Unit 10: Quantum Statistics & Condensed Matter Thermodynamics

In quantum mechanics, all particles are classified into two fundamental families based on their intrinsic spin angular momentum \(S\) (the Spin-Statistics Theorem of Wolfgang Pauli):

The Two Quantum Families

1. Bosons (Integer Spin: \(S = 0, 1, 2, \dots\)):

• Governed by Bose-Einstein (BE) Statistics.
• Examples: Photons (\(S = 1\)), Gluons (\(S = 1\)), \(^4\text{He}\) atoms (\(S = 0\)), \(^{87}\text{Rb}\) atoms (\(S = 1\)), Phonons, Cooper pairs.
• Symmetric many-body wavefunctions: \(\hat{P}_{i j} \Psi = +\Psi\).
• No restriction on occupation numbers: Any number of bosons can occupy the exact same single-particle quantum state (\(n_k = 0, 1, 2, 3, \dots, \infty\)).

2. Fermions (Half-Integer Spin: \(S = 1/2, 3/2, 5/2, \dots\)):

• Governed by Fermi-Dirac (FD) Statistics.
• Examples: Electrons (\(S = 1/2\)), Protons (\(S = 1/2\)), Neutrons (\(S = 1/2\)), \(^3\text{He}\) atoms (\(S = 1/2\)), Quarks.
• Antisymmetric many-body wavefunctions: \(\hat{P}_{i j} \Psi = -\Psi\).
• Pauli Exclusion Principle: The occupation number of any single-particle quantum state is strictly restricted to 0 or 1:

Grand Partition Function Formulation

Using the grand canonical ensemble where the grand partition function factors into independent single-state terms:

• For Fermions (\(n_k \in \{0, 1\}\)):

• For Bosons (\(n_k \in \{0, 1, 2, \dots\}\)):

This simple structural distinction generates the contrasting physics of matter: fermions create stable atomic shells, rigid matter, and the periodic table, while bosons mediate forces, create lasers, and undergo Bose-Einstein condensation.

The average occupation number of a single-particle state \(k\) of energy \(\varepsilon_k\) is:

Derivation of the Bose-Einstein Distribution

Using \(\Xi_k^{\text{BE}} = [1 - e^{-\beta (\varepsilon_k - \mu)}]^{-1}\):

Differentiating:

where \(\beta = \frac{1}{k_B T}\).

Constraint on Chemical Potential

For the geometric series to converge and for \(\langle n_k \rangle\) to remain positive for all states:

Setting the ground-state energy to zero (\(\varepsilon_0 = 0\)):

The chemical potential of an ideal Bose gas can never be positive! As temperature drops, \(\mu\) approaches zero from below: \(\mu \rightarrow 0^-\).

Special Case: Photons and Phonons (\(\mu = 0\))

For particles whose total number is not conserved (photons in cavity radiation, phonons in a lattice), particles can be freely created and destroyed. The thermodynamic equilibrium condition \(\left(\frac{\partial A}{\partial N}\right)_{T, V} = 0\) requires:

The distribution simplifies to the celebrated Planck Distribution Law:

High-Temperature Classical Limit

When \(e^{\beta (\varepsilon_k - \mu)} \gg 1\) (low density and high temperature, \(\Lambda \ll d\)):

recovering the Maxwell-Boltzmann distribution.

For fermions, the grand partition function is \(\Xi_k^{\text{FD}} = 1 + e^{-\beta (\varepsilon_k - \mu)}\).

Derivation of the Fermi-Dirac Distribution

Differentiating:

Notice that for any energy \(\varepsilon_k\) and any temperature \(T\):

The Pauli exclusion principle is automatically satisfied for all conditions.

The Fermi Energy \(E_F\) at Absolute Zero (\(T = 0\text{ K}\))

At \(T = 0\), \(\beta = \frac{1}{k_B T} \rightarrow \infty\). The chemical potential at \(T = 0\) defines the Fermi Energy:

Evaluate \(\langle n(\varepsilon) \rangle\) as \(T \rightarrow 0\):

• For \(\varepsilon < E_F\): \(\varepsilon - E_F < 0 \implies \beta (\varepsilon - E_F) \rightarrow -\infty \implies e^{-\infty} = 0 \implies \langle n \rangle = \frac{1}{0 + 1} = 1\)
• For \(\varepsilon > E_F\): \(\varepsilon - E_F > 0 \implies \beta (\varepsilon - E_F) \rightarrow +\infty \implies e^{+\infty} = \infty \implies \langle n \rangle = \frac{1}{\infty + 1} = 0\)

At absolute zero, the Fermi-Dirac distribution is a sharp step function:

All quantum states with \(\varepsilon \le E_F\) are 100% filled, while all states with \(\varepsilon > E_F\) are completely empty. The boundary in momentum space separating filled and empty states is the Fermi Surface.

