Renyi-Shannon Entropy andBoundary Field Theory

Masaki Oshikawa(ISSP, University of Tokyo) 1

Quantum Matter and Information?
Quantum Condensed Matter Physics
Realization ofquantum computation/
quantum information
processing

New understanding/formulation in foundation
Efficient numericalalgorithm(Quantum) Information
2

How to distinguish phases?
Different orders (or their absence) characterizeeach phase— what is “order”?
Ferromagnet: magnetic order
spontaneously breaks
Z2 symmetry (Ising model),SU(2) symmetry (Heisenberg model)…..

Superfluid: off-diagonal long-range order
spontaneously breaks U(1) symmetry“order”

LandauSpontaneous Symmetry Breaking
3Beyond Landau Paradigm
Many quantum phases, which are distinct but cannot
be characterized by a (conventional) SSB have been found“topological phases”- quantum Hall states- Haldane gap phase- topological insulators/topological superconductors……
How to characterize them? new tools will be useful!
→ “information theoretic” measurese.g. entanglement entropy4

In this talk, I will discuss an“information theoretic” measure of a quantum state
which is different from (but also related to)entanglement entropy.

Based on a collaboration with
Grégoire Misguich and Vincent Pasquier,also on several earlier works
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Rényi-Shannon Entropy
Define Rényi/von Neumann entropy from the prob. dist.
SvN =Xpi log pii= lim Sn
n!1

This DOES depend on the choice of the basis
Anyway, Rény-Shannon Entropy can be defined fora given quantum state (and choice of basis)
can be used as a new characterization??
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XXZ chain in 1+1D

The ground state is critical (gapless) for-1 ≦ Δ < 1
Effective theory: Tomonaga-Luttinger liquid(free boson field theory in 1+1D)

Bethe Ansatz
exact solution →pgR =7s12⇡✓11cos 1⇡◆

Rényi-Shannon Entropyof the S=1/2 XXZ chain in the Sz basis:zSj ⇠1
j@x + const.( 1) cos
2⇡RRSz basis

Φ-basisτ

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RSE of XXZ chain
Stephan-Furukawa-Misguich-Pasquier 2009, MO 2010

universal
non-universal“area (=system length) law”
Rényi-Shannon entropy does contain the universal
characteristics of the quantum state (TL liquid)
9Relation to EE
Entanglement Entropy of “Quantum Lifschitz” state in2 spatial dimensions

RSE of free boson field theory in 1+1 dimensions(=Tomonaga-Luttinger Liquid / S=1/2 XXZ chain)

10Quantum Lifshitz Field Theory
Ardonne-Fendley-Fradkin 2004
e.g. critical point ofquantum dimer model
on a square lattice

Groundstatewavefunction
EE of this state in replica formalism= same formula as RSE of XXZ chain (TL liquid)
Fradkin-Moore 2006, Hsu-Mulligan-Fradkin-Kim 200911

Phase Transition in RSE
Stephan-Misguich-Pasquier 2011

numericaltheory(discussed earlier)discrepancy
for n>nc ?

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Boundary PerturbationsThe effective field theory, in general, contains
all the possible perturbations which are not forbiddenby the symmetries
Bulk: all the perturbations are irrelevant in the gapless
regime, irrespective of the Renyi parameter n
Boundary: n replica fields are coupled, and we need toconsider possible perturbations w.r.t.
“center of mass” fieldnX
10 = pn j=113j

Boundary Perturbations
compactification of individual fields:
replica condition at the boundary:j ⇠1 =2 = ... =effective compactification of the c.o.m. field
1 X
0 = pn jj ⇠0 + 2⇡

pnR

Possible boundary perturbations:
m 0cos pnR

m 0
sin pnR

scaling dim.

m = 1, 2, 3, . . .

m2
x=ngR2

relevant if < 114j + 2⇡Rn

Boundary Phase Transition
m=1 : forbidden by the translation symmetryzSj ⇠1j@x + const.( 1) cos
2⇡RR

m=2: most relevant perturbation in generic caserelevant if

2n > nc =gR2

2gR
=1SU(2) AF Heisenberg (Δ=1):

XY (Δ=0):

