Interplay of exciton condensation and quantum spin Hall effect in InAs/GaSb bilayers
IInterplay of exciton condensation and quantum spin Hall effect in InAs/GaSb bilayers
D. I. Pikulin and T. Hyart Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
We study the phase diagram of the inverted InAs/GaSb bilayer quantum wells. For small tunnel-ing amplitude between the layers, we find that the system is prone to formation of an s -wave excitoncondensate phase, where the spin-structure of the order parameter is uniquely determined by thesmall spin-orbit coupling arising from the bulk inversion asymmetry. The phase is topologically triv-ial and does not support edge transport. On the contrary, for large tunneling amplitude, we obtain atopologically non-trivial quantum spin Hall insulator phase with a p -wave exciton order parameter,which enhances the hybridization gap. These topologically distinct insulators are separated by aninsulating phase with spontaneously broken time-reversal symmetry. Close to the phase transitionbetween the quantum spin Hall and time-reversal broken phases, the edge transport shows quan-tized conductance in small samples, whereas in long samples the mean free path associated with thebackscattering at the edge is temperature independent, in agreement with recent experiments. PACS numbers: 71.10.Pm, 73.20.-r, 73.63.Hs
Introduction.–
Two-dimensional quantum spin Hall(QSH) insulators are topologically distinguishable fromconventional insulators due to a non-trivial topologicalinvariant arising from band inversion [1–3]. The conduct-ing and valence bands in QSH insulators are connected bygapless helical edge modes, which are protected againstelastic backscattering from time-reversal symmetric per-turbations. Recent experimental advances have revealedtwo materials, HgTe/CdTe [4–6] and InAs/GaSb [7, 8]quantum wells, where the existence of helical edge stateshave been confirmed. In addition to the unique electricalproperties arising due to edge modes [1, 2, 4–8], thesematerials in proximity to superconductors are interest-ing as a platform for Majorana zero-modes [9, 10] andflux-controlled quantum information processing [11, 12].The recent observation of the QSH effect in InAs/GaSbbilayers [8] is theoretically puzzling, because conductancequantization was found up to magnetic field on the orderof 10 T in short samples. On the other hand, even inthe absence of magnetic field the longitudinal resistancein long samples increased linearly with the device length.While inelastic processes can in principle give rise to afinite mean free path associated with the backscatteringat the edge, the existing theoretical models [13–20] donot explain the observation that the mean free path wasfound to be temperature independent at least for a tem-perature range 20 mK - 4.2 K [8, 21, 22]. Therefore, weexpect that non-perturbative effects due to disorder orinteractions beyond the existing approaches are impor-tant. In particular, the temperature-independent meanfree path indicates that the dominating backscatteringprocess might be an elastic one, which is allowed if time-reversal symmetry is either dynamically or spontaneouslybroken. While mechanisms resulting in dynamical time-reversal symmetry breaking have been proposed [23, 24],it is unlikely that they could account for the experimen-tally observed mean free path ∼ µ m.In this Letter, we consider the influence of exciton condensation on the QSH effect in InAs/GaSb bilayers.While exciton condensation is theoretically predicted ininverted type II electron-hole bilayers [25–31], such asInAs/GaSb quantum wells, an unambiguous observationof a thermodynamically stable exciton condensate phasein these systems is a long-standing problem. Indeed, sofar the best studied exciton condensate phase is the quan-tum Hall bilayer state at half-filled Landau levels [32, 33],where the ability to separately contact the two layers hasallowed to probe the order parameter in terms of coun-terflow superfluidity along the layers and Josephson-liketunneling between the layers [34–38]. Here, we go be-yond the earlier theoretical models for exciton conden-sates [25, 28–31] by studying the spin-structure of theorder parameter, when the relevant spin-orbit and tun-neling terms for the InAs/GaSb bilayers are taken intoaccount. For small tunneling amplitude we find a topo-logically trivial s -wave exciton condensate phase, whereasfor relatively large tunneling we obtain a topologicallynon-trivial QSH insulator phase. These topologically dis-tinct insulators are separated by an insulating phase withspontaneously broken time-reversal symmetry. Close tothe phase transition between the QSH and time-reversalbroken phases, the conductance is quantized in smallsamples, whereas the mean free path in long samples istemperature-independent for a wide range of tempera-ture, in agreement with the recent experiments [8]. Model.–
We consider a bilayer electron-hole systemsdescribed by Hamiltonian ˆ H = ˆ H + ˆ H I , where ˆ H issingle particle Hamiltonian for the InAs/GaSb quantumwells (see below) and the Coulomb interaction betweenthe electrons is described by the Hamiltonianˆ H I = 12 (cid:88) a,a (cid:48) ,s,s (cid:48) (cid:88) k , k (cid:48) , q V aa (cid:48) ( q ) c † k sa c † k (cid:48) s (cid:48) a (cid:48) c k (cid:48) + q s (cid:48) a (cid:48) c k − q sa , (1)where V aa ( q ) = e F aa ( q ) / (2 (cid:15)(cid:15) L q ), V ( q ) = V ( q ) = e F ( q ) e − qd / (2 (cid:15)(cid:15) L q ) and F ab are the structure factors a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y for the layers of thicknesses W , , see [29].By performing a mean field approximation for the in-teraction term ˆ H I and including the terms of the single-particle Hamiltonian ˆ H [3], we arrive to a mean-fieldHamiltonianˆ H mf = (cid:88) k ψ † k H ( k ) ψ k , H ( k ) = (cid:18) H ( k ) H ( k ) H † ( k ) H ( k ) (cid:19) , (2)where the Hamiltonian for each layer is given by H ( k ) = (cid:20) (cid:126) k m e − E G + (cid:15) mf1 ( k ) − µ (cid:21) σ − (cid:88) i =1 h mf1 ,i ( k ) σ i +∆ e ( k x σ − k y σ ) + ξ e ( k y σ − k x σ ) ,H ( k ) = (cid:20) E G − (cid:126) k m h + (cid:15) mf2 ( k ) − µ (cid:21) σ − (cid:88) i =1 h mf2 ,i ( k ) σ i +∆ h ( k x σ + k y σ ) (3)and the coupling between the layers is described by H ( k ) = A ( k x σ + ik y σ ) − i ∆ z σ − ∆ mf ( k ) . (4)Here E G is the inverted band gap, µ is the chemical po-tential and m e ( h ) are the effective masses. Because ofthe s - and p -like natures of the conduction and valencebands, respectively, the tunneling term A ( k x σ + ik y σ )must be odd in momentum. The spin-orbit couplings ∆ e ,∆ h and ∆ z arise due to bulk inversion asymmetry, and ξ e is the Rashba coupling. The mean field potentials shouldbe solved self-consistently from equations [31] (cid:15) mf a ( k ) = − (cid:88) s, k (cid:48) V aa ( k − k (cid:48) )[ ρ aass ( k (cid:48) ) − ρ aa ( k (cid:48) )] , (5) h mf a ( k ) = 12 (cid:88) s,s (cid:48) , k (cid:48) V aa ( k − k (cid:48) ) ρ aass (cid:48) ( k (cid:48) ) (cid:126)σ s,s (cid:48) (6)and ∆ mf s,s (cid:48) = (cid:88) k (cid:48) V ( k − k (cid:48) ) ρ s (cid:48) s ( k (cid:48) ) . (7)Here ρ aa (cid:48) ss (cid:48) ( k ) = (cid:104) c † k sa c k s (cid:48) a (cid:48) (cid:105) is the Hartree-Fock densitymatrix and ρ aa ( k ) is the density matrix for full valenceband in the hole layer and empty conduction band in theelectron layer [31]. The mean field potentials (cid:15) mf a ( k ) de-scribe the renormalization of the band structure, whereas h mf a ( k ) can account for spontaneous magnetization andthe renormalization of the spin-orbit couplings. For ourpurposes, the most interesting mean field potentials are∆ mf ( k ) = (cid:80) i =0 ∆ mf i ( k ) σ i , which describe the full spinstructure of the exciton condensate order parameter.The natural length d and energy E scales of the prob-lem can be determined from the relation E = ( m − e + m − h ) (cid:126) / d = e / (4 π(cid:15)(cid:15) d ) . For InAs/GaSb bilayers typical parameters in the regime of band inversion areexpected to be E /k B ∼
100 K, d ∼
10 nm, m e /m h ∼ A/ ( E d ) ∼ .
