No Bose-Einstein Condensation in 1D or 2D

Welcome back. In our previous lesson, we explored the thermodynamic signature of Bose-Einstein condensation in three dimensions, culminating in the derivation of the characteristic cusp in the heat capacity at the critical temperature $T_{c}$ . This entire discussion was predicated on the system being in a 3D space.

Today, we will challenge that assumption and investigate one of the most fundamental results in the study of phase transitions. Our objective is to explain why Bose-Einstein condensation does not occur in 1D or 2D for a non-interacting, non-relativistic gas. We will see that dimensionality is not a mere parameter but a crucial determinant of a system's collective behavior. Our analysis will hinge on the mathematical properties of the density of states, connecting a concrete calculation to the profound Mermin-Wagner theorem.

1. The General Condition for Condensation

Let's begin by recalling the essential condition for BEC. Condensation occurs when the ground state, which we take to have energy $E_{0} = 0$ , becomes macroscopically occupied. This happens if, at a finite temperature $T > 0$ , the maximum number of particles that all the excited states ( $E > 0$ ) can collectively hold, $N_{e x}^{ma x}$ , is less than the total number of particles $N$ .

The number of particles in excited states is given by the integral of the Bose-Einstein distribution over the density of single-particle states, $g (E)$ :

N_{e x} (T, μ) = \int_{0^{+}}^{\infty} \frac{g ( E )}{e ^{β (E - μ)} - 1} d E

The upper limit on this quantity, $N_{e x}^{ma x}$ , is found by setting the chemical potential to its maximum possible value, $μ = 0$ .

N_{e x}^{ma x} (T) = \int_{0^{+}}^{\infty} \frac{g ( E )}{e ^{β E} - 1} d E

A phase transition to a BEC occurs if and only if this integral converges to a finite value. If it converges, we can define a critical temperature $T_{c}$ by solving $N_{e x}^{ma x} (T_{c}) = N$ . For $T < T_{c}$ , the excited states are saturated, and the remaining $N_{0} = N - N_{e x}^{ma x}$ particles must occupy the ground state.

Conversely, if the integral for $N_{e x}^{ma x} (T)$ diverges for any $T > 0$ , it means the excited states have an infinite capacity. For any finite number of particles $N$ , we can always find a suitable negative chemical potential ( $μ < 0$ ) to satisfy the equation $N_{e x} (T, μ) = N$ . The ground state never needs to become macroscopically occupied.

Therefore, our entire investigation boils down to a single question: for which dimensions does this integral converge?

2. The Density of States and Its Dimensionality Dependence

The fate of the integral is determined by the behavior of the integrand, which depends critically on the density of states $g (E)$ . For non-relativistic particles with a quadratic dispersion relation $E = p^{2} /2 m \propto k^{2}$ , the density of states in $d$ dimensions scales as:

g_{d} (E) \propto E^{d /2 - 1}

This gives us the following behavior:

3D: $g_{3} (E) \propto E^{1/2}$
2D: $g_{2} (E) \propto E^{0} = constant$
1D: $g_{1} (E) \propto E^{- 1/2}$

The image below, which you may find familiar from your condensed matter studies, provides a clear summary of these relationships.

A comparison of the density of states $g(E)$ as a function of energy $E$ for non-relativistic particles in 3, 2, 1, and 0 dimensions. The functional form of $g(E)$ is the key factor in determining the existence of BEC.

To refresh your memory on the derivation of these forms, you may find the first part of this video helpful.

Density of States - Statistical Physics - University Physics

This video from Pazzy Boardman provides a pedagogical derivation of the density of states.

Focus on the logic of converting the sum over discrete states in k-space into a continuous integral. You can watch the full derivation for 3D from this segment to see all the steps. The key result, highlighting the different energy dependencies in 1D, 2D, and 3D, is summarized succinctly in the passage from 2D and 1D systems.

