🔬 materials science

Vacancy Effect on the Ground-State Energy of a Bose Gas Trapped by 1D Imperfect Artificial Crystal

This paper numerically investigates how randomly introduced vacancies in a one-dimensional artificial Dirac comb potential affect the ground-state energy and boson distribution of a weakly interacting Bose gas, revealing an exponential energy decrease with increasing vacancy count and the emergence of localization features around defects.

Original authors: E. I. Guerrero-Cruz, O. A. Rodríguez-López, M. A. Solís

Published 2026-07-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: E. I. Guerrero-Cruz, O. A. Rodríguez-López, M. A. Solís

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded dance floor where thousands of tiny, invisible dancers (bosons) are moving in perfect unison. In a perfect world, they'd glide smoothly across a floor made of a repeating pattern of sticky spots (a "Dirac comb" potential). But what happens if we randomly rip a few of those sticky spots out of the floor, creating empty "vacancies"? That's exactly what this paper explores, using a super-computer simulation to see how these missing spots change the energy of the whole dance troupe.

The Main Discovery: The Energy Drop
The researchers found that when you start removing these sticky spots (vacancies), the total energy of the dance floor drops dramatically. It doesn't just go down a little; it plummets in a very specific way: exponentially.

Think of it like a rollercoaster. If you have a perfect track with 1,000 bumps, the ride has a certain high energy. If you randomly remove 10% of the bumps, the energy drops. Remove 50%, and it drops even further. The paper shows that as you remove more and more deltas (the sticky spots), the energy slides down a steep curve until it hits the bottom: the energy of the dancers moving on a completely flat, empty floor with no bumps at all.

They ran these simulations for different numbers of dancers (system sizes) and different "stickiness" levels of the interactions between the dancers (represented by the number g).

For a system with 1,000 initial deltas and no interaction between dancers (g = 0), removing just 10% of the deltas dropped the energy from 6.8942 down to 1.8338 (in units of ℏ²/2ma²).
When they removed 50% of the deltas in that same setup, the energy fell to 0.1056.
If they removed 100% of the deltas (a completely free gas), the energy hit 0.

The "Gap" and the "Freeze"
Here is the most exciting part: when the crystal is perfect, the dancers can easily jump to higher energy levels. But when you introduce a vacancy (a missing spot), a mysterious "energy gap" opens up. It's like a moat appearing between the ground floor and the first floor of a castle.

The paper notes that this gap is most obvious when the dancers interact weakly (g ≤ 0.1) and the sticky spots are strong (P₀ = 10). In this specific case, the gap is about 4.656 ℏ²/2ma². The authors suggest this gap is a big deal because it might help the dancers "freeze" into a single, synchronized state (Bose-Einstein condensation) even at temperatures that aren't absolute zero, which is usually hard to do in one dimension.

Where Do the Dancers Go?
The paper also looked at where the dancers actually stand. In a perfect crystal, they spread out evenly, like a perfectly tiled floor. But when a vacancy appears, the dancers start to huddle around the empty spot. It's as if the missing sticky spot acts like a magnet, pulling the probability of finding a dancer right next to it.

However, this huddling behavior is fragile. The paper shows that as the dancers start pushing against each other more strongly (increasing g), they stop caring about the missing spots.

At g = 2, the dancers still show some tendency to cluster near the vacancies.
But as g gets larger, that clustering disappears, and the dancers spread out again, effectively ignoring the holes in the floor.

How Sure Are We?
It's important to know that these results come from numerical simulations, not a physical experiment in a lab. The authors used a method called "Gradient Flow with Discrete Normalization" (which is basically a fancy way of saying they let the system "cool down" in a computer until it found its lowest energy state).

They tested their computer model against a famous, mathematically proven theory called the Lieb-Liniger model (which describes free interacting bosons). Their simulation matched this known theory with an error of less than 17%, which gives them confidence that their results for the imperfect crystals are reliable.

They also looked at what happens as the crystal gets infinitely large. By simulating crystals with 50, 100, 500, and 1,000 deltas and then mathematically "extrapolating" to infinity, they predicted the energy for a giant, infinite crystal with just one missing spot. For example, with g = 1, the energy for an infinite system with one vacancy is predicted to be 8.742 ℏ²/2ma², which is slightly lower than the perfect infinite crystal.

What They Didn't Find
The paper explicitly rules out the idea that the energy drops in a straight line. It's not a simple slide; it's an exponential curve. They also found that for strong interactions (high g), the specific number of deltas in the system matters less—the energy values for different system sizes tend to group together, meaning the "size" of the crystal becomes less important than the strength of the push between the dancers.

In short, the paper suggests that if you poke holes in a 1D crystal of quantum dancers, you can drastically lower their energy and create a gap that might help them sync up, but only if they aren't pushing each other too hard. If they push too hard, the holes stop mattering.

