Bose–Einstein Condensation

Bose–Einstein condensation is the appearance of a one-particle mode whose occupation is proportional to the total number of bosons. In a sequence of systems approaching a thermodynamic limit,

$$N_0\sim f_0N,\qquad f_0>0,$$

or, more sharply,

$$\lim_{N\to\infty} \frac{N_0}{N} > 0.$$

For a uniform ideal gas, the macroscopically occupied mode is the zero-momentum ground state. That familiar case is important but not the general definition. In an interacting or inhomogeneous system, the condensate mode is identified as an eigenvector of the one-body density matrix. It need not be a plane wave or the bare one-particle ground orbital.

The ideal-gas mechanism is an excited-state capacity limit. As the chemical potential approaches the lowest one-particle energy, the excited modes can hold only a finite density of particles in suitable dimensions. Any particles beyond that capacity accumulate macroscopically in the lowest mode.

The ideal Bose gas provides the box spectrum and thermodynamics. Here we explain what counts as condensation and how its criterion differs from broken symmetry and superfluid response. The historical account describes the prediction and experimental realization.

Helpful background. The ideal Bose gas supplies state counting and Bose integrals; occupation numbers supplies natural orbitals and mean populations.

Bose occupation, saturation, and condensation

It is useful to separate three statements.

Statistical permission.
Bosonic statistics allows

$$n_i = 0,1,2,\ldots$$

in one mode. This makes macroscopic occupation possible but does not guarantee it.

Thermodynamic mechanism.
For some spectra and thermodynamic limits, the excited states have finite capacity at fixed temperature. A macroscopic occupation then becomes necessary below a critical scale.

General many-body criterion.
Condensation is diagnosed by an extensive eigenvalue of the one-body density matrix. This definition remains meaningful with interactions and without translation invariance.

Confusing these levels leads to statements such as “all bosons condense because they can share a state” or “BEC always means zero momentum.” Neither is generally correct.

Penrose–Onsager Criterion

For field operators $\psi(\mathbf r)$ and $\psi^\dagger(\mathbf r)$, define the equal-time one-body density matrix

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

At finite mean particle number this is a positive trace-class kernel and has a spectral decomposition into orthonormal natural orbitals:

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

The eigenfunctions $\phi_\alpha$ are natural orbitals and the eigenvalues $N_\alpha$ are natural occupations. Their sum is the mean particle number:

$$\sum_\alpha N_\alpha = \langle\hat N\rangle.$$

A conventional, or simple, condensate has one eigenvalue that is extensive:

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

while all other $N_\alpha$ are subextensive. The condensate fraction is

$$f_0 = \frac{N_0}{N}.$$

If two or more eigenvalues are extensive, the condensate is called fragmented. Fragmentation can arise from symmetry, degeneracy, strong correlations, or constraints. It is not captured by saying that one bare orbital is occupied.

The Occupation Numbers page develops the distinction between integer Fock-state counts, mean mode occupations, and natural occupations.

The Ideal Uniform Benchmark

Consider $N$ spinless, noninteracting bosons in a cubic box of volume $V$ with periodic boundary conditions. Measure energies relative to the unique ground mode:

$$\epsilon_{\mathbf k} = \frac{\hbar^2k^2}{2m}, \qquad \epsilon_{\mathbf0} = 0.$$

In the grand-canonical description the mean occupation of mode $\mathbf k$ is

$$\overline n_{\mathbf k} = \frac{1}{ z^{-1}e^{\beta\epsilon_{\mathbf k}}-1 }, \qquad z = e^{\beta\mu}.$$

For a normalizable finite-volume grand-canonical state at positive temperature,

$$0 < z < 1, \qquad \mu < 0.$$

Separate the ground occupation:

$$N = N_0 + N_{\mathrm{ex}},$$

where

$$N_0 = \frac{z}{1-z},$$

and

$$N_{\mathrm{ex}} = \sum_{\mathbf k\neq\mathbf0} \frac{1}{ z^{-1}e^{\beta\epsilon_{\mathbf k}}-1 }.$$

Here $N$ and $N_0$ are mean occupations. The leading fixed-density condensate fraction also describes a canonical gas in the thermodynamic limit, but the finite-system product occupations and fluctuations need not agree. The ground mode must remain discrete when the excited modes are replaced by an integral.

