Table Of Content

Induced gauge potentials in reduced density matrix dynamics Ryan Requist∗ SISSA, via Bonomea 265, 34136 Trieste, Italy (Dated: May 20, 2014) The combination of interactions and nonadiabaticity in many body systems is shown to induce magneticgaugepotentialsintheequationofmotionfortheone-bodyreduceddensitymatrixaswell as the effective Schro¨dinger equation for the natural orbitals. The consequences of induced gauge geometry for charge and energy transfer are illustrated in the exact nonlinear dynamics of a three- site Hubbard ring ramped into a Floquet state by a time dependent circulating electric potential. Remarkably, the pumped charge flows against the driving in the strongly interacting regime, and the quasienergy level shift, which defines the work done on the system, can become negative. 4 1 PACSnumbers: 03.65.Vf,71.10.-w,72.10.-d,31.15.ee 0 2 y Gauge geometry is inherent to physical theories whose though induced gauge potentials and geometric phases a equations are phrased in terms of redundant variables. are equally valid for interacting systems, they are diffi- M Theclassicalelectromagneticgaugepotentials(V,A)are culttocomputeifthecomplexityofthemanybodywave redundant because infinitely many of them generate the function scales exponentially with the number of parti- 7 same electromagnetic fields, yet they acquire a degree of cles. For this reason, it is desirable to identify geomet- 1 observable significance in quantum physics through the ric structures at the finer level of n-body reduced den- ] Aharonov-Bohm effect [1]. The phase factor [2] sity matrices (rdms), defined through the partial trace el (cid:18) (cid:73) (cid:19) ρn = (cid:0)Nn(cid:1)Trn+1...Nρ. Reduced geometric phases for n- r- exp i(cid:126)e Aµdxµ , (1) body rdms are one example [28]. t s ThepurposeofthisLetteristopointouttheexistence t. responsible for Aharonov-Bohm interference, is the fiber of induced gauge geometries in the equations of motion a m bundleholonomyoftheconnection i(cid:126)eAµdxµassociated for n-body rdms ρ1, ρ2, ρ3, ..., which are organized into − a chain-like structure called the quantum Bogoliubov- with local gauge invariance (gauge symmetry) [3, 4]. - Born-Green-Kirkwood-Yvon(BBGKY)hierarchy. These d Induced, as opposed to primitive, gauge geometries n have gained attention only relatively recently. Induced multifarious gauge structures are associated with the o gauge freedom induced by separating the rdm variables vector potentials were first found in the coupled equa- c intoahierarchyoflevelsρ ρ ρ ...,inagreementwith [ tionsforelectronicandnuclearwavefunctions[5],andthe 1\ 2\ 3\ associatedAharonov-Bohmphasegivesanalternativeex- Berry’snotionthatinducedgaugegeometriesresultfrom 2 thedivisionofacompositesystemintotwoparts[10]. For planationforthesignchangeuponpseudorotationintri- v example, at the first level of the hierarchy, the marginal atomic molecules [6]. The discovery of geometric phase 9 densityρ actslikethenuclearwavefunctionΦ(R)inthe 1 [7–9] established the geometric origin and observability 1 Born-Oppenheimer approximation, while the density ρ 7 of induced gauge geometries. Induced vector potentials 2 3 are in fact only the magnetic part of a general geometric (conditional on ρ1) acts like the electronic factor ψR(r). 