Core Objects and Notation

Many-body notation compresses large physical structures into a few symbols. That compression is useful only when every index, normalization, ensemble, operator ordering, and limiting convention is recoverable from the page.

This page establishes the default notation for this volume. A specialized article may choose another convention, but it must declare the translation explicitly.

The Notation Contract

Every calculation should identify:

1. the retained degrees of freedom;
2. the Hilbert space or Fock-space sector;
3. particle, site, mode, spin, and internal indices;
4. the state or statistical ensemble;
5. the Hamiltonian and conserved quantities;
6. boundary conditions and geometry;
7. Fourier-transform normalization;
8. operator ordering and time ordering;
9. units and constants set to one;
10. the finite-size or thermodynamic limits being taken.

Symbols alone do not supply this information. For example, $N$ may mean particle number, number of sites, an internal symmetry rank, or the size parameter in a large-$N$ approximation.

Typography and Indices

The default visual distinctions are:

Object	Typical notation	Comment
scalar number	$N$, $T$, $\mu$, $t$	Meaning and units must be stated
spatial vector	$\mathbf r$, $\mathbf k$, $\mathbf q$	Boldface distinguishes vectors
abstract operator	$H$, $N$, $S^\alpha$	Hats are used when ambiguity remains
state vector	$	\Psi\rangle$
density operator	$\rho$	Positive, trace one for a normalized state
local site	$i,j,\ell$	Often integer lattice labels
one-particle mode	$p,q,k$	May denote momentum or a generic mode
internal component	$\sigma,\alpha,\beta$	Spin, flavor, band, or species
spatial component	$a,b$	Cartesian or tensor component
eigenstate label	$n,m$	Not automatically a particle label
imaginary time	$\tau$	Usually $0\le\tau<\beta\hbar$

The same Greek letter should not serve simultaneously as a spin index, a coupling, and a critical exponent on one page.

System Size and Geometry

Common size symbols are:

Symbol	Default meaning
$N$	number of particles
$N_s$	number of lattice sites
$M$	number of retained modes or orbitals
$V$	physical volume
$L$	linear system size
$d$	spatial dimension
$a$	lattice spacing
$n=N/V$	particle-number density
$\nu=N/N_s$	filling per lattice site

For a hypercubic lattice with $N_s$ sites,

$$N_s = \left( \frac{L}{a} \right)^d$$

when every direction has length $L$ and the endpoints are counted according to a compatible periodic convention.

For an anisotropic system, list $L_1,\ldots,L_d$ rather than hiding the shape in one $L$.

Particle number versus site number

Particle number and site number are independent. The filling

$$\nu = \frac{N}{N_s}$$

can be less than, equal to, or greater than one for bosons. For spinless fermions with one orbital per site,

$$0\le\nu\le1,$$

while spin or orbital degeneracy changes the maximum occupancy.

States and Expectations

A normalized pure state is denoted $|\Psi\rangle$, with

$$\langle\Psi|\Psi\rangle=1.$$

A normalized mixed state is a density operator $\rho$ satisfying

$$\rho\ge0, \qquad \operatorname{Tr}\rho=1.$$

Expectation values are

$$\langle A\rangle_\Psi = \langle\Psi|A|\Psi\rangle$$

or

$$\langle A\rangle_\rho = \operatorname{Tr}(\rho A).$$

When the state is clear, the subscript is omitted:

$$\langle A\rangle.$$

The omission must not conceal an ensemble change.

Reduced states

For a subsystem $A$ of a joint state $\rho_{AB}$,

$$\rho_A = \operatorname{Tr}_B\rho_{AB}.$$

The label $A$ may denote a spatial region, a set of sites, a species, or a mode collection. State the partition, especially for identical particles where particle and mode partitions are not interchangeable.

The canonical finite-system definitions live in Reduced States and Partial Trace.