Thermal Wavelength and Excited-State Density

Define the thermal de Broglie wavelength

$$\lambda_T = \sqrt{ \frac{2\pi\hbar^2}{ mk_{\mathrm B}T } }.$$

In the three-dimensional thermodynamic limit, the excited-state sum becomes

$$\frac{N_{\mathrm{ex}}}{V} = \int \frac{d^3k}{(2\pi)^3} \frac{1}{ z^{-1}e^{\beta\hbar^2k^2/(2m)}-1 }.$$

Evaluating the integral gives

$$n_{\mathrm{ex}} = \frac{1}{\lambda_T^3} \operatorname{Li}_{3/2}(z).$$

The polylogarithm increases with $z$, but the bosonic constraint prevents $z$ from exceeding one. Therefore

$$n_{\mathrm{ex}} \leq \frac{\zeta(3/2)}{ \lambda_T^3 }.$$

The maximum excited-state density is finite:

$$n_{\mathrm{ex,max}}(T) = \frac{\zeta(3/2)}{ \lambda_T^3 }.$$

This finite capacity is the mathematical core of ideal-gas condensation.

Critical Temperature in a Box

At the critical temperature, the total density just equals the maximum excited-state density:

$$n = \frac{\zeta(3/2)}{ \lambda_{T_c}^3 }.$$

Equivalently,

$$n\lambda_{T_c}^3 = \zeta(3/2) \simeq 2.612.$$

Solving for temperature gives

$$T_c^{\mathrm{box}} = \frac{2\pi\hbar^2}{ mk_{\mathrm B} } \left[ \frac{n}{ \zeta(3/2) } \right]^{2/3}.$$

This formula assumes:

• a uniform three-dimensional gas;
• a quadratic nonrelativistic dispersion;
• one independent internal component;
• a unique ground mode;
• no interactions;
• the thermodynamic limit at fixed $n=N/V$.

Changing any of these ingredients can change the formula or the existence of a sharp transition.

Condensate Fraction

Below $T_c$, the thermodynamic-limit fugacity reaches

$$z \to 1, \qquad \mu \to 0^-.$$

The excited-state population is then

$$N_{\mathrm{ex}}(T) = \frac{V}{\lambda_T^3} \zeta(3/2).$$

Since

$$\lambda_T^{-3} \propto T^{3/2},$$

comparison with the number equation at $T_c$ yields

$$\frac{N_{\mathrm{ex}}}{N} = \left( \frac{T}{T_c} \right)^{3/2}.$$

The ideal uniform condensate fraction is therefore

$$\frac{N_0}{N} = 1 - \left( \frac{T}{T_c} \right)^{3/2}, \qquad 0 \leq T \leq T_c.$$

Above $T_c$, the thermodynamic-limit condensate fraction vanishes and $z<1$ is fixed by

$$n\lambda_T^3 = \operatorname{Li}_{3/2}(z).$$

Uniform three-dimensional ideal gas in the thermodynamic limit. Below $T_c$, the thermal cloud is saturated and its capacity scales as $T^{3/2}$; the remaining particles occupy the ground mode. A finite system rounds the corner near $T_c$.

The formula does not mean that particles are divided into two distinguishable species. “Condensate” and “thermal cloud” refer to contributions to the one-body density matrix and mode occupations.

Why Saturation Occurs

At low momentum and $z=1$,

$$\frac{1}{ e^{\beta\hbar^2k^2/(2m)}-1 } \sim \frac{ 2mk_{\mathrm B}T }{ \hbar^2k^2 }.$$

In three dimensions, the momentum-space measure contributes $k^2\,dk$. The low-momentum excited-number integral therefore behaves as

$$\int_0 dk\, k^2 \frac{1}{k^2} \sim \int_0 dk,$$

which is finite at the lower limit.

