Off-Diagonal Long-Range Order

Off-diagonal long-range order (ODLRO) is macroscopic coherence encoded in a reduced density matrix. Its basis-independent signature is an eigenvalue with a nonzero ratio to particle number in the thermodynamic limit:

$$\begin{gathered} \text{bosonic one-body condensation} \\ \Longleftrightarrow \\ N_0/N\longrightarrow f_0>0, \\ \text{fermion-pair condensation} \\ \Longleftrightarrow \\ \Lambda_0/N\longrightarrow\lambda_0>0. \end{gathered}$$

Here $N_0$ is an eigenvalue of the one-body reduced density matrix, whereas $\Lambda_0$ is an eigenvalue of the fermionic two-body density matrix on antisymmetric pair space. The two criteria have the same logic but act on different body ranks because Pauli exclusion prevents a fermionic one-body eigenvalue from becoming macroscopic.

The word “off-diagonal” does not mean that some matrix element happens to be nonzero in a chosen basis. It refers historically to matrix elements between configurations that remain macroscopically separated. The spectral criterion is safer: eigenvalues survive basis changes, while individual diagonal and off-diagonal entries do not.

ODLRO supplies a number-conserving language for condensates. A fixed-particle-number state can have

$$\langle\psi(x)\rangle = 0$$

or

$$\langle\psi_\downarrow(x)\psi_\uparrow(x')\rangle = 0$$

and still possess a macroscopic density-matrix eigenvalue. This distinction connects exact finite systems, symmetry-preserving thermodynamic states, and broken-symmetry approximations without identifying them.

Helpful background. One-Body Operators and Two-Body Operators define the density matrices used here. Long-Range Order explains spatial limits, while Bose–Einstein Condensation develops the bosonic occupation criterion and ideal-gas examples.

This page connects one-body and pair coherence. Superfluidity and Superconductivity explain the additional response measurements needed to characterize flow or electromagnetic rigidity.

Macroscopic eigenvalues and thermodynamic limits

ODLRO is a statement about a sequence of systems. Unless stated otherwise:

• $N$ particles occupy a region of volume $V=L^d$;
• $N,V\to\infty$ at fixed density $n=N/V$;
• the body rank of the reduced density matrix remains fixed;
• reduced density matrices are unnormalized, so their traces count particles or ordered particle tuples;
• the thermodynamic limit is taken before the large-separation limit;
• a macroscopic eigenvalue means an eigenvalue proportional to $N$, with a nonzero limiting ratio.

At finite size, every density matrix merely has finite eigenvalues. It is therefore more precise to say that a finite system shows scaling consistent with ODLRO than to claim a strict long-range limit from one size.

For variable particle number, $N$ in ratios should be replaced by $\langle N\rangle$ and number fluctuations should be reported. Comparing a canonical calculation at fixed $N$ with a grand-canonical calculation at fixed chemical potential can otherwise confuse ensemble effects with condensation.

The Reduced-Density-Matrix Hierarchy

For one field species, define the unnormalized $p$-body kernel

$$\Gamma^{(p)} (x_1,\ldots,x_p; x_1',\ldots,x_p') ={} \left\langle \psi^\dagger(x_1') \cdots \psi^\dagger(x_p') \psi(x_p) \cdots \psi(x_1) \right\rangle .$$

Each $x$ may include position, spin, species, band, or another one-particle label. In a fixed-$N$ sector,

$$\operatorname{Tr}\Gamma^{(p)} = N(N-1)\cdots(N-p+1).$$

The first two levels are

$$\gamma = \Gamma^{(1)}, \qquad \Gamma^{(2)}.$$

They answer different questions:

Kernel	Natural modes	Macroscopic coherence diagnosed here
bosonic $\gamma$	one-particle orbitals	Bose condensation
fermionic $\gamma$	one-particle orbitals	no one-body ODLRO because occupations are Pauli bounded
fermionic $\Gamma^{(2)}$	antisymmetric pair wavefunctions	pair condensation
bosonic $\Gamma^{(p)}$	symmetric $p$-particle modes	higher-body coherence, with normalization-dependent powers of $N$

The last row warns against a common overgeneralization. In the fixed-number product condensate with all $N$ bosons in one orbital, the leading $p$-body eigenvalue equals the falling factorial $N(N-1)\cdots(N-p+1)$, not merely as $N$. The $O(N)$ two-body criterion discussed below is specifically the conventional fermion-pair criterion on normalized antisymmetric pair space.

Bosonic One-Body ODLRO

One-body kernel

The one-body reduced density matrix is

$$\gamma(x,x') = \left\langle \psi^\dagger(x') \psi(x) \right\rangle .$$

In an orthonormal one-particle basis,

$$\gamma_{ij} = \left\langle a_j^\dagger a_i \right\rangle .$$

It is Hermitian:

$$\gamma(x,x')^* = \gamma(x',x).$$

It is positive semidefinite. For a normalized orbital $f$ with

$$a_f = \int d^d x\, f^*(x)\psi(x),$$

one has

$$\langle f\vert\gamma\vert f\rangle = \langle a_f^\dagger a_f\rangle \geq 0.$$

Its trace is the mean particle number:

$$\operatorname{Tr}\gamma = \int d^d x\, \gamma(x,x) = \langle N\rangle .$$

These properties make $\gamma$ a positive one-particle operator even though it is not normalized to unit trace.

Natural orbitals and occupations

The spectral problem is

$$\int d^d x'\, \gamma(x,x') \phi_\alpha(x') = N_\alpha\phi_\alpha(x).$$

The orthonormal eigenfunctions $\phi_\alpha$ are natural orbitals, and the eigenvalues $N_\alpha$ are natural occupations. The spectral decomposition is

$$\gamma(x,x') = \sum_\alpha N_\alpha \phi_\alpha(x) \phi_\alpha^*(x').$$

Positivity and the trace rule imply

$$N_\alpha \geq 0, \qquad \sum_\alpha N_\alpha = \langle N\rangle .$$

A change of one-particle basis conjugates $\gamma$ by a unitary operator. It changes the entries and coordinates of the natural orbitals but not the multiset of eigenvalues.

