Common Many-Body Hamiltonians

A many-body Hamiltonian is not fully specified by one line of algebra. The same formula can describe different physics when the Hilbert space, statistics, lattice, boundary conditions, filling, sign conventions, cutoff, or ensemble changes.

If the unresolved question is which family can predict a material observable, start with Choosing a Model for Quantum Matter. This page begins after that choice and supplies convention-aware formulas and canonical model links.

This page collects eleven standard Hamiltonian families:

1. continuum bosons with contact interactions;
2. continuum fermions with two-body interactions;
3. tight-binding particles;
4. the spinless $t$–$V$ chain;
5. the fermionic Hubbard model;
6. the Bose–Hubbard model;
7. the Heisenberg model;
8. the transverse-field Ising model;
9. the reduced BCS pairing model;
10. the Anderson impurity model;
11. the Kondo impurity model.

It is a convention-aware lookup sheet. The Model Encyclopedia owns convention-complete dossiers, comparisons, solution-status labels, and benchmark handoffs. The linked teaching pages own derivations, phase structure, methods, and applications. The Model-to-Volume Cross-Link Index records which page owns each layer and where planned applications will live.

Read the Formula with Its Data

Before using any row, state:

• the Hilbert space and local degrees of freedom;
• bosonic, fermionic, spin, or constrained statistics;
• continuum or lattice geometry and spatial dimension;
• open, periodic, twisted, or other boundary conditions;
• fixed particle number or grand-canonical controls;
• which bonds or ordered pairs are included in each sum;
• whether spin operators or Pauli matrices are used;
• the ultraviolet regulator or lattice spacing;
• the approximation and energy window in which the model is valid.

The notation

$$K = H-\mu N$$

denotes a grand Hamiltonian. Many papers call $K$ simply “the Hamiltonian.” This sheet keeps $H$ and $K$ distinct whenever the distinction prevents a convention error.

Quick Comparison

Model	Degrees of freedom	Competing scales	Canonical continuation
continuum contact bosons	bosonic field $\psi(\mathbf x)$	kinetic energy, $g$, density	Field Operators in Many-Body Models
continuum interacting fermions	spinful fermionic fields	kinetic energy, interaction, density	Field Operators in Many-Body Models
tight binding	particles in localized orbitals	hopping, onsite energies	Tight-Binding Chain
spinless $t$–$V$ chain	one spinless fermion mode per site	$t$, $V$, filling	Spinless Fermion Chains
Hubbard	spin-$1/2$ lattice fermions	$t$, $U$, filling	Hubbard Model
Bose–Hubbard	lattice bosons	$t$, $U$, $\mu$	Bose–Hubbard Model
Heisenberg	localized spins	exchange $J_{ij}$, fields	Heisenberg Model
transverse-field Ising	spin-$1/2$ sites	Ising exchange, transverse field	Transverse-Field Ising Model
reduced BCS	time-reversed fermion pairs	attraction, cutoff, density of states	Exact dossier; BCS Mean-Field Theory
Anderson impurity	one correlated orbital and bath fermions	$\epsilon_d$, $U$, hybridization function	model dossier; teaching preview
Kondo	impurity spin and conduction fermions	$J_K$, bandwidth, density of states	model dossier; screening and RG

Continuum Bosons with Contact Interaction

Hilbert space and algebra

Use bosonic Fock space over a continuum one-particle space, with equal-time algebra

$$[\psi(\mathbf x),\psi^\dagger(\mathbf y)] = \delta^{(d)}(\mathbf x-\mathbf y).$$

The density and total number are

$$n(\mathbf x) = \psi^\dagger(\mathbf x)\psi(\mathbf x), \qquad N = \int d^d x\, n(\mathbf x).$$

Hamiltonian

A common effective Hamiltonian is

$$\begin{aligned} H_B = \int d^d x\, \bigg[ & \psi^\dagger \left( -\frac{\hbar^2\nabla^2}{2m} + V_{\mathrm{ext}} \right) \psi \\ & + \frac{g}{2} \psi^\dagger \psi^\dagger \psi \psi \bigg](\mathbf x). \end{aligned}$$

The corresponding grand Hamiltonian is

$$K_B = H_B-\mu N.$$

The factor $1/2$ avoids double counting pairs. The displayed contact term is normal ordered.

Convention and validity notes

• $g>0$ denotes repulsion in the displayed convention.
• Coincident-point products require a regulator and matching prescription.
• In a dilute three-dimensional pseudopotential regime,

$$g = \frac{4\pi\hbar^2a_s}{m},$$

where $a_s$ is the $s$-wave scattering length.

• This relation is not a cutoff-independent microscopic identity at arbitrary density, range, or dimension.
• The model conserves $N$ when no source or conversion term is added.

