Untangling Quantum Correlations in Fermi Gases

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Author: Denis Avetisyan

New theoretical work clarifies the interplay of spin, symmetry, and the Pauli exclusion principle in understanding correlated behavior across the BCS-BEC crossover.

The spatial behavior of correlation functions-specifically <span class="katex-eq" data-katex-display="false">g_{ss^{\prime}}(\delta{\bf r})</span> and <span class="katex-eq" data-katex-display="false">g_{nn}(\delta{\bf r})</span>-reveals distinct characteristics of translationally and rotationally invariant systems dependent on interaction parameters, exhibiting behavior consistent with either a deep BCS regime (<span class="katex-eq" data-katex-display="false">\ln k_{\rm F}a=2.15</span>) or a crossover region (<span class="katex-eq" data-katex-display="false">\ln k_{\rm F}a=0.36</span>), as modulated by the level of approximation detailed in Table 1.
The spatial behavior of correlation functions-specifically g_{ss^{\prime}}(\delta{\bf r}) and g_{nn}(\delta{\bf r})-reveals distinct characteristics of translationally and rotationally invariant systems dependent on interaction parameters, exhibiting behavior consistent with either a deep BCS regime (\ln k_{\rm F}a=2.15) or a crossover region (\ln k_{\rm F}a=0.36), as modulated by the level of approximation detailed in Table 1.

This review details a theoretical framework for calculating equal-spin and opposite-spin density-density correlations, emphasizing the role of two-particle irreducibility and its connection to experimental observations.

Understanding the collective behavior of interacting Fermi gases remains a significant challenge, particularly when bridging the Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes. This work, ‘Equal-spin and opposite-spin density-density correlations in the BCS-BEC crossover: Gauge Symmetry, Pauli Exclusion Principle, Wick’s Theorem and Experiments’, presents a general theoretical framework for spin-dependent density-density correlations, rigorously incorporating constraints from gauge invariance and the Pauli exclusion principle. We demonstrate that accurate descriptions of experimental observations-specifically, the emergence of a minimum in opposite-spin correlations-require accounting for two-particle irreducible contributions beyond standard mean-field approaches. How will these insights into many-body correlations advance our ability to design and interpret quantum simulations using ultracold atomic gases?

Unveiling the Symphony of Many-Body Physics

The behavior of many-particle systems presents a fundamental challenge across numerous branches of physics, extending far beyond the realm of quantum mechanics. From the superconductivity exhibited by certain materials to the complex dynamics of fluids and the properties of stellar interiors, understanding how countless particles interact is crucial. These interactions, often subtle and emergent, dictate macroscopic properties in ways that are frequently impossible to predict from the characteristics of individual components. Consequently, research into many-body physics isn’t merely an academic exercise; it underpins advancements in materials science, condensed matter physics, and even cosmology, driving the development of new technologies and a deeper understanding of the universe itself. The difficulty arises because the sheer number of interacting particles creates a computational bottleneck, requiring innovative theoretical frameworks and experimental techniques to untangle the resulting complexities.

The challenge of describing many-body systems – those comprising a vast number of interacting particles – frequently overwhelms conventional theoretical methods. These approaches, while successful in simpler scenarios, often falter when faced with the exponential increase in computational demands as particle number grows. This limitation stems from the intricate correlations that emerge between particles, requiring calculations that scale unfavorably with system size. Consequently, physicists are increasingly reliant on advanced computational techniques, such as quantum Monte Carlo simulations and density functional theory, alongside innovative experimental platforms. These combined efforts aim to circumvent the theoretical bottlenecks and provide insights into the collective behavior arising from these complex interactions, ultimately pushing the boundaries of condensed matter physics and beyond.

Ultracold Fermi gases represent a significant advancement in condensed matter physics, offering an unprecedented level of experimental control over interacting quantum particles. By cooling gases of fermionic atoms to temperatures just above absolute zero – on the order of nanokelvins – researchers can dramatically reduce the kinetic energy of the atoms, effectively isolating and magnifying the influence of interatomic interactions. This precise manipulation, often achieved using laser and magnetic trapping techniques, allows scientists to tune the strength of these interactions – from weakly bound to strongly correlated regimes – and observe the resulting quantum phenomena with remarkable clarity. Unlike traditional solid-state systems where interactions are fixed, these gases provide a highly tunable platform, enabling the exploration of a wide range of many-body effects, including the formation of Cooper pairs and the emergence of novel superfluid phases, and offering valuable insights into the behavior of complex systems across diverse fields like materials science and nuclear physics.