Penrose–Onsager criterion

The complete bosonic treatment and its ideal-gas applications belong to Bose–Einstein Condensation. The criterion is recorded here in the notation needed to compare one-body and pair ODLRO.

A simple bosonic condensate has one extensive natural occupation:

$$\lim_{N\to\infty} \frac{N_0}{N} = f_0 > 0,$$

while

$$\frac{N_{\alpha>0}}{N} \longrightarrow 0.$$

The number $f_0$ is the condensate fraction in this convention. This definition does not require the condensate orbital to be an eigenstate of the bare one-particle Hamiltonian. In an interacting, disordered, rotating, or trapped system, $\phi_0$ is determined by the many-body state through $\gamma$.

A large occupation at one finite size is not yet a limiting statement. One must test whether $N_0$ remains proportional to $N$ along a controlled thermodynamic sequence.

From a Macroscopic Eigenvalue to a Spatial Limit

Homogeneous zero-momentum condensate

In a translation-invariant box, write

$$\gamma(\mathbf r) = \left\langle \psi^\dagger(\mathbf0) \psi(\mathbf r) \right\rangle .$$

With plane-wave occupations $n_{\mathbf k}$,

$$\gamma(\mathbf r) = \frac{1}{V} \sum_{\mathbf k} n_{\mathbf k} e^{i\mathbf k\cdot\mathbf r}.$$

If the $\mathbf k=\mathbf0$ mode has

$$\frac{n_{\mathbf0}}{V} \longrightarrow n_0 > 0$$

and the noncondensed momentum distribution is sufficiently regular, its Fourier transform decays at large separation. Therefore

$$\lim_{\lvert\mathbf r\rvert\to\infty} \lim_{V\to\infty} \gamma(\mathbf r) = n_0.$$

Since $n=N/V$,

$$f_0 = \frac{n_0}{n}.$$

The real-space plateau and the extensive plane-wave occupation are then equivalent descriptions of the same simple homogeneous condensate.

Finite-momentum condensate

If the macroscopic orbital is a plane wave at $\mathbf k_0$, then

$$\gamma(\mathbf r) = n_0 e^{i\mathbf k_0\cdot\mathbf r} + \gamma_{\mathrm{reg}}(\mathbf r),$$

and the phase-demodulated limit is

$$\lim_{\lvert\mathbf r\rvert\to\infty} \lim_{V\to\infty} e^{-i\mathbf k_0\cdot\mathbf r} \gamma(\mathbf r) = n_0.$$

Condensation therefore need not mean zero momentum. Rotation, synthetic gauge fields, multiple band minima, or driven settings can select nonzero momentum. The ordering wavevector must be diagnosed rather than assumed.

Why the order of limits matters

At fixed finite $V$, periodic images, exact recurrences, a largest available separation, and symmetry-induced level splittings prevent a literal infinite-distance limit. The meaningful order is

$$\lim_{\lvert\mathbf r\rvert\to\infty} \lim_{V\to\infty},$$

not

$$\lim_{V\to\infty} \lim_{\lvert\mathbf r\rvert\to\infty}.$$

This is the same thermodynamic discipline required for other forms of long-range order. A plateau observed only near $L/2$ can be a boundary or periodic-image artifact.

Inhomogeneous systems

In a trap or an explicitly inhomogeneous state, translation averaging is unavailable. A simple condensate contributes

$$\gamma(x,x') = N_0 \phi_0(x) \phi_0^*(x') + \gamma_{\mathrm{reg}}(x,x').$$

The condensate orbital may decay toward the edge, so no position-independent plateau exists even when $N_0=O(N)$. The spectrum of $\gamma$ is the canonical diagnostic; a separation-only fit is not.

The thermodynamic limit for a trap must also be declared. Holding the trapping frequency fixed while increasing $N$ is not the same sequence as weakening the trap so the cloud size grows at fixed central density.

Simple and Fragmented Condensation

A condensate is fragmented when two or more natural occupations remain extensive:

$$\lim_{N\to\infty} \frac{N_\alpha}{N} = f_\alpha > 0$$

for more than one $\alpha$. The macroscopic eigenspace, rather than any one vector inside a degenerate subspace, is then the invariant object.

Fragmentation can reflect:

• exact symmetries or conserved quantities;
• degenerate one-particle minima;
• multicomponent structure;
• rotation or finite-momentum sectors;
• strong correlations and interaction-driven constraints;
• averaging over symmetry-related condensates.

Two large occupations in an arbitrary orbital basis do not prove fragmentation. One must diagonalize $\gamma$. Conversely, a nearly degenerate pair of large finite-size eigenvalues may collapse to one extensive eigenvalue after a symmetry-selecting field is applied. Genuine fragmentation and finite-size symmetry restoration must therefore be distinguished by controlled limits.

For a simple condensate, the ratio

$$\frac{N_1}{N_0} \longrightarrow 0$$

is often informative. It is not universal: critical, low-dimensional, or nearly degenerate systems can have several subextensive eigenvalues with nontrivial scaling.

Fixed Number and Broken-Symmetry Fields

A fixed-number condensate

Let

$$\lvert N:\phi_0\rangle = \frac{ \left(a_0^\dagger\right)^N }{ \sqrt{N!} } \lvert0\rangle$$

be a number state with every boson in $\phi_0$. Number conservation gives

$$\langle N:\phi_0\vert \psi(x) \vert N:\phi_0\rangle = 0.$$

Nevertheless,

$$\gamma(x,x') = N \phi_0(x) \phi_0^*(x'),$$

so $\gamma$ has the eigenvalue $N$. The state has maximal one-body ODLRO even though its field expectation vanishes exactly.

