Symbols and Conventions

This page is the rapid lookup and translation layer for notation in many-body and statistical quantum mechanics. It answers questions such as “Is $N$ a number or an operator?”, “Does $\mathbf k$ mean momentum or wave number?”, and “Which factors of $N_s$, $2\pi$, $\hbar$, and $k_{\mathrm B}$ are in this formula?”

The deeper explanatory home is Core Objects and Notation. Sitewide choices for bras, operators, density matrices, Fourier transforms, spin, and units live in the Conventions Overview. This page repeats only the minimum data needed for lookup and translation.

A specialized article may override any default below when another convention is standard for that subject. The override must be stated near the first affected formula and used consistently.

Minimum Declaration Block

A reusable calculation is not specified by its Hamiltonian alone. Before interpreting a formula, recover the following data:

Item	Questions that must be answerable
degrees of freedom	particles, sites, modes, spins, bands, flavors, or fields?
state space	fixed-particle Hilbert space, Fock space, symmetry sector, or truncated basis?
statistics and algebra	bosonic commutators, fermionic anticommutators, spin algebra, or an effective algebra?
indices	which labels denote sites, particles, modes, components, and eigenstates?
state or ensemble	pure state, mixed state, microcanonical, canonical, or grand canonical?
geometry	dimension, lattice, physical volume, boundary conditions, and metric if relevant?
normalization	state, field, Fourier-transform, structure-factor, and trace normalization?
ordering	ordinary, normal, time, imaginary-time, contour, or out-of-time ordering?
units	are $\hbar$, $k_{\mathrm B}$, lattice spacing, mass, or a coupling set to one?
limits	which variables become large, small, continuous, or late, and in which order?

The essential rule is:

A symbol identifies an object only together with its index domain, units, normalization, and local definition.

For example, $N$ can denote a particle-number eigenvalue, the number of sites, the number of field components, or the control parameter of a large-$N$ limit. Those meanings are not interchangeable.

Typography at a Glance

Object	Default notation	Interpretation
scalar parameter	$T$, $\mu$, $t$, $J$	a number with stated units
spatial vector	$\mathbf r$, $\mathbf k$, $\mathbf q$	boldface marks a spatial vector
operator	$H$, $A$, $S_i^\alpha$	hats are omitted when operator status is unambiguous
ambiguous number operator	$\hat N$	the hat distinguishes it from $N$ or $\langle\hat N\rangle$
pure state	$	\Psi\rangle$
density operator	$\rho$	positive and trace one for a normalized state
identity	$I$ or $I_{\mathcal H}$	subscript names the space when needed
complex conjugate	$z^*$	scalar conjugation
adjoint	$A^\dagger$	Hilbert-space adjoint
transpose	$A^{\mathsf T}$	basis-dependent transpose
expectation value	$\langle A\rangle$	state or ensemble must be recoverable

Bold symbols are reserved for geometric vectors, not merely for collections of components. An arrow notation such as $\vec k$ may appear in cited literature, but a page should not alternate between $\vec k$ and $\mathbf k$ without reason.

Index Register

These are defaults, not universal laws:

Index	Usual domain	Common role
$i,j,\ell$	lattice sites	local operators and hopping links
$a,b$	orbitals, bands, species, or Cartesian components	must be declared locally
$\sigma,\sigma'$	$\uparrow,\downarrow$ or an internal set	spin or pseudospin component
$\alpha,\beta$	internal or Cartesian components	flavor, spin component, or tensor index
$n,m$	integers or eigenstate labels	energy level, occupation, or Matsubara integer
$\mathbf k,\mathbf q$	reciprocal-space grid or continuum	wave vector and transferred wave vector
$p,q$	generic one-particle modes	not necessarily momentum
$A,B$	subsystems or observables	distinguish by typography and context
$s$	spin projection, species, or contour branch	define before use

Repeated indices are not summed automatically unless an article declares an Einstein convention. Lattice and mode sums are normally written explicitly. Cartesian tensor formulas may use implicit summation after saying so.

Composite labels compress several indices. For example,

$$1 \equiv (\mathbf r_1,\sigma_1,t_1)$$

is acceptable in a Green-function derivation only after the measure associated with $\int d1$ has also been defined.