The ability to finely tune interactions in ultracold Fermi gases unlocks access to strongly correlated systems – those where particles’ behaviors are inextricably linked, defying prediction by treating them independently. These systems, prevalent in materials like high-temperature superconductors, present a significant challenge to conventional theoretical methods. Through precise control of interatomic forces-using magnetic fields, for instance-researchers can experimentally realize and probe regimes previously inaccessible, effectively creating a ‘tabletop laboratory’ for many-body physics. This experimental prowess allows for direct comparison with advanced theoretical models, validating or refining predictions and fostering a synergistic relationship between computation and observation. The resulting insights aren’t merely academic; they promise to illuminate the fundamental principles governing complex materials and potentially pave the way for novel technologies.

Figure 2:Numerical results forg↑↑​(δ​𝐫)g\_{\uparrow\uparrow}(\delta{\bf r})ranging from0and11(top panel),g↑↓​(δ​𝐫)g\_{\uparrow\downarrow}(\delta{\bf r})(middle panel) ranging from0.70.7to1.41.4andgn​n​(δ​𝐫)g\_{nn}(\delta{\bf r})(bottom panel) ranging from0.50.5to1.51.5versus distancekF​|δ​𝐫|k\_{\rm F}|\delta{\bf r}|ranging from0to55. The dashed black (solid blue) lines illustrate caseI{\rm I}(IV) in Table1. The gray region in the middle panel shows thatg↑↓​(δ​𝐫)≥1g\_{\uparrow\downarrow}(\delta{\bf r})\geq 1with no minimum for the dashed black line, whileg↑↓​(δ​𝐫)g\_{\uparrow\downarrow}(\delta{\bf r})has a clear minimum below 1 for the solid blue line. The interaction parameterln⁡kF​a\ln k\_{\rm F}aranges from2.152.15(BCS regime) to0.360.36(crossover region) and are closely related to the experimental values[63].
Figure 2:Numerical results forg↑↑​(δ​𝐫)g\_{\uparrow\uparrow}(\delta{\bf r})ranging from0and11(top panel),g↑↓​(δ​𝐫)g\_{\uparrow\downarrow}(\delta{\bf r})(middle panel) ranging from0.70.7to1.41.4andgn​n​(δ​𝐫)g\_{nn}(\delta{\bf r})(bottom panel) ranging from0.50.5to1.51.5versus distancekF​|δ​𝐫|k\_{\rm F}|\delta{\bf r}|ranging from0to55. The dashed black (solid blue) lines illustrate caseI{\rm I}(IV) in Table1. The gray region in the middle panel shows thatg↑↓​(δ​𝐫)≥1g\_{\uparrow\downarrow}(\delta{\bf r})\geq 1with no minimum for the dashed black line, whileg↑↓​(δ​𝐫)g\_{\uparrow\downarrow}(\delta{\bf r})has a clear minimum below 1 for the solid blue line. The interaction parameterln⁡kF​a\ln k\_{\rm F}aranges from2.152.15(BCS regime) to0.360.36(crossover region) and are closely related to the experimental values[63].

Mapping the Transition: The BCS-BEC Crossover

The BCS-BEC crossover details a continuous morphological change in a fermionic system. In the Bardeen-Cooper-Schrieffer (BCS) regime, fermions form weakly bound Cooper pairs with large spatial extent, exhibiting properties similar to a conventional superconductor. Conversely, in the Bose-Einstein condensate (BEC) regime, fermions form tightly bound molecules that behave as bosons and condense into a Bose-Einstein condensate. The crossover signifies a transition where the pairing size decreases, evolving from large, diffuse Cooper pairs to smaller, localized molecules, with the system’s behavior changing continuously between these two limits without a sharp phase transition. This allows exploration of a quantum many-body system where the fundamental constituents transition from being weakly interacting fermions to strongly interacting bosons.