Useful limits

Limit	Result
$g=0$	ideal continuum Bose gas
weak depletion and low temperature	Bogoliubov expansion around a condensate
strong short-range repulsion in one dimension	Lieb–Liniger and Tonks–Girardeau regimes, with convention changes
periodic potential and localized Wannier basis	Bose–Hubbard model after projection

For the detailed operator form, momentum-space normalization, and regularization cautions, use Field Operators in Many-Body Models.

Continuum Fermions with Two-Body Interaction

Hilbert space and algebra

For components $\sigma$,

$$\{ \psi_\sigma(\mathbf x), \psi_{\sigma'}^\dagger(\mathbf y) \} = \delta_{\sigma\sigma'} \delta^{(d)}(\mathbf x-\mathbf y).$$

The total number is

$$N = \sum_\sigma \int d^d x\, \psi_\sigma^\dagger(\mathbf x) \psi_\sigma(\mathbf x).$$

General number-conserving Hamiltonian

$$\begin{aligned} H_F ={}& \sum_\sigma \int d^d x\, \psi_\sigma^\dagger(\mathbf x) h_{0,\sigma} \psi_\sigma(\mathbf x) \\ &+ \frac12 \sum_{\sigma,\sigma'} \int d^d x\,d^d y\, \psi_\sigma^\dagger(\mathbf x) \psi_{\sigma'}^\dagger(\mathbf y) \\ &\qquad\qquad{}\times V_{\sigma\sigma'}(\mathbf x,\mathbf y) \psi_{\sigma'}(\mathbf y) \psi_\sigma(\mathbf x). \end{aligned}$$

A standard one-body operator is

$$h_{0,\sigma} = -\frac{\hbar^2\nabla^2}{2m_\sigma} + V_{\mathrm{ext},\sigma}(\mathbf x).$$

The grand Hamiltonian is

$$K_F = H_F - \sum_\sigma \mu_\sigma N_\sigma.$$

Convention and validity notes

• Hermiticity requires the interaction kernel to obey the appropriate exchange-conjugation relation.
• The factor $1/2$ assumes the integration counts both ordered particle pairs.
• For identical single-component fermions, a zero-range $s$-wave contact term vanishes because two fields at the same point anticommute.
• A two-component contact interaction can be written

$$H_{\mathrm{int}} = g \int d^d x\, n_\uparrow(\mathbf x) n_\downarrow(\mathbf x),$$

with the sign of $g$ setting attraction or repulsion in the displayed convention.

• Coulomb, dipolar, finite-range, and effective interactions require their own kernels and regularization.
• Random Phase Approximation owns the self-consistent density response and screened interaction generated from a specified direct kernel.

Useful limits

Limit	Result
$V=0$	ideal Fermi gas
weak short-range attraction	Cooper instability and BCS reduction near the Fermi surface
periodic potential and one-band projection	tight-binding or Hubbard descriptions
low-energy quasiparticle regime	Landau Fermi-liquid description when applicable

Tight-Binding Hamiltonian

The derivation, finite boundaries, graph form, and band connection are developed in Tight-Binding Model. This section is the compact convention reference.

Hilbert space

Choose localized orbitals $\lvert i,\sigma\rangle$. The same one-body form can act on a one-particle Hilbert space or on a bosonic or fermionic Fock space. Filling and statistics must be specified before assigning many-body meaning.

General form

$$H_{\mathrm{tb}} = - \sum_{i,j,\sigma} t_{ij} c_{i\sigma}^\dagger c_{j\sigma} + \sum_{i,\sigma} v_{i\sigma} n_{i\sigma}.$$

Hermiticity requires

$$t_{ji} = t_{ij}^*, \qquad v_{i\sigma}\in\mathbb R.$$

For real nearest-neighbor hopping with each unordered bond counted once,

$$H_{\mathrm{tb}} = -t \sum_{\langle i,j\rangle,\sigma} \left( c_{i\sigma}^\dagger c_{j\sigma} + c_{j\sigma}^\dagger c_{i\sigma} \right) + \epsilon_0 \sum_{i,\sigma} n_{i\sigma}.$$

One-dimensional dispersion

For a periodic chain with lattice spacing $a$,

$$\epsilon(k) = \epsilon_0 - 2t\cos(ka).$$

The sign of $t$ fixes where the band minimum occurs in this convention. It can be changed by a sublattice gauge transformation only for selected bipartite hopping networks.

What is absent

The tight-binding term alone contains no interaction. Calling a partially filled tight-binding band a metal, insulator, magnet, or superconductor requires filling, dimensionality, symmetry, interactions, disorder, and boundary conditions.

