BKT Transition in 2D Exciton Systems

Hello! In our last lesson, we derived the critical temperature for Bose-Einstein Condensation (BEC) in a 3D ideal exciton gas. We saw that condensation is a phase transition driven by the saturation of excited quantum states, leading to a macroscopic population of the ground state. A natural and crucial question follows: what happens if the excitons are confined to a two-dimensional plane, as is common in many semiconductor heterostructures?

This lesson addresses that question, and in doing so, we will explore a completely different, and in many ways richer, type of phase transition. We will see that the rules of the game change fundamentally in 2D.

Today's learning outcome is to: Describe the Berezinskii-Kosterlitz-Thouless (BKT) transition for interacting 2D exciton systems. We will discover that while conventional BEC is forbidden, a remarkable topological phase transition into a "quasi-condensate" state still occurs, driven not by particle statistics alone, but by the interplay of energy and entropy of topological defects.

1. The Mermin-Wagner Theorem: A Roadblock to 2D Condensation

In our 3D derivation, the integral for the number of particles in excited states, $N_{e x}$ , converged even when we set the chemical potential $μ = 0$ . This meant there was a finite maximum number of particles the excited states could hold at a given temperature, $N_{e x, ma x} (T)$ . The story is different in 2D.

Let's re-examine the key integral, but with a 2D density of states. In 2D, the density of states $D (ϵ)$ is constant with respect to energy (for $ϵ > 0$ ). Let's call it $A_{g}$ , where $A_{g} = g \frac{A M _{e x}}{2 π ℏ ^{2}}$ for a system of area $A$ .

The number of particles in excited states is:
$N_{e x} = \int_{0}^{\infty} \frac{D ( ϵ )}{e ^{β (ϵ - μ)} - 1} d ϵ = A_{g} \int_{0}^{\infty} \frac{1}{e ^{β ϵ} e ^{- β μ} - 1} d ϵ$

As we approach the critical point from high temperatures, $μ \to 0^{-}$ . The integral becomes:
$N_{e x, ma x} = A_{g} \int_{0}^{\infty} \frac{d ϵ}{e ^{β ϵ} - 1}$
Let's analyze the behavior of the integrand near $ϵ = 0$ . For small $ϵ$ , $e^{β ϵ} \approx 1 + β ϵ$ , so the integrand behaves like $1/ (β ϵ)$ . The integral is of the form $\int d ϵ / ϵ$ , which is logarithmically divergent at the lower limit $ϵ = 0$ .

This divergence means that for any finite temperature $T > 0$ , the excited states can hold an infinite number of particles. There is no "saturation" of excited states. No matter how many excitons are in the system, they can always be accommodated in the low-energy excited states without needing to macroscopically occupy the ground state.

This is a key consequence of the Mermin-Wagner (or Hohenberg-Mermin-Wagner) theorem, which states that a continuous symmetry (in our case, the global U(1) phase symmetry of the wavefunction) cannot be spontaneously broken at any finite temperature in systems with short-range interactions in dimensions $d \leq 2$ . Intuitively, long-wavelength thermal fluctuations (Goldstone modes) are so powerful in 2D that they always destroy any long-range phase coherence. The result is that true long-range order, and thus conventional BEC, is impossible.

Two-dimensional Bose fluids: An atomic physics perspective & Nonequilibrium BKT transition in microcavity polaritons

For a formal statement of this fundamental theorem and its implications for 2D Bose gases, please read the following brief sections from two excellent resources.

First, read section 1.1 'Absence of true long-range order in 2d' in the review by Hadzibabic and Dalibard. It provides a clear, conceptual explanation of the Mermin-Wagner theorem.

Nonequilibrium BKT transition in microcavity polaritons

Next, read section 2.2.3 'The Hohenberg-Mermin-Wagner theorem' in the thesis by Dagvadorj. This section reinforces the concept and provides the mathematical reasoning based on the particle number integral we just discussed (see Eq. 2.74 and the surrounding text).

2. Topological Order and Vortices

If long-range order is impossible, does that mean a 2D exciton gas is always a disordered, classical gas? No. The system can still undergo a phase transition into a state with quasi-long-range order (QLRO). This is a subtle, intermediate state of matter where correlations decay not exponentially (like a gas) but as a much slower power law.

The transition into this state is not driven by the statistical mechanics of occupying energy levels, but by the emergence and interaction of topological defects. For a 2D superfluid or exciton gas, these defects are vortices.

A vortex is a point-like singularity in the system where the density of the exciton gas goes to zero and the phase of the macroscopic wavefunction $Ψ (r) = n (r) e^{i ϕ (r)}$ winds by an integer multiple of $2 π$ around it.

