Floquet-Bloch Oscillations and Intraband Zener Tunneling in an Oblique Spacetime Crystal
FFloquet-Bloch Oscillations and Intraband Zener Tunneling in an Oblique SpacetimeCrystal
Qiang Gao and Qian Niu
Department of Physics, University of Texas at Austin, Texas, USA (Dated: December 29, 2020)We study an oblique spacetime crystal realized by a monoatomic crystal in which a sound wavepropagates, and analyze its quasienergy band structure starting from a tight-binding Bloch bandfor the static crystal. We investigate Floquet-Bloch oscillations under an external field, which showdifferent characteristics for different band topologies. We also discover intraband Zener tunnelingbeyond the adiabatic limit, which effectively converts between different band topologies. Our resultsindicate the possibility of a quantum acoustoelectric generator that converts energy between thesound wave and a DC electric field in quantized units.
Introduction .— Periodically driven quantum systemshave a long history of study in physics, and have emergedin recent years as a new playground for novel topologicalproperties [1–3] and quantum materials engineering [4].There are also discussions on the tantalizing possibilityof the spontaneous formation of time crystals [5–8] andspacetime crystals [9], adding new excitements to thisfield. However, there is still much to be explored, andthere are challenges in understanding the electronic dy-namics in such systems [10, 11].According to a recently reported symmetry classifica-tion [12], spacetime crystals fall into rectangular andoblique categories, depending on whether the systemhas separate translational symmetries in space and time.Here we present an example for the latter, a monoatomiccrystal in which a single mode of sound wave propa-gates. One can still make a Floquet-Bloch analysis, butquasienergies and momenta are now defined modulo anoblique Brillouin zone, and the usual concepts of Blochoscillations and Zener tunneling for Bloch bands can beessentially modified. We find Floquet-Bloch oscillationsunraveling unusual types of band topologies. We thendiscuss intraband Zener tunneling, which cannot occurfor a rectangular spacetime crystal, and the adiabaticconditions for the validation of realizing one particularband topology. Our results indicate a novel mechanismfor a quantum acoustoelectric generator that converts en-ergy between the sound wave and a dc electric field.
Floquet-Bloch analysis for an oblique spacetimecrystal .— The oblique spacetime crystal considered hereis a monoatomic crystal with sound waves propagatingthrough it: H ( x , t ) = − (cid:126) M ∇ x + (cid:88) R V (cid:16) x − ˜ R (cid:17) (1)with the atomic position being time-dependent ˜ R = R − A cos( κ · R − Ω t ) . Here the ( κ , Ω) is the momentum andfrequency of that sound wave, | A | is the oscillation ampli-tude and R = n a + n a + n a labels the lattice sites.This Hamiltonian has the following translational symme-tries: H ( x , t + 2 π/ Ω) = H ( x , t ) = H ( x + R , t + κ · R / Ω), which defines an oblique spacetime lattice with non-orthogonal lattice vectors: ( , π Ω ) and ( R , κ · R Ω ). Thosevectors determine the reciprocal lattice structure to bealso oblique, characterized by vectors: ( κ , Ω) and ( G , G is the reciprocal lattice vector of the correspond-ing static crystal [14]. (When the sound wave vector κ is rationally related to G , one may adopt a superlat-tice point of view with a folded Brillouin zone, so thatthe system may be taken as a rectangular spacetime su-perlattice, but there will be seemingly ‘mysterious’ bandcrossings due to the band folding.)The Floquet-Bloch band theory of the oblique space-time crystal goes quite parallel to that for a rectangularspacetime crystal [13]. The eigenstates that respect pe-riodicities of the Hamiltonian satisfy the time-dependentSchrodinger equation: H ( x , t ) | Ψ( x , t ) (cid:105) = i (cid:126) ∂ t | Ψ( x , t ) (cid:105) . (2)We consider that | A | (cid:28) lattice constants and the effectof lattice vibration to leading orders in the amplitude A : H ( x , t ) = H ( x ) + A · H ( x , t ) + · · · , where H is the Hamiltonian of the corresponding static crystal.To simplify matters, we assume that the electrons areall in the lowest Bloch band of the static crystal, whichis well-separated from all other bands energetically sothat mixing with them can be ignored when the latticevibration is turned on.Under these conditions, lattice vibrations can still mixa Bloch state {| ψ k ( x ) (cid:105)} of energy ω g ( k ) in the lowestband with its phononic “sidebands”, which differ witheach other by an integer multiple of the phonon energyand momentum (Ω , κ ). In other words, we can choose thebasis states to be the phonon replica of a Bloch state: {| Φ n, k ( x , t ) (cid:105) ≡ e − in Ω t | ψ k + n κ ( x ) (cid:105)} , (3)where n is the replica index. This basis can be madeorthonormal under a new inner product defined as (cid:104)(cid:104) φ ( x , t ) | ψ ( x , t ) (cid:105)(cid:105) ≡ T (cid:90) T dt (cid:90) d x φ ∗ ( x , t ) ψ ( x , t ) (4)where T = 2 π/ Ω is the Floquet time period. a r X i v : . [ c ond - m a t . o t h e r] D ec Having the phonon replica basis, we can then expandthe eigenstate in Eq.(2) as: | Φ ω, k ( x , t ) (cid:105) = (cid:88) n e − iωt f n k | Φ n, k ( x , t ) (cid:105) , (5)where ( ω, k ) are the quasienergy and quasimomentum,respectively. Since those two quantities are conservedmodulo a Brillouin zone due to the periodicity in thereciprocal space, we can use them as characterizationsfor the eigenstates. Utilizing the orthonormal condi-tions of the basis functions, we then find that the coef-ficients f n k satisfy the following matrix eigenvalue equa-tion: (cid:80) n H m,n ( k ) f n k = (cid:126) ωf m k where the elements of theKernel matrix H ( k ) are given by H m,n ≡ (cid:104)(cid:104) Φ m, k ( x , t ) | H ( x , t ) | Φ n, k ( x , t ) (cid:105)(cid:105)− (cid:126) n Ω δ mn . (6)In the static limit of A = , the matrix H is diagonal witheigenvalues ω n ( k ) = ω g ( k + n κ ) − n Ω, meaning that thequasienergy bands are just the original Bloch band ω g ( k )shifted by the reciprocal lattice vector ( κ , Ω). When lat-tice vibration is turned on, off-diagonal elements of theHamiltonian will appear, which can open gaps at placeswhere the Bloch band crosses with its phonon replicas.From the calculation, we also find a general relation be-tween different quasienergy bands: ω n + m ( k ) = ω m ( k + n κ ) − n Ω , (7)which reflects the periodicity in the reciprocal space.These general features of the Floquet-Bloch bandstructure are illustrated in Fig.1 for the case of a (1+1)Doblique crystal. The dashed curves are the unperturbedbands with no oscillation, and we can see they are noth-ing but replicas of the original cosine-shape Bloch band.The solid curves are the band dispersion under time-dependent perturbation. The red shaded area stands forthe Brillouin zone of the oblique spacetime crystal char-acterized by two reciprocal lattice vectors: ( G = 2 π/a, κ, Ω). Without loss of physics, we take the regioncontaining the ( n = 0) band ω ( k ) as our first Brillouinzone and all others as replicas. Floquet-Bloch oscillations and band topology .— Very interesting phenomena such as Bloch oscillationsand Zener tunneling can occur for Bloch bands in thepresence of an external electric field E , and it is nat-ural to ask what can happen to the quasienergy bandsin a spacetime crystal. If we represent the field by avector potential and treat its time dependence adiabati-cally, i.e., as very slow compared to all other time scalesin the problem, we can still use the quasienergy states assolutions provided that the quasimomentum is replacedas k → k − eEt/ (cid:126) . Then by using the perturbationmethod, we can find an expression for the group velocity˙ x . Together with the time-dependent quasimomentum, Brillouin Zone
