Superconductivity on a quasiperiodic lattice: Extended-to-localized crossover of Cooper pairs

Abstract

We study a possible superconductivity in quasiperiodic systems, by portraying the issue within the attractive Hubbard model on a Penrose lattice. Applying a real-space dynamical mean-field theory to the model consisting of 4181 sites, we find a superconducting phase at low temperatures. Reflecting the nonperiodicity of the Penrose lattice, the superconducting state exhibits an inhomogeneity. According to the type of the inhomogeneity, the superconducting phase is categorized into three different regions which cross over each other. Among them, the weak-coupling region exhibits spatially extended Cooper pairs, which are nevertheless distinct from the conventional pairing of two electrons with opposite momenta. 71.23.Ft, 67.85.Lm Quasicrystal is a crystal without translational symmetry. Prominent spots observed in its diffraction pattern manifest an orderly structure while they do not conform to any periodicity. An example of such structures holds the icosahedral pointgroup symmetry, as first discovered by Shechtman et al. [1], and various other structures have hitherto been reported [2][3][4]. These structures may originate novel electronic properties distinct from those of conventional periodic crystals. In fact, previous theoretical works revealed various nontrivial properties, such as the presence of a confined state [5,6], fractal dimensions [7][8][9], singular continuous spectral measure [8,10,11], pseudogap in the density of states [12], and a conductance decaying in power of system size [13,14], for free electrons on the quasiperiodic lattices. Moreover, recent observation of quantum critical behavior in Au 51 Al 34 Yb 15 [15] has stimulated theoretical studies [16][17][18][19][20][21][22] on the role of electron correlations in these systems.Another interesting recent observation is a superconductivity in approximant crystals (i.e., periodic crystals with the same local structure as the quasicrystals), Au 64 Ge 22 Yb 14 and Au 63.5 Ge 20.5 Yb 16 [23]. A superconductivity has also been reported in Al-Cu-(Mg, Li) quasicrystalline alloys [24,25]. These observations raise fundamental questions about a possible superconductivity in quasicrystals: How can a superconductivity emerge in a system without translational symmetry? If it exists, what differs from the superconductivity in periodic systems? These questions also have a relevance to experiment of ultracold atomic gases, for which optical quasiperiodic lattices have been available [26][27][28].According to an early consideration by Anderson [29] about the impurity effect on superconductivity, Cooper pairs can exist in principle even in the absence of the translational symmetry. In this case, an electron finds its partner in the time-reversed state, which is a generalization of the standard pairing of k ↑ and −k ↓. However, as a matter of course, this does not guarantee the presence of superconductivity in quasiperiodic systems. This many-body problem requires an explicit calculation taking into account both the pairing interaction and the lattice geome...

Antiferromagnetic order in the Hubbard model on the Penrose lattice

“…One of the simplest ones with geometrical quasiperiodicity is the Penrose lattice in two dimensions [9][10][11][12][13][14][15][16][17]. On this lattice, a few strongly correlated models have been studied: the Ising model for classical spins [18][19][20], the Heisenberg model for quantum spins [21,22], and the Hubbard [23,24] and Anderson lattice models [25,26] for correlated electrons. One important feature of the Penrose lattice is the presence of thermodynamically degenerate oneparticle states, and they are called confined states, since the their wave functions are strictly confined in a finite region in space [9,14].…”

Exotic pairing state in quasicrystalline superconductors under a magnetic field

“…In Ref. 23, we found that, even without a Fermi surface, there exists a superconducting state with spatially extended Cooper pairs in a weak-coupling regime of the attractive Hubbard model. This Cooper pairing cannot be straightforwardly captured by Bardeen-Cooper-Schrieffer picture in the sense that the latter is based on a description in a momentum space absent in quasiperiodic systems.…”

“…Eventually, above h c2 0.85 the normal phase appears. Note that, unlike in periodic systems, the superconducting state in quasiperiodic systems is always inhomogeneous with respect to the amplitude of ∆ i even without magnetic field [23] while its phase (sign) is uniform. The finding here is an inhomogeneous sign structure in the superconducting state, which emerges under the magnetic field.…”

“…Our model is based on the attractive Hubbard model on a Penrose tiling [22], introduced in Ref. 23 as a simple theoretical model to discuss a superconductivity in quasiperiodic systems. The Penrose tiling [Inset to Fig.…”

Propriedades eletrônicas e supercondutividade em quase cristais

“…4 Também mostram que a supercondutividade neste tipo de sistema parece ser convencional 43 e estudos teóricos corroboram com a afirmativa. 44 Também há proposição de estados superfluidos, 45 mas ainda não foi relatada uma realização experimental.…”