Spinless t–V Chain

Hilbert space and algebra

Retain one spinless fermionic mode per site,

$$\{c_i,c_j^\dagger\} = \delta_{ij}, \qquad n_j = c_j^\dagger c_j \in \{0,1\}.$$

For $L$ sites, the fixed-$N$ sector has dimension $\binom{L}{N}$.

Centered Hamiltonian

For an open chain,

$$\begin{aligned} K_{tV} = {}& -t \sum_{j=1}^{L-1} \left( c_j^\dagger c_{j+1} + c_{j+1}^\dagger c_j \right) \\ &+ V \sum_{j=1}^{L-1} \left(n_j-\frac12\right) \left(n_{j+1}-\frac12\right) \\ &- \mu \sum_{j=1}^{L} \left(n_j-\frac12\right). \end{aligned}$$

The centered convention makes particle-hole symmetry occur at $\mu=0$ on an even bipartite chain. Replacing the interaction by $V\sum_jn_jn_{j+1}$ requires a corresponding chemical-potential and constant shift.

For a ring, include the $L$th bond and specify

$$c_{L+1} = e^{i\phi}c_1.$$

The twist $\phi$, the filling $N/L$, and whether the model is defined directly or obtained through the Jordan–Wigner Transformation are part of the Hamiltonian data.

Standard one-dimensional facts

At $V=0$,

$$\epsilon(k) = -2t\cos(ka)-\mu.$$

At half filling for $t>0$, the uniform thermodynamic chain is phase separated for $V<-2t$, is a gapless Luttinger liquid for $-2t<V<2t$, and has gapped period-two charge order for $V>2t$. The endpoint at $V=2t$ is a Berezinskii–Kosterlitz–Thouless transition.

After a staggered hopping-sign transformation, the XXZ parameters are

$$J = 2t, \qquad \Delta = \frac{V}{2t}.$$

Spinless Fermion Chains owns the free solution, correlators, phase interpretation, transport twist, and finite-size parity caveats. XXZ Spin Chain owns the canonical spin-chain treatment.

Fermionic Hubbard Model

Hilbert space

Each site has four local states,

$$\lvert0\rangle, \qquad \lvert\uparrow\rangle, \qquad \lvert\downarrow\rangle, \qquad \lvert\uparrow\downarrow\rangle.$$

For $L$ sites, the unrestricted Fock space has dimension $4^L$.

Fixed-number Hamiltonian

$$\begin{aligned} H_{\mathrm{Hub}} = & -t \sum_{\langle i,j\rangle,\sigma} \left( c_{i\sigma}^\dagger c_{j\sigma} + \mathrm{h.c.} \right) \\ & + U \sum_i n_{i\uparrow}n_{i\downarrow} + \sum_i v_i n_i. \end{aligned}$$

Here

$$n_i = n_{i\uparrow} + n_{i\downarrow}.$$

The grand Hamiltonian is

$$K_{\mathrm{Hub}} = H_{\mathrm{Hub}} - \mu \sum_i n_i.$$

Sign and filling conventions

• $t>0$ in the displayed nearest-neighbor convention.
• $U>0$ is repulsive and $U<0$ attractive.
• The filling is

$$n = \frac{N}{L}.$$

• Half filling is $n=1$, because each site has two spin modes.
• Each unordered bond appears once when the Hermitian conjugate is written explicitly.

Useful limits

Limit	Result
$U=0$	spinful tight-binding fermions
$t=0$	independent atomic sites
repulsive $U\gg t$ near half filling	low-energy spin model with superexchange
$U<0$	onsite pairing tendency

At half filling in the large-repulsion regime, the leading exchange scale is

$$J_{\mathrm{ex}} = \frac{4t^2}{U}$$

for the standard two-site hopping convention. Higher-order and lattice-dependent terms are omitted.

The full conventions, symmetries, limits, and strong-coupling derivation live in Hubbard Model.

Bose–Hubbard Model

Hilbert space

Each lattice site is a bosonic mode with

$$n_i = b_i^\dagger b_i \in \{0,1,2,\ldots\}.$$

Numerical calculations often impose a maximum occupation as a truncation. That cutoff is not part of the exact model.

Grand Hamiltonian

$$\begin{aligned} K_{\mathrm{BH}} = & -t \sum_{\langle i,j\rangle} \left( b_i^\dagger b_j + b_j^\dagger b_i \right) \\ & + \frac{U}{2} \sum_i n_i(n_i-1) - \mu \sum_i n_i. \end{aligned}$$

For a fixed-$N$ calculation, omit the chemical-potential term and restrict the Hilbert space to

$$\sum_i n_i = N.$$

Convention notes

• $U>0$ is onsite repulsion.
• The factor $n_i(n_i-1)/2$ counts unordered onsite pairs.
• $t=0$ is the atomic limit.
• The competition between $t/U$ and $\mu/U$ organizes the standard Mott-lobe diagram.
• At integer filling, large $U/t$ suppresses number fluctuations; large $t/U$ favors phase coherence.

