Anisotropic Long Range Spin Systems Nicol` o Defenu Scuola - - PowerPoint PPT Presentation
Anisotropic Long Range Spin Systems Nicol` o Defenu Scuola Internazionale Superiore di Studi Avanzati, Trieste (Italy). ndefenu@sissa.it July 26, 2016 Nicol` o Defenu (SISSA) Functional RG July 26, 2016 1 / 21 Overview Long Range
Nicol`
Scuola Internazionale Superiore di Studi Avanzati, Trieste (Italy). ndefenu@sissa.it
July 26, 2016
Nicol`
Functional RG July 26, 2016 1 / 21
1
Long Range Interactions Spin systems
2
LR Spin Systems Traditional results Controversy Effective Dimension
3
The Anisotropic case Dimensional Analysis Three regimes Effective dimension Critical exponents
Nicol`
Functional RG July 26, 2016 2 / 21
Nicol`
Functional RG July 26, 2016 3 / 21
Why spin systems
spin systems are the testbed of statistical mechanics. Various Monte Carlo (MC) and perturbative results available. Diverse interesting physical problems in a single formalism.
Issues:
Phase diagram for diverse interaction shapes. Description of different symmetry groups. Description of high order critical points.
Nicol`
Functional RG July 26, 2016 4 / 21
Lattice Hamiltonian
H = −J 2
1 |i − j|d+σ SiSj
Nicol`
Functional RG July 26, 2016 5 / 21
Lattice Hamiltonian
H = −J 2
1 |i − j|d+σ SiSj
Mean Field Propagator
G(q)−1 = J(q) =
Nicol`
Functional RG July 26, 2016 5 / 21
Lattice Hamiltonian
H = −J 2
1 |i − j|d+σ SiSj
Mean Field Propagator
G(q)−1 = J(q) =
Leading momentum term
lim
q→0 G −1(q) ∝ qσ
if σ ≤ 2 lim
q→0 G −1(q) ∝ q2
if σ > 2
Nicol`
Functional RG July 26, 2016 5 / 21
Traditional Results
Three regimes: 0 < σ < d/2 Mean field exponents (η = 2 − σ and ν = σ−1). d/2 < σ < 2 Long range exponents (η ≡ η(σ) and ν ≡ ν(σ)). σ > 2 Short range exponents (η = ηSR and ν = νSR).a
aM.E. Fisher et al. PRL 29,14 Nicol`
Functional RG July 26, 2016 6 / 21
Traditional Results
Three regimes: 0 < σ < d/2 Mean field exponents (η = 2 − σ and ν = σ−1). d/2 < σ < 2 Long range exponents (η ≡ η(σ) and ν ≡ ν(σ)). σ > 2 Short range exponents (η = ηSR and ν = νSR).a
aM.E. Fisher et al. PRL 29,14
Peculiar Long Range Behavior
Using ǫ-expansion technique with ǫ = 2σ − d or 1/N expansion is possible to calculate the critical exponent η. η = 2 − σ + O(ǫ3) Exact at any order in ǫ. η = 2 − σ for all σ < 2. Discontinuity in σ = 2.
Nicol`
Functional RG July 26, 2016 6 / 21
Sak’s Results
The anomalous dimension cannot be less than ηSR, η = 2 − σ σ < σ∗ η = ηSR σ > σ∗ where σ∗ = 2 − ηSR. No discontinuity is present.
1.0 1.2 1.4 1.6 1.8 2.0 2.2 σ 2.0 2.5 3.0 3.5 4.0 d
M e a n F i e l d Long Range Short Range
Nicol`
Functional RG July 26, 2016 7 / 21
Sak’s Results
The anomalous dimension cannot be less than ηSR, η = 2 − σ σ < σ∗ η = ηSR σ > σ∗ where σ∗ = 2 − ηSR. No discontinuity is present.
