Renormalization of random quantum magnets Ferenc Igl oi (Budapest) - - PowerPoint PPT Presentation

Renormalization of random quantum magnets Ferenc Igl´

• i (Budapest)

in collaboration with Istv´ an Kov´ acs, R´

• bert Juh´

asz Cargese, August 25- September 6, 2014

1

Introduction

• Quantum phase transitions

– takes place at T = 0 – due to quantum fluctuations – by varying a quantum control pa- rameter

• Examples

– rare-earth magnetic insulators – heavy-fermion compounds – high-temperature superconductors – two-dimensional electron gases – quantum magnets: LiHoF4

• Disorder often play an important rˇ
• le

– Anderson localization – many-body localization – quantum spin glasses: LiHoxY1−xF4

• Paradigmatic model: random transverse

Ising model (RTIM) – in 1d exact results, also through strong disorder RG method ∗ infinite disorder scaling at the critical point ∗ dynamical (Griffiths-McCoy) sin- gularities

• utside

the critical point – in 2d numerical implementation of the SDRG method ∗ infinite disorder scaling ∗ but contradictionary results through the quantum cavity ap- proach 2

Aim of the present talk

• Improve the numerical algorithm of the

SDRG method

• Study the critical behavior of the RTIM

for d > 2

• Study Erd˝
• s-R´

enyi random graphs (d → ∞)

• Study the boundary critical behavior at

surfaces, corners, edges

• Study the entanglement entropy and its

singularity at the critical point

• Study systems with long-range interac-

tions 3

Random transverse Ising model H = −∑i, jJijσz

i σz j −∑ihiσx i

.

• Ji j couplings
• independent random numbers from the

distribution p(J)

• hi transverse fields
• independent random numbers from the

distribution q(h) .

• box-h disorder

– p(J) = Θ(J)Θ(1−J) (Θ(x) : Heaviside step-function) – q(h) = 1 hb Θ(h)Θ(hb −h)

• fix-h disorder

– p(J) = Θ(J)Θ(1−J) – q(h) = δ(h−h f)

Quantum control parameter: θ = log(hb) or θ = log(h f).

4

Strong disorder RG approach

(Ma, Dasgupta, Hu 1979, D.S. Fisher 1992, F.I. & Monthus 2005)

• sort the couplings and transverse fields, Ω = max(Ji,hi)
• eliminate the largest parameter - reduce the number of spins by one
• generate new effective parameters between the remaining spins

– Ω = Ji j i and j form a ferromagnetic cluster - aggregation in an effective field: hi j = hih j

Jij

having a moment: µi j = µi + µj – Ω = hi site i is decimated out - annihilation new effective couplings between sites j and k: Jjk = JijJik

hi

• repeate the transformation
• at the fixed point Ω is reduced to Ω∗ = 0.
• final result: set of connected clusters with different masses, µ,

decimated at different energies, Ω. 5

Exact results in 1D Infinite disorder fixed point (IDFP)

• Critical point: log(J) = log(h)

– distribution of the effective parame- ters is logarithmically broad – asymptotically exact decimation steps – strongly anisotropic scaling lnΩL ∼ Lψ, ψ = 1/2 – large effective spin clusters size: ξ ∼ |θ −θc|−ν ν = 2 moment: µ ∼ [ln(Ω0/Ω)]φ ∼ Ldf d f = φψ = d −x, φ =

√ 5+1 2

, x = 3−

√ 5 4

• In the Griffiths phase: θ −θc > 0

– non-singular static behaviour: ξ < ∞ – singular dynamical behaviour: Ω ∼ L−z dynamical exponent: z = z(θ) χ(T) ∼ T −1+d/z, Cv(T) ∼ T d/z 6

Numerical implementation of the RG procedure for D > 1

• Phys. Rev. B83, 174207 (2011), J. Phys.: Cond. Mat. 23, 404204 (2011)
• differences with the 1D procedure

– change in the topology – application

• f

the maximum rule (valid at IDFP)

J′ = max(Jai, Jaj)

d d J′ Jad a a i Jij hi i b c a b c i h′ J′ j a

J′ = JaiJbi/hi Ω = Jij Ω = hi ˜ J = max(Jad, JaiJdi/hi) h′ = hihj/Jij

˜ J

• problems with the na

¨ ıve implementation – h-decimations induce several new bonds – the lattice transforms to a fully con- nected cluster – slow algorithm: for N sites works in O(N3) time

