Ferromagnetic Phase in Nonequilibrium Quantum Dots

By nonperturbatively solving the nonequilibrium Anderson two-impurity model with the hierarchical equations of motion approach, we report a robust ferromagnetic (FM) phase in series-coupled double quantum dots, which can suppress the antiferromagnetic (AFM) phase and dominate the phase diagram at finite bias and detuning energy in the strongly correlated limit. The FM exchange interaction origins from the passive parallel spin arrangement caused by the Pauli exclusion principle during the electrons transport. At very low temperature, the Kondo screening of the magnetic moment in the FM phase induces some nonequilibrium Kondo effects in magnetic susceptibility, spectral functions and current. In the weakly correlated limit, the AFM phase is found still stable, therefore, a magnetic-field-free internal control of spin states can be expected through the continuous FM–AFM phase transition.

The ferromagnetism intrinsically origins from the spin-independent Coulomb interaction and the Pauli exclusion principle (PEP), as initially proposed by Heisenberg 1 . The Hubbard model 2 , which includes both two elements with on-site electron-electron (e − e) interaction U, is regarded as the minimal model for ferromagnetic (FM) states. Unfortunately, it has not been well addressed whether the Hubbard model has a general FM phase, except under some special conditions [3][4][5][6] . The Hartree-Fock approximation once predicted an itinerant Stoner-like FM phase 7 , but we now know that the mean-field theory deduces incorrect results and the FM region has been overestimated 3 . Besides the Hubbard model, the Anderson (multi)-impurity model 8 may act as another minimal model for magnetic phase in a bottom-up fashion, with the advantage of implementation simplicity in quantum dots (QDs). For example, the antiferromagnetic (AFM) correlation J AF due to nearest-neighbour electron hopping or tunneling t (J AF ~ 4t 2 ) has been well understood experimentally in series-coupled double QDs (SDQDs) [see Fig. 1(a)] 9 . Theoretically, J AF is responsible for the AFM ground state at half filling in the Hubbard model, while it induces the spin singlet competing with the Kondo singlet at temperature T < T K (T K being the Kondo temperature) in the Anderson two-impurity model [9][10][11][12][13] .
Does there exist a FM phase in the Anderson two-impurity model or in SDQDs? That issue may help to understand Heisenberg's original idea and to determine the FM phase in various strongly correlated models. Please be noted that the sign-indefinite Ruderman-Kittel-Kasuya-Yosida (RKKY) magnetic order, whose implementation must through a third mediated dot in experiments 14 , is not our concern here. What we are seeking is a stable FM phase strong enough to compete with the AFM one in SDQDs, which has not been explicitly determined yet in the phase diagrams of SDQDs 9 and other two-impurity systems 15 .
The FM phase in SDQDs also has great application potential in solid-state quantum computing. QDs-based spin qubit is one of the most possible physical realization of scalable qubit put forward so far, which has been extensively studied in last two decades 16,17 since its original proposal in SDQDs 18 . It has the advantages of fast operation and long coherence times but the disadvantage of seriously dependence on magnetic fields. The technical difficulties caused by magnetic fields are transparent: (i) The localized oscillating magnetic fields required in qubit or quantum gate manipulation are very hard to realize in practice; (ii) The Zeeman energy is an inefficient way to control spin states; and (iii) The magnetic fields are incompatible with present large-scale integrated circuit. If a stable FM phase in SDQDs does exist, these difficulties may be overcome by possible magnetic-field-free manipulations.
In the present work, by nonperturbatively solving the Anderson two-impurity model, we will firstly verify no FM phase in the range of parameters investigated under the equilibrium condition in SDQDs. Then, we will report a robust FM phase under nonequilibrium conditions at finite bias and detuning energy, which are strong enough to suppress the AFM phase in the strongly correlated limit  t U ( ) . We will demonstrate that the FM exchange interaction origins from the passive parallel spin arrangement caused by the PEP during the electrons transport [see Fig. 1(b)]. At large t, the AFM phase keeps stable, which defines a tunnel-barrier control of spin states through the FM-AFM transition in SDQDs, similar to the initial proposal in ref. 18 but no magnetic field (or auxiliary FM-dots) needed any more. The FM phase is the effect of PEP on magnetic order, which shows different properties from another effect of PEP called Pauli spin blockade (PSB). The PSB was first observed in vertically coupled GaAs/AlGaAs double quantum dots in 2002 19 , and then received extensive experimental and theoretical studies in various quantum dot systems [20][21][22][23][24] . Fundamentally, the hopping of electrons between two dots can be influenced by their spin configuration. When the total excess electrons of the SDQD is N T = N 1 + N 2 = 2 with occupation state (N 1 , N 2 ) = (2, 0), (1, 1) or (0, 2), the probability of formation of the spin triplet state T(1, 1) may be much larger than that of singlet S(1, 1) under one direction of bias of voltage, and then the transport of electrons is blocked due to the PEP which is unfavor of T(0, 2) or T(2, 0). The story does not happen under the other direction of bias. As a consequence, the current-voltage (I − V) curve will show a rectification behaviour. As an effect of the PEP on electric current, the PSB is mainly measured and manipulated in the boundary of Coulumb blockage (CB) 20,24 . Basing on previous results of PSB in literatures, we would like to discuss the following two issues: what is the effect of PEP on magnetic order? and what happens in the deep CB area?

