Periyodik potansiyelin hermitsel olmayan fermiyonik süperakışkanlık üzerine etkisi
Effects of periodic potentials on the non-hermitian fermionic superfluidity
- Tez No: 955587
- Danışmanlar: PROF. DR. AHMET LEVENT SUBAŞI
- Tez Türü: Yüksek Lisans
- Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2025
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Lisansüstü Eğitim Enstitüsü
- Ana Bilim Dalı: Fizik Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Fizik Mühendisliği Bilim Dalı
- Sayfa Sayısı: 103
Özet
Periyodik potansiyelin fermiyonik sistemler üzerindeki etkileri, yoğun madde fiziğinin ana konularından birisidir. Periyodik potansiyel altındaki açık kuantum sistemlerinde, esnek olmayan çarpışmalar yüzünden, kayıplar meydana gelir. Bu kayıplar, hermitsel olmayan bir hamiltonyen ile modellenebilir ve artan kayıp oranı, süperakışkanlık üzerine monoton olmayan etkiler meydana getirebilir. Bu çalışmada, özellikle bu kayıpların, süperakışkan durum üzerine etkilerini incelemek istiyoruz. Bunun için bir boyutlu optik örgüde, iki parçacık problemini ve buna bağlı olan Cooper çifti problemini ele alıyoruz. Hesabı kolaylaştırmak için sadece ilk enerji bandını dikkate aldığımız“tek bant yaklaşımı”altında, sıkı bağ modelini kullanıyoruz. Parçacık kayıplarına sebep olan esnek olmayan çarpışmaları, kompleks etkileşim parametresi ile modelliyoruz. Nümerik çözümümüzde, hem konum hem de momentum bazında, hamiltoniyen matrisini oluşturup köşegenleştiriyoruz. Tek bant yaklaşımında; bağlı durum enerjilerini, kütle merkezi kuazimomentumu $K$'ya göre elde ediyoruz. Kompleks etkileşim olduğunda, belli enerji değerlerinde, bağlı durum eğrisinin sürekli bölge içinde oluştuğunu (BIC) ve kompleks etkileşim sabitinin artışının, bağlı durum eğrisinin $K$'ya göre varyasyonunu azaltarak eğriyi düz bir banda dönüştürdüğünü görüyoruz. Etkileşimin sanal kısmının; bağlanma enerjilerini azalttığını, sonlu ömürler oluşturduğunu ve yüksek sönüm oranlarında, bağlı durumların etkin kütlelerini arttırdığını gözlüyoruz. Belli bir limitte süperakışkan faz, Cooper çiftlerinin oluşturduğu, bir Bose Einstein yoğuşması (BEC) olarak düşünülebilir. Dolayısıyla bu çalışmada kayıp mekanizmasını, Bardeen-Cooper-Schrieffer süperakışkanlığın (BCS) temelini oluşturan, Cooper problemine taşıyoruz ve aynı hesapları orada, momentum uzayında tekrarlıyoruz. Sonuçlarımız, daha önce ortalama alan teorisiyle elde edilen, sanal kısma bağlı süperakışkan düzen parametresindeki monoton olmayan davranışın; iki parçacık fiziğinden kaynaklanabileceğini göstermektedir.
