† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11774418 and 11374363).
We investigate symmetrically coupled double quantum dots via the hierarchical equations of motion method and propose a novel zero-energy mode (ZEM) at a temperature above the spin singlet–triplet transition temperature. Owing to the resonance of electron quasi-particle and hole quasi-particle, ZEM has a peak at ω = 0 in the spectral density function. We further examine the effect of the magnetic field on the ZEM, where an entanglement of spin and charge has been determined; therefore, the magnetic field can split the ZEM in the spectra.
Recently, it is under debate whether the half-quantized conductance plateau could be used as an evidence for chiral Majorana modes in a millimeter-size quantum anomalous Hall–niobium hybrid device.[1,2] As a symmetric solution for the Dirac equation, Majorana fermion is its own antiparticle.[3,4] In solid-state physics, Majorana fermionic quasi-particle produces a zero-energy mode (ZEM), which is present in differential conductance and spectral density functions.[5–9] The Majorana ZEM has some remarkable physical properties that could lead to advances in quantum computing.[10–12] ZEMs have received considerable attention owing to their special performance. Ferreira et al. have studied ZEMs resilient to localization in graphene subjected to chiral-symmetric disorder and reported accurate quantum transport calculations. Ganeshan et al. have determined that the commensurate off-diagonal Aubry–André or Harper models support ZEMs. Fan et al. have proposed ZEMs and tunable electronic band gap in periodic heterosubstrate-induced graphene superlattices.[13]
Quantum dot (QD), which is also referred to as an artificial atom, is a small structure with electrons that are restricted in all three dimensions. It is convenient to study quantum phenomena in QDs, particularly those with many-body properties, which are vital for miniaturizing electronic devices.[14,15] Double quantum dots (DQDs), which are frequently used in laboratory experiments,[16–18] have a simple configuration that allows to study the effects of inter-dot interactions. In this study, we adopt hierarchical equations of motion (HEOM) to calculate the spectra of symmetrically serially coupled DQD. We propose a novel ZEM which occurs at finite temperature (approximately 7.5Δ, Δ is the hybridization of the QD system and electron reservoir, see details in next section). Owing to accessible temperature, this ZEM is easy to achieve, and the corresponding quasi-particle may provide another candidate for quantum computation.
The paper is arranged as follows. First, we introduce the HEOM numerical method. Then, we report the characters and parameters of ZEM using a simple explanation. Finally, we present conclusions and expectations.
By using the time derivative path integral influence functional expression,[19–24] the HEOM approach allows to address open systems. On the basis of the linear response theory of quantum open systems,[25] HEOM can accurately and efficiently produce the dynamical observables of strongly correlated quantum impurity systems.[26] The total Hamiltonian can be regarded as the system-plus-reservoir composite, Htotal = Hsys + Hres + Hsys-res. We consider a coupled DQD described by the Anderson impurity model, where the system Hamiltonian Hsys is
![](cpb_29_6_067302/cpb_29_6_067302_ieqn1.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn2.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn3.gif)
We treat the reservoirs as a bath environment. The Hamiltonian is written as
![](cpb_29_6_067302/cpb_29_6_067302_ieqn4.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn5.gif)
In this study, we consider a serially coupled DQD for which the coupling Hamiltonian is written as
![](cpb_29_6_067302/cpb_29_6_067302_ieqn6.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn7.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn8.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn9.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn10.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn11.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn12.gif)
HEOM has a universal formalism for an arbitrary system Hamiltonian; the coupled DQD we choose in this study is an application. The form of the HEOM formalism reads
![](cpb_29_6_067302/cpb_29_6_067302_ieqn13.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn14.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn15.gif)
![](cpb_29_6_067302/cpb_29_6_067302_ieqn16.gif)
The HEOM formalism, Eq. (
Numerically, because the maximum hierarchical level for the full HEOM theory is difficult in practice, we adopt the Padé spectrum decomposition (PSD) scheme[31,32] to minimize the computational expenditure while maintaining the quantitative accuracy. We determine that a relatively low truncation tier level is sufficient to yield quantitatively converged results for this system–reservoir coupling.[26] In this work, the truncation tier L = 4.
For the dynamical quantities, the correlation function
HEOM numerical evaluation focuses on the reduced system density [cf. Eq. (
Therefore, QD1 and QD2 are equivalent to each other and symmetrically serially coupled to reservoirs L and R, respectively. We write Gs(ω) = G1s(ω) = G2s(ω) and omit the subscript s at spin degeneracy so that the spectral density function is written as
In this work, we focus on the equilibrium scenario where μL = μR = 0. We also set ΔL1s = ΔR2s = Δ, and use Δ as the unit of energy.
For DQD, at extremely low temperatures (T ≪ TK, TK is the Kondo temperature of individual QD),[33,34] the nonzero t produces an effective anti-ferromagnetic coupling, J = 4t2/U, between the local spin moments on both QDs.[35] When J is weak, the two spin moments are almost independent of each other, and itinerant electrons separately screen the local spin at each QD. Thus, we can observe one zero-energy peak in the spectral function A(ω), which is recognized as a many-body Kondo resonant peak.[26,36,37] In contrast, when J is sufficiently strong, spin-singlet states are formed, which span over both impurities. Then, the zero-energy peak will split into two.[26,36] At the low temperature of approximately T ∼ TK, the Kondo effect remains dominant but cannot fully develop owing to the finite temperature.[34]
At medium temperature T ≫ TK, we observe another ZEM, which indicates a new type of resonance processes other than the Kondo one. Using the HEOM method, we systematically study DQD [Eq. (
By referring to Fig.
We observe that ZEM returns to the Hubbard two-peak structure at t, T, or U; however, the Hubbard two-peaks correspond to different ground states, for which we show the schematic of phase diagram in Fig.
![]() | Fig. 2. (a) Energy levels in an isolate DQD with tuning t. (b) Schematic of temperature T–the interdot tunneling coupling t phase diagram of DQDs coupled to reservoirs. |
We further analyze the mechanism of ZEM from an energy-level view illustrated in Fig.
In the t–T phase diagram of the coupled DQDs, the newly observed ZEM is schematically identified by the green marked position in Fig.
In this scenario with spin symmetry, between the cross levels, both electrons and holes have the same energy gap. Thus, in the ZEM regime, the reflection on spectral function A(ω) shows the symmetry of electrons and holes.
Both the asymmetrically composed single-electron level and the spin triplet level are sensitive to the magnetic field. We examine the scenario at the finite magnetic field B = 1.5μBΔ. For the energy levels in the isolate DQD, shown in Figs.
![]() | Fig. 3. Energy levels in an isolated DQD at the finite magnetic field of B = 1.5μBΔ: (a) spin up, (b) spin down. Corresponding spectral functions at different t of (c) spin up and (d) spin down. |
For the excitation of spin up, the energy levels in Fig.
By tuning J via t, at medium temperature above Tst, we can observe a phase transition from the spin singlet regime into ZEM regime at finite magnetic field in spectral functions shown in Fig.
We adopt the HEOM formalism on a symmetrically coupled DQD. At the temperature above the spin singlet–triplet transition temperature, we propose a novel zero-energy mode, which arises from the resonance of electron quasi-particle and hole quasi-particle; this zero-energy mode shows a ZEM peak in the spectral function. By further examining the effect of the magnetic field, because the degrees of freedom of spin and charge are entangled, it is determined that the magnetic field can split ZEM in the spectra.
The new reported ZEM in symmetrically serially coupled DQD is available for the evaluation at finite temperature. Therefore, the new ZEM could be a potential quantum bit candidate for quantum calculations.
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