• improved algorithm

– concept of local maxima - which can be decimated independently – concept of optimal RG trajectory - along which the time is minimal – filtering out irrelevant bonds - get- ting rid of latent couplings – improved algorithm works in O(N logN) time 7

Bulk critical behavior

• Finite-size critical points - θc(S,L)

– two-copies of the same sample (S and S′) are coupled together

S S’

– continuously increase θ and monitor the clusters, which are built of iden- tical sites in the copies – at θc(S,L) the last correlated cluster disappears, thus for θ > θc(S,L) we are in the paramagnetic phase

• Distribution of pseudocritical points

10-4 10-3 10-2 10-1 100

• 0.1
• 0.09
• 0.08
• 0.07
• 0.06
• 0.05
• 0.04

p θc

L=16 24 32 48 64 96 10-4 10-3 10-2 10-1

• 0.4
• 0.2

0.2 0.4 p ~ y

10-4 10-3 10-2 10-1 100

• 0.07
• 0.06
• 0.05
• 0.04
• 0.03
• 0.02

p θc 4D

L=8 12 16 24 32 10-4 10-3 10-2 10-1

• 0.2

0.2 p ~ y

• Finite-size scaling

– shift of the mean:

• θc −θc(L)
• ∼ L−1/νs

– width of the distribution: ∆θc(L) ∼ L−1/νw – numerical estimates: νs = νw like in a conventional random fixed point 8

Scaling at the critical point

• Cluster structure

correlation (left) and energy (right) clusters

• Correlation clusters → magnetization

– mass: µ = N# of connected sites – typical mass: µ ∼ Ldf – distribution function: P

L(µ) = Ld f ˜

P(µL−df) – power-law tail for large µL−d f = u ˜ P(u) ∼ u−τ, with τ = 1+ d df

10-8 10-6 10-4 10-2 100 1 10 100 1000 P µ 2D 3D 4D

0.1 0.2 0.3 0.4 0.5 102 103 104 105 106 df/d N

9

Energy clusters → dynamical scaling

• energy scale:

εL smallest gap associ- ated with the energy cluster

• We use: γL = logεL
• Critical point

– typical value: γL ∼ Lψ – scaling combination ˜ γ = (γL −γ0)L−ψ – Infinite disorder scaling

10-4 10-3 10-2 10-1 0 10 20 30 40 50 60 70 p γ

3D

10-4 10-3 10-2 10-1 1 2 3 4 5 6 7 8 9 p ~ γ ~ L=24 32 48 64 96 128 10 20 30 40 50 γ

4D

2 3 4 5 6 7 8 9 10 γ ~ L=8 12 16 24 32 48

• (disordered) Griffiths phase

– typical value: γL ∼ zlog(L) [εL ∼ L−z] – distr.: log p(γ) ≈ −(d/z)γ, (γ ≫ 1) – scaling comb.: ˜ γ = (γL −zln(L)−γ0

10-4 10-3 10-2 10-1 5 10 15 20 25 30 p γ 3D L=24 L=48 L=96

10-4 10-3 10-2 10-1

• 10
• 5

5 10 p ~ γ-z ln(L)

10

Erd˝

• s-R´

enyi (ER) random graphs D → ∞

• Construction

– N sites – kN/2 edges in random positions – k > 1 random graph is percolating

• Distribution of the pseudocritical points

10-4 10-3 10-2 10-1 1 2 3 4 5 6 p θc ER

N=210 N=212 N=214 N=216 N=218

10-4 10-3 10-2 10-1

• 5

5 10 15 p ~ y

• log-energy scaling

10-4 10-3 10-2 10-1 5 10 15 20 25 30 35 40 p γ ER

N=212 N=214 N=216 N=218 N=220 N=222

0.3 0.6 0.9 1.2 1.5 101 102 103 104 105 106 w N

inset: fixed-h +, box-h ⊡

• logarithmically infinite disorder scaling
• width of the distribution:

W ≈ W0 +W1logε N ε = 1.3(2) 11

Bulk critical parameters

1D 2D 3D 4D ER Lmax 2048 128 48 Nmax 4.2×106 2.1×106 5.3×106 4.2×106 θ (b)

c

1.6784(1) 2.5305(10) 3.110(5) 2.775(2) θ ( f)

c

−1. −0.17034(2) −0.07627(2) −0.04698(10) −0.093(1) dνw 2. 2.48(6) 2.90(15) 3.3(1) 7.(2) dνs 2.50(6) 2.96(5) 2.96(10) 5.(1) x/d

3− √ 5 4

0.491(8) 0.613(3) 0.68(3) 0.81(2) ψ/d 1/2 0.24(1) 0.15(2) 0.11(2)

• 0. (log)

.

