Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review B

Quantum phases of superconducting quantum dot arrays

Sujit Sarkar

Published Jan 26, 2007• 13 citations
Publisher

Abstract

We analyze the quantum phases of a superconducting quantum dot array. Superconducting quantum dots are connected through Josephson junction. We study the effect of finite range Josephson coupling and also finite range interactions between the dots. Our field-theoretical study predicts different kinds of phases near the charge degeneracy point. We predict two different kinds of repulsive Luttinger liquid phases originate from different sources: one from the commensurate-incommensurate transition and the other from the $XY$ criticality. Apart from the charge-density wave, we also predict the possibility of dimer-order density wave from the interplay of long-range interaction between the dots and Cooper-pair density. We observe different multicritical points with interesting phases and also two special points. The renormalization group flow diagram also shows many interesting phases.

DOI
10.1103/physrevb.75.014528

Related papers

Quantum phase diagram of a superconducting quantum dots array

S. Sarkar

We present the quantum phase diagram of a superconducting quantum dots array. Superconducting quantum dots are connected through Josephson junctions. We study the effect of finite-range Josephson coupling and also finite-range interactions between the dots. Our field-theoretical study predicts different kinds of phase near the charge degeneracy point. We predict two different kinds of repulsive Luttinger liquid phase and also the possibilities of different density wave instabilities from the inter-play of long-range interaction between the dots and Cooper pair density. We observe different multicritical points with interesting phases.

Europhysics Letters (EPL)•Dec 1, 2006•3 citations

Quantum Phase Analysis of Extended Bose-Hubbard Model

Manoranjan Kumar, Sujit Sarkar, S. Ramasesha

We have obtained the quantum phase diagram of one dimensional extended Bose-Hubbard model using the density-matrix renormalization group and Abelian bosonization methods for different commensurabilities. We describe the nature of different quantum phases at the charge degeneracy point. We find a direct phase transition from Mott insulating phase to superconducting phase for integer band fillings of bosons. We predict explicitly the presence of two kinds of repulsive Luttinger liquid phases, apart from the charge density wave and superconducting phases for half-integer band fillings. Our study reveals that extended range interactions are necessary to get the correct phase boundary of an one-dimensional interacting bosons system.

arXiv:0803.0245•Mar 3, 2008•0 citations

Josephson effect through a quantum dot array

F. S. Bergeret, A. Levy Yeyati, A. Martín‐Rodero

We analyze the ground state properties of an array of quantum dots connected in series between superconducting electrodes. This system is represented by a finite Hubbard chain coupled at both ends to BCS superconductors. The ground state is obtained using the Lanczos algorithm within a low energy theory in which the bulk superconductors are replaced by effective local pairing potentials. We study the conditions for the inversion of the sign of the Josephson coupling ($\ensuremath{\pi}$-junction behavior) as a function of the model parameters. Results are presented in the form of phase diagrams, which provide a direct overall view of the general trends as the size of the system is increased, exhibiting a strong even-odd effect. The analysis of the spin-spin correlation functions and local charges give further insight into the nature of the ground state and how it is transformed by the presence of superconductivity in the leads. Finally, we study the scaling of the Josephson current with the system size and relate these results with previous calculations of Josephson transport through a Luttinger liquid.

arXiv:0707.3897•Physical Review B•Nov 15, 2007•33 citations

Entanglement entropy and quantum phase transitions in quantum dots coupled to Luttinger liquid wires

Moshe Goldstein, Yuval Gefen, Richard Berkovits

We study a quantum phase transition that occurs in a system composed of two impurities (or quantum dots), each coupled to a different interacting (Luttinger liquid) lead. While the impurities are coupled electrostatically, there is no tunneling between them. Using a mapping of this system onto a Kondo model, we show analytically that the system undergoes a Berezinskii-Kosterlitz-Thouless quantum phase transition as a function of the Luttinger liquid parameter in the leads and the dot-lead interaction. The phase with low values of the Luttinger liquid parameter is characterized by an abrupt switch of the population between the impurities as a function of a common applied gate voltage. However, this behavior is hard to verify numerically since one would have to study extremely long systems. Interestingly, though, at the transition the entanglement entropy drops from a finite value of $\mathrm{ln}(2)$ to zero. The drop becomes sharp for infinite systems. One can employ finite-size scaling to extrapolate the transition point and the behavior in its vicinity from the behavior of the entanglement entropy in moderate size samples. We employ the density matrix renormalization-group numerical procedure to calculate the entanglement entropy of systems with lead lengths of up to 480 sites. Using finite-size scaling, we extract the transition value and show it to be in good agreement with the analytical prediction.

arXiv:1203.2820•Physical Review B•Jun 21, 2011•13 citations

Hybrid quantum dot–superconducting systems: Josephson current and Kondo effect in the narrow-band limit