FIG. 1: Floquet-Bloch band structure for the (1+1)D obliquespacetime crystal modeled by Eq.(6), originating from thelowest Bloch band (dashed curves repeated over the Brillouinzone) of the unperturbed system. The quasienergy dispersion(solid curves) is calculated when a single mode of the soundwave is turned on to a finite amplitude [15]. The labels on theright vertical axis are the indices of the replicas of the samesingle band as defined within the Brillouin zone. the electronic motion in a quasienergy band can be sum-marized as ˙ k = − eE (cid:126) , ˙ x = ∂ω n ( k ) ∂k , (8)which leads to a similar motion as Bloch oscillations thatwe call Floquet-Bloch oscillations in the present con-text [17]. Indeed, for the band structure in Fig.1, thequasienergy is periodic in momentum, ω n ( k + 2 π/a ) = ω n ( k ), which implies, according to the equations of mo-tion, that the velocity of electrons is also periodic in timewith period (cid:126) eE G [18]. Similar discussion in the time-driven system can be found in Ref.[19].However, in the oblique spacetime crystal, quasienergybands can also, in principle, exhibit nontrivial periodicitylike ω n ( k + κ ) = ω n ( k )+Ω or even more exotically ω n ( k +2 π/a ± κ ) = ω n ( k ) ± Ω, which will lead to new oscillationperiods of (cid:126) eE κ and (cid:126) eE ( G ± κ ), respectively. Those uniquebehaviors suggest different unusual band topologies [20].To appreciate the possibilities of different topologies,we project the Brillouin zone onto a torus by shearing itinto a rectangle and wrapping around to join the oppositeedges. A quasienergy dispersion is then characterizedby a pair of winding numbers N ω and N k around thetwo topologically distinct directions represented by thereciprocal lattice vectors ˜ ω = ( κ, Ω) and k = ( G, N ω = 0 , N k = 1 for panel (a), N ω = 1 , N k = 0 in panel(b), and N ω = − , N k = 1 in panel (c), correspondingto the Floquet-Bloch oscillations with periods ∝ N k G + N ω κ . FIG. 2: The topology of band dispersions (blue curves) asseen in the Brillouin zone and on the torus after shearingit and wrapping around along the reciprocal lattice vectors: ˜ ω = ( κ, Ω) and k = ( G, Although we do not yet find the possibilities illus-trated in panels (b) and (c) within the model studied inthis work, we discover a system called Oscillating DiracComb that can possess a gapless band structure for aspecific oscillation amplitude (with more details given inthe S.M.[16]), as shown schematically in panel (d). Wehave intentionally plotted the band in red and blue cor-responding to the topologies in panels (b) and (c), re-spectively. Taking the band structure as two differenttopologies requires that the electron cannot be in a su-perposition of the two segments plotted in blue and redand must remain consistently on one of them when pass-ing through the crossing point.However, an exact gap closing in the oscillating Diraccomb requires a fine-tuning of parameters [16], which isnot robust under any other perturbations and thus un-realistic in real experiments. So, we have to allow suchsystem to have a tiny gap. In the next section, we willsee how the joint topology shown in panel (d) is possibleeven with a tiny gap opened at the crossing point by dis-cussing the intraband Zener tunneling and the adiabaticconditions.
Intraband Zener tunneling and adiabaticcondition .— Zener tunneling refers to the breakdownof adiabaticity when the rate of parameter changecannot be regarded as small compared to the gapbetween the energy levels, and there is also an analog ofthe phenomenon between quasienergy levels in Floquetsystems [22–24]. In crystals under an electric field,the crystal momentum becomes a time-dependent parameter, and interband Zener tunneling has been wellstudied. Here, due to the fact that in oblique spacetimecrystals, the gaps can be opened between a quasienergyband and its periodic replicas (as shown in Fig. 1), wecan actually anticipate an intraband Zener tunnelinghappening through such gaps.The analysis of the intraband Zener tunneling is quitesimilar to that of normal Zener tunneling between dif-ferent Bloch bands. The key idea is that we considertunneling between two eigenstates | ψ ( k ) (cid:105) and | ψ ( k ) (cid:105) : | ψ ( k ) (cid:105) = (cid:88) n f nk | Φ n,k ( x, t ) (cid:105)| ψ ( k ) (cid:105) = (cid:88) n f nk − κ | Φ n − ,k ( x, t ) (cid:105) , (9)sitting on two adjacent bands (replicas) labeled by 1and 2, which have quasienergies (cid:15) = ω ( k ) and (cid:15) = ω ( k − κ ) + Ω, respectively, with a direct gap ∆ . Onecan check that | ψ ( k ) (cid:105) and | ψ ( k ) (cid:105) are orthonormal( (cid:104)(cid:104) ψ i ( k ) | ψ j ( k ) (cid:105)(cid:105) = δ ij ). The reason why such tunnelingis indeed an intraband process is that | ψ ( k ) (cid:105) is equiva-lent to | ψ ( k − κ ) (cid:105) since they differ by a reciprocal latticevector. The transition between | ψ ( k ) (cid:105) and | ψ ( k − κ ) (cid:105) then involves a shift in momentum, which is associatedwith absorption or emission of a quantum of sound mode(Ω , κ ).To make the tunneling happen, we apply an electricfield E , so that, from the equations of motion in Eq.(8),electrons will move adiabatically along k axis. The realwavefunction can be approximated as a linear combi-nation of two eigenstates: | Φ( t ) (cid:105) = C ( t ) | ψ ( k ( t )) (cid:105) + C ( t ) | ψ ( k ( t )) (cid:105) . In the context of oblique spacetime crys-tals, such expansion is valid under a vertical (or irra-tional) basis representation [16].Now, plugging the wavefunction | Φ( t ) (cid:105) into the time-dependent Schrodinger Equation, we obtain the followingdifferential equation regarding C , : i (cid:126) ∂∂t (cid:20) C C (cid:21) − eE (cid:20) A A A A (cid:21) (cid:20) C C (cid:21) = (cid:20) (cid:15) (cid:15) (cid:21) (cid:20) C C (cid:21) (10)where A ij ≡ (cid:104)(cid:104) ψ i ( k ( t )) | ( i∂ k + x ) | ψ j ( k ( t )) (cid:105)(cid:105) is the multi-band Floquet-type Berry connection. This result has thesame form as in Bloch crystals but with modified Berryconnections. For spacetime crystal, A ij generically hastwo contributions: A ij = i (cid:88) n ( f j ) ∗ ∂ k f i + (cid:88) n ( f j ) ∗ f i A k + nκ (11)where f → f nk and f → f n +1 k − κ . The first term is theFloquet contribution, while the second term is the modi-fied Bloch contribution with A k + nκ being the usual Berryconnection. For the Floquet-Bloch system generated bya single Bloch band well-separated from all other bands,this Bloch contribution is numerically small and negligi-ble. Then A ij ( k ) has only the Floquet contribution that FIG. 3: (a) Floquet-Bloch band structure with a very small gap of avoided crossing, calculated for the example of an oscillatingDirac comb with oscillation amplitude A = 0 .
15 [16], with A , B and C labeling three typical points. The blue and red dottedcurves represent two paths with topologies also illustrated in Fig.2(d). (b) The numerical results of the tunneling processbetween two bands near the point B in panel (a) by solving Eq.(10) with initial conditions C = 1 and C = 0 under differentexternal field strengths. (c) The numerical results (stars) and the theory from Eq.(13) (curve) of the tunneling rate of theelectron from band 1 to 2. The green shade highlights the area where the tunneling rate is approximately one. comes solely from the time variations, which allows us toconsider only the kernel H ( k ) in Eq.(6).We again use the oscillating Dirac comb but now witha small gap as an example to show some numerical re-sults. Fig.3(a) shows the band structure of such system,which resembles a so-called Landau-Zener grid [19, 26].We then numerically solve Eq.(10) near the gap at point B in Fig.3(a), with the electron initially sitting on band (cid:15) ( C = 1 , C = 0). The squared moduli | C | and | C | as functions of k under different external field strengthsare plotted in Fig.3(b) using dashed and solid curves, re-spectively. We can see that when | eE | = 10 − eV / ˚ A (bluecurves), the evolutions of | C , | are close to step func-tions indicating total tunneling through the gap, whilefor larger | eE | , the electron is in a superposition of twobands, violating the adiabaticity. Such violation comesfrom a larger direct gap near the gap at point B , whichmixes two bands too early. This tells us that when | eE | is small enough, we can just ignore the influences of thatlarger gap and only consider the behavior of electron atthe vicinity of point B , allowing us to have an analyticdiscussion.The system near the point B can be asymptoticallyapproximated by a 2-level system as h ( k ) = (cid:20) E ( k ) ∆ / / E ( k ) (cid:21) , (12)where ∆ is the gap at k = k B , and E , ( k ) = µ , ( k − k B )are the asymptotes of bands (cid:15) and (cid:15) near the gap. Thus,we end up with Zener’s original tunneling model with atransition rate [25]:Γ = exp (cid:18) − π ∆ eE | µ − µ | (cid:19) . (13)In Fig.3(c), we compare the numerical results with theEq.