Symmetry-induced even/odd parity in charge and heat pumping
SSymmetry-induced even/odd parity in charge and heat pumping
Miguel A. N. Ara´ujo,
1, 2, 3
Pedro Ninhos, and Pedro Ribeiro
1, 3 CeFEMA, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Departamento de F´ısica, Universidade de ´Evora, P-7000-671, ´Evora, Portugal Beijing Computational Science Research Center, Beijing 100084, China CeFEMA, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
It is shown that the presence of discrete symmetries in Floquet systems connected to metallicreservoirs imprints a definite parity on the charge and heat pumping as a function of the reservoir’schemical potential, µ . In particular, when particle-hole symmetry (PHS) holds, the pumping ofcharge (heat) is an odd (even) function of µ . Whereas, if only the product of PHS and paritysymmetry is present, pumping of charge (heat) is even (odd) in µ . Our results also extend to thepresence of other unitary symmetries and provide a simple criterion for reversing (or maintaining)the direction of the flow. We illustrate our findings using two variants of the Su-Schrieffer-Heegermodel under a time-periodic perturbation. Quantum pumping consists of coherently transportingheat or charge between macroscopic reservoirs using aperiodic drive. By exchanging energy with the drive,carriers can be transported across reservoirs even in theabsence of any temperature or voltage bias.The pumping of charge was first understood in the so-called adiabatic limit [1, 2], where the ratio between thedriving period, T = 2 π/ Ω, and the time taken by carriersto traverse the sample, τ , is large, i.e. 2 π/ Ω τ (cid:29)
1. As τ is proportional to the linear sample size, L , the adia-batic approximation eventually breaks down and one hasto resort to a fully time dependent modeling using scat-tering Floquet theory [3, 4] or a non-equilibrium Green’sfunctions approach [4–9]. Since their introduction, quan-tum charge pumps have found applications in quantummetrology [10, 11], single-photon or electron emitters[12, 13] and quantum information processing [14].It has long been noted that pumping is only allowedwhen parity symmetry is broken [3, 15]. However, asystematic understanding of how symmetry affects thepumping of heat and charge is still incomplete. In thisLetter, we take a step further in this direction by show-ing that some symmetry properties leave a signatureon the oven/odd dependence of the charge and heatpumped, within a cycle, on the reservoirs chemical po-tential. Specifically, we show that if inversion symmetryalong the direction of the current is broken and particle-hole symmetry (PHS) holds, or if only the product ofthe two holds, charge and heat pumping then dependon the chemical potential in qualitatively different ways:the pumped charge or heat in one cycle can either bean even or odd function of the chemical potential. Weextend these findings to compositions of the above sym-metries with unitary symmetries. This result provides avery simple criterion for reversing (or maintaining) thedirection of the flow.We illustrate our findings in two variants of the peri-odically driven Su-Schrieffer-Heeger (SSH) chain [16, 17]coupled to two wide-band reservoirs [15]. Symmetrybreaking can either be induced by the coupling to the leads or by explicitly breaking spatial symmetry, for in-stance, through a spatially non-uniform drive of the 1Dconductor. As an example of a composite symmetry, weintroduce a special form of PHS, acting in space-timedomain, which affects differently the two model variants. Charge and energy pumping. — We consider a typicaltransport setup consisting of two macroscopic metallicleads connected by a mesoscopic system, S, to which thedriving is applied. The total Hamiltonian is given by H ( t ) = H S ( t ) + X l =R,L ( H l + H l − S ) , (1)where the Hamiltonian of the system, H S ( t ) = P αβ ˆ c † α H S; αβ ( t ) ˆ c β , is assumed to be quadratic, with ˆ c † α and ˆ c α the fermionic creation and annihilation operators,with a periodic single-particle Hamiltonian, H S ( t ) = H S ( t + T ). The Hamiltonians, H l , for the right andleft ( l = R , L ) leads are time independent and non-interacting, and the same applies to the system-lead cou-pling term, H l − S . Under these conditions, the retardedGreen’s function of the system verifies Dyson’s equa-tion, [ i∂ t − H S ( t )] G R S = P l Σ Rl .G R S , where Σ Rl ( t, t ) = R dω π e − iω ( t − t )Σ Rl ( ω ) is the time-translational invariantretarded self-energy induced by lead l . Under periodicdriving, it is convenient to define the Floquet Green’sfunction [15]: G ( m ) ( (cid:15) ) = 