The full conventions, limits, optical-lattice reduction, and mean-field boundary live in Bose–Hubbard Model. Bose–Hubbard Dimer owns the exact two-mode fixed-number specialization, while Bose–Hubbard Chain owns the one-dimensional soft-core model and its hard-core and Luttinger limits. The compact model card remains a lookup entry, while Quantum Phase Transitions owns the generic critical interpretation.

Heisenberg Model

Hilbert space

Place a spin-$s$ representation on each site:

$$\mathcal H = \bigotimes_{i=1}^{L} \mathbb C^{2s+1}.$$

Let $\mathbf s_i$ be dimensionless spin operators satisfying

$$[s_i^a,s_j^b] = i\delta_{ij} \epsilon^{abc} s_i^c.$$

Isotropic exchange Hamiltonian

$$H_{\mathrm{Heis}} = \sum_{\langle i,j\rangle} J_{ij} \mathbf s_i\cdot\mathbf s_j - \sum_i \mathbf h_i\cdot\mathbf s_i.$$

Each unordered bond is counted once. The field $\mathbf h_i$ is measured in energy units.

For uniform nearest-neighbor exchange,

$$H_{\mathrm{Heis}} = J \sum_{\langle i,j\rangle} \mathbf s_i\cdot\mathbf s_j.$$

Sign convention

With the displayed plus sign:

• $J>0$ is antiferromagnetic;
• $J<0$ is ferromagnetic.

For two spin-$1/2$ sites,

$$\mathbf s_1\cdot\mathbf s_2 = \begin{cases} -3/4, & \text{singlet},\\ +1/4, & \text{triplet}. \end{cases}$$

Common variants

$$H_{XYZ} = \sum_{\langle i,j\rangle} \left( J_xs_i^xs_j^x + J_ys_i^ys_j^y + J_zs_i^zs_j^z \right)$$

is anisotropic and generally has less symmetry than the isotropic model. Dzyaloshinskii–Moriya exchange, longer-range bonds, single-ion anisotropy, and multi-spin interactions are distinct extensions.

For the uniform spin-$1/2$ chain with $J_x=J_y=J$ and $J_z=J\Delta$, use XXZ Spin Chain for phase structure, Jordan–Wigner fermions, and integrability.

Use Heisenberg Model for exchange spectra, symmetries, limiting cases, and approximation methods.

Transverse-Field Ising Model

Hilbert space and convention

For spin-$1/2$ sites with Pauli matrices,

$$H_{\mathrm{TFIM}} = -J \sum_{\langle i,j\rangle} \sigma_i^z\sigma_j^z - h \sum_i \sigma_i^x.$$

The Ising interaction and transverse-field terms do not commute.

Symmetry

The global spin flip

$$P = \prod_i \sigma_i^x$$

satisfies

$$[H_{\mathrm{TFIM}},P] = 0.$$

It changes

$$\sigma_i^z \longmapsto -\sigma_i^z.$$

Pauli versus spin operators

If

$$s_i^\alpha = \frac{\sigma_i^\alpha}{2},$$

then

$$\sigma_i^z\sigma_j^z = 4s_i^zs_j^z, \qquad \sigma_i^x = 2s_i^x.$$

Couplings must therefore be rescaled when changing notation.

One-dimensional benchmark

For the infinite nearest-neighbor chain in the displayed convention, the critical point is

$$|h| = |J|.$$

Boundary conditions and parity sectors matter in the finite Jordan–Wigner solution. Use Ising Chain for the compact model card, Transverse-Field Ising Model for the model treatment and exact-solution preview, and Quantum Phase Transitions for general critical scaling.

Reduced BCS Pairing Hamiltonian

Hilbert space

Use spinful fermionic momentum modes. The reduced interaction scatters pairs of time-reversed states,

$$(\mathbf k\uparrow,-\mathbf k\downarrow) \longrightarrow (\mathbf k'\uparrow,-\mathbf k'\downarrow).$$

Define

$$b_{\mathbf k}^\dagger = c_{\mathbf k\uparrow}^\dagger c_{-\mathbf k\downarrow}^\dagger.$$

Grand Hamiltonian

$$\begin{aligned} K_{\mathrm{BCS}} = & \sum_{\mathbf k,\sigma} \xi_{\mathbf k} c_{\mathbf k\sigma}^\dagger c_{\mathbf k\sigma} \\ & - V \sum_{\mathbf k,\mathbf k'\in\mathcal S} b_{\mathbf k}^\dagger b_{\mathbf k'}. \end{aligned}$$

Here

$$\xi_{\mathbf k} = \epsilon_{\mathbf k}-\mu,$$

$V>0$ denotes attraction in the displayed convention, and $\mathcal S$ is the pairing shell or model space.