(13), which are in good agreement with each other when | eE | ≤ . × − eV / ˚ A . However, as | eE | get-ting bigger, the discrepancies occur due to the non-adiabaticity of the states before reaching the gap at point B .As discussed in last section, for band structure inFig.3(a) (or Fig.2(d) if ∆ = 0) to have separate topolo-gies, we need the adiabatic condition when electrons areaway from the gap (or the band crossing point) and totaltunneling when passing through the gap (or the crossingpoint), which requires the tunneling rate to be one at thepoint B and zero elsewhere. That can be realized when | eE | is in the green shaded area in Fig.3(c). Discussion .— The uniqueness of the gapless bandstructure in Fig.2(d) or the similar one but with a tinygap in Fig.3(a) turn out to have non-trivial electric prop-erties that a non-zero DC current can be induced by anexternal electric field, and the energy can be transferredbetween the electric field and the sound wave in quan-tized unit Ω. Thus we can design a prototypical quantumacousto-electric generator based on such system.Given the band structure depicted in Fig.3(a), we ap-ply an electric field with strength | eE | lying preciselywithin the area where Γ ∼
1. The electrons can thenmove freely along the red or the blue dotted curvesin Fig.3(a) depending on their initial positions. Thosetwo paths correspond to the band dispersion depicted inFig.2(b,c), which means that the electrons are oscillatingwith periods of (cid:126) eE κ and (cid:126) eE ( G − κ ). However, unlike theordinary Bloch oscillation, the electrons moving alongthe red or the blue paths actually have nonzero displace-ments in real space due to the energy change after eachperiod. To see that, we have∆ x = (cid:90) ˙ xdt = (cid:90) ∂ω ( k ) ∂k dk ˙ k = − (cid:126) eE ∆ ω (14)where we have applied the equations of motion Eq.(8).The displacements then give a gain of electric energy∆ E = − ( − e ) E ∆ x = − (cid:126) ∆ ω .To see how the energy is transferred between the elec-tric field and the sound wave in a quantized unit Ω, wenow restrict our consideration within the Brillouin zonewhich is the energy dispersion from point A to B and to C shown in Fig.3(a). We have to keep in mind that thissystem only has one band and all others are just replicas.Imaging one electron sitting initially on the segment BC and driven adiabatically from point B to C by | eE | , theenergy gain from the electric field is − (cid:126) ( ω C − ω B ). Thenat point C , the electron will tunnel through the gap to anadjacent state on the lower band which is equivalent tothe state at point B , since they differ by a reciprocal lat-tice vector. In other words, this is tunneling from point C to point B on a single band associated with absorptionof a quantum of sound mode (changes in both energy andmomentum), which is the essence of the intraband Zenertunneling. By oscillating through B → C → B , the en-ergy is continuously transferred from the sound wave tothe electric energy of the electron. When the electron isinitially at AB , the process is similar but reversed.The oscillation periods of those two processes can alsobe used as experimental signatures to determine whetherone of those is happening. Similarly, due to the relationbetween electronic current and electron velocity: j = eρ ˙ x ( ρ being the electron density), the two different periodsalso correspond to different frequencies in the AC part ofthe current j . Acknowledgment.—