Thermal Broadening at Finite Temperature (\(T > 0\))

When \(T > 0\):

• At \(\varepsilon = \mu\): \(\langle n(\mu) \rangle = \frac{1}{e^0 + 1} = \frac{1}{2}\). The chemical potential is the energy level with exactly 50% occupation probability.
• The step function rounds off over a narrow thermal energy window of width \(\sim 4 k_B T\) centered around \(\mu\).

Because typical Fermi energies in metals are \(E_F \sim 5 - 10\text{ eV}\) (\(T_F = E_F / k_B \sim 50,000 - 100,000\text{ K}\)), room temperature (\(k_B T \approx 0.026\text{ eV} \ll E_F\)) represents an extreme degenerate quantum limit.

In the Sommerfeld free electron model, conduction electrons in a metal are treated as a non-interacting gas of fermions confined within volume \(V\).

Density of States for Free Electrons in 3D

For a free particle in a 3D box, \(\varepsilon = \frac{\hbar^2 k^2}{2 m_e}\). Accounting for the electron spin degeneracy \(g_s = 2s + 1 = 2\), the number of quantum states in spherical shell \(k\) to \(k + dk\) is:

Converting from wavevector \(k = \frac{\sqrt{2 m_e \varepsilon}}{\hbar}\) to energy \(\varepsilon\):

Calculation of Fermi Energy \(E_F\)

At \(T = 0\), all \(N\) electrons occupy states up to \(E_F\):

Solving for \(E_F\):

where \(n = N/V\) is the conduction electron density. The Fermi wavevector is \(k_F = (3\pi^2 n)^{1/3}\), and the Fermi velocity is \(v_F = \frac{\hbar k_F}{m_e} \sim 10^6\text{ m/s}\).

Electronic Heat Capacity of Metals

Classically, equipartition predicted that conduction electrons should contribute \(\frac{3}{2} R\) to heat capacity, which was contradicted by experiment (\(C_V \approx 3 R\) for the whole metal, with virtually zero electronic contribution at room temperature). Resolution via Fermi Statistics: Only electrons within \(\sim k_B T\) of the Fermi surface can be thermally excited to empty states above \(E_F\). The fraction of active electrons is roughly \(\frac{k_B T}{E_F} = \frac{T}{T_F} \sim \frac{300}{50,000} \approx 0.6\%\). The thermal energy is:

Differentiating yields the linear electronic heat capacity:

At room temperature, the electronic heat capacity is suppressed by two orders of magnitude, beautifully explaining the experimental puzzle.

Consider an ideal gas of \(N\) non-interacting bosons of mass \(m\) in volume \(V\). The total number of particles is:

where \(N_0\) is the population of the \(\varepsilon = 0\) ground state, and the 3D density of states is \(g(\varepsilon) = \frac{2\pi V}{h^3} (2m)^{3/2} \varepsilon^{1/2}\).

The Critical Temperature \(T_c\)

Because \(\mu \le 0\), the integral over excited states reaches its maximum possible value when \(\mu = 0\):

Substituting \(x = \beta \varepsilon\):

where \(\zeta(3/2) = \sum_{k=1}^\infty k^{-3/2} \approx 2.612\) is the Riemann zeta function. If \(N > N_{\text{excited, max}}(T)\), the excited states cannot accommodate all particles! The critical temperature \(T_c\) is defined when excited states can just barely hold all \(N\) particles:

Solving for \(T_c\):

The Condensate Fraction Below \(T_c\)

For \(T < T_c\), the excess particles are forced into the single zero-momentum ground state \(\varepsilon = 0\):

• At \(T = T_c\): \(N_0 = 0\) (Condensation begins).
• At \(T \rightarrow 0\): \(N_0 \rightarrow N\) (100% of particles occupy the single macroscopic quantum ground state).

A macroscopic fraction of particles condenses into an identical spatial wavefunction, creating a macroscopic quantum state characterized by off-diagonal long-range order, superfluidity, and phase coherence (demonstrated experimentally in 1995 by Eric Cornell, Carl Wieman, and Wolfgang Ketterle with rubidium and sodium vapors).

In 1819, Pierre Dulong and Alexis Petit discovered empirically that all elemental solids exhibit the same molar heat capacity at room temperature:

By the late 1800s, low-temperature cryogenic measurements revealed that \(C_V(T)\) drops sharply toward zero as \(T \rightarrow 0\), completely violating classical equipartition.