System regimes
1.0 1.2 1.4 1.6 1.8 2.0 2.2 σ 2.0 2.5 3.0 3.5 4.0 d
M e a n F i e l d Long Range Short Range
Nicol`
Functional RG July 26, 2016 7 / 21
Luijte and Blotea results (2002) seemed to confirm Sak results, but new, more complete, results (2013)b question on Sak validity
Figure: MC 2002
VS
Figure: MC 2013
Nicol`
Functional RG July 26, 2016 8 / 21
Ginzburg-Landau Free Energy
ΦSR =
+ · · · ΦLR =
σ 2 ψ − Z2,kψ∆ψ + µψ2 + gψ4
+ · · ·
Nicol`
Functional RG July 26, 2016 9 / 21
Ginzburg-Landau Free Energy
ΦSR =
+ · · · ΦLR =
σ 2 ψ − Z2,kψ∆ψ + µψ2 + gψ4
+ · · ·
Effective dimension results
Zk = Z2,k = 1 → dSR = 2dLR σ Z2,k = 1 → dLR = (2 − ηSR)dLR σ
Nicol`
Functional RG July 26, 2016 9 / 21
I Approximation Level: No anomalous dimension
dSR = 2dLR
σ : Exact N → ∞, Correct σ ranges, σ∗ = 2
Nicol`
Functional RG July 26, 2016 10 / 21
I Approximation Level: No anomalous dimension
dSR = 2dLR
σ : Exact N → ∞, Correct σ ranges, σ∗ = 2
II Approximation Level: Pure Long range case
dSR = (2−ηSR)dLR
σ
: Exact N → ∞, Correct σ ranges, σ∗ = 2 − ηSR
Nicol`
Functional RG July 26, 2016 10 / 21
I Approximation Level: No anomalous dimension
dSR = 2dLR
σ : Exact N → ∞, Correct σ ranges, σ∗ = 2
II Approximation Level: Pure Long range case
dSR = (2−ηSR)dLR
σ
: Exact N → ∞, Correct σ ranges, σ∗ = 2 − ηSR
III Approximation Level: Mixed theory space
Competition between Short and Long range fixed points: ✟✟
✟ ❍❍ ❍
dSR
Fixed Points Solutions and Stability
1 2 d 2σ∗ 2 −2 −1 1 θ
d 2σ∗ 2 ηSR 1 η2 0.8σ∗ 0.9σ∗ σ∗ 0.004 0.008 0.01 |νLR(d, σ) − 2
σνSR(Deff)|Nicol`
Functional RG July 26, 2016 10 / 21
Short Range Corrections
Short Range corrections spoil dimensional equivalence. Small every- where but at σ ≃ σ∗.
Correlation Length Exponent
1.0 1.2 1.4 1.6 1.8 2.0 0.0 0.2 0.4 0.6 0.8 1.0
Σ 1 ΝLR
1.0 1.2 1.4 1.6 1.8 2.0 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.5 1.6 1.7 1.8 1.9 2.0 1.0 1.1 1.2 1.3 1.4 1.5 1.6
Σ 1 ΝLR
Nicol`
Functional RG July 26, 2016 11 / 21
Lattice Hamiltonian
H = −
J 2 SiSj rd1+σ
,ij
δ(r ⊥,ij) −
J⊥ 2 SiSj rd2+τ
⊥,ij
δ(r ,ij).
Nicol`
Functional RG July 26, 2016 12 / 21
Lattice Hamiltonian
H = −
J 2 SiSj rd1+σ
,ij
δ(r ⊥,ij) −
J⊥ 2 SiSj rd2+τ
⊥,ij
δ(r ,ij).
Mean Field Propagator
lim
q→0 G(q)−1 = lim q→0 J(q) = Zqσ + Z⊥qτ ⊥ + µ + O(q2)
Nicol`
Functional RG July 26, 2016 12 / 21
Lattice Hamiltonian
H = −
J 2 SiSj rd1+σ
,ij
δ(r ⊥,ij) −
J⊥ 2 SiSj rd2+τ
⊥,ij
δ(r ,ij).