The work is supported by NSF(EFMA-1641101) and Welch Foundation (F-1255). [1] Rechtsman, Mikael C., et al. Photonic Floquet topologi-cal insulators. Nature , 196–200 (2013).[2] Lindner, Netanel H and Refael, Gil and Galitski, Victor.Floquet topological insulator in semiconductor quantumwells. Nat. Phys. , 490–495 (2011).[3] Yao, Shunyu and Yan, Zhongbo and Wang, Zhong. Topo-logical invariants of Floquet systems: General formula-tion, special properties, and Floquet topological defects.Phys. Rev. B , 195303 (2017).[4] Oka, Takashi and Kitamura, Sota. Floquet engineering ofquantum materials. Annual Review of Condensed MatterPhysics , 387–408 (2019).[5] Wilczek, Frank. Quantum time crystals. Phys. Rev. Lett. , 160401 (2012).[6] Else, Dominic V and Bauer, Bela and Nayak, Chetan.Floquet time crystals. Phys. Rev. Lett. , 090402(2016).[7] Zhang, Jiehang, et al. Observation of a discrete time crys-tal. Nature , 217–220 (2017).[8] Autti, S and Eltsov, VB and Volovik, GE. Observationof a time quasicrystal and its transition to a superfluidtime crystal. Phys. Rev. Lett. , 215301 (2018).[9] Li, Tongcang, et al. Spacetime crystals of trapped ions.Phys. Rev. Lett. , 163001 (2012). [10] Genske, Maximilian and Rosch, Achim. Floquet-Boltzmann equation for periodically driven Fermi sys-tems. Phys. Rev. A , 062108 (2015).[11] Messer, Michael, et al. Floquet dynamics in driven Fermi-Hubbard systems. Phys. Rev. Lett. , 233603 (2018).[12] Xu, Shenglong and Wu, Congjun. Spacetime Crystal andSpacetime Group. Phys. Rev. Lett. , 096401 (2018).[13] G´omez-Le´on, Alvaro and Platero, Gloria. Floquet-Blochtheory and topology in periodically driven lattices. Phys.Rev. Lett. , 200403 (2013).[14] Due to the convention of quantum mechanics that wealways write the propagating phase factor as e i k · x − iωt ,the space and time have naturally different signaturesencoded. In this paper, the signature is set to be(+ , + , +; − ) where the last entry stands for time. Thenwe can check that the reciprocal lattice vectors indeedfulfill: − ( , π Ω ) · ( κ , Ω) = ( R , κ · R Ω ) · ( G ,
0) = 2 π and( , π Ω ) · ( G ,
0) = ( R , κ · R Ω ) · ( κ , Ω) = 0[15] The system for obtaining the typical Floquet-Bloch bandstructure in Fig.1 is a kernel Matrix in Eq.(6) with thediagonal elements being shifted ‘cos’-shape band disper-sion: H n,n = 2 β cos[( k + nκ ) a ] − n Ω, and with the onlynonzero off-diagonal terms being H n,n ± = δ . The pa-rameters used are a = 2, κ = Ω = 1, β = 0 .
4, and δ = 0 . , 4508 (1996).[19] Gagge, Axel and Larson, Jonas. Bloch-like energy oscil-lations. Phys. Rev. A , 053820 (2018).[20] This is equivalent to say that ω n ( k ) = ω n ( k + κ ) − Ω = ω n +1 ( k ) which means that ω n ( k ) ≡ ω ( k ) for all n . Later,we will discuss that it requires a very unique band topol-ogy depicted in Fig.2(b).[21] Fig.2(b) represents a band topology with ω n ( k + κ ) = ω n ( k ) + Ω. If κ = 0 which corresponds to a rectangu-lar spacetime crystal, then there must be a point on theband such that ˙ x = ∂ω n ( k ) ∂k → ∞ , which is forbidden bymechanics.[22] Breuer, Heinz Peter and Holthaus, Martin. Quantumphases and Landau-Zener transitions in oscillating fields.Physics Letters A , 507–512 (1989).[23] Hijii, Keigo and Miyashita, Seiji. Symmetry for the nona-diabatic transition in Floquet states. Phys. Rev. A ,013403 (2010).[24] Rodriguez-Vega, M and Lentz, Meghan and Seradjeh,Babak. Floquet perturbation theory: formalism and ap-plication to low-frequency limit. New Journal of Physics , 093022 (2018).[25] Zener, Clarence Non-adiabatic crossing of energy levels.Proc. R. Soc. Lond. A , 696–702 (1932).[26] Demkov, Yu N and Ostrovsky, VN Non-adiabatic cross-ing of energy levels. Journal of Physics B , 403 (1995). FIG. 4: (a) The color map of the logarithm of the bandgap ∆ (log ∆ / ( U/L )) as a function of the oscillating amplitude A andthe phonon wavelength κ , where we have fixed the phonon frequency to be Ω = 0 . U/L . The blue dashed line emphasizesthe parameters which give gapless band structures.
U/L = 7.62eV. (b) The band structure corresponding to the parameterslabeled by the red star in panel (a), which is gapless.
