Classical & Quantum Field Theory of Matter
Classical & Quantum Field Theory of Matter
Statistical field theory extends the Ising model to continuous fields, revealing deep connections with quantum field theory. Path integrals unify equilibrium statistical mechanics with imaginary-time quantum mechanics.
Definition
Ginzburg-Landau theory: free energy functional \(F[\phi] = \int d^dr[a\phi^2 + b\phi^4 + c(\nabla\phi)^2]\). Minimizing gives the GL equation governing the order parameter field near a transition.
Key Result
Partition function as path integral: \(Z = \int\mathcal{D}\phi\, e^{-\beta F[\phi]}\). This is formally identical to a quantum field theory in \(d\)-dimensional Euclidean space, establishing the quantum-classical correspondence.
Example 1
Correlation function in Ginzburg-Landau: \(\langle\phi(\mathbf{r})\phi(0)\rangle \sim e^{-r/\xi}/r^{d-1}\) where \(\xi = \sqrt{c/|a|}\) is the correlation length. As \(T\to T_c\), \(\xi\to\infty\).
Example 2
Wilson's renormalization group in \(4-\epsilon\) dimensions gives critical exponents as power series in \(\epsilon\). Setting \(\epsilon=1\) recovers 3D Ising exponents to within 1% — a triumph of modern statistical mechanics.
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Practice
- Describe the Mermin-Wagner theorem and its consequences.
- What is the role of topological defects in the Berezinskii-Kosterlitz-Thouless transition?
- How does the quantum-classical mapping work for a 1D quantum spin chain?
- Explain the universality hypothesis using RG fixed points.
Show Answer Key
1. Mermin-Wagner theorem: in dimensions $d \leq 2$, continuous symmetries cannot be spontaneously broken at finite temperature for short-range interactions. Consequence: no long-range magnetic order in 2D Heisenberg or XY models at $T>0$. However, the 2D Ising model (discrete $\mathbb{Z}_2$ symmetry) can order. The 2D XY model has a BKT transition instead.
2. In the 2D XY model, topological defects are vortices (winding number $\pm 1$). Below $T_{BKT}$, vortex-antivortex pairs are bound (logarithmic interaction energy), and the system has quasi-long-range order (algebraic correlation decay). Above $T_{BKT}$, pairs unbind, destroying order. This is a topological phase transition — no symmetry breaking, no local order parameter.
3. A $d$-dimensional quantum system at $T=0$ maps to a $(d+1)$-dimensional classical system (the extra dimension is imaginary time $\tau \in [0,\beta\hbar]$). For a 1D quantum spin chain: the transfer matrix along the chain becomes the classical partition function of a 2D model. Quantum phase transitions ($T=0$, tuned by a parameter) correspond to classical thermal transitions in one higher dimension.
4. RG fixed points govern critical behavior. Systems flowing to the same fixed point share the same critical exponents (universality class). The fixed point depends only on symmetry of order parameter ($\mathbb{Z}_2$, $O(n)$, etc.), spatial dimension $d$, and range of interactions — not microscopic details. Example: liquid-gas and Ising ferromagnet transitions are in the same ($\mathbb{Z}_2$, $d=3$) universality class.