Abstract
Bandinverted electronhole bilayers support quantum spin Hall insulator and exciton condensate phases. Interest in quantum spin Hall effect in these systems has recently put them in the spotlight. We investigate such a bilayer in an external magnetic field. We show that the interlayer correlations lead to formation of a helical quantum Hall exciton condensate state. Existence of the counterpropagating edge modes in this system results in formation of a ground state spintexture not supporting gapless singleparticle excitations. The charged edge excitations in a sufficiently narrow Hall bar are confined: a charge on one of the edges always gives rise to an opposite charge on the other edge. Magnetic field and gate voltages allow the control of a confinementdeconfinement transition of charged edge excitations, which can be probed with nonlocal conductance. Confinementdeconfinement transitions are of great interest, not least because of their possible significance in shedding light on the confinement problem of quarks.
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Introduction
The role of the electron–electron interactions for the experimentally accessible topological media is best appreciated in quantum Hall (QH) systems. The fractionally charged quasiparticles have been studied at the fractional filling factors^{1,2}, and the nonabelian excitations of more exotic QH states may eventually lead to a revolution in quantum computing^{3,4,5,6,7}. However, Coulomb interactions play a crucial role also in the case of the integer filling factors^{1,8,9,10,11,12,13}. Remarkably, interactions create a QH ferromagnetic ground state at total filling fraction v=1 even in the absence of Zeeman energy. In such systems, the SU(2) spin rotation symmetry is spontaneously broken, resulting in the lowenergy excitations being spin waves and charged topological spin textures, skyrmions^{1,8,12,13}. The presence of a small symmetrybreaking Zeeman field does not change the lowenergy excitations qualitatively.
In QH bilayer systems the role of spin is played by the layer index (pseudospin)^{8,10,11,13}. In this case the SU(2) pseudospin rotation symmetry is explicitly broken by the interactions, as they are larger within the layers than between the layers. The interactions favour the pseudospin orientations in (x, y)plane, where the direction is chosen spontaneously (spontaneous U(1) symmetry breaking) so that the QH bilayers realize an easyplane ferromagnet. Since the spontaneously chosen direction in the (x, y)plane corresponds to a spontaneous interlayer phase coherence, this easyplane ferromagnetic state is equivalent to an exciton condensate^{11,13}.
The QH ferromagnet and QH exciton condensate in electron–electron bilayers support a single chiral edge mode. However, the two Landau levels may also support counterpropagating edge modes. The natural hosts of such kind of QH states are systems supporting quantum spin Hall (QSH) effect^{14,15,16,17,18,19} due to inverted electronhole bandstructure. In these materials the magnetic field allows tuning through the Landau level crossing^{20,21}, where we expect to find a QH state with spontaneously broken (pseudo)spinrotation symmetry. Thus, we argue that there exist four different experimentally accessible pseudospin ferromagnetic states, determined by spontaneously broken symmetry (SU(2) in single layer and U(1) in bilayer systems) and the edge structure (chiral or helical). All these possibilities are illustrated in Fig. 1.
In this paper, we concentrate on the helical QH exciton condensate state (broken U(1) symmetry and helical edge structure). Remarkably, we find that in this system the charged edge excitations in a sufficiently narrow Hall bar are confined: a charge on one of the edges is always connected to the opposite charge on the other edge through the bulk by a stripe of rotated pseudospins, and thus lowenergy isolated charged excitations cannot be observed. The gapless singleparticle excitations are prohibited since the electron–electron interactions lead to an edge reconstruction and opening of a singleparticle gap^{22,23}. However, unlike it happens in the existing examples, the helical exciton condensate creates longrange correlations between edges. We show that a magnetic field and gate voltages can be used to tune in and out of the exciton condensate phase. Thus this system provides a unique opportunity to study a confinementdeconfinement transition, similar to the one which is hypothesized to liberate the quarks from their color confinement at extremely high temperatures or densities^{24}. Finally, we show that the confined and deconfined phases can be distinguished using nonlocal conductance.
