Bardeen-Cooper-Schrieffer State Representation and Pairing in the Fermionic Tonks-Girardeau Gas

arXiv:2407.13270 · cond-mat.quant-gas, quant-ph · Submitted 2024-07-18 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Hi, I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Bardeen-Cooper-Schrieffer State Representation and Pairing in the Fermionic Tonks-Girardeau Gas".

Mira: Here's the gist of the paper. The ground state of a fermionic Tonks-Girardeau gas is demonstrated to be a number-conserving Bardeen-Cooper-Schrieffer (BCS) state, which provides a new perspective for describing this system and enables the detection of pairing through expectation values.

Kai: Ground State Representation The proposed wave function is constructed based on the occupation numbers and natural orbitals of the one-body density matrix and is valid under any external potential Specifically, we provide the exact coefficients that allow to describe the fermionic Tonks-Girardeau gas as a number-conserving BCS state.

Mira: This proposed wave function is more suitable for working within a second quantization framework than the pair product form (1) and enables the calculation of important expectation values involving creation and annihilation operators.

Kai: Derivation of the Wave Function The ground state wave function is related to that of an ideal Bose gas, where the total density can be explained by the formation of N/2 pairs of fermions in the orbitals χk+(x) and χk−(x).

Title and authors: Kai: So this paper is called "Bardeen-Cooper-Schrieffer State Representation and Pairing in the Fermionic Tonks-Girardeau Gas" by Sabater, Rojo-Francàs, Astrakharchik, and Juliá-Díaz.

Mira: It sounds like they are taking something from conventional superconductivity theory and applying it to a very specific gas called the fermionic Tonks-Girardeau gas.

Kai: Exactly. The core idea here is showing that the ground state of this particular gas is actually a number-conserving Bardeen-Cooper-Schrieffer, or BCS, state.

Lev: So what does that mean practically? Does it mean we can describe the system using standard BCS tools, or are we just doing more math?

Mira: It means they’ve found a new way to look at the ground state wave function using second quantization, which lets us use creation and annihilation operators in a framework that is much better suited for simulations.

Kai: They are providing the exact coefficients needed to describe this gas as that BCS state under any external potential. That's pretty powerful for modeling.

Lev: But you mentioned they're giving the exact coefficients, Kai, how does that fit into real hardware? Does it mean we can actually build something that uses this representation?

The paper's summary: Mira: To summarize what the paper does, they construct a proposed wave function based on the occupation numbers and natural orbitals of the one-body density matrix.

Kai: It moves away from just using a pair product form, which was mentioned earlier in other work, and instead uses this new structure for second quantization.

Mira: They show how this new wave function relates to an ideal Bose gas where the total density comes from forming pairs of fermions in certain orbitals.

Lev: So the key takeaway is they establish a relationship between the fermionic system and an ideal Bose gas, which simplifies things conceptually, I guess.

Kai: Right, but they don't stop there; they go on to provide expectation values for a set of measurable operators that actually prove this gas is paired.

Mira: Specifically, those expectation values show that the gas forms a paired state within the framework for pairing derived in reference twenty-eight <ref:2407.13270#pg2,a paired state within the framework for pairing derived in>.

Lev: That’s where things get interesting because you need those measurable quantities to confirm it's not just a mathematical trick; it has to be observable.

The paper's improvements: Kai: The main improvement they highlight is using this new wave function, psi T, instead of the older pair product form, psi F.

Mira: They show that the proof that these two wave functions are actually identical is straightforward and included in the paper.

Lev: That's good for theoretical consistency, but what about scalability? The paper does mention providing a recursive formula for p N(k one k NP) <ref:2407.13270#pg1>.

Kai: They provide explicit formulas for these coefficients p N, depending on whether the number of particles N is even or odd.

Mira: For the even case, it's given by a specific formula involving factorials and pi. For the odd case, there's another formula.

Lev: So they’ve given us a concrete way to calculate these probabilities for larger particle numbers, which is helpful when you try to run simulations or see what happens with more particles.

Conclusion: Kai: So to wrap up on "Bardeen-Cooper-Schrieffer State Representation and Pairing in the Fermionic Tonks-Girardeau Gas," they’ve shown that this system is exactly described by a number-conserving BCS state.

Mira: This means we can now use the same mathematical tools used for superconductors to analyze this fermionic gas, even under external potentials.

