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arXiv:cond-mat/0202115 · Physical Review Letters

Pairing of Cooper Pairs in a Fully Frustrated Josephson-Junction Chain

Benoît Douçot, Julien Vidal

Published May 17, 2002• 177 citations
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Abstract

We study a one-dimensional Josephson-junction chain embedded in a magnetic field. We show that, when the magnetic flux per elementary loop equals half the superconducting flux quantum phi0 = h/2e, a local Z2 symmetry arises. This symmetry is responsible for a nematic Luttinger liquid state associated with bound states of Cooper pairs. We analyze the phase diagram and discuss some experimental possibilities to observe this exotic phase.

DOI
10.1103/physrevlett.88.227005

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We investigate uniform one-dimensional arrays of small Josephson junctions [${\mathit{E}}_{\mathit{J}}$\ensuremath{\ll}${\mathit{E}}_{\mathit{C}}$, ${\mathit{E}}_{\mathit{C}}$=(2e${)}^{2}$/2C] with a realistic Coulomb interaction U(x)=${\mathit{E}}_{\mathit{C}}$\ensuremath{\lambda}exp(-|x|/\ensuremath{\lambda}) (here \ensuremath{\lambda}\ensuremath{\gg}1 is the screening length in units of the lattice constant of the array). At low energies this system can be described in terms of interacting Bose particles (extra single Cooper pairs) on the lattice. With increasing concentration \ensuremath{\nu} of extra Cooper pairs, a crossover from the Bose gas phase to the Wigner crystal phase and then to the superlattice regime occurs. The phase diagram in the superlattice regime consists of commensurable insulating phases with \ensuremath{\nu}=1/l (l is integer) separated by superconducting regions where the current is carried by excitations with fractional electric charge q=\ifmmode\pm\else\textpm\fi{}2e/l. The Josephson current through a ring-shaped array pierced by magnetic flux is calculated for all of the phases. \textcopyright{} 1996 The American Physical Society.

