interpretation:
(G Evenbly &
G Vidal; S Yang, ZC Gu, XG Wen)
MERA, RENORMALIZATION & WAVELETS
4d⌧
imag
inar
y tim
e
ev
olut
ion
…
-
Tensor network renormalization interpretation:
(G Evenbly &
G Vidal; S Yang, ZC Gu, XG Wen)
For classical stat mech systems: can also be done using
non-negative matrix factorization → M. Bal et al, PRL 118, 250602
(2017)
MERA, RENORMALIZATION & WAVELETS
4d⌧
imag
inar
y tim
e
ev
olut
ion
…
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MERA, RENORMALIZATION & WAVELETS
Wavelets and renormalization: multiscale analysis
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MERA, RENORMALIZATION & WAVELETS
Wavelets and renormalization: multiscale analysis
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MERA, RENORMALIZATION & WAVELETS
Wavelets and renormalization: multiscale analysis
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MERA, RENORMALIZATION & WAVELETS
Wavelets and MERA: G Evenbly & S White, PRL 116, 140403
(2016)
A free fermion MERA (unitaries generated by quadratic operators)
implements a wavelet transform at the single particle level.
|0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i|1i
|0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i|1i
scaling coefficients
wavelet coefficients
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MERA, RENORMALIZATION & WAVELETS
Wavelets and MERA: G Evenbly & S White, PRL 116, 140403
(2016)
A free fermion MERA (unitaries generated by quadratic operators)
implements a wavelet transform at the single particle level.
Free fermion ground state: fill all negative energy modes
→ fill
set of modes that span the negative energy subspace (Fermi/Dirac
sea)→ construct wavelets that are completely supported in either
positive or negative energy subspace
|0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i |0i|1i
wavelet coefficients
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Massless Dirac fermions on the lattice: staggering
(Kogut-Susskind)
1+1 DIRAC FERMIONS
0 e�ik � 1
eik � 1 0
� 1 1
ieik/2 �ieik/2�=
1 1
ieik/2 �ieik/2�
sin(k/2) 00 � sin(k/2)
�, k 2 [�⇡,+⇡)
0 e�ik � 1
eik � 1 0
�u(k) = u(k)
�| sin(k/2)| 0
0 | sin(k/2)|
�, k 2 [�⇡,+⇡)
u(k) =1p2
1 1
�isign(k)eik/2 isign(k)eik/2�=
1 00 �isign(k)eik/2
�1p2
1 11 �1
�
HD = �X
n
b†1,nb2,n � b†2,nb1,n+1 + b
†2,nb1,n � b
†1,n+1b2,n
HD =
Z +⇡
�⇡
dk
2⇡
b1(k)b2(k)
�† 0 e�ik � 1
eik � 1 0
� b1(k)b2(k)
�
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1+1 DIRAC FERMIONS
u(k) =1p2
1 1
�isign(k)eik/2 isign(k)eik/2�=
1 00 �isign(k)eik/2
�1p2
1 11 �1
�
A pair of wavelet transforms such that wavelet filters in
Fourier domain have equal magnitude and a relative phase difference
.
Scaling filters should have phase difference
(half shift or
half delay condition) → same phase difference for wavelets from
higher levels of the transform (scale invariance).
Impossible with filters of finite support ⇒ approximation?
�isign(k)eik/2(gw(k), hw(k))
(gs(k), hs(k)) eik/2
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1+1 DIRAC FERMIONS
K = 4
L = 6
Problem considered by Selesnick et al: a family of solutions,
satisfying , in terms of two parameters K and L, leading to filters
of width 2(K+L):
hs(k) = ei✓(k)gs(k)
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FERMI SURFACES
Non-relativistic fermions hopping at half filling:
H1 = �X
n2Za†nan+1 + a
†n+1an
b1,n = (�1)na2n, b2,n = (�1)na2n+1 HD
H2 = �X
(m,n)2Z2a†m,nam+1,n + a
†m+1,nam,n + a
†m,nam,n+1 + a
†m,n+1am,n
b1,x,y = (�1)x+yax+y,x�y, b2,x,y = (�1)x+yax+y+1,x�y
H =
Z
[�⇡,⇡)2dk
x
dky
b1(kx, ky)b2(kx, ky)
�† 0 (1� e�ikx)(1� e�iky )
(1� eikx)(1� eiky ) 0
� b1(kx, ky)b2(kx, ky)
�
u(kx
, ky
) =
1 00 �isign(k
x
)eikx/2
� 1 00 �isign(k
y
)eiky/2
�1p2
1 11 �1
�
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FERMI SURFACES
Branching MERA (Evenbly & Vidal)R Shankar, RG approach to
interacting fermions (RMP 66, 129)
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FERMI SURFACES
S(R) c2R+ 2(R/2)S1d MERA(R/2) + S(R/2) . . . 2c1R log2 R+O(R)
11where
N trz̄ ≡z̄−1∑
z=0
Rz(
(lz + 2)D − (lz)
D)
(41)
≈ 2D(l0)D−1
z̄−1∑
z=0
Rz
(
1
2D−1
)z
(42)
= 2D(l0)D−1f(l0) (43)
where in Eq. 42 we have only kept leading order in l0,and
where
f(l0) ≡z̄−1∑
z=0
Rz
(
1
2D−1
)z
. (44)
Thus we see that N trz̄ scales as the boundary law (l0)D−1
times a multiplicative correction f(l0) that depends onthe
branching structure of the underlying holographictree through Rz.
