Irradiated three-dimensional Luttinger semimetal: A factory for engineering Weyl semimetals
IIrradiated three-dimensional Luttinger semimetal: A factory for engineering Weylsemimetals
Sayed Ali Akbar Ghorashi, Pavan Hosur, Chin-Sen Ting
Texas Center for Superconductivity and Department of Physics,University of Houston, Houston, Texas 77204, USA
We study the interaction between elliptically polarized light and a three-dimensional Luttingersemimetal with quadratic band touching using Floquet theory. In the absence of light, the touchingbands can have the same or the opposite signs of the curvature; in each case, we show that simplytuning the light parameters allows us to create a zoo of Weyl semimetallic phases. In particular,we find that double and single Weyl points can coexist at different energies, and they can be tunedto be type I or type II. We also find an unusual phase transition, in which a pair of Weyl nodesform at finite momentum and disappear off to infinity. Considering the broad tunability of lightand abundance of materials described by the Luttinger Hamiltonian, such as certain pyrochloreiridates, half-Heuslers and zinc-blende semiconductors, we believe this work can lay the foundationfor creating Weyl semimetals in the lab and dynamically tuning between them.
Introduction .- Topological phases of matter have at-tracted tremendous interest since the discovery of topo-logical insulators. Topological protection of their edgeand surface states is the hallmark of these systems, andleads to applications ranging from quantum computationto robust transport and exotic superconductivity [1, 2].In contrast to topological insulators, which are gappedphases of matter like most topological phases, it hasbeen shown recently that gapless phases of matter can betopological as well [3–9]. Among them, Weyl semimetals(WSMs) have been particularly attractive due to theirunconventional properties such as the chiral anomaly[11, 12], negative magnetoresistance [10, 11] and anoma-lous Hall effect [5, 8]. Experimental observation of thesephases in TaAs [11, 13, 14] and photonic crystals [15] hasignited further interest in exploring these systems.Very recently, new types of WSMs, namely, type-IIand multi-WSMs were also discovered [16, 18–20]. Thedefining feature of type-II Weyl points is that the disper-sion around them is strongly anisotropic, such that theslope changes sign along some directions. As a result, theWeyl nodes become the touching points between electronand hole Fermi surfaces, and result in properties differentfrom those of type–I WSMs. For example, there are indi-cations that the chiral anomaly depends on the relativedirection of the magnetic field and the tilt of the cone,but the issue is still under debate [16, 17]. Moreover,unlike in type-I WSMs, the anomalous Hall effect cansurvive in type-II WSMs under certain conditions evenwhen the nodes are degenerate [21]. On the other hand,multi-WSMs occur when the monopole charges of Weylpoints are higher than 1, and can be either type-I or type-II [18–20]. In general, the search for Weyl semimetallicphases has been a vigorous field of research lately, andproposals have been put forth to engineer these phasesin a tunable way by shining light on Dirac semimetals[28, 29], band insulators [30], stacked Graphene [31], line-nodal semimetals [29, 32] and crossing-line semimetals[25, 26]. Finally, proposals have been made to create tunable WSMs in pyrochlore iridates with Zeeman fields.[43, 52].
Phase III - ( ) &2 TPD
Phase I - ( ) &2 Critical line ( Isotropic limit ) P ha s e I l P ha s e I l A x A y FIG. 1: Phase diagram for 3D Floquet Luttinger semimetal.Critical line (Isotropic limit): diagonal red line with twolower ( w l ) and two higher ( w h ) Weyl points on the k z axis;phase I: 4 w l + 2 w h , blue, where for bands bending oppo-sitely (similarly), the 4 w l are type-I (type-II) denoted by phaseI − w h + 2 w l , orange,where for bands bending oppositely (similarly), the 2 w l aretype-I (type-II) and denoted by phaseIII − k z -direction ( k x − k y plane) forbands bending in opposite (same) directions, as well as merg-ing and splitting of lower and upper nodes in k y − k z and k x − k z planes, respectively, occur. ”TPD” denotes the triplydegenerate point, which exists only for circular light. In this work, we expand the horizons for creating tun-able WSMs, by computing the band structure of a three-dimensional Luttinger semimetal with quadratic band a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n touching irradiated by elliptically polarized light usingFloquet theory. We find Weyl nodes of different charges( ± ± k . In a reg-ularized lattice model, this pair would form at the edge ofthe Brillouin zone. Crucially, given the bare band struc-ture, all these phases can be accessed by simply changingthe properties of the light, making this system highly tun-able. Fig. 1 summarizes the results of this paper. Weexpect these results to hold for real systems described bythe Luttinger Hamiltonian [33], such as the zinc-blendsemiconductors GaAs, HgTe, α -Sn etc. and a class ofpyrochlore iridates [34–37] studied recently. Model and Formalism .