Roger Melko

Renyi entropiesin

Theory, Numerics, and Experiment

Why are we interested in entanglement?• To characterize equilibrium phases (including topological order)

Lucy Reading-Ikkanda/Quanta Magazine, adapted from figure by Xiao-Gang Wen

• To distinguish thermalized and localized states

• To characterize quantum critical points and QFTs

Bardarson, Pollmann, Moore, Phys. Rev. Lett. 109, 017202 (2012)

S[�] =

Z

d

3x

1

2

n

(@�)2 �m

2�

2oT

g

�S

• To determine how much entanglement exists in matter (e.g. for QIP)

an area law is heuristically related to short-range correlations:

-( )1p2

=

Entanglement in quantum many-body systems

• A powerful probe of many-body properties

• Entanglement entropy “area law” is the paradigm for groundstate wavefunctions

• Violations, and subleading scaling terms may provide universal quantities useful for identifying phases and phase transitions

S = a `d�1

S = a `d�1log `

S = a `+ �

S =

c

3

log `

Fermi surface

Topological order in two dimensions

1+1 conformal field theory

d+1 quantum critical pointS = a`d�1+ �(geom.)

Entanglement and Algorithms

• The area law has emerged as a powerful paradigm underlying the success of DMRG in one-dimensional systems

• Higher dimensional extensions are called tensor networks, and hold great promise both as theoretical and numerical tools

• A brief introduction to Renyi entropies in condensed matter physics

• Renyi entropies in classical Monte Carlo

• Renyi entropies in quantum Monte Carlo

• An example of universal Renyi entropies in 2+1

`

Outline

von Neumann entanglement entropy

S1(⇢A) = �Tr(⇢A ln ⇢A)

• Quantifies the entanglement between spins A and B

• Positive

• if A and B are unentangled

• (if the full density matrix describes a pure state)

• Basis independent

• Strongly subadditive

⇢A = TrB(⇢)⇢ = | ih |

`

S1(⇢A) = 0

S1(⇢A) = S1(⇢B)

S1(⇢A) � 0

S1(⇢ABC) + S1(⇢B) S1(⇢AB) + S1(⇢BC)

• Quantifies the entanglement between spins A and B

• Positive

• if A and B are unentangled

• (if the full density matrix describes a pure state)

• Basis independent

• Strongly subadditive

Renyi entanglement entropies

Sn(�A) =1

1� nln

�Tr

��n

A

��

S2(�A) = � ln�Tr(�2

A)�

`

Sn(⇢A) � 0

Sn(⇢A) = Sn(⇢B)

Sn(⇢A) = 0

S1(⇢ABC) + S1(⇢B) S1(⇢AB) + S1(⇢BC)

A B| i = cos(✓)| i| i+ sin(✓)| i| i

⇢A =

✓cos

2(✓) 0

0 sin

2(✓)

◆S1 = � cos

2(✓) ln cos2(✓)� sin

2(✓) ln sin2(✓)

Sn

| i| i | i| i✓

1p2| i| i+ 1p

2| i| i

• Renyi entropies satisfy the inequalities

• They are all equal in the minimal and maximal entangled case

• No Renyi entropies are generally considered to be “observables” due to their non-linear dependence on the reduced density matrix… (but let’s take a closer look at n>1 integers Renyis)

Sm � Sn m < n

Sn =1

1� nln

dX

i=1

�ni =

1

1� nln

dX

i=1

1

dn

=1

1� nln

d

dn=

1

1� nln

1

dn�1

= ln(d)

S2, S3, S4, etc.

�i =1

d

dX

i=1

�i = 1

max. entanglement

• Topological entanglement entropy is Renyi-index independent

Sn = A`+ �n �n = � ln(2)Kitaev and Preskill - Phys. Rev. Lett. 96, 110404 (2006) Levin and Wen, - Phys. Rev. Lett. 96, 110405 (2006) Flammia, Hamma, Hughes, Wen Phys. Rev. Lett. 103, 261601 (2009)

Are Renyis “as good” as vN?

• In critical systems, universal quantities may depend on n

x

S1

S2Sn =

c

6

⇣1 +

1

n

⌘log

hL

⇡

sin

⇡x

L

iCalabrese and Cardy. J. Phys. A: Math. Theor., 42:504005, 2009

How do we calculate the Renyi entropies?