1 T Z T dt Z + ∞−∞ dτ e i ( m (cid:126) Ω t − (cid:15)τ ) G R S ( t, t + τ ) . (2)Assuming there are no bound-states, at large times afterthe periodic drive has been turned on, a recurrent stateis attained and observables become periodic with driv-ing period [6]. Here, we are concerned with the averagecharge J cl , and energy, J el , currents leaving lead l over onedriving cycle, defined as J c/el = lim τ →∞ R τ + Tτ dtT J c/el ( t ).In terms of the Floquet Green’s function, average cur- a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec rents are given by [6, 15] J cl = − | e | h X m ∈ Z Z d(cid:15) n T ( m ) l ¯ l ( (cid:15) ) f ¯ l ( (cid:15) ) − T ( m )¯ ll ( (cid:15) ) f l ( (cid:15) ) o , (3) J el = 1 h X m ∈ Z Z d(cid:15) n ( m (cid:126) Ω + (cid:15) ) T ( m ) l ¯ l ( (cid:15) ) f ¯ l ( (cid:15) ) − (cid:15)T ( m )¯ ll ( (cid:15) ) f l ( (cid:15) ) + m (cid:126) Ω T ( m ) ll ( (cid:15) ) f l ( (cid:15) ) o , (4)where, T ( m ) ll ( (cid:15) ) = tr h G ( m ) ( (cid:15) ) Γ l ( (cid:15) ) G † ( m ) ( (cid:15) ) Γ l ( (cid:15) + m (cid:126) Ω) i is the transmission probability for a fermion leaving l with energy (cid:15) and arriving at l after absorbing m energy quanta (photons) from the driving field.Γ l ( (cid:15) ) = i (cid:2) Σ Rl ( (cid:15) ) − Σ Al ( (cid:15) ) (cid:3) is the hybridization matrixof lead l , and we introduced the notation ¯R = L and¯L = R. f l ( (cid:15) ) denotes de Fermi-Dirac distributionfunction at lead l at temperature 1 /β . The first termof Eq.(4) describes the energy absorbed by l when anelectron leaves ¯ l with energy (cid:15) and absorbs m photons;the second term is the energy lost by l when a electronwith energy (cid:15) is transmitted to ¯ l ; and the last term isthe energy gained when an electron is reflected back to l having absorbed m photons.In the following, we set the reservoirs to the samechemical potential, µ l = µ , and consider the total chargetransferred in one cycle between the leads, Q = π Ω J cL .Charge conservation ensures that J c L = − J c R . Wealso consider the total energy generated in one cycle, E t = π Ω ( J e L + J e R ), and the heat pumped between leads, E ∆ = π Ω ( J e L − J e R ) , as functions of the leads’ chemicalpotential. For convenience, we study the derivatives ofthese quantities with respect to µ , Q ( µ ) = Z βd(cid:15) h β ( (cid:15) − µ )2 i Q (0) ( (cid:15) ) , (5) E t/ ∆ ( µ ) = Z βd(cid:15) h β ( (cid:15) − µ )2 i E (0) t/ ∆ ( (cid:15) ) , (6)where the corresponding zero-temperature expressionsread Q (0) ( (cid:15) ) = − | e | Ω (cid:126) X m ∈ Z h T ( m )LR ( (cid:15) ) − T ( m )RL ( (cid:15) ) i , (7) E (0) t/ ∆ ( (cid:15) ) = 1 h X m ∈ Z (cid:26) X l h T ( m )L l ( (cid:15) ) ± T ( m )R l ( (cid:15) ) i m + (1 ∓ (cid:15) (cid:126) Ω h T ( m )LR ( (cid:15) ) ± T ( m )RL ( (cid:15) ) i (cid:27) . (8)The heat current follows from (4) as the transport of (cid:15) − µ instead of (cid:15) , J hl = J el − µ | e | J cl . (9) So, the heat transported per cycle is given by Q t = E t and Q ∆ = E ∆ − µ | e | Q . Symmetries. — To study the implications of symme-try for pumping, we analyze the properties of the Green’sfunctions under symmetry transformations, which fol-low from those of the Hamiltonian [18–23]. We con-sider a generic transformation, H ( t ) → H X ( t ), of the fullHamiltonian (of S and leads), under the symmetry trans-formation, X . Specifically, time-reversal and particle-hole transformations read H T ( t ) = U † T H ∗ ( − t ) U T and H C ( t ) = − U > C H > ( t ) U ∗ C , respectively, where U X aresuitable unitary matrices. We also consider x -axis inver-sion, h x | H P ( t ) | x i = U † P h− x | H ( t ) |− x i U P , where x isthe coordinate along the propagation direction of the cur-rent. As in one dimension, x -axis inversion coincides withparity symmetry (PS), we denote this transformation as P . However, the arguments below are valid for any sym-metry transformation that inverts the x -axis [15, 24, 25].Since the total total system evolves unitarily, thetransformed Green’s function is given by G RX ( t, t ) = − i Θ ( t − t ) T e − i R tt dτ H X ( τ ) , with T the time orderingoperator. Using the definition of H X , the transformedGreen’s functions read [26] G RT ( t, t ) = U † T (cid:2) G R ( − t , − t ) (cid:3) > U T , (10) G RC ( t, t ) = − U > C (cid:2) G R ( t, t ) (cid:3) ∗ U ∗ C , (11) h x | G RP ( t, t ) | x i = h− x | U † P G R ( t, t ) U P |− x i . (12)Note that the equalities H X = H and G RX ( t, t ) = G R ( t, t ) hold whenever the symmetry X is present [27].For transport, we are interested in the Green’s func-tions restricted to degrees of freedom solely