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1/d x/d ψ 1/(dνs) 1/(dνw)

Conclusions at this point

• Infinite disorder fixed point at any di-

mensions

• Strong disorder RG approach is asymp-

totically exact

• Spin glass and random ferromagnet are

in the same universality class 12

Boundary critical behavior

• Phys. Rev. B 87, 024204 (2013)
• Different finite geometries

. slab wedge pyramid

N L L L L L N N L

• Different local exponents

. surface edge corner

• magnetization profiles

– from the fixed boundary: (Fisher-de Gennes) ml ∼ l−xb, x = xb, l ≪ L – from the free part ml′ ∼ (l′)xαb, l′ = Lα −l +1 ≪ Lα xαb = xα −x, α : s, c, e – interpolation formula ml = A Lx [sin(πλ)]x[cos(πλ/2)]xα λ = l/Lα

• results in 3D

10-5 10-3 10-1 0.2 0.4 0.6 0.8 1 m λ fixed box

0.8 0.9 1 1.1 1.2 0.2 0.4 0.6 0.8 1 r λ

slab geometry, inset: ratio with the interpolation formula 13

• results in 3D

10-5 10-3 10-1 3 10 30 m l’+l’0

fixed box xcb=2.65

3 3.5 4 4.5 0.04 0.08 0.12 xc 1/L

pyramid geometry, inset: corner exp.

• results in 4D

10-5 10-3 10-1 1 2 5 10 20 m l’+l’0

fixed box xsb=0.85

3.6 3.8 4 0.04 0.08 0.12 xs 1/L

slab geometry, inset:surface exp.

Boundary critical exponents - summary

bulk surface corner edge x xb xs xsb xc xcb xe xeb 1D (3− √ 5)/4 0.5 2D 0.982(15) 0.98(1) 1.60(2) 0.65(2) 2.3(1) 1.35(10) 3D 1.840(15) 1.855(20) 2.65(15) 0.84(7) 4.2(2) 2.65(25) 3.50(15) 1.75(15) 4D 2.72(12) 2.72(10) 3.7(1) 0.85(15) 14

Entanglement entropy

EPL 97, 67009 (2012)

• entanglement entropy between a subsys-

tem: A and the enviroment: B: SA = −Tr(ρAlog2ρA)

• ρA = TrB|0 >< 0|:

reduced density matrix with |0 > the ground state of the com- plete system: which is a set of indepen- dent clusters

• each connected cluster with c number of

spins is in a GHZ-state: 1 √ 2 (|↑↑ ··· ↑

c times

+|↓↓ ··· ↓

c times

)

• for the GHZ-state

– ρA =

• 1/2

1/2

• – SGHZ = 1
• each cluster contained both in A and B

gives 1 contribution to SA

• SA ∼ Ld−1 : area law
• corrections at the critical point?

– are they singular? – form: ∼ Ld−1lnlnL or ∼ Ld−1 +blnL ? – origin: corner and/or bulk? – are they universal? – related to a diverging ξ? 15

Numerical calculation by the SDRG method

• for a sample obtain the clusters through

renormalization L = 64 box-h (left) fixed-h (right)

• Calculation in different geometries

A B B B A

. square (cube) strip (slab)

S3 S3 1 S3 2

cube column slab 16

Slab geometry

• the same area (term) for different ℓ:

S (d)

slab(L,ℓ) = ad−1 fd−1 +corr(ℓ), fd−1 = Ld−1

• the finite difference is related to the cor-

rection δS (d)

slab(L,ℓ) = S (d) slab(L,ℓ+1)−S (d) slab(L,ℓ)

10-5 10-3 10-1 101 103 1 10 100 1000 δS(d)

slab

l d=2 d=3 d=4 D=0 D=1 D=2

• numerical data at the critical point:

δS (d)

slab(L,ℓ) ∼ ℓ−d

• no singular corrections!
• phenomenological explanation

– domains which contribute to the en- tropy are of size ξ ≤ ℓ – finite-size corrections are for ξ ≈ ℓ – number of these blobs are nbl ∼ (L/ℓ)d−1 – each blob has the same O(1) correc- tion – total correction: S (d)

slab(L,ℓ)−ad−1 fd−1 ∼ nbl ∼ ℓ−d+1

• singular contributions are due to corners!