Ensemble Notation

The principal equilibrium ensembles are:

Ensemble	Fixed macroscopic data	Density operator
microcanonical	energy window, particle number, volume	normalized projector onto the allowed shell
canonical	temperature, particle number, volume	$e^{-\beta H}/Z$
grand canonical	temperature, chemical potential, volume	$e^{-\beta(H-\mu\hat N)}/\Xi$

The inverse temperature is

$$\beta = \frac{1}{k_{\mathrm B}T}.$$

The canonical partition function is

$$Z(\beta) = \operatorname{Tr} \left( e^{-\beta H} \right),$$

and the canonical state is

$$\rho_\beta = \frac{e^{-\beta H}} {Z(\beta)}.$$

The grand partition function is

$$\Xi(\beta,\mu) = \operatorname{Tr} \left[ e^{-\beta(H-\mu\hat N)} \right],$$

with state

$$\rho_{\beta,\mu} = \frac{ e^{-\beta(H-\mu\hat N)} }{ \Xi(\beta,\mu) }.$$

Use $\hat N$ for the number operator when it could be confused with its expectation or eigenvalue. Once the distinction is established, many-body texts often write $N$ for both; this volume keeps the hat where ambiguity matters.

Thermodynamic Quantities

Common symbols are:

Symbol	Quantity	Defining relation
$U$ or $E$	internal energy	$\langle H\rangle$
$S$	thermodynamic entropy	$-k_{\mathrm B}\operatorname{Tr}(\rho\ln\rho)$
$F$	Helmholtz free energy	$-k_{\mathrm B}T\ln Z$
$\Omega$	grand potential	$-k_{\mathrm B}T\ln\Xi$
$\mu$	chemical potential	conjugate to particle number
$P$	pressure	$-(\partial F/\partial V)_{T,N}$
$C_V$	heat capacity	$(\partial U/\partial T)_{V,N}$

The symbol $S$ may also denote spin, an action, or a structure factor. Context or a subscript should distinguish these meanings.

For a canonical ensemble,

$$U = -\frac{\partial}{\partial\beta} \ln Z$$

when $H$ has no implicit $\beta$ dependence.

For a grand-canonical ensemble,

$$\langle\hat N\rangle = \frac{\partial}{\partial(\beta\mu)} \ln\Xi$$

when $\beta$ is held fixed in that derivative.

Equivalent formulas written as derivatives with respect to $\mu$ must carry the factor of $\beta$ correctly.

Fock-Space Operators

Bosonic annihilation and creation operators are denoted

$$b_i, \qquad b_i^\dagger,$$

and satisfy

$$\begin{aligned} [b_i,b_j^\dagger] &= \delta_{ij}, \\ [b_i,b_j] &=0, \\ [b_i^\dagger,b_j^\dagger] &=0. \end{aligned}$$

Fermionic operators are denoted

$$c_i, \qquad c_i^\dagger,$$

and satisfy

$$\begin{aligned} \{c_i,c_j^\dagger\} &= \delta_{ij}, \\ \{c_i,c_j\} &=0, \\ \{c_i^\dagger,c_j^\dagger\} &=0. \end{aligned}$$

The generic symbol $a_i$ may denote either statistics only after the algebra has been stated.

The mode occupation operators are

$$n_i^{\mathrm B} = b_i^\dagger b_i$$

and

$$n_i^{\mathrm F} = c_i^\dagger c_i.$$

The total number operator is

$$\hat N = \sum_i n_i.$$

Occupation numbers are eigenvalues of $n_i$, not probabilities. Bosonic occupations are nonnegative integers; a single spinless fermionic mode has occupation zero or one.

The operator construction belongs to Creation, Annihilation, and Second Quantization.

Spin and Pauli Notation

For spin-$1/2$ degrees of freedom,

$$S_i^\alpha = \frac{\hbar}{2} \sigma_i^\alpha, \qquad \alpha\in\{x,y,z\}.$$

The Pauli matrices $\sigma^\alpha$ are dimensionless. The spin operators $S^\alpha$ carry units of angular momentum unless $\hbar=1$ has been declared.

The raising and lowering operators may be defined as

$$S_i^\pm = S_i^x \pm iS_i^y,$$

or in dimensionless form using $\sigma_i^\pm$. A page must not switch between $S^\pm$ and $\sigma^\pm$ without the corresponding factor of $\hbar$.

See Spin and Pauli-Matrix Conventions.

Field Operators

Continuum bosonic or fermionic fields are denoted

$$\psi_\sigma(\mathbf r), \qquad \psi_\sigma^\dagger(\mathbf r),$$

where $\sigma$ labels spin, species, flavor, or another internal component.