At $\mu=0$ the occupation diverges as $k\to0$; a fixed nonzero momentum retains a finite occupation. The phase-space volume near zero shrinks quickly enough that the integrated excited density remains finite. Condensation depends on this competition between the occupation factor and the density of states.

General Density-of-States Criterion

Suppose the low-energy density of excited one-particle states is

$$D(\epsilon) \simeq C\epsilon^{\alpha-1}, \qquad \epsilon \to 0^+.$$

At the condensation boundary,

$$\mu \to \epsilon_0,$$

so, measuring $\epsilon$ from $\epsilon_0$,

$$N_{\mathrm{ex,max}} \simeq \int_0^\infty d\epsilon\, \frac{ C\epsilon^{\alpha-1} }{ e^{\beta\epsilon}-1 }.$$

The low-energy integrand behaves as

$$\epsilon^{\alpha-2}.$$

The integral converges at zero exactly when

$$\alpha > 1.$$

If this power law describes the full thermally populated spectrum, the continuum expression is

$$N_{\mathrm{ex,max}} = C \Gamma(\alpha) \zeta(\alpha) (k_{\mathrm B}T)^\alpha.$$

The ideal critical temperature is

$$k_{\mathrm B}T_c = \left[ \frac{N}{ C\Gamma(\alpha)\zeta(\alpha) } \right]^{1/\alpha},$$

and below it,

$$\frac{N_0}{N} = 1 - \left( \frac{T}{T_c} \right)^\alpha.$$

The infrared exponent decides convergence, but does not fix the full capacity or critical temperature when higher energies have a different density of states. The coefficient $C$ must scale with the system size in a stated thermodynamic limit. Ordinary condensation into a finite set of lowest modes also assumes that no growing family of anomalously low levels absorbs the excess; strongly anisotropic limits can require a generalized condensation analysis. For fixed-shape boxes and traps the usual criterion applies.

Dimension and Dispersion

For a uniform $d$-dimensional system with low-momentum dispersion

$$\epsilon_{\mathbf k} \propto k^s,$$

the density of states scales as

$$D(\epsilon) \propto \epsilon^{d/s-1}.$$

Thus

$$\alpha = \frac{d}{s},$$

and the ideal excited-state integral converges when

$$d > s.$$

For ordinary massive particles,

$$s = 2,$$

so a uniform ideal gas has a finite-temperature condensation transition only for

$$d > 2.$$

In three dimensions, $\alpha=3/2$. In two dimensions, the low-momentum integral diverges logarithmically. In one dimension, it diverges more strongly.

This is a thermodynamic-limit statement. A finite one- or two-dimensional box can show a large ground-state population and a pronounced crossover, but no exact finite-temperature nonanalytic transition survives the standard uniform thermodynamic limit.

For interacting homogeneous Bose systems with short-range forces, long-wavelength phase fluctuations also forbid ordinary one-body long-range order at finite temperature in one and two dimensions under the hypotheses of Hohenberg-type results. A two-dimensional interacting fluid can nevertheless undergo a Berezinskii–Kosterlitz–Thouless transition with algebraically decaying order and nonzero superfluid response. That phenomenon is not the ideal-gas condensation derived here.

Low-Dimensional Quantum Gases develops the explicit one- and two-dimensional densities of states, finite-size qualifications, phase-fluctuation argument, dimensional-freezing criteria, and bridges to BKT and Luttinger-liquid physics.

Harmonic Traps

Laboratory atomic gases are often confined by an anisotropic harmonic potential:

$$V(\mathbf r) = \frac{m}{2} \left( \omega_x^2x^2 + \omega_y^2y^2 + \omega_z^2z^2 \right).$$

Define the geometric mean frequency

$$\overline\omega = (\omega_x\omega_y\omega_z)^{1/3}.$$

At excitation energies large compared with the level spacings, the three-dimensional trap density of states is

$$D_{\mathrm{trap}}(\epsilon) \simeq \frac{\epsilon^2}{ 2\hbar^3 \omega_x\omega_y\omega_z }.$$

