Math Needed for Many-Body QM

Many-body quantum mechanics begins when several degrees of freedom must be described together. You need tensor products to build their state space, exchange symmetry for identical particles, and traces to calculate averages in mixed or thermal states. Learn these tools as the physical problem requires them; you do not need every item below before starting.

Tensor products, occupations, and thermal averages

For $N$ distinguishable spin-$1/2$ sites, the state space is $(\mathbb C^2)^{\otimes N}$ and has dimension $2^N$. A product basis labels all sites at once. This is why a Hamiltonian that is short to write can become a very large matrix. Begin with Tensor Products and then Scaling of Hilbert Space.

For identical particles, an occupation basis labels how many particles occupy each mode. The state $|n_1,n_2,\ldots\rangle$ is a basis vector, not a list of probabilities. Bosonic occupations can be any nonnegative integers; a fermionic mode has occupation zero or one. Here a mode includes every one-particle label, including spin. Fixing total particle number selects the sector $\sum_j n_j=N$ inside Fock space.

For a thermal calculation, the basic object is instead a density operator. In a canonical ensemble,

$$\rho_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}e^{-\beta H},\qquad \langle A\rangle=\operatorname{Tr}(\rho_\beta A),$$

where $\beta=1/(k_BT)$ and the trace is over the chosen particle-number sector. Assume $Z$ is finite; a finite-dimensional model automatically satisfies this. Thermal Density Operators develops the physical interpretation.

Choose the tools for your calculation

If you want to…	Read and practice…
Build a spin-chain Hamiltonian	Operators on Composite Systems for $I\otimes A$ and $A\otimes I$, then Pauli Matrices
Describe identical particles	Symmetric Group for permutations, then Symmetric and Antisymmetric Wavefunctions
Use second quantization	Occupation-Number Basis → Creation and Annihilation Operators → Many-Particle Hamiltonians
Calculate thermal or subsystem averages	Trace Rule and Partial Trace
Understand connected correlations	Variance and Covariance for subtracting products of means, then Green Functions for propagation and response
Compute a few low-energy levels	Sparse Matrices → Sparse Eigensolvers, with Convergence Tests

Fourier transforms become useful for translation-invariant lattices and frequency-dependent response. Complex analysis and contour integration can wait until a calculation actually uses them. For a first calculation, two spins or two fermionic modes are small enough to check every matrix element by hand.

Two distinctions that prevent confusion

Thermal entropy and entanglement entropy answer different questions. A pure state of a whole spin chain has zero von Neumann entropy, yet a reduced state of part of the chain can have nonzero entropy. Read Subsystem Entropy when you need to distinguish the full state from its restriction to a subsystem.

A finite-size change is not automatically a phase transition. Exact diagonalization can show a level crossing, small gap, or rapidly changing correlation. To infer thermodynamic behavior, compare several sizes and specify boundary conditions and symmetry sectors. A small eigensolver residual checks the finite matrix problem; it does not establish convergence with system size.

Check your understanding

1. Why do $N$ spin-$1/2$ sites have $2^N$ basis states rather than $2N$?

Solution

Each of the $N$ sites independently has two basis labels. Counting all simultaneous choices multiplies the dimensions. A direct sum would describe alternatives between spaces, rather than a system containing all $N$ sites.

2. Two fermionic modes have Fock basis $|00\rangle,|10\rangle,|01\rangle,|11\rangle$. What is the dimension of the one-particle sector? How does this differ from two bosonic modes with exactly two particles?

Solution

The fermionic one-particle sector contains $|10\rangle$ and $|01\rangle$, so its dimension is two. Two bosons in two modes have three basis states: $|20\rangle,|11\rangle,|02\rangle$. The differences come from both the specified particle number and the allowed occupations of each mode.

Continue with Quantum Many-Body Theory and Statistical Mechanics for physical models, or Math Needed for Quantum Matter for lattice and band calculations.

References

• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press, 2015 — occupation notation, correlations, and many-body methods.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971 — second quantization and response theory.
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press, 2011 — finite-size and thermodynamic reasoning.