At equal time, bosonic fields satisfy

$$\left[ \psi_\sigma(\mathbf r), \psi_{\sigma'}^\dagger(\mathbf r') \right] = \delta_{\sigma\sigma'} \delta^{(d)} (\mathbf r-\mathbf r'),$$

while fermionic fields satisfy

$$\left\{ \psi_\sigma(\mathbf r), \psi_{\sigma'}^\dagger(\mathbf r') \right\} = \delta_{\sigma\sigma'} \delta^{(d)} (\mathbf r-\mathbf r').$$

The local number density is

$$n(\mathbf r) = \sum_\sigma \psi_\sigma^\dagger(\mathbf r) \psi_\sigma(\mathbf r).$$

The field operators are operator-valued distributions. Expressions at coincident points may require regularization or a lattice cutoff.

The canonical definitions and mode-expansion construction live in Field Operators. Their use in continuum and lattice many-body models is developed in Field Operators in Many-Body Models.

One-Body and Two-Body Hamiltonians

In a discrete one-particle basis, a number-conserving one-body operator is

$$H_1 = \sum_{ij} h_{ij} a_i^\dagger a_j.$$

A two-body interaction is written

$$H_2 = \frac{1}{2} \sum_{ijkl} V_{ijkl} a_i^\dagger a_j^\dagger a_l a_k.$$

The factor $1/2$ and index ordering depend on how the matrix elements are defined. Antisymmetrized fermionic matrix elements can move numerical factors between $V_{ijkl}$ and the operator sum.

In the continuum, a common form is

$$\begin{aligned} H = {}& \int d^dr\, \psi^\dagger(\mathbf r) h_0 \psi(\mathbf r) \\ &+ \frac{1}{2} \int d^dr\,d^dr'\, \psi^\dagger(\mathbf r) \psi^\dagger(\mathbf r') \\ &\qquad\times V(\mathbf r-\mathbf r') \psi(\mathbf r') \psi(\mathbf r). \end{aligned}$$

Spin or species sums are implicit only when the page says so.

Lattice Operators

On a lattice, $i,j$ label sites and $\langle i,j\rangle$ usually denotes a set of neighboring unordered pairs. A hopping Hamiltonian may be written

$$H_t = -t \sum_{\langle i,j\rangle,\sigma} \left( c_{i\sigma}^\dagger c_{j\sigma} + c_{j\sigma}^\dagger c_{i\sigma} \right).$$

If the sum instead runs over oriented pairs, the Hermitian-conjugate term and counting convention change. The notation $\langle i,j\rangle$ is not self-defining.

For the Hubbard interaction,

$$H_U = U \sum_i n_{i\uparrow}n_{i\downarrow}.$$

The sign of $t$ can sometimes be changed by a gauge transformation on a bipartite lattice, but not in every geometry or flux sector.

Fourier Transforms on a Finite Lattice

For $N_s$ sites at positions $\mathbf r_j$ with periodic boundary conditions, this volume uses the symmetric discrete normalization

$$c_{\mathbf k\sigma} = \frac{1}{\sqrt{N_s}} \sum_j e^{-i\mathbf k\cdot\mathbf r_j} c_{j\sigma},$$

with inverse

$$c_{j\sigma} = \frac{1}{\sqrt{N_s}} \sum_{\mathbf k} e^{i\mathbf k\cdot\mathbf r_j} c_{\mathbf k\sigma}.$$

The allowed $\mathbf k$ values depend on lattice vectors and boundary conditions. With these conventions,

$$\{c_{\mathbf k\sigma}, c_{\mathbf k'\sigma'}^\dagger\} = \delta_{\mathbf k\mathbf k'} \delta_{\sigma\sigma'}.$$

Some references put the entire $1/N_s$ factor in one direction. Formulas for momentum sums and Kronecker deltas must be translated consistently.

