Many-Particle Hamiltonians

A many-particle Hamiltonian describes how many identical particles or excitations move, interact, and exchange energy among modes. In first quantization, a standard fixed-$N$ Hamiltonian has the form

$$H_N = \sum_{\alpha=1}^N h^{(\alpha)} + \sum_{\alpha<\beta} v^{(\alpha\beta)}.$$

The first sum contains one-body terms: kinetic energy, external potentials, spin couplings, or hopping in a single-particle basis. The second sum contains pair interactions. In second quantization, the same number-conserving Hamiltonian is written

$$H = \sum_{ij} h_{ij}d_i^\dagger d_j + \frac12 \sum_{ijkl} V_{ij;kl} d_i^\dagger d_j^\dagger d_l d_k.$$

Here $d_i,d_i^\dagger$ denote bosonic or fermionic mode operators with the appropriate algebra. The visible structure is simple: one-body terms have one creation and one annihilation operator; two-body terms have two creations and two annihilations. In the displayed form the operator strings are already normal ordered: creation operators stand to the left of annihilation operators.

Helpful background. One-Body Operators and Two-Body Operators define the matrix elements and pair-counting factors.

General Number-Conserving Hamiltonian

Let $\{\lvert\varphi_i\rangle\}$ be an orthonormal one-particle mode basis. A general number-conserving Hamiltonian with up to two-body interactions is

$$H = H_1+V+E_0,$$

with

$$H_1 = \sum_{ij} h_{ij}d_i^\dagger d_j,$$

and

$$V = \frac12 \sum_{ijkl} V_{ij;kl} d_i^\dagger d_j^\dagger d_l d_k.$$

A single constant $E_0I$ shifts all sectors equally, so it cancels from energy differences between them. It shifts absolute free energies, and a parameter-dependent constant must be retained when differentiating those free energies. An N-dependent term is not a common constant and can change particle-addition energies. Effective reductions can also generate three-body or higher terms beyond the two-body model displayed here.

The one-body matrix elements are

$$h_{ij} = \langle\varphi_i\vert h\vert\varphi_j\rangle,$$

and the two-body matrix elements are

$$V_{ij;kl} = \langle\varphi_i\otimes\varphi_j\vert v \vert\varphi_k\otimes\varphi_l\rangle.$$

The Hamiltonian preserves total particle number when every term contains the same number of creation and annihilation operators:

$$[H,N_{\mathrm{tot}}]=0.$$

It may still change individual mode occupations. Interactions and off-diagonal one-body terms scatter particles between modes even when total number is fixed.

Field-Operator Form

For continuum problems, the same Hamiltonian is often written in field-operator notation:

$$H = \int d^3x\, \psi^\dagger(\mathbf x) h_{\mathbf x} \psi(\mathbf x) + \frac12 \int d^3x\,d^3y\, \psi^\dagger(\mathbf x) \psi^\dagger(\mathbf y) v(\mathbf x,\mathbf y) \psi(\mathbf y) \psi(\mathbf x).$$

For spinful particles, spin labels are included in the fields and summed. For example, a spin-independent interaction has the form

$$V = \frac12 \sum_{st} \int d^3x\,d^3y\, \psi_s^\dagger(\mathbf x) \psi_t^\dagger(\mathbf y) v(\mathbf x,\mathbf y) \psi_t(\mathbf y) \psi_s(\mathbf x).$$

The field form is compact and basis-independent in appearance. The mode form is better for computations in a chosen basis. The mode expansion connects the two.

Noninteracting Particles

If the Hamiltonian contains only a one-body term, then

$$H_0 = \sum_{ij} h_{ij}d_i^\dagger d_j.$$

Choose the mode basis to diagonalize $h$:

$$h\lvert\epsilon_i\rangle = \epsilon_i\lvert\epsilon_i\rangle.$$

Then

$$H_0 = \sum_i \epsilon_i N_i.$$

This formula describes noninteracting bosons, noninteracting fermions, oscillator modes, trap eigenmodes, and single-particle band modes. The statistics enter through the allowed occupations:

$$n_i=0,1,2,\ldots \quad \text{for bosons}, \qquad n_i\in\{0,1\} \quad \text{for fermions}.$$

A noninteracting Bose gas and a noninteracting Fermi gas can therefore share the same one-particle spectrum $\epsilon_i$ while having very different many-particle spectra because their occupation rules differ.

Interacting Particles

Interactions add terms with two creation and two annihilation operators:

$$V = \frac12 \sum_{ijkl} V_{ij;kl} d_i^\dagger d_j^\dagger d_l d_k.$$

The operator annihilates particles in modes $k,l$ and creates particles in modes $i,j$. In a momentum basis, this is naturally interpreted as scattering between incoming and outgoing momenta. In a spatial orbital basis, it encodes Coulomb, contact, dipolar, exchange, or effective interactions through the matrix elements.

For a translationally invariant two-body interaction, momentum conservation appears in the matrix elements. A common schematic form is

$$V = \frac{1}{2\Omega} \sum_{\mathbf k,\mathbf p,\mathbf q} \widetilde v(\mathbf q)\, d_{\mathbf k+\mathbf q}^\dagger d_{\mathbf p-\mathbf q}^\dagger d_{\mathbf p} d_{\mathbf k},$$

Here $\Omega$ is the periodic-box volume, the discrete mode operators obey Kronecker-delta normalization, and $\widetilde v(\mathbf q)=\int_\Omega d^3r\,e^{-i\mathbf q\cdot\mathbf r}v(\mathbf r)$. Thus $\widetilde v$ has units of energy times volume. The transfer $\mathbf q$ is a wave vector; momentum transfer is $\hbar\mathbf q$. On a lattice, crystal momentum is conserved modulo a reciprocal-lattice vector.