A phase-selected description

In a source-selected or symmetry-breaking state, define

$$\Phi(x) = \langle\psi(x)\rangle.$$

If the selected phase clusters,

$$\gamma(x,x') \longrightarrow \Phi(x)\Phi^*(x')$$

when the two points separate through the bulk. The outer product generates the macroscopic eigenvalue.

A phase average can restore

$$\langle\psi(x)\rangle = 0$$

while retaining the same number-neutral kernel $\gamma$. Thus the anomalous one-point field is a useful representative in a selected phase, whereas the one-body density matrix is the number-conserving diagnostic.

What is and is not equivalent

For a simple homogeneous condensate under appropriate clustering assumptions, the following descriptions agree in the thermodynamic limit:

$$\begin{gathered} N_0/N\longrightarrow f_0>0, \\ \gamma(\mathbf r) \longrightarrow n_0e^{i\mathbf k_0\cdot\mathbf r}, \\ \Phi(\mathbf r) \neq 0 \quad \text{in a selected phase}. \end{gathered}$$

The qualifications matter. Fragmentation, traps, gauge fields, finite-size symmetry restoration, and algebraic order can break a naive one-to-one translation among these statements.

Why Fermions Need a Pair Density Matrix

For a normalized fermionic orbital $f$,

$$n_f = c_f^\dagger c_f$$

is a projector. Therefore

$$0 \leq \langle n_f\rangle \leq 1.$$

The variational characterization of the largest eigenvalue then gives

$$0 \leq \nu_\alpha \leq 1$$

for every eigenvalue $\nu_\alpha$ of the fermionic one-body density matrix. Since

$$\sum_\alpha\nu_\alpha = N,$$

fermions need at least $N$ occupied one-particle natural orbitals in a fixed-$N$ state rather than placing $O(N)$ particles in one orbital. A normal Fermi sea can have a slowly decaying one-body kernel and a sharp Fermi surface, but it cannot have bosonic one-body ODLRO.

Pairing changes the body rank of the question. A pair mode can be macroscopically occupied even though neither constituent one-particle orbital is.

Fermionic Pair ODLRO

Pair matrix on antisymmetric space

Choose an orthonormal one-particle basis and label an unordered fermion pair by

$$A = (a,b), \qquad a<b.$$

Define

$$P_A = c_b c_a, \qquad P_A^\dagger = c_a^\dagger c_b^\dagger.$$

The pair density matrix is

$$\mathcal M_{AB} = \left\langle P_A^\dagger P_B \right\rangle.$$

It is positive semidefinite because, for

$$B_u = \sum_A u_A P_A,$$

one has

$$u^\dagger\mathcal M u = \langle B_u^\dagger B_u\rangle \geq 0.$$

In a fixed-$N$ sector its trace on unordered pair space is

$$\operatorname{Tr}\mathcal M = \frac{N(N-1)}{2}.$$

The full ordered-index convention used for $\Gamma^{(2)}$ has trace $N(N-1)$. In a normalized antisymmetric basis, the kernel $\Gamma^{(2)}$ has matrix $2\mathcal M^T$ with the index order used here. Its eigenvalues are therefore twice those of $\mathcal M$. Transposition also conjugates the eigenvectors; it does not change eigenvalues.

Yang criterion

Diagonalize

$$\sum_B \mathcal M_{AB} u_B^{(\alpha)} = \Lambda_\alpha u_A^{(\alpha)}.$$

Fermionic pair ODLRO is present when

$$\lim_{N\to\infty} \frac{\Lambda_0}{N} = \lambda_0 > 0.$$

The leading eigenvector $u$ specifies the pair annihilator $B_u=\sum_Au_AP_A$. The corresponding pair creation wavefunction has coefficients $u_A^*$ in the normalized states $P_A^\dagger|0\rangle$. Its labels contain both constituent orbitals, so it carries the internal spin, orbital, band, and relative-coordinate structure of the pair.

Although $\operatorname{Tr}\mathcal M=O(N^2)$, ordinary uncorrelated pairs distribute that trace over $O(N^2)$ pair modes with eigenvalues of order one. Pair ODLRO concentrates an order-$N$ weight into one pair eigenmode. That concentration is the nontrivial statement.

In the present unordered-pair convention, Yang’s bound gives $\Lambda_0\leq N/2$, so $2\Lambda_0/N$ lies in $[0,1]$. For even particle number $N$ in an even-dimensional one-particle space of dimension $D$,

$$\Lambda_0\leq\frac{N(D-N+2)}{2D}.$$

The normalization bound is exact, although interpreting the ratio as a literal probability for distinguishable molecules would be misleading for overlapping Cooper pairs. With the ordered kernel, divide its leading eigenvalue by $N$, not by $N/2$. Yang’s theorem and its normalization are derived in Solovej’s notes, Theorem 8.12.

Worked example: fixed-number coherent pairs

Take $K$ disjoint orbital pairs and a normalized equal-amplitude superposition of all configurations with $m$ occupied pairs, so $N=2m$. In the block of those $K$ pair modes, combinatorial counting gives

$$\langle P_i^\dagger P_i\rangle=\frac mK, \qquad \langle P_i^\dagger P_j\rangle= \frac{m(K-m)}{K(K-1)}\quad(i\ne j).$$

The uniform vector has eigenvalue

$$\Lambda_0=\frac mK+(K-1)\frac{m(K-m)}{K(K-1)} =\frac{m(K-m+1)}K.$$

At fixed $m/K=f$ with $0<f<1$, $\Lambda_0/N\to(1-f)/2>0$, even though every $\langle P_i\rangle$ vanishes by exact number conservation. For $K=4,m=2$, $\Lambda_0=3/2$ and $2\Lambda_0/N=3/4$. This state saturates Yang’s sharper bound with $D=2K$; at full filling $m=K$, its largest eigenvalue is only $1$.