1, ∆ z /E ∼ . ξ e / ( E d ) ∼ − . d/d (cid:28) W a /d ∼ e , ∆ h ∼ .
001 [39]. Theparameters E G and µ describe the densities in the layers,and can be controlled with gate voltages. The tunnelingterms A and ∆ z are exponentially sensitive to width andheight of an insulating barrier between the layers. Results.–
For A = ∆ z = ∆ e = ∆ h = ξ e = µ = 0,the only non-zero mean-field potentials are (cid:15) mf ( k ) and∆ mf ( k ). The main effect of (cid:15) mf ( k ) is the renormaliza-tion of E G to E RG . Because the densities are controlledby the gate voltages, we express our results in terms of E RG . For realistic densities of electrons and holes the sys-tem undergoes a second order phase transition as func-tion of temperature, and below the critical temperature T c ∼ . E /k B an s -wave exciton condensate order pa-rameter ∆ mf ( k ) appears due to spontaneous symmetrybreaking. Because the Hamiltonian has lot of symmetriesin the absence of tunneling and spin-orbit couplings, alltime-reversal symmetric s -wave order parameters ∆ mf ( k )with equal total magnitude (cid:80) i =0 | ∆ mf i ( k ) | are degener-ate solutions of problem.Two most important parameters concerning the exci-ton order parameter are the tunnel couplings A and ∆ z ,because they act as a symmetry breaking terms in theHamiltonian turning the second order phase transition tothe exciton condensate phase into a crossover. It is intu-itively clear, that ∆ z and A favor an even parity excitoncondensate order parameter i ∆ s σ and an odd-parity ex-citon order parameter − ∆ p ( k x σ + ik y σ ) (∆ s , ∆ p ∈ R ),respectively. Therefore, there is a competition betweeneven and odd parity exciton condensate which can bedescribed by studying the parity of order parameter P = ∆ eventot − ∆ oddtot ∆ eventot + ∆ oddtot , (8)where ∆ eventot = (cid:113)(cid:82) dk k (cid:80) i | ∆ mf i, ( k ) | and ∆ oddtot = (cid:113)(cid:82) dk k (cid:80) i,n = ± | ∆ mf i,n ( k ) | are obtained using expan-sion ∆ mf i ( k ) = (cid:80) n ∆ mf i,n ( k ) e inθ k in terms of azimuthalangle θ k in momentum space. We obtain the phase dia-gram in parameters A and E RG as they can be changed ina controlled way in the experiments. For small A we finda topologically trivial s -wave exciton condensate phase,whereas for large A we obtain a topologically non-trivialQSH insulator phase with a p -wave exciton order param-eter [Fig. 1(a)], in agreement with our expectations. In-terestingly, we find that these phases are separated by aninsulating phase with spontaneously broken time-reversalsymmetry. This time-reversal symmetry broken phase isshown in Fig. 1(b), where we have characterized the time-reversal symmetry breaking with a parameter T br = ∆ brtot ∆ tstot + ∆ brtot . (9) (a) -11 (b) (c) Figure 1. (a) Parity of order parameter P as a function of E RG and A for ∆ z /E = 0 . m e /m h = 1, W , /d = 0 . µ = d = ∆ e = ∆ h = ξ e = 0. (b) The time-reversal symmetrybreaking order parameter T br as a function of E RG and A forthe same parameters. (c) Line cut showing P and T br as afunction of E RG for A/E d = 0 .
06. The insulating phase withbroken time-reversal symmetry is separated from the trivialand QSH insulators by second order phase transitions.
