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NL2678 Renormalization groups

Systems undergoing phase transitions at a critical temperature Tc are convenientlydescribed in terms of order parameters and critical exponents. Above Tc, for example,

the specific heat is found experimentally to vary as where (T Tc)/Tc, and

below Tc, it varies as (). Thus and are called critical exponents for specific

heat. Similar critical exponents are found for order parameters, isothermal susceptibility,

response to an external field, the correlation length , and the pair correlation function

(r). These are denoted as , , , , and , respectively.

Relations among these critical exponents can be derived from scaling assumptions

such as the static scaling hypothesis, which asserts that the Gibbs potential G(T, H)

for magnetic systems (as an example) is a generalized homogeneous function

G(a, aHH) = G(, H) . (1)

With an arbitrary value of and selecting a, aH appropriately, relations among

critical exponents are found through thermodynamic identities, for example, M(, H) =

dG(, H)/dH. Thus, Equation (1) leads to the following relations:

=1 aH

a, =

aH1 aH

, =2aH 1

a, =

2aH 1

a, = 2

1

a. (2)

These can be recast as critical exponent equations:

+ ( + 1) = 2 ,

+ 2+ = 2 ,( + 1) = (2 )( 1) ,

= ( 1) ,

= ,

= ,

reducing the number of independent critical exponents to just four (a, aH, , and ).

The two exponents that describe spatial correlations ( and ) also involve a scaling

relationship as was shown through dynamic scaling arguments by Leo Kadanoff and

Michael Fisher, among others. In the two-dimensional (2-D) Ising model aboveTc

, for

example, spin-spin correlations show short-range order, whereas below Tc the system

exhibits long-range order. As T tends to Tc from either side, (r r) becomes long

ranged and decays slowly as |r r| 1, with the correlation length diverging. This is

typical of most critical systems for which long-range fluctuations accompany the onset

of a second-order phase transition. However, in the 2-D XY model, the correlation

function exhibits an algebraic fall-off as (r) r(T), demonstrating lack of long-

range order in an unusual critical systemthe so-called KosterlitzThouless transition

whose hallmark is the formation of vortex-antivortex pairs. Mermin and Wagner (1966)

proved that any 2-D system with short-range interactions whose ordered phase has a

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Figure 1. A schematic illustration of the Kadanoff construction.

continuous symmetry does not support long-range order. This is because at least one

branch of collective excitationscalled Goldstone bosonshas energy that tends to zero

continuously as its wave vector (crystal momentum) vanishes.

With the exception of such unusual systems, spatial scaling applies to criticality,

which according to Kadanoff is to be understood as coarse graining. This means that the

essential features of the system remain unchanged as the lattice length scale is increased

by a factor such that 1 /a, where is the correlation length and a is the

lattice spacing (see Figure 1).IfN is the number of lattice sites and d the dimensionality of the system, then m =

N/d is the number of blocks obtained in the first rescaling process. Simultaneously, the

lattice variables (e.g., spins) and the interaction parameters are redefined. The effective

spins now represent each block and interaction parameters refer to the interacting blocks.

Taking the Ising model as an example, the cell Hamiltonian is:

Hcell = JN

i,j

SiSj hN

i

Si. (3)

Then the block Hamiltonian becomes

Hblock = J

m

,SS h

m

S, (4)

where the tilde quantities refer to blocks. Crucial assumptions are that thermodynamic

potentials scale with block size as:

Fblock(, h) = dFcell(, h), (5)

where h = xh, = y, and that F is a homogeneous function. Hence the exponents

x and y are calculated as y = ad and x = aHd. The Kadanoff construction applied to

the pair correlation function,

(e, ) = (Si S)(Sj S), (6)

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gives y = 1, and since y = da and a = (2)1 we find that: = and d = 2.

Using x = aHd with aH = /( + 1) results in (2 ) = .

Thus static and dynamic scaling hypotheses reduce the number of independent

critical exponents to just two. The renormalization group (RG) theory was developed

to calculate the values of these exponents for particular models by carrying out the

scaling procedure up to , and because as T Tc, scaling should continue ad

infinitum as T approaches Tc.