FM phase in SDQDs
The SDQDs we study here can be described by the nonequilibrium Anderson two-impurity model. The total Hamiltonian reads H total = H S + H res + H sys−res , where the isolated QD part is here ˆ † c i s , (ĉ i s , ) is the operator that creates (annihilates) an s-spin (s = ↑, ↓) electron with energy ∈ i,s in the dot i , , corresponds to the s-spin electron number operator of dot i. As mentioned above, U (U = U 1 = U 2 ) is the on-dot Coulomb interaction between s-and s-spin electrons (s being the opposite spin of s), and t is the interdot coupling strength.
The Hamiltonians of reservoirs are ks denotes the creation (annihilation) operator of an electron in the s-spin state in the α-reservoir with wave vector k. We set the Fermi energy at equilibrium and μ L /e =− μ R /e = V/2 at nonequilibrium. The system-reservoir coupling is = ∑ + . .

H t c c h c sys res kis kis is ks
The hybridization function is assumed to be a Lorentzian form 2 . We adopt the hierarchical equations of motion (HEOM) approach 13,25 to numerically solve the nonequilibrium Anderson two-impurity model in a nonperturbative fashion. The HEOM can achieve the same level of accuracy as the latest high-level numerical renormalization group (NRG) 26 for both static and dynamical quantities under equilibrium conditions 13 . Under nonequilibrium conditions, the HEOM has many advantages above other approaches in the prediction of dynamical properties 24,27-32 . The details of the HEOM formalism and In order to figure out whether there exists a FM state, we calculate the spin-spin correlation function between QD1 and 2, where → S i is the quantum spin operator at dot i. In Fig. 1(c)-(e), we depict the phase diagram at bias V = 0, 0.5 and 1.0 mV, characterized by the sign and value of C 12 in the Δ − t plane. Under the equilibrium condition, as shown in Fig. 1(c), the sign of C 12 keeps always negative, which indicates a single AFM phase independent of t (t > 0) and Δ. It is understandable. From the second-order perturbation, one can obtain ] AF 2 2 2 at finite Δ, seeming a negative J AF included. However, the condition for that equation (  t U and Δ < U/2) makes J AF < 0 impossible, even under nonequilibrium conditions. Thus, the following FM phase can not result from this mechanism. As shown in Fig. 1(c), with increasing t, C 12 positively increases, and finally an AFM QD-molecule forms in the large t limit 34 , as an analogue of hydrogen molecule.
When a positive bias applied, as shown in Fig. 1(d) and (e), our results reveal a FM phase appearing in the region of <  t U 0 and 0.2U < Δ < 0.7U. In view of the phase changes from Fig. 1(d,e), the FM phase can be seen as growing from the AFM background at finite bias. The FM-AFM phase boundary (where C 12 changing its sign) seems quite smooth with no abrupt phase transition occurring, instead, a continuous crossover behaviour is clearly visible. With increasing bias, the area of FM phase is enlarged and the strength of exchange interaction enhanced, as C 12 positively increases. In the strongly correlated limit <  t U (0 ), the FM phase can well suppress the AFM one and dominate the phase diagram at finite V and Δ, as shown in Fig. 1(e). However, the AFM molecular state will survive at large t and very small Δ, which respectively determine the right and bottom boundary of FM phase. If Δ is too large to destroy the single occupation of any dot, C 12 will decrease to zero rapidly, which determines the upper boundary. The left boundary is naturally at t ~ 0. As a comprehensive result, the FM phase forms a closed irregular circle area in the phase diagram, as shown in Fig. 1(d) and (e).
