Fractional Solitons in Excitonic Josephson Junctions

The Josephson effect is especially appealing to physicists because it reveals macroscopically the quantum order and phase. In excitonic bilayers the effect is even subtler due to the counterflow of supercurrent as well as the tunneling between layers (interlayer tunneling). Here we study, in a quantum Hall bilayer, the excitonic Josephson junction: a conjunct of two exciton condensates with a relative phase ϕ0 applied. The system is mapped into a pseudospin ferromagnet then described numerically by the Landau-Lifshitz-Gilbert equation. In the presence of interlayer tunneling, we identify a family of fractional sine-Gordon solitons which resemble the static fractional Josephson vortices in the extended superconducting Josephson junctions. Each fractional soliton carries a topological charge Q that is not necessarily a half/full integer but can vary continuously. The calculated current-phase relation (CPR) shows that solitons with Q = ϕ0/2π is the lowest energy state starting from zero ϕ0 – until ϕ0 > π – then the alternative group of solitons with Q = ϕ0/2π − 1 takes place and switches the polarity of CPR.

The ↑ (↓ ) denotes the which layer degree of freedom and X the guiding center quantum number of the lowest Landau level. Vacuum state 0 indicates no electron in either layer. When representing Ψ as a vector in the Bloch sphere, θ(X) and φ(X) become the associated polar and azimuthal angles. Such a vector is hereafter referred to as the psuedospin → m (classical) -the exciton system is ultimately mapped to a pseudospin ferromagnet. In the limit of smooth textures, the energy functional can be expressed by pseusospin The first term is the capacitive energy deduced from the Hartree and intralayer exchange interactions. The positive anisotropic parameter β means an energy cost for the pseudospin out-of-plane excursion. In the counterflow geometry (upper half), a relative phase φ 0 is generated between the two excitonic condensates, EC1 and EC2. The navy arrows indicate the current flows. Note that currents are allowed to flow between layers while the net current should conserve. Lower half of Fig. 1 demonstrates the pseudospin picture of the excitonic Josephson effect. The relative phase φ 0 corresponds to an angle difference in two pseudospin Zeeman fields (red arrows). The Zeeman fields attempt to work against the pseudospin stiffness to align the local pseudospins (pink arrows) with themselves. There are two configurations, a direct solution (DS) and a complementary solution (CS) that correspond to incline angles equal φ 0 and φ 0 − 2π, respectively. This term is neglected from now on since we focus on the condensation regime in this report. The second term is essentially the kinetic energy of excitonic condensates which arises from the interlayer exchange. In the pseudospin language, the positive ρ s tends to align neighboring pseudospins in parallel and serves as the pseudospin stiffness.
The third term, namely the interlayer tunneling, is the key to our work. Here n and Δ t are the pseudospin density and the interlayer tunneling strength respectively. Aside from the ordinaries, this term contains an extra φ 0 Θ (x) [Θ (x) is the standard Heaviside function]. Note that in arriving at this expression we have implicitly taken the absolute phase of EC1 to be 0, without losing generality. The relative phase φ 0 is originated from the Aharnov-Bohm phase when the electrons circle around a loop with effective magnetic flux. This form of interlayer tunneling can be interpreted as the pseudospin Zeeman term: , which attempts to align the local pseudospins → m with itself.

Equation of motion.
We derive the Landau-Lifshitz-Gilbert (LLG) equation for pseudospins by (2). Pseudospin in general has two degrees of freedom; we pick φ and m ⊥ to describe. Two coupled equations are then solved numerically by our LLG solver, detailed in ref. 50.
In parallel, we minimize the energy functional in Eq. (2) with respect to φ to acquire the static properties. An equation that resembles the sine-Gordon equation is obtained: t is the Josephson length that sets the spatial scale of topology. The phase φ is apparently a function of the relative phase φ 0 . However for a given φ 0 , the solution is not unique: replacing φ 0 by φ 0 + 2Nπ in a known solution gives rise to a set of distinct valid solutions (not all are stable). To describe the situation, we denote the phase profile φ as a function of the incline angle φ φ π ≡ + N 2 i 0 instead of φ 0 itself. Among all, the direct solution (DS) and complementary solution (CS) are energetically most relevant. Here we illustrate the two in the pseudospin picture: a given relative phase φ 0 determines the direction of the pseudospin Zeeman field → h as depicted in Fig. 1 (the absolute phase of EC1 is set to zero). The same Zeeman field is however associated with various incline angles Taking φ i = φ 0 , the local pseudospins then wind counter-clockwise (for positive φ 0 ), seeking to be commensurate with both the Zeeman fields and yields the "direct solution" (plotted in row marked as "DS"). Taking φ i = φ 0 − 2π, on the other hand, the local pseudospins then wind clockwise to yield the "complementary solution" (CS). As shown in Fig. 1, the two solutions exhibit opposite polarities and even more, are generally not mirror symmetric.