S1 =Length(�A)

4G(3)N

=c

3ln

x

a

J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998)S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006)

S1 =Area(�A)

4G(d+2)N

• Using holographic formulas

• Directly from the density matrix

Sn(�A) =1

1� nln⇤Tr

��nA

⇥⌅- exact diagonalization- density matrix renormalization group- quantum state tomography

S1 = � limn!1

d

dnTr⇢nA

J. Callan, Curtis G. and F. Wilczek, Phys. Lett. B333 , 55–61 (1994) P. Calabrese and J. L. Cardy, J. Stat. Mech. 0406, P002 (2004)

- field theory - Monte Carlo methods- Cold atom experiments

• From a multi-sheeted Riemann surface

• A brief introduction to Renyi entropies in condensed matter physics

• Renyi entropies in classical Monte Carlo

• Renyi entropies in quantum Monte Carlo

• An example of universal Renyi entropies in 2+1

`

Outline

(one-slide) reminder of Monte Carlo strategies

hOi = 1

ZTr

⇥e��HO⇤

=

Px

O(x)WxP

x

W

x

xt�1

xt+1

xt

W

x

P (x ! y) = W

y

P (y ! x)

Classical:Quantum:

A multi-sheeted Riemann surface can be used in Monte Carlo: a “replica trick”

piA =�

iBe��E(iA,iB)

Z

p2iA

= Z�2

��

iB

e��E(iA,iB)

��

��

jB

e��E(iA,jB)

�

�

S2(A) = � ln

�

�Z�2�

iA

�

iB

�

jB

e��(E(iA,iB)+E(iA,jB))

�

� = � ln�

Z[A, 2, T ]Z2

�

• Define the entropy of a subregion A:

~ two copies of B atT/2 ~ one copy of A at T

V. Alba, J. Stat. Mech. P05013 (2013)Iaconis, Inglis, Kallin, RGM: PRB 87, 195134 (2013)

Classical “entanglement” entropy of a bipartition

For general n, a n-sheeted “book” gives you the Renyi entropy

=1

1� nln

Z[A, n,�]

Zn

�

Sn(A) =1

1� nln

⇥Tr

�⇢nA

�⇤

Stéphan, Misguich, Pasquier PRB 82, 125455 (2010)

• This general picture gives the basic idea how integer Renyi entropies are amenable to “measurement” in both Monte Carlo (and experiment)

Cardy, PRL 106, 150404 (2011)Abanin, Demler, PRL 109, 020504 (2012)Pichler, Bonnes, Daley, Läuchli, Zoller, NJP. 15 (2013) 063003

Measuring the Renyi entropy of a two-site Fermi-Hubbard model on a trapped ion quantum computerLinke et. al arXiv:1712.08581

Rajibul Islam: Tomorrow 8:30am

Calculation of the entropy of a bipartition is technically equivalent to calculating the difference of two free energies:

• Thermodynamic integration

S2 = � lnTr(�2A) = � ln

�Z[A, 2,�]

Z(�)2

�

= �SA(� = 0) +

Z �

0hEiAd� + 2S0(� = 0)� 2

Z �

0hEi0d�

= � lnZ[A, 2,�] + 2 ln Z(�)

- Wang-Landau & multicanonical methods

• Extended ensemble methods

- Ratio trick

Humeniuk, Roscilde. Phys. Rev. B, 86, 235116 (2012)

Inglis, RGM, Phys. Rev. E 87, 013306 (2013)

Z[A1, 2,�]

Z[A2, 2,�]=

⌧N1

N2

�

MC

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

Sn(A) �= Sn(B)

In general:

each becomes an extensive quantity at finite temperature (volume-law)

TOne can consider the Mutual Information

In(A;B) = Sn(A) + Sn(B)� Sn(A �B)

The Mutual Information is the amount of information about A contained in B: is sensitive to all correlations

AB A

B

Wolf, Verstraete, Hastings, Cirac, PRL 100, 070502 (2008)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