within thesystem, G R S = P α,β ∈ S | α i h α | G R | β i h β | . If the unitarymatrices, U X , do not mix degrees of freedom of the sys-tem with those of the leads, G R S follows the same trans-formation rules (10-12) [26] and obeys[ i∂ t − H X, S ( t )] G RX, S ( t, t ) = Z dτ X l =R,L Σ RX,l ( t, τ ) G RX, S ( τ, t ) , (13)where Σ RX,l is the transformed self-energy of lead l , thatalso conforms with Eq. (10-12).We note that when Σ Rl ( t, t ) = δ ( t − t ) Σ Rl ( t ),the Green’s function can be written as G R S ( t, t ) = − i Θ ( t − t ) T e − i R tt dτ Z ( τ ) , where Z ( t ) = H S ( t ) + P l Σ Rl ( t ) can be identified with an effective non-hermitian Hamiltonian, which transforms as Z T ( t ) = U † T Z > ( − t ) U T and Z C ( t ) = − U > C Z ∗ ( t ) U ∗ C . The thesetransformation properties also found in systems withMarkovian environments [28], and were recently studiedunder the nomenclature TRS † and PHS † , respectively, inRef. [29].Using Eq.(2) and the transformation rules (10-12) it isstraightforward to show that G T ;( m ) ( (cid:15) ) = U † T (cid:2) G ( − m ) ( (cid:15) + m (cid:126) Ω) (cid:3) > U T , (14) G C ;( m ) ( (cid:15) ) = − U > C (cid:2) G ( − m ) ( − (cid:15) ) (cid:3) ∗ U ∗ C , (15) h x | G P ;( m ) ( (cid:15) ) | x i = U † P h− x | G ( m ) ( (cid:15) ) |− x i U P . (16)In turn, Eqs. (14-16) can be used to deduce the trans-formation properties of the transmission probabilities: T ( m ) T ; ll ( (cid:15) ) = T ( − m ) l l ( (cid:15) + m (cid:126) Ω) , (17) T ( m ) C ; ll ( (cid:15) ) = T ( − m ) ll ( − (cid:15) ) , T ( m ) P ; ll ( (cid:15) ) = T ( m )¯ l ¯ l ( (cid:15) ) . (18)The details of the derivation are given in [26] where itis also shown that Eqs. (17-18) can alternatively be ob-tained in the Floquet scattering matrix approach [3, 30]. Implications for transport. — We now derive the im-plications of the above symmetries for transport. It haslong been known that charge, energy, or heat pump-ing requires inversion symmetry breaking [3, 15]. Ifthe x -axis inversion leaves the full Hamiltonian, H , in-variant, then the Green’s function remains invariant, G P ;( m ) ( (cid:15) ) = G ( m ) ( (cid:15) ), whereas the hybridizations are in-terchanged, Γ P ; l ( (cid:15) ) = Γ ¯ l ( (cid:15) ). In this case T ( m ) ll ( (cid:15) ) = T ( m ) T ; ll ( (cid:15) ) = T ( m )¯ l ¯ l ( (cid:15) ), yielding Q ( µ ) = Q ∆ ( µ ) = 0, so notransport of charge or heat occurs.However, if the system plus reservoirs are invariant un-der PHS, then G C ;( m ) ( (cid:15) ) = G ( m ) ( (cid:15) ) and Γ C ; l ( (cid:15) ) = Γ l ( (cid:15) ),which implies T ( m ) ll ( (cid:15) ) = T ( m ) C ; ll ( (cid:15) ) = T ( − m ) ll ( − (cid:15) ) [26].From Eqs.(5)-(8), we now obtain Q ( µ ) = Q ( − µ ) , Q t/ ∆ ( µ ) = −Q t/ ∆ ( − µ ) . (19)Then, Q ( µ ) is an odd function and Q t/ ∆ ( µ ) are even. Q ( µ ) even implies Q ( µ ) odd plus a constant. That thisconstant is zero can be seen by considering either thelimit µ → −∞ , where no available particles exist, or theopposite limit µ → ∞ , where the fermionic states are alloccupied and, therefore, Pauli blocked. Composition of symmetries. — It may happen thatboth x -axis inversion and PHS are broken while theirproduct, PC , still holds as a symmetry. In that case, T ( m ) ll ( (cid:15) ) = T ( m ) P C ; ll ( (cid:15) ) = T ( − m )¯ l ¯ l ( − (cid:15) ), and from Eqs.(5)-(8), Q ( µ ) = − Q ( − µ ) , (20) Q t ( − µ ) = −Q t ( µ ) , Q ∆ ( µ ) = Q ∆ ( − µ ) . (21)In this case the pumped charge (heat) is an even (odd)function of µ and the total heat absorbed is even.Composition with other unitary symmetry leads tothe same dependence on µ . Consider, for instance,the half-period time translation [18, 19], H Π ( t ) = U † Π H ( t + T / U Π , and a generic unitary symmetry, U ,implemented by an unitary operator, U , that acts locally on the unit cells, H U ( t ) = U † H ( t ) U . Under the compo-sition Π C , we have G Π C ;( m ) ( (cid:15) ) = ( − m U † Π G C ;( m ) ( (cid:15) ) U Π and Γ Π C ( (cid:15) ) = U † Π Γ C ( (cid:15) ) U Π , implying T ( m )Π C ; ll ( (cid:15) ) = T ( m ) C ; ll ( (cid:15) ). Therefore, for a system invariant under Π C ,both Q ( µ ) and Q t/ ∆ ( µ ) have the same properties un-der µ → − µ as a system invariant under C . In the sameway, one can show that invariance under the combinationΠ P C , yields the same results as invariance under
P C .More generally, for X = U , Π , U Π, invariance underthe combination XP , XC and XP C , yields the sameresults as invariance under P , C and P C , respectively.An account of the symmetries and their effects of thedifferent pumping quantities is given in Table I.