Results
Helical quantum Hall exciton condensate phase
We consider bilayer QSH systems, such as InAs/GaSb^{17,18,19}, described by the BHZ Hamiltonian^{15,17} (Supplementary Note 1). The important property of these systems is that there is a crossing of electron and hole Landau levels as a function of magnetic field at B=B_{cross} (refs 20, 21; Fig. 2a and Supplementary Fig. 1), where the band inversion is removed. Near this crossing the singleparticle Hamiltonian is (Supplementary Note 1):
where is the energy–momentum dispersion of Landau levels and are the electron creation operators for the lowest electron and hole Landau levels. Here we have fixed the total filling factor of the Landau levels , and used the fact that the momentum k in the Landau level wavefunctions is directly connected to the position y in the real space. Importantly, the spin and layer degrees of freedom are locked with each other, so that the pseudospin means simultaneously up (down) spin and upper (lower) layer. The Fermi level is set to be at zero energy.
In the bulk the energy E_{G}(y)=E_{Gb} is independent of the momentum (E_{Gb}<0 for B<B_{cross} and E_{Gb}>0 for B>B_{cross}). When approaching the edge the Landau level originating from the electron (hole) band always disperses upwards (downwards) in energy. The spatial variation of E_{G}(y) occurs within a characteristic length scale , which depends on the details of the edge, but due to topological reasons E_{G}(y)>0 reaches extremely large values (on the order of the energy separation between the bulk Landau levels) close to the edge (Supplementary Note 1). Therefore, for the magnetic fields B<B_{cross}, E_{G}(y) goes through zero near the edge, yielding to the helical edge states (Fig. 2b). On the other hand for B>B_{cross} the edge is gapped in the noninteracting theory (Fig. 2c).
The electron–electron interactions are described by
where is obtained by projecting the Coulomb interactions to the subspace generated by the wavefunctions of the lowest Landau levels (Supplementary Note 2). Here we assumed that the higher Landau levels are energetically separated from the lowest ones by an energy gap larger than the characteristic energy scale of the Coulomb interactions . We find that this assumption can be satisfied with the material parameters corresponding to InAs/GaSb bilayers^{25}.
To find the ground state of the Hamiltonian , we consider states where the local direction of the pseudospin h(r) (h(r)=1) varies in space. Because the Hamiltonian is translationally invariant in the x direction, we assume that h(r) is independent of x (It is known that for sufficiently large interlayer separation the quantum Hall bilayers display an instability towards formation of a charge density wave ground state^{26}. Here we assume that the interlayer separation is small enough that such kind of instability does not occur.). By further noticing that the ydependence translates to a momentum dependence of the pseudospin , we can express our ansatz for the ground state manyparticle wavefunction as a Slater determinant , where for each momentum k we create an electron with pseudospin pointing along . To compute the ground state, we need to minimize the energy functional for such kind of pseudospin texture^{13}. For the energy functional we obtain (Supplementary Note 2)
Here and are the interaction coefficients, which characterize the anisotropy of the interactions within a layer and between the layers.
We start by considering an infinite system. In this case, the pseudospin direction h(r) is spatially homogeneous. By minimizing the energy functional (3), we find that the pseudospin direction h_{b} is determined by the parameters E_{Gb} and (Supplementary Note 3). Here E_{Gb} acts as an effective magnetic field preferring the pseudospin direction along . On the other hand, the interactions prefer the pseudospin directions within the (x,y)plane (), and the energy cost to rotate the pseudospin so that it points along the z direction is proportional to . Thus, as a balance between these two competing effects, the direction of the pseudospin is tilted away from the (x, y)plane, resulting in the three distinct phases of the system, which are summarized in Fig. 3. For sufficiently large E_{Gb}, we see that h_{zb}=1, meaning that only one layer is occupied. The phases h_{zb}=1 (uncorrelated helical QH phase) and h_{zb}=−1 (trivial QH phase) are topologically distinct from each other. For and h_{zb}=1 the system supports helical edge modes (the spinresolved Chern numbers are ). On the other hand, in the regime and h_{zb}=−1, the edge is completely gapped (the spinresolved Chern numbers are ). Between these two phases is the helical QH exciton condensate phase, where h_{zb}<1 and thus . In this phase the direction of the pseudospin projection onto the (x, y)plane is determined spontaneously. Because the pseudospin in this system labels spin and layer index simultaneously, this phase has simultaneously spontaneous inplane spin polarization and spontaneous interlayer phase coherence.
