Monte Carlo Renormalization Flows with Dangerously Irrelevant - - PowerPoint PPT Presentation

Anders W Sandvik Boston University and Institute of Physics, Chinese Academy of Sciences, Beijing

Monte Carlo Renormalization Flows with Dangerously Irrelevant Operators

Computational Approaches to Quantum Many-Body Problems ISSP , U of Tokyo, July 22, 2019

Hui Shao (Beijing Normal University) Wenan Guo (Beijing Normal University) arXiv:1905.13640

Motivations

• I. To better understand emergent U(1) symmetry in quantum magnets

Antiferromagnet 4 − fold degenerate VBS

L = 12 L = 24 max

VBS orderparameter distribution P(Dx, Dy)

L = 64

Emergent U(1) symmetry close to deconfined quantum-critical point

• a new length scale 휉’ > 휉
• role of 휉’ in finite-size scaling

(Shao, Guo, Sandvik, Science 2016)

• II. Better general understanding of dangerously irrelevant perturbations
• in classical and quantum systems
• how to best analyze them in Monte Carlo simulations?

q-fold clock perturbation of the 3D XY model

Cross-over from XY ordering to Zq ordering at length scale 흃’q

H = −J X

hiji

cos(Θi − Θj) − h X

i

cos qΘi

q = 6

Jose, Kadanoff, Kirkpatrick, Nelson, PRB 1977

Clock models Fixed points: P = paramagnet X = 3D XY critical point Y = XY symmetry breaking Q = Zq symmetry breaking

Okubo et al, PRB 2015

RG flows can be observed in MC simulations “phenomenological renormalization”

Dangerously irrelevant perturbation

• irrelevant at Tc, relevant for T<Tc
• correlation length and emergent U(1) length

ξ ∝ (g − gc)−ν ξ0 ∝ (g − gc)ν0

ν0 > ν

MC simulations of classical 3D clock model

q = 6

H = −J X

hiji

cos(Θi − Θj) q clock angles (hard clock model)

mx = 1 N

N

X

i=1

cos(Θi)

my = 1 N

N

X

i=1

sin(Θi)

Standard order parameter (mx,my) Probability distribution P(mx,my) shows cross-over from U(1) to Zq for T<Tc

Lou, Balents, Sandvik, PRL 2007

Can be quantified with “angular order parameter”: 흋q > 0 only if q-fold anisotropy Finite-size scaling of 흋q can be used to extract length scale 휉’ > 휉 and associated scaling dimension yq

φq = Z 2π dθ cos(qθ)P(θ)

→ global angle θ

H = −J X

hiji

cos(Θi − Θj) − h X

i

cos qΘi (soft clock model)

Relevant field accessed through the Binder cumulant: Um = 2 hm4i hm2i2 Angular order parameter 흋q reflects the dangerously irrelevant field

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 4
• 2

2 4 6 8 10

• 10

3D, q=6 (a) L=32 L=48 L=64 L=96 L=128

Okubo et al. (PRB 2015) The exponent 휈’ can be directly extracted from 휑q when it is large

• follows from scaling function

L = 2, 3, . . .

MC RG flows in the plane (Um,휑q)

0.02 0.04 0.06 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 G XY NG Z6 U φ6 T = 1.0 T = Tc T = 3.0

Entire RG flow can be explained by phenomenological scaling function with two relevant arguments:

q = LyqΦ(tL1/ν, tL1/ν0

q)

Conventional finite-size scaling (general)

A(t, L) = L−κ/νf(tL1/ν, λ1L−ω1, λ2L−ω2, . . .)

Finite-size scaling form:

A ∝ tκ

Quantity with thermodynamic-limit critical form

t ∝ T − Tc

Take infinite-size limit: A(t, L → ∞) ∝ L−κ/ν(tL1/ν)κ + . . . ∝ tκ + . . .

