Entanglement Entropy inExotic Phases of Quantum

Matter

Roger Melko

outlinecollaborators

Ann B Kallin

Ivan Gonzalez

Valence Bond and von Neumann Entanglement Entropy in Heisenberg Ladders Phys. Rev. Lett., 103, 117203 (2009)

Measuring Renyi Entanglement Entropy with Quantum Monte Carlo Phys. Rev. Lett. 104, 157201 (2010)

Finite-size scaling of mutual information in Monte Carlo simulations: Application to the spin-1/2 XXZ model Phys. Rev. B, 82, 180504 (2010)

Rajiv Singh

Sergei Isakov

Topological Entanglement Entropy of a Bose-Hubbard Spin Liquid Nature Physics 7, 772 (2011)

Finite temperature critical behavior of Mutual Information Phys. Rev. Lett. 106, 135701 (2011)

Anomalies in the Entanglement Properties of the Square Lattice Heisenberg Model arXiv:1107.2840 (2011)

Station-Q

Stephen Inglis

Matt Hastings

Wang-Landau calculation of Renyi entropies in finite-temperature quantum Monte Carlo simulations In preparation

DMRG

QMC

ED

outlineoutline

• Motivation: the role of QMC in condensed matter theory, and the interest in entanglement in quantum materials

• T=0 projector valence bond QMC

• Finite-T partition-function based QMC

• Topological entanglement in a Bose-Hubbard spin liquid

• Simulations perform importance sampling of d+1 dimensional configurations

• Finite and zero-temperature methods available

• The “sign problem” inhibits the simulation of frustrated spins or fermions

• Don’t have access to the groundstate wavefunction

Z =�

α

�α|e−βH |α� |ψ� ≈ (−H)m|ψtrial�

Quantum Monte Carlo

S1(ρA) = −Tr(ρA ln ρA)

N3 Nto

• Comparison to experiment

• Scaling crossovers

• critical points (quantum and classical)

outlinethe need for large systems sizes

H = J

�

�ij�

�b†i bj + h.c.

�+ K

�

�ijkl�

�b†i bjb

†kbl + h.c.

�

H = −t

�

�ij�

�b†i bj + bib

†j

�+

U

2

�

i

ni(ni − 1) + · · ·

(troyer)

H = J

�

�ij�

Si · Sj

(sandvik)

ξ ξ →∞L

CMT interest in entanglement

• Entanglement scaling gives a picture analogous to symmetry breaking in “non-Landau” phases

�m�γ

• It can also be used to distinguish conventional from “non-Landau” (e.g. deconfined) quantum critical points

cXY∗ = cXY + γswingle, senthil, arXiv:1109.3185:

Sn = a�− c log(�) + · · ·

• Subleading corrections to the area law are believed to harbour new universal physics.

• These can be used as a resource to diagnose new phases and phase transitions in condensed matter systems.

topological spin liquid Sn = a�− γ + · · ·

area law Sn = a� + · · · gapped wavefunctions, local Hamiltonians

Area law + corrections

fermi surface Sn = c� ln(�) + · · ·

criticalSn = a�− cn + · · ·{Sn = a�− dn ln(�) + · · ·

no “corners”

“corners”

H. Casini and M. Huerta, Nucl. Phys. B 764, 183 (2007)M. A. Metlitski, et.al, Phys.Rev. B 80, 115122 (2009)

A B

ρA = TrB(ρ)

ρ =�

i

λi|Ψi��Ψi|

Renyi Entanglement Entropies

S1(ρA) = −Tr(ρA ln ρA)

S2(ρA) = − ln�Tr(ρ2

A)�

Sn(ρA) =1

1− nln

�Tr

�ρn

A

��

• Goal: to calculate Renyi entropies in typical models that are studied in condensed matter theory (evaluate/benchmark area law and corrections)

• Use as a practical tool to study new phases and phase transitions in QMC simulations

DMRG

QMC

ED

outlineoutline

• Motivation: the role of QMC in condensed matter theory, and the interest in entanglement in quantum materials

• T=0 projector valence bond QMC

• Finite-T partition-function based QMC

• Topological entanglement in a Bose-Hubbard spin liquid

(i, j) =1√2(| ↑i↓j� − | ↓i↑j�)