I2

L

H = �J�

�ij�

Szi Sz

j

T/J

A B

Example: Mutual Information of the 2D Ising model

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4

0.1

0.2

0.3

0.4

0.5

0.6

H = �J�

�ij�

Szi Sz

j

T/J

I2

L

A B

Example: Mutual Information of the 2D Ising model

“Geometric” Mutual Information at the 2D Ising critical point

Gn

=

c

2

✓n

n� 1

◆log

✓f(L

A

/Lx

)f(LB

/Lx

)pLx

f(Ly

/Lx

)

◆

In(A;B) = anL+ Gn + · · ·

J.-M. Stephan, S. Inglis, P. Fendley, RGM, Phys. Rev. Lett. 112, 127204 (2014)

�0.2

�0.1

0

0 1/4 1/2

I 2(`,L

)�I 2(L

/2,L

)

`/L

�0.25

0

�0.5 02J(c=1)(`/L)

c = 1/2

L = 16L = 32L = 48

LA/L` =

Lx

= Ly

Gives a simple way to extract the central charge c=1/2 of the CFT (with no non-universal velocities)

c.f. DMRG on the 1D quantum Ising model

• If studying universal properties of entanglement entropy at a quantum critical point, can look instead at a classical (thermal) phase transition in a d+1 model

H = �JX

hi,ji

�z

i

�z

j

� hX

i

�x

i

(h/J)c = 1

H = �JX

hiji

Szi S

zj

(T/J)c = 2.269

General quantum-classical correspondence

• Since EE depends on subregion geometry, once must take care in how the d+1 classical system is defined (c.f. QMC, next)

Exercise: Classical Z2 Ising gauge theory

pi

j

k

l

Sn = A`+ �

H = �X

p

�zi �

zj�

zk�

zl � � 2��1

Castelnovo, Chamon Phys. Rev. B 76, 184442 (2007)

0 1 2 3 4 5 6

�

�0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Sto

po,

LW

L = 8

L = 16

L = 24

L = 32

log(2)

2�

�

Exercise: Classical Z2 Ising gauge theory

2� = �SAn + SB

n + SCn � SD

n

A B C D

pi

j

k

l

Levin and Wen, Phys. Rev. Lett. 96, 110405 (2006)

� = � ln(2)/2

H = �X

p

�zi �

zj�

zk�

zl � � 2��1

Sn = A`+ �

• A brief introduction to Renyi entropies in condensed matter physics

• Renyi entropies in classical Monte Carlo

• Renyi entropies in quantum Monte Carlo

• An example of universal Renyi entropies in 2+1

`

Outline

Quantum Monte Carlo

space (d-dimensional)

imaginarytime

• d+1 dimensional simulation cell

• can have periodic or open boundaries in imaginary time

• sign problem may exist

• no direct access to the reduced density matrix

Ground-state projector QMC

(�H)m|↵i = c0|E0|m|0i+ c1

c0

✓E1

E0

◆m

|1i · · ·�

= c0|E0|m|0i as m ! 1

|↵i

|↵i

|0i|0i

Z = h0|0i

hOi = 1

Zh0|O|0i

A

B0 |0i,,

⇢AB, |0ih0| ,0 0

TrB(⇢AB) = ⇢A0 0 ,

0 0 0 0,Tr(⇢2A)

The “replica” trick

0

,Tr(⇢2A)

0

0 0

hSwapBi

0

,Tr(⇢2A)

0

0 0 ,

S2 = � ln Tr(⇢2A)

Tr(⇢2A) = h0 ⌦ 0|SwapB|0 ⌦ 0i

= hSwapBi = hSwapAi

Tr(⇢2A) = h0 ⌦ 0|SwapB|0 ⌦ 0i

= hSwapBi = hSwapAi

0

,Tr(⇢2A)

0

0 0 , hSwapAi

S2 = � ln Tr(⇢2A)

Can be used in to obtain the higher-order Renyi entropies, and hence the

- entanglement spectrum

- logarithmic negativity

h⇧(4)A i,Tr(⇢4

A) ,

00

0 0

0 0

00

Chung, Alba, Bonnes, Chen, and Lauchli, arXiv:1312.1168V Alba, J. Stat. Mech. P05013 (2013)

