Topics#

Key Concepts#

• Boltzmann distribution from max entropy: \(p_n = \mathrm{e}^{-\beta E_n}/Z\) with \(\beta = 1/(k_B T)\).

• Partition function: \(Z(\beta) = \sum_n \mathrm{e}^{-\beta E_n} = \mathrm{Tr}\,\mathrm{e}^{-\beta\hat{H}}\) — generating function of equilibrium statistical mechanics.

• Wick rotation: \(t \to -\mathrm{i}\tau\); action \(S \to +\mathrm{i}S_E\); weight \(\mathrm{e}^{\mathrm{i}S/\hbar}\to\mathrm{e}^{-S_E/\hbar}\).

• Euclidean action: \(S_E = \int(\frac{m}{2}\,x'(\tau)^2 + V)\,\mathrm{d}\tau\) — both terms positive.

• Thermal circle: \(Z(\beta) = \oint_{x(\hbar\beta)=x(0)}\mathcal{D}[x]\,\mathrm{e}^{-S_E/\hbar}\); the trace forces imaginary time to be periodic with period \(\hbar\beta\), set by temperature.

• Quantum-classical duality (sketch): \(d\)-dim QM at temperature \(T\) ⇔ \((d{+}1)\)-dim classical stat mech with one periodic direction of length \(\hbar\beta\).

• Schwarzschild metric: \(\mathrm{d}s^2 = -f c^2\,\mathrm{d}t^2 + f^{-1}\,\mathrm{d}r^2 + r^2\,\mathrm{d}\Omega^2\) with \(f = 1 - r_s/r\), \(r_s = 2GM/c^2\).

• No conical singularity: smoothness of the Euclidean horizon forces \(\tau\) to have period \(\Delta\tau = 8\pi GM/c^3\).

• Hawking temperature: \(T_H = \hbar c^3/(8\pi G M k_B)\); Bekenstein–Hawking entropy \(S_{\mathrm{BH}} = k_B c^3 A/(4 G\hbar)\).

• Inverted potential and instantons: Euclidean Newton’s law \(m\,x'' = +V'(x)\); finite-action kink solutions cross barriers and produce non-perturbative splittings \(\Delta \propto \mathrm{e}^{-S_{\mathrm{inst}}/\hbar}\).

Learning Objectives#

• Derive the Boltzmann distribution from the maximum-entropy principle and define the partition function in both spectral and trace forms.

• Explain how the identification \(t = -\mathrm{i}\hbar\beta\) motivates the Wick rotation; perform it inside the path integral and obtain the Euclidean action with the correct signs.

• State and use the central result \(Z(\beta) = \oint\mathcal{D}[x]\,\mathrm{e}^{-S_E/\hbar}\) over closed loops of period \(\hbar\beta\) in imaginary time.

• Wick-rotate the Schwarzschild metric, switch to proper-distance coordinates near the horizon, and recognize the resulting flat 2D polar form.

• Use the no-conical-singularity condition to derive the period of imaginary time and read off the Hawking temperature; integrate the first law to obtain the Bekenstein–Hawking entropy.

• Construct the instanton solution for a double-well potential, compute its Euclidean action, and use the dilute-instanton-gas argument to obtain the energy splitting.

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.