1. electromagnetism [10–12], unifying induced electric and In the simplest (Abelian) case, the gauge variables are 0 magnetic fields in a quantum geometric tensor [10, 13] U(1)phasescorrespondingtotheunitarytransformation 4 over the space of slow variables. The effective Hamil- V†ρ2V,whereV =eiφknk andnk isthenumberoperator 1 for a ρ eigenstate. Our results suggest an extension of tonian for the slow variables generally also contains a 1 : v geometric induced inertia tensor [14]. geometric electromagnetism to many body systems and i establish the BBGKY hierarchy as a framework for ap- X Abelianandnon-Abeliangaugegeometrieshavefound plying differential geometry to many body dynamics. many applications in condensed matter physics, among r a which are adiabatic charge transport [15], the theory The physical effects of induced gauge potentials are of macroscopic polarization [16], the anomalous veloc- exemplified here in a three-site Hubbard ring ramped ityandothergeometriceffectsofBlochelectrons[17–20], into a Floquet state by a circulating potential well. In- andthequantumHalleffect[15,21,22]. Recentworkhas duced gauge potentials mediate energy transfer through studiedtheBerrycurvatureingradientexpansionsofthe the electromotive force implied by dynamical variations quantum kinetic equations of Fermi liquids [19, 23]. An- oftheinducedmagneticflux(Faraday’slaw). Wefindan other line of research aims to simulate condensed matter intriguing many body effect whereby the pumped charge phasesbyrealizingartificialgaugepotentialsfortrapped flowsbackwardsagainstthedrivingfieldswhenHubbard ultracold neutral atoms [24–26]. interactionsaresufficientlystrong. Theworkdoneonthe The above gauge geometries were formulated for non- systembythedrivingfieldsduringtheadiabaticramping interacting systems or at the mean field level. Al- is given by the quasienergy level shift, and surprisingly, 2  whichdescribeselectronsthathopwithamplitudest i,i+1   among three sites (we set t = 1). Coulomb interac- i,i+1   tions are approximated by a local on-site Hubbard form  0.4 U = U(nˆ1↑nˆ1↓+nˆ2↑nˆ2↓+nˆ3↑nˆ3↓). The sites can repre- 1.0 sent atomic orbitals, quantum dots, impurities, etc. For example, thisthree-siteHubbardringwasusedtomodel 0.8 0.3 the quantum electric dipole moment (a geometric effect) 0.6 oftriatomicmoleculesandtriplequantumdots[30]. Our Ω Ω0.2 main result is that the reduced equations of motion con- 0.4 tain magnetic gauge potentials even though Eq. (5) has 0.1 0.2 no external magnetic fields, since ti,i+1 are real and (cid:15)i represent purely electric driving. We demonstrate the 0.0 0.0 0 5 10 15 20 0 2 4 6 8 10 12 existence of induced gauge potentials in two quantities: U U (I)theoperatoruinEq.(2)and(II)theeffectiveHamil- FIG. 1: (a) Three-site Hubbard ring, (b) pumped charge Q tonian h for the natural orbitals (defined below). and (c) ratio of the real to imaginary part of the time aver- Gauge geometry type I — The quantity agedgauge-invariantloopquantityu u u asafunctionof 12 23 31 π (U,ω0) for ramping speed α=0.11, well depth (cid:15)=4, U =7 Φ =Argu u u (6) u 12 23 31 andinitialphaseφ =2π/3. Scale: black=0,lightgreen=0.4. − 2 0 (cid:80) with u = (2U/i) (δ δ )ρ is gauge invariant jk l jl− kl 2,jlkl like Φ in Eq. (4), cf. Φ = Argt t t . If Φ = 0, the it can become negative. 12 23 31 u (cid:54) operator u contains a magnetic-type flux, implying non- Our starting point is the first equation of the BBGKY trivialgaugegeometryassociatedwithclosedloopsinco- hierarchy, the dynamical equation for the operator ρ , 1 ordinatespace. LikethebaremagneticfluxpresentinH i∂ ρ =[v,ρ ]+iu, (2) ifΦ=0, itcannotberemovedbyanygaugetransforma- t 1 1 (cid:54) tion. Figure 1c shows the ratio of the real to imaginary where v =p2/2m+V(rt) and the Hermitian operator u (cid:82)t+T/6 partof u u u dsatlongtimesforadrivenring is nonlocal in coordinate/spin space t 12 23 31 with two electrons in a spin singlet. This proves that rur(cid:48) = 2(cid:90) d3z(cid:20) e2 e2 (cid:21)ρ (rzr(cid:48)z), (3) u12u23u31 is not purely imaginary, and