Accessing localization properties of many - body systems with - - PowerPoint PPT Presentation

Accessing localization properties of many-body systems with quantum Monte Carlo

Fabien Alet

In collaboration with : David Luitz, Nicolas Laflorencie

Laboratoire de Physique Théorique Toulouse

Statistical physics of quantum matter, Taipei, July 2013

• Ref. :

arXiv:1308.1916

• Question : How much a wave-function is localized

in a given (computational) basis ?

Stanley (1988)

|Ψi =      . . . . . .     

{|ii}

• V

arious motivations :

• Localization physics : Anderson (single-particle, disorder), many-

body localization

• Complexity theory : how many states needed to describe

correctly phenomena ? (see variational methods, computational complexity)

• Relation to multifractal analysis in various fields (physics,

chemistry, finance...)

Introduction

This talk

• 1. Introduction : Measures of localization, what

to look for

• 4. Preliminary results on 1d and 2d quantum spin models
• Will concentrate on wave-functions ground-states of

quantum many-body lattice problems

• 2. Review : results in spin chains
• 3. Localization and quantum Monte Carlo

Sq

|ii

. . . . .

β

τ = 0

Measures of localization

Definitions

• Moments = typical tools for measuring localization
• Historically : Inverse Participation Ratio (IPR)

|Ψi = X

i

ai|ii

• Assume normalized wave-function

X

i

|ai|2 = 1 IPR = X

i

|ai|4

• More generally, define

Renyi entropy pi = |ai|2 Shannon entropy

S1 = lim

q→1 Sq = −

X

i

pi ln(pi)

q 6= 1 Sq = 1 1 − q ln X

i

pq

i

Simple expectations

|Ψi = X

i

ai|ii

• Denote by the size of configuration space

H

• Consider the simple wave-function ai =

(

1 √ N

∀i ∈ 1...N

• therwise
• One simply obtains Sq = ln(N)
• Scaling : Sq ∝ ln(H) : delocalized

Sq = O(1) : localized

• Remark 1 : Many-body problem : , in general expect

H = αN

Sq ∝ N

• Remark 2 : Obviously, is basis-dependent !

Sq Is there something else beyond these remarks ?

Review in quantum spin chains

Disclaimer : all results obtained mostly by Stéphan, Misguich, Pasquier

Stéphan et al., PRB 80, 184421 (2009) Stéphan et al., PRB 82, 125455 (2010) Stéphan et al., PRB 84, 195128 (2011)

Shannon-Renyi Entropy of spin chains

• Periodic chains : Universal term is constant :
• Results in 1d provide evidence for scaling:
• Open chains : Universal term is :

bq

Sq = aqN + universal termq + cq/N + · · ·

Stéphan et al.

lq ln(N) + ˜ bq

Essentially two models considered : H = J X

i

Sx

i Sx i+1 + Sy i Sy i+1 + ∆Sz i Sz i+1

XXZ (XX) chain T ransverse- field Ising chain

H = J X

i

σx

i σx i+1 − hσz i

1st example : periodic XX chain (+equiv. models)

5 10 15 20 5 10 15 20 25 30 35 40 45 carr´ e ρ = 1/2 hexagonal ρ = 1/4 hexagonal ρ = 1/3 hexagonal ρ = 1/2 S

XX chain honeycomb p=1/3 honeycomb p=1/4 square dimers p=1/2

S1

N

Different a1 Sameb1

• 0.51
• 0.5
• 0.49
• 0.48
• 0.47
• 0.46

0.01 0.02 0.03 0.04 carr´ e ρ = 1/2 hexagonal ρ = 1/4 hexagonal ρ = 1/3 hexagonal ρ = 1/2

N −1

b1

• Shannon entropy
• Renyi entropy

1 2 [ln (n) − 1]

Numerical observation

bq = ln q 2(1 − q)

(1 − q∂q)((1 − q)bq)

q

Stéphan et al.

The Renyi book

• Idea : Conformal field theory (CFT) usually good at computing

ratio of partitions functions with different boundary conditions

• Replica point of view (q half-integer) : “Renyi book”

Zq = X

i

pq

i =

X

i

|ai|2q = Z2q

book

Zq

|ii

j = 1j = 2j = 3 j = 2n

. . . . .