Number conservation and mean field

The reduced interaction has two creation and two annihilation operators, so the exact reduced Hamiltonian conserves total particle number:

$$[K_{\mathrm{BCS}},N] = 0.$$

The usual BCS mean-field approximation introduces

$$\Delta = V \sum_{\mathbf k\in\mathcal S} \left\langle c_{-\mathbf k\downarrow} c_{\mathbf k\uparrow} \right\rangle$$

and obtains quasiparticle energies

$$E_{\mathbf k} = \sqrt{ \xi_{\mathbf k}^2 + |\Delta|^2 }.$$

The unprojected mean-field state is not an eigenstate of $N$. Do not confuse that approximation with number nonconservation in the exact reduced pairing Hamiltonian.

Use Reduced BCS Model for seniority sectors, Richardson equations, and the exact finite-level benchmark; BCS Mean-Field Theory for the Cooper instability, gap equation, and quasiparticle derivation; and BCS Model for the compact model card.

Anderson Impurity Hamiltonian

Hilbert space

The model combines:

• one spinful localized fermion orbital with states $\lvert0\rangle$, $\lvert\uparrow\rangle$, $\lvert\downarrow\rangle$, and $\lvert2\rangle$;
• a Fock space of noninteracting bath fermions;
• coherent bath–impurity tunneling.

Hamiltonian

With all one-particle energies measured relative to the equilibrium chemical potential,

$$\begin{aligned} H_{\mathrm A} &= \sum_{k,\sigma} \epsilon_k c_{k\sigma}^{\dagger}c_{k\sigma} + \epsilon_d \sum_\sigma n_{d\sigma} \\ &\quad+ U n_{d\uparrow}n_{d\downarrow} + \sum_{k,\sigma} \left( V_kc_{k\sigma}^{\dagger}d_\sigma + V_k^*d_\sigma^{\dagger}c_{k\sigma} \right). \end{aligned}$$

The atomic impurity energies are

$$E_0=0, \qquad E_\uparrow=E_\downarrow=\epsilon_d, \qquad E_2=2\epsilon_d+U.$$

For repulsive $U$, the atomic local-moment window is

$$\epsilon_d<0<\epsilon_d+U.$$

Hybridization means that impurity charge is not conserved even though total charge is.

Bath convention

The bath and tunneling amplitudes enter through

$$\Delta(z) = \sum_k \frac{\lvert V_k\rvert^2}{z-\epsilon_k}.$$

For the retarded boundary value,

$$\Delta^R(\omega) = \Lambda(\omega)-i\Gamma(\omega),$$

with

$$\Gamma(\omega) = \pi \sum_k \lvert V_k\rvert^2 \delta(\omega-\epsilon_k).$$

Thus an unweighted bath density of states is not enough to specify the impurity problem. In the common wide-band convention, constant $\Gamma$ is the half-width of the noninteracting impurity resonance.

The Anderson Impurity Model dossier owns the compact specification, hybridization conventions, regime and exact-status map, observable–method dictionary, and finite benchmark. Anderson Impurity Model Preview owns the resonant-level derivation, impurity spectra, thermodynamics, transport, solution-method narrative, and DMFT connection. Effective Hamiltonians in Many-Body Systems owns the comparative Kondo reduction and its validity audit.

Kondo Impurity Hamiltonian

Hilbert space

The model combines:

• a localized impurity spin $\mathbf S$;
• a Fock space of conduction electrons;
• a local exchange interaction at the impurity position.

Hamiltonian

$$H_{\mathrm K} = \sum_{\mathbf k,\sigma} \epsilon_{\mathbf k} c_{\mathbf k\sigma}^\dagger c_{\mathbf k\sigma} + J_K \mathbf S\cdot \mathbf s_c(\mathbf 0),$$

where the local conduction-electron spin density is

$$\mathbf s_c(\mathbf 0) = \frac12 \sum_{\alpha,\beta} \psi_\alpha^\dagger(\mathbf 0) \boldsymbol\sigma_{\alpha\beta} \psi_\beta(\mathbf 0).$$

The normalization of $\psi(\mathbf0)$ determines the units assigned to $J_K$. A bandwidth or ultraviolet cutoff and conduction density of states are part of the model.

Sign convention and physics

With the displayed plus sign:

• $J_K>0$ is antiferromagnetic and favors local singlet formation;
• $J_K<0$ is ferromagnetic.