Ingredients of the BKT Transition — This diagram illustrates the essence of a topological defect. A closed loop in the physical space of the material maps to a non-trivial loop in the order parameter space (here, the phase angle of the XY model, which is analogous to our exciton gas). This 'winding' cannot be smoothly undone, giving the defect its topological stability.

We can have vortices with a phase winding of $+ 2 π$ and anti-vortices with a winding of $- 2 π$ . As we will see, the behavior of these vortices governs the properties of the 2D gas.

3. The BKT Transition: An Unbinding of Vortex Pairs

The Berezinskii-Kosterlitz-Thouless (BKT) theory describes a phase transition driven by the proliferation of these vortices. The mechanism is a competition between the energy it costs to create a vortex and the entropy gained by having it.

To understand this, we can draw a powerful analogy to a 2D Coulomb gas of charged particles. It turns out that a vortex and an anti-vortex attract each other with a potential that is logarithmic with their separation, $U (r) \propto ln (r)$ , just like the 2D electrostatic potential between opposite charges.

Let's explore the BKT mechanism with this in mind.

Two-dimensional Bose fluids: An atomic physics perspective

The following paper provides a superb and intuitive explanation of the BKT transition, using a simple physical picture based on a free energy argument. It is one of the clearest introductions to the topic.

Please read section 3, 'The Berezinskii-Kosterlitz-Thouless (BKT) transition in a 2d Bose gas', focusing especially on subsections 3.1 and 3.2. These parts explain the role of vortices and walk through the simple free-energy argument for the transition.

Let's summarize the key ideas from that reading.

The Two Phases:

Low Temperature (T < T_BKT): Ordered Phase
At low temperatures, the energy cost to separate a vortex-antivortex pair is too high. Vortices can only exist as tightly bound pairs. A bound pair acts like a small electric dipole; its influence is local and falls off quickly with distance. These pairs do not disrupt the overall phase coherence of the system on long length scales. The system is a superfluid with quasi-long-range order. The first-order correlation function decays as a power-law:
$g^{(1)} (r) = ⟨ Ψ^{*} (r) Ψ (0)⟩ \propto r^{- α (T)}$
High Temperature (T > T_BKT): Disordered Phase
Above a critical temperature, the entropy gain from having free vortices moving around the system outweighs the energy cost of creating them. The vortex-antivortex pairs unbind, and the system fills with a "plasma" of free vortices and anti-vortices. These free vortices move randomly, completely scrambling the phase of the wavefunction over long distances. This destroys the superfluidity and the system behaves like a normal fluid with only short-range, exponential correlations:
$g^{(1)} (r) \propto e^{- r / ξ}$

The Free Energy Argument

The reading you did presents a beautifully simple argument to find the transition point. It considers the free energy $F = E - T S$ for creating a single, isolated free vortex in a system of size $R$ .

The energy cost $E$ to create a vortex is dominated by the kinetic energy of the swirling superflow around it. This integral yields $E \propto n_{s} ln (R / ξ)$ , where $n_{s}$ is the superfluid density and $ξ$ is the vortex core size.
The entropy $S$ is related to the number of places the vortex can be placed, which is proportional to the area $π R^{2}$ . This gives $S = k_{B} ln ((R / ξ)^{2}) = 2 k_{B} ln (R / ξ)$ .

Combining these, the free energy change is:
$F \approx (\frac{π ℏ ^{2} n _{s}}{m} - 2 k_{B} T) ln (R / ξ)$
In the thermodynamic limit $R \to \infty$ , the sign of the term in the parentheses determines everything.

If $F > 0$ , free vortices are energetically forbidden.
If $F < 0$ , the system is unstable to the proliferation of free vortices.

The transition occurs precisely when this term is zero:
$\frac{π ℏ ^{2} n _{s} ( T _{B K T} )}{m} = 2 k_{B} T_{B K T}$
Rearranging this and using the thermal de Broglie wavelength $λ_{T} = h / 2 π m k_{B} T$ , we get the famous universal jump condition:
$n_{s} (T_{B K T}) λ_{T_{B K T}}^{2} = 4$
This is a remarkable result. It states that at the moment of the transition, the superfluid density $n_{s}$ doesn't smoothly go to zero (like the condensate fraction in 3D BEC). Instead, it jumps discontinuously from a finite value, $4/ λ_{T}^{2}$ , to zero. This universal jump is a key experimental signature of the BKT transition.

Test your understanding!

