Planar Order in a 3DPlaquette Ising model
Marco Mueller, Wolfhard Janke, Des JohnstonMECO40, Esztergom, March 2015
Mueller, Janke, Johnston Planar Order 1/20
Plan of talk
A 3D plaquette Ising model with a highly degenerate low-Tphase
Consequences for order parameter (main focus here)
Consequences for FSS at first order transition (in passing)
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A 3D Plaquette Ising action
3D cubic lattice, spins on vertices
H = −∑�
σiσjσkσl
Not edges
H = −∑�
UijUjkUklUli
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Plaquette Hamiltonian Groundstates:Single cube
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Ground states, Low T: Lattice
Persists into low temperature phase (Wegner, Pietig)
Degeneracy 23L
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Anisotropic (“Fuki-Nuke”) Model
Consider anisotropic variantSuzuki 1973, Jonsson/Savvidy 2000, Castelnovo et.al.2010
Haniso = −Jx∑�yz
σiσjσkσl − Jy∑�xz
σiσjσkσl
Set Jz = 0, Jx = Jy = 1. No horizontal plaquettes
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Anisotropic Model
Define new spins τ from pairs of σ’s
τx,y,z = σx,y,zσx,y,z+1
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Anisotropic Model = Stack of 2D Ising
Hfuki−nuke = −L∑x=1
L∑y=1
L∑z=1
(τx,y,zτx+1,y,z + τx,y,zτx,y+1,z) ,
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Anisotropic Model Order Parameter
Single layer
m2d,z =
⟨1
L2
L∑x=1
L∑y=1
τx,y,z
⟩In terms of original variables
m2d,z =
⟨1
L2
L∑x=1
L∑y=1
σx,y,zσx,y,z+1
⟩
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Isotropic case
Hypothesis: Same order parameter works
Hashizume and Suzuki (2011)
Take layers, add ’em up
mzabs =
⟨1
L3
L∑z=1
∣∣∣∣∣∣L∑x=1
L∑y=1
σx,y,zσx,y,z+1
∣∣∣∣∣∣⟩
mzsq =
⟨1
L5
L∑z=1
L∑x=1
L∑y=1
σx,y,zσx,y,z+1
2⟩
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Isotropic case: Effect of flips
mzabs =
⟨1
L3
L∑z=1
∣∣∣∣∣∣L∑x=1
L∑y=1
σx,y,zσx,y,z+1
∣∣∣∣∣∣⟩
+
+
−
−
+
−
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Isotropic case: numericalinvestigation
Strong first order PT
Multicanonical simulation
Correct order parameter: predicts PT point? isotropic?
FSS properties?
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FSS for 1st order PTs
Pirogov-Sinai Theory (Borgs/Kotecký)
Z(β) =[e−βL
dfd + qe−βLdfo]
[1 + . . .]
Fixed boundaries (1/L leading term)
Z(β) =[e−β(L
dfd+Ld−1fo) + qe−β(L
dfo+Ld−1fd)]
Exponential degeneracy (1/L2 in d = 3 )
Z(β) =[e−βL
dfd + 23Le−βLdfo]
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FSS for 1st order PTs: degeneratecase
With q ∝ 23L = e(3 ln 2)L
βCmaxV (L) = β∞ − ln q
Ld∆e+ . . .
becomeβC
maxV (L) = β∞ − 3 ln 2
Ld−1∆e+ . . .
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Numerical results: order parameters
Order parameter mxabs Order parameter mx
sq
my and mz identical to mx - isotropy restored
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Numerical results: susceptibilities
χ(β) = βL3(〈m2〉(β)− 〈m〉(β)2
)Susceptibility for mx
abs Susceptibility for mxsq
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Numerical results: scaledsusceptibilities
βχmx
abs (L) = 0.551 37(3)− 2.46(3)/L2 + 2.4(3)/L3
Susceptibility for mxabs Susceptibility for mx
sq
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Flipping
Running averages, magnetization and standard susceptibility,various starting configs (green ordered, blue intermediate, reddisordered):
Magnetization Susceptibility
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Conclusions
Suzuki was right - isotropic model displays Fuki-Nuke order
....which is a curious “2.5d” order
Scaling is non-standard (c/o degeneracy)
Results agree well with energetic observables (Spec heat,Binder etc)
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References
Y. Hashizume and M. Suzuki, Int. J. Mod. Phys. B 25(2011) 73; Int. J. Mod. Phys. B 25 (2011) 3529.
M. Mueller, W. Janke and D. A. Johnston,Phys. Rev. Lett. 112 (2014) 200601.
M. Mueller, D. A. Johnston and W. Janke,Nucl. Phys. B 888 (2014) 214.
M. Mueller, W. Janke and D. A. Johnston,Nucl. Phys. B 894 (2015) 1.
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