For an isolated pair of spin-$1/2$ objects,

$$\mathbf S\cdot\mathbf s = \begin{cases} -3/4, & \text{singlet},\\ +1/4, & \text{triplet}. \end{cases}$$

The many-electron Kondo problem is not merely this two-spin spectrum. Repeated low-energy scattering produces logarithmic scale dependence and, for antiferromagnetic exchange, a Kondo scale schematically of the form

$$T_K \sim D \exp \left[ -\frac{1}{\rho_{\mathrm{tot}} J_K} \right],$$

up to convention-dependent prefactors and definitions. Here $D$ is a conduction-band cutoff and $\rho_{\mathrm{tot}}$ is the spin-summed local density of states. With a per-spin density $\rho_0$, the same exponent is $-1/(2\rho_0J_K)$. This formula is an asymptotic weak-coupling scale, not an exact universal equality.

Common extensions include potential scattering, anisotropic exchange, multiple channels, multiple impurities, and lattice versions. Each changes the symmetry and possible low-energy behavior.

The Kondo Model dossier owns the compact specification, normalization conventions, solvability map, controlled limits, observable–method dictionary, and finite benchmark. The Kondo Model Preview owns the one-loop flow, screening interpretation, thermodynamics, screening cloud, and low-energy local Fermi-liquid narrative.

Conserved Quantities at a Glance

Model as displayed	Total particle number	Spin symmetry	Translation symmetry
continuum bosons	conserved	depends on components	if $V_{\mathrm{ext}}$ and regulator permit
continuum fermions	conserved	depends on $V_{\sigma\sigma'}$	if kernels are translation invariant
tight binding	conserved	often spin independent	only for a periodic uniform lattice
spinless $t$–$V$ chain	conserved	no retained spin degree of freedom	only for a periodic uniform chain
Hubbard	conserved	$SU(2)$ spin for the standard spin-independent form	only for a uniform lattice
Bose–Hubbard	conserved	not applicable for one component	only for a uniform lattice
Heisenberg	no particle number	global $SU(2)$ without anisotropy or field	only for a uniform exchange network
TFIM	no particle number	discrete $\mathbb Z_2$	only for uniform couplings
reduced BCS	conserved exactly	reduced by the pairing structure	momentum-space form assumes translation invariance
Anderson impurity	total bath-plus-impurity number conserved	$SU(2)$ for spin-independent parameters	broken by the impurity
Kondo impurity	conduction number conserved	$SU(2)$ for isotropic exchange	broken by the impurity

Additional conserved quantities can arise from integrability, point-group symmetry, bipartite structure, or special parameter choices.

How the Models Connect

Continuum to lattice

Expanding a continuum field in localized orbitals,

$$\psi(\mathbf x) \approx \sum_i w_i(\mathbf x)a_i,$$

turns kinetic and periodic-potential terms into hopping matrix elements and short-range interactions into onsite and extended lattice couplings. Truncating to one band is an approximation controlled only when higher bands remain sufficiently separated.

Tight binding to Hubbard

Adding an onsite interaction to spinful tight-binding fermions gives the standard single-band Hubbard structure:

$$H_{\mathrm{Hub}} = H_{\mathrm{tb}} + U \sum_i n_{i\uparrow}n_{i\downarrow}.$$

Spinless fermions to XXZ

In one dimension, the Jordan–Wigner Transformation maps nearest-neighbor spinless hopping and density interaction to transverse and longitudinal spin exchange. With centered density and a staggered hopping-sign convention,

$$s_j^z = n_j-\frac12, \qquad J=2t, \qquad \Delta=\frac{V}{2t}.$$

For a periodic chain, the spin boundary induces a fermionic twist tied to total parity. The local bulk parameter map does not remove that finite-size boundary condition.

Hubbard to Heisenberg

At exact half filling in the no-doublon sector and for $U\gg |t|$, virtual hopping through doubly occupied states produces antiferromagnetic exchange:

$$H_{\mathrm{eff}} = \frac{4t^2}{U} \sum_{\langle i,j\rangle} \left( \mathbf s_i\cdot\mathbf s_j - \frac14 n_i n_j \right) + \cdots.$$

Inside the singly occupied subspace, $n_i n_j=1$ on occupied bonds, so the density term is an additive constant.

Effective Hamiltonians in Many-Body Systems derives the projection and explains why doped systems also retain projected hopping and same-order three-site terms.

Fermions to reduced BCS

Projecting an attractive interaction onto time-reversed states near a Fermi surface yields the reduced pairing model. The projection discards other scattering channels and introduces a cutoff or model shell.

Anderson impurity to Kondo

When

$$-\epsilon_d, \quad \epsilon_d+U \gg \Gamma,$$

the empty and doubly occupied impurity states are virtual at low energy. A Schrieffer–Wolff transformation projects onto the singly occupied doublet and generates

$$J_K \simeq 2\lvert V\rvert^2 \left( \frac1{-\epsilon_d} + \frac1{\epsilon_d+U} \right) >0$$

for the normalized local-orbital convention. The reduction fails near a charge degeneracy or in mixed valence.