hence Φu (cid:54)= 0. (cid:104) | | (cid:105) i r z − r(cid:48) z 2 | The implied gauge geometry is due to the cooperation | − | | − | of interactions and nonadiabaticity. This is intuitively suppressing spin indices. Although rur(cid:48) is not invari- clear since in the noninteracting limit, u 0 in Eq. (2) (cid:104) | | (cid:105) → ant to local gauge transformations, it nevertheless con- and v has no magnetic fields by assumption. At the tains gauge invariant information. This is easily seen in same time, the nonadiabatic transitions between instan- thecontextoflatticemodels,wherevectorpotentialsare taneouseigenstatesresponsibleforinducingthemagnetic represented by Peierls phases on the links between sites, fluxvanishinthelimitω 0(see[31]forthederivation 0 i.e. tij tijeiAij. A given lattice Hamiltonian has mag- of an adiabatic effective m→any body Hamiltonian). Fig- → netic fields if and only if the flux through a plaquette, a ure 1c provides numerical support for these conclusions, gauge-invariant quantity, is nonzero, showing that the time averaged real part of u u u 12 23 31 vanishes faster than imaginary part in both the nonin- N (cid:88) teracting and adiabatic limits. The (cid:15) driving is param- Φ= A =0, (4) i n,n+1 (cid:54) eterized by (cid:15) = V /2+V /2√3, (cid:15) = V /2+V /2√3, n=1 1 3 8 2 − 3 8 and (cid:15) = V /√3 with (notations explained in [31]) 3 8 where the sum runs over a circuit of sites n=1,2,...N − (cid:104) (cid:105) andsiteN+1isthesameassite1. ThephasefactoreiΦ V = √3(cid:15) sin φ(t)+φ 3 0 is analogous to the Wilson loop phase factor in lattice − (cid:104) (cid:105) gauge theory [29]. For simplicity, we restrict our atten- V =+√3(cid:15) cos φ(t)+φ , (7) 8 0 tion to U(1) lattice gauge theory to avoid complications associated with path ordering. where φ(t) = ω (t+ 1 log2coshαt)/2, corresponding to 0 α The minimal model realizing nontrivial induced gauge the frequency ω(t) = dφ/dt = ω (1+tanhαt)/2. This 0 potentialsisathree-siteHubbardring(Fig.1b)withthe describesapotentialwell,localizedatsite1att= if Hamiltonian φ = 2π/3, which slowly increases its rate of circu−la∞tion 0 H = (cid:88)(cid:16)t c† c +H.c.(cid:17)+(cid:88)(cid:15) (t)nˆ aroundtheringultimatelyreachingaconstantrotational − i,i+1 iσ i+1σ i iσ speed ω0. In all cases, we choose the initial state to be i,σ i,σ the ground state. The ground state energy depends on +U(nˆ nˆ +nˆ nˆ +nˆ nˆ ), (5) 1↑ 1↓ 2↑ 2↓ 3↑ 3↓ φ . For small U, the most stable ground state occurs 0 3 for φ =0 mod 2π/3, since for that value both electrons factor [35, 36] 0 can lower their energy by occupying the potential well; however,asU isincreasedanditbecomesunfavorablefor Ψ(t) =e−i(cid:82)−t∞Ω(s)ds ξ(t) . (8) | (cid:105) | (cid:105) This factorization is not unique, and for convenience we 1.6 have chosen ξ(t) = e−iArg(cid:104)ψ0|ψ(t)(cid:105) ψ(t) . This choice 1.2 v +6 gives an oscil|lator(cid:105)y Ω(t) = i∂ log ψ| ψ(t(cid:105)) as shown in hvi +6 t (cid:104) 0| (cid:105) 0.8 hhUiiT Fig. 2b, but the running time average ΩT(t) approaches 0.4 hhhUiiiT a constant asymptotic quasienergy 0 ξ H(t) i∂ ξ t Ω= (cid:104) | − | (cid:105). (9) ξ ξ 0 (cid:104) | (cid:105) -0.1 ThequasienergyofaFloquetstateisonlydefinedmodulo -0.2 ∆E ω0,implyingthatthesetofquasienergieshaveaBrillioun ∆E zonestructure[0,ω ],[ω ,2ω ]... Itispossibletomakea -0.3 HT E 0 0 0 -0.4 hhhHiii−T−0E0 dapiffperroeancthgeasuΩge+cnhωoicefofroarn|ξy(nt).