Technical Summary: Vacancy Effect on the Ground-State Energy of a Bose Gas Trapped by 1D Imperfect Artificial Crystal

Problem Statement
This study investigates the ground-state properties of a weakly interacting Bose gas confined within a one-dimensional (1D) artificial crystal. The system is modeled using an external Dirac comb potential, where "vacancies" (punctual defects) are introduced by randomly removing a predetermined fraction of the delta-function scattering centers. The primary objective is to quantify how these vacancies affect the ground-state energy (GSE), the chemical potential, and the spatial distribution of the bosons, particularly in comparison to both the perfect crystal and the free gas limits. The work addresses the gap in understanding how translational symmetry breaking via point defects influences Bose-Einstein condensation (BEC) conditions and thermodynamic properties in low-dimensional interacting systems.

Methodology
The authors employ the mean-field approximation, describing the system dynamics via the time-dependent Gross-Pitaevskii equation (GPE). To determine the ground state, they utilize the "Gradient Flow with Discrete Normalization" (GFDN) method, also known as the imaginary time method.

Key methodological steps include:

Dimensionless Formulation: The GPE and associated Hamiltonian are converted into dimensionless units based on a characteristic length and frequency derived from a quantum harmonic oscillator analogy.
Numerical Solution: The stationary GPE is solved using a two-step numerical scheme:
- Gradient Flow: The wave function evolves in imaginary time ( $\tau = it$ ) according to an unconstrained gradient flow equation to minimize the free energy.
- Discrete Normalization: At each time step, the wave function is explicitly renormalized to satisfy the particle number constraint.
Spectral Approach: The spatial derivatives are handled using a Discrete Fourier Transform (DFT) based pseudo-spectral approach combined with a Backward Euler finite-difference discretization in time. This ensures efficient handling of periodic boundary conditions.
System Modeling: The artificial crystal is defined by a Dirac comb potential $V(x) = v_0 \sum \delta(x-ja)$ . Vacancies are simulated by randomly removing a specific percentage of these deltas. The study covers various system sizes (50 to 1000 initial deltas) and interaction strengths ( $g$ ), extrapolating results to the thermodynamic limit (infinite system size) for the case of a single vacancy.

Key Results

Ground-State Energy (GSE) vs. Vacancy Percentage: For an ideal Bose gas ( $g=0$ ) and weakly interacting gases ( $g=1, 2$ ), the GSE decreases exponentially as the percentage of randomly removed deltas increases. The energy transitions from the value of the perfect crystal to that of a free Bose gas as the vacancy ratio approaches 100%.
System Size Dependence: For a fixed vacancy percentage, the GSE values for different system sizes (50, 100, 500, 1000 deltas) converge as the system size increases. The authors extrapolate the GSE for a single vacancy to the infinite system limit, finding that the infinite system GSE represents a maximum value toward which finite systems tend as their size grows.
Energy Gap and BEC: The introduction of a single vacancy breaks translational symmetry, creating an energy gap between the ground state and the first excited state. This gap is most pronounced for weak interactions ( $g \leq 0.1$ ) with a delta strength $P_0 = 10$ . The authors note that this gap promotes BEC at finite temperatures in dimensions $d > 0$ , a condition not met by the perfect periodic potential in 1D.
Interaction Dependence:
- The GSE increases almost linearly with the interaction magnitude $g$ .
- The difference in energy between the perfect and imperfect (one vacancy) systems is maximal at $g=0$ and decreases exponentially as $g$ increases. For $g \geq 0.2$ , the energy difference becomes negligible, indicating that strong inter-particle interactions render the system less sensitive to single punctual defects.
- The results for the free interacting gas ( $P_0=0$ ) align closely with the analytical Lieb-Liniger solution, with errors less than 17%.
Chemical Potential: The chemical potential is derived directly from the GSE. It increases proportionally with interaction strength and delta strength. Imperfections cause the chemical potential to decrease relative to the perfect system, though it remains higher than the GSE.
Particle Distribution: The probability density functions reveal that vacancies act as attractive centers, causing particle localization around the missing deltas. This localization feature diminishes and eventually disappears as the interaction strength $g$ increases.

Significance and Claims
The paper claims to provide a numerical characterization of how punctual defects influence the ground-state properties of a Bose gas in a 1D artificial crystal. The authors assert that their work:

Demonstrates the exponential decay of ground-state energy as a function of vacancy concentration, bridging the gap between perfect crystal and free gas behaviors.
Confirms the existence of an energy gap induced by a single vacancy, which theoretically facilitates BEC in 1D systems at finite temperatures, addressing limitations of perfect periodic potentials.
Validates the numerical approach by reproducing the Lieb-Liniger results for free interacting gases and establishing a baseline for more complex defect scenarios.
Highlights the competition between interaction strength and defect localization, showing that strong interactions effectively "wash out" the impact of single vacancies on the ground-state energy and density distribution.

The study concludes that while vacancies significantly alter the ground state of ideal and weakly interacting gases, their influence becomes imperceptible as boson-boson interactions become dominant.

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