Pair Wavefunctions in Real Space

Two-body kernel

Using the first-pair-as-annihilation convention,

$$\Gamma^{(2)} (1,2;1',2') ={} \left\langle \psi^\dagger(1') \psi^\dagger(2') \psi(2) \psi(1) \right\rangle .$$

The pair eigenproblem is

$$\int d1'\,d2'\, \Gamma^{(2)} (1,2;1',2') \times \Phi_\alpha(1',2') = 2\Lambda_\alpha \Phi_\alpha(1,2).$$

The $\Phi_\alpha$ are normalized over both ordered coordinates; the factor of two matches the preceding unordered-pair convention. For fermions,

$$\Phi_\alpha(1,2) = - \Phi_\alpha(2,1).$$

The antisymmetry can be carried by spin, spatial, orbital, or combined internal structure. Saying only “the pair wavefunction” is incomplete until the one-particle labels and normalization are declared.

Relative and center-of-mass coordinates

For a continuum pair, introduce

$$\mathbf R = \frac{\mathbf r_1+\mathbf r_2}{2}, \qquad \mathbf r = \mathbf r_1-\mathbf r_2.$$

The internal coordinate $\mathbf r$ describes pair size and symmetry. The center coordinate $\mathbf R$ describes the collective motion of pairs. A projected pair annihilator is

$$\Delta_f(\mathbf R) ={} \int d^d r\, f(\mathbf r) \times \psi_\downarrow \left( \mathbf R-\frac{\mathbf r}{2} \right) \times \psi_\uparrow \left( \mathbf R+\frac{\mathbf r}{2} \right).$$

The form factor $f$ selects an internal channel. On a lattice, the integral becomes a sum over onsite, bond, plaquette, or longer-range relative coordinates.

Pair-correlation limit

Define

$$P_f(\mathbf R,\mathbf R') = \left\langle \Delta_f^\dagger(\mathbf R) \Delta_f(\mathbf R') \right\rangle .$$

For a homogeneous zero-momentum pair condensate,

$$\lim_{\lvert\mathbf R-\mathbf R'\rvert\to\infty} \lim_{V\to\infty} P_f(\mathbf R,\mathbf R') = \mathcal P_f > 0$$

when $f$ has nonzero overlap with the condensed internal mode. The pair size is held microscopic while the separation between pair centers becomes macroscopic.

A finite-momentum pair condensate instead has

$$P_f(\mathbf R,\mathbf R') \longrightarrow \mathcal P_f e^{i\mathbf Q\cdot(\mathbf R'-\mathbf R)}.$$

This includes pair-density-wave and $\eta$-pairing-type structures. Demodulating by the candidate $\mathbf Q$ or diagonalizing the full pair matrix avoids averaging the signal to zero.

The limit is channel dependent. Projecting onto an $s$-wave form factor can miss $d$-wave order, and summing bonds with the wrong relative signs can cancel the leading eigenvector.

The fixed-number example has one unordered-pair eigenvalue $3/2$ and twenty-seven eigenvalues $1/6$, summing to $\binom42=6$. The ordered kernel doubles these eigenvalues and its trace. This finite example checks normalization; true pair ODLRO follows from the increasing-size sequence shown at the bottom.

The BCS Benchmark

The BCS state provides a transparent illustration, not the definition. Let

$$b_{\mathbf k} = c_{-\mathbf k\downarrow} c_{\mathbf k\uparrow},$$

and define

$$F_{\mathbf k} = \langle b_{\mathbf k}\rangle, \qquad n_{\mathbf k} = \left\langle c_{\mathbf k\uparrow}^\dagger c_{\mathbf k\uparrow} \right\rangle .$$

For the standard spin-balanced Gaussian BCS state, Wick’s theorem gives

$$\begin{aligned} M_{\mathbf k\mathbf k'} &= \left\langle b_{\mathbf k}^\dagger b_{\mathbf k'} \right\rangle \\ &= F_{\mathbf k}^* F_{\mathbf k'} + \delta_{\mathbf k\mathbf k'} n_{\mathbf k}^2. \end{aligned}$$

The first term is rank one. Its nonzero eigenvalue is

$$\sum_{\mathbf k} \lvert F_{\mathbf k}\rvert^2.$$

When $F_{\mathbf k}$ remains appreciable over an extensive set of momentum modes,

$$\sum_{\mathbf k} \lvert F_{\mathbf k}\rvert^2 = O(V) = O(N).$$

The diagonal term has bounded entries, so it changes the leading eigenvalue by at most an order-one amount under the standard thermodynamic assumptions. The pair matrix therefore has one extensive eigenvalue.

This calculation separates several objects that are often conflated:

Object	Role
$F_{\mathbf k}$	anomalous pair amplitude in a broken-number Gaussian state
$\Delta_{\mathbf k}$	self-consistent mean field entering the BCS Hamiltonian
leading eigenvector of $M$	number-conserving condensed pair mode
quasiparticle gap	spectral excitation scale
$\Lambda_0$	occupation scale of the condensed pair mode

These objects are related within BCS mean field, but they are not definitions of one another. In particular,

$$\sum_{\mathbf k} \lvert F_{\mathbf k}\rvert^2$$

is generally not $N/2$. Weak-coupling Cooper pairs overlap strongly; they are not a gas of distinguishable, tightly bound molecules whose number can be read off by dividing the fermion count by two.

In an exact fixed-$N$ state or a number-projected BCS state,

$$F_{\mathbf k} = 0$$

by the number selection rule. The neutral matrix

$$M_{\mathbf k\mathbf k'} = \langle b_{\mathbf k}^\dagger b_{\mathbf k'}\rangle$$

can still retain its extensive eigenvalue. This is precisely why the density-matrix criterion is more fundamental than the anomalous average.