The role of time reversal symmetry (TRS). — Forthe transmission probability, TRS implies T ( m ) ll ( (cid:15) ) = T ( − m ) l l ( (cid:15) + m (cid:126) Ω) [see Eq.(17)]. Although these proba-bilities are the same, these two processes will happen atdifferent rates due to the different occupation numbersof energies (cid:15) and (cid:15) + m (cid:126) Ω in the equilibrium distributionof the leads. Therefore, only for f ( (cid:15) + m (cid:126) Ω) = f ( (cid:15) ),which requires infinite temperature ( f ( (cid:15) ) = ), can TRSbe used to infer qualitative features of transport quanti-ties. It is then easy to show that breaking TRS allowsfor pumping between infinite temperature leads whereasall currents vanish in the time-symmetric case [26]. Symmetry PumpingPHS PS PC Q ( µ ) Q ∆ /t ( µ ) Model example C P P C Z xy (hom) U Π C P P U Π C Z zx ] ν =0 (hom) C - - odd even/even Z xy (inhom) U Π C - - odd even/even [ Z xy ] ν =0 (inhom) U Π C - - odd even/even [ Z zx ] ν =0 (inhom)- - P C even odd/even [ Z zx ] ν =0 (hom)TABLE I. Model symmetries and parity of charge and heatpumping. PS denotes x -axis inversion, and PC the compo-sition of PHS and PS. Examples are included where PHS isimplemented by a transformation XC with X = U Π [SeeEqs.(28)-(29)] .
Examples. — To illustrate the results above, we con-sider two versions of the SSH model for spinless fermions,illuminated by monochromatic radiation with angularfrequency Ω. In momentum space, the models read H xy ( k, t ) = [cos k + ν + A cos(Ω t ) , sin k, · ~σ , (22) H zx ( k, t ) = [sin k, , cos k + ν + A cos(Ω t )] · ~σ , (23)where the three Pauli matrices ~σ act on sublattice spaceand k is the Bloch wave vector over lattice cells [31]. Notethat, for an infinite chain, both Hamiltonians obey PS as U P H ( − k, t ) U † P = H ( k, t ), with U P = σ for H xy , and U P = σ for H zx . Here, we consider a finite chain with N/ − i γ L | ih | − i γ R | N ih N | , (24)where | j i denotes the state at site j . We note that | i isthe first site of the cell at x = 1 and | N i is the secondsite of the cell at x = N/
2. In the following we take γ L = γ R = γ .A finite chain described by Z xy ( t ) = H xy ( t ) + Σ stillenjoys PS because U P = σ [32] and U P Σ( − x ) U † P = − i γσ [ | N − ih N − | + | ih | ] σ = − i γ ( | N ih N | + | ih | ) = Σ( x ) . (25)It also has PHS with U C = σ . Therefore, charge pump-ing does not occur in an homogeneous xy chain.For the finite zx chain, described by Z zx ( t ) = H zx ( t ) +Σ, Σ breaks both PS and PHS. This is because, although H zx admits U P = σ and U C = σ , we have U P Σ( − x ) U † P = Σ( x ) , U C Σ ∗ U † C = − Σ . (26)Nevertheless, the product of PS and PHS holds: U P U C Σ ∗ ( − x ) U † C U † P = − Σ( x ) . (27)This PC symmetry then ensures that Eqs. (20)-(21) hold.So, Q ( µ ) and Q t ( µ ) are even, while Q ∆ ( µ ) is odd.We now consider the inhomogeneous system sketchedin Fig.1-(upper-right panel), where two halves of thechain are illuminated with different amplitudes. Here,both parity and the PC symmetry are explicitly brokenby the non-uniform illumination of the chain. In the caseof the xy chain, PHS still holds, and Q ( µ ) is an oddfunction while Q t/ ∆ ( µ ) is even.The inhomogeneous zx chain is invariant under a XC transformation when ν = 0, with X = U Π, where, inreal space, U | x i = σ ( − x | x i . In momentum space, U = σ ⊗ ( k → k + π ). Setting U C = 1, XC transformsthe Hamiltonian as X [ H ∗ zx ( − k, t )] ν X − = − [ H zx ( k, t )] − ν , (28) X Σ ∗ X − = − Σ . (29)Therefore, PHS, implemented by XC , holds for ν = 0and renders Q ( µ ) odd and Q t/ ∆ ( µ ) even. Because thehomogeneous zx chain, for ν = 0, enjoys both the above PC symmetry and the PHS of Eqs. (28) and (29), nocharge or heat pumping occurs. We note, for the sakeof completeness, that the inhomogeneous xy chain alsoenjoys a similar PHS for ν = 0, with U | x i = ( − x | x i .Table I summarizes these results for the model systemsconsidered. Some representative cases of charge and heat pumping are also illustrated in Figure 1, exhibiting theeven/odd parity identified above. Note that for the in-homogeneous zx chain with ν = 0, neither the charge(black line in the second row right panel) or heat (bot-tom panels black line) pumping are odd or even, as noneof the above discussed symmetries exist. Q ( μ ) - - - = ℍ zx = ℍ zx = ℍ xy Q ( μ ) - - - - - - = ( left ) A = ( right ) t ( μ ) - - - = Δ ( μ ) - - - - - = = t ( μ ) - - - μ zx chainA = ( left ) , A = ( right ) = = Δ ( μ ) - - - μ FIG. 1. Homogeneous (left upper panel) and inhomoge-neous (right upper panel) setups, consisting of illuminatedSSH chains coupled to wide-band leads. Second row: chargepumped in the homogeneous zx chain (left panel) and inho-mogeneous (right panel) setups. Third row: heat pumpingin the homogeneous zx setup. Bottom row: heat pumping inthe inhomogeneous zx setup. (cid:126) Ω = 2 π/ . γ = 0 . N = 10. In summary, we have discussed the role of discrete sym-metries on the pumping of charge, energy or heat. PHScauses the charge (heat) pumping to be an odd (even)function of the chemical potential. On the other hand,the composition of PS and PHS causes the charge (heat)pumping to be an even (odd) function, and the totalheat absorbed to be even. These results provide simplepractical criteria to control the direction of the charge orheat flows, following the symmetry properties of physicalsetups.