(cid:105)Tsuhcehinthteagtetrhne,pwahthichΩTco(tr)- 0 -0.5 responds to a winding number of ξ(t) , is a topological | (cid:105) quantity in the sense that any two paths that end in dif- -30 -20 -10 0 10 20 30 40 50 60 t(1/t ) ferent zones at t = cannot be smoothly transformed 12 ∞ into each other. Nevertheless, there is a class of gauge FIG. 2: (a) Instantaneous one-body (cid:104)v(cid:105) and interaction (cid:104)U(cid:105) choices for which Ω (t) remains close to the adiabati- T energies,(b)thequasienergylevelshift∆Eandinstantaneous cally continued quasienergy Ω (t) of the instantaneous 0 energy (cid:104)H(cid:105); all shown with running time averages. Same Floquet state ξ (t) , thereby defining a unique Ω [31]. 0 parameters as Fig. 1. | (cid:105) The constant asymptotic level shift ∆E = Ω E 0 − represents the work done on the system in the course both electrons to occupy the same site, the ground state of ramping on the perturbation. Evidence of that work undergoesatransitiontoadelocalizedstateforwhichthe is done is visible in the running time average H(t) T most stable value of φ0 is π mod 2π/3. For φ0 = π, the shown in Fig. 2b, which changes from E0 to a(cid:104)c(cid:104)onsta(cid:105)(cid:105)nt potential well is halfway between sites 1 and 2, so they value close to Ω. The persistent oscillations in H(t) are initially degenerate. Cyclic driving protocols similar represent the continuous exchange of energy, bac(cid:104)k and(cid:105) to Eq. (7) have been realized in trapped Bose-Einstein forth, between the system and its environment, i.e. the condensates [32, 33] and could be implemented in triple collection of charges, currents and fields responsible for quantumdots(seeRef.[34]andreferencescitedtherein). producingthegivenelectricdriving(weneglectthesmall The operator u describes how interactions affect the associated magnetic fields). In order for ∆E to provide dynamicsofρ1 bymediatingenergytransferbetweencol- a consistent definition of work, it is imperative to keep lective variables and internal interaction energy. To see track of its zone. In second-order perturbation theory, this, consider the time derivative of the one-body energy ∆E as observable in the ac Stark shift [35]. Figure 3 shows how ∆E changes as a function of U. d v The quasienergy level shift can assume negative values. (cid:104) (cid:105) =Tr(ρ ∂ v)+Tr(uv) 1 t dt First, consider small U. If φ = 0, the system starts (cid:88) 0 = n (cid:15)˙ +(4+(cid:15)3cos3(φ+φ ))u u u sinΦ in the stable ground state and ∆E is positive because i i 0 12 23 31 u | | i the driving does work on the system by increasing its time-averaged kinetic plus potential energy v . On T The first term is the power applied to the whole system (cid:104)(cid:104) (cid:105)(cid:105) theotherhand,ifφ =π,thesystemstartsintheunsta- 0 bytheexternaldriving,andthesecondtermisthepower ble ground state and ∆E is negative because the system applied on the one-body variables by two-body interac- lowers v bystartingtorotate. Now,considerlargeU. T tions;thelatterismodulated bythefluxΦ andvanishes (cid:104)(cid:104) (cid:105)(cid:105) u The situation is reversed, and ∆E is negative for φ =0 0 when Φ = 0. Figure 2a illustrates energy exchange be- u andpositiveforφ =π. Forφ =0,theworkdoneonthe 0 0 tween v and ;alsoshownaretherunningtimeaver- system is negative because the stabilization of the inter- (cid:104) (cid:105) (cid:104)U(cid:105) (cid:82)t+T/2 ages, e.g. (cid:104)(cid:104)v(t)(cid:105)(cid:105)T =(1/T) t−T/2(cid:104)v(s)(cid:105)ds; T =2π/ω0. actionenergy(cid:104)U(cid:105)morethancompensatesfortheincrease TheHamiltonianin(5)becomesT-periodicast . in v , as shown in Fig. 2a. Another physical mechanism →∞ (cid:104) (cid:105) For sufficiently small α the system evolves adiabatically thatstabilizestherotatingstateisgeometricphase. The from the ground state ψ to a Floquet state ψ(t) [31]. geometricphasecontributiontoΩis i1 (cid:82)t+T