The bulk gap for single particle excitations E_{gap,s} can be calculated using HartreeFock linearization (Supplementary Note 3)
In addition to the single particle excitations, the helical QH exciton condensate supports collective excitations^{13}: the neutral pseudospin waves (Goldstone mode) give rise to spin and counterflow charge superfluidity, and the lowest energy charged excitations are topological pseudospin textures, which carry fractional charge . Here are the pseudospin resolved filling factors of the different Landau levels. The energy required to create these charged excitations is slightly lower than E_{gap,s} (ref. 13).
We point out that although h_{zb}=±1 are topologically distinct phases, the bulk gap for creating charged excitations never closes, when one tunes from one phase to the other by controlling E_{Gb}. This is possible because the pseudospin rotation symmetry, which protects the existence of spinresolved Chern numbers as topological numbers, is spontaneously broken in the helical exciton condensate phase. The interacting BHZ model for bilayers shows somewhat similar behaviour also at zero magnetic field, where a trivial insulator phase can be connected to a quantum spin Hall insulator phase without closing of the bulk gap, because of an intermediate phase where the timereversal symmetry is spontaneously broken^{27}. It is also experimentally known that the exciton condensate phase with can be smoothly connected to uncorrelated QH state with and in conventional QH bilayers^{28}. Experimental investigations of InAs/GaSb bilayers in the QH regime^{18,29} are consistent with this prediction, because no gap closing has been observed as a function of magnetic field.
Confinementdeconfinement transition of edge excitations
We now turn to the description of the ground state pseudospin texture h_{z}(y)=cos[θ_{0}(y)], h_{x}(y)=sin[θ_{0}(y)]cos (ϕ) and h_{y}(y)=sin[θ_{0}(y)]sin(ϕ) at the edge. (The ground state will be degenerate with respect to the choice of ϕ.) As discussed above close to the edge E_{G}(y)>0 takes large values, because the edge states are topologically protected to exist at all energies between the lowest Landau levels and higher ones. Therefore close to the edge θ_{0}(y)=π. On the other hand, in the bulk θ_{0}(y)=θ_{b}=arccos(h_{zb}). This means that there always exists a domain wall, where θ_{0} rotates from π to θ_{b}. Although the existence of the domain wall is a robust topological property of the system, the detailed shape of θ_{0}(y) and the length scale l_{dw}, where this rotation happens, depend on the details of the sample (Supplementary Note 4). The ground states in the uncorrelated helical QH phase (θ_{b}=0) and helical QH exciton condensate phase (θ_{b}≠0, π) are illustrated in Fig. 4a,b, respectively. It turns out that the existence of spontaneous interlayer phase coherence, which distinguishes the two different phases of matter, also has deep consequences on the nature of the lowenergy excitations in this system.
By using a HartreeFock linearization for the ground state, we find that the single particle excitations are gapped also close to the edge, and the magnitude of the energy gap is determined by the Coulomb energy scale . However, similarly as for the case of a coherent domain wall in QH ferromagnetic state in graphene^{23}, the lowest energy edge excitations are not the singleparticle ones. Namely, the ground state is degenerate with respect to the choice of ϕ, and therefore in accordance with the Goldstone’s theorem the system supports lowenergy excitations described by spatial variation of . Due to the general relationship between the electric and topological charge densities in QH ferromagnets^{13,23} (Supplementary Note 5)
these excitations also carry charge, which is localized at the edges of the sample. In this section we illustrate these excitations in a closed system obtained by connecting the ends of the sample to form a narrow cylinder with width W and circumference L (Alternatively, instead of comparing the energies to create elementary excitations in a closed system one could compare the energies needed to create a fixed charge density on the edge. This generalization allows the possibility to consider open systems.). We point out that in addition to the topological contribution (5) there can also be nontopological contributions to the electric charge density of pseudospin textures due to excitation of higher Landau levels. However, these contributions are small for the pseudospin textures which are smooth on the scale of l_{B} (Supplementary Note 8 and Supplementary Fig. 2).