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Supplementary Materials for:Symmetry-induced even/odd parity in charge and heat pumping
In this Supplemental Material, we provide additional details on the derivations in the main text. In particular, wediscuss the transformation of the Green’s functions (Sec. S1), and transmission probabilities (Sec. S2). In Sec. S3,we discuss the role of time reversal in detail. We provide the explicit construction of the Green’s function for theexamples used in the main text in Sec. S4. Finally, in Sec. S5 we present a discussion on the consequences of symmetryon transport, based on the Floquet scattering matrix approach.
CONTENTS
References 5S1. Green’s function under symmetrytransformations SM - 1S2. Transmission Probabilities under symmetrytransformations SM - 3S2.1. T , C and P SM - 3S3. The Role of Time Reversal SM - 3S4. Calculation of the Floquet Green’s functionin the wide-band limit SM - 3S5. Symmetries of the Floquet scatteringmatrix SM - 4
S1. GREEN’S FUNCTION UNDER SYMMETRYTRANSFORMATIONS
We first consider the total Hamiltonian, i.e. system+ leads, H ( t ) = C † H ( t ) C , with H † ( t ) = H ( t ),and obtain the transformation properties of the totalGreen’s function. In this case, the evolution of the op-erator C under the transformed Hamiltonian, H X ( t ) = C † H X ( t ) C , is ∂ t C X ( t ) = i [ H X ( t ) , C ( t )] = − i H X ( t ) C ( t ) , (S1)and thus C X ( t ) = U X ( t, C , (S2)with U X ( t, t ) = ( T e − i R tt dτ H X ( τ ) for t > t U X ( t , t ) † for t > t , (S3)yielding the retarded Green’s function G RX ( t, t ) = − i Θ ( t − t ) U X ( t, t ) . (S4) Under time reversal, H T ( t ) = U † T H ∗ ( − t ) U T , (S5)and therefore G RT ( t, t ) = − i Θ ( t − t ) T e − i R tt dτ H T ( τ ) = U † T " − i Θ ( t − t ) T t Y τ = t e − i H > ( − τ )∆ t U T = U † T − i Θ ( t − t ) ¯ T − t Y τ = − t e − i H > ( τ )∆ t U T = U † T − i Θ ( t − t ) T − t Y τ = − t e − i H ( τ )∆ t > U T = U † T " − i Θ ( t − t ) T e − i R − t t dτ H ( τ ) > U T = U † T (cid:2) G R ( − t , − t ) (cid:3) > U T , (S6)where T and ¯ T are, respectively, the forward-time or-dered and backward-time ordered operators.Under charge conjugation, H C ( t ) = − U > C H > ( t ) U ∗ C , (S7)we get G RC ( t, t ) = − i Θ ( t − t ) T e − i R tt dτ H C ( τ ) = U > C " − i Θ ( t − t ) T t Y τ = t e i H ∗ ( t )∆ t U ∗ C = − U > C " − i Θ ( t − t ) T t Y τ = t e − i H ( t )∆ t ∗ U ∗ C = − U > C (cid:2) G R ( t, t ) (cid:3) ∗ U ∗ C . (S8)Under time translation by half a period we have, H Π ( t ) = U † Π H (cid:18) t + T (cid:19) U Π , (S9)M - 2which yields G R Π ( t, t ) = − i Θ ( t − t ) T e − i R tt dτ H Π ( τ ) = U † Π " − i Θ ( t − t ) T t Y τ = t e i H ( t + T ) ∆ t U Π = U † Π (cid:20) G R (cid:18) t + T , t + T (cid:19)(cid:21) U Π . (S10)Finally, under x -coordinate inversion, h x | G RR ( t, t ) | x i = h− x | U † R G RR ( t, t ) U R |− x i . (S11)If the unitary matrices, U X , do not mix degreesof freedom of the system with those of the leads, wemay consider Green’s functions restricted to degrees offreedom solely within the system by defining G R S = P α,β ∈ S | α i h α | G R | β i h β | . However, rather than respect- ing Eq.