Basis Invariance and Gauge Covariance

Ordinary basis changes

Under a one-particle unitary transformation $U$,

$$\gamma \longmapsto U\gamma U^\dagger$$

up to the chosen index convention. The eigenvalues are unchanged. The induced transformation on antisymmetric pair space similarly conjugates $\mathcal M$, so its eigenvalues are unchanged.

A single entry such as

$$\langle a_i^\dagger a_j\rangle$$

can become diagonal, off diagonal, or zero after a basis rotation. ODLRO cannot be defined by that entry alone.

Global number rotations

Under

$$\psi \longmapsto e^{i\theta}\psi,$$

the one-point field and pair amplitude transform as

$$\langle\psi\rangle \longmapsto e^{i\theta}\langle\psi\rangle, \qquad \langle\psi\psi\rangle \longmapsto e^{2i\theta}\langle\psi\psi\rangle.$$

The neutral kernels

$$\langle\psi^\dagger(x')\psi(x)\rangle$$

and

$$\langle\Delta^\dagger(\mathbf R)\Delta(\mathbf R')\rangle$$

are invariant under a spatially uniform number rotation. This is why they can diagnose coherence in fixed-number states.

Local gauge transformations

For a field of charge $q$, a local gauge transformation acts as

$$\psi(x) \longmapsto e^{iq\chi(x)/\hbar} \psi(x).$$

The one-body kernel transforms covariantly:

$$\gamma(x,x') \longmapsto e^{iq[\chi(x)-\chi(x')]/\hbar} \gamma(x,x').$$

For a prescribed classical background field, this is unitary conjugation, so the spectrum is invariant when the one-particle Hilbert space and background fields are transformed consistently. With a dynamical gauge field, a gauge-averaged bare charged kernel can vanish; a dressed operator and a specified physical state are then needed. Classical gauge covariance alone does not settle that case. The phase of a spatial matrix element is not itself gauge invariant.

For a compact charged pair,

$$P(\mathbf R,\mathbf R') = \langle \Delta^\dagger(\mathbf R) \Delta(\mathbf R') \rangle$$

also acquires an endpoint phase under a local gauge transformation. Define the line integral along a declared path by

$$\mathcal I_A(\mathbf R,\mathbf R') = \int_{\mathbf R}^{\mathbf R'} \mathbf A\cdot d\boldsymbol\ell,$$

and the charge-$2q$ parallel transporter by

$$W_{2q}(\mathbf R,\mathbf R') = \exp\left[ -\frac{2iq}{\hbar} \mathcal I_A(\mathbf R,\mathbf R') \right] .$$

A gauge-invariant, path-dependent comparison is then, schematically,

$$P_{\mathrm{inv}}(\mathbf R,\mathbf R') ={} \bigl\langle \Delta^\dagger(\mathbf R) W_{2q}(\mathbf R,\mathbf R') \times \Delta(\mathbf R') \bigr\rangle ,$$

for the convention

$$\mathbf A \longmapsto \mathbf A+\boldsymbol\nabla\chi.$$

The path, electromagnetic fluctuations, and nonlocal internal pair structure can matter. In a fixed gauge one may study the ordinary pair kernel, but its phase plateau should not be presented as a standalone gauge-invariant observable.

Local gauge redundancy is not an ordinary physical symmetry that must literally break. Superconductivity is established through a consistent collection of number-conserving coherence and electromagnetic-response diagnostics, not by assigning physical meaning to a gauge-dependent field expectation alone.

Finite-Size Scaling

Leading-eigenvalue regimes

Let $K(\mathbf R-\mathbf R')$ denote a translation-invariant one-body or projected pair kernel over center coordinates. If

$$K(\mathbf R) \sim \frac{1}{R^p},$$

then its largest long-wavelength eigenvalue scales as the integrated correlation:

$$\lambda_{\max}(L) \sim \int_a^L dR\, R^{d-1-p}.$$

Thus

$$\lambda_{\max}(L) \sim \begin{cases} L^d, & p=0, \\ L^{d-p}, & 0<p<d, \\ \log L, & p=d, \\ O(1), & p>d. \end{cases}$$

Since $N\propto L^d$,

$$\frac{\lambda_{\max}}{N} \sim \begin{cases} \text{const.}>0, & p=0, \\ L^{-p}, & 0<p<d, \\ L^{-d}\log L, & p=d, \\ L^{-d}, & p>d. \end{cases}$$

The first line is true ODLRO, the middle two are algebraic or marginal coherence, and the last is integrable short-range order. This table applies to the center-coordinate kernel after fixing or resolving the internal channel. It should not be applied blindly to the full $O(V^2)$ pair-coordinate space.

Spectral indicators

Useful finite-size quantities include

$$f_0(L) = \frac{N_0(L)}{N(L)}$$

for bosons and

$$\lambda_0(L) = \frac{\Lambda_0(L)}{N(L)}$$

for fermion pairs. Also inspect

$$\frac{\lambda_1(L)}{\lambda_0(L)}$$

or the normalized separation $[\Lambda_0(L)-\Lambda_1(L)]/N(L)$ for the unnormalized pair eigenvalues. Do not divide an already normalized $\lambda_0=\Lambda_0/N$ by $N$ again. A stable isolated leading eigenvalue supports simple condensation; several extensive eigenvalues support fragmentation or multiple condensed channels.

An apparent eigenvalue gap at one size is not decisive. Near a transition, a finite correlation length larger than the simulated box can mimic extensive scaling over a short range of $L$.

Real-space and spectral cross-check

In a homogeneous simple condensate, compare:

$$\frac{N_0}{V}$$

with the large-distance one-body plateau, or compare

$$\frac{\Lambda_0}{V}$$

with a channel-projected pair plateau after matching normalizations. Agreement catches Fourier factors, pair-counting factors, and center-of-mass averaging errors.