Denoting the cell Hamiltonian by H0 and the Hamiltonian after the nth step of

rescaling as Hn, the chain of scaling transformations R is

R(H0) = H1, R(H1) = H2, . . . , R(Hn) = Hn+1, . . . , R(H) = H, (7)

where H denotes a fixed point Hamiltonian characteristic of the critical state. Each

step in the RG transformation chain reduces the number of degrees of freedom by d.

In statistical mechanics we need to ensure that the partition function retains the samesymmetry and ground-state, and hence we must re-scale the coupling constant at each

step, for example, K = J/kT in the Ising model. We then develop a recursion relation

to compute the partition function.

An iterative solution of this recursion relation for the partition function yields

roots or fixed points which correspond to resultant critical behaviors in the model.

The partition function is preserved in the RG procedure via the condition

ZN(Hn) = Zm(Hn+1) (8)

where m = N/d.

It is conjectured that the values of critical exponents are characteristics not ofindividual Hamiltonians but their sets, with numerous models leading to the same fixed

point. The universality hypothesis states that any two physical systems with the same

dimensionality, d, and the same number of order parameter components, n, belong to

the same universality class, and each fixed point corresponds to one universality class.

Table 1 is a summary of the critical exponent values obtained from RG calculations for

key theoretical models.

RG ideas extend into many areas of physics, chemistry, biology, and engineering.

Based on the work of Kadanoff, Kenneth Wilson proposed an algorithmic approach to

the scaling problem by formulating it in reciprocal space. Although less intuitive than

the real-space RG of Kadanoff, it leads to exact results for the removal of divergenciesin theories of elementary particles. For this work, Wilson was awarded the 1982 Nobel

Prize in Physics.

Jack A. Tuszynski

See also Critical phenomena; Ising model; Order parameters; Phase transitions

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Model

Classical (MFT) 0 (disc.) 12 1 3

Spherical: d = 3 12 2 1 5 0

Spherical: > 0 /(2 ) 1

2

2/(2 ) 1/(2 ) 1 + 4/(2 ) 0

Ising: d = 2 (exact) 0 (log) 18

74

1 15 14

Ising: d = 3 0.12 0.33 1.25 0.64 4.8 0.04

Heisenberg: d = 3 0.12 0.36 1.39 0.71 4.8 0.04

S4-model: d > 4 /2 12 /4 112 3 + 0

S4-model: d = 4 0 12

1 12

3 0

S4-model: d < 4 /6 12 /6 1 + /612 + /12 3 + 0

S4-model: d = 3 0.17 0.33 1.17 0.58 4 0

XY-model: d = 3 0.01 0.34 1.30 0.66 4.8 0.04

Table 1. Summary of Critical Exponents for Key Models. Note that = (4 d).

Further Reading

Creswick, R.J., Farach, H.A. & Poole, C.P. 1992. Introduction to Renormalization Group

Methods in Physics, New York: Wiley

Kosterlitz, J.M. & Thouless, D.J. 1973. Ordering, metastability and phase transitions

in two-dimensional systems. Journal of Physics C, 6: 11811203

Ma, S.-K. 1976. Modern Theory of Critical Phenomena, New York: Benjamin

Mermin, N.D. & Wagner, H. 1966. Absence of ferromagnetism or antiferromagnetism

in one- or two-dimensional isotropic Heisenberg models. Physics Review Letters, 17:11331136

Reichl, L.E. 1979. A Modern Course in Statistical Physics, Austin, Texas: University of

Texas Press

Stanley, H.E. 1972. Introduction to Phase Transitions and Critical Phenomena, Oxford:

Clarendon Press and New York: Oxford University Press

Wilson, K.G. 1972. Feynman-graph expansion for critical exponents. Physics Review

Letters, 28: 548551

Wilson, K.G. 1983. The renormalization group and critical phenomena. Reviews of

Modern Physics, 55: 583600

Yeomans, J.M. 1992. Statistical Mechanics of Phase Transitions, Oxford: Clarendon

Press and New York: Oxford University Press