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We study a one-dimensional Josephson-junction chain embedded in a magnetic field. We show that, when the magnetic flux per elementary loop equals half the superconducting flux quantum phi0 = h/2e, a local Z2 symmetry arises. This symmetry is responsible for a nematic Luttinger liquid state associated with bound states of Cooper pairs. We analyze the phase diagram and discuss some experimental possibilities to observe this exotic phase.
We use bosonization and renormalization group methods to determine the ground-state phase diagram of a one-dimensional frustrated Kondo-Heisenberg system consisting of a one-dimensional spin-12\frac{1}{2}21​ Luttinger liquid coupled by a Kondo exchange interaction JK{J}_{K}JK​ to a frustrated quantum antiferromagnetic Heisenberg chain, with a nearest-neighbor exchange coupling J1{J}_{1}J1​ and a next-nearest-neighbor (frustrating) exchange interaction J2{J}_{2}J2​. We analyze the interplay of quantum frustration in the antiferromagnetic chain with the Kondo exchange coupling JK{J}_{K}JK​ with the Luttinger liquid. We discuss the structure of the phase diagram of this system as a function of the ratios JK/J1{J}_{K}/{J}_{1}JK​/J1​, J2/J1{J}_{2}/{J}_{1}J2​/J1​, and of the parameters of the Luttinger liquid. In particular, we discuss in detail the regimes in which a pair-density-wave state may be realized and its relation with the spin correlations in the frustrated antiferromagnetic chain.
The ground-state phase diagram of the one-dimensional Hubbard chain with pair-hopping interaction is studied. The analysis of the model is performed using the continuum-limit field theory approach and exact diagonalization studies. At half-filling the phase diagram is shown to consist of two superconducting states with Cooper-pair center-of-mass momentum $Q=0 (\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ phase) and $Q=\ensuremath{\pi} ({\ensuremath{\eta}}_{\ensuremath{\pi}}$ phase) and four insulating phases corresponding to the Mott antiferromagnet, the Peierls dimerized phase, the charge-density-wave (CDW) insulator, and an unconventional insulating phase characterized by the coexistence of a CDW and a bond-located staggered magnetization. Away from half-filling the phase diagram consists of the superconducting $\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ and ${\ensuremath{\eta}}_{\ensuremath{\pi}}$ phases and the metallic Luttinger-liquid phase. The $\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ phase exhibits a smooth crossover from a weak-coupling BCS type to a strong-coupling local-pair regime. The ${\ensuremath{\eta}}_{\ensuremath{\pi}}$ phase shows the properties of the doublon (zero-size Cooper-pair) superconductor with Cooper-pair center-of-mass momentum $Q=\ensuremath{\pi}.$ The transition into the ${\ensuremath{\eta}}_{\ensuremath{\pi}}$-paired state corresponds to an abrupt change in the ground-state structure. After the transition the conduction band is completely destroyed and a new ${\ensuremath{\eta}}_{\ensuremath{\pi}}$-pair band, corresponding to the strongly correlated doublon motion, is created.
We explore the effects of geometric frustration within a one-dimensional Bose-Hubbard model using a chain of rhombi subject to a magnetic flux. The competition of tunneling, self-interaction, and magnetic flux gives rise to the emergence of a pair-superfluid (pair-Luttinger liquid) phase besides the more conventional Mott-insulator and superfluid (Luttinger liquid) phases. We compute the complete phase diagram of the model by identifying characteristic properties of the pair-Luttinger liquid phase such as pair correlation functions and structure factors and find that the pair-Luttinger liquid phase is very sensitive to changes away from perfect frustration (half-flux). We provide some proposals to make the model more resilient to variants away from perfect frustration. We also study the bipartite entanglement properties of the chain. We discover that, while the scaling of the block entropy pair-superfluid and of the single-particle superfluid leads to the same central charge, the properties of the low-lying entanglement spectrum levels reveal their fundamental difference.
Topological interfaces of two-dimensional conformal field theories contain information about symmetries of the theory and exhibit striking spectral and entanglement characteristics. While lattice realizations of these interfaces have been proposed for unitary minimal models, the same has remained elusive for the paradigmatic Luttinger liquid, i.e., the free, compact boson model. Here, we show that a topological interface of two Luttinger liquids can be realized by coupling special one-dimensional superconductors. The gapless excitations in the latter carry charges that are specific integer multiples of the charge of Cooper pairs. The aforementioned integers are determined by the windings in the target space of the bosonic fields --- a crucial element required to give rise to nontrivial topological interfaces. The latter occur due to the perfect transmission of certain number of Cooper pairs across the interface. The topological interfaces arise naturally in Josephson junction arrays with the simplest case being realized by an array of experimentally demonstrated $0\ensuremath{-}\ensuremath{\pi}$ qubits, capacitors and ordinary Josephson junctions. Signatures of the topological interface are obtained through entanglement entropy computations. In particular, the subleading contribution to the so-called interface entropy is shown to differ from existing field theory predictions. The proposed lattice model provides an experimentally realizable alternative to spin and anyon chains for the analysis of several conjectured conformal fixed points which have so far eluded ab initio investigation.
Superconductivity mediated by Cooper quartets of charge 4e is a phenomenon of key importance for both the understanding of certain exotic superconductors and the engineering of quantum memories protected at the hardware level by topological order. Josephson junction arrays in the shape of the dice lattice constitute one of the main candidates for the realization of this phase of matter. Here, we analyze numerical signatures of the emergence of this exotic phase when superconducting dice arrays are frustrated by inserting one third of a flux quantum per rhombic plaquette. We adopt simulations of relaxation dynamics to study the critical current of such devices and analyse the Fourier decomposition of their bulk supercurrents. A further characterization of these systems at finite temperature is obtained through Monte Carlo techniques and two-dimensional infinite tensor networks. Our results indicate that the observed peaks of the critical current and temperature at frustration 1/3 correspond to a superconductor-insulator phase transition compatible with the deconfinement of half-vortices, while the low-temperature correlation functions of the model confirm the onset of a 4e superconducting phase mediated by Cooper quartets. We finally address the effects of Josephson energy and flux disorder typical of experimental arrays, and comment on the role of charging energies in the corresponding two-dimensional quantum model.
We investigate uniform one-dimensional arrays of small Josephson junctions [EJ{\mathit{E}}_{\mathit{J}}EJ​\ensuremath{\ll}EC{\mathit{E}}_{\mathit{C}}EC​, EC{\mathit{E}}_{\mathit{C}}EC​=(2e)2{)}^{2})2/2C] with a realistic Coulomb interaction U(x)=EC{\mathit{E}}_{\mathit{C}}EC​\ensuremath{\lambda}exp(-|x|/\ensuremath{\lambda}) (here \ensuremath{\lambda}\ensuremath{\gg}1 is the screening length in units of the lattice constant of the array). At low energies this system can be described in terms of interacting Bose particles (extra single Cooper pairs) on the lattice. With increasing concentration \ensuremath{\nu} of extra Cooper pairs, a crossover from the Bose gas phase to the Wigner crystal phase and then to the superlattice regime occurs. The phase diagram in the superlattice regime consists of commensurable insulating phases with \ensuremath{\nu}=1/l (l is integer) separated by superconducting regions where the current is carried by excitations with fractional electric charge q=\ifmmode\pm\else\textpm\fi{}2e/l. The Josephson current through a ring-shaped array pierced by magnetic flux is calculated for all of the phases. \textcopyright{} 1996 The American Physical Society.