The two charge denominators, potential-scattering term, and normalization conventions are derived in Effective Hamiltonians in Many-Body Systems.

Lattice spins to continuum fields

Near a continuous transition, a spin Hamiltonian can produce a continuum order-parameter or quasiparticle field theory. That continuum theory is an effective long-wavelength description, not a literal replacement of Pauli matrices at every lattice scale.

Model-Specification Checklist

For reproducible work, record:

1. the exact Hamiltonian or grand Hamiltonian;
2. the operator algebra and local Hilbert space;
3. lattice graph, dimension, and boundary conditions;
4. bond-counting convention;
5. parameter signs and units;
6. particle number, filling, or chemical potential;
7. conserved symmetry sector;
8. ultraviolet cutoff, basis truncation, or maximum occupation;
9. energy, temperature, and momentum regime;
10. approximation method and omitted terms;
11. observable definitions;
12. finite-size and thermodynamic-limit procedure.

Common Mistakes

• Calling $H-\mu N$ a fixed-$N$ Hamiltonian without stating the convention.
• Counting each lattice bond twice while also writing an explicit Hermitian conjugate.
• Switching between Pauli matrices and spin operators without rescaling couplings.
• Using the same symbol $J$ with opposite ferro- and antiferromagnetic sign conventions.
• Treating $g=4\pi\hbar^2a_s/m$ as a universal bare coupling at every cutoff.
• Writing a same-component fermion $s$-wave contact term without addressing antisymmetry.
• Calling a one-body tight-binding band a correlated many-body model.
• Treating the spinless $t$–$V$ chain as free when $V\ne0$.
• Mixing centered and uncentered $t$–$V$ conventions without shifting $\mu$.
• Forgetting the parity-dependent boundary twist when the chain comes from periodic spins.
• Forgetting that half filling in the spinful Hubbard model is $N/L=1$.
• Omitting the onsite-pair factor $1/2$ in the Bose–Hubbard interaction.
• Treating $J_{\mathrm{ex}}=4t^2/U$ as exact outside the large-$U$ low-energy regime.
• Assuming the reduced BCS Hamiltonian itself violates number conservation.
• Treating the impurity charge as fixed in an Anderson model with nonzero hybridization.
• Specifying an Anderson bath without the weighted hybridization function.
• Replacing an Anderson impurity by a Kondo spin outside the local-moment regime.
• Quoting a Kondo temperature without defining the bandwidth, density of states, and coupling convention.
• Applying a model beyond the band, density, temperature, or interaction regime used to derive it.

Cross-Links

• Many-Body and Quantum Statistical Mechanics
• Choosing a Model for Quantum Matter
• Symbols and Conventions
• Common Spin Hamiltonians
• Operator Identities
• Occupation-Number Representation
• Field Operators in Many-Body Models
• Many-Particle Hamiltonians
• Hubbard Model
• Heisenberg Model
• XXZ Spin Chain
• Spinless Fermion Chains
• Anderson Impurity Model
• Anderson Impurity Model Preview
• Kondo Model
• Kondo Model Preview
• Quantum Phase Transitions
• Tight-Binding Chain
• Bose–Hubbard Model
• Bose–Hubbard Model Card
• BCS Mean-Field Theory
• BCS Model
• Ising Chain
• Many-Body Model Cards
• Spin Model Cards
• Most-Used Hamiltonians
• Common Hamiltonians Table
• Ensemble Formula Sheet

References

1. A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
2. A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010).
3. P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015).
4. J. Hubbard, “Electron Correlations in Narrow Energy Bands”, Proceedings of the Royal Society A 276, 238–257 (1963).
5. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson Localization and the Superfluid-Insulator Transition”, Physical Review B 40, 546–570 (1989).
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Exercises

1. Hermiticity of hopping. Show that

$$H_t = - \sum_{i,j} t_{ij} c_i^\dagger c_j$$

is Hermitian when $t_{ji}=t_{ij}^*$.

Solution

Take the adjoint:

$$H_t^\dagger = - \sum_{i,j} t_{ij}^* c_j^\dagger c_i.$$

Relabel $i\leftrightarrow j$:

$$H_t^\dagger = - \sum_{i,j} t_{ji}^* c_i^\dagger c_j.$$

If

$$t_{ji}^* = t_{ij},$$

then

$$H_t^\dagger = H_t.$$

For a restricted bond sum, the same condition is implemented by writing the Hermitian-conjugate hopping explicitly.