Results
Phase profiles and fractional sine-Gordon Solitons. The phase profile φ φ ( , )  x i is a function of both the position  x and the incline angle φ i as mentioned. We first show the static phase profiles, which is obtained by solving LLG equation with φ i = π/2 for various system sizes (main body of Fig. 2). A widely accepted set of parameters for the total filling number v T = 1 QHB are used in this and in all later calculations (parameters expressed as dimensionless quantities): β β ≡ / = .
(the range of these parameters are nicely summarized in ref. 38). All lengths hereafter are also dimensionless quantities taken with respect to the Josephson length λ.
The phase profiles in the short and the long junctions are visibly different although both exhibit odd symmetry about =  x 0 and φ φ = /2 i (Fig. 2). The shape is practically linear in short junctions while saturates at both ends in long junctions. The zoom-in in the vicinity of =  x 0 (upper-left inset) further shows that the profiles from the two limits deviate already at small  x: the shorter junctions tend to possess gentler slopes. The slope vs. system size plot (lower-right inset) demonstrates that tendency and shows a smooth crossover at ∼  L 2. The behavior evolves from linear to constant with ascending  L. Below we analyze the detailed profiles in the two limits.
In the short junctions, ( < )  L 1 , the phase profiles do not extend fully from 0 to φ 0 and are reasonably described by a slanting line . By minimizing the system's total energy with respect to  C, we obtain the profile The short-junction assumption φ / C L 0 has been employed in reaching this expression. Note especially that the corresponding phase slope   L 2 1 (purple dotted line), = .
 L 2 11 (blue dash-dash-dotted line). The upper-left inset is the zoom-in of the small  x region that shows deviations of =  x 0 phase slope for short systems. The lower-right inset is the phase slope vs. the system size  L. For short systems <  L 2, the slopes are linear with  L while for long junctions ( > )  L 4 they are practically a constant. The two red dash lines are guide to eyes. We can deduce, from the above, difference between the CS and the DS energy is Δ t cos(φ 0 /2). The two come to degenerate when φ i equals multiples of π.
In the long junctions ( ) L 1 , the phase profiles show kinks that span between 0 and φ i [ Fig. 3(a)]. Those are fractional sine-Gordon solitons that can be constructed from the ordinary sine-Gordon solitons [ Fig. 3(b)]: The ultimate guideline are the boundary conditions, φ ( = − ∞) =  x 0 and φ φ ( = ∞) =  x i . With the ordinary sine-Gordon soliton at hand, we first shift its tail left to − x 0 (light blue curve) rightward by  x " " 0 . The tail right to  x 0 (light red curve) is then shifted first downward by "2π − φ 0 " then leftward by  x " " 0 . No shrinking or stretching should be involved in the entire process. The landmark The profile for the fractional soliton is then literally: can be assigned to characterize the solitons; this charge is closely related to the winding number that describes a vortex. It is interesting to note that similar geometric objects (fractional Josephson vortices) have been predicted also in an extended 0 − κ superconducting Josephson junction 46,47 .
The energy associated with the fractional soliton in a long junction is then given by Eqs. (2) and (6): With φ i = 2π, the energy E = 2Δ t is that of an ordinary soliton. This is a direct consequence of the equipartition theorem if interpreting Δ t as the potential energy. Finally we remark that although Eq. (7) might appear as a nonmonotonic function of φ i , it actually is the opposite -there is no soliton solution  for φ π >4 i . The construction above is impossible for φ i exceeding 4π because a meaningful  x 0 does not exist.
