Topics#

Key Concepts#

• Wick rotation: \(t \to -\mathrm{i}\tau\) transforms oscillatory path integral into convergent Euclidean integral.

• Partition function: \(Z = \text{Tr}(\mathrm{e}^{-\beta H})\) as imaginary-time path integral with periodic boundary conditions.

• Ground state projection: Long imaginary-time evolution \(\mathrm{e}^{-\tau H}\) projects out excited states, isolating the ground state.

• Quantum-classical correspondence: \(d\)-dimensional quantum system at temperature \(T\) maps to \((d+1)\)-dimensional classical system.

• Instantons: Tunneling amplitude computed as integral over non-perturbative “bounce” solutions in imaginary time.

Learning Objectives#

• Perform Wick rotation to transform the real-time path integral into the imaginary-time (Euclidean) formulation.

• Compute partition functions and thermal expectation values using imaginary-time path integrals.

• Explain how long imaginary-time evolution projects onto the ground state of a system.

• Relate quantum systems in \(d\) dimensions to classical statistical mechanics in \(d+1\) dimensions.

• Connect imaginary-time bounce solutions (instantons) to tunneling rates in quantum mechanics.

Project: Quantum Phase Transitions via Imaginary-Time Path Integrals#

Objective: Study the transverse-field Ising model using the path integral formulation in imaginary time, implement Monte Carlo sampling of Euclidean worldlines, and explore quantum phase transitions as a function of field strength.

Background: The path integral in imaginary time provides a bridge between quantum mechanics and classical statistical mechanics, revealing that quantum phase transitions can be understood through a classical lens: mapping the (1+1)D quantum Ising model to a 2D classical Ising model with defects (instantons). This classical mapping is powerful and intuitive: quantum tunneling between ordered states becomes a classical defect (kink) in the imaginary-time worldline. Quantum phase transitions—abrupt changes in ground state properties at zero temperature—correspond to classical phase transitions of the effective classical system. This perspective is essential for understanding quantum criticality and has applications from condensed matter to quantum field theory.

Suggested Approach:

• Set up the transverse-field Ising Hamiltonian: \(\hat{H} = -J\sum_i \hat{\sigma}_i^z \hat{\sigma}_{i+1}^z - \Gamma \sum_i \hat{\sigma}_i^x\), where \(\Gamma\) is the transverse field and \(J\) is the interaction strength.

• Construct the path integral: for each quantum spin configuration, discretize imaginary time into slices and form a (1+1)D classical Ising model on the lattice.

• Implement a path integral Monte Carlo (or classical MC on the effective 2D system): sample worldlines weighted by the effective partition function.

• Vary \(\Gamma/J\) from \(0\) (ordered, all spins aligned in \(z\)-direction) to \(\infty\) (disordered, field dominates). Compute the ground state energy, magnetization \(\langle \hat{\sigma}^z \rangle\), and correlation length.

• Identify the quantum phase transition at \(\Gamma/J \sim 1\): analyze how order parameter and correlation length diverge; extract critical exponents and compare to known universality class (Ising universality in 1D quantum = Ising universality in 2D classical).

• Visualize worldlines: show how instantons (domain walls in imaginary time) proliferate as the transition is approached.

Expected Deliverable: A research report (5–10 pages) with code and visualizations. Include: (i) path integral formulation and classical mapping, (ii) Monte Carlo algorithm pseudocode, (iii) phase diagram and transition point, (iv) order parameter, correlation length, and critical exponents, (v) visualization of worldlines and instanton dynamics, (vi) physical interpretation: how do instantons signal the transition?, (vii) connection to classical phase transitions and universality.

Key References: S. Sachdev, Quantum Phase Transitions (Cambridge, 2nd ed., 2011); J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge, 1996); path integral Monte Carlo methods for quantum spin systems.