Here the relative strength of the order parameter obey-ing the time-reversal symmetry is defined as (∆ tstot ) = (cid:82) dk k (cid:26) ( (cid:60) ∆ mf0 , ) + ( (cid:60) ∆ mf0 , −(cid:60) ∆ mf0 , − ) + ( (cid:61) ∆ mf0 , + (cid:61) ∆ mf0 , − ) + (cid:80) i =1 (cid:20) ( (cid:61) ∆ mf i, ) + ( (cid:60) ∆ mf i, + (cid:60) ∆ mf i, − ) + ( (cid:61) ∆ mf i, −(cid:61) ∆ mf i, − ) (cid:21)(cid:27) and the strength of time-reversal symmetry-breaking∆ brtot can be calculated by interchanging the real (cid:60) ∆ mf i,n and imaginary parts (cid:61) ∆ mf i,n of the order parameter in thisequation [42]. Second order phase transitions are clearlyseen at the two boundaries of the time-reversal symme-try broken phase [Fig. 1 (c)]. We have numerically con-firmed that our results are valid for chemical potentials | µ | /E (cid:46) .
05. With increasing | µ | the difference betweenthe densities of the electrons and holes increases, and thepreference for Fermi surface nesting gives rise to magne-tization [31, 40, 41]. Our results are robust against in-cluding small spin-orbit coupling ξ e / ( E d ) = − .
07 andasymmetry of effective masses m e /m h = 0 .
84 that areexpected to be present in InAs/GaSb bilayers [39], butthe locations of the phase boundaries depend stronglyon the parameter ∆ z [43]. This situation should be con-trasted to HgTe/CdTe QSH insulators described by thesame Hamiltonian [2], where the exciton condensationdoes not give rise to phase transitions, because the con-duction and valence bands are localized in the same quan-tum well so that A is an order of magnitude larger thanin InAs/GaSb bilayers [39].Our numerical results can also be interpreted in thelight of Ginzburg-Landau theory, which is obtained byexpressing the exciton order parameter as ∆ mf ( k ) = i ∆ s e iφ s σ − ∆ p e iφ p ( k x σ + ik y σ ) and expanding thefree-energy perturbatively using the tunnel couplings andexciton order parameter as a perturbation. We find,that similarly as in Ref. [30], the lowest order terms inthe free-energy are proportional to − ∆ z ∆ s cos φ s and − A ∆ p cos φ p favoring the order parameter with φ s = φ p = 0. On the other hand, the fourth order expan-sion contains terms which are proportional to differentcombinations of ∆ z , A , ∆ s and ∆ p . These terms try totwist the phases of the order parameters φ s and φ p awayfrom zero, and are thus responsible for the spontaneousbreaking of the time-reversal symmetry in the regime ofthe phase diagram where both s - and p -wave order pa-rameters are simultaneously large. Transport.–
To calculate the influence of the excitoncondensation on the transport properties of the QSHinsulator, we perform a k · p expansion of the meanfield potentials, and calculate the conductance using atight-binding Hamiltonian constructed from the result-ing continuum model. The band structures in differentparameter regimes are shown in Fig. 2 and the resultsfor the disorder-averaged differential conductance (cid:104) G (cid:105) inFig. 3. In the QSH regime, the edge states are protectedfrom elastic back-scattering, and therefore we find perfectconductance quantization for all disorder strengths V dis shown in the figure. On the other hand, in the regime ofweakly-broken time-reversal symmetry the Born approx-imation gives a mean free path (cid:96) = a (cid:126) v k F / ( V ξ ∆ ),where we have assumed uncorrelated disorder potential (cid:104) V ( x ) V ( x (cid:48) ) (cid:105) = V ξδ ( x − x (cid:48) ) along the edge, ∆ br is theenergy gap in the edge state spectrum due to the time-reversal symmetry breaking order parameter, and a ∼ (cid:104) G (cid:105) are in good agreement with the Born approxi-mation (Fig. 3). Importantly, the mechanism of elasticscattering due to the spontaneous breaking of the time-reversal symmetry remains effective for T (cid:28) T c . Since (a) (b) (c) (d) Figure 2. Band structures for (a) E RG = 0 . E , (b) E RG = 0 . E , (c) E RG = 0 . E and E RG = 1 . E and the other parameterssame as in Fig. 1(c). Protected helical edge states appear only in the QSH phase. By decreasing E RG , the time-reversal symmetrybreaking opens a gap in the edge state spectrum, and finally the edge states disappear when one approaches the trivial phase. QSH regime
Figure 3. Disorder-averaged differential conductance (cid:104) G (cid:105) asa function of V dis for a device with length L = 100 d anddifferent values of E RG and voltage eV . In the QSH regime E RG = 1 . E , the conductance is quantized to G = 2 G ( G = e /h ). In the regime of weakly-broken time-reversalsymmetry E RG = 0 . E , numerically calculated (cid:104) G (cid:105) (thicklines) are in agreement with G = 2 G (1 − L/(cid:96) ) (dashed lines),where (cid:96) is obtained from Born approximation with a fittingparameter a = 3 .