β = 1/8 ν = 1

Tc/J = 2 ln( √ 2 + 1) ≈ 2.269

∝ |t|2β

hm2i / L−2 L−2β/νf(tL1/ν) ∝ L−2 L−2β/ν(tL1/ν)x ∝ L−2 T > Tc x = 2(β − ν) = −7/4

Example: 2D Ising model, squared order parameter hm2i / |t|2β (T < Tc)

Relevant and irrelevant perturbations H = H0 + hP

i mi = hM (≡ hNm = hLdm)

RG description of effects of hM at a critical point. Free energy density:

fs(t, h, L) = LdFs(tL1/ν, hLy).

yd

Irrelevant perturbation if y<0.

fs(t, h, L) = LdFs(tL1/ν, tL1/ν0, hLy, λLω),

Proposal to describe all scaling regimes (L < ξ, ξ0,

ξ < L < ξ0, ξ, ξ0 < L)

Two relevant fields tuned by same parameter

• first introduced for deconfined criticality (Shao, Guo, Sandvik, Science 2016)
• Here: detailed tests and new insights for classical clock models

Relationship between 휈’ and y?

h p y = d ∆

→

h ξ / |t|ν. L increases i ale ξ0 / tν0

(let t = Tc − T now)

How about dangerously irrelevant?

h

• to leading order,

e fs = hhmi / hL∆;

From Hamiltonian:

• s

s fs / hLyd

Taylor expand at t=0:

h

0.02 0.04 0.06 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 G XY NG Z6 U φ6 T = 1.0 T = Tc T = 3.0

Scaling at Tc (XY critical point)

φq = LyqΦ(tL1/ν, tL1/ν0, hLyq, λL−ω) ∝ Lyq(1 + hLyq + λL−ω + . . .)

10 100 L 10

• 4

10

• 2

φq q=4, h=1 q=5, hard q=5, h=5 q=5, h=2 q=6, hard

yq q 4 5 6

• Ref. [8]
• 0.2
• 1.5
• 3.0
• Ref. [11]
• 0.114
• 1.16
• 2.29
• Refs. [10, 14]
• 0.108(6)
• 1.25
• 2.5

This work

• 0.114(2)
• 1.27(1)
• 2.55(6)

Scaling dimensions for q = 4, 5, 6

[8] Oshikawa, PRB 2000 [10] Okubo et al., PRB 2015 [11] Leonard & Delamotte, PRL 2015 [14] Hasenbusch & Vicari, PRB 2011

Normally the scaling dimension is extracted from the corresponding correlation function in the h=0 model

• noisy because fast decay

0.02 0.04 0.06 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 G XY NG Z6 U φ6 T = 1.0 T = Tc T = 3.0

Distance to the XY critical point

φq = LyqΦ(tL1/ν, tL1/ν0, hLyq, λL−ω)

q / Lyq(1 + tL1/ν) ) = UXY + tL1/ν + Lω,

U(tL1/ν, L−ω)

Distance: d1 =

q φ2

q + (U − UXY)2

d1 / q (tL1/ν + Lω)2 + L2yq(1 + tL1/ν)2.

q Since ! ⌧ |y6| /

→

! ⌧ |y6|, d1 is dom ; d1 / tL1/ν + Lω.

Predicted minimum distance and corresponding system size

D1 / t

ω 1/ν+ω = t0.39(2),

L

, L1 / t

1 1/ν+ω = t0.412(4),

e ⌫ = 0.6717(1) Known exponent:

e ! = 0.94(3),

“Effective 휔”:

0.001 0.01 t 2 3 4 D1×10

2

5 10 15 L1 10 20 L 2 3 4 5 d1×10

2

(a) (b)

Fits to data give exponents:

0.372(1) [D1], − 0.404(4) [L1] (no scaling corrections included)

0.02 0.04 0.06 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 G XY NG Z6 U φ6 T = 1.0 T = Tc T = 3.0

Minimum value of 흋q

φq = LyqΦ(tL1/ν, tL1/ν0, hLyq, λL−ω)

q / Lyq(1 + tL1/ν)

Minimize as function of L at fixed t: D2 ∝ t−y6ν = t1.71(4), L2 ∝ t−ν = t0.6717(1)

y6 = −2.55(6)

e ⌫ = 0.6717(1) Used exponents:

0.05 0.1 t 2 4 6 D2×10

3

10 12 14 16 L2 10 20 L 4 8 d2×10

3

(a) (b)

From MC fits

1.88(2) [D2], − 0.60(3) [L2]

tL1/ν, tL1/ν0 ⌧ 1, tL1/ν0 ⌧ tL1/ν

q = LyqΦ(tL1/ν, tL1/ν0

q)

Relationship between ힶq’ and yq

Consider the case: tL1/ν large,

tL1/ν0 still small (ξ ⌧ L ⌧ ξ0)