Ly 2 ! 1

The 2q half-infinite cylinders / strips are forced to have the same bottom configuration

Z2q

book

Z

Partition function of infinite cylinder / strip = Normalization

j = 1 · · · 2q

→ ∞

• Caution : Straight-forward replica fail !

Stéphan et al.

Free field calculation (replica-free)

• Final prediction

b1

Stéphan et al.

• Agrees with XX chain
• Offers an easy way to compute

Luttinger parameter !

• Numerics on XXZ chain
• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.6 0.8 1 1.2 1.4 1.6 Lanczos M=0 Lanczos M=1/5 Lanczos M=1/4 Lanczos M=1/2 ln R − 1/2

• Further: phase transition as a function of q

bq = −1 2(ln K + ln q q − 1)

K−1/2 = (2 − 2/π arccos(∆))1/2

T ransverse-field Ising chain

• Single critical point in a different universality class (c=1/2)

hc = 1 H = J X

i

σx

i σx i+1 − hσz i

• Two natural configurations bases : x (spins) and z (field)
• Duality :

S(x)

q

(h) = S(z)

q (1/h) + ln(2)

• Simple points :

h = 0 : S(x)

q

= ln 2 S(z)

q

= (N − 1) ln 2

Parity

h = ∞ : S(x)

q

= N ln 2 S(z)

q

= 0

T ransverse-field Ising chain

• Duality :

H = J X

i

σx

i σx i+1 − hσz i

S(x)

q

(h) = S(z)

q (1/h) + ln(2)

• Simple points :

h = 0 : S(x)

q

= ln 2 S(z)

q

= (N − 1) ln 2

h = ∞ : S(x)

q

= N ln 2 S(z)

q

= 0

• Another “Renyi duality” based on

CFT : “Free” state CFT : “Fixed” state Most likely state

S(x/z)

1/2

= N ln(2) − S(z/x)

∞

X

i

|iiz/x = N 1/2(| "" · · · "ix/z)

equal-superposition

Field-induced phase transition

• 0.5

0.5 1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

s1(µ) µ

ln(2) L=8..12 L=16..20 L=24..28 L=30..34 L=34..38

• 0.5

0.5 1

• 2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(µ-1)L

High field : b(x)

1

1/h b(x)

q

→ 0 ∀q b(x)

q

→ ln(2) ∀q Low field : Critical field : Non-trivial value (at least q=1) b(x)

q

→ b∗

q

Stéphan et al.

q-induced phase transition

• Now sit at critical point and vary q

q acts as a relevant pertubation q b(x)

q (h = 1)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 1 1.5 2

sn( =1) n

ln(2) CICT L=4..10 CICT L=8..14 CICT L=22..28 CICT L=38..44 Triangulaire L=6..12 Carre L=6..12

0.1 0.2 0.3 0.4 0.5 0.6 0.7

• 1
• 0.5

0.5 1

(n-1)L0.25

Ising chain 2d Ising triangular 2d Ising square

“Fixed” state “Free” state q > 1 : b∗

q → ln(2)

q < 1 : b∗

q → 0

Exactly marginal

Stéphan et al.

• Same results for models in the same universality class

q = 1 : b∗

1 = 0.2543925(5)

• (in)dependance on basis

| ", φi = cos(φ)| "ix + sin(φ)| #ix

| #, φi = sin(φ)| "ix + cos(φ)| #ix

b∗

1(φ) = b∗ 1

0 ≤ φ < π/4 = b∗

1 − ln(2)

φ = π/4

x basis “stable” fixed point z basis “unstable”

Summary of 1d results

• How much survives beyond the 1d tractable (exact) cases ?
• Periodic chains:

Luttinger liquids

q ≤ qc = d2/2κr2

Ising universality class

bq>qc = q ln(R) − ln(d) q − 1

b∗(x)

q

=      q < 1 0.25439 · · · q = 1 ln(2) q > 1

Sq = aqN + bq + cq/N + · · ·

bq = ln R − ln q 2(q − 1)

• Open chains (not shown) : Sq = aqN + lq ln(N) + ˜

bq + cq/N · · ·

LL :

lq<qc = −1/4

lq>qc = q q − 1(R2/4 − 1/4)

Ising : ...