We start by considering this kind of closed system, where (see Fig. 4). This geometry is topologically equivalent to a Corbino ring, which has been experimentally realized for QH exciton condensates^{30,31}. Using equations (3) and (5) we find that the lowest energy excitations correspond to rotation of ϕ(x) by 2π and carry a net charge within one of the edges (Supplementary Note 6). They have an energy in the uncorrelated phase and deep in the helical exciton condensate phase. Here characterizes the cost of exchange energy caused by ∇ϕ(x) (Supplementary Note 4).
In the uncorrelated phase these excitations have a charge ±e. By inspecting Fig. 4, we notice that because the spin points along zdirection in the bulk, there is no rotation happening in the bulk. This means that we can choose separate fields ϕ_{1}(x) and ϕ_{2}(x) for the two edges, so that these excitations can be created independently on the different edges much as in graphene^{23}.
The situation is dramatically different in the helical QH exciton condensate phase. There, the elementary excitations in a closed system have a charge . Moreover, as illustrated in Fig. 4, a charge on one of the edges is always connected to the opposite charge on the other edge by a stripe of rotated bulk pseudospins. Breaking the bulk pseudospin configuration costs an energy comparable to the Coulomb energy, and thus isolated charges cannot be observed at low energies. This means that this type of charged edge excitations in the helical QH exciton condensate phase are confined.
It is illustrative to consider what happens to the excitations in the helical QH exciton condensate, when the width of the sample is increased. Namely, the excitation energy increases proportionally to the width of the sample and eventually for W∼L it becomes energetically favourable instead of having a large area of rotating spins between two edges to create a bulk meron (Fig. 5). This resembles the physics of quarks, where the growing separation of a quarkantiquark pair eventually results in the creation of a new quarkantiquark pair between them.
Luttinger liquid theory and nonlocal transport
To predict experimentally measurable consequences of the charge confinement, we consider nonlocal transport in an open system. By considering the timedependent field theory for the pseudospin for a reasonably narrow sample in the helical QH exciton condensate phase we arrive at an effective onedimensional Hamiltonian (Supplementary Note 7)
where is the pseudospin stiffness and describes the interlayer capacitance per unit area, which is strongly enhanced from the electrostatic value by the exchange interactions. The onedimensional charge densities in the different edges (labelled 1 and 2) are always opposite and determined by a single field ϕ(x), highlighting the confinement of the charged edge excitations. The onedimensional theory describes a Luttinger liquid, and the socalled Luttinger parameter K in the convention used in ref. 32, is given by
The Luttinger parameter in quantum Hall systems determines the conductance for ideal contacts G_{cf}=Ke^{2}/h (ref. 32, 33). Because the pseudospin waves are charge neutral in the bulk, conductance decreases with W as G_{cf}∝1/W. It is important to notice that in this system G_{cf} describes the conductance for a counterflow/drag geometry, where opposite currents are flowing in the two edges. The helical QH exciton condensate phase does not support net transport current as long as the voltages eV are small compared with ħvπ/W (Supplementary Note 7). This automatically leads to a remarkable transport property that characterizes the helical QH exciton condensate phase. Namely, by considering a nonlocal transport geometry shown in Fig. 6a, where a drive current is applied on one of the edges and a resulting drag current is measured on the opposite edge, we find that necessarily I_{drag}=I_{drive} at small voltages. This should be contrasted to the uncorrelated helical phase, where the charged edge excitations are deconfined. In that case, one has two independent Luttinger liquid theories for the two edges (Supplementary Note 7), and therefore one expects only a weak drag current due to the Coulomb force acting between the charges. For , we expect that this effect is negligible compared with drag current in the confined phase.
