Disorder-induced decoupling of attracting identical fermions: transfer matrix approach

Abstract

We consider a pair of identical fermions with a short-range attractive interaction on a finite lattice cluster in the presence of strong site disorder. This toy model imitates a low density regime of the strongly disordered Hubbard model. In contrast to spinful fermions, which can simultaneously occupy a site with a minimal energy and thus always form a bound state resistant to disorder, for the identical fermions the probability of pairing on neighboring sites depends on the relation between the interaction and the disorder. The complexity of `brute-force' calculations (both analytical and numerical) of this probability grows rapidly with the number of sites even for the simplest cluster geometry in the form of a closed chain. Remarkably, this problem is related to an old mathematical task of computing the volume of a polyhedron, known as NP-hard. However, we have found that the problem in the chain geometry can be exactly solved by the transfer matrix method. Using this approach we have calculated the pairing probability in the long chain for an arbitrary relation between the interaction and the disorder strengths and completely described the crossover between the regimes of coupled and separated fermions.

Figures (6)

• Figure 1: Dependence of the probability to observe a bound state $P_\text{b}$ on the interaction $U$ (in units of the disorder width) for the number of sites $N =4$.
• Figure 2: Crossover between the coupling and decoupling regimes. Dependencies of the probability $P_\text{b}$ on the interaction $U$ calculated in the numerical experiments for the number of sites $N=30$ and $N=100$ are presented.
• Figure 3: Dependence of the bound state probability $P_\text{b} (U)$ on the relative interaction strength $U$ in the range of $U \ll 1$ for the number of sites $N =30$.
• Figure 4: The areas $\Sigma_1$ and $\Sigma_2$, determined by the conditions (\ref{['eq:region1']}) and (\ref{['eq:region2']}), on the plane of the 'external' variables $V_1$ and $V_k$ (in the sector $V_1 < V_k$) for the interaction strength $U < 1/2$ (a) and $U \geq 1/2$ (b).
• Figure 5: Graphical representation of the characteristic equation (\ref{['eq:tan']}). It corresponds to the intersection of the curve $Y_1 = \tan[ \pi/4 - \mu \left(U+V_1 - V_k \right)/2]$ and the straight line $Y_2 = \mu (1 - U - V_1)$.