Localization and Monte Carlo

arXiv:1308.1916

Measuring Sq with Monte Carlo

• Importance sampling actually does the exact job !
• Probability of seeing configuration in Monte Carlo

Classical MC

pMC

i

∝ e−βEi |ii

• Measure Histogram and normalize to obtain all

and therefore all H(|ii) pi Sq

Quantum MC

pMC

i

/ hi|e−βH|ii

β→∞

= a2

i = pi

β

τ = 0

|ii = |"##"#"i pi = h|iihi|i

Further computational tricks

• Replica trick for integer : Simulate q independent copies

q ≥ 2 Estimator for X

i

pq

i = hδ|i1i,|i2i,··· ,|iqii

|i1i = |"##"#"i · · · S∞

• If possible, do simulations in different basis and use

S(x/z)

1/2

= N ln(2) − S(z/x)

∞

• is easily measured as S∞ = − ln(pmax)

|i2i = |"#"##"i |iqi = |"##"#"i

d=1 quantum systems

arXiv:1308.1916

First example : spin ladders in a field

• Luttinger parameter extracted already on very small sizes !

5 10 15 20 25

N

5 10

S1

0.46146*N - 0.5*[1+ln(1.1474))] + 0.6345/N (χ

2=7x10

• 7)

0.43238*N - 0.5*[1+ln(0.81687)] + 0.252/N (χ

2=2x10

• 8)

Jrung = 4 Jrung = 1

• Agrees with independent estimates

H = X

i,α=1,2

Si,α · Si+1,α + J⊥ X

i

Si,1 · Si,2 − hSz

i

J⊥

Sz = 1/2

(J⊥ = 1)

K ' 1.15 K ' 0.82

(J⊥ = 4)

Intercept gives

Hikihara, PRB 2001

Discrete symmetry breaking d>1

arXiv:1308.1916

2d transverse-field Ising model

H = J X

hi,ji

σx

i σx j − h

X

i

σz

i

• Quantum phase transition between a ferromagnet and a

polarized phase at hc ' 3.044J

• QMC data well fitted by
• Most simulations in the z basis (a few corroborative results

in the x basis too), up to N = 242 = 576 Sq = aqN + bq + cq/N + · · ·

2d transverse-field Ising model

• QMC data well fitted by Sq = aqN + bq + cq/N + · · ·

1.5 2 2.5 3 3.5 4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 N=4

2-10 2

N=6

2-12 2

N=8

2-14 2

N=10

2-16 2

N=12

2-18 2

N=14

2-20 2

N=16

2-22 2

N=18

2-24 2

b∞ h high-field

b(z)

q

→ 0

low-field

b(z)

q

→ − ln(2)

Marked anomaly close to hc

2d transverse-field Ising model

• QMC data well fitted by Sq = aqN + bq + cq/N + · · ·

b∞ h Convergence to a non-trivial value

2.95 3 3.05 3.1 3.15 3.2 3.25

• 0.1

0.1 0.2 N=4

2-10 2

N=6

2-12 2

N=8

2-14 2

N=10

2-16 2

N=12

2-18 2

N=14

2-20 2

N=16

2-22 2

N=18

2-24 2

hc

b∗

∞ ' 0.18

2d transverse-field Ising model

• QMC data well fitted by Sq = aqN + bq + cq/N + · · ·
• Similar behavior for b2, b3, b4 · · ·

b2

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 N=4

2-8 2

N=6

2-10 2

N=8

2-12 2

N=10

2-14 2

N=12

2-16 2

N=14

2-18 2

h

2d transverse-field Ising model

• QMC data well fitted by Sq = aqN + bq + cq/N + · · ·
• Similar behavior for b2, b3, b4 · · ·

b2 h

3 3.1 3.2

• 0.2
• 0.1

0.1 0.2 0.3 N=4

2-8 2

N=6

2-10 2

N=8

2-12 2

N=10

2-14 2

N=12

2-16 2

N=14

2-18 2

b∗

2 ' 0.36

b∗

3 ' 0.26

b∗

4 ' 0.22

(not shown)