R. Allub, César R. Proetto

The case of a quantum dot connected to two superconducting leads is studied by using the narrow-band limit to describe the superconducting degrees of freedom. The model provides a simple theoretical framework, almost analytical, to analyze the interplay between the Kondo effect, superconductivity, and finite temperature. In the quantum dot Kondo regime, the model is completely characterized by the ratio $\ensuremath{\Delta}/J$, with $\ensuremath{\Delta}$ the superconducting gap and $J$ an effective antiferromagnetic exchange coupling between the dot and the leads. The model allows us to calculate, at any temperature $T$, the equilibrium Josephson current through the dot in a very straightforward way as a function of $\ensuremath{\Delta}/J$. The behavior of the current allows us to distinguish the four types of hybrid junctions: 0, ${0}^{\ensuremath{'}},{\ensuremath{\pi}}^{\ensuremath{'}}$, and $\ensuremath{\pi}$. The presence of the 0- and ${0}^{\ensuremath{'}}$-junction configurations are intrinsically linked to the Kondo effect in the quantum dot, while the $\ensuremath{\pi}$- and ${\ensuremath{\pi}}^{\ensuremath{'}}$-junction configurations are driven by the superconductivity in the leads. The Josephson critical current has a non-monotonic behavior with temperature, that may be used for the experimental characterization of the fundamental $0\ensuremath{-}\ensuremath{\pi}$ transition. The model allows us to obtain easily a phase diagram $\ensuremath{\Delta}/J$ vs temperature, from where we can obtain an overall picture on the stability of the different types of junctions. From the explicit analytical expressions for the ground-state, low-energy excitations, free energy, and Josephson current, it is easy to understand the physical nature of the main features of the critical current and the phase diagram. The results, obtained with a minimum of numerical effort, are in a good qualitative agreement with more demanding calculational approaches aimed to solve the full model.

Physical Review B•Jan 30, 2015•9 citations

Quantum phases of a weakly disordered Josephson ladder

Eyal Walach, Efrat Shimshoni

The interplay of interactions and disorder in low-dimensional superconductors supports the formation of multiple quantum phases as possible instabilities of the superconductor-insulator transition (SIT) at a singular quantum critical point. We explore a one-dimensional model which exhibits such a variety of phases in the strongly quantum fluctuations regime. Specifically, we study the effect of weak disorder on a two-leg Josephson ladder with comparable Josephson and charging energies $({E}_{J}\ensuremath{\sim}{E}_{C})$. An additional key feature of our model is the requirement of perfect ${\mathbb{Z}}_{2}$ symmetry, respected by all parameters including the disorder. Using a perturbative renormalization-group (RG) analysis, we derive the phase diagram and identify at least one intermediate phase between a full-fledged superconductor and a disorder-dominated insulator. Most prominently, for repulsive interactions on the rungs we identify two distinct mixed phases: In both of them the longitudinal charge mode is a gapless superconductor, however one phase exhibits a dipolar charge density order on the rungs, while the other is disordered. This latter phase is characterized by coexisting superconducting (phase-locked) and charge-ordered rungs, and encompasses the potential of evolving into a Griffith's phase characteristic of the random-field Ising model in the strong disorder limit.