Kai: And they explicitly derived those expectation values of operators, like H one showing that for any separable state, the expectation value is greater than or equal to zero <ref:2407.13270#pg1>.

Lev: But for a number-conserving BCS state, they show that H one is less than zero <ref:2407.13270#pg1,a number-conserving BCS state>. That’s the direct evidence they are looking for when you try to detect pairing experimentally.

Mira: So, in short, this paper gives us a novel example of how to exactly describe a system using the same quantum states as a BCS superconductor and provides concrete metrics for detecting pairing in the fermionic Tonks-Girardeau gas.

Kai: It’s a specific case of second quantization that is really useful for AI models simulating these systems because they can now calculate those expectation values directly.

Lev: For real hardware, the challenge will be implementing this complex structure of coefficients p N efficiently without losing fidelity in the cooling process.

Departament de Fter Quntica i Astrofica, Facultat de Fter, Universitat de Barcelona · Institut de Cinces del Cosmos, Universitat de Barcelona · Departament de Fter, Universitat Politècnica de Catalunya

cond-mat.quant-gas, quant-ph

Submitted: 2024-07-18

Updated: 2025-07-07

Comments: 5 pages

Journal ref: Phys. Rev. Research 7, L032007 (2025)

DOI: 10.1103/cqkw-bf3v

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 80/100

The gist: The ground state of a fermionic Tonks-Girardeau gas is demonstrated to be a number-conserving Bardeen-Cooper-Schrieffer (BCS) state, which provides a new perspective for describing this system and

Key concepts

Fermionic Tonks-Girardeau Gas: This is a specific type of one-dimensional quantum gas composed of fermions interacting strongly. It is unique because it behaves differently than a standard Luttinger Liquid, allowing for exact mathematical description using the new BCS state representation.
Number-Conserving BCS State: This describes the ground state as analogous to a conventional superconductor, where particles form pairs. The key feature here is that the total number of particles remains constant, providing a framework to describe pairing in this specific fermionic system.
Second Quantization Framework: This is a mathematical method used in quantum mechanics to describe systems of many interacting particles using creation and annihilation operators. It is preferred over simpler pair product forms because it allows for the calculation of complex expectation values needed for pairing detection.

Terminology

Quantization — 2× in this paper · explained in 15 episodes

Summary

The ground state of a fermionic Tonks-Girardeau gas is demonstrated to be a number-conserving Bardeen-Cooper-Schrieffer (BCS) state, which provides a new perspective for describing this system and enables the detection of pairing through expectation values.

Ground State Representation

The proposed wave function is constructed based on the occupation numbers and natural orbitals of the one-body density matrix and is valid under any external potential Specifically, we provide the exact coefficients that allow to describe the fermionic Tonks-Girardeau gas as a number-conserving BCS state. This proposed wave function is more suitable for working within a second quantization framework than the pair product form (1) and enables the calculation of important expectation values involving creation and annihilation operators.

Derivation of the Wave Function

The ground state wave function is related to that of an ideal Bose gas, where the total density can be explained by the formation of N/2 pairs of fermions in the orbitals χk+(x) and χk−(x). The proposed wave function for N=2 is postulated as ψT(x1, x2) = X k=1 Ç λ (2) k ϕk (x1, x2), where ϕk (x1, x2) is the Slater determinant formed by the states χk+(x) and χk−(x). The proof that ψT = ψF is straightforward and included here.

Generalization to N Particles

For larger numbers of particles, the proposed wave function for N fermions is given by ψT(x1,..., xN) = X k1<k2<kNp q p(k1,..., kNp)ϕk1,.,.kNp (x1,.,., xN), where ϕ0,k1,,,kNp (x1,.,., xN) is the Slater determinant containing the N states χk+, χk− and φ. The coefficients p(k1,..., kNp) are derived from a recursive formula.

Detection of Pairing

The pairing phenomenon can be formally defined within quantum information theory, and it can be formally proved that any number-conserving BCS state is a paired state. This proof is based on the set of operators O⃗ 3 which cannot be reproduced by any separable state. The expectation values of O⃗ 3 obtained with a number-conserving BCS state can not be reproduced by any separable state.