Current-phase relation. The supercurrent density j s can further be acquired from the phase profile by j s = (e/h -) ρ φ ∇ s ; in this report, we focus on the supercurrent density at the interface. In Fig. 4 we plot the CPRs obtained from the numerical LLG calculation. The analytical result for short junction is obtained side-by-side from Eq. (4) (in dimensionless expression): This dimensionless current density ˜j s is taken with respect to eE 0 /h λ. The above exhibits a critical current density of ρ = ( / )  j L 1 4 c s ; also, with φ i = φ 0 , the expression corresponds to DS while with φ i = φ 0 − 2πsgn(φ 0 ) to CS. In Fig. 4, we plot the supercurrent densities of analytic solution for both the DS (blue dash line) and the CS (black dash line), as well as the numerical calculation (red open circles). The sin(φ 0 /2) dependence in the DS is in great accordance with the numerical result (Fig. 4) for φ π < 0 (Note that in plotting j s we have intentionally set back the units to make direct comparison with the numerical results). The case of φ π > 0 will be discussed jointly with that in the long junction. In the long junction, the fractional solitons in Eq. (6) give rise to the supercurrent density This is distinct from the current density in the short junction. The critical current is size-independent and is given by ρ = j 2 c s . The CPR of sin(φ i /4) dependence is also qualitatively different from that of sin(φ i /2) in the short junction. The numerical and the analytical results for DS are consistent for φ π < 0 , when DS is the lowest energy state. What appears to be different starts at φ π = 0 , when the DS and CS become degenerate in both the short and the long junctions. As φ 0 exceeds π, the CS replaces the DS to be the lowest energy state. The polarity of the CPR swiches. In either the long or the short junction, an abrupt commensurate-incommensurate (linear-soliton) transition as that in the Pokrovsky-Talapov model 5,7,51,52 is not seen. Topological objects are present for arbitrarily small φ 0 .
The switch of the lowest energy state can best be visualized in the pseudospin picture; here a long junction is chosen for illustration in Fig. 5 but the description is general. The pseudospin Zeeman fields in EC1 and in EC2 align the local pseudospin differently and give rise to two different domains, and a wall in between (soliton). When the two domains are not completely anti-aligned, the local pseudospins do not wind a full circle within the wall -this corresponds to a fractional soliton. While the local pseudospins can be arranged into different configurations with the same φ 0 , the one with the slowest variation costs least energy. For 0 < φ 0 < π, the DS (winding counter-clockwise) has the slower variation and is preferred. The DS and CS become mirror symmetric when φ 0 equals π. For φ 0 > π, the CS (winding clockwise) has slower variation [ Fig. 5(a)] and takes over the lowest energy state. Also we recognize that the CS for any φ 0 is genuinely the DS for another φ φ π φ ′ = − ( ) 2 sgn 0 0 0 [ Fig. 5(b)]. The full CPR (lowest in energy) should be the periodic replications of that between − π and π.

Conclusion
To summarize, the excitonic Josephson junction is mapped to a pseudospin ferromagnet and described by the Landau-Lifshitz-Gilbert equation. The phase profile and current-phase relation are calculated. We find distinct behaviors in the long-and the short-junction limits. In the short junctions, the phases are essentially slanting lines with the slopes proportional to the system size. In the long junctions most interestingly, we recognize the static fractional sine-Gordon soliton -the soliton fraction (topological charge) can be tuned continuously by the relative phase φ 0 . In addition, there are two relevant solutions, a direct solution (DS) and a complementary solution (CS); both are present for any junction length. These two solutions are opposite in polarity and carry different energies. The DS is the lowest energy state for φ 0 up to φ π = 0 . The CS then takes over as the lowest energy state for φ π > 0 . If the system can rapidly relax the excess energy, a sudden jump should appear in the current-phase relation. There is still plenty to explore in excitonic Josephson junctions: interaction of fractional solitons in multiple junctions, exotic materials as weak links and much more. This work can open up new physics in the context of excitonic superfluid.