Chung, Bonnes, Chen, Lauchli, arXiv:1305.6536Tubman, Yang, arXiv:1402.0503

Renyi entropies can be easily accessed in any flavor quantum Monte Carlo that estimates the ground-state wavefunction

T. Grover, Phys. Rev. Lett. 111, 130402 (2013)Assaad, Lang, Toldin, Phys. Rev. B 89, 125121 (2014)Broecker and Trebst, J. Stat. Mech. (2014) P08015

Zhang, Grover, and Vishwanath, PRB 84, 075128 (2011)

McMinis, Tubman, Phys. Rev. B 87, 081108(R) (2013)

Herdman, Roy, RGM, Del Maestro, Phys. Rev. B 89, 140501 (2014)Phys. Rev. E 90, 013308 (2014)

Kallin, Gonzalez, Hastings, RGM, PRL., 103, 117203 (2009)

SWAP SWAP

- Valence-bond basis QMC

- Variational MC

- Path Integral Ground State (PIGS)

- Auxiliary Field/Determinental QMC

H = J�

�ij�

Si · Sj

x

Example:ground-state Renyi entropy QMC

x

x

S1

S2

` = 2L

�

` = 2L

Example: Area Law at T=0

S2 = a`

�+O

⇣�`

⌘+ · · ·

H =X

hiji

(Sx

i

Sx

j

+ Sy

i

Sy

j

)

S2 � a`

S2 = a`+NG log(`⇢s/c) + �

`

Kulchytskyy et al. Phys. Rev. B 92, 115146 (2015)

Metlitski and Grover arXiv:1112.5166

Example: log correction from a broken continuous symmetry

Finite-temperature QMC

SWAP SWAP

Z = Tr⇥e��H

⇤=

X

↵

h↵|e��H |↵i

need periodic boundary conditions in imaginary time

Finite-temperature QMC

Z = Tr⇥e��H

⇤=

X

↵

h↵|e��H |↵i

SWAP SWAP

�

need periodic boundary conditions in imaginary time

�

�

2�

Z[A, 2, T ]

S2 = � log

Z[A, 2, T ]

Z2

Multi-sheeted Riemann surface

�

�

2�

Z[A, 2, T ]

S2 = � log

Z[A, 2, T ]

Z2

Multi-sheeted Riemann surface

Z[A, 2, T ]

S2 = � log

Z[A, 2, T ]

Z2

Multi-sheeted Riemann surface

RGM, Kallin, Hastings Phys. Rev. B 82, 100409(R) (2010)

- Stochastic Series Expansion - Path Integral Monte Carlo

Del Maestro: Thursday 1:30pm

T

S2(T )

SA = SB extensive scaling

Sn(A) �= Sn(B)

Like in the classical case:in the presence of thermal mixing.

Example: Mutual information of quantum XXZ model

In(A :B) = Sn(A) + Sn(B)� Sn(A �B)

Tc

I2/L

� = 4

2Tc

A

� = L/2

Singh, Hastings, Kallin, RGM, Phys. Rev. Lett. 106, 135701 (2011)

• picks up all correlations (classical and quantum)

• exhibits critical scaling at finite-T phase transitions H =�

�ij�

��Sz

i Szj + Sx

i Sxj + Sy

i Syj

�

Example: Topological Entanglement Entropy of a Bose-Hubbard Spin Liquid

SuperfluidTopologicalSpin Liquid

QuantumCritical

Fan

T/t

V/t(V/t)c � 7

H = �t�

�ij�

(b†i bj + bib†j) + V

�(n )2

n =�

i�(ni � 1/2)

Isakov, Hastings, RGM Nature Physics 7, 772 (2011)Science 335, 193 (2012)

A B C D

2� = �SAn + SB

n + SCn � SD

n

SuperfluidTopologicalSpin Liquid

QuantumCritical

Fan

T/t

V/t(V/t)c � 7

H = �t�

�ij�

(b†i bj + bib†j) + V

�(n )2

n =�

i�(ni � 1/2)

Isakov, Hastings, RGM Nature Physics 7, 772 (2011)Science 335, 193 (2012)

Example: Topological Entanglement Entropy of a Bose-Hubbard Spin Liquid