Pauling, Anderson, Liang, Sandvik, BeachSandvik, Phys. Rev. Lett. 95, 207203 (2005)

|Vr� = |(ir,1, jr,1)(ir,2, jr,2) · · · (ir,N/2, jr,N/2)�

Valence Bond Basis

Pr = Hbr1Hbr

2Hbr

3· · ·

Projector QMC

(C −H)m|Ψ� → co(C − E0)m

�|0�+

c1

c2

�C − E1

C − E0

�m

|1�+ · · ·�

(C −H)m =

��

b

Hb

�m

=�

r

Pr

the sum is done through importance sampling

• a fixed (large) m is chosen• a sequence of m bond operators propagates an initial state• some number of operators is swapped each Monte Carlo step through a Metropolis algorithm

Estimators

�A� =�

rl WrWl�Vl|A|Vr��rl WrWl�Vl|Vr�

the “weights” depend on the Hamiltonian being studied

• estimators can be constructed for a variety of quantities (energies, order parameters, correlation functions)

• In order to measure entanglement, an expectation value based on this QMC sampling procedure has to be constructed

Ps|Ψ� = Ws|Vs�

Ss(qx, qy) =1N

�

k,l

ei(rk−rl).q�Sk · Sl�

Hastings, Gonzalez, Kallin, RGM Phys. Rev. Lett. 104, 157201 (2010)

|α�|β�

swap operator

|Ψ0�

swap operator

|α�|β�

|α�|β�“ancillary”

“real”

Two non-interacting copies

|Ψ0 ⊗Ψ0�

swap operator

�Ψ0 ⊗ Ψ0|SwapA|Ψ0 ⊗ Ψ0�

Two non-interacting copies

|Ψ0 ⊗Ψ0�

S2(ρA) = − ln(Tr(ρ2A)) = − ln(�SwapA�)

|α1�

“ancillary”

“real”

|α2�|β2�

|β1�

=�

α1,α2

�α1|ρA|α2��α2|ρA|α1� = Tr(ρ2A)

|ψ� =�

V

|V �

Hamiltonians of interest

H = J

�

�ij�

Si · Sj

• Heisenberg model (Néel groundstate)

• RVB spin liquids

• J-J’ models - conventional quantum critical points

• J-Q models - unconventional (deconfined) quantum critical points

Heisenberg model groundstate�

� = 12� = 2L

S2 = a� + c ln(�) + d

• Large finite sizes allow us to access important subtleties

x

S2 = a� + c ln(�) + b ln�L/π · sin(xπ/L)

�+ d

DMRG

QMC

ED

outlineoutline

• Motivation: the role of QMC in condensed matter theory, and the interest in entanglement in quantum materials

• T=0 projector valence bond QMC

• Finite-T partition-function based QMC

• Topological entanglement in a Bose-Hubbard spin liquid

Z =�

α

�α|e−βH |α�

Stochastic Series Expansion

=�

α

�

n

(−β)n

n!�α|Hn|α�

= Szi Sz

j

= (S+i S−j + S−i S+

j )

Hbi

=�

α

�

n

�

Sn

(−β)n

n!�α|

n�

i=1

Hbi |α�

Sandvik

• Spin and Boson models

• Scales as N and

• Various ways of sampling/updating the world-lines

β

Replica Trick

where is the partition function of the systems having special topology - the n-sheeted Riemann surface.

A AB B

2β

Z[A,n, T ]

Z[A, 2, T ] Z[A, 3, T ]

3ββ

Calabrese and Cardy, J. Stat. Mech. 0406, P002 (2004).Fradkin and Moore, Phys. Rev. Lett. 97, 050404 (2006)Nakagawa, Nakamura, Motoki, and Zaharov, arXiv:0911.2596Buividovich and Polikarpov, Nucl. Phys. B, 802, 458 (2008)M. A. Metlitski, et.al, Phys.Rev. B 80, 115122 (2009).