It follows that the entanglement en-tropy S(ρ0) is bounded above
by
S(ρ0) ! kD(l0)D−1f (l0) , (45)
where the constant kD depends on χ and D (but is in-dependent of
l0). Here we have used that the first termon the rhs of Eq. 40,
which also depends on l0 throughRz̄, can be seen to be of
subleading order in l0, whencompared to (l0)D−1f(l0), for any
relevant choice of Rz.Next we evaluate function f(l0) for two
classes of holo-
graphic trees.
C. Regular holographic trees
Let us evaluate the above upper bound on entangle-ment entropy
for branching MERA with a regular holo-graphic tree with branching
ratio b, where each node ofthe tree has exactly b child nodes.
Notice that for thisfamily of trees the number of branches at depth
z scalesas Rz = bz. Then the function f(l0) of Eq.44,
whichdescribes the multiplicative correction to the boundarylaw,
becomes
f (l0) =z̄−1∑
z=0
(
b
2D−1
)z
. (46)
Notice that this is a geometric series with common ratior =
b21−D and, as such, can be summed explicitly. Thissum takes has a
different functional dependance on l0contingent on whether the
branching b is such that thecommon ratio is greater than, equal to
or less-than unity.In these three cases, to leading order in l0 the
functionf(l0) reads
f (l0) ≈
⎧
⎪
⎨
⎪
⎩
c1 b < 2D−1
c2log2(l0) b = 2D−1
c3(l0)(1−D+log
2(b)) b > 2D−1
(47)
FIG. 9. (a) A depiction of part of a branching MERA inD = 2
dimensions. The density matrix ρz is obtained bycombining two
copies of ρz+1 with isometries/decouplers wand disentanglers u, and
then tracing out ntrz = 20 indices.(b-e) A branch of the branching
MERA in D = 2 dimensionscan split into b = 1, 2, 3, 4 sub-branches
at each level. Diagram(a) corresponds to the case of b = 2.
for some constants c1, c2, and c3 that depend on D andb (but are
independent of l0). These, together with Eq.45, lead to the
following upper bounds for the scalingof entanglement in the
branching MERA with a regularholographic tree
Sl ≤
⎧
⎪
⎨
⎪
⎩
c̃1 lD−1 b < 2D−1
c̃2 lD−1 log2(l) b = 2D−1
c̃3 llog2(b) b > 2D−1(48)
for some constants c̃α = cαkD that depend on D, b, andχ. A
subset of these results can be found on table III.Notice in
particular that for b = 2D−1 we obtain a loga-rithmic correction to
the boundary law for all dimensionsD, whereas b = 2D produces a
bulk law.
[
˜S(R) c2R+ 2˜S(R/2) · · · c2R log2(R) +O(R)]
S1d MERA(R) c1 + S1d MERA(R/2) . . . c1 log2(R) +O(1)
S2d MERA(R) c2R+ S2d MERA(R/2) . . . 2c2R+O(1)
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RIGOROUS APPROXIMATION RESULT
f 2 `2(Z) : a(f) =X
n2Zf [n]an
Given pair of scaling filters:
B = k�sk1
Let:
G({fi}) = h |a†(f1) . . . a†(fN )a(fN+1) . . . a(f2N )| i
|G({fi})⌦ �G({fi})⌦MERA | 24pNqC2�L/2 + 6✏(log2(C/✏))
2Then:
Where: C = 23/2p
D({fi})B(K + L)L : number of layers in the MERA
|hs(k)� eik/2gs(k)| ✏, 8k 2 [�⇡,+⇡)
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RIGOROUS APPROXIMATION RESULT
K=L=1 K=L=3
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OUTLOOK AND EXTENSIONS
Extensions
Massive theories
Pairing term:
fermion number → fermion parity conservation
Outlook
Dirac cones, topological insulators,…?
Relevance for interacting theories?
(e.g. with asymptotic
freedom)
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QUESTIONS?