Condensed Matter Physics

Prerequisites: Canonical Quantum Mechanics, Statistical Mechanics. Full path (20 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Condensed matter physics studies matter made of very many particles that are close together and interact strongly: crystals, metals, magnets, liquids, superconductors, superfluids. The microscopic rules are known. Electrons and nuclei obey quantum mechanics and attract or repel each other through the Coulomb force. The difficulty is the number of particles. A gram of copper contains about ten thousand billion billion atoms, and nobody can solve the equations for all of them.

What makes the field possible is that large collections of particles often behave in a simpler way than their parts. The sound waves in a crystal can be described as particles of their own, called phonons. An electron moving through a metal drags a cloud of disturbance with it and behaves like a free particle with a different mass. Such collective excitations are called quasiparticles. Water and ice are made of the same molecules but are different phases, and a magnet differs from a non-magnet only in how the spins are arranged. Condensed matter physicists classify these phases, find the right collective variables for each, and study the transitions between them.

Many ideas that later became central in particle physics, such as spontaneous symmetry breaking, the Higgs mechanism in the form found in superconductivity, and the renormalization group, were developed or understood first in condensed matter systems.

Concrete

The starting point is the Hamiltonian of electrons (mass $m$, positions $\mathbf r_i$) and nuclei (masses $M_I$, charges $Z_I e$, positions $\mathbf R_I$) interacting through the Coulomb force: $$H = \sum_i \frac{\mathbf p_i^2}{2m} + \sum_I \frac{\mathbf P_I^2}{2M_I} + \frac{1}{2}\sum_{i\neq j}\frac{e^2}{4\pi\varepsilon_0|\mathbf r_i-\mathbf r_j|} - \sum_{i,I}\frac{Z_I e^2}{4\pi\varepsilon_0|\mathbf r_i-\mathbf R_I|} + \frac{1}{2}\sum_{I\neq J}\frac{Z_I Z_J e^2}{4\pi\varepsilon_0|\mathbf R_I-\mathbf R_J|} .$$ Almost every calculation simplifies this in a few standard steps.

Born-Oppenheimer approximation. Nuclei are thousands of times heavier than electrons, so one first solves the electron problem with the nuclei held fixed and then treats the slow nuclear motion separately. Small oscillations of the nuclei around a crystal lattice are quantized as phonons. For a one-dimensional chain of masses $M$ with spring constant $K$ and lattice spacing $a$, the phonon dispersion is $\omega(k) = 2\sqrt{K/M}\,|\sin(ka/2)|$.

Bloch's theorem and bands. In a perfect crystal the potential is periodic, $V(\mathbf r + \mathbf R) = V(\mathbf r)$ for every lattice vector $\mathbf R$. Single-electron eigenstates can then be chosen to satisfy $$\psi_{n\mathbf k}(\mathbf r + \mathbf R) = e^{i\mathbf k\cdot\mathbf R}\,\psi_{n\mathbf k}(\mathbf r) ,$$ with crystal momentum $\mathbf k$ in the Brillouin zone and band index $n$. The energies $\varepsilon_n(\mathbf k)$ form bands. Electrons fill them up to the Fermi energy, following the Pauli principle. If the Fermi energy lies in a gap between bands, the material is an insulator or semiconductor. If it cuts through a band, it is a metal.

Second quantization and lattice models. Interacting problems are written with creation and annihilation operators $c^\dagger_{i\sigma}, c_{i\sigma}$ for an electron of spin $\sigma$ on lattice site $i$. The Hubbard model is the standard minimal example: $$H = -t\sum_{\langle ij\rangle,\sigma}\left(c^\dagger_{i\sigma}c_{j\sigma} + c^\dagger_{j\sigma}c_{i\sigma}\right) + U\sum_i n_{i\uparrow}n_{i\downarrow} ,$$ where $\langle ij\rangle$ runs over nearest neighbours, $t$ is the hopping amplitude, $U$ the on-site repulsion and $n_{i\sigma} = c^\dagger_{i\sigma}c_{i\sigma}$. For $U = 0$ on a one-dimensional chain the Fourier transform gives the band $\varepsilon(k) = -2t\cos(ka)$. For large $U$ at one electron per site, the electrons localize and the low-energy physics is that of a Heisenberg antiferromagnet with exchange $J = 4t^2/U$.