Continuum Fourier Transforms

For a continuum field in $d$ dimensions, a symmetric wave-number convention is

$$\psi_\sigma(\mathbf r) = \int \frac{d^dk} {(2\pi)^{d/2}} e^{i\mathbf k\cdot\mathbf r} a_{\mathbf k\sigma},$$

with inverse

$$a_{\mathbf k\sigma} = \int \frac{d^dr} {(2\pi)^{d/2}} e^{-i\mathbf k\cdot\mathbf r} \psi_\sigma(\mathbf r).$$

This uses wave number $\mathbf k$. Momentum is

$$\mathbf p = \hbar\mathbf k.$$

The site-wide Fourier-Transform Conventions page uses momentum-normalized wavefunctions. Translate the measure and normalization when moving between $\mathbf p$ and $\mathbf k$.

Finite-volume box normalization replaces integrals by sums. State the replacement, for example

$$\frac{1}{V} \sum_{\mathbf k} \longrightarrow \int \frac{d^dk}{(2\pi)^d}$$

in a periodic continuum limit.

Correlation Functions

For Heisenberg-picture operators,

$$A(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}.$$

An unordered two-point function is

$$C_{AB}(t) = \langle A(t)B(0)\rangle.$$

The connected function is

$$C_{AB}^{\mathrm c}(t) = \langle A(t)B(0)\rangle - \langle A(t)\rangle \langle B(0)\rangle.$$

For a stationary state commuting with $H$, the correlator depends only on the time difference.

The superscript $\mathrm c$ means connected here. It should not be confused with a creation operator or complex conjugation.

Real-Time Green and Response Functions

A retarded response function is commonly defined by

$$\chi_{AB}^{\mathrm R}(t) = -\frac{i}{\hbar} \theta(t) \langle[A(t),B(0)]\rangle.$$

Some fields use the symbol $G^{\mathrm R}_{AB}$ for the same structure and reserve $\chi$ for a specific physical response. The perturbation convention determines possible signs.

A time-ordered Green function for fermionic fields is often

$$G(1,2) = -i \langle \mathcal T \psi(1) \psi^\dagger(2) \rangle$$

when $\hbar=1$. Other conventions include $1/\hbar$ or change the sign for bosons. Every Green-function page must state the definition before using a diagram or Fourier transform.

The volume will use:

• $G$ for Green functions;
• $\chi$ for response or susceptibility;
• $C$ for generic correlations;
• $S(\mathbf q,\omega)$ for a dynamic structure factor.

Structure Factors

For local density fluctuations

$$\delta n_j = n_j-\langle n_j\rangle,$$

a finite-lattice static structure factor may be defined as

$$S(\mathbf q) = \frac{1}{N_s} \sum_{j\ell} e^{-i\mathbf q\cdot (\mathbf r_j-\mathbf r_\ell)} \langle \delta n_j \delta n_\ell \rangle.$$

The prefactor may instead involve particle number $N$. State the normalization because it changes the scaling of Bragg peaks and sum rules.

A dynamic structure factor adds a time transform and an operator-ordering convention:

$$S(\mathbf q,\omega) \propto \int dt\, e^{i\omega t} \langle \delta n_{\mathbf q}(t) \delta n_{-\mathbf q}(0) \rangle.$$

The proportionality sign is intentional here: the exact $2\pi$, volume, and normalization factors belong on the dedicated definition page.

Imaginary Time

Imaginary time is denoted $\tau$. A thermal Heisenberg operator is

$$A(\tau) = e^{\tau H} A e^{-\tau H}$$

when $\hbar=1$ and $\tau$ has inverse-energy units.

If $\tau$ instead has units of time, the exponent is

$$e^{\tau H/\hbar}.$$

Do not mix these conventions.

Imaginary Time gives the operator-semigroup, heat-kernel, projection, and thermal-trace dictionary with $\hbar$ explicit.

For inverse-temperature units with $\hbar=1$, bosonic and fermionic Matsubara frequencies are

$$\begin{aligned} \omega_n^{\mathrm B} &= \frac{2\pi n}{\beta}, \\ \omega_n^{\mathrm F} &= \frac{(2n+1)\pi}{\beta}. \end{aligned}$$

With explicit $\hbar$ and time-valued $\tau$, the corresponding angular frequencies carry $1/\hbar$ relative to the energy convention.