2d transverse-field Ising model

• QMC data well fitted by Sq = aqN + bq + cq/N + · · ·
• Summary for b(z)

q b∗

2 ' 0.36

b∗

3 ' 0.26

b∗

4 ' 0.22

b∗

1 ∼ 1.2?

bq(h > hc) = 0

b(z)

q

h

• Non-trivial (universal ?) values at , different from 1d

hc

• Speculation : boundary-induced phase transition at ?

0.5 ≤ qc ≤ 1

b∗

∞ ' 0.18

bq(h < hc) = − ln(2)

b∗

1/2 ∼ − ln(2)?

Universality

• Same model on triangular lattice
• Same constants found at and out of criticality

hc

b∗

2 ' 0.36

b∗

b∗

4 ' 0.22

b∗

b∗

bq(h > hc) = 0

bq(h < hc) = ln(2)

b(z)

q

h

4.6 4.7 4.8 4.9 5 5.1 5.2 0.05 0.1 0.15 0.2 N=4

2-10 2

N=6

2-12 2

N=8

2-14 2

N=10

2-16 2

N=12

2-18 2

h b∞

hc

b∗

1/2 ∼ − ln(2)?

2d transverse-field Ising model

• QMC data well fitted by
• Signal of phase transition also detected in leading term
• Clear change of slope at

2.5 3 3.5 4 0.02 0.04 0.06 0.08 0.1 0.12 N=4

2 - 8 2

N=6

2-10 2

N=8

2-12 2

N=10

2-14 2

N=12

2-16 2

N=14

2-18 2

N=16

2-20 2

N=18

2-22 2

N=20

2-24 2

hc

a∞ h Sq = aqN + bq + cq/N + · · ·

2d transverse-field Ising model

• QMC data well fitted by
• Signal of phase transition also detected in leading term
• Detection of in (similar signal for other q)

hc h

2.9 2.95 3 3.05 3.1 3.15

• 0.14
• 0.12
• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

N=4

2-8 2

N=6

2-10 2

N=8

2-12 2

N=10

2-14 2

N=12

2-16 2

N=14

2-18 2

N=16

2-20 2

N=18

2-22 2

N=20

2-24 2

hc

da∞/dh da∞/dh Sq = aqN + bq + cq/N + · · ·

Continuous symmetry breaking d>1

arXiv:1308.1916

2d XXZ antiferromagnetic model

• Long range order in ground-state

QMC data well fitted by Sq = aqN + lq log(N) + bq + · · ·

H = J X

hi,ji

Sx

i Sx j + Sy i Sy j + ∆Sz i Sz j

• continuous symmetry breaking

0 ≤ ∆ ≤ 1

• discrete symmetry breaking : Log vanish

∆ > 1

5 10 15 20 100 200 300 400 Entropy S∞ System size N a∞=0.0699(3) a

∞

= . 2 5 2 3 6 ( 7 ) a

∞

= . 2 5 8 5 ( 2 ) a

∞

= . 1 4 2 4 ( 7 ) A Sz

∞, ∆ = 1.0

Sz

∞ ∆ = 1.5

Sz

∞, ∆ = 2.0

Sx

∞, XY

20 50 400 100 ln(2) 1 1.5 2 2.5 3 Subleading entropy S∞ − a∞N System size N l∞=0.51(6) l∞ = . 2 6 9 ( 1 ) l∞=0.001(5) B

Conclusions & outlooks

• Message 2 : QMC is well suited when wave-function is

“reasonably” localized.

• Message 1 : Universality sits in subleading terms of Shannon -

Renyi localization entropies

• Extension of localization at finite temperature is possible

(many-body localization ?)

• Multifractal analysis of
• Check universality for different phases of matter (gapless :

can we count Goldstone modes ? gapped : topological phases ?)

• Outlooks :

aq

• Beyond entropies : Statistical properties of “spectrum”

pi