In order to better understand the details of the AFM-FM transition, we theoretically lift the spin degeneracy in QD1 by applying a local magnetic field B 1 , with its direction paralleling to ↓-spins. B 1 is chosen to be strong enough to push ∈ 1↑ much higher than μ L but left ∈ 1↓ = −1.0 meV + Δ, which can be achieved by simultaneously adjusting the gate voltage on QD1. By fixing t = 0.2 meV and Δ = 0.75 meV, we calculate both static and dynamical quantities as functions of V and summarize the results in Fig. 2, where Fig. 2(a) depicts some typical static quantities (n 1↓ , n 2↑ , n 2↓ and C 12 ) and Fig. 2(b)-(e) show the spectral functions [A 1↓ (ω), A 2↑ (ω) and A 2↓ (ω)] at V = 0, 0.14 (V c , AFM-FM phase crossover point), 0.2, 0.5, 1.0 mV, respectively. As a starting point, the AFM phase at V = 0 is clearly shown in Fig. 2(a), where the magnetic moments m 1 ≡ n 1↑ − n 1↓ ≈ −n 1↓ < 0 and m 2 ≡ n 2↑ − n 2↓ > 0. Accordingly, the degeneracy of A 2↑ (ω) and A 2↓ (ω) is lifted due to the AFM exchange interaction J AF , as shown in Fig. 2(b), where the singly-occupation transition peak of A 2↑ (ω) is higher than that of A 2↓ (ω). Under nonequilibrium conditions, ↓ -spin electrons irreversibly flow from L-to R-reservoir through interdot tunneling. During the transport process, the PEP affects both electrical 19,23,24 and magnetic properties, of which the latter is our focus here. In Fig. 2(a), the continuous crossover from AFM to FM phase is shown in detail. With increasing V, n 2↑ gradually decreases while n 2↓ increases, thus m 2 positively decreases. At V ~ 0.14 mV, . As a consequence, C 12 ~ 0, which defines an AFM-FM phase crossover point, V c , as shown in Fig. 2(a). By checking the spectral functions, we find the singly-occupation transition peak of A 2↑ (ω) almost overlaps with that of A 2↓ (ω) at V = V c with a little splitting [see Fig. 2(c)]. With further increasing V at V > V c , m 2 becomes to negatively increase and C 12 positively increase, as shown in Fig. 2(a), thus the FM phase is gradually enhanced. At V ~ 0.9 mV, both m 2 and C 12 reach their saturation values of 0.9 and 0.21, respectively. The continuous increase of C 12 with a smooth sign change indicates the competition between AFM and FM phases is far from intense.
Fundamentally, finite bias injects ↓ -spin electrons from L-reservoir into QD1, followed by interdot tunneling to QD2. In the next step, the PEP prohibits the double occupation of two ↓ -spin electrons, and electrons can only flow out through off-resonance cotunneling 35 or many-body tunneling 29 into R-reservoir, both of which produce small current. As shown in Fig. 1(b), for electrons in QD2, increasing V and/or Δ will enhance their inflowing probability and meanwhile decrease their off-resonance outflowing probability. When the former becomes much larger than the latter at V > V c and Δ > 0.2U, ↓ -spin electrons will accumulate within QD2, which induces a positive to negative sign change of m 2 . As a consequence, the exchange of → S 2 produces a FM order characterized by C 12 > 0. The spectral functions shown in Fig. 2(d) at V = 0.2 mV verifies this FM correlation (although still weak), where the singly-occupation transition peak of A 2↑ (ω) becomes lower than A 2↓ (ω).
With further increasing V, the FM exchange interaction becomes stronger. In spectral functions, this trend is represented by the gradually increasing of the singly-occupation transition peak of A 2↓ (ω) and decreasing of that of A 2↑ (ω) [see Fig. 2(e)]. At V > 0.9 mV, the former reaches its maximum value and the latter almost disappears, as shown in Fig. 2(f). By summarizing Fig. 2(a-f), one can see that the FM phase in SDQDs origins from the passive parallel spin arrangement caused by the PEP during the electrons transport in the presence of e − e interactions. That mechanism is universal, which should play roles in other strongly correlated models including the Hubbard model.