Size, Geometry, and Density

Symbol	Default meaning
$N$	number of particles or a fixed particle-number eigenvalue
$\hat N$	total particle-number operator
$N_s$	number of lattice sites
$M$	number of retained one-particle modes or orbitals
$N_f$	number of flavors or field components
$V$	physical volume
$L$	linear system size
$d$	spatial dimension
$a$	lattice spacing
$n$	particle density $N/V$
$\nu$	filling $N/N_s$

For an isotropic hypercubic lattice with compatible periodic counting,

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

Density and filling answer different questions:

$$\begin{aligned} n &= \frac{N}{V}, \\ \nu &= \frac{N}{N_s}. \end{aligned}$$

The filling is dimensionless. The density has units of inverse volume. A continuum limit usually changes $a$, $N_s$, and the couplings together; it is not obtained by replacing $N_s$ with $V$ in a lattice formula.

For anisotropic boxes, use $L_1,\ldots,L_d$ and state

$$V = \prod_{a=1}^{d}L_a$$

rather than concealing the shape in one length $L$.

States, Traces, and Ensembles

A pure-state and density-operator expectation are

$$\begin{aligned} \langle A\rangle_\Psi &= \langle\Psi|A|\Psi\rangle, \\ \langle A\rangle_\rho &= \operatorname{Tr}(\rho A). \end{aligned}$$

The subscript may be omitted only when the state or ensemble remains unambiguous. A change from a ground-state expectation to a thermal or disorder-averaged expectation must be visible.

The principal equilibrium symbols are:

Symbol	Meaning
$\beta$	inverse temperature $1/(k_{\mathrm B}T)$
$Z$	canonical partition function
$\Xi$	grand partition function
$\rho_\beta$	canonical Gibbs state
$\rho_{\beta,\mu}$	grand-canonical Gibbs state
$K$	grand Hamiltonian $H-\mu\hat N$ when declared
$\overline{X}$ or $[X]_{\mathrm{dis}}$	disorder average, never a thermal average by default

The canonical and grand-canonical states are

$$\begin{aligned} \rho_\beta &= \frac{e^{-\beta H}}{Z}, & Z &= \operatorname{Tr}e^{-\beta H}, \\ \rho_{\beta,\mu} &= \frac{e^{-\beta K}}{\Xi}, & K &= H-\mu\hat N. \end{aligned}$$

Here $\beta H$ and $\beta\mu\hat N$ are dimensionless. If $K$ is called a Hamiltonian, remember that physical real-time evolution is generated by $H$, not automatically by $K$. The Ensemble Formula Sheet owns the thermodynamic derivative identities and their assumptions.

Use distinct notation for nested averages. In a disordered thermal system, for example,

$$\overline{ \langle A\rangle_J }$$

means: compute the thermal or quantum expectation for one realization $J$, then average over realizations. Reversing those operations generally defines a different object.

Many-Body Operator Register

Symbol	Default role	Required declaration
$b_i,b_i^\dagger$	bosonic mode operators	the mode or site indexed by $i$
$c_i,c_i^\dagger$	fermionic mode operators	the mode, site, spin, and ordering convention
$a_i,a_i^\dagger$	statistics-neutral placeholder	the algebra must be stated first
$n_i$	mode or site occupation operator	species sums and allowed eigenvalues
$\hat N$	total number operator	which species it counts
$\psi_\sigma(\mathbf r)$	continuum field operator	statistics, normalization, and components
$S_i^\alpha$	spin operator	spin representation and units
$\sigma_i^\alpha$	Pauli operator	dimensionless; not the same as $S_i^\alpha$
$H$	physical Hamiltonian	degrees of freedom, parameters, and sector

Bosonic and fermionic mode algebras are, respectively,

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

For either statistics,

$$n_i = a_i^\dagger a_i, \qquad \hat N = \sum_i n_i,$$

but the occupation spectrum depends on the algebra. An occupation number is an eigenvalue of $n_i$, not the probability that the mode exists.

Continuum fields obey a delta-normalized algebra. The nonvanishing equal-time bracket is

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

where the upper commutator choice applies to bosons and the lower anticommutator choice to fermions. Because fields are operator-valued distributions, products at the same point may require a regulator or lattice definition.