1. The other parameters same as in Fig. 1(c). typically T c ∼
10 K, and we estimate that (cid:96) ∼ µ m al-ready for a reasonably weak disorder, we conclude thatthis mechanism is a viable candidate for the explanationof the temperature-independent mean free path observedin recent experiments [8, 21, 22]. We also predict thatthe resistance is peaked at the crossing point of the edgestate spectrum – in agreement with the recent experi-ment [21], where the maximum resistance was observeddeep inside the topological gap. Summary and discussion.–
In summary, we have stud-ied the exciton condensation in inverted electron-hole bi-layers, where the s -like conduction band and p -like holeband are localized in different quantum well layers. Wehave calculated the phase diagrams, which show compe-tition between a topologically trivial s -wave exciton con-densate phase and a non-trivial QSH phase. These topo-logically distinct phases are separated by an insulatingphase with spontaneously broken time-reversal symme-try, which is energetically favoured, because it keeps thesystem gapped when it experiences a transition between the topologically distinct insulators. Our results can ex-plain the unexpected temperature-independent mean freepath observed in InAs/GaSb bilayers [8].We also point out that a more detailed experimentalstudy can confirm that the backscattering at the edgehappens due to the spontaneous time-reversal symmetrybreaking. The phase diagram we discuss can be studiedas function of the tunneling amplitudes by controllingthe width and height of an insulating barrier betweenthe layers [33], and the exciton order parameter can beprobed via the collective modes and vortex excitations.In quantum Hall bilayers, the exciton order parameterhas been studied in terms of counterflow superfluidityand Josephson-like tunneling anomaly [34–38], and therethese properties are known to be strongly influenced bydisorder-induced fractionally charged vortices [45–56]. Acontrollable way to open an energy gap in the edge statespectrum by tuning the gate voltage may also be use-ful for studying Majorana zero modes and for electronicapplications. Acknowledgements.–
The conductance and band struc-tures were calculated using the kwant [57]. Wehave benefited from discussions with I. Knez andC. W. J. Beenakker. This work was supported bythe Foundation for Fundamental Research on Matter(FOM), the Netherlands Organization for Scientific Re-search (NWO/OCW), and the European Research Coun-cil.
Note added.