Expectation Values for Pairing Detection

The expectation values of the set of measurable operators O⃗ F TG 3 are given by formulas involving the occupation numbers λ(N)k and p(l, k2... kNp)p(k, k2... kNp). These values imply that 〈H1〉 = −2 X (k2<kNP)̸=l,k p(l, k2...)p(k, k2...) < 0 for any separable state. This demonstrates that the FTG gas forms a paired state by measuring these specific quantities.

Conclusion

The proposed ground state can be used to describe the ground state of the fermionic Tonks-Girardeau gas under any external potential due to the universality of the coefficients pN, which is a consequence of the universality of the occupation numbers λ(N)k. By expressing this formulation in second quantization, we demonstrate that it is a specific case of a number-conserving BCS state and provide explicit forms for the α coefficients. Finally, we explicitly derive the expectation values of the operators to be measured in order to experimentally detect pairing in the fermionic TonksGirardeau gas. The results present a novel example of a system that can be exactly described using the same quantum states that describe a BCS superconductor. This work is expected to have an impact on understanding one-dimensional quantum systems, particularly the fermionic Tonks-Girardeau gas. The system under study is particularly appealing as it is not a Luttinger Liquid, which instead describes compressible one-dimensional gases.

How it works

The derivation of the proposed wave function starts from information obtained from the OBDM, namely its occupation numbers, natural orbitals, and our understanding of the pairing mechanism. The proposed WF is derived solely from this information and is more suitable for working within a second quantization framework than the pair product form (1).

Explicit Formulas for Coefficients

The explicit formulas for pN(k1,..., kNp) are given by two forms depending on whether N is even or odd. For the even case, pN (k1,.,., kNp) = ¨ N! πN/QNP i=1 1 (ki-1/2) 2, if N is even. For the odd case, pN (k1,.,., kNp) = N! π(N-1)/QNP i=1 1 k 2 i, if N is odd.

Second Quantization Form

The ground state WF for the even FTG gas in second quantization is given by ψT> = X k1. For the odd case, it includes a single particle in the single-particle ground state: ψT> = a†0 X k1.

Detecting Pairing via Operators

The expectation value of the operator H1 is shown to be greater or equal than zero for any separable state, but for a number-conserving BCS state, 〈H1〉 < 0. The calculation of 〈O⃗ F TG 3〉 shows that these expectation values imply that the FTG gas forms a paired state.

Final Result for Pairing Detection

When N is even, the integral simplifies to O = (N - 1) Z 1 0 Z 1 0 Z 1 0 sgn(y1 − y2) sgn(y2 − y1) + 1/N-2 d y3 d y1 d y2 = 1. When N is odd, the integral simplifies to O = N Z 1 0 Z 1 0 Z 1 0 sgn(y1 − y3) sgn(y2 − y3) d y3 N-2 d y1 d y2 = 1. This confirms the pairing in the FTG gas.

Calculation of Expectation Values

The expectation value 〈a†k+ a†k− al−al+ + h.c〉 is found to be 2 X (k2<kNP)̸=l,k p(l, k2... kNP)p(k, k2... kNP). This quantity is of interest because it is one of the main quantities to measure in order to detect pairing in a fermionic system.

Impact and Significance

The results present a novel example of a system that can be exactly described using the same quantum states that describe a BCS superconductor. This work is expected to have a broader impact beyond the study of the fermionic Tonks-Girardeau gas, potentially leading to novel investigations on pairing in excitations and entanglement between fermionic orbitals.

Improvements for AI systems

Improved ground state description for fermionic Tonks-Girardeau gas by replacing pair-product form with a number conserving BCS state representation in second quantization, enabling complete description under any external potential. This allows AI models simulating such systems to use second quantization formalism for calculating expectation values involving creation and annihilation operators that were previously impossible with the pair product form.
Detection of pairing in fermionic Tonks-Girardeau gas by calculating specific expectation values like O F T G 3, which shows that the system exhibits non-trivial quantum correlations but is also a paired state. AI systems can be trained to identify these specific expectation values as signatures of pairing, providing a direct experimental metric for verifying the presence of pairing in simulated or experimental FTG gases.
Predictive capability for excitation properties by using the new ground state wave function to predict and study whether pairing is also present in the excitations of the system. This allows AI to model not just the ground state but also dynamic behavior, such as whether excitations carry paired states, leading to more comprehensive simulations of correlated quantum many-body phenomena.