Mean Field Propagator
lim
q→0 G(q)−1 = lim q→0 J(q) = Zqσ + Z⊥qτ ⊥ + µ + O(q2)
Effective field theory
2 φ(x)∆σ/2φ(x) − Z 2 φ(x)∆τ/2φ(x) + ... + U(φ(x))
Functional RG July 26, 2016 12 / 21
Quantum Lattice Hamiltonian
H = −J 2
σ(z)
i
σ(z)
j
|i − j|d+σ − h
σ(x)
i
,
Nicol`
Functional RG July 26, 2016 13 / 21
Quantum Lattice Hamiltonian
H = −J 2
σ(z)
i
σ(z)
j
|i − j|d+σ − h
σ(x)
i
,
Mapping between Quantum LR and Anisotropic LR
σi → Si d1 = d d2 = z σ = σ τ = 2. The Quantum LR Ising is obtained for z = 1
Nicol`
Functional RG July 26, 2016 13 / 21
Asymptotic propagators
G(q1, q1) ≈ q−σ+ησ
1
G(1, q2q−θ
1 ) ≈ q−τ+ητ 2
G(q1q
− 1
θ
2
, 1)
Correlation Lengths
ξ ≈ |T − Tc|−ν1 ξ⊥ ≈ |T − Tc|−ν2,
Nicol`
Functional RG July 26, 2016 14 / 21
Asymptotic propagators
G(q1, q1) ≈ q−σ+ησ
1
G(1, q2q−θ
1 ) ≈ q−τ+ητ 2
G(q1q
− 1
θ
2
, 1)
Correlation Lengths
ξ ≈ |T − Tc|−ν1 ξ⊥ ≈ |T − Tc|−ν2,
Anisotropy index
σ − ησ τ − ητ = ν2 ν1 = θ.
Nicol`
Functional RG July 26, 2016 14 / 21
Asymptotic propagators
G(q1, q1) ≈ q−σ+ησ
1
G(1, q2q−θ
1 ) ≈ q−τ+ητ 2
G(q1q
− 1
θ
2
, 1)
Correlation Lengths
ξ ≈ |T − Tc|−ν1 ξ⊥ ≈ |T − Tc|−ν2,
Anisotropy index
σ − ησ τ − ητ = ν2 ν1 = θ.
Mean field Results
ησ = ητ = 0, ν1 = σ−1, ν2 = τ −1.
Nicol`
Functional RG July 26, 2016 14 / 21
Mean Field Regions
0.0 0.5 1.0 1.5 2.0 2.5 3.0
σ
0.0 0.5 1.0 1.5 2.0 2.5 3.0
τ
I IIa IIb III
1 2
τ
1 2 3
z
MF WF
Three regimes
Region I: Anisotropic pure LR system (σ < σ∗ and τ < τ∗). Region II a\b: Anisotropic S-LR system (σ < σ∗ and τ > τ∗). Region III: Isotropic SR system (σ > σ∗ and τ > τ∗).
Nicol`
Functional RG July 26, 2016 15 / 21
In region I the system is equivalent to an isotropic LR one
D = d1 + θd2, θ = σ τ .
Nicol`
Functional RG July 26, 2016 16 / 21
In region I the system is equivalent to an isotropic LR one
D = d1 + θd2, θ = σ τ .
Exact in the spherical limit
ν1 = τ τd1 + σd2 − τσ, ν2 = σ τd1 + σd2 − τσ.
Nicol`
Functional RG July 26, 2016 16 / 21
In region I the system is equivalent to an isotropic LR one
D = d1 + θd2, θ = σ τ .
Exact in the spherical limit
ν1 = τ τd1 + σd2 − τσ, ν2 = σ τd1 + σd2 − τσ.
Independent confirmation: the spherical ANNNI model
ν1 = L (d1 − 2)L + d2 , ν2 = 1 (d1 − 2)L + d2 .
