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Chinese Physics B

Phase diagram characterized by transmission in a triangular quantum dot

Jin Huang, Weizhong Wang

Published Nov 1, 2018• 1 citation
Publisher

Abstract

We propose a theoretical model to detect the quantum phase transition in a triangular quantum dot molecule with frustration. The boundaries of the phase diagram are accurately determined by the transmission. For small frustration t , as the interdot Coulomb repulsion V increases, the system undergoes a Kosterlitz–Thouless (KT) transition from the Kondo resonance state with a transmission peak at zero energy to the Coulomb blocked state with zero transmission, which is followed by a first transition to the V -induced resonance (VIR) state with unitary transmission. For large frustration t , as V increases, the orbital spin singlet without transmission transits to the VIR state through a KT transition.

DOI
10.1088/1674-1056/27/11/117303

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We propose a theoretical model to detect the quantum phase transition in a triangular quantum dot molecule with frustration. The boundaries of the phase diagram are accurately determined by the transmission. For small frustration t , as the interdot Coulomb repulsion V increases, the system undergoes a Kosterlitz–Thouless (KT) transition from the Kondo resonance state with a transmission peak at zero energy to the Coulomb blocked state with zero transmission, which is followed by a first transition to the V -induced resonance (VIR) state with unitary transmission. For large frustration t , as V increases, the orbital spin singlet without transmission transits to the VIR state through a KT transition.
The quantum phase transition and the electronic transport in a triangular quantum dot system are investigated using the numerical renormalization group method. We concentrate on the interplay between the interdot capacitive coupling V and the interdot tunnel coupling t . For small t , three dots form a local spin doublet. As t increases, due to the competition between V and t , there exist two first-order transitions with phase sequence spin-doublet-magnetic frustration phase-orbital spin singlet. When t is absent, the evolutions of the total charge on the dots and the linear conductance are of the typical Coulomb-blockade features with increasing gate voltage. While for sufficient t , the antiferromagnetic spin correlation between dots is enhanced, and the conductance is strongly suppressed for the bonding state is almost doubly occupied.
By means of the numerical renormalization group method, I study the Kondo effect and quantum phase transitions in a double quantum dot system with on-site repulsion UUU, interdot repulsion V,V,V, and magnetic field BBB. At zero temperature and B=0B=0B=0, there is a local spin triplet-singlet transition of the Kosterlitz-Thouless type when VVV increases to a critical value ${V}_{c}\ensuremath{\approx}U$. With increasing BBB, a singlet-triplet transition of the second order occurs at a critical magnetic field ${B}_{c}\ensuremath{\approx}V\ensuremath{-}U$. The magnetic susceptibility and the first derivatives of the spin correlation and double occupation are expected to diverge at Bc{B}_{c}Bc​ and to have different critical exponents, which are universal for different parameters. An appropriate magnetic field can restore the Kondo effect.
We investigate electronic transport through a parallel double quantum dot (DQD) system with strong on-site Coulomb interaction and capacitive interdot coupling. By applying the numerical renormalization group (NRG) method, the ground state of the system and the transmission probability at zero temperature have been obtained. For a system of quantum dots with degenerate energy levels and small interdot tunnel coupling, the spin correlations between the DQDs is ferromagnetic and the ground state of the system is a spin-1 triplet state. The linear conductance will reach the unitary limit (2e(2)/h) due to the underscreened Kondo effect at low temperature. As the interdot tunnel coupling increases, there is a quantum phase transition from ferromagnetic to antiferromagnetic spin correlation in DQDs and the linear conductance is strongly suppressed.
We investigate electronic transport through a parallel double quantum dot (DQD) system with strong on-site Coulomb interaction, as well as the interdot tunnelling. By applying numerical renormalization group method, the ground state of the system and the transmission probability at zero temperature are obtained. For a system of quantum dots with degenerate energy levels and small interdot tunnel coupling, the spin correlations between the DQDs is ferromagnetic, and the ground state of the system is a spin-1 triplet state. The linear conductance will reach the unitary limit (2 e 2 / h ) due to the Kondo effect at low temperature. As the interdot tunnel coupling increases, there is a quantum phase transition from ferromagnetic to anti-ferromagnetic spin correlation in DQDs and the linear conductance is strongly suppressed.
Quantum Ising model on a triangular lattice hosts a finite temperature Berezinskii-Kosterlitz-Thouless (BKT) phase with emergent U(1) symmetry, and it will transit into an up-up-down (UUD) phase with C3{C}_{3}C3​ symmetry breaking upon an infinitesimal external field along the longitudinal direction, but the overall phase diagram spanned by the axes of external field and temperature remains opaque due to the lack of systematic investigations with controlled methodologies. By means of quantum Monte Carlo at finite temperature and ground state density matrix renormalization group simulations, we map out the phase diagram of triangular quantum Ising model. Starting from the upper BKT temperature at zero field, we obtain the phase boundary between the UUD and paramagnetic phases with its 2D q=3q=3q=3 Potts universality at weak field and weakly first order transition at strong field. Originated from the lower BKT temperature at zero field, we analyze the low temperature phase boundary between the clock phase and the UUD phase with Ising symmetry breaking at weak fields and the quantum phase transition between the UUD and fully polarized phases at strong fields. The accurate many-body numerical results are consistent with our field theoretical analysis. The experimental relevance towards the BKT magnet TmMgGaO4{\mathrm{TmMgGaO}}_{4}TmMgGaO4​ and programmable quantum simulators are also discussed.
The spectral properties and the quantum phase transitions in parallel triple quantum dot systems are investigated using the numerical renormalization-group method. For small interdot hopping ttt, the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction between the dots is identified in the spectra and is consistent with that obtained by the perturbation theory. With increasing ttt, due to the competition between the RKKY exchange and the direct exchange between dots, there exist two first-order quantum phase transitions between the phases with the local spin Sdot=3∕2{S}_{\mathit{\text{dot}}}=3∕2Sdot​=3∕2, 1∕21∕21∕2, and 0 on the dots. At low temperature, the local spin Sdot=3∕2{S}_{\mathit{\text{dot}}}=3∕2Sdot​=3∕2 is partially screened to a residual spin S=1S=1S=1, while the local spin Sdot=1∕2{S}_{\mathit{\text{dot}}}=1∕2Sdot​=1∕2 is totally screened. In both local quadruplet and doublet states, the Kondo resonance accompanied with a conductance of the unitary limit 2e2∕h2{e}^{2}∕h2e2∕h is observed, while no Kondo effect is found in the local spin-singlet state.