Computational Complexity and High Energy Physics August 1st, … · 2017. 8. 23. · Rigorous free...

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

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

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

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, …

… 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)






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)






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)




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




Can be extended to (2+1)d (and (3+1)d?):  first explorations by L Tagliacozzo, E Zohar, …

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


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

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

Tensor network renormalization interpretation:  (G Evenbly & G Vidal; S Yang, ZC Gu, XG Wen)

 


imag

inar

y tim

e ev

olut

ion

…

d⌧


 


2d⌧

imag

inar

y tim

e ev

olut

ion

…


 


4d⌧

imag

inar

y tim

e ev

olut

ion

…


 

For classical stat mech systems: can also be done using non-negative matrix factorization → M. Bal et al, PRL 118, 250602 (2017)


4d⌧

imag

inar

y tim

e ev

olut

ion

…


Wavelets and renormalization: multiscale analysis


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


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

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)

�

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

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)

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

�

FERMI SURFACES

Branching MERA (Evenbly & Vidal)R Shankar, RG approach to interacting fermions (RMP 66, 129)

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)

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 [�⇡,+⇡)

RIGOROUS APPROXIMATION RESULT

K=L=1 K=L=3

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)

QUESTIONS?

Computational Complexity and High Energy Physics August 1st, … · 2017. 8. 23. · Rigorous free...

Documents

Transcript of Computational Complexity and High Energy Physics August 1st, … · 2017. 8. 23. · Rigorous free...