The crucial aspect to analyze is the behavior of the integral for $N_{e x}^{ma x}$ at its lower limit, as $E \to 0$ , since this is where a potential divergence lies. In this limit, the denominator simplifies: $e^{β E} - 1 \approx β E$ . The integrand thus behaves as:

\frac{g _{d} ( E )}{e ^{β E} - 1} \propto \frac{E ^{d /2 - 1}}{E} = E^{d /2 - 2}

A definite integral $\int_{0}^{a} x^{p} d x$ converges if and only if $p > - 1$ . Applying this to our case, the integral for $N_{e x}^{ma x}$ converges if:

\frac{d}{2} - 2 > - 1 ⟹ \frac{d}{2} > 1 ⟹ d > 2

This simple and powerful result, known as the Mermin-Wagner-Hohenberg theorem for this specific context, dictates that BEC in an ideal, non-relativistic, uniform gas is only possible for spatial dimensions greater than two.

3. Proof for 2D and 1D Systems

Let's now explicitly demonstrate this conclusion. The following text from a University of British Columbia course provides the mathematical framework we need.

Quantum ideal gases: bosons 1 The model

This document lays out the standard grand canonical treatment of the ideal Bose gas and introduces the key mathematical functions.

First, review the setup in Section 2. This introduces the bosonic integral J_\nu^{(-)} and the Riemann-Dirichlet function g_\nu(z), where z=e^{\beta\mu}. Pay close attention to the statement: "The essential thing to remember is that g_\nu(1) ... is finite if \nu > 1, and infinite ... if \nu \le 1". This property is the core of the entire argument. Next, read the argument presented in the section "Low-temperature limit". This section shows how in 3D, the number of particles in excited states is related to g_{3/2}(z). Since the index \nu=3/2 > 1, g_{3/2}(1) = \zeta(3/2) is finite, meaning the excited states can be saturated. Finally, read the short concluding paragraph that begins "In conclusion". It explicitly states that changing the dimensionality alters the index \nu of the relevant Riemann-Dirichlet function, which changes the outcome.

Let's use this framework to analyze the 2D and 1D cases.

The 2D Case

In two dimensions, the density of states is constant: $g_{2} (E) = A_{2}$ , where $A_{2} = m S / (π ℏ^{2})$ for a surface of area $S$ . The maximum number of excited particles is:

N_{e x}^{ma x} (T) = A_{2} \int_{0^{+}}^{\infty} \frac{1}{e ^{β E} - 1} d E

Let $x = β E$ , so $d E = (k_{B} T) d x$ . The integral becomes:

N_{e x}^{ma x} (T) = A_{2} k_{B} T \int_{0^{+}}^{\infty} \frac{1}{e ^{x} - 1} d x

As we established, the integrand behaves as $1/ x$ near $x = 0$ . The integral $\int (1/ x) d x = ln (x)$ diverges at the lower limit. The excited states can hold an infinite number of particles at any temperature $T > 0$ . Thus, no BEC occurs in 2D.

This exact argument is laid out very clearly in the following resource.

[PDF] Two-dimensional Bose fluids: An atomic physics perspective(∗)

This review article by Dalibard and Hadzibabic is a standard reference on 2D Bose gases. Section 2.1 deals specifically with the ideal gas.

Please read the subsection "The ideal 2d Bose gas". It walks through the calculation of the number of particles (Eq. 4) and shows that a valid solution for the chemical potential (Eq. 6) always exists, preventing condensation.

The 1D Case

In one dimension, $g_{1} (E) = A_{1} E^{- 1/2}$ . The integral for $N_{e x}^{ma x}$ is:

N_{e x}^{ma x} (T) = A_{1} \int_{0^{+}}^{\infty} \frac{E ^{- 1/2}}{e ^{β E} - 1} d E

Near $E = 0$ , the integrand behaves like $E^{- 1/2} / E = E^{- 3/2}$ . The integral $\int E^{- 3/2} d E = - 2 E^{- 1/2}$ diverges even more strongly at the lower limit than in the 2D case. Therefore, no BEC occurs in 1D.

The divergence of the integral in low dimensions is a direct consequence of the high density of available low-energy states. These states are so numerous that they can always accommodate the particles being cooled without needing to "spill over" into the ground state.

4. A Deeper Analogy: Dimensionality and Bound States

This dimensional dependence of collective phenomena is a profound and recurring theme in physics. A beautiful analogy can be drawn from a fundamental quantum mechanics problem: the formation of a two-particle bound state in an attractive potential.

