Path Integrals

• First formulated by R. P. Feynman

• Introduces concept of ”imaginary time”, τ = \frac{1}{k_{B}T}

The Density Matrix, ρ

• ”All static properties and, in principle, dynamic properties of a many-body system in thermal equilibrium are obtainable from the density matrix.”^{1}

• H = ∑_{i=1}^{N} −\frac{h̄^{2}}{2m}∇_{i}^{2} + ∑_{i<j} v(| r_{i} − r_{j} |)

• ρ(R, R′; β) = ⟨R | e^{−βH} | R′⟩

• e^{−βH} = (e^{− \frac{β}{m} H})^{m}

• ρ(R, R′; β) = ∫ ... ∫ ρ(R, R_{1}; \frac{β}{m})ρ(R_{1}, R_{2}; \frac{β}{m})...ρ(R_{m−1}, R_{m}; \frac{β}{m})ρ(R_{m}, R′; \frac{β}{m})dR_{1}dR_{2}...dR_{m}

• ρ(R, R′; β) = ⟨R_{0} | e^{− \frac{β}{m} H} | R_{1}⟩...⟨R_{m−1} | e^{− \frac{β}{m} H} | R_{m}⟩

1D. M. Ceperley and E. L. Pollock, in Monte Carlo Methods in Theoretical Physics, 35 (1992).

Figure 1: Sketch of a ”Ring Polymer”

Monte Carlo

• Based on random numbers

• New particle positions are proposed

• New positions can be chosen randomly or with some specific distribution, as long as each particle can move to any point in configuration space with a finite number of moves

• Probability of new positions are calculated in a systematic manner

• Random number drawn to determine if move occurs

• Process is repeated until relaxes

3 Figure 2: PIMC simulation

4