MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � � � � V. Position Space Renormalization Group V.A Lattice Models While Wilson’s perturbative RG provides a systematic approach to probing critical properties, carrying out the ǫ-expansion to high orders is quite cumbersome. Models de-fined on a discrete lattice provide a number of alternative computational routes that can complement the perturbative RG approach. Because of universality, we expect that all models with appropriate microscopic symmetries and range of interactions, no matter how simplified, lead to the same critical exponents. Lattice models are convenient for visualiza-tion, computer simulation, and series expansion purposes. We shall thus describe models in which an appropriate “spin” degree of freedom is placed on each site of a lattice, and the spins are subject to simple interaction energies. While such models are formulated in terms of explicit ‘microscopic’ degrees of freedom, depending on their degree of complexity, they may or may not provide a more accurate description of a specific material than the Landau–Ginzburg model. The point is that universality dictates that both descriptions describe the same macroscopic physics, and the choice of continuum or discrete models is a matter of computati onal convenience. Some commonly used la ttice models are described here: 1. The Ising Model is the simplest and most widely applied paradigm in statistical me-chanics. At each site i of a lattice, t here is a spin σi which takes the two values of +1 or −1. E ach state may correspond to one of two species in a binary mixture, or to empty and occupied cells in a la ttice approximation to a n interacting gas. The simplest possible short range interaction involves only neighbo ring spins, and is described by a Hamiltonian H = Bˆ(σi, σj), (V.1) <i,j> where the notation < i, j > is commonly used to indicate the sum over all nearest neighbor pairs on the lattice. Since σi 2 = 1, the most general interaction between two spins is hˆBˆ(σ, σ ′ ) = −gˆ − (σ + σ ′ ) − Jσσ ′ . (V.2) z For N spins, there are 2N possible micro-states, and the (Gibbs) partition function is  Z = e −βH= exp Kσiσj + hσi + g , (V.3) {σi} {σi} <i,j> i 75� � � � � � � � � � where we have set K = βJ, h = βhˆ, and g = βgˆ (β = 1/kBT , and z is the number of bonds per site, i.e. the coordination number of the latt ice). For h = 0 at T = 0, the ground state has a two fold degeneracy with all spins pointing up or down (K > 0). This order is destroyed at a critical Kc = J/kBTc with a phase transition to a disordered state. The field h breaks the up–down symmetry and removes the phase transiti on. The parameter g merely shifts t he origi n of energy, and has no effect on the relative weights of microstates, or the macroscopic properties. All the following models can be regarded as generalizations of the Ising model. 2. The O(n) model: Each lattice site is now occupied by an n -component unit vector, i.e n Si ≡ (Si 1, Si 2 , , Sin), with (Siα)2 = 1. (V.4) ··· α=1 A nearest–neighbor i nteraction can be written as H = −J S~i · S~j −~hˆ· S~i. (V.5) <i,j> i In fact, the most general interaction consistent with spherical symmetry is f(S~i S~j) for · an arbitrary function f. Similarly, the rotational symmetry can be broken by a number of “fields” such as i(~hp S~i)p. Specific cases are the Ising model (n = 1), the XY model · (n = 2) , and the Heisenberg model (n = 3). 3. The Potts model: Each site of the lattice is occupied by a q-valued spin Si ≡ 1, 2, , q.··· The interactions between the spins are described by the Hamiltonian H = −J δSi,Sj − hˆδSi,1. (V.6) <i,j> i The field h now breaks the permutation symmetry amongst the q-states. The Ising model is recovered for q = 2, since δσ,σ′ = (1 + σσ ′ )/2. The 3 state Potts model can for example describe the distortion of a cube along one of its faces. Potts models with q > 2 represent new universality classes not covered by the O(n) model. Actually, t he transitions for q ≥ 4 in d = 2, and q > 3 in d = 3 are discontinuous. 4. Spin s-models: The spin at each site takes the 2s + 1 values, si = −s, −s + 1, , +s.··· A general nearest–neighb or Hamiltonian is H = J1sisj + J2(sisj)2 + ··· + J2s(sisj)2s − hˆsi. (V.7) <i,j> i 76The Ising model corresponds to s = 1/ 2, while s = 1 is known as the Blume–Emery–Griffith (BEG) model. It describes a mixture of non-magnetic (s = 0), and magnetic (s = ±1) elements. This model exhibits a tricritical point separating continuous and discont inuous transitions. However, since the ordered phase breaks an up–down symmetry, the phase transition belongs to the Ising universality class for all values of s. Some of the computational tools employed in the study of discrete models are: 1. Position space renormalizations: These are implementations of Kadanoff’s RG scheme on lattice models. Some approximati on is usually necessary to keep the space of inter-actions tractable. Most such approximations are uncontrolled; a number of them wi ll be discussed here. 2. Series expansions: Low-temperature expansions start with the ordered ground state and examine the l owest energy excita tions around it. High temperatures expansions begin with the collection of non-interacting spins at infinite temperature and include the interactions between spins perturbatively. Critical behavior is then extracted from the singularities of such series. 3. Exact solutions can be obtained for a very limited subset of lat tice models. These include one