PHYSICAL REVIEW A 95, 022129 (2017) Born-Kothari condensation in an ideal Fermi gas Arnab Ghosh 1, 2 , * and Deb Shankar Ray 2 1 Weizmann Institute of Science, 76100 Rehovot, Israel 2 Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India (Received 13 November 2016; published 28 February 2017) “Condensation” in Fermi-Dirac statistics [D. S. Kothari and B. Nath, Nature 151, 420 (1943)], which appears as a natural consequence of Born’s reciprocity principle [M. Born, Proc. R. Soc. London A 165, 291 (1938); Nature 141, 328 (1938)], is examined from a theoretical perspective. Since fermions obey the Pauli exclusion principle, it is conceptually different from Bose-Einstein condensation, which permits macroscopic occupation of bosons at the single-particle level below a critical temperature. Yet, in accordance with the Cahill and Glauber [Phys. Rev. A 59, 1538 (1999)] formulation for fermionic fields, and in close kinship to bosonic fields, we have shown that in analogy to Bose-Einstein condensation, it is possible to associate an intrinsic notion of symmetry breaking and the thermodynamic “order parameter” to characterize the foregoing hitherto unexplored phenomenon in an ideal Fermi-Dirac gas as condensation-like coherence within fermions. DOI: 10.1103/PhysRevA.95.022129 I. INTRODUCTION Trapped dilute, atomic gases have proven to be remarkable model systems for the realization of quantum statistical effects at the fundamental level, such as the direct observation of Bose-Einstein condensation (BEC) [1–3] at ultracold temperatures. Perhaps not as dramatic as the phase transition of bosons, the behavior of trapped Fermi gases also merits attention in its own right, both as a degenerate quantum system and as a possible precursor to a paired Fermi condensate at very low temperatures [4–6]. The use of mean-field theory, the pseu- dopotential, and consideration of fluctuations around the mean field has already led to interesting advancements in the understanding of basic physics of ultracold Fermi gases [7–9]. Here, instead, we invoke a seldom used but reasonable basis for the possible existence of a “condensed phase” for an ideal Fermi-Dirac (FD) gas [10] that is based on the notion of Born’s reciprocity theory [11], which is considered one of the cornerstones for the development of the theory of elementary particles [12,13] and other related fields [14,15]. The attempt was made many decades ago by Kothari and Nath [10] in the course of examining the relationship between Born’s reciprocity principle [11] with FD statistics. In the present study we put forward a convenient description of such states following Cahill and Glauber [16], on the basis of close parallelism between the expressions found for fermionic fields and the more familiar ones for bosonic fields that resulted in a unification between the two seemingly distinct seminal works. Our approach, however, is not in contradiction to the standard pairing theory for fermions [17]. A key element of our formulation is the fermionic coherent state, defined as a unitary displaced state of all singly occupied filled up modes. The transformation with this displacement operator displaces the fermionic field operators over anticom- muting (Grassmann) numbers [18–20]. This resembles the harmonic oscillator coherent state, for which the displacement operator displaces the bosonic field operators over classical commuting variables [21–23]. The basic question behind the * arnab.ghosh@weizmann.ac.il present approach is whether a state of macroscopic coherence for an FD gas can be described as a fermionic coherent state. This approach gives an equivalent result to the problem of a fixed number of particles N in the limit N −→ ∞ for the BEC case [24–26]. In the same spirit, here, we have extended the coherent-state approach of Cahill and Glauber [16] to its fermionic counterpart. It forms an essential ingredient for demonstration of the thermodynamic limit, fermionic order parameter, and spontaneous symmetry breaking of the state comprising FD statistics. The paper is organized as follows: In Sec. II we revisit the Born-Kothari approach to stimulate the motivation for the present work. Since the fermionic coherent state plays a crucial role in the formulation of the problem, in Sec. III we briefly review the relevant parts of the coherent state of fermions as developed by Cahill and Glauber [16]. The aspect of the thermodynamic limit, spontaneous symmetry breaking and the fermionic analog of the order parameter are then introduced to comprehend the Born-Kothari criterion for so-called “condensation” as a state of macroscopic coherence that can be depicted as a fermionic coherent state. The paper is concluded in Sec. IV. II. REVISITING “CONDENSATION” IN FD STATISTICS: THE BORN-KOTHARI APPROACH In the spirit of condensation phenomena for a Bose-Einstein gas (where a condensed phase is formed by the particles in the lowest energy state), Kothari and Nath have shown condensation in FD statistics [10] as a direct consequence of Born’s reciprocity principle [11–15], where the condensed phase is formed by particles in the highest energy state. According to Born’s reciprocity theory [11], the number of wave functions of a particle of weight factor g (due to its internal degrees of freedom) within the momentum range p to p + dp is given by a(p)dp = 4πVg (2π ¯ h) 3 p 2 dp (1 − p 2 /b 2 ) 1/2 . (1) It is indeed essential to limit the momentum p in the above equation by an absolute constant b (whose existence is ensured 2469-9926/2017/95(2)/022129(6) 022129-1 ©2017 American Physical Society