Here

$$\alpha = 3,$$

so the maximum excited population is

$$N_{\mathrm{ex,max}} = \zeta(3) \left( \frac{ k_{\mathrm B}T }{ \hbar\overline\omega } \right)^3.$$

The leading ideal-gas critical temperature is

$$T_c^{\mathrm{trap}} = \frac{ \hbar\overline\omega }{ k_{\mathrm B} } \left[ \frac{N}{ \zeta(3) } \right]^{1/3}.$$

Below this scale,

$$\frac{N_0}{N} = 1 - \left( \frac{T}{ T_c^{\mathrm{trap}} } \right)^3.$$

The exponent is $3$, not $3/2$, because a harmonic trap has a different density of states from a uniform box.

The approximation requires

$$k_{\mathrm B}T \gg \hbar\omega_i$$

for thermally relevant directions. A sharp trapped-gas thermodynamic limit is constructed by taking

$$N \to \infty, \qquad \overline\omega \to 0,$$

while keeping

$$N\overline\omega^3$$

fixed. At finite $N$, discrete-level corrections shift and round the crossover. Interactions also change the density profile and transition region.

The condensate in a trap occupies a spatial orbital. It is not generally a zero-momentum plane wave.

Example: Box and Trap Scales

Uniform rubidium-87 estimate

Take an ideal uniform gas of rubidium-87 atoms with

$$m \simeq 1.443\times10^{-25}\ {\rm kg},$$

and density

$$n = 10^{20}\ {\rm m}^{-3}.$$

The box formula gives

$$T_c \simeq 3.98\times10^{-7}\ {\rm K} = 398\ {\rm nK}.$$

This is an order-of-magnitude benchmark. A real trapped, interacting cloud must be analyzed with its actual confinement, component populations, and scattering properties.

Harmonic-trap estimate

For

$$N = 10^5$$

atoms in a trap with

$$\overline\omega = 2\pi\times100\ {\rm Hz},$$

the leading ideal result is

$$T_c^{\mathrm{trap}} \simeq 2.10\times10^{-7}\ {\rm K} = 210\ {\rm nK}.$$

At half this temperature, the ideal trap fraction is

$$\frac{N_0}{N} = 1-\left(\frac12\right)^3 = \frac78.$$

For a uniform box at $T/T_c=1/2$, the corresponding ideal fraction is

$$1-\left(\frac12\right)^{3/2} \simeq 0.646.$$

The contrast comes entirely from the different density-of-states exponents.

Real-Space Coherence

For the homogeneous ideal gas condensed at zero momentum, with the thermodynamic limit taken before the large-separation limit,

$$\gamma(\mathbf r,\mathbf r') = \frac{N_0}{V} + \int \frac{d^3k}{(2\pi)^3} \overline n_{\mathbf k} e^{i\mathbf k\cdot(\mathbf r-\mathbf r')},$$

where the integral contains the noncondensed modes. At large separation, the thermal contribution decays, leaving

$$\lim_{ |\mathbf r-\mathbf r'| \to \infty } \gamma(\mathbf r,\mathbf r') = n_0,$$

with

$$n_0 = \frac{N_0}{V}.$$

This nonzero asymptote is off-diagonal long-range order. For this zero-momentum condensate it is the real-space form of the extensive natural occupation. A condensate at nonzero momentum instead contributes an oscillating phase, and fragmented states can contain several macroscopic terms. The off-diagonal-order discussion treats these distinctions.

In an inhomogeneous trap there is no translation-invariant separation limit of this simple form. Diagonalizing $\gamma(\mathbf r,\mathbf r')$ provides the more general criterion.

Long-range one-body coherence is not complete state tomography. Density correlations, phase fluctuations, depletion, and higher-order correlations contain additional information.

Condensation and Symmetry Breaking

The microscopic Hamiltonian of a number-conserving Bose gas is invariant under the global transformation

$$\psi(\mathbf r) \longrightarrow e^{i\theta} \psi(\mathbf r).$$

In an exact fixed-$N$ state,

$$\langle\psi(\mathbf r)\rangle = 0$$

because $\psi$ changes particle number. Nevertheless, the one-body density matrix can have an extensive eigenvalue. The Penrose–Onsager criterion therefore defines condensation without assuming broken symmetry.

