Accessing localization properties of many - body systems with quantum Monte Carlo Fabien Alet Laboratoire de Physique Théorique Toulouse In collaboration with : David Luitz, Nicolas Laflorencie Ref. : arXiv:1308.1916 Statistical physics of quantum matter, Taipei, July 2013
Introduction . • Question : How much a wave - function is localized . in a given ( computational ) basis ? | Ψ i = . . . • V arious motivations : . {| i i } • 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... ) Stanley ( 1988 )
This talk • Will concentrate on wave - functions ground - states of quantum many - body lattice problems S q 1. Introduction : Measures of localization, what to look for j = 1 j = 2 j = 3 . . . . . j = 2 n L y 2 ! 1 2. Review : results in spin chains | i i β 3. Localization and quantum Monte Carlo τ = 0 4. Preliminary results on 1d and 2d quantum spin models
Measures of localization
Definitions X | Ψ i = a i | i i i | a i | 2 = 1 X • Assume normalized wave - function i • Moments = typical tools for measuring localization X | a i | 4 • Historically : Inverse Participation Ratio ( IPR ) IPR = i p i = | a i | 2 • More generally, define 1 X p q Renyi entropy S q = 1 − q ln q 6 = 1 i i X Shannon entropy S 1 = lim q → 1 S q = − p i ln( p i ) i
Simple expectations X | Ψ i = a i | i i • Denote by the size of configuration space H i ( 1 ∀ i ∈ 1 ... N √ N • Consider the simple wave - function a i = 0 otherwise • One simply obtains S q = ln( N ) • Scaling : S q ∝ ln( H ) : delocalized S q = O (1) : localized H = α N • Remark 1 : Many - body problem : , in general expect S q ∝ N • Remark 2 : Obviously, is basis - dependent ! S q 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 • Results in 1d provide evidence for scaling: Stéphan et al. S q = a q N + universal term q + c q /N + · · · • Periodic chains : Universal term is constant : b q l q ln( N ) + ˜ • Open chains : Universal term is : b q Essentially two models considered : X i +1 + S y i S y S x i S x i +1 + ∆ S z i S z XXZ ( XX ) H = J i +1 chain i T ransverse - X σ x i σ x i +1 − h σ z H = J field Ising i chain i
1st example : periodic XX chain ( +equiv. models ) • Shannon entropy Stéphan et al. -0.46 20 e ρ = 1 / 2 carr´ hexagonal ρ = 1 / 4 Di ff erent a 1 -0.47 hexagonal ρ = 1 / 3 hexagonal ρ = 1 / 2 15 -0.48 S 1 b 1 10 S -0.49 square dimers p=1/2 carr´ e ρ = 1 / 2 -0.5 5 hexagonal ρ = 1 / 4 honeycomb p=1/4 hexagonal ρ = 1 / 3 honeycomb p=1/3 -0.51 hexagonal ρ = 1 / 2 XX chain 0 Same b 1 0 0 . 01 0 . 02 0 . 03 0 . 04 0 5 10 15 20 25 30 35 40 45 N − 1 N (1 − q ∂ q )((1 − q ) b q ) • Renyi entropy Numerical observation ln q b q = 2(1 − q ) 1 2 [ln ( n ) − 1] q
The Renyi book • Replica point of view ( q half - integer ) : “Renyi book” Stéphan et al. | a i | 2 q = Z 2 q X p q X book Z q = i = Z q i i j = 1 · · · 2 q j = 1 j = 2 j = 3 The 2q half - infinite . . . . . j = 2 n Z 2 q cylinders / strips are forced book to have the same bottom → ∞ L y 2 ! 1 configuration Partition function of infinite | i i Z cylinder / strip = Normalization • Idea : Conformal field theory ( CFT ) usually good at computing ratio of partitions functions with di ff erent boundary conditions • Caution : Straight - forward replica fail !