SUPPLEMENTARY MATERIALOscillating Dirac Comb
Here we show a toy model called an Oscillating Dirac Comb where the electron is trapped in one-dimensionalperiodic Dirac potentials that are also shaking periodically in time. The Hamiltonian for such system is: H ( x, t ) = − (cid:126) m ∂ x − U (cid:88) p δ ( x − pa + A cos( κpa − Ω t )) (15)where U is a parameter. The corresponding static Hamiltonian at p th site is H p = − (cid:126) m ∂ x − U δ ( x − pa ) (16)which has only one bound state with energy E = − U/ L and wavefunction | φ ( x ; p ) (cid:105) = (1 / √ L ) e −| x − pa | /L ( L ≡ (cid:126) /mU being typical length). Now we can evaluate the kernel matrix H m,m up to the second order in A using the tight-bindingmethod: H m,m ( k ) = − (cid:126) m Ω + 2 cos(( k + mκ ) a ) β + a , (17) H m,m +1 ( k ) = A (cid:16) δ ∗ + e i ( k + mκ ) a γ ∗− a, + e − i ( k + mκ ) a γ ∗ + a, (cid:17) (18) H m,m +2 ( k ) = A (cid:16) σ ∗ + e i ( k + mκ ) a η ∗− a, + e − i ( k + mκ ) a η ∗ + a, (cid:17) (19)where we have introduced the spacetime hopping integrals β , δ , γ in first order and σ , η in the second order. Theyare evaluated as β + a = − UL e − aL ; δ ≈ − i UL sin( κa ) e − a/L ; γ + a, ≈ − U L e − aL ( e iκa −
1) = e iκa γ − a, ; σ = g ( L, b ) UL − UL cos(2 κa ) e − aL ; η + a, = (cid:18) g ( L, b ) U L − U L (cid:19) e − aL (1 + e i κa ) = e i κa η − a, , (20)where g ( L, b ) is a parameter that corrects the divergence induced by the unrealistic δ -shape potential. The reasonwhy we need to discuss the second order in A is that in this specific model, the leading order is actually the secondorder.In Fig.4(a), we show how the bandgap of the oscillating Dirac comb depends on the oscillating amplitude A andthe phonon wavelength κ , where we have fixed other parameters: a = 4˚ A , L = 1˚ A , U = 7 . · ˚ A , (cid:126) Ω = 0 . U/L , b = 0 . g = 1 .
2. As we can see, the bandgap is most of the time none-zero, but there is a curve (depicted using ablue dashed line) in such parameter space along which the gap is zero. In Fig.4(b), a typical gapless band structureis depicted with the parameters labeled by a red star in Fig.4(a), which is A = 0 . A and κ = 0 . A − . We note thatthis gap-closing feature is unique in the oblique spacetime crystals where κ (cid:54) = 0, given that the blue dashed curvenever intersects with the line κ = 0 which corresponds to rectangular spacetime crystals. In the main text, we alsoconsider a slightly gaped system in the discussion about the intraband Zener tunneling, where we have changed theoscillating amplitude from A = 0 . A to A = 0 . A . Vertical Basis and Irrational Sampling in k -space In the (1+1)D system, after solving the Hamiltonian by diagonalizing the kernel matrix H m,n , we find all theeigenenergies and correspondingly the eigenstates. Then we can use these eigenstates as our new basis (perturbedbasis, as opposed to the unperturbed Floquet-Bloch basis discussed in the main text). Due to the unique structureof the oblique spacetime crystal, we can have two different choices for representing the basis: the normal basis andthe vertical basis.As shown in Fig. 5, two different K -space sampling methods are depicted. In the left panel, we use the most usualway of sampling which is to choose a set of equally spacing K -points within the first Brillouin zone (in this case, the n = 0 band), and then an arbitrary state can be expanded as | Ψ (cid:105) = (cid:88) k ∈N C k ( t ) | Φ ω ( k ) ( x, t ) (cid:105) , (21)where N is the set of equally spacing K -points, and C k ( t ) is the time-dependent coefficient. However, if the systemhas an irreducible oblique structure where κ/G is an irrational number, we can have another way of sampling the K -space, and it is shown in the right panel of Fig. 5. A vertical line intersects with all bands with different index n giving the crossing points exactly the same momentum k (an arbitrarily selected k ). But because the periodicity ofthe oblique structure, we can transform all crossing points back into the first Brillouin zone by doing ( k + nκ ) mod G for all n (see, for