In a symmetry-breaking description, one writes the complex order parameter

$$\Phi(\mathbf r) = \langle\psi(\mathbf r)\rangle,$$

with

$$\int d^d r\, |\Phi(\mathbf r)|^2 \simeq N_0.$$

The identification with $N_0$ assumes a simple condensate and an appropriate phase-selected state with clustering; a single mean field does not represent every fragmented condensate. This representation is powerful in the thermodynamic limit and in mean-field theory. It selects a phase and replaces the condensate operator partly by a classical field. A number-conserving treatment and a broken-symmetry treatment can agree on local observables in their common regime while organizing the calculation differently.

The condensate phase is not an absolute observable. Interference experiments reveal relative phases, and measurement can establish a relative phase even between initially number-defined clouds.

The general logic of fixed-number symmetry, source-selected phase states, and their local thermodynamic equivalence is developed in Spontaneous Symmetry Breaking.

Finite Systems

At finite $N$ and finite volume:

• the spectrum is discrete;
• the partition function is analytic;
• for the finite ideal grand-canonical gas at positive temperature, $\mu$ remains below the lowest one-particle energy;
• the condensate fraction changes smoothly;
• there is no exact thermodynamic singularity.

For the ideal ground mode,

$$N_0 = \frac{z}{1-z}.$$

Solving for fugacity gives

$$z = \frac{N_0}{ N_0+1 }.$$

Therefore

$$\beta\mu = \ln \left( \frac{N_0}{ N_0+1 } \right) \simeq - \frac{1}{N_0}$$

for macroscopic $N_0$. The statement $\mu=0$ below $T_c$ is a thermodynamic-limit shorthand. A finite ideal grand-canonical gas at positive temperature has $\mu<0$ in this energy convention; a finite canonical addition free energy is a different definition.

A finite cloud may still have a clearly dominant natural occupation and experimentally useful crossover temperature. “No exact phase transition” does not mean “no observable condensate.” The general distinction among finite-$N$ rounding, mode resolution, trap scales, and a bulk comparison belongs to Finite-Size Effects.

Ensemble Dependence of Condensate Fluctuations

The leading thermodynamic-limit mean condensate fraction can agree across ensembles, while condensate-number fluctuations do not.

For a single ideal bosonic mode in the grand-canonical ensemble,

$$\operatorname{Var}(N_0) = N_0(1+N_0).$$

If $N_0$ is extensive,

$$\frac{ \sqrt{ \operatorname{Var}(N_0) } }{ N_0 } \to 1.$$

These order-one relative fluctuations are often called the grand-canonical condensate-fluctuation problem. They arise from coupling an exactly zero-energy ideal mode to an unlimited particle reservoir.

In a canonical ensemble, fixed total number couples the condensate and thermal cloud:

$$N_0 = N - N_{\mathrm{ex}}.$$

Total-number fluctuations vanish, and condensate fluctuations are controlled by fluctuations of the excited population. Interactions modify the result again.

Ensemble equivalence must therefore be stated for a specified observable and limit. Bulk pressure can be ensemble equivalent while a zero-mode fluctuation is not.

Thermodynamic Signatures of the Ideal Transition

Below $T_c$ in the uniform ideal gas,

$$P = \frac{ k_{\mathrm B}T }{ \lambda_T^3 } \zeta(5/2),$$

and

$$U = \frac32 PV.$$

The condensate carries no kinetic energy in the convention $\epsilon_0=0$. Hence

$$U \propto T^{5/2}$$

below $T_c$, and

$$\frac{C_V}{ Nk_{\mathrm B} } = \frac{15}{4} \frac{ \zeta(5/2) }{ \zeta(3/2) } \left( \frac{T}{T_c} \right)^{3/2}.$$

Along fixed-density cooling, the entropy and energy are continuous at the ideal transition; the heat capacity approaches the same finite value from either side. Higher thermodynamic derivatives become nonanalytic only in the thermodynamic limit. These statements concern the fixed-density transition; the ideal gas has a singular isobaric response because its condensed pressure is independent of density. The general distinction between first-order and continuous thermal transitions, including finite-size rounding, is developed in Finite-Temperature Phase Transitions.