(S4), the Green’s function obeys[ i∂ t − H X, S ( t )] G RX, S ( t, t ) = Z dτ X l =R,L Σ RX,l ( t, τ ) G RX, S ( τ, t ) , (S12)where Σ RX,l is the transformed self-energy obeying thesame G RX, S . In the following, as in the main text, we dropthe the label S and refer to G R as the Green’s functionof the system.For periodically driven systems, the Floquet retardedGreen’s function is defined as G Rmm ( (cid:15) ) = (S13)1 T Z T dte i (Ω m − Ω m ) t Z dτ G R ( t, t + τ ) e − i ( (cid:15) +Ω m ) τ , where Ω m = m Ω, m ∈ Z . There is some re-dundancy in the definition of this quantity, since G Rm + k,m + k ( (cid:15) − Ω m ) = G Rm,m ( (cid:15) ), which is lifted bydefining the Floquet Green’s function given in the maintext by G ( m ) ( (cid:15) ) = G Rm ( (cid:15) ) . (S14)Nevertheless, it is useful to consider G Rmm ( (cid:15) ) for thederivations in the Supplemental Material.Under the time reversal transformation we have G RT ; mm ( (cid:15) ) = 1 T Z > dte i (Ω m − Ω m ) t Z dτ G RT ( t, t + τ ) e − i ( (cid:15) +Ω m ) τ = 1 T Z T dte i (Ω m − Ω m ) t Z dτ U † T (cid:2) G R ( − t − τ, − t ) (cid:3) > U T e − i ( (cid:15) +Ω m ) τ = 1 T Z T dte − i (Ω m − Ω m ) t Z dτ U † T (cid:2) G R ( t − τ, t ) (cid:3) > U T e − i ( (cid:15) +Ω m ) τ = 1 T Z T dt e − i (Ω m − Ω m ) ( t + τ ) Z dτ U † T (cid:2) G R ( t + nT, t + τ + nT ) (cid:3) > U T e − i ( (cid:15) +Ω m ) τ = 1 T Z T dt e i (Ω m − Ω m ) t Z dτ U † T (cid:2) G R ( t , t + τ ) (cid:3) > U T e − i ( (cid:15) +Ω m ) τ = U † T (cid:2) G Rm m ( (cid:15) ) (cid:3) > U T , (S15)using t − τ = t + nT , n ∈ Z . Similarly, for chargeconjugation G RC ; mm ( (cid:15) ) = − U > C (cid:2) G R − m − m ( − (cid:15) ) (cid:3) ∗ U ∗ C , (S16)and for the time translation by half a period, G R Π; mm ( (cid:15) ) = ( − m + m U † Π G Rmm ( (cid:15) ) U Π . (S17) Using the definition of the Floquet Green’s function, weobtain the transformations given in the main text.M - 3 S2. TRANSMISSION PROBABILITIES UNDERSYMMETRY TRANSFORMATIONSS2.1. T , C and P Using the transformation properties of G ( m ) ( (cid:15) ) andΓ l ( (cid:15) ), the transmission probabilities under T transformas T ( m ) T ; ll ( (cid:15) ) ==tr h G T, ( m ) ( (cid:15) ) Γ T,l ( (cid:15) ) G † T ( m ) ( (cid:15) ) Γ T,l ( (cid:15) + Ω m ) i =tr h G > ( − m ) ( (cid:15) + Ω m ) Γ > l ( (cid:15) ) G † T ( − m ) ( (cid:15) + Ω m ) G > l ( (cid:15) + Ω m ) i =tr h G † T ( − m ) ( (cid:15) + Ω m ) Γ > l ( (cid:15) + m (cid:126) Ω) G > ( − m ) ( (cid:15) + Ω m ) Γ > l ( (cid:15) ) i =tr h G ( − m ) ( (cid:15) + Ω m ) Γ l ( (cid:15) + m (cid:126) Ω) G † ( − m ) ( (cid:15) + Ω m ) Γ l ( (cid:15) ) i ∗ = T ( − m ) l l ( (cid:15) + Ω m ) ∗ , (S18)and, similarly, under C , T ( m ) C ; ll ( (cid:15) ) ==tr h G C, ( m ) ( (cid:15) ) Γ C,l ( (cid:15) ) G † C ( m ) ( (cid:15) ) Γ C,l ( (cid:15) + Ω m ) i =tr h G ∗ C, ( − m ) ( − (cid:15) ) Γ ∗ C,l ( − (cid:15) ) G †∗ C ( − m ) ( − (cid:15) ) Γ ∗ C,l ( − (cid:15) − Ω m ) i =tr h G C, ( − m ) ( − (cid:15) ) Γ C,l ( − (cid:15) ) G † C ( − m ) ( − (cid:15) ) Γ C,l ( − (cid:15) − Ω m ) i ∗ = T ( − m ) ll ( − (cid:15) ) . (S19)Invariance under x -axis inversion implies that theGreen’s function is invariant but the hybridization ma-trices are mapped onto each other, i.e., G P ;( m ) ( (cid:15) ) = G ( m ) ( (cid:15) ) , (S20)Γ P,l ( (cid:15) ) =Γ ¯ l ( (cid:15) ) . (S21)In this case, T ( m ) ll ( (cid:15) ) = tr h G ( m ) ( (cid:15) ) Γ ¯ l ( (cid:15) ) G † ( m ) ( (cid:15) ) Γ ¯ l ( (cid:15) + m (cid:126) Ω) i = T ( m )¯ l ¯ l ( (cid:15) ) . (S22) S3. THE ROLE OF TIME REVERSAL