Entropy Notation

Thermodynamic entropy is

$$S_{\mathrm{th}} = -k_{\mathrm B} \operatorname{Tr} \left( \rho\ln\rho \right).$$

The dimensionless von Neumann entropy is

$$S_{\mathrm{vN}}(\rho) = -\operatorname{Tr} \left( \rho\ln\rho \right).$$

For a reduced state $\rho_A$, entanglement entropy of a global pure state is

$$S_A = -\operatorname{Tr} \left( \rho_A\ln\rho_A \right).$$

Thus

$$S_{\mathrm{th}} = k_{\mathrm B} S_{\mathrm{vN}}$$

for the same density operator, but thermodynamic and entanglement interpretations depend on which state and partition are used.

The logarithm base is natural unless stated. Base-two entropy should be labeled explicitly and is measured in bits.

Limits and Scaling Symbols

The thermodynamic limit is written

$$N,V\to\infty, \qquad \frac{N}{V}=n \quad\text{fixed}.$$

For a lattice,

$$N_s\to\infty$$

with filling, geometry, and coupling scaling specified.

The symbols have distinct meanings:

Symbol	Meaning
$f\sim g$	ratio approaches a stated constant, often one
$f\propto g$	proportional with an unspecified constant
$f\simeq g$	approximate equality in a stated regime
$f=O(g)$	bounded in magnitude by a constant times $g$ asymptotically
$f=o(g)$	$f/g\to0$
$\Delta$	gap, difference, or fluctuation, as locally defined

Do not use $\sim$ merely to mean “roughly similar.”

Intensive and Extensive Quantities

An extensive quantity $X$ scales with system size:

$$X(\lambda N,\lambda V) \simeq \lambda X(N,V)$$

in an appropriate thermodynamic regime.

The associated density

$$x = \frac{X}{V}$$

is intensive. Boundary contributions can scale as $L^{d-1}$ and finite-size corrections can violate exact proportionality.

For long-range interactions, the energy may be superextensive unless the coupling is normalized with system size. Extensive and Intensive Quantities is the canonical scaling audit for limiting densities, additivity, boundary terms, fluctuations, and Kac normalization.

Units

This volume keeps $\hbar$ and $k_{\mathrm B}$ explicit on orientation and derivation pages unless setting them to one materially clarifies advanced notation.

If natural many-body units are used, state them near the first formula:

$$\hbar=1, \qquad k_{\mathrm B}=1, \qquad a=1$$

are three independent choices.

Setting $a=1$ makes momentum dimensionless in lattice units but does not make physical momentum dimensionless after units are restored. Setting $k_{\mathrm B}=1$ makes temperature carry energy units.

The combinations in an equilibrium exponent must be dimensionless:

$$\beta H, \qquad \beta\mu\hat N.$$

Boundary Conditions

Use:

• OBC for open boundary conditions;
• PBC for periodic boundary conditions;
• APBC for antiperiodic boundary conditions;
• twisted boundary conditions with an explicit phase $\theta$.

For a one-dimensional periodic lattice,

$$c_{j+N_s} = c_j.$$

For a twist,

$$c_{j+N_s} = e^{i\theta}c_j.$$

Boundary conditions determine the momentum grid and can change finite-size degeneracies, topology probes, and fermionic sign conventions.

Commonly Overloaded Symbols

Symbol	Possible meanings	Preferred repair
$N$	particles, sites, flavors, large-$N$ rank	use $N_s$, $N_f$, or a sentence
$L$	length, number of sites, Liouvillian	use $N_s$ for sites and $\mathcal L$ for Liouvillian
$S$	entropy, spin, action, structure factor	use $S_{\mathrm{th}}$, $\mathbf S$, $\mathcal S$, $S(\mathbf q)$
$\Omega$	grand potential, angular frequency, volume	use $\omega$ for frequency and $V$ for volume
$\Delta$	gap, detuning, order parameter, finite difference	add a descriptive subscript
$\Gamma$	rate, linewidth, vertex	define units and role
$\chi$	susceptibility, response, spinor	use arguments and indices
$G$	Green function, conductance, Gibbs free energy	use $G^{\mathrm R}$, $\mathcal G$, or descriptive text
$T$	temperature, time ordering, transpose	use $\mathcal T$ for time ordering
$\sigma$	Pauli matrix, spin index, conductivity	distinguish $\boldsymbol\sigma$, index position, or $\sigma(\omega)$

Overloading is acceptable across different pages when local definitions are unambiguous. It is dangerous within one derivation.