The ideal uniform gas also becomes anomalously compressible as $z\to1^-$. Its pathological softness below the transition is another sign that interactions are essential for a realistic stable superfluid.

The complete ideal-gas thermodynamics and its assumptions belong to the Ideal Bose Gas.

Interactions change the state, not the definition

For a stable dilute three-dimensional gas with short-range repulsion, the leading low-energy coupling is $g=4\pi\hbar^2a/m$, where $a>0$ is the scattering length and $na^3\ll1$. The contact description means a regularized low-energy interaction matched to scattering, not an unregulated three-dimensional delta potential used at every momentum.

Interactions make the condensate orbital self-consistent, give a bulk inverse compressibility and sound, produce collisions, and shift the transition region. They also generate zero-temperature quantum depletion. The leading homogeneous Bogoliubov result is

$$\frac{N-N_0}{N}=\frac{8}{3\sqrt\pi}\sqrt{na^3}+\cdots.$$

Thus $N_0$ need not equal $N$ even at zero temperature. The Penrose–Onsager criterion remains unchanged: interactions alter the one-body density matrix and its eigenvalues. The weakly interacting gas derives these approximation-dependent results; they do not follow from the ideal saturation mechanism.

Condensate Fraction Is Not Superfluid Fraction

The condensate fraction is

$$f_0 = \frac{N_0}{N},$$

defined from the largest eigenvalue of the one-body density matrix.

The superfluid fraction is a response coefficient. For a neutral equilibrium fluid it can be defined through a phase-twist or transverse-current response with the geometry, thermodynamic scaling, and order of limits stated. Slow rotation and persistent-flow experiments probe related properties. A nonzero curvature in one finite system alone is insufficient to establish a bulk superfluid fraction.

The two quantities are related in many weakly interacting Bose gases but are not identical:

• the uniform ideal Bose gas condenses, yet its quadratic excitation spectrum gives zero Landau critical velocity;
• strongly interacting liquid helium can have a superfluid fraction much larger than its condensate fraction;
• an infinite homogeneous two-dimensional fluid can have BKT superfluidity with algebraic one-body order but no conventional finite-temperature condensate;
• disorder, lattices, and finite geometry can further separate one-body coherence from transport response.

“BEC” is therefore not a synonym for “superfluid,” and neither term should be inferred from the other without assumptions.

Condensation in state space

The condensate orbital can extend across the whole sample. Macroscopic occupation is not a claim that atoms form a droplet or sit at one point. Kinetic energy, the trap, and interactions determine the spatial profile. Evaporative cooling explains how selective loss and rethermalization prepare degenerate gases; ultracold atoms discusses interaction tuning and dimensional confinement.

Experimental Diagnostics

No single measurement is universally sufficient in every system. Common diagnostics include:

Momentum-space bimodality.
After release from a trap, a narrow low-momentum component can appear on top of a broad thermal distribution.

One-body coherence.
Interference and correlation measurements probe the spatial coherence encoded in $\gamma(\mathbf r,\mathbf r')$.

Condensate-mode fitting.
In trapped gases, density profiles can be decomposed into a condensate contribution and a thermal cloud using a model appropriate to the interaction and expansion regime.

Macroscopic natural occupation.
When the one-body density matrix can be reconstructed or computed, its leading eigenvalue gives the basis-independent condensate fraction.

Response and collective behavior.
Sound, vortices, collective modes, and superfluid response provide important information about interactions and superfluidity, but they probe more than condensation alone.

The experimental sequence, 1995 dilute-gas observations, and historical interpretation belong to Bose–Einstein Condensation as Historical and Modern Topic and the compact experiment card.