Özet (Çeviri)
The aim of this work is to investigate the simultaneous effects of a periodic potential and a complex-valued interaction for the two-body problem in a periodic potential. This study focuses on how periodicity and non-Hermitian interactions interplay in forming bound states and shaping their physical properties. With the gained understanding of the bound-state formation in this problem and the established computational approach, we also consider the Cooper problem in the same context. The standard approach to two-body problems is to use the so-called center-of-mass and relative coordinates instead of the individual particle coordinates. When the problem is separable, the equations decouple and the relative coordinate involving the interaction becomes effectively a one-body problem which is independent of the center-of-mass motion. However, unlike the textbook examples, the two-body problem in a periodic potential is not separable in the center-of-mass and relative coordinates. The single particle problem in a one-dimensional periodic potential, the band structure for the one-body problem has stationary states that can be labeled by their quasi-momentum $k$ and band index $n$. In order to simplify the calculations, one can further make a single-band approximation where only one energy band is considered. This allows one to describe the system in terms of a tight-binding Hamiltonian where each unit cell has one orbital. Such a model is described by a hopping matrix element between nearest-neighbor unit-cells and an on-site interaction term on a lattice. The two-body problem within a single-band approximation has been studied before. The interaction leads to bound states with both repulsive and attractive interactions. Here, we focus on the binding energy of two-particles with an attractive interaction. The Cooper problem in a one-dimensional lattice has also been studied with a single band tight-binding approximation. For the two-body problem in the lattice, a center-of-mass quasi-momentum $K_\mathrm{CM}$ can still be defined as a conserved quantity. It appears in the eigenvalue equation for the relative coordinate in the solution of the energy eigenstates as a constant and the Schrödinger equation for the relative coordinate has to be solved for each value of the center-of-mass momentum. This symmetry consideration leads to smaller problems for each $K_\mathrm{CM}$ and all solutions can be obtained in this way avoiding the larger Hilbert space to two-particles. Our numerical approach involves constructing the Hamiltonian matrix in both position and momentum bases and diagonalizing it. We consider fermionic particles in a spin singlet state so that the spatial wave functions are symmetric under the exchange of particle coordinates. Following previous work in the literature, we consider the problem described above with a complex-valued interaction which makes the Hamiltonian describing the system non-Hermitian. (Non-Hermitian Hamiltonians lead to complex energy eigenvalues.) The complex-valued interaction models the inelastic collision with particle losses. The mechanism for non-Hermiticiy enters in the two-particle problem as opposed to other non-Hermitian mechanisms acting the on the single particle dynamics. Since we focus on fermions in a spin singlet configuration, which requires the spatial part of the wave function to be symmetric under particle exchange. The inclusion of complex interactions introduces qualitative differences compared to Hermitian systems. For example, in the Hermitian case, the binding energy depends monotonically on the interaction strength. In contrast, the non-Hermitian interaction introduces non-monotonic behavior, especially in regimes where the bound-state energy overlaps with the continuum of extended two-particle states. Considering the effects of a complex-valued interaction on the bound state energies of two-particles in a periodic potential within a single band approximation, we find that the imaginary part of the interaction leads to lower binding energies, finite life-times and increased effective masses for the bound states. As expected, the bound state with zero center-of-mass momentum has the longest lifetime and the states with maximum center-of-mass quasi-momentum have the largest decay rates for a given interaction. Finally, looking at the curvature of the real part of the bound state energy curves it is seen that the effective mass of the bound state increases with dissipation. The behavior of the dependence of these quantities on the imaginary part of the interaction is non-monotonic for weak interactions when the bound state band can be partly within the continuum of extended two particle states. This gives rise to bound states in continuum (BICs) for this model. Finally, we present the Cooper problem spectrum at half filling. The formation of Cooper pairs, which are bound-states in the presence of a Fermi sea, is crucial for Bardeen-Cooper-Schrieffer type fermionic superfluidity and superconductivity. In particular, the condensed phase in the strongly interacting limit can be described by the Bose-Einstein condensation of Cooper pairs which behave like tightly-bound dimers. The behavior of such a system in the presence of a periodic potential, which can be realized in the laboratory with ultracold atomic gases in optical lattices, leads to a large effective mass among other effects. The peculiar band structure of the quasiparticle energy spectrum in the lattice can reduce the effects of pair-breaking excitations. Similarly, when the pairing gap is large compared to the strength of the lattice, the effects of the band structure can be suppressed which can play an important role for neutron superfluid density in the inner crust of neutron stars. Therefore, the effects of the periodic potential in superfluid systems are significant for various physical systems with vast differences in their properties such as densities and absolute temperatures. In the presence of inelastic collisions, it has been suggested that the superfluidity in ultracold atoms can show reentrant behavior with increasing dissipation for weak attractive interactions and should be enhanced by dissipation due to an interplay between the BCS-BEC crossover and the quantum Zeno effect. In addition, models with non-Hermitian dynamics and losses can host bound states in continuum for the two body problem. With this motivation, our work offers multiple directions for further investigation. A complete analysis of the Cooper problem can be pursued for different filling values of the band. The inclusion of additional single-particle bands is also possible. The effects of a multiband problem can be studied as well as the effects of multiple dimensions. We also plan consider the consequences resulting from two-body physics for superfluid systems in a periodic potential with particle losses.
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