17

Cube (square) geometry in 2d

• corner correction to the area-law:

S (2)

cube(ℓ) = a1 f1 +S (2) cr (ℓ)

• calculate the difference:

δS (2)(ℓ) ≡ S (2)(ℓ)−2S (2)(ℓ/2) ≈ S (2)

cr (ℓ)−2S (2) cr (ℓ/2)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 1 2 3 4 5 6 7 8 δS(2) ln(l)

0.02 0.03 0.04 0.05 0.05 0.1

• b(2)

ln(l)/l

fixed-h (+) and box-h ()

• numerical data:

δS (2)(ℓ) ≃ S (2)

cr (ℓ)+cst ≃ −b(2)lnℓ+cst

• universal logarithmic correction:

b(2) = −0.029(1)

• direct calculation of the corner con-

tribution for ℓ = L/2 ← the same total area → . corners no corners!

• relation with the cluster geometry

D = 0 D = 1 1 1 1 1/2 1/2 3/4 1 1/2

1. 2. 3.* 4.* 5.

18

Cube geometry in d > 2

• edge and corner corrections to the area-

law: S (d)

cube(ℓ) = ad−1 fd−1 +∑d−2 E=1aE fE +S (d) cr (ℓ)

• 1 ≤ E < d − 1:

dimension of the edge, fE ∼ LE

• aE −aE(ℓ) ∼ ℓ−E, as for the surface term
• direct calculation of the corner con-

tribution for ℓ = L/2 S (d)

cr

=

d−1

∑

D=0

• −1

2 Dd D

• S (d)

D

0.1 0.2 0.3 1 2 3 4 5 6 7 8 |S(d)

cr |

ln(l)

0.01 0.02 0.03 0.04 0.1 0.2 0.3 |b(d)| ln(l)/l

fixed-h (+) and box-h () 2d,3d and 4d

• numerical data:

S (d)

cr (ℓ) ≃ −b(d)lnℓ+cst

• universal logarithmic correction:

b(2) = −0.029(1), b(3) = 0.012(2) and b(4) = −0.006(2) 19

Phenomenological explanation

• consider 2-site clusters (at the end of

the RG)

• “corner clusters” have points at hyper-

cubes connected by the main diagonal

• relative coordinates of the 2-site clus-

ters: 0 ≤ xj ≤ L/2, j = 1,2,...,d) (periodic b.c.)

• the corner entropy (averaging over all

positions): −2∏d

j=1(−x j/L)

• probability of a 2-site cluster of length r

is: the average pair-correlation function: Cav(r) ≈ n2

r

nr ∼ r−d: the density of non-decimated sites

• average contribution:

S (d)

cr (ℓ) ∼ −

ℓ

1 dx1...

ℓ

1 dxd ∏d j=1(−xj/r2)

∼ (−1)d+1 ℓ

1(rd−1rd)/r2ddr

∼ (−1)d+1lnℓ 20

Corner correction outside the critical point

0.05 0.1 0.15

• 0.4 -0.2 0 0.2 0.4

|S(d)

cr |

δ

d=2

• 0.12
• 0.08
• 0.04
• 10
• 5

5

|S(d)

cr |-|b(d)|ln(L)

δL1/ν 128 256 512

0.01 0.02 0.03

• 0.4 -0.2 0 0.2 0.4

δ

d=3

• 0.04
• 0.03
• 0.02
• 8
• 4

δL1/ν 48 64 96 0.003 0.006 0.009

• 0.4 -0.2 0 0.2

δ

d=4

• 0.017
• 0.015
• 0.013
• 0.011
• 8
• 4

δL1/ν 16 24 32

The singularity is related to a diverging correlation length!

21

Effect of long-range interactions

EPL 107, 47008 (2014) Hamiltonian with power-law decaying in- teractions

H = −∑i=j

bij rα

ij σx

i σx j −∑ihiσz i

• with α = d +σ
• we take d = 1 (chain)
• and σ > 0 (extensive energy)
• bi j and hi i.i.d. random numbers
• in d = 3 possible relation with LiHoxY1−xF4

Conclusions of numerical renormalization

• anomalous scaling of the pseudocritical

points

0.0001 0.001 0.01 0.1 0.5 1 1.5 2 p θs

L=1024 2048 4096

0.0001 0.001 0.01 0.1

• 3 -2 -1 0 1 2

p (θs

c-θc)ln(L/L0)

• KT-like scaling of the correlation length

ξ ∼ exp(const/|θ −θc|)