Interactions generally make occupation of the noninteracting modes nonconserved:

$$[H,N_i]\ne 0$$

for individual $i$, even when $[H,N_{\mathrm{tot}}]=0$.

Lattice Hamiltonians

In lattice models, modes are often localized orbitals on sites. Lattice Models introduces local Hilbert spaces and boundaries, while the Tight-Binding Model diagonalizes a typical one-body Hamiltonian

$$H_t = \sum_{rs} t_{rs}d_r^\dagger d_s.$$

The one-particle two-site version is Tight-Binding Dimer, where this one-body term reduces to a two-by-two Hamiltonian.

For spinful fermions, the Hubbard model is

$$H_{\mathrm{Hub}} = -t \sum_{\langle r,s\rangle,\sigma} \left( c_{r\sigma}^\dagger c_{s\sigma} + c_{s\sigma}^\dagger c_{r\sigma} \right) + U\sum_r n_{r\uparrow}n_{r\downarrow}.$$

Each nearest-neighbor bond $\langle r,s\rangle$ is counted once in these formulas. The hopping term is one-body: it moves a fermion between neighboring sites. The on-site interaction is two-body: it counts opposite-spin pairs on the same site.

For bosons on a lattice, the Bose-Hubbard Hamiltonian is often written

$$H_{\mathrm{BH}} = -J \sum_{\langle r,s\rangle} \left( a_r^\dagger a_s + a_s^\dagger a_r \right) + \frac U2 \sum_r n_r(n_r-1) + \sum_r \epsilon_r n_r.$$

The factor $n_r(n_r-1)/2$ counts unordered pairs of bosons occupying the same site. This is the lattice analog of a local contact interaction. The full conventions, limits, and optical-lattice reduction live in Bose–Hubbard Model.

Pairing Terms Preview

Some effective Hamiltonians include terms that do not conserve particle number:

$$H_{\Delta} = \sum_{\mathbf k} \left( \Delta_{\mathbf k} c_{\mathbf k\uparrow}^\dagger c_{-\mathbf k\downarrow}^\dagger + \Delta_{\mathbf k}^* c_{-\mathbf k\downarrow} c_{\mathbf k\uparrow} \right).$$

These terms create or annihilate pairs, so

$$[H_{\Delta},N_{\mathrm{tot}}]\ne 0.$$

They appear in mean-field descriptions of superconductivity and paired superfluids. This does not mean microscopic electric charge conservation has disappeared. Rather, the effective mean-field Hamiltonian describes a subsystem coupled to a condensate or uses an approximation in which the phase of the pair field is treated as a classical parameter.

These pairing terms preserve number parity exactly:

$$(-1)^{N_{\mathrm{tot}}},$$

even though they do not preserve $N_{\mathrm{tot}}$ itself.

A mean-field reduction with its constant retained

For a density interaction between two distinct modes, write $n_i=\bar n_i+\delta n_i$ and $n_j=\bar n_j+\delta n_j$. Dropping only the product of fluctuations gives

$$n_in_j\approx\bar n_i n_j+n_i\bar n_j-\bar n_i\bar n_j.$$

The constant prevents double counting the mean interaction energy. Determine $\bar n_i=\langle n_i\rangle$ self-consistently in the approximate state. Fermionic exchange or pairing contractions require additional channels when relevant; this density-only formula is not a universal decoupling of every quartic operator.

Mean-Field Theory and Hartree–Fock develop the approximation and its limits. Neglecting fluctuations can fail in low dimensions, near criticality, or when correlations determine the observable of interest.

Exercises

1. Noninteracting spectrum. If $H_0=\sum_i\epsilon_iN_i$, what is the energy of the occupation state $\lvert n_1,n_2,\ldots\rangle$?

Solution

Since $N_i\lvert n_1,n_2,\ldots\rangle=n_i\lvert n_1,n_2,\ldots\rangle$,

$$H_0\lvert n_1,n_2,\ldots\rangle = \left( \sum_i \epsilon_i n_i \right) \lvert n_1,n_2,\ldots\rangle.$$

The allowed values of $n_i$ depend on the statistics.

2. Bose-Hubbard pair counting. Show that $n_r(n_r-1)/2$ counts unordered boson pairs on one site.

Solution

On a number state with $n_r$ bosons at site $r$,

$$\frac12 n_r(n_r-1)$$

is the number of unordered pairs that can be chosen from $n_r$ objects. For example, it gives $0$ for $n_r=0,1$, $1$ for $n_r=2$, and $3$ for $n_r=3$.

3. **A retained mean-field constant.** Under the displayed density decoupling, evaluate the mean-field expectation of $Un_in_j$. What error results from omitting the constant?

Solution

The expectation is $U(\bar n_i\bar n_j+\bar n_i\bar n_j-\bar n_i\bar n_j)=U\bar n_i\bar n_j$. Omitting the constant gives twice that value. The constant does not alter eigenvectors at fixed trial means, but it matters for comparing self-consistent energies and minimizing the variational functional.

References

• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
• J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of Superconductivity”, Physical Review 108, 1175–1204 (1957).
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press, 2015.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
• J. Hubbard, “Electron correlations in narrow energy bands”, Proceedings of the Royal Society A 276, 238-257, 1963.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Kluwer Academic/Plenum, 2000.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley, 1988.