arXiv:2103.11501•Physical review. B./Physical review. B•Jun 1, 2021•2 citations
We analyze the quantum phases of a superconducting quantum dot array. Superconducting quantum dots are connected through Josephson junction. We study the effect of finite range Josephson coupling and also finite range interactions between the dots. Our field-theoretical study predicts different kinds of phases near the charge degeneracy point. We predict two different kinds of repulsive Luttinger liquid phases originate from different sources: one from the commensurate-incommensurate transition and the other from the XYXYXY criticality. Apart from the charge-density wave, we also predict the possibility of dimer-order density wave from the interplay of long-range interaction between the dots and Cooper-pair density. We observe different multicritical points with interesting phases and also two special points. The renormalization group flow diagram also shows many interesting phases.
We present the quantum phase diagram of a superconducting quantum dots array. Superconducting quantum dots are connected through Josephson junctions. We study the effect of finite-range Josephson coupling and also finite-range interactions between the dots. Our field-theoretical study predicts different kinds of phase near the charge degeneracy point. We predict two different kinds of repulsive Luttinger liquid phase and also the possibilities of different density wave instabilities from the inter-play of long-range interaction between the dots and Cooper pair density. We observe different multicritical points with interesting phases.
We have obtained the quantum phase diagram of one dimensional extended Bose-Hubbard model using the density-matrix renormalization group and Abelian bosonization methods for different commensurabilities. We describe the nature of different quantum phases at the charge degeneracy point. We find a direct phase transition from Mott insulating phase to superconducting phase for integer band fillings of bosons. We predict explicitly the presence of two kinds of repulsive Luttinger liquid phases, apart from the charge density wave and superconducting phases for half-integer band fillings. Our study reveals that extended range interactions are necessary to get the correct phase boundary of an one-dimensional interacting bosons system.
We analyze the ground state properties of an array of quantum dots connected in series between superconducting electrodes. This system is represented by a finite Hubbard chain coupled at both ends to BCS superconductors. The ground state is obtained using the Lanczos algorithm within a low energy theory in which the bulk superconductors are replaced by effective local pairing potentials. We study the conditions for the inversion of the sign of the Josephson coupling ($\ensuremath{\pi}$-junction behavior) as a function of the model parameters. Results are presented in the form of phase diagrams, which provide a direct overall view of the general trends as the size of the system is increased, exhibiting a strong even-odd effect. The analysis of the spin-spin correlation functions and local charges give further insight into the nature of the ground state and how it is transformed by the presence of superconductivity in the leads. Finally, we study the scaling of the Josephson current with the system size and relate these results with previous calculations of Josephson transport through a Luttinger liquid.
We study a quantum phase transition that occurs in a system composed of two impurities (or quantum dots), each coupled to a different interacting (Luttinger liquid) lead. While the impurities are coupled electrostatically, there is no tunneling between them. Using a mapping of this system onto a Kondo model, we show analytically that the system undergoes a Berezinskii-Kosterlitz-Thouless quantum phase transition as a function of the Luttinger liquid parameter in the leads and the dot-lead interaction. The phase with low values of the Luttinger liquid parameter is characterized by an abrupt switch of the population between the impurities as a function of a common applied gate voltage. However, this behavior is hard to verify numerically since one would have to study extremely long systems. Interestingly, though, at the transition the entanglement entropy drops from a finite value of ln(2)\mathrm{ln}(2)ln(2) to zero. The drop becomes sharp for infinite systems. One can employ finite-size scaling to extrapolate the transition point and the behavior in its vicinity from the behavior of the entanglement entropy in moderate size samples. We employ the density matrix renormalization-group numerical procedure to calculate the entanglement entropy of systems with lead lengths of up to 480 sites. Using finite-size scaling, we extract the transition value and show it to be in good agreement with the analytical prediction.
The case of a quantum dot connected to two superconducting leads is studied by using the narrow-band limit to describe the superconducting degrees of freedom. The model provides a simple theoretical framework, almost analytical, to analyze the interplay between the Kondo effect, superconductivity, and finite temperature. In the quantum dot Kondo regime, the model is completely characterized by the ratio $\ensuremath{\Delta}/J$, with $\ensuremath{\Delta}$ the superconducting gap and JJJ an effective antiferromagnetic exchange coupling between the dot and the leads. The model allows us to calculate, at any temperature TTT, the equilibrium Josephson current through the dot in a very straightforward way as a function of $\ensuremath{\Delta}/J$. The behavior of the current allows us to distinguish the four types of hybrid junctions: 0, ${0}^{\ensuremath{'}},{\ensuremath{\pi}}^{\ensuremath{'}}$, and $\ensuremath{\pi}$. The presence of the 0- and ${0}^{\ensuremath{'}}$-junction configurations are intrinsically linked to the Kondo effect in the quantum dot, while the $\ensuremath{\pi}$- and ${\ensuremath{\pi}}^{\ensuremath{'}}$-junction configurations are driven by the superconductivity in the leads. The Josephson critical current has a non-monotonic behavior with temperature, that may be used for the experimental characterization of the fundamental $0\ensuremath{-}\ensuremath{\pi}$ transition. The model allows us to obtain easily a phase diagram $\ensuremath{\Delta}/J$ vs temperature, from where we can obtain an overall picture on the stability of the different types of junctions. From the explicit analytical expressions for the ground-state, low-energy excitations, free energy, and Josephson current, it is easy to understand the physical nature of the main features of the critical current and the phase diagram. The results, obtained with a minimum of numerical effort, are in a good qualitative agreement with more demanding calculational approaches aimed to solve the full model.
The interplay of interactions and disorder in low-dimensional superconductors supports the formation of multiple quantum phases as possible instabilities of the superconductor-insulator transition (SIT) at a singular quantum critical point. We explore a one-dimensional model which exhibits such a variety of phases in the strongly quantum fluctuations regime. Specifically, we study the effect of weak disorder on a two-leg Josephson ladder with comparable Josephson and charging energies $({E}_{J}\ensuremath{\sim}{E}_{C})$. An additional key feature of our model is the requirement of perfect Z2{\mathbb{Z}}_{2}Z2​ symmetry, respected by all parameters including the disorder. Using a perturbative renormalization-group (RG) analysis, we derive the phase diagram and identify at least one intermediate phase between a full-fledged superconductor and a disorder-dominated insulator. Most prominently, for repulsive interactions on the rungs we identify two distinct mixed phases: In both of them the longitudinal charge mode is a gapless superconductor, however one phase exhibits a dipolar charge density order on the rungs, while the other is disordered. This latter phase is characterized by coexisting superconducting (phase-locked) and charge-ordered rungs, and encompasses the potential of evolving into a Griffith's phase characteristic of the random-field Ising model in the strong disorder limit.