We here consider the role of time reversal symmetry onthe charge and energy currents. Introducing the trans-formation of the transmission probability, given in themain text, into the expression for the particle current, we obtain J cl = − | e | h X m ∈ Z Z d(cid:15) n T ( m ) l ¯ l ( (cid:15) ) f ¯ l ( (cid:15) ) − T ( − m ) l ¯ l ( (cid:15) + m (cid:126) Ω) f l ( (cid:15) ) o = − | e | h X m ∈ Z Z d(cid:15) n T ( m ) l ¯ l ( (cid:15) ) [ f ¯ l ( (cid:15) ) − f l ( (cid:15) + m (cid:126) Ω)] o . (S23)Similarly, for the energy current, J el = 1 h X m ∈ Z Z d(cid:15) n ( m (cid:126) Ω + (cid:15) ) T ( m ) l ¯ l ( (cid:15) ) [ f ¯ l ( (cid:15) ) − f l ( m (cid:126) Ω + (cid:15) )]+ 12 m (cid:126) Ω T ( m ) ll ( (cid:15) ) [ f l ( (cid:15) ) − f l ( (cid:15) + m (cid:126) Ω)] (cid:27) . (S24)Therefore, for the pumping setup f l ( (cid:15) ) = f ( (cid:15) ), we findthat only in the infinite temperature case can the time-reversed processes happen with the same probability. Inthat case f l ( (cid:15) ) = and J cl = J el = 0 . S4. CALCULATION OF THE FLOQUETGREEN’S FUNCTION IN THE WIDE-BANDLIMIT
In the wide-band limit, the self-energy induced by theleads becomes frequency-independent. Recalling the def-inition of the effective Hamiltonian, we obtain Z ( t ) = H ( t ) − i (Γ L + Γ R ), where H ( t ) is the time-periodicsingle-particle Hamiltonian. The eigenstates of this op-erator obey the Floquet equation, i (cid:126) ∂ t | φ (cid:15) ( t ) i = [ Z ( t ) − (cid:15) ] | φ (cid:15) ( t ) i , (S25)with (cid:15) the quasi-energy. The time Fourier series for theFloquet state reads | φ (cid:15) ( t ) i = X n ∈ Z e − in Ω t | Φ n ( (cid:15) ) i . (S26)Expanding the effective Hamiltonian as Z ( t ) = P n Z n e in Ω t , the Fourier components of the Floquetstate, | Φ n ( (cid:15) ) i , satisfy the equation X n ∈ Z [ Z n − m − n (cid:126) Ω δ n,m ] | Φ n ( (cid:15) ) i = (cid:15) | Φ m ( (cid:15) ) i . (S27)Because Z ( t ) is not hermitian, the quasi-energies are, ingeneral, complex-valued. The Floquet states with (cid:15) and (cid:15) + (cid:126) Ω are physically the same, so it is assumed that − (cid:126) Ω / < < ( (cid:15) ) ≤ (cid:126) Ω /
2. One must also consider the lefteigenstates, φ + (cid:15) ( t ), satisfying the Floquet equation − i (cid:126) ∂ t h φ + (cid:15) ( t ) | = h φ + (cid:15) ( t ) | [ Z ( t ) − (cid:15) ] , (S28)whose Fourier time components obey X n ∈ Z h Φ + n ( (cid:15) ) | [ Z m − n − m (cid:126) Ω δ n,m ] = (cid:15) h Φ + m ( (cid:15) ) | , (S29)M - 4and satisfy the normalization condition P n h Φ + n ( (cid:15) ) | Φ n ( (cid:15) ) i = 1. The orthonormality andcompleteness of the right- and left- eigenvector basisworks out for the lattice sites | i i as X (cid:15) X m ∈ Z (cid:12)(cid:12) Φ + m ( (cid:15) ) (cid:11) h Φ m ( (cid:15) ) | = 1 . (S30)Using this relations, we obtain the Green’s function forFloquet systems as G ( m ) ( E ) = X (cid:15) X n (cid:12)(cid:12) Φ + m + n ( (cid:15) ) (cid:11) h Φ n ( (cid:15) ) | E − (cid:15) − n (cid:126) Ω . (S31) S5. SYMMETRIES OF THE FLOQUETSCATTERING MATRIX
As already stated in the main text, the even/odd be-havior of the transport properties can also be obtainedfrom the Floquet scattering matrix approach. Here, wepresent a discussion of the symmetry properties of theFloquet scattering matrix in general terms, where theasymptoptic form of the wave function far from a scat-terer assumes a plane-wave form.We view the Floquet function ψ ( x, t ) = X n φ n ( x ) e − i (cid:126) Et e − in Ω t , with quasi-energy E , as a superposition of states withenergies E + n (cid:126) Ω. For a scattering state, the spatial partof the Floquet functions, φ n ( x ), takes the form of planewaves far from the scatterer: φ n ( x → −∞ ) = A n e ikx + C n e − ikx (S32) φ n ( x → + ∞ ) = D n e ikx + B n e − ikx . (S33)The Floquet scattering matrix, S ( E + n (cid:126) Ω , E + n (cid:126) Ω),relates the Fourier amplitudes of the incoming waves withthe outgoing ones: " C n D n = S ( E + n (cid:126) Ω , E + n (cid:126) Ω) " A n B n . (S34)We may think of the column vectors as having all entries n ∈ Z . Then, the S matrix has four blocks: S = " S LL ( n , n ) S LR ( n , n ) S RL ( n , n ) S RR ( n , n ) , (S35)and we rewrite (S34) as " C ∗ D ∗ = S " A ∗ B ∗ . (S36)Probability conservation implies SS † = 1. The relationbetween the S matrix and the above transmission prob-abilities is T ( m ) ll ( (cid:15) ) = | S ll ( (cid:15) + m (cid:126) Ω , (cid:15) ) | . (S37)We now consider the role of symmetries. 