tags: - colorclass/phase transitions -⇒tell me more about Coarse-Graining in RG theory. I’m looking for technical details. illustrate with a demonstrative example, heavy on the technical details and math
Coarse-graining is a fundamental step in Renormalization Group theory that involves averaging or integrating out the microscopic details of a system to focus on its macroscopic or large-scale behavior. This process is essential for understanding how physical properties and the dynamics of a system change as one observes it at different scales. The coarse-graining approach allows us to study the critical behavior of systems, phase transitions, and the scaling laws that emerge near critical points. Here, I’ll describe the technical process of coarse-graining using the Ising model as an example, a classic system in statistical physics used to study phase transitions.
The Ising Model
The Ising model consists of spins ((s_i)) that can take values of (\pm 1), arranged on a lattice. Each spin interacts with its nearest neighbors, and the model is used to study magnetism and phase transitions. The Hamiltonian (energy function) for the Ising model in the absence of an external magnetic field is given by:
[ H = -J \sum_{\langle i,j \rangle} s_i s_j ]
where (J) is the coupling constant representing the interaction strength between neighboring spins, and the sum is over nearest-neighbor pairs (\langle i,j \rangle).
Coarse-Graining Process
1. Partition the Lattice: Divide the lattice into blocks of spins. The size of the blocks determines the scale of the coarse-graining. For simplicity, consider blocks that contain (b^d) spins each, where (d) is the dimensionality of the lattice, and (b) is the linear size of the block.
2. Define Block Spins: For each block, define a new “block spin” ((\Sigma)) that represents the average state of the spins within the block. There are different schemes to define block spins, but a common choice is the majority rule for simplicity, where the block spin takes the value of the majority of spins within the block. Another method is to take the sum or average of spins within the block, which is more common in theoretical analyses.
3. Effective Hamiltonian: The goal is to write down an effective Hamiltonian ((H’)) in terms of the block spins ((\Sigma)) that captures the essential physics of the original system at the new scale. This involves integrating out the degrees of freedom within each block, which is a non-trivial task generally requiring approximations or numerical techniques.
Mathematical Formulation
Let’s consider a simplification where each block spin (\Sigma) is the sum of spins within a block. The effective Hamiltonian aims to preserve the partition function’s form, ensuring thermodynamic properties remain consistent under the transformation. Mathematically, we aim to find (H’({\Sigma})) such that:
[ Z = \sum_{{s}} e^{-\beta H({s})} = \sum_{{\Sigma}} e^{-\beta H’({\Sigma})} ]
where (\beta = 1/(k_B T)), (k_B) is the Boltzmann constant, and (T) is the temperature.
Example: Simple Coarse-Graining in 1D Ising Model
Consider a 1D Ising model for simplicity, and let’s coarse-grain by grouping every two spins ((b=2)). If we simply sum the spins in each block, a block spin (\Sigma) could take values (-2, 0, +2), indicating the majority rule or the sum within each block.
The challenge lies in accurately determining the effective interactions between block spins (J’) to ensure the critical behavior of the model is preserved under coarse-graining. This usually involves approximations since exact solutions are often not feasible except in the simplest cases.
The Kadanoff block spin transformation is a concrete example where spins are grouped, and an effective Hamiltonian is derived, leading to a new expression for (J’) that depends on (J) and the coarse-graining scale. The analysis of how (J’) changes with scale (i.e., as you group more spins together) is what leads to the RG flow equations.
This simplified demonstration omits many technical details, especially in higher dimensions where the interactions and configurations become more complex. However, it captures the essence of coarse-graining in RG theory: reducing the system to a simpler, effective model that retains the critical properties of the original system, allowing us to study how these properties evolve across scales.