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Then, we summarize these results in a phase-diagram, which contains the different phases we found in the numerical calculations.For the sake of transparency, we use here our numerical observation that the order parameter can be written as∆ mf ( k ) = i ∆ s e iφ s σ − ∆ p e iφ p ( k x σ + ik y σ ) , (10)where ∆ s , ∆ p ∈ R , and we neglect the asymmetry of effective masses m e /m h = 1 and the spin-orbit couplings ∆ e ,∆ h and ξ e . All these assumptions can be numerically justified, because for realistic values of these parameters thephase-diagram does not change qualitatively. With these assumptions, the mean-field Hamiltonian for µ = 0 can bewritten as H = (cid:18) (cid:126) k m − E RG (cid:19) s σ + (∆ z + ∆ s cos φ s ) s σ + ( A + ∆ p cos φ p ) (cid:2) s σ k x − s σ k y (cid:3) +∆ s sin φ s s σ + ∆ p sin φ p (cid:2) − s σ k x − s σ k y (cid:3) , (11)where the pauli matrices s i and σ i describe the layer and the spin degrees of freedom, respectively.The time-reversal symmetry operator is defined as T = is σ K , where K is the complex conjugation operator. Itis easy to see by straightforward calculation that all the terms in the first line of Eq. (11) obey the time-reversalsymmetry and all the terms in the second line break it. Therefore, all values of phases φ s , φ p (cid:54) = 0 , π result inspontaneous time-reversal symmetry breaking. On the other hand, using the definition of the order parameter T br given in the main text and the exciton order parameter given by Eq. (10), we notice that T br = ∆ brtot ∆ tstot + ∆ brtot (∆ tstot ) = (cid:90) dk k (cid:26) (∆ p k cos φ p ) + (∆ s cos φ s ) (cid:27) (∆ brtot ) = (cid:90) dk k (cid:26) (∆ p k sin φ p ) + (∆ s sin φ s ) (cid:27) . (12)Clearly, T br (cid:54) = 0 if and only if φ s , φ p (cid:54) = 0 , π , and therefore T br can be used as an order parameter for description ofthe spontaneous time-reversal symmetry breaking.We now explain the connection between the order parameter P and Z topological invariant of the two-dimensionaltime-reversal symmetric insulators. The Z topological invariant is only well-defined in the presence of time-reversalsymmetry. Therefore, we set set φ s = φ p = 0, so that the Hamiltonian simplfies to a form H = (cid:18) (cid:126) k m − E RG (cid:19) s σ + (∆ z + ∆ s ) s σ + ( A + ∆ p ) (cid:2) s σ k x − s σ k y (cid:3) . (13)Furthermore, the order parameter P defines a relation between ∆ s and ∆ p as P = ∆ eventot − ∆ oddtot ∆ eventot + ∆ oddtot , (∆ eventot ) = (cid:90) dk k ∆ s (∆ oddtot ) = (cid:90) dk k (∆ p k ) . (14)These integrals seem to diverge. However, this is only because we have used the low- k expansion of the orderparameter instead of taking into account the full momentum dependence of ∆ s ( k ) and ∆ p ( k ). The relevant scalewhere the momentum dependent order parameters change must be determined by the parameters k F = (cid:113) mE RG / (cid:126) Figure 4. Phase-diagram for Hamiltonian (11) as a function of order parameters P and T br . The topological transitionbetween quantum spin Hall and trivial insulators happens at P = P cr and T br = 0. For T br (cid:54) = 0 the time-reversal symmetry isspontaneously broken. For typical parameters E RG = 0 . E , A = 0 . E /d , ∆ z = 0 . E and ∆ s + ∆ p k F = 0 . E , the criticalpoint is given by P cr ≈ . and (cid:126) /d , which are approximately equal to each other for typical parameters considered in the main text. Therefore,in order to obtain transparent expression for P we use √ k F as cut-off momentum in Eqs. (14). This way we obtain P = ∆ s − ∆ p k F ∆ s + ∆ p k F . (15)In order to describe the connection between the topological invariant, we need to also consider the other parameters E RG , A and ∆ z and the overall magnitude of the exciton order parameter ∆ s + ∆ p k F . We fix them to the typicalvalues considered in the main text E RG = 0 . E , A = 0 . E /d , ∆ z = 0 . E and ∆ s + ∆ p k F = 0 . E , and calculatethe Z topological invariant as a function of P for Hamiltonian (13). The results are summarized in Fig. 4, and can beunderstood by repeatedly utilizing the powerful result that Z topological invariant can only change when the energygap of the system closes.First, we notice