- We begin with an isotropic ver-sion of the Luttinger Hamiltonian [33], H = 12 (cid:90) k c † ( k ) (cid:18) ( λ + 52 λ ) k − λ ( J . k ) − µ (cid:19) c ( k ) , (1)where λ , are positive constants, k = { k x , k y , k z } , c ( k ) = ( c / k , c / k , c − / k , c − / k ) T , J = { J x , J y , J z } are effective spin-3/2 operators, and c m k denotes afermion annihilation operator with momentum k and J z quantum number m . The energy dispersions are E ( k ) = ( λ ∓ λ ) k − µ for the j = 3 / j = 1 / λ < λ ( λ > λ ), as depicted in Fig. 2. When bothbands bend the same way, Eq. (1) is widely used tomodel heavy- and light-hole bands in zinc-blende semi-conductors [37]. Many properties of such a dispersionhave been studied in the literature, including a recentstudy on the realization of fully gapped topologicalsuperconductivity with p -wave pairing which has stateswith exotic cubic and linear dispersions coexisting onthe surface [38, 39]. On the other hand when bandsbend oppositely, the above model is relevant for certainpyrochlore iridates as well as for some doped half-Heusleralloys such as LaPtBi [40–42]. Various aspects of thisscenario have been explored as well, such as the phasediagram in the presence of electronic interactions [43],the effect of anisotropy [44] and superconductivity[45, 46]. Systems with higher effective spins and windingnumbers have also attracted interest in the contextof multi-weyl phases [47] and the investigation of thespin quantum Hall plateau transition on the surfaceof topological superconductors with general windingnumbers [48]. - - - - - kz E ( k z ) (a) - - - kz E ( k z ) (b) FIG. 2: Energy dispersion of Eq. (1) for (a) λ = 0 . λ = 0 . J = 3 / J = 1 / λ = 1 . λ = 0 . Here, we study another aspect of this model. Byemploying machinery from Floquet theory, we inves-tigate light-matter interactions in this model in bothband-bending scenarios. We consider periodic drivinginduced by laser light with a general vector potential A ( t ) = ( A x cos( ωt ) , A y η sin( ωt ) ,
0) , where η = ± A i ∝ E i /ω , where E i is its electric field. The time-dependent Hamilto-nian can be written as H ( k , t ) = (cid:80) n H n ( k ) e inωt , where H ± n ( k ) = T (cid:82) T H ( k , t ) e ± inωt .The effective time-independent Hamiltonian in the highfrequency limit, as dictated by Floquet theory, is [24, 27], H eff ( k ) = H + (cid:88) n ≥ [ H + n , H − n ] nω + O (cid:18) ω (cid:19) . (2)where, H = ( λ + 52 λ ) k . A − λ { J . k , J . A } (3) H = 14 (( λ + 52 λ ) A − λ ( J . A ) ) (4) H − n = H † n , (5)and A = ( A x , iηA y , γ = λe E / (cid:126) ω , where λ iseither λ or λ which are of the same order of magnitudeand have units of inverse mass, E is the magnitude ofthe electric field of the incident light and c is the speedof light in the medium. Clearly, γ (cid:28) e = (cid:126) = 1. We first analyze the limit ofcircular polarization, which is the only case can be fullystudied analytically. Then, we analyze the general caseof elliptical polarization. Circularly polarized light .- Since H is quadratic in k , H n = 0 for n > n = ± k -dependent terms arising from n = ± A is insuf-ficient, and it is necessary to go to a higher order. For cir-cularly polarized light, rotational symmetry ensures thatWeyl points appear only on the k z axis, which makesextracting the salient features of the model analyticallypossible. For k x = k y = 0, the effective Hamiltonianreads H eff ( k z ) = H ( k z )+ 2 iηA λ ω (cid:18) − k z [ { J x , J z } , { J y , J z } ](6)+ A J y − J x , { J x , J y } ] (cid:19) , (7)with dispersions of E , ± = ( λ + 2 λ ) k z ± (3 A λ η ( A − k z )) / ω − µ and E , ± = ( λ − λ ) k z ± (3 A λ η ( A +8 k z )) / ω − µ . Note that introduction of circularly polar-ized light has broken time-reversal symmetry and liftedthe double degeneracy of the bands. Inversion symmetrysurvives, though, because only even powers of the lightamplitude enter H eff . The four non-degenerate bandsintersect in various pairs, giving rise to Weyl nodes at (cid:126)K = (0 , , ± A/ √