1
gR =2215nc = 4nc = 2

RSE above nc
2 0 relevantcos pnR
Sn>nc =boundary condition is
Φ=const. i.e. “Néel state”
11nnlog [2(pmax ) ]pmax = |h+ + + . . . |ZD
1==pZ2gR

160 i|

2

Phase Transition in RSE
Stephan-Misguich-Pasquier 2011

theoryn>nc

theoryn<nc17

RSE of XXZ model in 2+1D

On a bipartite lattice (square etc.) there is nogeometric frustration; the ground state has along-range (Néel) order|Δ|>1: Z2 symmetry is spontaneously broken
Δ=1 : SU(2) symmetry is spontaneously broken|Δ|<1 : U(1) symmetry is spontaneously broken
These states have conventional order, but let us
first study RSE in these well understood states18

Numerical Approach
Luitz-Alet-Laflorencie 2013
Plot: RSE inzx
S - or S -basis
n!1

Efficient Quantum Monte Carlo evaluation of the RSE:
dominant “area law” contribution19

Subleading term in RSE?

subleading log(N) contribution to RSE
only exist when a continuous (SU(2) or U(1))
symmetry is broken spontaneously;
universal?
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Effective Field Theory
Nambu-Goldstone modeorder parameter
For the antiferromagnet, NG modeis described by free boson in 2+1D

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Simplest case: S∞

Q: Which state (in the Sz-basis) has the maximum
amplitude (probability)?
A: Néel state!

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Boundary FormulationτBoundary condition at τ=0:
no fluctuation of NG mode

Dirichlet boundary condition

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What is pmax?
cf.) single harmonic oscillator

1 2 m! 2 2
H=p +x2m2

NG modes: ∞ collection of harmonic oscillatorslabelled by the wave number k!k = c|k|24

Determinant of Laplacian

X2log k = log det

Determinant ofLaplacian
M. Kac 1966 etc.

0k6=0

χ: Euler characteristics of the spatial manifold
1osc
2
log (pmax ) = log L
425

for torus (χ=0)

“GS degeneracy” factor
Finite-size ground state: generally symmetric(even in the SSB phase!)
X nearly-degenerate|symmetry-broken ground statei ⇠|finite-size ground states

SSBi

existence of nearly degenerate ground statesin finite size (“Anderson tower of states”)

Finite-size GS(almost) linearly independent
symmetry-brokenGSs26

How many ground states?
U(1) SSB phase:pNQ ⇠ O( p ) ⇠ O( N )N
N = L227

How many ground states?
SU(2) SSB phase
2NQ ⇠ O( p 2 ) ⇠ O(N )N
N = L228

Universal term in S∞log (pmax ) ⇠osclog (pmax )1log ( )QNN GNN G⇠
log N +log N
4
2NN G⇠+log N4

NNG :number of Nambu-Goldstone mode(= number of broken symmetry generators)
1 for XY / XXZ ( |Δ| < 1 )2 for Heisenberg AF (XXX) (Δ=1)※ here we consider “relativistic” case (type-A NG modes) only29

Comparison with Numerics

Our theory:
Δ=1: NNG = 2
Δ=0: NNG = 1
Δ>1: NNG = 0

l∞ = 2/4 = 0.5
l∞ = 1/4 = 0.25
l∞ = 0 30

RSE for general n
Boundary phase transition?
recall in 1+1D, boundary phase transition at n=nc
what about 2+1D?
Nambu-Goldstone mode in a SSB phase:“small fluctuation” around the ordered state
Leading boundary perturbation:

02“boundary mass”: always relevant!!
“fixed phase” at least for n>1
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Sn>1 ⇠⇠11nnn

1

log (pmax )nnNN G⇠log N4(n 1)

S132Conclusions- Basis-dependent Rényi-Shannon Entropy (RSE) exhibits
universal behaviors (shown analytically & numerically)- could be useful in characterizing ground states ofquantum many-body systems- some similarity (and relation) to entanglement entropy,-in some respects “simpler” than EE (good for practicalapplications??)
So far elucidated only for “conventional” phases(Tomonaga-Luttinger liquid in 1+1D, SSB phase in 2+1D)— can we apply to more exotic phases?
Numerical approach: Exact Diagonalization /Quantum Monte Carlo… tensor networks??
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