The eigenvector should also have the expected symmetry. A large eigenvalue in an unexpected edge-localized, boundary-pinned, or phase-separated mode may not diagnose the proposed bulk condensate.

Finite-Size and Inhomogeneity Caveats

Shape and boundary conditions

Keep aspect ratio, boundary conditions, filling, and flux sector controlled. Open boundaries can localize the leading orbital; periodic boundaries can force commensurate momenta; twisted boundaries can split or move condensate minima.

Traps and density profiles

A global eigenvalue can be dominated by a dense central region. Compare natural-orbital support with the density profile, and define the trap thermodynamic sequence before extrapolating.

Translation averaging

Pair-center averaging improves statistics in a homogeneous state:

$$\overline P_f(\mathbf r) = \frac{1}{V} \int d^d R\, P_f(\mathbf R,\mathbf R+\mathbf r).$$

It can hide domains, interfaces, disorder, or boundary localization in an inhomogeneous state. Inspect unaveraged data before imposing translation symmetry.

Pair size versus box size

The internal pair extent must remain small compared with $L$ for a clean separation between relative and center coordinates. In weak-coupling BCS systems, the coherence length can be large, so boxes smaller than the pair size can strongly distort the eigenvector and scaling.

Competing and degenerate channels

Nearly degenerate $s$-, $d$-, triplet, valley, or finite-momentum channels can rotate under small perturbations. Track the macroscopic subspace and symmetry quantum numbers rather than matching eigenvectors by raw component overlap alone.

Positivity and noisy estimators

Exact reduced density matrices are positive semidefinite. Sampling noise can produce small negative eigenvalues and unstable leading eigenvectors. Symmetrize only in ways justified by exact symmetries, propagate covariance where possible, and report whether positivity was imposed during estimation.

Ensemble and number fluctuations

In a grand-canonical calculation, large number fluctuations can affect both the trace and the apparent leading eigenvalue. Scale against $\langle N\rangle$, record

$$\langle(\Delta N)^2\rangle,$$

and compare ensembles when fluctuations are anomalously large.

Low-Dimensional Systems

Algebraic order is not true ODLRO

In many one-dimensional ground states and two-dimensional finite-temperature superfluids,

$$\gamma(r) \sim r^{-\eta}$$

or

$$P_f(r) \sim r^{-\eta}.$$

For $0<\eta<d$ and a nonoscillatory or correctly demodulated leading channel, the leading eigenvalue diverges as

$$\lambda_{\max} \sim L^{d-\eta},$$

but

$$\frac{\lambda_{\max}}{N} \sim L^{-\eta} \longrightarrow 0.$$

This is quasi-long-range order or algebraic coherence, not true ODLRO. A finite system can nevertheless have a very large leading eigenvalue and a visually flat correlation function when $\eta$ is small.

Two dimensions at nonzero temperature

For short-range neutral systems with a continuous $U(1)$ symmetry, ordinary finite-temperature symmetry breaking and a constant off-diagonal plateau are excluded under the standard assumptions. A Berezinskii–Kosterlitz–Thouless phase can retain algebraic coherence and nonzero stiffness.

Therefore

$$\rho_s > 0$$

does not imply

$$\lim_{r\to\infty}\gamma(r) > 0$$

in two dimensions at nonzero temperature.

Zero temperature and three dimensions

The finite-temperature restriction does not automatically forbid a two-dimensional ground-state condensate. Three-dimensional systems can support true ODLRO at nonzero temperature. In both cases, the result remains model and phase dependent; dimensionality permits rather than guarantees condensation.

Low-Dimensional Quantum Gases develops the infrared mechanisms and trap qualifications.

What ODLRO Does Not Establish by Itself

Diagonal long-range order

Magnetic or density-wave order is diagnosed by correlations of local number-neutral operators. ODLRO instead concerns reduced-density kernels that transfer a particle or pair between distant configurations. A phase may have either, both, or neither.

Pair binding

A negative two-particle binding energy or enhanced short-range pair correlation says that two particles favor being near each other. Pair ODLRO says that pair transfer remains coherent across macroscopic distances. A gas of localized, mutually incoherent bound pairs can have strong binding and no extensive pair-matrix eigenvalue.

Spectral gap and pseudogap

A single-particle gap can arise from band structure, Mott localization, symmetry breaking, or pairing without global phase coherence. A pseudogap regime can retain local pairing correlations while the long-distance pair matrix remains short ranged. Conversely, some condensed phases possess gapless excitations.

Superfluid stiffness

The superfluid density or helicity modulus measures the free-energy response to phase twists or flow. Condensate fraction and superfluid fraction coincide only in special limits. Liquid helium, lattice bosons, disordered systems, and low-dimensional phases make the distinction operationally important.

Meissner response

For charged matter, magnetic-field expulsion and electromagnetic rigidity are response properties. Pair ODLRO is an important coherence diagnostic, but a complete superconducting claim must handle gauge covariance and electromagnetic coupling. Ginzburg–Landau Theory gives the material-scale amplitude and field functional, while the Josephson Effect makes a gauge-invariant phase difference operational through weak-link current and interference. One should not infer those response coefficients from one matter-only density-matrix eigenvalue without additional assumptions.

An anomalous average

An anomalous average is representation and phase-sector dependent:

$$\langle\psi\psi\rangle \neq 0$$

can be useful in a selected mean-field state, while it vanishes in every exact fixed-number state. ODLRO can reside in a number-neutral density matrix after number projection, as the fixed-number example shows. Projection onto an atypical sector need not preserve the original condensate; for example, projection onto the vacuum removes it.

Fermi-surface coherence

A normal Fermi sea has algebraic one-body correlations at zero temperature and natural occupations bounded by one. Slow decay alone does not turn its fermionic one-body density matrix into a condensate.