Fano-Feshbach resonances are utilized to experimentally access the BCS-BEC crossover by manipulating the scattering length between fermionic atoms. These resonances occur when a bound state associated with an interatomic interaction becomes degenerate with a free particle state, allowing external magnetic fields to tune the strength of the interaction. Specifically, applying a magnetic field near a resonance value alters the s-wave scattering length, effectively controlling the attraction or repulsion between the fermions. By sweeping the magnetic field, the interaction strength can be continuously varied, enabling researchers to explore the transition between the weakly coupled BCS regime, where fermions form loosely bound Cooper pairs, and the strongly coupled BEC regime, where fermions coalesce into tightly bound bosonic molecules. Precise control of the magnetic field is essential for maintaining the system at specific points along the crossover and for quantitative comparison with theoretical models.

Manipulation of interatomic interactions allows researchers to systematically explore the BCS-BEC crossover regime. This is achieved by varying parameters such as magnetic field or atomic density to control the scattering length, a_s, which characterizes the interaction strength between fermionic atoms. As a_s is tuned from negative to positive values, the system transitions from a weakly bound BCS state, where Cooper pairs have a large separation, to a tightly bound BEC state composed of bosonic molecules. Importantly, many physical properties, such as the superfluid transition temperature and the momentum distribution, exhibit universal scaling behavior across this entire crossover, independent of specific atomic species, demonstrating the robustness of the underlying physics.

Fano-Feshbach resonances enable precise control over interatomic interactions in ultracold fermionic gases, a capability vital for testing theoretical models of the BCS-BEC crossover. These resonances, arising from the coupling of closed and open scattering channels, allow researchers to manipulate the s-wave scattering length a_s using external magnetic fields. By systematically varying the magnetic field near a resonance, the strength of the interaction potential can be tuned from weak attraction to strong attraction, and even to repulsion. This level of control is essential because many theoretical predictions regarding the crossover-such as the evolution of the superfluid gap, the condensate fraction, and the Tan’s function-are highly sensitive to the interaction strength. Experimental verification of these predictions relies on the ability to accurately prepare and characterize samples at specific values of a_s, a task facilitated by the precise tuning afforded by Fano-Feshbach resonances.

Numerical results demonstrate that correlation functions <span class="katex-eq" data-katex-display="false">g\_{\uparrow\uparrow}(\delta{\bf r})</span>, <span class="katex-eq" data-katex-display="false">g\_{\uparrow\downarrow}(\delta{\bf r})</span>, and <span class="katex-eq" data-katex-display="false">g\_{nn}(\delta{\bf r})</span> vary with <span class="katex-eq" data-katex-display="false">k\_{F}|\delta{\bf r}|</span>, exhibiting behavior consistent with both the BCS (at <span class="katex-eq" data-katex-display="false">\ln k\_{\rm F}a = 3.01</span>) and BEC (<span class="katex-eq" data-katex-display="false">\ln k\_{\rm F}a = -0.46</span>) regimes as indicated by the dashed black and solid blue lines representing cases I and IV from table 1.
Numerical results demonstrate that correlation functions g\_{\uparrow\uparrow}(\delta{\bf r}), g\_{\uparrow\downarrow}(\delta{\bf r}), and g\_{nn}(\delta{\bf r}) vary with k\_{F}|\delta{\bf r}|, exhibiting behavior consistent with both the BCS (at \ln k\_{\rm F}a = 3.01) and BEC (\ln k\_{\rm F}a = -0.46) regimes as indicated by the dashed black and solid blue lines representing cases I and IV from table 1.

Deciphering Interactions: Density-Density Correlations

The density-density correlation function, denoted as \chi(\mathbf{r}, \mathbf{r'}) , quantifies the fluctuations in particle density at position \mathbf{r} given a perturbation at position \mathbf{r'} . It mathematically expresses the probability of finding a particle at \mathbf{r} when the density at \mathbf{r'} is altered. A non-zero value indicates a correlation – either positive, where fluctuations tend to align, or negative, where they oppose each other – revealing information about the collective behavior of particles within the system. Specifically, \chi(\mathbf{r}, \mathbf{r'}) measures how the density at one point responds to changes in density at another, thereby mapping the spatial extent and strength of correlations present in the many-body system.

Determining the density-density correlation function in many-body systems is computationally complex due to the inherent interactions between particles; direct calculation scales exponentially with system size. Consequently, approximations are essential. Commonly employed techniques include perturbation theory, where the many-body problem is treated as a series of single- and two-particle interactions, and the use of density functional theory (DFT), which maps the interacting many-body system onto an effective single-particle problem. Other advanced methods involve coupled cluster theory, quantum Monte Carlo simulations, and diagrammatic techniques to systematically account for many-body effects, each with its own limitations and applicable regimes. The choice of method depends heavily on the specific system and the desired level of accuracy, with ongoing research focused on developing more efficient and accurate approximations to overcome the computational challenges.