1. TRS : There exists a unitary matrix, U T , such that U T ψ ∗ ( x, − t ) has the same quasi-energy, E . Com-plex conjugation with t → − t does not change thetime-dependent exponentials, but the spatial partis modified as U T φ ∗ n ( x → −∞ ) = A ∗ n U T e − ikx + C ∗ n U T e ikx (S38) U T φ ∗ n ( x → + ∞ ) = D ∗ n U T e − ikx + B ∗ n U T e ikx , (S39)(here it is assumed that U T acts on the spinor formof the plane waves). This operation inverts the di-rection of propagation of the plane waves. We thenwrite " A ∗ n B ∗ n = S ( E + n (cid:126) Ω , E + n (cid:126) Ω) " C ∗ n D ∗ n , (S40)Then, from (S40) and (S34) we see that " A ∗ B ∗ = S " C ∗ D ∗ ⇔ S > " C ∗ D ∗ = S " C ∗ D ∗ . (S41)Thus, S > = S .2. PHS : There exists a unitary matrix, U C , such thatthe state U C ψ ∗ ( x, t ) has quasi-energy − E . Notethat complex conjugation changes both the timeand spatial dependence of the exponentials, there-fore, the direction of propagation of the waves is notchanged. The state U C ψ ∗ ( x, t ) has the asymptoticbehavior: X n (cid:0) A ∗ n U C e − ikx + C ∗ n U C e ikx (cid:1) e i (cid:126) Et e in Ω t as x → −∞ , X n (cid:0) D ∗ n U C e − ikx + B ∗ n U C e ikx (cid:1) e i (cid:126) Et e in Ω t as x → ∞ . (S42)The waves ‘ n ” have energy − E − n (cid:126) Ω. Taking theenergy labels into account and the definition of the S matrix, we write " C ∗ n D ∗ n = S ( − E − n (cid:126) Ω , − E − n (cid:126) Ω) " A ∗ n B ∗ n , (S43)and comparing with (S34) we get S ∗ ( E + n (cid:126) Ω , E + n (cid:126) Ω) = S ( − E − n (cid:126) Ω , − E − n (cid:126) Ω) . M - 5In particular, for the case n = 0, S ∗ ( E + n (cid:126) Ω , E ) = S ( − E − n (cid:126) Ω , − E ) . (S44)3. Symmetry under U Π C operator: this is the modi-fied PHS for the model H zx at ν = 0, as discussedin the main text.The state U Π φ ∗ (cid:15) ( x, t ) has quasi-energy − E in theHamiltonian H − ν ( t ) + Σ. Its asymptotic behaviorreads X n σ (cid:16) A ∗ n e − i ( k + π ) x + C ∗ n e i ( k + π ) x (cid:17) e i (cid:126) Et e in Ω t + inπ as x → −∞ , X n σ (cid:16) D ∗ n e − i ( k + π ) x + B ∗ n e i ( k + π ) x (cid:17) e i (cid:126) Et e in Ω t + inπ as x → ∞ . (S45)Like in the case of PHS, we see that the waves “ n ” have energies − E − n (cid:126) Ω. We then can write( − n " C ∗ n D ∗ n =( − n S ( − ν ) ( − E − n (cid:126) Ω , − E − n (cid:126) Ω) " A ∗ n B ∗ n , (S46)which, compared with the definition (S34) yields S ( ν ) ∗ ( E + n (cid:126) Ω , E + n (cid:126) Ω) = S ( − ν ) ( − E − n (cid:126) Ω , − E − n (cid:126) Ω)( − n + n . (S47)Taking n = 0, then S ( ν ) ∗ ( E + n (cid:126) Ω , E ) = S ( − ν ) ( − E − n (cid:126) Ω , − E )( − n . Parity : There exists a unitary matrix, U P , suchthat the function P ψ ( − x, t ) has the same quasi-energy, E . The function P ψ ( − x, t ) then obeys U P φ n ( x → −∞ ) = D n U P e ikx + B n U P e − ikx (S48) U P φ n ( x → + ∞ ) = A n U P e ikx + C n U P e − ikx , (S49)so, we write " DC = S " BA (S50) ⇔ σ Sσ = S , (S51)where σ acts on the ( L, R ) subspace. The blocksof the S matrix in (S35) then obey S LL ( n , n ) = S RR ( n , n ) , (S52) S LR ( n , n ) = S RL ( n , n ) ..