As you may recall from your quantum mechanics courses, in 1D and 2D, an arbitrarily weak attractive potential is sufficient to form a bound state. In 3D, however, the potential must exceed a certain minimum depth to bind two particles. Why? The reason is precisely the same: the availability of low-energy kinetic states. In 3D, there are so many ways for a particle to "escape" in different directions (a larger phase space for kinetic energy) that a weak potential can't confine it. In 1D and 2D, the restricted kinetic phase space makes it easier for the potential to "capture" the particle.

The following video by Prof. Wolfgang Ketterle provides a superb explanation of this concept and its surprising connection to Cooper pairing in superconductors.

20. Fermi gases, BEC-BCS crossover

In this lecture from his MIT course, Prof. Ketterle explains why the formation of bound states is dimension-dependent.

Watch the segment from 1D and 2D, where he poses and answers the question of why an infinitesimal potential creates a bound state in 1D/2D but not 3D. The key is the boundary condition on the radial wave function at the origin. Next, watch the more formal argument from one equation. Here, he shows that the condition for forming a bound state is equivalent to the divergence of an integral involving the density of states. This directly parallels our argument for BEC. The logarithmic divergence in 2D appears as the marginal case.

This analogy reinforces the idea that the divergence of the density of states at low energies in 1D and 2D is the fundamental reason for the absence of BEC.

5. An Important Caveat: The Role of Trapping

The results we have derived apply to a uniform gas in a box-like potential. Modern experiments with ultracold atoms, which you've worked with in the context of photoluminescence, almost always use harmonic trapping potentials, like $V (r) = \frac{1}{2} m ω^{2} r^{2}$ . This changes the game completely.

In a 2D harmonic trap, the single-particle energy levels are $E_{n_{x}, n_{y}} = (n_{x} + n_{y} + 1) ℏ ω$ . The degeneracy of the $j$ -th energy level $E_{j} = (j + 1) ℏ ω$ is $j + 1$ . This leads to a density of states that is linear in energy: $g (E) \propto E$ .

If we now re-evaluate our integral for $N_{e x}^{ma x}$ with $g (E) \propto E$ , the integrand near $E = 0$ behaves like $E / E = E^{0}$ . The integral $\int E^{0} d E = E$ converges at the lower limit! Consequently, BEC can and does occur in a 2D harmonically trapped gas. This is a crucial distinction between uniform and trapped systems and is the basis for many modern experiments on 2D physics.

[PDF] Two-dimensional Bose fluids: An atomic physics perspective(∗)

The same review article discusses the crucial modification introduced by harmonic trapping.

Please read the subsection "The ideal case" in Section 5.1. It directly shows how the modified density of states in a harmonic trap leads to a finite critical number of particles, allowing for condensation.

Conclusion

In this lesson, we have demonstrated why Bose-Einstein condensation, a hallmark of quantum statistics in 3D, is absent for a non-interacting, non-relativistic gas in one and two spatial dimensions.

Key Takeaways:

The existence of BEC hinges on whether the total number of particles that can occupy excited states, $N_{e x}^{ma x}$ , is finite at $T > 0$ .
This is determined by the convergence of the integral $\int g (E) / (e^{β E} - 1) d E$ , which depends on the behavior of the density of states $g (E)$ as $E \to 0$ .
For a uniform gas, $g (E) \propto E^{d /2 - 1}$ . The integral diverges for $d \leq 2$ due to the abundance of low-energy states, thus forbidding BEC. This is a specific manifestation of the Mermin-Wagner theorem.
This result is fundamentally altered by external potentials. In a 2D harmonic trap, the density of states becomes $g (E) \propto E$ , which allows the integral to converge and BEC to occur.

This exploration of dimensionality highlights how profoundly the large-scale properties of a many-body system can depend on the geometry of the space it inhabits. In our next lesson, we will apply the Bose-Einstein statistics to a different type of particle: phonons, the quantized vibrations of a crystal lattice. This will lead us to the Debye model for the low-temperature specific heat of solids, another triumph of quantum statistical mechanics.

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