2) and (cid:126)K = (0 , , ∓ A (cid:112) λ /ω/ H k ∝ n ( k ) · σ and using, W n = (cid:90) S d k (cid:15) ijk n . ( ∂ j n × ∂ k n ) (8)where n is a unit vector and the integration is over asurface S surrounding the node. We obtain W n = ± W n = ± K and K respectively. This is aremarkable result, that single and double-Weyl nodescoexist at different energies, thus allowing us to accessboth dynamically by tuning the chemical potential. Asis clear, the positions of single Weyl points are only afunction of the light parameters while the locations of thedouble-Weyl points also depend on the band structureparameter, λ . Moreover, for circularly polarized light,there is a special point in parameter space, namely, A m = ± (cid:112) ω/ λ where the two types of nodes mergeand form a triply degenerate point (TDP).Fig. 3(a-d) shows the evolution of the band struc-ture with the light intensity, for a representative set ofparameters with ηλ > A x = A y line in Fig. 1. Two pairs of nodes appear(Fig. 3a) as soon as light is turned on. The nodes higher(lower) in energy are type-II (type-I), have monopolecharge ± ±
2) and occur at K ( K ). On increasing A ,the lower nodes flatten along k z (not shown) and tran-sition into type-II nodes, before merging with the uppernodes at the TDP at A = A m (Fig. 3b). On furtherincreasing A , the bands cross, and the charge ± -π π/2 0 3π/2 π - - k z E ( k z ) -π π/2 - - k z E ( k z ) E + E - E - E + -π π/2 0 3π/2 π - - k z E ( k z ) -π π/2 0 3π/2 π - - k z E ( k z ) (a) (b)(c)(d) -π π/2 0 3π/2 π - - k z E ( k z ) -π π/2 0 3π/2 π - - k z E ( k z ) -π π/2 0 3π/2 π - - k z E ( k z ) -2π π 0 π 2π - k z E ( k z ) (e) (f)(g) (h) FIG. 3: Evolution of the Weyl nodes with light amplitude A . (a)-(d) show A = 2 , . , . A = 2 , . , . λ = 0 . , λ = 0 . λ = 1 . , λ = 0 .
5) are usedfor bands bending oppositely (similarly). ω = 20, µ = 0 and η = 1 is used for all of the plots. end up being higher in energy than the charge ± ± ±
2) for the upper (lower)nodes for low intensity, and the correspondence gets re-versed as A is tuned across the TDP.Fig. 3(e-h) show the evolution when the bare bandsbend the same way. It shows the same trend as the casewhere the bare bands bend oppositely, except that all thenodes are type-II. Moreover, there is a type-II to type-Itransition at high intensity, as shown in Figs. 3(g) and3(h). Elliptically polarized light .-Now, we turn to the moregeneral case of elliptically polarized light, i.e, A x (cid:54) = A y . - - - - - k z E ( k z ) - - - - - k z E ( k z ) - - - - - k z E ( k z ) - - - - - k z E ( k z ) (a) (b)(c) (d) - - - - - k z E ( k z ) (e) FIG. 4: (a)-(e) show the k x = k y = 0 cut for A y = 4 and A x = 1 , . , . , . , λ = 0 . , λ = 0 . ω = 20, η = 1 and µ = 0 ( µ = 4) for (a)-(d)((e)) The phase diagram is much richer when the incident lightis anisotropic in the field’s amplitudes. In the following,we analyze various driven phases in the two band-bendingpossibilities shown in Fig. 2.Let us first study Eq. (4) when the bands are bentoppositely. The phase evolution for this case is depictedin Fig. 4. In describing the evolution, we keep A y fixedat a high or a low value, and tune A x from 0 to A y .Let us first look at high A y . For large anisotropy with A y (cid:29) A x , there are 4 type-I nodes of unit monopolecharge in the k y − k z plane and 2 type-II nodes of charge ± k z -axis (Fig. 4a and 6a). Onincreasing A x , the two higher nodes split into four type-II nodes of unit charge in the k x − k z plane (Fig. 4b and6b). On further increasing A x a pair of type-II nodes ofunit charge come in on the k z axis from k z = ±∞ whilethere are still 4 nodes in k y − k z plane (Fig. 4c and 5a).The new node at k z > k y − k z plane at k z >
0; an analogous condition holds for k z < ± k z axis. These nodes change charac-ter from type-II to type-I, accompanied by the flatteningof one of the bands participating in the nodes (Fig. 4d),and survive in this form up to the circularly-polarizedlimit (Fig. 4e). In the meantime, the four higher nodesremain type-II with unit charge, but merge into two type-II, charge ± A x = A y . Therefore (a) (b) (c) FIG. 5: Appearance of type-II node from infinity (a) whilethere are two nodes on the k y − k z plane. The node frominfinity moves towards origin (b) and merges with two othernodes on the k z axis (c). (a)-(c) we used fixed A y = 4 and A x = 1 . , . , .