Linear response. Experiments measure how a system reacts to a weak probe: conductivity, magnetic susceptibility, neutron or X-ray scattering. To first order in the perturbation these responses are given by retarded correlation functions of the unperturbed system (the Kubo formula), so much of the theory consists of computing such correlation functions, often with Green's functions and Feynman diagrams adapted to finite density and temperature.

• Condensed Matter Field Theory by A. Altland and B. Simons (book)

Abstract

A precise way to phrase the central question of the field is the classification of phases. A phase is an equivalence class of many-body systems, usually local Hamiltonians on a lattice in the thermodynamic limit, whose members can be deformed into each other without a phase transition, that is without a singularity in the free energy or, at zero temperature, without closing the energy gap above the ground state.

Symmetry breaking. If $H$ is invariant under a symmetry group $G$ but the ground states (or the equilibrium state) are invariant only under a subgroup $H_0 \subset G$, the symmetry is spontaneously broken. Locally, the inequivalent ground states are labeled by points of the order parameter manifold $\mathcal M = G/H_0$. For a Heisenberg ferromagnet, $G = SO(3)$, $H_0 = SO(2)$ and $\mathcal M = S^2$. The low-energy theory is a nonlinear sigma model with target $\mathcal M$, and broken continuous symmetries give gapless Goldstone modes. In Lorentz-invariant systems there is one mode per broken generator. Without Lorentz invariance the counting can differ: the ferromagnet has two broken generators but a single magnon with quadratic dispersion.

Topological defects of the order parameter are classified by homotopy groups of $\mathcal M$. In three spatial dimensions, domain walls are classified by $\pi_0(\mathcal M)$, line defects (vortices, dislocations) by $\pi_1(\mathcal M)$ and point defects (hedgehogs) by $\pi_2(\mathcal M)$. For example $\pi_1(U(1)) = \mathbb Z$ labels the winding number of superfluid vortices, and $\pi_2(S^2) = \mathbb Z$ labels hedgehogs in a ferromagnet.

Renormalization group and universality. Near a continuous transition the correlation length diverges and the long-distance behaviour is controlled by a fixed point of the renormalization group flow. Phases correspond to basins of attraction of stable fixed points, critical points to fixed points with relevant directions. Critical exponents depend only on the fixed point, which depends on dimension, symmetry and range of interactions. Microscopically different systems that flow to the same fixed point therefore share the same exponents (universality).

Phases beyond symmetry breaking. Gapped phases can be distinct even when no symmetry is broken. Topologically ordered phases, such as fractional quantum Hall states, have a ground-state degeneracy that depends on the topology of space (for example on a torus) and excitations with anyonic braiding statistics, described at low energies by topological quantum field theories such as Chern-Simons theory. Symmetry-protected topological phases, such as topological insulators, are trivial without a symmetry but cannot be deformed to a product state while the symmetry is preserved and the gap stays open. For non-interacting fermions these phases are classified by K-theory, organized by the ten Altland-Zirnbauer symmetry classes and the spatial dimension.

Why is it interesting?

It is possible to present condensed-matter physics as interesting, compelling, and even inspiring. Emergence, universality, and symmetry are powerful, amazing ideas. The same essential physics that holds up a white dwarf star is a key ingredient in what makes solids solid, whether we’re talking about a diamond or a block of plastic. Individual electrons seem simple, but put many of them together with a magnetic field in the right 2D environment and presto: excitations with fractional charges. Want electrons to act like ultrarelativistic particles, or act like their own antiparticles, or act like spinning tops pointing in the direction of their motion, or pair up and act together coherently? No problem, with the right crystal lattice. This isn’t dirt physics, and it isn’t squalid.https://physicstoday.scitation.org/do/10.1063/PT.6.3.20181219a/full/

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Hilbert Space
6. Legendre Transformation
7. Newton's Second Law
8. Statistics
9. Vector Calculus
10. Entropy
11. Newtonian Mechanics
12. Variational Calculus
13. Configuration Space
14. Euler-Lagrange Equations
15. Statistical Mechanics
16. Lagrangian Formalism
17. Phase Space
18. Hamiltonian Formalism
19. Schrödinger Equation
20. Canonical Quantum Mechanics