Low temperature properties
We are now on the position to elucidate the temperature effect, especially the low temperature properties of the FM phase. In what follows, we recover the spin degeneracy in QD1 and fix V = 1.0 mV, t = 0.2 meV and Δ = 0.75 meV. The dependence of the inverse of magnetic susceptibility 1/χ on temperature T is depicted in Fig. 3(a), which shows an unambiguous Curie-Weiss behaviour at high temperature, χ = C/(T −T c ), with a fitted Curie point T c ~ 0.15 meV (~1.75 K). We also find a upward deviation at very low temperature T < 0.02 meV, resulting from the Kondo screening of the FM phase at T < T K . Under equilibrium conditions, this kind of S = 1 Kondo screening induces a 'singular Fermi liquid state' [36][37][38] . Here, some nonequilibrium Kondo features are expected.
The present HEOM approach can not directly determine T K as NRG does, but it can easily obtain spectral functions and current at sufficient low temperature to elucidate nonequilibrium Kondo characteristics. The HEOM results of A is (ω) s and current-voltage (I − V) curve at T = 0.01 meV are respectively shown in Fig. 3(c) and (d), where the I − V curve at T = 0.1 meV (T > T K ) is also shown for comparison. As shown in Fig. 3(b), one small Kondo peak is developed at ω = μ L in A 1s (ω), and another developed at ω = μ R in A 2s (ω). It can be seen as the DQD extension of the bias-induced Kondo peak splitting in single QDs 39 . Although the Kondo peaks in A is (ω) s seem not high in Fig. 2(b), their effects are quite significant on both of the magnetic and transport properties. For the latter, the nonequilibrium Kondo resonance assists the electrons transport, which is characterized by the low-temperature current enhancement shown in Fig. 3(c), when the FM phase dominates at V > 0.25 mV.

FM phase in stability diagrams
We can further elucidate the effect of PEP on magnetic order in stability diagrams by expanding the parameter Δ to the V 1 − V 2 plane, where V 1 /V 2 is the gate voltage applied onto QD1/QD2. It will help us to directly compare the parameter regions of FM state and PSB. The results at (V, t) = (0.5 mV, 0.15 meV) and (V, t) = (1.0 mV, 0.25 meV) are summarized in Fig. 4(a) and (b), respectively. In the figure, the AFM phase is shown in the colour of dark gray, and the dashed gray lines schematically mark off the boundary of CB (or degenerate lines in stability diagrams). By referring the figure, one can see that the FM phase expands into the deep CB area. At V = 1.0 mV, as shown in Fig. 4(b), the FM phase occupies almost all of the stability diagram of 0 ≤ V 1 ≤ 2.0 mV(+U) and −2.0 mV (−U) ≤ V 2 ≤ 0. Basing on experimental observations and theoretical results in literatures 19,20,24 , the range of PSB is approximately within the dotted blue circle (with the radius of V/2) in Fig. 4. Obviously, the range of FM phase is much larger than that of PSB.
Taking the case of V = 0.5 mV as an example, as shown in Fig. 1(a), if we start from the center of (1, 1) occupation state (V 1 = V 2 = 0), we will first reach a weak FM phase after a AFM-FM transition at V 1 (V 2 ) ~ 0.3 mV. Then, we will follow the enhancement of FM phase with C 12 gradually increasing. Only when V 1 (V 2 ) ~ 0.75 mV, we can observe the PSB effect. It thus indicates that the effect of PEP on magnetic order (FM phase) is prior to that on electric current (PSB). Our HEOM calculations have precisely captured the main features of the former.

Summary
In summary, we have theoretically reported a robust ferromagnetic phase under nonequilibrium conditions in series-coupled double quantum dots by nonperturbatively solving the Anderson two-impurity model. The ferromagnetic exchange interaction origins from the passive parallel spin arrangement caused by the Pauli exclusion principle during the electrons transport. The ferromagnetic phase can conduce to understand the Heisenberg's initial idea of ferromagnetic order. In addition, it also predicts a convenient way to internally control spin states without magnetic field.

Method
The serially coupled DQD system constitutes the open system of primary interest, and the surrounding reservoirs of itinerant electrons are treated as environment. The total Hamiltonian for the system is H total = H S + H res + H sys-res , where the interacting DQD In what follows, the symbol μ is adopted to denote the electron orbital (including spin, space, etc.) in the system for brevity, i.e., μ = {s, i...}. The device leads are treated as noninteracting electron reservoirs and the Hamiltonian can be written as ks ks ks ks res and the term of dot-electrode coupling is