Past exam of the advanced Quantum Condensed Matter Physics course of the University of Cambridge / 2015 / Paper 1 / 1

The Hartree-Fock method minimizes the expectation of an interacting-electron Hamiltonian over normalized Slater determinants of orthonormal one-particle spin orbitals. It is a variational, self-consistent mean-field approximation. The determinant enforces fermionic antisymmetry exactly within that class, while each orbital responds to the mean charge density and a nonlocal exchange operator generated by the occupied orbitals.

For an electron gas, this accounts for kinetic pressure, the average electrostatic field, same-spin exchange and the associated exchange hole, and possible spin polarization. It does not account for general dynamical electron correlation, such as the correlated avoidance of opposite-spin electrons, and cannot capture all screening or collective fluctuations merely by optimizing one determinant. Uniform plane waves are an additional restriction, not the definition of Hartree-Fock itself.

In the energy functional, the one-body term is the sum of electron kinetic energy and ionic-potential energy. The direct two-body term is the Hartree energy: the classical interaction of the mean electron density with itself. Its factor $1/2$ avoids counting every pair twice. The negative Fock term comes from exchanging the two indistinguishable electron labels in the antisymmetric state. For a spin-independent interaction, spin orthogonality makes this exchange act only between equal spin components. For an orbital interacting with itself, direct and exchange contributions cancel exactly; the exchange term removes the spurious one-electron self-interaction. Ionic background terms and their prescribed cancellation must be included consistently with the direct term.

Use normalized plane-wave spin orbitals $V^{-1/2}{e}^{i\mathbf{k}\cdot \mathbf{r}}{\chi }_{\sigma }$ and fill a separate Fermi sphere for each spin species. Each spin state counts once, soIts kinetic energy per volume isThe spin-independent contact interaction has uncancelled direct energy $E_H/V=\gamma n^2/2$, with $n=n_{\uparrow }+n_{\downarrow }$. By the imposed background assumption this direct contribution cancels. For the Fock term the contact delta function sets the two spatial positions equal; its spin factors then select equal spins. The local one-body density matrix for each spin has diagonal value $n_{\sigma }$, givingCombining these results gives the contact-interaction Hartree-Fock spin energyHere $\gamma$ has units of energy times volume. The result is the model with the stated background subtraction; an isolated contact-interacting gas without that subtraction retains its direct contribution.

For the unpolarized gas write $x=n/2$. The positive direct Hartree energy before background cancellation and the kinetic energy areTheir ratio is thereforeThis compares the separate direct and kinetic terms. The net Hartree-plus-background contribution after the stipulated cancellation is zero, so it would be incorrect to interpret $r_s$ as that residual contribution divided by $T$. The exchange-energy magnitude in the unpolarized state is $E_H/2$, not $E_H$.

At fixed total density, the unpolarized and fully polarized candidate energies arePolarization increases the kinetic energy because one spin species must occupy a larger Fermi sphere, but gains negative exchange energy. The fully polarized candidate wins this endpoint comparison ifUsing the defined $r_s$ givesThus the specified two-state comparison indeed predicts a polarized state below the unpolarized state's energy for $r_s>1.175$.

The endpoint and continuous polarization thresholds of a contact electron gas are different. To check the actual onset within the same uniform Hartree-Fock model, allow $P=(n_{\uparrow }-n_{\downarrow })/n$ to vary continuously. In units $2A(n/2{)}^{5/3}$,The unpolarized state loses local stability already at $r_s=10/9\simeq 1.111$. Full polarization becomes the minimizing endpoint only when $f^{\prime }(1)\le 0$, or $r_s\ge (5/6)2^{2/3}\simeq 1.323$. Between those values the optimum is partly polarized. Consequently $1.175$ is the requested endpoint crossing, not the exact onset of spontaneous polarization after all spin populations are allowed.

The central limitation is that a Hartree-Fock electron gas includes exchange but omits the correlation energy that is not representable by one determinant. Opposite-spin electrons can lower their repulsion by avoiding one another in a correlated unpolarized state. Hartree-Fock misses that avoidance while retaining the exchange advantage of aligned spins, so it need not place a magnetic instability at a reliable interaction strength. The relevant energy differences between competing spin states can be much smaller than their total energies. Correlation calculations of the electron gas explicitly address this competition.^[ref]

There are further model restrictions. A constant contact coupling discards the momentum dependence of the Coulomb interaction and the resulting nontrivial screening. Real materials have a lattice, band structure and material-dependent effective interactions. Spatial magnetic textures or other competing states are excluded by a comparison restricted to uniform Fermi spheres. Even before those changes, part c's two endpoints do not find the continuous polarization onset.

The symbol $r_s$ also needs care. Here it is an energy ratio proportional to $\gamma n^{1/3}$. The usual Dimensionless Wigner-Seitz radius of a three-dimensional Coulomb electron gas is $r_s^{\mathrm{WS}}=[3/(4\pi n){]}^{1/3}/a_B$, proportional to $n^{-1/3}$. They are not the same density parameter. A quoted experimental value of the latter cannot be compared numerically to $1.175$ without a model-specific mapping. The contact interaction, missing correlations, restricted variational class and parameter convention prevent the number $1.175$ from being a quantitative experimental prediction. The paper supplies no particular experimental threshold, so none is invented here.