Sn(ρA) =1

1− nln

�Tr

�ρn

A

��=

11− n

lnZ[A,n, T ]

Z(T )n

2β

β

β

SSE simulation cell

Z[A, 2, T ]

S2 = − lnTr(ρ2A) = − ln

�Z[A, 2,β]

Z(β)2

�

= −SA(β = 0) +

� β

0�E�Adβ + 2S0(β = 0)− 2

� β

0�E�0dβ

= − lnZ[A, 2,β] + 2 ln Z(β)

Finite-T Renyi Entropy

T

S2(T )

SA = SB

∆ = 4B

A

L = 8

� = 4

A

T

�E� A

,B

B

A

H = ∆�

�ij�

(ni − 1/2)(nj − 1/2)−�

�ij�

�b†i bj + bib

†j

�

• Coincides with the entanglement entropy at zero temperature

• Provides a bound on some two-point correlation functions

In(A :B) = 2Sn(A) = 2Sn(B)T = 0

T

I 2(A

:B)1

1− nln

Z(nβ)Z(β)n

In(A :B) = Sn(A) + Sn(B)− Sn(A ∪B)

Wolf et al. PRL 100, 070502 (2008)

Mutual Information

Finite-Size Scaling at a phase transition

A

� = L/2

Tc

I2/L

∆ = 4

2Tc

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

H = ∆�

�ij�

(ni − 1/2)(nj − 1/2)−�

�ij�

�b†i bj + bib

†j

�

DMRG

QMC

ED

outlineoutline

• Motivation: the role of QMC in condensed matter theory, and the interest in entanglement in quantum materials

• T=0 projector valence bond QMC

• Finite-T partition-function based QMC

• Topological entanglement in a Bose-Hubbard spin liquid

• Fluid-like states of matter, where constituent spins are highly correlated, but also highly fluctuating

• Support exotic collective phenomena: emergent gauge fields, fractional particle excitations, topological order

• Spin liquids are scarce

Spin Liquids

Leon Balents Nature 464, 11 (2010)

“physical” models

• There exists a class of kagome-lattice Bose-Hubbard models know to support the simplest Z2 quantum spin liquid - and do not have a sign problem

• These are starting to be studied routinely with large-scale QMC simulations Dang, Inglis, RGM

arXiv:1106.0761, Phys. Rev. B (2011)Isakov, Kim, Paramekanti Phys. Rev. Lett. 97, 207204 (2006) Phys. Rev. B, 76, 224431 (2007)

Isakov, Hastings, RGM Nature Physics 7, 772 (2011)

Balents, Fisher, Girvin, Phys. Rev. B, 65, 224412

n =�

i∈(ni − 1/2)

fractional charges

“loop gas”

fractional charges

“loop gas”

Hamma, Ionicioiu, Zanardi - Phys. Lett. A 337, 22 (2005) - Phys. Rev. A 71, 022315 (2005)

Kitaev and Preskill - Phys. Rev. Lett. 96, 110404 (2006)Levin and Wen, - Phys. Rev. Lett. 96, 110405 (2006)

• Topological order of the loop gas is manifest as a universal correction in the entanglement entropy

γ = ln(2)for a Z2 spin liquid

Topological EE

Sn = a�− γ + · · ·

2γ = −SAn + SB

n + SCn − SD

n

A B C D

Flammia, Hamma, Huges, Wen, Phys. Rev. Lett. 103, 261601 (2009)

Bp =�

i∈p

σzi

H = −λB

�

p

Bp − λA

�

v

Av

Av =�

j∈v

σxj

kitaev’s toric code

Claudio Castelnovo and Claudio Chamon, PRB 76, 184442 2007

Finite-size systems retain a statistical contribution to the topological EE:

∝ Ne−βλA

Probability of having a defect in the loop gas

∝ Ne−βλB

2γ

Super!uidTopologicalSpin Liquid

QuantumCritical

Fan

T/t

V/t(V/t)c ≈ 7

Topological EE Isakov, Hastings, RGM Nature Physics 7, 772 (2011)

discussion

• Renyi entropies are measurable in many variants of QMC

• Detects Z2 topological order in a very non-trivial model

• Finite-size effects can be overcome with QMC

• Gives us insight into the nature of excitations

• Our work on gapless phases and critical points has just begun