Spin and Pauli Translation

For spin $1/2$,

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

Consequently,

$$\mathbf S_i\cdot\mathbf S_j = \frac{\hbar^2}{4} \boldsymbol\sigma_i\cdot\boldsymbol\sigma_j.$$

If a source sets $\hbar=1$ and calls $\mathbf S=\boldsymbol\sigma/2$ “dimensionless spin,” its exchange constant is tied to that choice. Translate the operator and the coupling together. Do not replace $S^\alpha$ by $\sigma^\alpha$ while leaving all numerical coefficients unchanged.

Raising and lowering operators are

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

Some sources instead define $\sigma^\pm=(\sigma^x\pm i\sigma^y)/2$. The factor of $1/2$ is part of the definition, not a cosmetic choice.

Spatial Fourier Transforms

Finite periodic lattice

For $N_s$ sites at positions $\mathbf r_j$, the default symmetric discrete transform is

$$\begin{aligned} c_{\mathbf k\sigma} &= \frac{1}{\sqrt{N_s}} \sum_j e^{-i\mathbf k\cdot\mathbf r_j} c_{j\sigma}, \\ c_{j\sigma} &= \frac{1}{\sqrt{N_s}} \sum_{\mathbf k} e^{i\mathbf k\cdot\mathbf r_j} c_{\mathbf k\sigma}. \end{aligned}$$

This preserves the canonical mode algebra without an extra volume factor. The allowed $\mathbf k$ grid is fixed by the lattice vectors and boundary conditions.

Continuum wave-number convention

For continuum fields in $d$ dimensions,

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

Here $\mathbf k$ is wave number and

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

is physical momentum. A source using $d^dp/(2\pi\hbar)^d$ has made a different but equivalent choice. Translate the integration measure, delta function, and operator normalization together.

For a periodic box, the continuum replacement is

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

The arrow describes a limit under stated regularity assumptions. It is not an exact identity for a finite box.

Time and Frequency Conventions

For real time, the default transform pair for an ordinary function is

$$\begin{aligned} A(\omega) &= \int_{-\infty}^{\infty} dt\, e^{i\omega t} A(t), \\ A(t) &= \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} e^{-i\omega t} A(\omega). \end{aligned}$$

With this sign convention, the Heisenberg operator is

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

A source may reverse both exponential signs. Translate the transform pair before comparing pole locations or response conventions.

Two imaginary-time unit conventions occur frequently:

Convention	Interval	Evolution operator	Matsubara quantity
time-valued $\tau$	$0\le\tau<\beta\hbar$	$e^{\tau H/\hbar}$	angular frequency in inverse time
inverse-energy $\tau$	$0\le\tau<\beta$	$e^{\tau H}$	energy-valued frequency, usually with $\hbar=1$

For time-valued $\tau$, the angular frequencies are

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

The bosonic or fermionic label follows the periodicity of the correlator, not necessarily the microscopic statistics of every operator appearing in it. The dedicated Matsubara-frequency page owns the derivation and endpoint subtleties.

Correlation, Response, and Ordering Symbols

Symbol	Usual role	What still must be stated
$C_{AB}(t)$	generic correlation	ordering, state, and connectedness
$C_{AB}^{\mathrm c}$	connected correlation	subtraction convention
$G$	Green function	fields, sign, factor of $i$ or $\hbar$, and ordering
$G^{\mathrm R},G^{\mathrm A}$	retarded and advanced Green functions	Fourier sign and bracket convention
$G^>,G^<$	greater and lesser Green functions	operator order and statistics sign
$G^{\mathsf T}$ or $G^{\mathrm T}$	time-ordered Green function	avoid confusing superscript with transpose
$\chi_{AB}^{\mathrm R}$	retarded response or susceptibility	perturbation sign and normalization
$S(\mathbf q,\omega)$	dynamic structure factor	Fourier and volume normalization
$\mathcal T$	real-time ordering	equal-time prescription
$\mathcal T_\tau$	imaginary-time ordering	bosonic or fermionic exchange sign
$\theta(t)$	step function	value at $t=0$ if it matters

A generic unordered correlator and its connected part are

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

One common retarded-response convention is

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

This formula is not a universal definition independent of the perturbation. If the source writes the perturbation as $H'(t)=+f(t)B$ rather than $H'(t)=-f(t)B$, the response sign changes.