2. Onsite boson pair counting. Explain why the Bose–Hubbard interaction energy on a site with $n$ bosons is $Un(n-1)/2$ rather than $Un^2/2$.

Solution

The number of unordered pairs among $n$ particles is

$$\binom n2 = \frac{n(n-1)}{2}.$$

If each onsite pair costs energy $U$, then

$$E_{\mathrm{int}}(n) = U \binom n2 = \frac{U}{2} n(n-1).$$

The expression vanishes for $n=0$ and $n=1$, as it should because no pair is present. The alternative $Un^2/2$ contains an unphysical one-particle self-interaction unless compensated by a linear term.

3. Number conservation in the reduced BCS model. Show by operator counting that the pair-scattering term commutes with total number.

Solution

For total fermion number $N$,

$$[N,c_i^\dagger] = c_i^\dagger, \qquad [N,c_i] = -c_i.$$

The pair-scattering monomial

$$c_{\mathbf k\uparrow}^\dagger c_{-\mathbf k\downarrow}^\dagger c_{-\mathbf k'\downarrow} c_{\mathbf k'\uparrow}$$

contains two creation and two annihilation operators. Using the commutator product rule, its net number change is

$$+1+1-1-1 = 0.$$

Therefore the exact reduced BCS Hamiltonian commutes with $N$. Number uncertainty enters the usual unprojected mean-field ansatz, not the exact pair-scattering operator.

4. Exchange sign for two spins. For two spin-$1/2$ sites with

$$H = J \mathbf s_1\cdot\mathbf s_2,$$

find the singlet and triplet energies and identify which is favored for each sign of $J$.

Solution

Use

$$\mathbf s_1\cdot\mathbf s_2 = \frac12 \left( \mathbf s_{\mathrm{tot}}^2 - \mathbf s_1^2 - \mathbf s_2^2 \right).$$

For two spin-$1/2$ objects,

$$\mathbf s_1^2 = \mathbf s_2^2 = \frac34.$$

The singlet has total spin $0$, so

$$E_s = -\frac{3J}{4}.$$

The triplet has total spin $1$, so

$$E_t = \frac{J}{4}.$$

Thus $J>0$ favors the singlet and is antiferromagnetic, while $J<0$ favors the triplet and is ferromagnetic in the displayed convention.

5. Hubbard superexchange scale. Explain why the leading low-energy exchange near half filling scales as $t^2/U$ and state the standard coefficient.

Solution

In the large-$U$ singly occupied subspace, one hop creates a virtual doubly occupied site and an empty site. That intermediate state costs energy of order $U$. A second hop returns the system to the low-energy subspace, possibly exchanging the two spins.

Second-order perturbation theory therefore produces an amplitude proportional to

$$\frac{t^2}{U}.$$

For the standard nearest-neighbor Hubbard hopping convention, adding the allowed virtual processes gives

$$J_{\mathrm{ex}} = \frac{4t^2}{U}.$$

For $U>0$, this is positive and therefore antiferromagnetic in the convention

$$H_{\mathrm{Heis}} = J_{\mathrm{ex}} \sum_{\langle i,j\rangle} \mathbf s_i\cdot\mathbf s_j.$$

The result assumes $U\gg t$, near-half filling, and projection to the low-energy subspace.

6. Kondo exchange sign. Treat the impurity and one local conduction spin as two spin-$1/2$ objects. What energy splitting does $J_K\mathbf S\cdot\mathbf s$ produce?

Solution

The same two-spin eigenvalues apply:

$$\mathbf S\cdot\mathbf s = \begin{cases} -3/4, & \text{singlet},\\ +1/4, & \text{triplet}. \end{cases}$$

Therefore

$$E_s = -\frac{3J_K}{4}, \qquad E_t = \frac{J_K}{4},$$

and the splitting is

$$E_t-E_s = J_K.$$

Antiferromagnetic $J_K>0$ favors the singlet. The full Kondo effect additionally involves the continuum of conduction states and logarithmic renormalization; it is not exhausted by this two-spin calculation.

7. Anderson impurity charge window. In the atomic limit, compare the energies of $\lvert0\rangle$, $\lvert\sigma\rangle$, and $\lvert2\rangle$. For $U>0$, determine when the singly occupied states have the lowest energy.

Solution

The atomic energies are

$$E_0=0, \qquad E_\sigma=\epsilon_d, \qquad E_2=2\epsilon_d+U.$$

Single occupation lies below the empty state when

$$\epsilon_d<0.$$

It lies below the doubly occupied state when

$$\epsilon_d < 2\epsilon_d+U,$$

or

$$\epsilon_d+U>0.$$

Therefore the atomic local-moment window is

$$-U<\epsilon_d<0.$$

Finite hybridization rounds the charge boundaries and makes the impurity occupancy fluctuate.