Worked Translation: Hubbard Model

Consider

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

The notation contract is:

• $i,j$ label lattice sites;
• $\sigma\in\{\uparrow,\downarrow\}$ labels spin;
• $\langle i,j\rangle$ is an unordered nearest-neighbor bond set;
• $t$ has energy units and is taken real here;
• $U$ is the onsite interaction energy;
• $c_{i\sigma}$ obeys fermionic anticommutation relations;
• $n_{i\sigma}=c_{i\sigma}^\dagger c_{i\sigma}$;
• boundary conditions and lattice geometry remain to be stated;
• particle number

$$\hat N = \sum_{i,\sigma} n_{i\sigma}$$

is conserved.

For a translationally invariant one-dimensional chain with lattice spacing $a$ and PBC,

$$c_{k\sigma} = \frac{1}{\sqrt{N_s}} \sum_j e^{-ikaj} c_{j\sigma}.$$

The kinetic term becomes diagonal:

$$H_t = \sum_{k,\sigma} \epsilon_k c_{k\sigma}^\dagger c_{k\sigma},$$

with

$$\epsilon_k = -2t\cos(ka).$$

The interaction is local in real space but couples several momenta in momentum space. Representation choice exposes one structure while obscuring another.

For the Hilbert space, parameter conventions, exact limits, symmetries, and physical diagnostics behind this notation, see Hubbard Model.

Page-Level Checklist

Before using a many-body formula, ask:

1. Are indices particles, sites, modes, bands, spin, or species?
2. Is the state pure, canonical, grand canonical, or nonequilibrium?
3. Is particle number fixed or fluctuating?
4. Are operators in Schrödinger, Heisenberg, interaction, or imaginary-time form?
5. Are Fourier variables momentum or wave number?
6. Are sums normalized by $N$, $N_s$, or $V$?
7. Is a correlation connected, ordered, symmetrized, retarded, or time ordered?
8. Are $\hbar$, $k_{\mathrm B}$, and $a$ explicit?
9. Which boundary conditions determine the mode grid?
10. Which limit is taken first?

Canonical Boundaries

This page owns the default notation contract for the Many-Body and Quantum Statistical Mechanics volume.

• Operator Conventions owns site-wide abstract-operator notation.
• Units and Constants owns site-wide unit policy.
• Fourier-Transform Conventions owns the wavefunction transform convention.
• Tensor-Product Ordering owns subsystem ordering.
• Density-Matrix Conventions owns density-operator notation.
• Fock Space and Occupation Number owns Fock-space definitions.
• Creation, Annihilation, and Second Quantization owns creation and annihilation algebras.
• Dedicated many-body pages own derivations of ensembles, Green functions, response, structure factors, and model Hamiltonians.

Common Mistakes

• Using $N$ for both sites and particles in the same equation.
• Leaving spin, band, or species sums implicit without warning.
• Confusing an occupation eigenvalue with a probability.
• Omitting the factor relating $S^\alpha$ and $\sigma^\alpha$.
• Moving between wave number and momentum without $\hbar$.
• Combining Fourier formulas with incompatible normalizations.
• Forgetting whether a bond sum is oriented or unordered.
• Double counting a two-body interaction.
• Using a Green function before defining its sign and ordering convention.
• Comparing structure factors normalized by $N$ and by $N_s$ as if identical.
• Mixing energy-valued and time-valued imaginary time.
• Taking a thermodynamic derivative while an implicit parameter also varies.
• Setting $\hbar$, $k_{\mathrm B}$, or $a$ to one silently.
• Writing a thermodynamic limit without density, geometry, or coupling scaling.

Exercises

Dimensionless grand-canonical exponent

Show that $\beta(H-\mu\hat N)$ is dimensionless and state the units of $\beta$ and $\mu$ when $k_{\mathrm B}$ is explicit.

Solution

The operator $H$ has energy units. The number operator $\hat N$ is dimensionless, so $\mu$ has energy units and $\mu\hat N$ also has energy units.