Finally, to estimate the critical current I_{c}, where the relation I_{drag}=I_{drive} breaks down, we notice that the maximum voltage is determined by the gap eV_{max}≈ħvπ/W. By using reasonable estimates , , θ_{b}=π/2, W=20 l_{B}, l_{B}=10 nm, v=14 km s^{−1} (refs 34, 35), we find I_{c}=G_{cf}V_{max}≈0.1 nA.
Discussion
In summary, we have predicted the existence of a helical QH exciton condensate state in bandinverted electronhole bilayers. We have shown that the counterpropagating edge modes give rise to a ground state pseudospin texture, where the polar angle of the pseudospin magnetization θ(y) rotates from the boundary value π to the bulk value θ_{b} along the direction perpendicular to the edge. Lowenergy charged excitations can be created by letting the azimuthal angle of the pseudospin polarization ϕ(x) to rotate along the edge. Remarkably, in a sufficiently narrow Hall bar these charged edge excitations are confined in the presence of spontaneous interlayer phasecoherence (θ_{b}≠0, π): a charge on one of the edges always gives rise to the opposite charge on the other edge, and thus isolated charges cannot be observed at low energies. Moreover, we predict the possibility to control θ_{b} with a magnetic field and gate voltages. This allows to study a confinementdeconfinement transition, which occurs simultaneously with the bulk phasetransition between the helical QH exciton condensate phase (θ_{b}≠0, π) and the uncorrelated helical QH phase (θ_{b}=0).
The helical QH exciton condensate phase can be experimentally probed using Josephsonlike interlayer tunnelling and counterflow superfluidity^{1,8,9,13,30,31,34,35}. Moreover, because the pseudospin in this system describes simultaneously both the spin and the layer degrees of freedom, the helical QH exciton condensate phase can also be probed using the spin superfluidity and the NMR techniques^{8}. Perhaps it is even possible to use local probe techniques to image the confinementdeconfinement transition and the confinement physics as illustrated in Figs 4 and 5. Finally, we have shown that the charge confinement also gives rise to a remarkable new transport property. Namely, a drive current applied on one of the edges gives rise to exactly opposite drag current I_{drag}=I_{drive} at the other edge.
Our results for the confinement of the edge excitations may also be applicable to the socalled canted antiferromagnetic phase, which is predicted to appear in graphene^{36}. Similarly to the helical QH exciton condensate state considered in this paper, the canted antiferromagnetic phase is characterized by a spontaneously broken U(1)symmetry in the bulk and a single edge supports only gapped meronantimeron excitations^{36}.
The phenomena of confinement stemming from the particle physics models^{24} has been studied also in condensed matter systems^{37,38,39,40,41}. However, we expect that the combination of the different techniques for probing the helical QH exciton condensate phase will provide a more intuitive understanding and new perspectives on the confinement physics.
We also point out that InAs/GaSb bilayers is a promising system for superconducting applications, and edgemode superconductivity has already been experimentally demonstrated in the QSH regime^{42}. In the presence of superconducting contacts, the helical QH exciton condensate may provide a new route for realizing exotic nonlocal Josephson effects and nonAbelian excitations, such as parafermions^{5,6}.
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How to cite this article: Pikulin, D. I. et al. Confinementdeconfinement transition due to spontaneous symmetry breaking in quantum Hall bilayers. Nat. Commun. 7:10462 doi: 10.1038/ncomms10462 (2016).
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Acknowledgements
This work was supported by the Academy of Finland Center of Excellence program, the European Research Council (Grant No. 240362Heattronics), the Dutch Science Foundation NWO/FOM, NSERC, CIfAR, Max Planck—UBC Centre for Quantum Materials, and the DFG grant RE 2978/11.
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Pikulin, D., Silvestrov, P. & Hyart, T. Confinementdeconfinement transition due to spontaneous symmetry breaking in quantum Hall bilayers. Nat Commun 7, 10462 (2016). https://doi.org/10.1038/ncomms10462
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DOI: https://doi.org/10.1038/ncomms10462
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