Variants and Boundaries

Internal components

If $g$ internal states are independent, degenerate, and share one chemical potential, the normal-phase excited density acquires a factor $g$. For a uniform ideal gas,

$$n_{\mathrm{ex,max}} = \frac{ g\zeta(3/2) }{ \lambda_T^3 }.$$

Then

$$T_c \propto \left( \frac{n}{g} \right)^{2/3}.$$

But the factor cannot be inserted blindly. If component numbers are separately conserved, if levels are split, or if the ground manifold is degenerate, several chemical potentials or fragmented condensation may be relevant.

Lattices

On a lattice, the low-energy band dispersion and density of states replace the continuum quadratic formulas. Interactions can drive a superfluid-to-Mott transition, where a simple ideal-band condensation picture is inadequate.

Quasiparticles

Photons, magnons, polaritons, and other bosonic quasiparticles may have nonconserved number, driven-dissipative dynamics, effective masses, or finite lifetimes. Their condensation criteria require the actual conservation laws and equilibration mechanism. The atomic fixed-$N$ formulas should not be transplanted automatically.

Degenerate minima

If several one-particle minima are exactly degenerate, the condensate can occupy a superposition, choose one minimum through symmetry breaking, or fragment across modes. The one-body density matrix decides which description applies.

Exercises

General low-energy criterion

Suppose

$$D(\epsilon) = C\epsilon^{\alpha-1}$$

near the lowest energy. Determine the infrared convergence criterion. Derive the full capacity and its temperature scaling only under the additional assumption that this power law describes the entire thermally occupied spectrum.

Solution

If the power law can be used over the thermally occupied spectrum, at the saturation boundary

$$N_{\mathrm{ex,max}} = C \int_0^\infty d\epsilon\, \frac{ \epsilon^{\alpha-1} }{ e^{\beta\epsilon}-1 }.$$

For small $\epsilon$,

$$e^{\beta\epsilon}-1 \simeq \beta\epsilon,$$

so the integrand behaves as

$$\frac{C}{\beta} \epsilon^{\alpha-2}.$$

The lower-limit integral converges when

$$\alpha-2 > -1,$$

or

$$\alpha > 1.$$

With $x=\beta\epsilon$,

$$N_{\mathrm{ex,max}} = C (k_{\mathrm B}T)^\alpha \int_0^\infty dx\, \frac{ x^{\alpha-1} }{ e^x-1 }.$$

The remaining integral is

$$\Gamma(\alpha)\zeta(\alpha),$$

so

$$N_{\mathrm{ex,max}} = C \Gamma(\alpha) \zeta(\alpha) (k_{\mathrm B}T)^\alpha.$$

Infrared divergence in two dimensions

Show directly that a uniform two-dimensional ideal Bose gas with quadratic dispersion has no finite excited-state capacity at nonzero temperature.

Solution

At $\mu=0$,

$$n_{\mathrm{ex}} = \int \frac{d^2k}{(2\pi)^2} \frac{1}{ e^{\beta\hbar^2k^2/(2m)}-1 }.$$

For small $k$,

$$\frac{1}{ e^{\beta\hbar^2k^2/(2m)}-1 } \simeq \frac{ 2mk_{\mathrm B}T }{ \hbar^2k^2 }.$$

The two-dimensional measure is proportional to $k\,dk$, so the infrared contribution is

$$n_{\mathrm{ex}} \sim \int_0 \frac{dk}{k}.$$

This diverges logarithmically. The excited modes can absorb arbitrarily large density as $\mu\to0^-$, so there is no standard finite-temperature ideal-gas BEC in the uniform two-dimensional thermodynamic limit.

Harmonic-trap result

Use

$$D_{\mathrm{trap}}(\epsilon) = \frac{\epsilon^2}{ 2(\hbar\overline\omega)^3 }$$

to derive the ideal three-dimensional harmonic-trap critical temperature and condensate fraction.