85, respectively. To increase the resolutiononly evolution of the nodes between two relevant bands isdepicted. Also, only one side of plot is shown. we end up of 2 higher (type-II, ±
2) and lower (type-I, ±
1) nodes as explained in the previous section (Fig. 4e).It should be noted that the TDPs are absent for ellipticalpolarization.However, for lower A y , situation is different. In this casethe situations of Fig. 4(b-d) do not happen. In an-other words, for lower A y , the upper nodes do not split,while the merging of lower four points happens near theisotropic limit. Moreover, no flat-line occurs, and the up-per nodes (type-II) have charge ± ± ( a ) ( c ) FIG. 6: Representative 3D plots: (a) shows a representativeplot in phase (I) illustrating four nodes in the k y − k z planewith A y = 4 and A x = 1, (b) shows a representative plot inphase (III) illustrating four nodes in k x − k z plane with A y = 4and A x = 2. Red rings denote nodes in the upper bands. (c)shows a representative plot of the flat-bands in k x − k y in thescenario with both bands bending in same direction (Fig. 2b)with A y = 4 and A x = 1 .
6. We used ω = 20, η = 1 and µ = 0for (a)-(c). are couple of differences. Firstly, as we mentioned in theprevious section, nodes are type-II for most amplituderanges. The second, instead of a flat line along k z , thereis a ”flat-band” in the k x − k y plane for the lower nodes(Fig. 6c). The flat-line along k z does happen, but onlyfor very large A y . This is consistent with the type-II totype-I transition that was found to occur at high inten-sities in the isotropic limit for bare bands bending in thesame direction. Discussion, experimental considerations and concludingremarks .- In this work we have studied the Floquet the-ory of the three-dimensional Luttinger semimetal withquadratic band touching points. We have found that de-pending on the orientation of bands and light parameters,both type I and II Weyl nodes with single and doublemonopole charges at different energies can be generated.In particular we arrive at the following main results: • When the incident light is circularly polarized, wehave solved the problem analytically and have ob- tained two nodes with charge ± ± • For the elliptically polarized light, we have onlysolved system numerically. For both bands bend-ing scenarios, for large anisotropy, A x (cid:28) A y , thereare two higher nodes on the k z axis and four lowernodes in k y − k z plane. Then, when the A y is heldfixed at a small value and A x is increased, the fourlower nodes merge around A x ∼ A y . However, forhigh enough A y , on increasing A x , the lower Weylnodes merge and then tilt back and turn into aflat-line (flat-band) for bare bands bending oppo-sitely (similarly) and make two nodes. On the otherhand the higher nodes deform to nearly flat linesand then split to four nodes in k x − k z plane whichfinally merge at isotropic limit.Therefore, we conclude that the Luttinger semimetalwith parabolic dispersion provides a master platform forrealizing various types of WSMs from type I to type IIwith four nodes or two nodes, as well as single and dou-ble monopoles. Remarkably, we found that single anddouble-Weyl can coexist at different energies, so eitherones can be accessed through controlled doping of thesystem and tuning of laser light. To the best of ourknowledge this is the only system that reported so farwith this level of tunability for broad range of possibleWeyl phases. In addition, irradiated Luttinger semimet-als is the only example so far discovered with Weyl nodewith different monopole charges coexist, making it feasi-ble for the possible applications of both single and doubleWSMs. To the best of our knowledge our work is the onlyexample that can generate such a broad range of WSMsfrom a system with no Weyl nodes. There have beensome recent studies [25, 26], where photo-induced multi-Weyl phases were generated from crossing-line systems.The Luttinger Hamiltonian describes a wide range ofmaterials from semiconductors to pyrochlore iridates andhalf-Heuslers which are accessible experimentally, un-like the other semimetals such as Dirac, loop-node, orlinked semimetals, where experimental examples are rareor non-existent. Therefore, this work might facilitatethe experimental realizations of photoinduced WSMs.Using λ = 4 . /m for HgTe, where m is the bareelectron mass, (cid:126) ω = 120 meV and an electric field of E = 2 . × V /m – typical values for pump-probe ex-periments [50] – we estimate the perturbation parameter γ = λe E / (cid:126) ω ∼ − , so the Floquet expansion iscertainly well-controlled. The only word of caution isthat, as with all three-dimensional Floquet systems, ourproposal only works for films thin enough for the electricfield to penetrate the system substantially. ACKNOWLEDGEMENTS
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