that for P = 1 the s -wave exciton order parameter strongly dominates the tunneling terms and the p -wave exciton order parameter. Therefore, in this case it is possible to adiabatically set A = ∆ p = 0 without closingthe energy gap. By inspecting the Hamiltonian (13) in this limit, we notice that it is topologically indistinguishablefrom the usual BCS s-wave superconductor, which is well-known to be topologically trivial i.e. it does not supportedge states.Second, in the opposite limit P = − s = ∆ z = 0 without closing the energy gap. In this limit the system is described by the Bernevig-Hughes-Zhang Hamiltonian in the topologically nontrivial phase.Finally, we find that if we continuously change P from 1 to − P , where the energygap closes. This means that the Z topological invariant must change at this point, and we have obtained the fullphase-diagram shown in Fig. 4. Phase diagrams for different ∆ z The two most important parameters concerning the exciton order parameter are the tunnel couplings A and ∆ z ,because they act as symmetry breaking terms in the Hamiltonian, turning the second order phase transition to theexciton condensate phase into a crossover. As we explained in the main text, these parameters fix the spin structureof the exciton order parameter. However, additionally, they control the phase-boundaries between the different typesof exciton condensate phases. In the main text, we showed the phase diagram of our model for ∆ z /E = 0 . z affects thephase-boundaries between these phases.The phase-diagrams for different values of ∆ z are shown in Fig. 5. Based on these figures we conclude that increasing∆ z shifts the phase-boundaries to larger values of A as one would have expected. However, this is a reasonably weakeffect as long as ∆ z (cid:28) k B T c ∼ . E , because in this limit the higher order terms in exciton order parameters ∆ s and ∆ p in the Ginzburg-Landau theory dominate effect of tunneling. Secondly, the area of the time-reversal brokeninsulating phase shrinks with increasing ∆ z . We can understand this in the framework of Ginzburg-Landau theory "data_35_parity.dat" -11 (a) "data_35_tr.dat" (b) -0.81 (c) (d) Figure 5. Parity of order parameter P and the time-reversal symmetry breaking order parameter T br the as a function of E RG and A for m e /m h = 1, W , /d = 0 . µ = d = ∆ e = ∆ h = ξ e = 0, and different values of ∆ z . (a),(b) ∆ z /E = 0 .
01 and(c),(d) ∆ z /E = 0 . by noticing that the terms − ∆ s ∆ z cos φ s and − A ∆ p cos φ p try to pin the phases of the s- and p-wave exciton orderparameters to zero, and thus for large enough ∆ z and A it becomes energetically less favorable for the system tospontaneously break the time-reversal symmetry. Temperature dependence of the conductance
In this section we calculate the temperature dependence of the conductance given by elastic backscattering due tospontaneous time-reversal symmetry breaking.As found in the main text the (disorder-averaged) energy-dependent transmission for short samples is given by T ( E ) = (1 − L/(cid:96) ( E )) , (16)where (cid:96) ( E ) = 4 a (cid:126) v E ξV ∆ , (17) E is the energy with respect to the crossing of the edge state spectrum, and L is the length of the sample.Temperature-dependent (disorder-averaged) differential conductance can be obtained from G ( V, T ) = 2 G (cid:90) + ∞−∞ dE k B T E − eV k B T T ( E ) (18)Then, by assuming k B T (cid:28) eV , we get G ( V, T ) = 2 G (cid:18) − ξV ∆ L a (cid:126) v e V − π ξV ∆ Lk B T a (cid:126) v e V (cid:19) = G − ∆ G ( T = 0) − ∆ G ( T = 0) (cid:18) πk B TeV (cid:19) . (19)This shows for short samples the temperature independent part of the backscattering dominates at low temperatures,and temperature-dependent corrections can be neglected as long as k B T (cid:28) eV .In the experiments, the mean free path is measured using long samples. In this case, the Coulomb interactions giverise to an inelastic mean free path and the distribution functions of the electrons have to be calculated from a solutionof a kinetic equation [1]. We expect that the temperature dependence at small temperatures is further suppressed inlong samples, because of the Joule heating. [1] D. A. Bagrets, I. V. Gornyi, and D. G. Polyakov, Phys. Rev. B80