• ratio of the frequency of the bond and

field decimations at the critical point: rθc(L) ∼ 1/ln2(L) 22

• average magnetic moment of the last re-

maining cluster is µ(L) ∼ ln2L

1 10 100 1000 5 10 15 20 25 30 d i

structure of the largest non-decimated cluster: µ = 32 and L = 8192

• energy scaling

0.001 0.01 0.1 p a)

L=256 512 1024 2048 4096

0.001 0.01 0.1 5 10 15 20 25 p ln h ~ b)

θ=1.0 1.1 1.2 1.3

ε ∼ L−z(θ)

• strong disorder fixed point with

zc ≡ z(θc) ≃ α

• in the Griffiths phase z(θ) < zc
• critical point entlangement entropy

SL = S∞ +O(ln−3L) saturates, in contrast with the short- range model: SL ∼ lnL. 23

Analytical derivation of the results - Primary model

• observations in the RG procedure
• 1. almost always transverse fields are

decimated

• 2. after a field decimation, the max-

imum rule leads almost always to ˜ Jjk = Jjk

• 3. the extension wi of (non-decimated)

clusters are typically much smaller than the distances between them

• construction of the primary model

– take bi j = b = 1, but let hi random – according to 2 we have: ˜ J−1/α

i−1,i+1 = J−1/α i−1,i +J−1/α i,i+1 +wi

and wi is neglected due to 3. – Using reduced variables ζ = Ω J 1/α −1 β = 1 α ln Ω h – the RG equations reads: ˜ ζ = ζi−1,i +ζi,i+1 +1 ˜ β = βi +βi+1

• equivalent to a 1d disordered O(2) quan-

tum rotor model (E. Altman, Y. Kafri,

• A. Polkovnikov, and G. Refael (2004))

with – grain charging energy Ui ↔ J1/α

i,i+1

– Josephson coupling Ji,i+1 ↔ h1/α

i

• Note, that site and bond variables are

interchanged in the two problems. 24

Solution of the primary model

• Let us change the log-energy-scale: Γ ≡ 1

α ln Ω0 Ω → Γ+dΓ

• the distributions gΓ(β) and fΓ(ζ) will follow the equations:

∂gΓ(β) ∂Γ = ∂gΓ(β) ∂β + f0(Γ)

• dβ ′gΓ(β ′)gΓ(β −β ′)+gΓ(β)[g0(Γ)− f0(Γ)]

∂ fΓ(ζ) ∂Γ = (ζ +1)∂ fΓ(ζ) ∂ζ +g0(Γ)

• dζ ′ fΓ(ζ ′)gΓ(ζ −ζ ′ −1)+ fΓ(ζ)[ f0(Γ)+1−g0(Γ)],
• the fixed-point solutions (Γ → ∞) are:

gΓ(β) = g0(Γ)e−g0(Γ)β, fΓ(ζ) = f0(Γ)e−f0(Γ)ζ

• which satisfy the ordinary differential

equations dg0(Γ) dΓ = − f0(Γ)g0(Γ), d f0(Γ) dΓ = f0(Γ)(1−g0(Γ)),

• thus f0(Γ) = g0(Γ)−lng0(Γ)−1+ε

ε = −a+ln(1+a)

• The boundary conditions in the Γ → ∞

limit: f0(Γ) → 0, g0(Γ) → 1+a – paramagnetic phase a > 0 – critical point a = 0

• The solutions in leading order in Γ:

g0(Γ) ≃ 1+acoth[(Γ+C)a/2], f0(Γ) ≃ a2 2sinh2[(Γ+C)a/2] , 25

Solution of the primary model

RG-flow

• the fraction of non-decimated sites n

satisfies dn dΓ = −n(g0 + f0)

• the length scale, l = 1/n is given by:

l ≃ eΓa−2sinh2[(Γ+C)a/2] ∼ Ω0 Ω 1+a

α

• the dynamical exponent

z = α/(1+a)

• z continuously varies with a
• maximal, at the critical point: zc = α.
• in the vicinity of the critical point: Γa =C′
• the characteristic length scale

ξ ∼ exp(Γ) ∼ exp(C′/a)

• explanation of the magnetization and

the entanglement entropy 26

Conclusions

• Short-range models

– Infinite disorder fixed point at any dimensions – Strong disorder RG approach is asymptotically exact – Universal (disorder independent) bulk and local exponents – Entanglement entropy: logarithmic correction to the area law due to corners at the critical point

• Long-range models

– strong disorder fixed point – unusual critical properties

27

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