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.

Required background. Many-particle Hamiltonians explains one- and two-body terms; field operators translates them into mode and position space. For choosing a model for a material, see Choosing a Model.

Particle number and grand Hamiltonians

Here $K=H-\mu\hat N$ includes chemical-potential control, while $H$ denotes the physical energy. A fixed-number calculation restricts the trace or state space to one sector. Every lattice bond is counted once unless an ordered-pair sum is displayed. The table links each formula to a fuller explanation.

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

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

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

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

Component-conserving two-body form

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

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

• Take the displayed density-interaction kernel real and symmetric under $(\sigma,\mathbf x)\leftrightarrow(\sigma',\mathbf y)$. Only this exchange-symmetric part contributes. More general spin-changing or nonlocal scattering requires a four-index kernel.
• 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 explains self-consistent screening from a direct interaction 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,

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

At $\mu=0$ the centered open-chain Hamiltonian is particle–hole symmetric. An exactly half-filled fixed-number sector requires even $L$. Expanding the interaction gives $V\sum_{\langle ij\rangle}n_i n_j-(V/2)\sum_i z_i n_i+V N_b/4$, where $z_i$ is the site degree and $N_b$ the bond count. On an open chain the endpoints have different linear shifts from the interior; one uniform chemical-potential shift is insufficient.

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 explains the correlations and phases; XXZ Spin Chain develops the spin representation and its boundary conditions.

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

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

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 $U\gg t$ at exact half filling	spin model with superexchange; doping retains mobile projected carriers
$U<0$	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

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

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.

See the Bose–Hubbard model for the optical-lattice reduction and phase boundary, the dimer for an exact two-mode calculation, and the chain for one-dimensional limits.

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 with nonzero $J$, 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

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

Here

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

$V>0$ denotes the pair-scattering matrix element, with energy units, and $\mathcal S$ is the pairing shell. In a bulk limit it scales inversely with volume. Assume $\epsilon_{\mathbf k}=\epsilon_{-\mathbf k}$ and an inversion-symmetric shell for the spin-singlet form.

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, for paired modes inside $\mathcal S$, quasiparticle energies

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

Outside a sharp pairing shell the gap is zero and the excitation energy is $|\xi_{\mathbf k}|$. The unprojected mean-field state is generally 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, this is the grand Hamiltonian (often called the Anderson Hamiltonian):

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

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 at energy $E$ (angular frequency $E/\hbar$),

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

with

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

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 compares charge and screening regimes. Its physical introduction explains spectra and transport; effective Hamiltonians derives the low-energy Kondo reduction.

Kondo Impurity Hamiltonian

Hilbert space

The model combines:

• a dimensionless 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

$$k_{\mathrm B}T_K \sim D \exp \left[ -\frac{2}{\rho_{\mathrm{tot}} J_K} \right],$$

up to convention-dependent prefactors and definitions. Here $D$ is a conduction-band cutoff energy and $\rho_{\mathrm{tot}}$ is the spin-summed local density of states. With a per-spin density $\rho_0$, the same exponent is $-1/(\rho_0J_K)$. 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 gives the coupling and density-of-states conventions. Kondo physics develops scaling, screening and thermodynamics.

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	spin $SU(2)$ for the stated singlet shell	uniform momentum-pair 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.

Relating the models

Projection onto localized orbitals connects continuum gases to lattice Hamiltonians; it requires a separated band and interaction matrix elements small enough to control omitted bands. At exact half filling, a repulsive Hubbard model with $U\gg|t|$ reduces to Heisenberg exchange. Away from half filling, projected hopping and additional terms remain: see effective Hamiltonians.

For an Anderson impurity, both charge costs $-\epsilon_d$ and $\epsilon_d+U$ must be positive and large compared with the relevant hybridization, temperature and excitation energies before replacing the fluctuating orbital by a spin. The Kondo reduction derives exchange and potential scattering together. Pairing and screening reductions select particular low-energy processes; they do not preserve every observable of the original model.

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. 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.

3. 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 pure spin Hamiltonian assumes exact half filling and $U\gg|t|$. Doped low-energy states retain projected carrier hopping, so exchange alone is incomplete.

4. 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.

References

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