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Notes on Statistical Field Theory Lecture Notes on Statistical Field Theory Kevin Zhou [email protected] These notes cover statistical field theory and the renormalization group. The primary sources were: • Kardar, Statistical Physics of Fields. A concise and logically tight presentation of the subject, with good problems. Possibly a bit too terse unless paired with the 8.334 video lectures. • David Tong's Statistical Field Theory lecture notes. A readable, easygoing introduction covering the core material of Kardar's book, written to seamlessly pair with a standard course in quantum field theory. • Goldenfeld, Lectures on Phase Transitions and the Renormalization Group. Covers similar material to Kardar's book with a conversational tone, focusing on the conceptual basis for phase transitions and motivation for the renormalization group. The notes are structured around the MIT course based on Kardar's textbook, and were revised to include material from Part III Statistical Field Theory as lectured in 2017. Sections containing this additional material are marked with stars. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Phonons...........................................3 1.2 Phase Transitions......................................6 1.3 Critical Behavior......................................8 2 Landau Theory 12 2.1 Landau{Ginzburg Hamiltonian.............................. 12 2.2 Mean Field Theory..................................... 13 2.3 Symmetry Breaking.................................... 16 3 Fluctuations 19 3.1 Scattering and Fluctuations................................ 19 3.2 Position Space Fluctuations................................ 20 3.3 Saddle Point Fluctuations................................. 23 3.4 ∗ Path Integral Methods.................................. 24 4 The Scaling Hypothesis 29 4.1 The Homogeneity Assumption............................... 29 4.2 Correlation Lengths.................................... 30 4.3 Renormalization Group (Conceptual)........................... 32 4.4 Renormalization Group (Formal)............................. 34 4.5 The Gaussian Model.................................... 36 4.6 ∗ Appendix......................................... 38 5 Perturbative Renormalization Group 42 5.1 Expectation Values..................................... 42 5.2 Perturbative RG...................................... 44 5.3 Wilson{Fisher Fixed Point................................. 47 6 Lattice Models 51 6.1 Introduction......................................... 51 6.2 Ising Model in d = 1.................................... 52 6.3 Ising Model in d = 2.................................... 55 6.4 Series Expansions...................................... 59 6.5 Duality in the Ising Model................................. 61 6.6 Summing over Phantom Loops.............................. 64 6.7 Monte Carlo......................................... 67 7 Continuous Spins 70 7.1 The Nonlinear σ Model.................................. 70 7.2 ∗ The Nonlinear σ Model, Redux............................. 72 7.3 The XY Model....................................... 78 7.4 The Coulomb Gas..................................... 81 7.5 ∗ The Sine-Gordan Model................................. 83 3 1. Introduction 1 Introduction We've learned to use statistical mechanics to go between microscopic properties of matter (i.e. the degrees of freedom for each particle) and macroscopic thermodynamic variables, such as pressure and temperature. We were able to handle noninteracting systems exactly, and account for weak interactions using perturbation theory. In statistical field theory, we investigate situations where the perturbative approach breaks down, often at a phase transition. We would like to answer the following questions: 1. In the thermodynamic limit, how do distinct macroscopic phases emerge from microscopic dynamics? What are the thermodynamic variables that describe the macroscopic state of these phases, and what are their identifying signatures in the measurement of bulk response functions (e.g. heat capacity, susceptibility)? 2. What are the characteristic low energy excitations of the system? In many cases, such as phonons in solids or superfluid helium, they are collective modes which involve the motion of many microscopic degrees of freedom. These low-lying modes are easily excited by thermal fluctuations, and probed by scattering experiments. In general, a true ab initio calculation is intractable. However, in some cases, the description may be simplified by averaging over many particles in the thermodynamic limit. At long wavelengths, we may coarse-grain the system into a field (the `hydrodynamic limit'), and apply classical field theory techniques. 