Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2013 / Paper 1 / 2

A periodic harmonic chain with $N$ masses $m_0$, spacing $a$ and nearest-neighbour spring constant $K$ hasHamilton's equations give $m_0{\ddot{u}}_j=K(u_{j+1}+u_{j-1}-2u_j)$. The discrete Fourier transform $u_j=N^{-1/2}{\sum }_k{Q}_k{e}^{ikja}$, $p_j=N^{-1/2}{\sum }_k{P}_k{e}^{ikja}$, with $k=2\pi l/(Na)$ in one Brillouin zone, diagonalizes the quadratic form:The harmonic-chain normal-mode reality condition means $Q_k^{†}=Q_{-k}$, not that $Q_k$ and $Q_{-k}$ are independent real coordinates. For canonical quantization, impose $[u_j,p_l]=i\hslash {\delta }_{jl}$, equivalently $[Q_k,P_{k^{\prime }}]=i\hslash {\delta }_{k,-k^{\prime }}$. For $k\ne 0$ setThen the quantum harmonic chain is a set of independent phonon oscillators:The centre-of-mass zero mode of a harmonic chain is the free coordinate $X=N^{-1}{\sum }_j{u}_j$, with $[X,P_{\mathrm{tot}}]=i\hslash$; it cannot be normalized by dividing by $\sqrt{{\omega }_0}$. Retaining it giveswith the corresponding momentum expansion and uniform term $P_{\mathrm{tot}}/N$. The nonzero modes alone give the site commutator $i\hslash ({\delta }_{jl}-1/N)$; the uniform term supplies the missing $1/N$. Each oscillator occupation is a nonnegative integer, and adding one phonon costs $\hslash {\omega }_k$ and crystal momentum $\hslash k$ modulo a reciprocal-lattice vector. ^[ref]

In the continuum limit of a harmonic chain, $\rho =m_0/a$, $Y=Ka$ andThe wave equation and linear low-$k$ dispersion relation follow. The lattice gives an ultraviolet cutoff; the acoustic approximation does not reproduce the zone-edge flattening.

With the centre of mass fixed, the oscillator partition function and Bose-Einstein distribution areFor a long chain, above its finite-size level spacing and below the zone-edge scale, integrating the two acoustic propagation directions gives $C/(Na)=\pi k_{\mathrm{B}}^{2}T/(3\hslash c)$. At high temperature each retained oscillator instead contributes $k_{\mathrm{B}}$ to the heat capacity.

For $H={\hat{p}}^2/(2m)+V(\hat{x})$, divide elapsed time $t$ into $J$ slices $\Delta t=t/J$ and apply the Trotter product formula. Inserting complete position and momentum bases gives the regulated phase-space path integralThe endpoint values are $x_0=x_i$, $x_J=x_f$. Each Gaussian momentum integration in a phase-space path integral can be performed explicitly, leaving the configuration-space path integralIts measure is the time-sliced limit of $(m/(2\pi i\hslash \Delta t){)}^{J/2}{\prod }_{j=1}^{J-1}d{x}_j$, with the usual infinitesimal convergence prescription. The integral sums amplitudes, not positive classical probabilities. Operator ordering remains part of the time-slice definition. Expanding a short-time kernel recovers the Schrödinger equation; composing kernels reproduces the time-evolution operator.

In the classical limit, rapid phase cancellation suppresses paths away from stationary action. Varying $S$ with fixed endpoints gives $m\ddot{x}=-V^{\prime }(x)$. The semiclassical result sums the classical-path phases multiplied by fluctuation functional determinants; for a quadratic Hamiltonian the fluctuation integral is Gaussian and this expansion is exact.