example, the black arrows in the right panel of Fig. 5). Because of the irrationality of κ/G , theprocess of ( k + nκ ) mod G can densely and almost evenly sample the K -space within the first Brillouin zone when n is large. That is called irrational sampling. As a result, the state | Ψ (cid:105) can also be expanded as | Ψ (cid:105) = (cid:88) n C n ( t ) | Φ ω n ( k ) ( x, t ) (cid:105) ≡ (cid:88) k (cid:48) ∈V k C (cid:48) k (cid:48) ( t ) | Φ ω ( k (cid:48) ) ( x, t ) (cid:105) , (22)where V k = { k (cid:48) | k (cid:48) = ( k + nκ ) mod G , for all n } is the set of K -points in the first Brillouin zone generated by irrationalsampling. We have to note that V k (cid:54) = N , but when n is large, those two are equivalent to each other.For some practical purposes, it is more convenient to use the vertical basis since it favors the conservation of thequasimomentum. In the main text, we used this vertical basis and only considered two coefficients that contributethe most to the intraband Zener tunneling. FIG. 5: Two different bases: (left panel) the normal basis with the blue dots denoting the equally spacing K -point samplingwithin one Brillouin zone; (right panel) the vertical basis with the red dots denoting the vertical (irrational) K -point samplingalong a vertical line. Intraband Zener Tunneling
Fig.6 shows a schematic plot of the intraband Zener tunneling process, where two states | ψ ( k ( t )) (cid:105) and | ψ ( k ( t )) (cid:105) setting on two adjacent replicas. After applying the electric field, the Hamiltonian becomes ˜ H = H + eEx where H is the original Floquet-Bloch Hamiltonian. By plugging the wave-function into the Schrodinger Equation ˜ H | Φ( t ) (cid:105) = i (cid:126) ∂ t | Φ( t ) (cid:105) , we get H | Φ( t ) (cid:105) = ( i (cid:126) ∂ t − eEx ) | Φ( t ) (cid:105) (23)Then, by utilizing the equation of motion ˙ k = − eE/ (cid:126) , we have C ( t ) ω ( k ( t )) | ψ ( k ( t )) (cid:105) + C ( t )( ω ( k ( t ) − κ ) + Ω) | ψ ( k ( t )) (cid:105) = i (cid:126) ∂C ∂t | ψ ( k ( t )) (cid:105) + C ( t ) eE (cid:18) − i ∂∂k − x (cid:19) | ψ ( k ( t )) (cid:105) + i (cid:126) ∂C ∂t | ψ ( k ( t )) (cid:105) + C ( t ) eE (cid:18) − i ∂∂k − x (cid:19) | ψ ( k ( t )) (cid:105) . (24)We have the orthonormal relation between two eigenstates: (cid:104)(cid:104) ψ i ( k ) | ψ j ( k ) (cid:105)(cid:105) = δ ij , but to avoid the integration overtime, we can see that the orthonormal relation also holds in space that (cid:104) ψ i ( k ) | ψ j ( k ) (cid:105) = δ ij due to the momentumconservation. Then we can obtain two differential equations for C , : C ( t ) ω ( k ( t )) = i (cid:126) ∂C ∂t + C ( t ) eE (cid:104) ψ ( k ( t )) | (cid:18) − i ∂∂k − x (cid:19) | ψ ( k ( t )) (cid:105) + C ( t ) eE (cid:104) ψ ( k ( t )) | (cid:18) − i ∂∂k − x (cid:19) | ψ ( k ( t )) (cid:105) C ( t )( ω ( k ( t ) − κ ) + Ω) = i (cid:126) ∂C ∂t + C ( t ) eE (cid:104) ψ ( k ( t )) | (cid:18) − i ∂∂k − x (cid:19) | ψ ( k ( t )) (cid:105) + C ( t ) eE (cid:104) ψ ( k ( t )) | (cid:18) − i ∂∂k − x (cid:19) | ψ ( k ( t )) (cid:105) , (25)where one can also show that due to the conservation of momentum, the factors in above equation actually satisfy (cid:104) ψ i ( k ( t )) | (cid:18) − i ∂∂k − x (cid:19) | ψ j ( k ( t )) (cid:105) = (cid:104)(cid:104) ψ i ( k ( t )) | (cid:18) − i ∂∂k − x (cid:19) | ψ j ( k ( t )) (cid:105)(cid:105) . (26) FIG. 6: Zener tunneling across a small gap ∆ occurs when the momentum k changes with time under an external field justlike the tunneling in Bloch states. However, since the different bands are really replica of the same Floquet-Bloch band, shiftedby the phonon energy and momentum (Ω , κ ), Zener tunneling in the present context is actually an intraband process indicatedby the dashed arrow. Thus, we can recast the Eq.(25) into a Matrix form: i (cid:126) ∂∂t (cid:20) C C (cid:21) − eE (cid:20) A A A A (cid:21) (cid:20) C C (cid:21) = (cid:20) (cid:15) (cid:15) (cid:21) (cid:20) C C (cid:21) (27)where A ij ≡ (cid:104)(cid:104) ψ i ( k ( t )) | ( i∂ k + x ) | ψ j ( k ( t )) (cid:105)(cid:105)(cid:105)(cid:105)