Scaling then requires the form

q = Lyq(tL1/ν)ag(tL1/ν0

q),

where the exponent a depends on the physics of the clock model this form should apply also when 흋q → 1

• scaling then demands g(x)=xb for some exponent b
• all L and t dependence must go away →

φq = Lptν(pyq)g(tL1/ν0

q)

• ν0

q = ν(1 yq/p) = ν(1 + |yq|/p),

Lou, Balents, Sandvik (PRL 2007): p=3 Okubo et al (PRB 2015), Leonard & Delmotte (PRL 2015): p=2

• how does 흋q depend on L at fixed t when 흋q is still small (g ~ 1)
• it should be some power of L; 흋q ~ Lp

φq = Lptν(pyq)g(tL1/ν0

q)

Determination of the exponent ힶq’

g ! (tL1/ν0

q)b[1 k(tL1/ν0 q)],

Can write g() asymptotically as

φq ! 1 k(tL1/ν0

q)

which gives 흋q in this regime: where k() must be dimensionless

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 4
• 2

2 4 6 8 10

• 10

3D, q=6 (a) L=32 L=48 L=64 L=96 L=128

Okubo et al. (PRB 2015) Determine 휈’ by data collapse

• or cross of 흋q(L) with constant

0.25 0.3 0.35 t 30 40 50 60 Lc

φ6=0.6 φ6=0.55 φ6=0.5

20 40 60 L 0.4 0.6 φ6 (a) (b)

d ν0

6 = 1.52(4).

atisfied if p =

Our fit:

ν0

q

ν = 1 + |yq| p

is satisfied with p=2 y6 = −2.55(6)e ⌫ = 0.6717(1)

0.02 0.04 0.06 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 G XY NG Z6 U φ6 T = 1.0 T = Tc T = 3.0

Near the Nambu-Goldstone point

0.02 0.04 0.06 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 G XY NG Z6 U φ6 T = 1.0 T = Tc T = 3.0

φq = Lptν(pyq)g(tL1/ν0

q)

Now g()~1 again Distance to the NG point:

• Need cumulant: U(tL1/ν) → 1

1 U / (tL1/ν)r,

10

• 4

10

• 2

1-U

L=16 L=32 L=64 L=128 L=256

(a) 10

• 1

10 10

1

10

2

tL

1/υ

What is the exponent r?

• surprisingly, it was not known!
• we find r=1.52(2) for 3D XY model

D3 / q t2r(R1) + t4(ν0

qRν),

L3 / tνR, q where R = (r+2ν0

q)/(r+2ν) 0.9(1)

e D3 / t0.9(1) and L3 / t1.07(3).

Using our q=6 exponents: 1.19(3) [D3], − 1.14(2) [L3] Data fit gives:

0.15 0.2 t 12 14 16 18 D3×10

3

14 16 18 L3 10 20 L 1 2 3 d3×10

2

(a) (b)

DQCP: In the field theory the VBS corresponds to condensation of topological defects (quadrupoled monopoles on square lattice)

Dx Dx Dy Dy

Analogy with 3D clock models: The topological defects should be dangerously irrelevant

r

AF

U(1) SL VBS DQCP

Graph from Matthew Fisher

Fugacity of topological defects ힴ4 MC RG flows for J-Q3 model

• work in progress

L = 4, 6, . . .

• 0.02

0.02 0.04 0.06 0.08 0.1 0.12 0.14

• 1
• 0.5

0.5 1 D4 UD-Um J/Q=0.0667 J/Q=0.2667 J/Q=0.3667 J/Q=0.4667 J/Q=0.5667 J/Q=0.6067 (J/Q)c=0.6667 J/Q=0.7667 J/Q=0.8667 J/Q=1.8667 J/Q=2.8667

J = Q3 model

Ratio 휈/휈’ plays important in finite-size scaling Shao, Guo, Sandvik (Science 2016)

Conclusions

L = 2, 3, . . .

0.02 0.04 0.06 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 G XY NG Z6 U φ6 T = 1.0 T = Tc T = 3.0

q = LyqΦ(tL1/ν, tL1/ν0

q)

U = U(tL1/ν)

MC RG flows expressed with: Consistent quantitative descriptions in all regimes:(L < ξ, ξ0,

ξ < L < ξ0, ξ, ξ0 < L

Method can be used for other models & dangerously irrelevant perturbations Should be useful for resolving the deconfined quantum-criticality puzzle

L = 64