Since

$$\beta = \frac{1}{k_{\mathrm B}T},$$

$\beta$ has inverse-energy units. Therefore

$$\beta(H-\mu\hat N)$$

is dimensionless, as required inside an exponential.

Fourier anticommutator

Using

$$c_{k\sigma} = \frac{1}{\sqrt{N_s}} \sum_j e^{-ik r_j} c_{j\sigma},$$

show that the momentum-space operators obey the canonical anticommutator for the allowed discrete momenta.

Solution

Compute

$$\begin{aligned} \{c_{k\sigma},c_{k'\sigma'}^\dagger\} &= \frac{1}{N_s} \sum_{j\ell} e^{-ikr_j} e^{ik'r_\ell} \\ &\qquad{}\times \{c_{j\sigma},c_{\ell\sigma'}^\dagger\} \\ &= \frac{\delta_{\sigma\sigma'}}{N_s} \sum_j e^{i(k'-k)r_j}. \end{aligned}$$

Discrete Fourier orthogonality gives

$$\frac{1}{N_s} \sum_j e^{i(k'-k)r_j} = \delta_{kk'}.$$

Hence

$$\{c_{k\sigma},c_{k'\sigma'}^\dagger\} = \delta_{kk'} \delta_{\sigma\sigma'}.$$

Number-operator commutator

For fermions, show that

$$[\hat N,c_j] = -c_j.$$

Solution

With

$$\hat N = \sum_i c_i^\dagger c_i,$$

only the $i=j$ term contributes. Using the fermionic algebra,

$$\begin{aligned} [c_j^\dagger c_j,c_j] &= c_j^\dagger c_jc_j - c_jc_j^\dagger c_j \\ &= 0 - (1-c_j^\dagger c_j)c_j \\ &= -c_j. \end{aligned}$$

Thus annihilation lowers total particle number by one. Similarly,

$$[\hat N,c_j^\dagger] = c_j^\dagger.$$

Grand-potential derivative

Given

$$\Omega = -\frac{1}{\beta} \ln\Xi,$$

show that

$$\langle\hat N\rangle = -\left( \frac{\partial\Omega}{\partial\mu} \right)_{\beta,V}.$$

Solution

At fixed $\beta$,

$$\frac{\partial\ln\Xi}{\partial\mu} = \beta \langle\hat N\rangle.$$

Therefore

$$\frac{\partial\Omega}{\partial\mu} = -\frac{1}{\beta} \frac{\partial\ln\Xi}{\partial\mu} = -\langle\hat N\rangle.$$

Rearranging gives the result.

Connected versus disconnected structure

Suppose $\langle A\rangle=2$, $\langle B\rangle=3$, and $\langle AB\rangle=7$. Find the connected correlation and explain what the disconnected contribution is.

Solution

The connected correlation is

$$\langle AB\rangle_{\mathrm c} = 7-(2)(3) =1.$$

The disconnected contribution is

$$\langle A\rangle \langle B\rangle =6.$$

It is the part already explained by the separate one-point expectations. The remaining value measures failure of the two-point expectation to factorize.

Cross-Links

• Many-Body and Quantum Statistical Mechanics
• Symbols and Conventions
• Why Many-Body Physics Is Different
• What Belongs Here vs Quantum Matter vs QFT.org
• Scaling of Hilbert Space
• Thermodynamic Limit
• Thermal Density Operators
• Canonical Ensemble
• Grand-Canonical Ensemble
• Bosonic and Fermionic Matsubara Frequencies
• Occupation-Number Representation
• Field Operators in Many-Body Models
• Hubbard Model
• Operator Conventions
• Units and Constants
• Fourier-Transform Conventions
• Tensor-Product Ordering
• Density-Matrix Conventions
• Spin and Pauli-Matrix Conventions
• Fock Space and Occupation Number
• Creation, Annihilation, and Second Quantization
• Canonical Field Operators
• Correlation Functions

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000).
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015).
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010).
• G. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid, Cambridge University Press (2005).
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011).
• M. Le Bellac, F. Mortessagne, and G. G. Batrouni, Equilibrium and Non-Equilibrium Statistical Thermodynamics, Cambridge University Press (2004).
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press (1998).