The density-density correlation function G(\mathbf{r}, \mathbf{r}') is often separated into a two-particle reducible component and an irreducible component. The reducible contribution represents correlations that can be explained by considering only single-particle effects and can be expressed in terms of the single-particle density matrix. Conversely, the irreducible contribution encapsulates the effects of many-body interactions, representing correlations that cannot be explained by independent particle motion. This decomposition allows for a systematic analysis of correlations; the reducible part is typically easier to calculate and provides a baseline, while focusing on the irreducible component isolates the influence of interactions and facilitates the application of advanced theoretical techniques to understand complex many-body phenomena.

The form of the density-density correlation function is not arbitrary, but is constrained by fundamental physical principles; the Pauli exclusion principle dictates the antisymmetry of the wavefunction for fermions, influencing correlation behavior, while gauge symmetry ensures invariance under transformations of the electromagnetic potential. Recent investigations have demonstrated the emergence of a minimum in the density-density correlation function specifically for opposite-spin electron pairs, indicating a suppression of correlations at intermediate distances due to exchange effects and the tendency to avoid double occupancy. This minimum occurs because, due to the Pauli principle, electrons with opposite spins are less likely to occupy the same spatial region, reducing the overall correlation strength at those distances, a feature directly attributable to the underlying fermionic nature of the system and the imposed symmetries.

Figure 3:Plots ofkF​rpk\_{\rm F}r\_{\rm p}(top panel),kF​|δ​𝐫|mink\_{\rm F}|\delta{\bf r}|\_{\rm min}(middle panel),hminh\_{\rm min}(bottom panel), versusln⁡kF​a\ln k\_{\rm F}a. The depth of the minimum ishmin=|g↑↓​(δ​𝐫min)−1|h\_{\rm min}=|g\_{\uparrow\downarrow}(\delta{\bf r}\_{\rm min})-1|orhmin=|gn​n​(δ​𝐫min)−1|h\_{\rm min}=|g\_{nn}(\delta{\bf r}\_{\rm min})-1|. The solid blue lines refer tog↑↓​(δ​𝐫)g\_{\uparrow\downarrow}(\delta{\bf r}), the dotted blue lines representg↑↓​(δ​𝐫)g\_{\uparrow\downarrow}(\delta{\bf r}), and the dashed blue line reflectsg↑↑​(δ​𝐫)g\_{\uparrow\uparrow}(\delta{\bf r}). The blue circles, diamonds and squares describe the scattering parameters used in Fig.2.
Figure 3:Plots ofkF​rpk\_{\rm F}r\_{\rm p}(top panel),kF​|δ​𝐫|mink\_{\rm F}|\delta{\bf r}|\_{\rm min}(middle panel),hminh\_{\rm min}(bottom panel), versusln⁡kF​a\ln k\_{\rm F}a. The depth of the minimum ishmin=|g↑↓​(δ​𝐫min)−1|h\_{\rm min}=|g\_{\uparrow\downarrow}(\delta{\bf r}\_{\rm min})-1|orhmin=|gn​n​(δ​𝐫min)−1|h\_{\rm min}=|g\_{nn}(\delta{\bf r}\_{\rm min})-1|. The solid blue lines refer tog↑↓​(δ​𝐫)g\_{\uparrow\downarrow}(\delta{\bf r}), the dotted blue lines representg↑↓​(δ​𝐫)g\_{\uparrow\downarrow}(\delta{\bf r}), and the dashed blue line reflectsg↑↑​(δ​𝐫)g\_{\uparrow\uparrow}(\delta{\bf r}). The blue circles, diamonds and squares describe the scattering parameters used in Fig.2.

Unveiling Collective Behavior: The Irreducible Contribution & Equation of State

The behavior of many-body systems, such as ultracold Fermi gases, isn’t simply the sum of individual particle interactions; correlations emerge that demand a more nuanced treatment. The two-particle irreducible (TPI) contribution specifically addresses effects beyond these basic two-particle correlations, encompassing the influence of all other particles on a given pair. This contribution isn’t about direct interactions between just two entities, but rather the screening and modification of that interaction due to the presence of the many-body environment. Essentially, it accounts for how a particle “feels” another, when the surrounding particles collectively alter the interaction potential. Ignoring this TPI contribution would lead to inaccurate predictions of system properties, as it misses the complex interplay between particles and the resulting emergent behavior – a critical component in accurately modeling the system’s collective response and thermodynamic characteristics.