Free field calculation ( replica - free ) b q = − 1 2(ln K + ln q • Final prediction q − 1) • Agrees with XX chain Stéphan et al. 0 b 1 -0.2 -0.4 • Numerics on XXZ chain -0.6 -0.8 Lanczos M=0 Lanczos M=1/5 -1 Lanczos M=1/4 • O ff ers an easy way to compute -1.2 Lanczos M=1/2 ln R − 1 / 2 -1.4 Luttinger parameter ! 0.6 0.8 1 1.2 1.4 1.6 K − 1 / 2 = (2 − 2 / π arccos( ∆ )) 1 / 2 • Further: phase transition as a function of q
T ransverse - field Ising chain X σ x i σ x i +1 − h σ z H = J i i • Single critical point in a di ff erent universality class ( c=1/2 ) h c = 1 • Two natural configurations bases : x ( spins ) and z ( field ) S ( x ) ( h ) = S ( z ) • Duality : q (1 /h ) + ln(2) q Parity h = 0 : S ( x ) S ( z ) = ln 2 = ( N − 1) ln 2 • Simple points : q q h = ∞ : S ( x ) S ( z ) = N ln 2 = 0 q q
T ransverse - field Ising chain X σ x i σ x i +1 − h σ z H = J i i S ( x ) ( h ) = S ( z ) q (1 /h ) + ln(2) • Duality : q h = 0 : S ( x ) S ( z ) = ln 2 = ( N − 1) ln 2 • Simple points : q q h = ∞ : S ( x ) S ( z ) = N ln 2 = 0 q q • Another “Renyi duality” based on X | i i z/x = N 1 / 2 ( | "" · · · "i x/z ) i Most likely state equal - superposition CFT : “Fixed” state CFT : “Free” state S ( x/z ) = N ln(2) − S ( z/x ) 1 / 2 ∞
Field - induced phase transition High field : 1 b ( x ) → 0 ∀ q ln(2) b ( x ) q 0.5 1 s 1 ( µ ) 1 Low field : 0.5 0 b ( x ) → ln(2) ∀ q 0 q L=8..12 L=16..20 -0.5 L=24..28 -0.5 L=30..34 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 L=34..38 ( µ -1)L Stéphan et al. 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 µ 1 /h b ( x ) Critical field : → b ∗ Non - trivial value ( at least q=1 ) q q
q - induced phase transition Stéphan et al. • Now sit at critical point and vary q q acts as a relevant 0.7 ln(2) 0.7 pertubation q ( h = 1) 0.6 0.6 0.5 q > 1 : b ∗ q → ln(2) 0.4 0.5 0.3 s n ( =1) “Fixed” state 0.4 0.2 0.1 b ( x ) 0.3 q < 1 : b ∗ 0 q → 0 -1 -0.5 0 0.5 1 (n-1)L 0.25 0.2 “Free” state CICT L=4..10 CICT L=8..14 0.1 Ising chain CICT L=22..28 CICT L=38..44 q = 1 : b ∗ 1 = 0 . 2543925(5) 2d Ising triangular Triangulaire L=6..12 0 2d Ising square Carre L=6..12 0 0.5 1 1.5 2 Exactly marginal n q • Same results for models in the same universality class • ( in ) dependance on basis x basis “stable” fixed point | " , φ i = cos( φ ) | "i x + sin( φ ) | #i x b ∗ 1 ( φ ) = b ∗ 0 ≤ φ < π / 4 1 | # , φ i = � sin( φ ) | "i x + cos( φ ) | #i x φ = π / 4 = b ∗ 1 − ln(2) z basis “unstable”
Summary of 1d results • Periodic chains: S q = a q N + b q + c q /N + · · · ln q q ≤ q c = d 2 / 2 κ r 2 Luttinger liquids b q = ln R − 2( q − 1) b q>q c = q ln( R ) − ln( d ) q − 1 0 q < 1 Ising universality class b ∗ ( x ) = 0 . 25439 · · · q = 1 q ln(2) q > 1 • Open chains ( not shown ) : S q = a q N + l q ln( N ) + ˜ b q + c q /N · · · q q − 1( R 2 / 4 − 1 / 4) l q<q c = − 1 / 4 l q>q c = LL : Ising : ... • How much survives beyond the 1d tractable ( exact ) cases ?
Localization and Monte Carlo arXiv:1308.1916
Measuring Sq with Monte Carlo • Importance sampling actually does the exact job ! | i i • Probability of seeing configuration in Monte Carlo Classical MC Quantum MC β | i i = | "##"#"i τ = 0 β →∞ p MC / h i | e − β H | i i a 2 i = p i p MC ∝ e − β E i = i i p i = h | i ih i | i • Measure Histogram and normalize to obtain all H ( | i i ) p i S q and therefore all
Further computational tricks • Replica trick for integer : Simulate q independent copies q ≥ 2 · · · | i q i = | "##"#"i | i 2 i = | "#"##"i | i 1 i = | "##"#"i X p q i = h δ | i 1 i , | i 2 i , ··· , | i q i i Estimator for i S ∞ • is easily measured as S ∞ = − ln( p max ) • If possible, do simulations in di ff erent basis and use S ( x/z ) = N ln(2) − S ( z/x ) 1 / 2 ∞
d=1 quantum systems arXiv:1308.1916
First example : spin ladders in a field X X S i, 1 · S i, 2 − hS z H = S i, α · S i +1 , α + J ⊥ J ⊥ i i, α =1 , 2 i 2 =7x10 -7 ) 0.46146*N - 0.5*[1+ln(1.1474))] + 0.6345/N ( χ 2 =2x10 -8 ) 0.43238*N - 0.5*[1+ln(0.81687)] + 0.252/N ( χ J rung = 4 10 J rung = 1 Intercept gives S 1 K ' 1 . 15 5 ( J ⊥ = 1) S z = 1 / 2 K ' 0 . 82 ( J ⊥ = 4) 0 0 5 10 15 20 25 N • Luttinger parameter extracted already on very small sizes ! • Agrees with independent estimates Hikihara, PRB 2001
Discrete symmetry breaking d>1 arXiv:1308.1916
2d transverse - field Ising model X X σ x i σ x σ z H = J j − h i h i,j i i • Quantum phase transition between a ferromagnet and a polarized phase at h c ' 3 . 044 J • Most simulations in the z basis ( a few corroborative results in the x basis too ) , up to N = 24 2 = 576 • QMC data well fitted by S q = a q N + b q + c q /N + · · ·
2d transverse - field Ising model • QMC data well fitted by S q = a q N + b q + c q /N + · · · 0.2 2 -10 2 N=4 2 -12 2 0 N=6 2 -14 2 N=8 high - field 2 -16 2 N=10 -0.2 2 -18 2 N=12 b ( z ) → 0 2 -20 2 N=14 b ∞ q 2 -22 2 -0.4 N=16 2 -24 2 N=18 -0.6 low - field -0.8 b ( z ) → − ln(2) q -1 1.5 2 2.5 3 3.5 4 h Marked anomaly close to h c
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