Entanglement

Reduced density matrices here are reduced by particle body rank, not necessarily by a spatial tensor-product partition. Their eigenvalues diagnose occupation and coherence, not an entanglement entropy. ODLRO can coexist with little or substantial many-body entanglement.

A coherent state

A Glauber coherent state has a nonzero field expectation and Poissonian number fluctuations. A fixed-$N$ condensate has the same rank-one one-body density matrix at leading order but a sharp particle number. Condensation is not synonymous with preparation of a coherent state.

Experimental Interpretation

Bosonic one-body coherence

For a homogeneous gas, the momentum distribution is the Fourier transform of the one-body kernel:

$$n(\mathbf k) = \int d^d r\, e^{-i\mathbf k\cdot\mathbf r} \gamma(\mathbf r).$$

A macroscopic natural occupation can produce a sharp momentum-space component, and matter-wave interference probes relative phase coherence. Finite imaging resolution, trap inhomogeneity, interactions during expansion, and finite system size broaden the signal. A narrow peak must be scaled before it is called a thermodynamic condensate.

Pair coherence

Pair momentum distributions, pair-transfer interference, noise correlations, and Josephson-type coupling can probe aspects of pair coherence. Most experiments access response functions or projected correlators rather than diagonalizing the complete two-body density matrix. The inferred channel, gauge convention, and dynamical assumptions should be stated.

Charged condensates

For superconductors, phase-sensitive junctions, flux quantization, magnetic response, and electrodynamics provide information not contained in a matter-only equal-time pair matrix. Combining coherence, spectroscopy, and response measurements constrains the interpretation more strongly than one proxy.

Common pitfalls

Off-diagonal entries depend on basis. Diagonalize the density matrix, inspect its natural modes, and compare several sizes. A narrow momentum peak or a large occupation at one size is insufficient.

Pair conventions change numerical factors and complex coordinates. Check the trace, Hermiticity, and positivity before comparing eigenvalues. With the definitions above, the ordered kernel has eigenvalues $2\Lambda_\alpha$.

Binding, condensation, and stiffness differ. Short-distance pairs can be mutually incoherent. A coherent pair channel needs macroscopic scaling; superfluid or electromagnetic response is an additional calculation.

Exercises

Exercise 1: Positivity, trace, and condensate fraction

Let

$$\gamma_{ij} = \langle a_j^\dagger a_i\rangle$$

for $N$ bosons. Prove that $\gamma$ is positive semidefinite and has trace $N$. Explain why one eigenvalue equal to $fN+o(N)$ with $f>0$ defines a basis-independent condensate fraction.

Solution

For an arbitrary vector $v$, define

$$a_v = \sum_i v_i^*a_i.$$

Then

$$\begin{aligned} v^\dagger\gamma v &= \sum_{ij} v_i^* \langle a_j^\dagger a_i\rangle v_j \\ &= \langle a_v^\dagger a_v\rangle \geq 0. \end{aligned}$$

Thus $\gamma$ is positive semidefinite. In a complete one-particle basis,

$$\operatorname{Tr}\gamma = \sum_i \langle a_i^\dagger a_i\rangle = \langle N\rangle = N.$$

Under a one-particle basis change,

$$\gamma \longmapsto U\gamma U^\dagger,$$

so its eigenvalues are unchanged. Hence

$$f = \lim_{N\to\infty} \frac{N_0}{N}$$

does not depend on which orbital coordinates were used to represent the state.

Exercise 2: Plane-wave occupation and spatial plateau

In a periodic translation-invariant box, suppose

$$n_{\mathbf0} = n_0V$$

and the remaining momentum distribution has a thermodynamic-limit density $\widetilde n_{\mathrm{reg}}(\mathbf k)$ whose Fourier transform vanishes at large distance. Show that

$$\gamma(\mathbf r) \longrightarrow n_0.$$

How does the result change for a condensate at $\mathbf k_0\neq\mathbf0$?

Solution

Separate the macroscopic mode:

$$\begin{aligned} \gamma(\mathbf r) &= \frac{n_{\mathbf0}}{V} + \frac{1}{V} \sum_{\mathbf k\neq\mathbf0} n_{\mathbf k} e^{i\mathbf k\cdot\mathbf r} \\ &\longrightarrow n_0 + \int \frac{d^d k}{(2\pi)^d} \widetilde n_{\mathrm{reg}}(\mathbf k) e^{i\mathbf k\cdot\mathbf r}. \end{aligned}$$

By the assumed regularity, the second term vanishes as $\lvert\mathbf r\rvert\to\infty$. Therefore the plateau is $n_0$.

For a condensate at $\mathbf k_0$,

$$\gamma(\mathbf r) \longrightarrow n_0e^{i\mathbf k_0\cdot\mathbf r}.$$

Equivalently,

$$e^{-i\mathbf k_0\cdot\mathbf r} \gamma(\mathbf r) \longrightarrow n_0.$$

Exercise 3: Number conservation does not remove ODLRO

For

$$\lvert N:\phi\rangle = \frac{(a_\phi^\dagger)^N}{\sqrt{N!}} \lvert0\rangle,$$

compute $\langle\psi(x)\rangle$ and $\gamma(x,x')$. Identify the natural occupations.

Solution

The field changes particle number by one, so orthogonality of different number sectors gives

$$\langle\psi(x)\rangle = 0.$$

Decompose

$$\psi(x) = \phi(x)a_\phi + \psi_\perp(x).$$

The orthogonal modes are empty, while

$$\langle a_\phi^\dagger a_\phi \rangle = N.$$

Therefore

$$\gamma(x,x') = N\phi(x)\phi^*(x').$$

The sole nonzero natural occupation is $N_0=N$, with natural orbital $\phi$. The state has maximal simple ODLRO despite its vanishing field expectation.