Determining the two-particle irreducible contribution to a many-body system frequently necessitates the application of advanced mathematical tools, prominently featuring the Lippmann-Schwinger equation. This integral equation provides a framework for rigorously calculating the scattering amplitude between particles, accounting for all orders of interaction. Solving it, however, is often computationally demanding, requiring careful consideration of boundary conditions and the choice of appropriate Green’s functions. The equation effectively sums over all possible scattering events, including those involving intermediate states, to reveal the correlated behavior of particles beyond simple pairwise interactions. This approach allows physicists to move beyond perturbative treatments and access regimes where strong correlations dominate, ultimately providing a more accurate description of the system’s fundamental properties and informing the development of a precise equation of state.

The two-particle irreducible contribution, crucial for understanding many-body interactions, isn’t simply a sum of individual particle effects; it’s profoundly shaped by collective excitations. These emergent modes, arising from the coordinated behavior of numerous particles, represent the system’s response to disturbances and fundamentally alter how particles interact. Instead of isolated pairings, the irreducible contribution reflects the influence of these propagating excitations – think of sound waves or density oscillations within the Fermi gas – on the effective interaction between particles. Calculating this contribution therefore requires accurately modeling these collective modes, as they screen or enhance interactions, and their energy and momentum dependence directly impact the overall many-body physics. The inclusion of these collective excitations provides a more complete picture of the system’s behavior, going beyond simple pairwise correlations and allowing for a more precise determination of the equation of state.

The equation of state for a Fermi gas, a fundamental descriptor of its thermodynamic behavior, hinges on a precise understanding of the correlations between its constituent particles. This work demonstrates that accurately capturing these many-body effects-particularly those beyond simple two-particle interactions-is crucial for predicting the gas’s properties. By explicitly incorporating the Pauli exclusion principle and gauge invariance into calculations, researchers have achieved a more robust and accurate description of these correlations than previously possible. This rigorous approach, constrained by the fluctuation-dissipation theorem, not only refines the prediction of the equation of state but also provides a pathway toward understanding more complex fermionic systems and their emergent phenomena, offering insights into areas like neutron stars and ultracold atomic gases.

The pursuit of understanding Fermi gases, as detailed in this work, necessitates a rigorous examination of their underlying structure. The theoretical framework presented prioritizes reproducibility through adherence to established principles like gauge invariance and the Pauli exclusion principle. This focus echoes Blaise Pascal’s sentiment: “The eloquence of the body is to move without speaking.” Just as Pascal highlights the power of unspoken truths revealed through action, this research unveils the hidden correlations within complex systems-the ‘movements’ within the Fermi gas-by meticulously applying fundamental physical constraints. The emphasis on two-particle irreducible contributions represents a deliberate effort to isolate the essential components, revealing the system’s core behavior rather than being obscured by extraneous effects.

Where Do the Patterns Lead?

The insistence on incorporating the Pauli exclusion principle and gauge symmetry into the description of density-density correlations isn’t merely a formal exercise. Rather, it highlights a persistent tension: how much of observed phenomena stems from fundamental constraints, and how much arises from emergent behavior within those boundaries. The current framework, while demonstrably successful across the BCS-BEC crossover, inevitably simplifies. Future work must grapple with the effects of imperfections – the inevitable deviations from ideal Fermi gases – and the impact of many-body interactions beyond the two-particle level.

Every deviation, every outlier in experimental data, presents an opportunity to uncover hidden dependencies. The two-particle irreducible contributions, so carefully considered here, may themselves be approximations of a more complex, irreducible reality. Quantum simulation offers a promising avenue, but even these simulations are constrained by computational limitations and the inherent difficulty of precisely characterizing many-body systems. The challenge isn’t just to reproduce experiments, but to anticipate them – to predict the qualitative behavior of these systems in unexplored regimes.

Ultimately, the goal isn’t simply a more accurate model, but a deeper understanding of the relationship between microscopic principles and macroscopic properties. The subtle dance between spin, density, and correlation reveals a system constantly testing the limits of its own coherence. Where those limits lie – and how they break – remains the most compelling question.

Original article: https://arxiv.org/pdf/2602.23019.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-02-28 08:04