Thermodynamic Register

Symbol	Quantity	Standard relation or warning
$U$ or $E$	internal energy	$\langle H\rangle$
$S_{\mathrm{th}}$	thermodynamic entropy	carries units of $k_{\mathrm B}$
$S_{\mathrm{vN}}$	von Neumann entropy	dimensionless with natural logarithm
$S_A$	subsystem entropy	partition and logarithm base must be stated
$F$	Helmholtz free energy	natural variables $T,V,N$
$\Omega$	grand potential	natural variables $T,V,\mu$
$\mu$	chemical potential	energy per particle added at fixed entropy and volume
$P$	pressure	do not confuse with parity or a projector
$C_V$	heat capacity	specify fixed variables and whether total or specific
$\kappa_T$	isothermal compressibility	normalization varies with density convention

For the same density operator,

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

but thermodynamic entropy and entanglement entropy answer different physical questions. A lowercase density such as $f=F/V$ or a per-site quantity such as $F/N_s$ should be introduced explicitly; lowercase is not an automatic “per particle” instruction.

Partial derivatives must display their held-fixed variables when more than one thermodynamic interpretation is possible. For example,

$$P = -\left( \frac{\partial F}{\partial V} \right)_{T,N}.$$

Limits and Approximation Symbols

Notation	Meaning
$f\sim g$	$f/g$ approaches a stated nonzero constant, often one
$f\propto g$	equality up to an unspecified multiplicative constant
$f\simeq g$	approximate equality in a declared regime
$f=O(g)$	asymptotic upper-order statement
$f=o(g)$	$f/g\to0$
$x\ll y$	a controlled or physically stated scale separation
$x\to x_0$	a limit, not an ordinary assignment

The thermodynamic limit for a continuum fluid is

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

while a lattice limit takes $N_s\to\infty$ with filling, shape, and coupling scaling specified.

Orders of limits may be physical data. In general,

$$\lim_{t\to\infty} \lim_{N_s\to\infty} C_{N_s}(t) \neq \lim_{N_s\to\infty} \lim_{t\to\infty} C_{N_s}(t).$$

The first can describe late-time behavior after a continuum spectrum emerges; the second probes the exact long-time behavior of each finite system before increasing its size. Write both limits when the distinction matters.

Boundary Conditions

Abbreviation	Meaning	Typical consequence
OBC	open boundary conditions	edges and standing-wave modes
PBC	periodic boundary conditions	discrete crystal momenta and translation symmetry
APBC	antiperiodic boundary conditions	shifted momentum grid
TBC	twisted boundary conditions	flux or stiffness probe through a phase $\theta$

For a one-dimensional lattice,

$$\begin{aligned} c_{j+N_s} &= c_j &&\text{(PBC)}, \\ c_{j+N_s} &= e^{i\theta}c_j &&\text{(twist)}. \end{aligned}$$

Boundary conditions affect the allowed momentum grid, symmetry sectors, finite-size degeneracies, and sometimes fermion-parity bookkeeping. They are not a numerical afterthought.

Units and Restoring Constants

Orientation and derivation pages keep $\hbar$ and $k_{\mathrm B}$ explicit unless suppressing them materially clarifies an advanced calculation. The declarations

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

are three independent choices.

• $\hbar=1$ identifies energy with angular frequency and momentum with wave number.
• $k_{\mathrm B}=1$ gives temperature energy units.
• $a=1$ measures lengths in lattice spacings and wave numbers in inverse lattice spacings.

To restore constants, first use dimensions and then verify the generating equation. Typical replacements include

$$\begin{aligned} e^{-iHt} &\longrightarrow e^{-iHt/\hbar}, \\ e^{-\beta H} &\longrightarrow e^{-H/(k_{\mathrm B}T)}, \\ \mathbf k &\longrightarrow \frac{\mathbf p}{\hbar}. \end{aligned}$$

Dimensional analysis alone may not recover every Fourier normalization or coupling convention. Check the canonical commutator, anticommutator, or partition-function exponent after translating.