The Einstein Model (1907)

Albert Einstein resolved this crisis by proposing that a crystalline solid of \(N\) atoms can be modeled as \(3N\) independent quantum harmonic oscillators, all vibrating at a single identical frequency \(\nu_E\). The total internal energy is:

Defining the Einstein Temperature \(\theta_E = \frac{h \nu_E}{k_B}\):

High and Low Temperature Limits

1. High-Temperature Limit (\(T \gg \theta_E\)):

Expanding \(e^{\theta_E/T} \approx 1 + \theta_E/T\):

recovering the classical Dulong-Petit law.

2. Low-Temperature Limit (\(T \ll \theta_E\)):

For \(T \rightarrow 0\), \(e^{\theta_E/T} \gg 1\):

Einstein's model correctly predicted that heat capacity must drop to zero as \(T \rightarrow 0\). However, experimental measurements revealed that \(C_V\) approaches zero as \(T^3\), whereas Einstein's formula decays exponentially (\(e^{-\theta_E/T}\)). This discrepancy occurs because real lattice atoms do not vibrate independently at a single frequency, but rather via collective coupled acoustic waves.

In 1912, Peter Debye replaced Einstein's independent oscillator assumption with a continuous elastic continuum of coupled vibrational waves (acoustic phonons).

Phonon Dispersion and Density of States

In an isotropic elastic solid, acoustic waves follow the linear dispersion relation \(\omega = v_s k\), where \(v_s\) is the speed of sound. Phonons possess three polarization modes (1 longitudinal, 2 transverse):

The density of vibrational modes in frequency space is:

Because an \(N\)-atom lattice possesses exactly \(3N\) vibrational degrees of freedom, the spectrum is cut off at a maximum Debye Cutoff Frequency \(\omega_D\):

Defining the Debye Temperature \(\theta_D = \frac{\hbar \omega_D}{k_B}\):

Total Internal Energy and Debye Heat Capacity

The total lattice vibrational energy is:

Differentiating with respect to \(T\) gives the Debye Heat Capacity:

The Celebrated Debye \(T^3\) Law

At low temperatures (\(T \ll \theta_D\)), the upper integration limit \(\theta_D / T \rightarrow \infty\). The definite integral evaluates to:

Substituting into \(C_V\):

The Debye \(T^3\) law reproduces experimental heat capacity measurements for all non-magnetic insulating solids at low temperatures with flawless mathematical precision.

In 1911, Heike Kamerlingh Onnes discovered that mercury cooled below \(T_c = 4.2\text{ K}\) loses all electrical resistance: the Superconducting Transition.

Fundamental Phenomonology

Superconductivity is not merely perfect electrical conductivity; it is an entirely distinct thermodynamic state of matter characterized by:

1. Zero Electrical Resistance: Direct current flows indefinitely without Joule dissipation (\(\rho = 0\)).

2. The Meissner-Ochsenfeld Effect: Complete expulsion of magnetic flux from the interior of the superconductor (\(\mathbf{B} = 0\)), establishing that superconductors are perfect diamagnets (\(\chi = -1\)).

3. Critical Magnetic Field \(H_c(T)\): Superconductivity is destroyed when an external magnetic field exceeds a critical value:

4. Second-Order Phase Transition: At \(T_c\) in zero magnetic field, there is no latent heat, but there is a sharp discontinuous jump in the electronic heat capacity:

The Microscopic BCS Theory (1957)

John Bardeen, Leon Cooper, and J. Robert Schrieffer formulated the microscopic theory of conventional superconductivity:

1. Cooper Pairing: A moving electron polarizes the surrounding positively charged ionic lattice, creating a local excess of positive charge. A second electron is attracted to this phonon-induced polarization cloud. This effective electron-phonon-electron attraction overcomes direct Coulomb repulsion, binding two electrons of opposite momenta and spins into a Cooper Pair:

2. Bose-Einstein Condensation of Cooper Pairs:

Because each Cooper pair consists of two fermions (\(s = 1/2\)), the composite pair has integer spin (\(S = 0\)) and behaves as an effective boson. Below \(T_c\), Cooper pairs condense into a single macroscopic quantum coherent state.

3. The Superconducting Energy Gap \(\Delta(T)\):

An energy gap \(2\Delta\) opens at the Fermi level, separating the paired condensate from single-particle (quasiparticle) excitations. At \(T = 0\text{ K}\), the BCS universal ratio is:

Because an energy of at least \(2\Delta\) is required to break a pair and scatter an electron, electrons move through the lattice without scattering, producing zero electrical resistance.