Exercise 4: Pauli obstruction

Show that every eigenvalue of a fermionic one-body density matrix lies in $[0,1]$. Why does this rule out fermionic one-body ODLRO but not pair ODLRO?

Solution

For a normalized orbital $f$,

$$n_f = c_f^\dagger c_f$$

satisfies

$$n_f^2 = n_f$$

by the canonical anticommutation relations. Its expectation therefore obeys

$$0 \leq \langle n_f\rangle \leq 1.$$

The largest natural occupation is the maximum of $\langle n_f\rangle$ over normalized $f$, so every one-body eigenvalue is at most one. None can scale as $N$.

A normalized pair mode is not a single fermionic orbital, and its occupation operator is not a one-particle projector. Antisymmetry permits the leading eigenvalue of the two-body pair matrix to scale as $N$, so pair ODLRO remains possible.

Exercise 5: Extensive eigenvalue in the BCS pair matrix

Suppose

$$M_{\mathbf k\mathbf k'} = F_{\mathbf k}^*F_{\mathbf k'} + \delta_{\mathbf k\mathbf k'}d_{\mathbf k},$$

where $0\leq d_{\mathbf k}\leq1$ and

$$\sum_{\mathbf k} \lvert F_{\mathbf k}\rvert^2 = cV$$

with $c>0$. Show that the largest eigenvalue is $cV+O(1)$.

Solution

Write

$$M = \lvert F^*\rangle\langle F^*\rvert + D,$$

where the components of $|F^*\rangle$ are $F_{\mathbf k}^*$, and $D$ is diagonal with

$$0 \leq D \leq I.$$

The rank-one matrix has eigenvalues

$$\langle F\vert F\rangle = cV$$

and zero otherwise. Weyl’s eigenvalue inequalities give

$$cV \leq \lambda_{\max}(M) \leq cV+1.$$

Hence

$$\lambda_{\max}(M) = cV+O(1).$$

At fixed density, $V\propto N$, so the eigenvalue is extensive.

Exercise 6: Algebraic coherence and pseudo-condensation

Let a translation-invariant center-coordinate kernel in $d$ dimensions obey

$$K(R) \sim R^{-p}, \qquad 0<p<d.$$

Estimate the leading eigenvalue and its ratio to $N\propto L^d$. Explain why a large finite-size occupation need not imply true ODLRO.

Solution

The largest long-wavelength eigenvalue scales as

$$\begin{aligned} \lambda_{\max}(L) &\sim \int_a^L dR\, R^{d-1-p} \\ &\sim L^{d-p}. \end{aligned}$$

Therefore

$$\frac{\lambda_{\max}}{N} \sim \frac{L^{d-p}}{L^d} = L^{-p} \longrightarrow 0.$$

The eigenvalue diverges and can be numerically large, but it is subextensive. This is algebraic coherence or pseudo-condensation rather than true ODLRO.

Exercise 7: Gauge covariance of a charged pair correlator

Let a compact pair field of constituent charge $q$ transform as

$$\Delta(\mathbf R) \longmapsto e^{2iq\chi(\mathbf R)/\hbar} \Delta(\mathbf R),$$

while

$$\mathbf A \longmapsto \mathbf A+\boldsymbol\nabla\chi.$$

Find the transformation of

$$\langle \Delta^\dagger(\mathbf R) \Delta(\mathbf R') \rangle$$

and verify that the Wilson-line-dressed correlator used above is invariant.

Solution

Write the undressed correlator as

$$C(\mathbf R,\mathbf R') = \left\langle \Delta^\dagger(\mathbf R) \Delta(\mathbf R') \right\rangle .$$

It transforms as

$$C(\mathbf R,\mathbf R') \longmapsto{} e^{2iq[\chi(\mathbf R')-\chi(\mathbf R)]/\hbar} \times C(\mathbf R,\mathbf R').$$

The line integral transforms according to

$$\int_{\mathbf R}^{\mathbf R'} \mathbf A\cdot d\boldsymbol\ell \longmapsto{} \int_{\mathbf R}^{\mathbf R'} \mathbf A\cdot d\boldsymbol\ell + \chi(\mathbf R') - \chi(\mathbf R).$$

Hence the exponential transporter acquires the inverse endpoint phase,

$$e^{-2iq[\chi(\mathbf R')-\chi(\mathbf R)]/\hbar},$$

which cancels the phase of the pair correlator. The dressed product is gauge invariant for the stated conventions.

Exercise 8: Bound but incoherent pairs

Consider a complete set of $M$ localized pair-center modes for a model of bound pairs, with

$$\left\langle P_R^\dagger P_{R'} \right\rangle = n_p\delta_{RR'},$$

where $n_p=O(1)$ as $M\to\infty$. Find the pair-matrix eigenvalues. Does the state have pair ODLRO even though every site can have strong local pair occupancy?

Solution

The pair matrix is

$$\mathcal M = n_p I_M.$$

All $M$ eigenvalues equal $n_p=O(1)$. Its trace is extensive in the number of pair locations,

$$\operatorname{Tr}\mathcal M = Mn_p,$$

but no eigenvalue in this pair-center model is extensive. The state can contain many localized bound pairs while lacking coherence between different centers. If these modes are only a projection of a more general fermionic pair space, this result excludes condensation in the tested subspace, not in every unmeasured channel.

Pair ODLRO would require off-diagonal coherence that concentrates weight into a collective mode, for example

$$\langle P_R^\dagger P_{R'} \rangle \longrightarrow \mathcal P$$

at large separation, producing an eigenvalue of order $M$.

Cross-Links

• Bose–Einstein Condensates Overview connects the one-body criterion to trapped-gas order parameters, interference, collective response, and experimental measurements.
• BCS Theory connects fermionic pair coherence to superconducting gaps, thermodynamics, stiffness, and material diagnostics without identifying those observables with one another.

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