Common Symbol Collisions

Symbol	Frequent meanings	Preferred repair on one page
$N$	particles, sites, flavors, large-$N$ rank	use $N_s$, $N_f$, or a descriptive subscript
$L$	length, site count, Liouvillian	use $N_s$ and $\mathcal L$ for the latter two
$S$	entropy, spin, action, structure factor	use $S_{\mathrm{th}}$, $\mathbf S$, $\mathcal S$, and $S(\mathbf q)$
$\Omega$	grand potential, volume, angular frequency	use $V$ for volume and $\omega$ for frequency
$\Delta$	gap, order parameter, detuning, finite difference	add $\Delta_{\mathrm{gap}}$, $\Delta_{\mathrm{sc}}$, or prose
$\Gamma$	rate, linewidth, hybridization, vertex	attach arguments or a descriptive subscript
$G$	Green function, Gibbs free energy, conductance	use $G^{\mathrm R}$, $G_{\mathrm{Gibbs}}$, or context-specific notation
$\chi$	susceptibility, field, spinor	expose arguments and indices
$T$	temperature, ordering, transpose	use $\mathcal T$ and $\mathsf T$ for the latter two
$P$	pressure, parity, projector, probability	use $\Pi$ for a projector or a descriptive subscript
$\sigma$	spin index, Pauli matrix, conductivity	distinguish index position, $\boldsymbol\sigma$, and $\sigma(\omega)$
$J$	exchange, hopping, current, disorder realization	use $t$ for hopping and label currents explicitly

Overloading across separate pages is manageable. Overloading inside one derivation is a source of false identities. The sitewide Notation Collisions page gives a broader repair guide.

Worked Convention Audit

Consider the finite-lattice density structure factor

$$\begin{aligned} S(\mathbf q,\omega) &= \frac{1}{2\pi N_s} \int_{-\infty}^{\infty} dt\, e^{i\omega t} \\ &\quad\times \sum_{j\ell} e^{-i\mathbf q\cdot(\mathbf r_j-\mathbf r_\ell)} \\ &\quad\times \langle \delta n_j(t) \delta n_\ell(0) \rangle, \\ \delta n_j &= n_j-\langle n_j\rangle. \end{aligned}$$

The formula is interpretable because its convention data are visible:

1. $j$ and $\ell$ are site indices, while $N_s$ is the site count.
2. $\mathbf q$ is a reciprocal-space wave vector fixed by the boundary conditions.
3. The $1/N_s$ normalization makes the quantity intensive away from Bragg scaling.
4. The $1/(2\pi)$ factor belongs to the chosen forward time transform.
5. The correlator is unordered and built from connected density fluctuations.
6. The expectation brackets still require a declared state or ensemble.
7. $n_j$ may include one species or a species sum; the model must say which.

Another source may omit $1/(2\pi)$, normalize by particle number $N$, or use $e^{-i\omega t}$. None of those choices is automatically wrong. A comparison must translate all three choices before applying a sum rule or comparing spectral weights.

Author Checklist

Before publishing a many-body formula, check that the page states:

• whether $N$, $N_s$, and $\hat N$ are numbers or operators;
• every index domain and every implicit sum;
• bosonic, fermionic, spin, or effective operator algebra;
• the state, ensemble, and any disorder average;
• whether $H$ or $K=H-\mu\hat N$ appears in imaginary-time evolution;
• the Fourier-transform signs and normalization;
• wave number $\mathbf k$ versus momentum $\mathbf p$;
• factors of $\hbar$, $k_{\mathrm B}$, $a$, $V$, and $N_s$ set to one;
• boundary conditions and momentum grid;
• operator ordering and equal-time prescription;
• connected versus full correlations;
• extensive, per-site, per-particle, or density normalization;
• every approximation regime and order of limits.

Common Mistakes

1. Using $N$ for particles and sites. Replace the site count by $N_s$ and define the filling $\nu=N/N_s$.
2. Calling $\mathbf k$ momentum after setting $\hbar=1$. State whether it is wave number and restore $\mathbf p=\hbar\mathbf k$ when dimensions matter.
3. Changing one Fourier factor. Measures, delta functions, inverse transforms, and mode algebras must change together.
4. Using $S^\alpha$ and $\sigma^\alpha$ interchangeably. For spin $1/2$, they differ by $\hbar/2$.
5. Hiding the ensemble. An unsubscripted bracket is safe only when the state remains fixed throughout the argument.
6. Confusing $H$ and the grand Hamiltonian. $K=H-\mu\hat N$ organizes grand-canonical weights; it is not automatically the physical generator of real time.
7. Treating $\sim$ as prose for “approximately.” State the limiting ratio or use $\simeq$ with a regime.
8. Taking limits silently. Finite size, continuum, zero temperature, large-$N$, and late time can fail to commute.