## Advanced Mathematical Supplement: Bogoliubov Canonical Transformations

Bogoliubov Canonical Transformations & The Landau Criterion for Superfluidity

To diagonalize interacting many-body Hamiltonians for weakly interacting Bose gases and BCS superconductors, Nikolay Bogoliubov introduced unitary transformations mixing creation and annihilation operators.

The Bogoliubov Transformation for Bosons

Consider a weakly interacting Bose gas with macroscopic condensate \(N_0 \approx N\). Replacing zero-momentum operators with numbers \(a_0, a_0^\dagger \rightarrow \sqrt{N_0}\), the effective Hamiltonian for non-zero momentum excitations \(\mathbf{k} \ne 0\) is quadratic:

We define new quasiparticle operators \(b_\mathbf{k}, b_\mathbf{k}^\dagger\) via the Bogoliubov Canonical Transformation:

Preserving bosonic commutation relations \([b_\mathbf{k}, b_{\mathbf{k}'}^\dagger] = \delta_{\mathbf{k} \mathbf{k}'}\) requires:

Choosing \(u_k, v_k\) to eliminate the off-diagonal pairing terms diagonalizes the Hamiltonian:

where the Bogoliubov Quasiparticle Dispersion Relation is:

where \(c_s = \sqrt{n_0 V_0 / m}\) is the speed of sound.

• At low momentum (\(k \rightarrow 0\)): \(\varepsilon(k) \approx \hbar c_s k\) (linear acoustic phonon dispersion).
• At high momentum (\(k \rightarrow \infty\)): \(\varepsilon(k) \approx \frac{\hbar^2 k^2}{2m} + n_0 V_0\) (free particle parabolic dispersion).

The Landau Criterion for Superfluidity

Lev Landau analyzed an object moving with velocity \(\mathbf{v}\) through a fluid. Creating an excitation of momentum \(\hbar \mathbf{k}\) and energy \(\varepsilon(k)\) is kinematically forbidden unless:

• For an ideal non-interacting Bose gas: \(\varepsilon(k) = \frac{\hbar^2 k^2}{2m} \implies \frac{\varepsilon(k)}{\hbar k} = \frac{\hbar k}{2m} \rightarrow 0\) as \(k \rightarrow 0\). The critical velocity is \(v_c = 0\); an ideal Bose gas is NOT superfluid!
• For an interacting Bose gas with Bogoliubov dispersion:

Because \(v_c = c_s > 0\), particles moving slower than the speed of sound cannot dissipate energy into the fluid, giving rise to frictionless macroscopic superfluidity.

## Research Monograph: Topological Quantum States, Anyons & Majorana Modes

Beyond the Landau Symmetry-Breaking Paradigm

For most of the twentieth century, all phase transitions were classified by Lev Landau's symmetry-breaking theory (e.g., liquid-to-solid breaks continuous translation symmetry, ferromagnetism breaks rotational spin symmetry). In 1980, Klaus von Klitzing discovered the Integer Quantum Hall Effect (IQHE): 2D electrons in a strong magnetic field at low temperature exhibit Hall conductance quantized to integer multiples of \(e^2/h\) with accuracy of 1 part in a billion:

Remarkably, the Hall plateaus occur without any broken spatial or gauge symmetry.

The TKNN Invariant and Chern Numbers

David Thouless, Mahito Kohmoto, M. Peter Nightingale, and Marcel den Nijs (TKNN) proved that the quantization integer \(\nu\) is a topological invariant—the First Chern Number of the Berry curvature integrated over the 2D magnetic Brillouin zone:

Because an integer cannot change continuously under smooth deformations, the Hall conductance is topologically protected against arbitrary non-magnetic impurities, disorder, and sample geometries!

Anyons and Fractional Statistics in 2D

In three dimensions, particles are strictly bosons or fermions because swapping two particles twice is topologically equivalent to looping one particle around the other, which can be continuously shrunk to zero. In two dimensions, particle trajectories form braids in \((2+1)\)-dimensional spacetime. Exchanging two particles can yield an arbitrary phase:

Particles with \(\theta \ne 0, \pi\) are Anyons. In the Fractional Quantum Hall Effect (\(\nu = 1/3\)), excitations carry fractional charge \(e^* = e/3\) and fractional exchange statistics \(\theta = \pi/3\).

Majorana Zero Modes & Fault-Tolerant Quantum Computing

In topological superconductors, zero-energy quasiparticle excitations can emerge as Majorana Fermions—particles that are their own antiparticles:

Spatially separated Majorana zero modes exhibit non-Abelian braiding statistics: exchanging two Majorana modes performs a unitary quantum gate operation that depends only on the topological braid knot, providing an inherently error-resistant hardware platform for topological quantum computing.