For quantum statistical mechanics, continue $t=-i\tau$ by a Wick rotation. The Euclidean action and thermal trace giveThe trace closes position-space paths. In a many-particle trace one also sums over endpoint permutations with the appropriate exchange signs. In a coherent state field formulation, bosonic fields are periodic and fermionic coherent states instead give antiperiodic Grassmann fields, hence bosonic and fermionic Matsubara frequencies $2\pi n/(\beta \hslash )$ and $(2n+1)\pi /(\beta \hslash )$. ^[ref]

The quantum-to-classical imaginary-time mapping views a local $d$-dimensional quantum problem as a classical-looking field theory in $d+1$ dimensions, with imaginary-time extent $\beta \hslash$. A genuine classical probability interpretation additionally requires a nonnegative weight; fermion signs and Berry terms can obstruct it.

For a smooth potential in the high-temperature or small-$\hslash$ limit, imaginary-time paths fluctuate only over the thermal de Broglie wavelength ${\lambda }_T=\sqrt{2\pi \beta {\hslash }^2/m}$. If this is short compared with the potential's variation scale, the diagonal free kernel givesThis classical phase-space limit of the thermal kernel is the classical phase space partition function for one degree of freedom. A dilute identical-particle gas also needs its $1/N!$ factor; when exchange is appreciable, classical statistics fails. As a check, a single quantum harmonic oscillator gives $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}\to 1/(\beta \hslash \omega )$, precisely the classical phase-space integral.

For conventional superconductivity, an attractive interaction pairs opposite-spin fermions. A complex pairing Hubbard–Stratonovich transformation introduces a Cooper pair field $\Delta$. The fermionic fields then appear quadratically, giving the pair-field effective action of a superconductorThe trace convention counts each independent spin-pair block once. The uniform saddle is the BCS gap equation; the fermionic excitations have $E_k=\sqrt{{\xi }_k^2+|\Delta {|}^2}$ in the fully gapped case. A slow-field expansion near the transition yields Ginzburg-Landau theory of superconductivity. Normalize its pair field $\Psi$ so $|\Psi {|}^2$ is an effective pair density, with pair mass $m_{\ast }$ and signed charge $q=-2e$:For $a<0$, the uniform minimum has ${\rho }_0=|\Psi {|}^2=-a/b$, with arbitrary constant phase. Writing $\Psi =(\sqrt{{\rho }_0}+h)e^{i\theta }$ separates the amplitude mode $h$ from the phase. The local potential has curvature term $2|a|h^2$ and coherence scale $\xi =\hslash /\sqrt{2m_{\ast }|a|}$. A neutral condensate has a gapless phase Goldstone mode when suitable dynamics is included.

The charged theory is invariant underThe gauge-invariant condensate momentum is $\hslash \nabla \theta -q\mathbf{A}$. Only $\hslash \nabla \theta -q\mathbf{A}$ and $\hslash \dot{\theta }+q\varphi$ enter physical phase dynamics. At nearly constant amplitude, varying $F$ givesFor a smooth phase, taking the curl and using Ampère's law gives ${\nabla }^2\mathbf{B}=\mathbf{B}/{\lambda }^2$: the Meissner effect screens a static bulk magnetic field over the London penetration depth $\lambda$. In a gauge with $\theta =0$, the phase stiffness is a quadratic term in $\mathbf{A}$, the superconducting version of the Higgs mechanism.

To identify the dynamical mode, a simple lossless long-wave superconducting phase-density action isHere ${\chi }_p$ is the pair compressibility coefficient. Eliminating the longitudinal electric potential givesThus the neutral sound mode becomes a gapped longitudinal plasma oscillation; transverse electromagnetic waves acquire ${\omega }^2=c_{\mathrm{vac}}^2{k}^2+c_{\mathrm{vac}}^2/{\lambda }^2$ in this simple bulk model. This redistribution of the phase degree of freedom into electromagnetic collective modes is the Anderson-Higgs mechanism in a superconductor. Local gauge symmetry is a redundancy, not an independently observable symmetry whose breaking creates another massless particle. ^[ref]

The static functional alone does not determine a frequency or damping rate. The displayed mode dispersions require the added dynamical assumptions; real screening, normal carriers, boundaries and dissipation modify them. The amplitude fluctuation is distinct from the phase mode and is not removed by the gauge choice.