Cross-Links

• Core Objects and Notation
• Ensemble Formula Sheet
• Quantum Gas Formula Sheet
• Fermi Gas Formula Sheet
• Bose Gas Formula Sheet
• Common Many-Body Hamiltonians
• Operator Identities
• Occupation-Number Representation
• Field Operators in Many-Body Models
• Correlation Functions Overview
• Thermodynamic Limit
• Bosonic and Fermionic Matsubara Frequencies
• Operator Conventions
• Commutator Conventions
• Fourier-Transform Conventions
• Spin and Pauli-Matrix Conventions
• Units and Constants

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003) — operator, Green-function, and many-particle notation.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000) — real- and imaginary-time Green functions and response conventions.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press (1998) — fields, thermal traces, and imaginary-time methods.
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010) — continuum, lattice, and field-theory translations.
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015) — lattice models, spin notation, and many-body scales.
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011) — ensemble and thermodynamic notation.

Exercises

1. Number, site count, and filling

A paper writes “take $N\to\infty$ at half filling” for a spin-$1/2$ fermion lattice with one orbital per site. Rewrite the statement so that particle number, site count, and filling cannot be confused. State what half filling means in this model.

Solution

Write the lattice thermodynamic limit as

$$N_s\to\infty, \qquad \nu = \frac{N}{N_s} = 1 \quad\text{fixed}.$$

There are two spin orbitals per site, so the maximum filling is $\nu=2$. Half filling therefore means one fermion per site on average, $N=N_s$, not $N=N_s/2$.

2. Fourier normalization and the mode algebra

Assume fermionic site operators obey $\{c_j,c_\ell^\dagger\}=\delta_{j\ell}$. Using

$$c_k = \frac{1}{\sqrt{N_s}} \sum_j e^{-ik r_j}c_j,$$

show that $\{c_k,c_{k'}^\dagger\}=\delta_{kk'}$ on the compatible periodic momentum grid.

Solution

Direct substitution gives

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

Discrete Fourier orthogonality on the periodic grid yields

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

which proves the claim. Moving the normalization factor to only one transform is possible, but the inverse transform and the orthogonality relation must then be changed consistently.

3. Spin versus Pauli exchange

Translate

$$H = J_S \sum_{\langle ij\rangle} \mathbf S_i\cdot\mathbf S_j$$

for spin $1/2$ into Pauli-operator notation. Give the relation between the coefficient $J_\sigma$ multiplying $\boldsymbol\sigma_i\cdot\boldsymbol\sigma_j$ and $J_S$.

Solution

Using $\mathbf S_i=(\hbar/2)\boldsymbol\sigma_i$,

$$H = \frac{J_S\hbar^2}{4} \sum_{\langle ij\rangle} \boldsymbol\sigma_i\cdot\boldsymbol\sigma_j.$$

Therefore

$$J_\sigma = \frac{\hbar^2}{4} J_S.$$

If a source defines dimensionless spin operators $\mathbf s=\boldsymbol\sigma/2$, then its coefficient convention differs again. The operator definition and coupling must be translated as one unit.

4. Restoring thermal units

An advanced calculation sets $\hbar=k_{\mathrm B}=1$ and writes

$$\begin{aligned} \rho &= \frac{e^{-(H-\mu\hat N)/T}}{\Xi}, \\ A(t) &= e^{iHt}Ae^{-iHt}. \end{aligned}$$

Restore $\hbar$ and $k_{\mathrm B}$.

Solution

The thermal exponent must be dimensionless, so

$$\rho = \frac{ \exp\!\left[-(H-\mu\hat N)/(k_{\mathrm B}T)\right] }{\Xi}.$$

Real-time evolution requires an action divided by $\hbar$:

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

No factor of $\hbar$ belongs in the equilibrium Gibbs exponent because $k_{\mathrm B}T$ already has energy units.

5. Order of limits

Explain why the statements “a finite isolated system has an exactly discrete spectrum” and “a thermodynamic correlation function decays irreversibly” need not contradict one another.

Solution

At fixed finite $N_s$, exact unitary dynamics has a discrete set of energy differences. Correlators can show recurrences, and an infinite-time limit may retain finite-size structure. Taking $N_s\to\infty$ first can produce a continuous set of frequencies and dephasing on all fixed observation times. Thus

$$\lim_{t\to\infty} \lim_{N_s\to\infty} C_{N_s}(t)$$

can differ from

$$\lim_{N_s\to\infty} \lim_{t\to\infty} C_{N_s}(t).$$

The apparent contradiction came from suppressing the order of limits.