Imagine you are conducting an experiment on an exciton gas that can be tuned from being 3D to 2D (e.g., by changing the thickness of a quantum well). Based on this lesson and the previous one, contrast the key experimental signatures you would look for to distinguish a 3D BEC transition from a 2D BKT transition as you measure the system properties versus temperature.

Show answer

Here are the key distinctions an experimentalist would look for:

Order Parameter / Coherence:
- 3D BEC: The signature is the emergence of true long-range order. An interference experiment would show high-contrast fringes that are stable across the entire sample below $T_{c}$ . The condensate fraction, $N_{0} / N$ , acting as an order parameter, grows continuously from zero as $1 - (T / T_{c})^{3/2}$ .
- 2D BKT: The signature is a change from exponential to power-law decay of spatial coherence ( $g^{(1)} (r)$ ). Interference fringes would be visible locally, but their contrast would decay with distance, even below $T_{B K T}$ . There is no macroscopic occupation of a single state.
Superfluid Density ( $n_{s}$ ):
- 3D BEC: The superfluid density $n_{s} (T)$ (which is approximately the condensate density $n_{0} (T)$ at low T) grows continuously from zero at $T_{c}$ .
- 2D BKT: The superfluid density exhibits a discontinuous, universal jump. It is finite just below $T_{B K T}$ and abruptly drops to zero right at the transition.
Direct Observation of Defects:
- 3D BEC: While vortices can exist as excitations, they do not drive the phase transition itself.
- 2D BKT: The transition is fundamentally driven by vortices. A direct imaging technique would reveal a proliferation of free, unbound vortices appearing suddenly as the temperature is raised above $T_{B K T}$ , whereas below $T_{B K T}$ only bound vortex-antivortex pairs would be seen.

4. BKT in Interacting Exciton Systems

The BKT theory is universal, applying to many 2D systems with a U(1) symmetry. This includes interacting 2D exciton gases and, as you might encounter in your research, exciton-polariton systems in microcavities.

The theory we've discussed is for a system in thermal equilibrium. Many exciton systems, especially those involving polaritons, are driven-dissipative and thus inherently non-equilibrium. A fascinating research area is understanding how BKT-like physics manifests in these settings.

Nonequilibrium BKT transition in microcavity polaritons

The thesis by Dagvadorj, from which you read a short excerpt earlier, provides a comprehensive theoretical and numerical study of a non-equilibrium BKT transition in exciton-polaritons. Reading the core theoretical summary and looking at the simulation results will give you a flavor of how these fundamental concepts apply at the research frontier.

First, read section 2.2.4, 'The Kosterlitz-Thouless transition'. This provides a more detailed theoretical summary, including the mapping to a 2D Coulomb gas and the result for the critical phase-space density (Eq. 2.83) which connects the transition temperature to the interaction strength ilde{g}. Then, skim section 4.3.4, 'Spatial correlation function', and look closely at Figures 4.14 and 4.15. These plots show exactly the transition from exponential to power-law decay in the correlation function from a numerical simulation, a direct visualization of the BKT transition's main signature. This will resonate with your experience in writing simulations.

As you saw, the key signatures of the BKT transition—vortex unbinding and a change in the functional form of correlation decay—persist even in complex, non-equilibrium systems, cementing the importance of this theoretical framework.

Conclusion

Today we have navigated the fascinating physics of 2D Bose gases, finding a stark contrast with their 3D counterparts.

Key Takeaways:

Due to the Mermin-Wagner theorem, true long-range order and conventional BEC are forbidden in 2D systems at finite temperatures.
Instead, an interacting 2D exciton gas can undergo a Berezinskii-Kosterlitz-Thouless (BKT) transition. This is a topological phase transition, not a statistical one.
The transition is driven by the unbinding of vortex-antivortex pairs. Below $T_{B K T}$ , vortices are bound, and the system exhibits quasi-long-range order (power-law correlations). Above $T_{B K T}$ , a plasma of free vortices destroys coherence, leading to short-range order (exponential correlations).
A key signature is the universal jump in the superfluid density, which drops from a finite value ( $n_{s} = 4/ λ_{T}^{2}$ ) to zero at the transition.

Preview of the Next Lesson

Both the 3D BEC and 2D BKT phenomena rely on the assumption of interactions between particles. In the BKT case, it is interactions that suppress density fluctuations and allow phase coherence to dominate. To properly describe these interactions and other phenomena like the dissociation of excitons at high density (the Mott transition), we need a more powerful formalism.

In the next lesson, we will begin our journey into the many-body physics of excitons, achieving the learning outcome: Formulate the electron-hole pair Hamiltonian using second quantization creation and annihilation operators. This will provide the rigorous foundation needed to understand the exciton as a true many-body quasiparticle.

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