Nicol`
Functional RG July 26, 2016 16 / 21
Anomalous dimension in the three regimes
Region I: η1 = 2 − σ and η2 = 2 − τ (ησ = ητ = 0). Region II a\b: η1 = η(τ) and η2 = 2 − τ Region III: η1 = η2 = ηSR
Anomalous dimension of the SR term in region II
0.8 1.0 1.2 1.4 1.6 1.8 2.0
τ
0.00 0.05 0.10 0.15 0.20 0.25
η d1 = d2 = 1
1.0 1.2 1.4 1.6 1.8 2.0
τ
0.00 0.01 0.02 0.03 0.04 0.05
η d1 = 1(2) and d2 = 2(1)
Nicol`
Functional RG July 26, 2016 17 / 21
Region I: Anisotropic pure LR system.
Deff = d1+θd2: Exact for N → ∞. σ∗ = 2−η(τ) and τ ∗ = 2−η(σ)
Nicol`
Functional RG July 26, 2016 18 / 21
Region I: Anisotropic pure LR system.
Deff = d1+θd2: Exact for N → ∞. σ∗ = 2−η(τ) and τ ∗ = 2−η(σ)
Region II: Anisotropic mixed S-LR region.
Deff = d1 + θd2: Exact for N → ∞. τ ∗ = 2 − ηSR. θ = 2−η(τ)
τ
.
Nicol`
Functional RG July 26, 2016 18 / 21
Region I: Anisotropic pure LR system.
Deff = d1+θd2: Exact for N → ∞. σ∗ = 2−η(τ) and τ ∗ = 2−η(σ)
Region II: Anisotropic mixed S-LR region.
Deff = d1 + θd2: Exact for N → ∞. τ ∗ = 2 − ηSR. θ = 2−η(τ)
τ
.
Region III: Isotropic SR case.
Universality class of an isotropic SR system in dimension d = d1 +d2.
Nicol`
Functional RG July 26, 2016 18 / 21
Region I: Anisotropic pure LR system.
Deff = d1+θd2: Exact for N → ∞. σ∗ = 2−η(τ) and τ ∗ = 2−η(σ)
Region II: Anisotropic mixed S-LR region.
Deff = d1 + θd2: Exact for N → ∞. τ ∗ = 2 − ηSR. θ = 2−η(τ)
τ
.
Region III: Isotropic SR case.
Universality class of an isotropic SR system in dimension d = d1 +d2.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
σ
0.0 0.5 1.0 1.5 2.0 2.5 3.0
τ
IMF I II MF II a II MF II b III
1.0 1.5 2.0 2.5 3.0
σ
1.0 1.5 2.0 2.5 3.0
τ
IMF I II MF II a II b III
1.95 1.96 1.97 1.94 1.95 1.96 1.97 Border region
Nicol`
Functional RG July 26, 2016 18 / 21
Short Range Corrections
Short Range corrections spoil dimensional equivalence. Small every- where but close to the boundaries.
Correlation Length Exponent
1 2 1 2 ν−1
2 σ = 0.4 σ = 0.8 σ = 1.2 σ = 1.6 σ > 2.01 2 τ 1 2 ν−1
1(c) N = 1
1 2 1 2 ν−1
2 σ = 0.4 σ = 0.8 σ = 1.2 σ = 1.6 σ > 2.01 2 τ 1 2 ν−1
1(d) N = 2
1 2 1 2 ν−1
2 σ = 0.4 σ = 0.8 σ = 1.2 σ = 1.6 σ > 2.001 2 τ 1 2 ν−1
1(e) N = 3
Nicol`
Functional RG July 26, 2016 19 / 21
Resident Advisors: Andrea Trombettoni & Stefano Ruffo External Advisor: Alessandro Codello
ndefenu@sissa.it
Nicol`
Functional RG July 26, 2016 20 / 21
Nicol`
Functional RG July 26, 2016 21 / 21