1.1 Phonons As a first example, we calculate the phonon contribution to the specific heat capacity of a solid using both approaches. First, consider an ab initio approach. The steps are as follows. • Starting with the Schrodinger equation, we perform approximations (i.e. density functional theory) to yield a many-body potential V (q1;:::; qN ). • We find the lattice ion positions at zero temperature by minimizing V . Typically, we find a Bravais lattice with ∗ ∗ q (l; m; n) = la + mb + nc = qr; r = (l; m; n): • We expand the potential to second order about this minimum. If the displacements are u(r), 1 X @2V ∆V = uα(r)uβ(r) 2 @qr,α@qr,β where α and β are spatial components. The full Hamiltonian for small deformations is found P 2 by adding pα(r) =2m where pα(r) is the conjugate momentum to uα(r). • By applying translational symmetry, 2 @ V 0 = Kαβ(r − r ) @qr,α@qr,β 4 1. Introduction the Fourier transform of the Hamiltonian becomes 2 X pα(k) H = + K (k)u (k)u (k)∗ m αβ α β k,α,β where the sum in k is over the Brillouin zone. • The matrix Kαβ(k) can now be diagonalized separately for each k, which can be done with the help of the crystal symmetry. This decomposes the Hamiltonian into 3N independent harmonic oscillators. • Quantizing each oscillator, we find X 1 H = ! (k) n (k) + ~ α α 2 k,α where the !α(k) are the three normal mode frequencies for each k. • Therefore the average energy at a temperature T is given by X 1 1 E(T ) = ~!α(k) hnα(k)i + ; hnα(k)i = 2 exp( !α=kT ) − 1 k,α ~ where we applied the Bose{Einstein distribution. We see that E(T ) has complex behavior, dependent on microscopic details through the !α. However, it is possible to extract universal behavior in the long-wavelength limit. To see this, we switch to a chain of particles in one dimension. • The most general quadratic translationally invariant potential energy for deformations is K1 X K2 X V = (u − u )2 + (u − u )2 + ··· : 2 n+1 n 2 n+2 n n n • The normal modes are just the Fourier components, with Z π=a −ikna X ikna un = d¯k e u(k); u(k) = a e un −π=a n where a is the equilibrium separation between the particles. • Plugging in the Fourier transform, doing the sum over n to get a delta function, and using ∗ u(−k) = u (k), which follows from un being real, we have Z π=a 1 2 V / d¯k (K1(1 − cos ka) + K2(1 − cos 2ka) + ··· ) ju(k)j : a −π=a • We thus find the frequency of the normal modes to be p !(k) / K1(1 − cos ka) + K2(1 − cos 2ka) + ···: In the limit k ! 0, the dispersion relation becomes !(k) = vjkj. While further neighbor interactions change the value of v, they don't change this fundamental feature. 5 1. Introduction • At low temperatures, only this part of the excitation spectrum is important, so Z 1 vjkj π2 E(T ) ≈ Na d¯k ~ = Na (kT )2: −∞ exp(~vjkj=kT ) − 1 6~v Therefore the heat capacity is linear in T . This is a universal property, independent of the detailed interatomic interactions. We can arrive at this conclusion without ever looking at microscopic details, using a phenomenological (field) approach. Since we only want to look at long-wavelength excitations, we ignore variations in the field on the scale of a, allowing us to replace the displacements un with a displacement field u(x). The kinetic energy term becomes ρ R dx u_(x)2=2, where ρ = m=a is the density. We now use general principles to constrain the form of the potential functional V [u]. • Locality: in most situations, interactions between particles are short-ranged, so we may write Z V [u] = dx Φ(u; du=dx; : : :) where Φ is a local potential energy density. • The above form isn't strong enough, since if we allow infinitely many derivatives, we can construct nonlocal interactions like u(x)u(x + a) by Taylor series. In general, higher derivative terms are `less local' (in a discrete lattice setup, the dnu=dxn takes n+1 lattice sites to compute), so locality really implies that higher derivative terms are less significant. • Translational symmetry: we have Φ[u(x) + c] = Φ[u(x)], so Φ cannot depend directly on u. • Stability: since the fluctuations are about an equilibrium position, there can be no linear term in u or its derivatives. Moreover, the quadratic part of V [u] must be positive definite. As a result, the most general allowed potential can be expanded as a power series. Each term is n1 2 n2 of the form (@xu) (@xu) :::. By locality and translational symmetry, the Fourier transform has `conservation of momentum', with derivatives becoming factors of k, e.g. Z Z 2 2 2 dx (@xu) (@xu) ! d¯k1d¯k2 k1k2(k1 + k2) u(k1)u(k2)u(−k1 − k2): In the limit k ! 0, higher order derivative terms become unimportant. Moreover, for small 2 deformations we may neglect terms beyond second order in u. Then we are left with only the (@xu) term, giving ρ Z H = dx (@ u)2 + v2(@ u)2 : 2 t x This is the Hamiltonian of a one-dimensional elastic string; its equation of motion is the wave equation. Thus ! = vjkj for small k, as seen earlier. Note. In higher dimensions, the result is slightly more complicated. In Fourier space, rotational invariance now allows dependence on k2, u2, and k · u. The final quantity allows us to distinguish longitudinal modes and transverse modes, which may have different velocities. In d dimensions, we find the universal result E(T ) / T d+1, giving a heat capacity proportional to T d.
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