Solution

At saturation,

$$\begin{aligned} N_{\mathrm{ex,max}} &= \frac{1}{ 2(\hbar\overline\omega)^3 } \int_0^\infty d\epsilon\, \frac{\epsilon^2}{ e^{\beta\epsilon}-1 } \\ &= \frac{ (k_{\mathrm B}T)^3 }{ 2(\hbar\overline\omega)^3 } \Gamma(3)\zeta(3). \end{aligned}$$

Since

$$\Gamma(3) = 2,$$

this becomes

$$N_{\mathrm{ex,max}} = \zeta(3) \left( \frac{ k_{\mathrm B}T }{ \hbar\overline\omega } \right)^3.$$

Setting this equal to $N$ gives

$$T_c = \frac{ \hbar\overline\omega }{ k_{\mathrm B} } \left[ \frac{N}{ \zeta(3) } \right]^{1/3}.$$

Below $T_c$,

$$\frac{N_{\mathrm{ex}}}{N} = \left( \frac{T}{T_c} \right)^3,$$

so

$$\frac{N_0}{N} = 1 - \left( \frac{T}{T_c} \right)^3.$$

Simple and fragmented condensation

Consider a two-mode one-body density matrix

$$\gamma = N \begin{pmatrix} 1/2 & c\\ c^* & 1/2 \end{pmatrix},$$

with $|c|\leq1/2$. Find its natural occupations. Classify the limits $|c|=1/2$ and $c=0$.

Solution

The eigenvalues are

$$N_\pm = N \left( \frac12 \pm |c| \right).$$

When

$$|c| = \frac12,$$

the occupations are

$$(N_+,N_-) = (N,0).$$

There is one macroscopically occupied natural orbital: simple condensation.

When

$$c = 0,$$

the occupations are

$$(N_+,N_-) = \left( \frac N2, \frac N2 \right).$$

Two orthogonal natural orbitals are macroscopically occupied, so the state is fragmented at the one-body level.

For fixed $0<|c|<1/2$ as $N\to\infty$, both eigenvalues remain proportional to $N$, which is also fragmented condensation.

Condensate and superfluid diagnostics

For each statement, identify whether it concerns condensation, superfluidity, both, or neither:

1. the largest eigenvalue of $\gamma$ is $0.4N$;
2. a properly scaled equilibrium phase-twist stiffness remains nonzero in the thermodynamic limit;
3. the Landau critical velocity is zero;
4. $\langle\psi\rangle=0$ in a fixed-number state.

Solution

1. An eigenvalue $0.4N$ is extensive, so it directly diagnoses condensation.
2. With the equilibrium protocol, boundary conditions, and limit specified, this probes superfluid response. Finite-size energy curvature or a nonequilibrium Drude weight alone would not establish the same claim.
3. Zero Landau critical velocity means the Landau criterion does not support stable superflow; it does not by itself decide whether there is a condensate. The uniform ideal Bose gas is the standard example with BEC but zero Landau critical velocity.
4. This decides neither. Exact particle-number conservation forces $\langle\psi\rangle=0$, but condensation can still be present in the one-body density matrix.

References

• F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose–Einstein Condensation in Trapped Gases”, Reviews of Modern Physics 71, 463–512 (1999) — authoritative review of trapped dilute-gas theory.
• P. C. Hohenberg, “Existence of Long-Range Order in One and Two Dimensions”, Physical Review 158, 383–386 (1967) — rigorous restrictions on long-range order in low-dimensional equilibrium systems.
• K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987) — standard ideal-gas derivation and thermodynamic behavior.
• W. Ketterle and N. J. van Druten, “Bose–Einstein Condensation of a Finite Number of Particles Trapped in One or Three Dimensions”, Physical Review A 54, 656–660 (1996) — finite-number and trap-dimensionality analysis.
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 4th ed., Academic Press (2021) — ideal Bose gas, condensation, dimensionality, and thermodynamic limits.
• O. Penrose and L. Onsager, “Bose–Einstein Condensation and Liquid Helium”, Physical Review 104, 576–584 (1956) — basis-independent definition through an extensive eigenvalue of the one-body density matrix.
• C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd ed., Cambridge University Press (2008) — dilute trapped gases, interactions, and condensate physics.
• L. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016) — condensation criteria, weak interactions, superfluidity, and finite-temperature theory.