self-consistent field = 1 1 Z X d d r h m ( ~ r ) 2 i = h m 2 i h - - PowerPoint PPT Presentation

S(φ) ≈ S0 + 1 2 X

i,j

⇢ (J−1)ijφiφj + kBT ⌧∂2 ln[2 cosh(βsφi] ∂φi∂φj

• φiφj
• Self-consistent field approximation

partition function from Gaussian transformation

expansion around

1

Z = C Z +∞

−∞

Y

i

dφi ! e−βS(φi,Hi)

S(φi, Hi = 0) = −1 2 X

i,j

(J−1)ijφiφj − 1 β X

i

ln[2 cosh(βsφi)]

C = 1 (2πkBT)N/2√ detJ

hφii = 0

Gaussian approximation

Self-consistent field approximation

2 Gaussian approximation Fourier transform:

φi = 1 N X

~ q

φ~

q ei~ q·~ ri

uniform

hφ2

i i = hφ2i

S(φ) ⇡ S0 + 1 2 X

i,j

⇢ (J−1)ij + βs2δij ⌧ 1 cosh2(βsφi)

• φiφj

⇡ S0 + 1 2 X

i,j

• (J−1)ij + βs2δij(1 β2s2hφ2i

φiφj .

S() = S0 + 1 2N X

~ q

⇢ 1 J(~ q) + s2(1 2s2h2i)

• ~

q−~ q

= S0 + ad 2J2z2N X

~ q

• q2 + A + 3Bs63h2i

~

q−~ q

Self-consistent field approximation

3

φ~

q = Jzm~ q

S = S0 + ad 2N X

~ q

• q2 + A + 3Bhm2i

m~

qm−~ q

= S0 + 1 2 X

~ q

G−1(~ q)m~

qm−~ q

Z0 = Z0 Y

~ q 0 Z

dm~

q dm~ q exp

• −G1(~

q)m~

q m~ q/2

= Z0 Y

~ q 0 Z

dm0

~ q dm00 ~ q exp

n −G1(~ q)(m0

~ q 2 + m00 ~ q 2)/2

• Y

~ q

runs only over a half-space of ~

q

e.g. {~

q | qz ≥ 0}

Self-consistent field approximation

self-consistent field 4 self-consistence equation for fluctuations hm2i

hm2i = 1 Ld Z ddr hm(~ r)2i = 1 N 2 X

~ q

hm~

qm~ qi

= Z0 Z0Ld Y

~ q 0 Z

dm~

q dm~ q m~ q m~ q exp

• G1(~

q)m~

q m~ q/2

Self-consistent field approximation - susceptibility susceptibility

5

Self-consistent field approximation - instabillity instability at

cutoff:

6

Self-consistent field approximation - critical exponents

temperature dependence of for

correlation length

cutoff necessary for

7

Self-consistent field approximation - critical exponents

Fisher scaling

mean field exponents

but

8

Self-consistent field approximation - critical exponents

regime 1:

dominant

"close" to

regime 2:

dominant

"far" from

mean field

9