ξ ∂ ξ ds, | 0(cid:105) | (cid:105) − T t (cid:104) | s (cid:105) The wavefunction can be split into a factor ξ(t) which and since it is negative, it stabilizes the rotating state. | (cid:105) becomesperiodicinthesteadystateandanoverallphase ThisstabilizationcanbeseeninthesmalloffsetofΩ (t) T 4 from H(t) in Fig. 2b. Remarkably, there is a small not be possible without induced magnetic fields. A simi- T (cid:104)(cid:104) (cid:105)(cid:105) regionnearU 3.3wherethesystemgivesupenergyby lar situation would occur if time dependent current den- ≈ adopting a rotating state for any initial phase φ . sityfunctionaltheory[38,39]wereappliedtothepresent 0 problem because the noninteracting Kohn-Sham system would contain an induced vector potential A even in 0.08 xc ∆E(φ =0) 0 the absence of externally applied magnetic fields. ∆E(φ =π) 0 Q/4e 0.04 Re(h h h ) 0.001 12 23 31 Im(h h h ) 12 23 31 0 0.0005 -0.04 0 -0.08 1 2 3 4 5 6 7 8 -0.0005 U FIG.3: Quasienergylevelshift∆E andpumpedchargeQas -0.001 a function of U for α=0.125, ω =0.3 and (cid:15)=4. 0 -10 0 10 20 30 40 t(1/t ) Gauge geometry type II — A second type of induced 12 gauge potential appears in the Hamiltonian h governing FIG.4: Realandimaginarypartsofthegaugeinvariantloop the dynamics of single-particle states ψ defined as | k(cid:105) quantity h12h23h31 for the same parameters as Fig 1. ψ =√n e−iζk φ , (10) | k(cid:105) k | k(cid:105) Including the ζk phases in the definition of the ψk | (cid:105) makes them properly gauge invariant and allows us to where φ isanaturalorbital(eigenstateofρ ),n isthe occupa|tiko(cid:105)n number and ζ is the phase conju1gatekto n define individual quasienergies Ωk by applying the same k k factorization as in Eq. (8). In order for the two-body [27]. The states in Eq. (10) were introduced in Ref. 37, state ξ to be periodic in the long-time regime, we must except without the factor √nk. The factor √nk was have Ω| (cid:105)=Ωmodω . Nevertheless, Ω (t) can have quite addedinRef.28forageometricreason,namelyi ψ dψ k 0 k (cid:104) k| k(cid:105) different time profiles within one period, and the phase constitutes a connection one-form whose holonomy is a variables ζ display nontrivial winding numbers. reduced geometric phase. For the two-electron system k The pumped charge Q is a decreasing function of U, consideredhere,theset ψ containsallofthedegrees {| k(cid:105)} as shown in Figs. 1b and 3, because the electric driving of freedom of Ψ in a compact form, which is apparent from the expr|ess(cid:105)ion |Ψ(cid:105) = (cid:80)k(cid:112)nk/2e−iµ−i2ζk|φkφk(cid:105). fieffeledcstibveecionmpeulmespsineffgecchtaivregefoornllayrginesUof.arEalescttrhiecrefiealrdesiamre- Since the induced magnetic flux enters h exactly as an balances between the site occupations, and for large U external magnetic flux does, it has a more straightfor- the amplitude of the periodic oscillations in the site oc- ward interpretation than the type I induced flux. cupancies is suppressed by strong two-body correlations The effective Schrödinger equation for the ψ is | k(cid:105) which inhibit double occupancy. Remarkably, the pumped charge becomes negative for i∂ ψ =hψ , (11) t| k(cid:105) | k(cid:105) large U. This is a many body effect related to the fact that for large U the site occupations are pinned to 1, wherehmustbenon-Hermitiansinceitchangesthemod- giving two singly occupied sites and one empty site. Al- ulusof ψ . Thecriterionforhtohaveinducedmagnetic | k(cid:105) thoughchargeflowswiththedrivingalongthelinksnear- fieldsisΦ =Argh h h =0. Figure4showsthereal h 12 23 31 (cid:54) esttothepotentialwell,thereisanevenlargerbackwards andimaginary partsof h h h . Theelementsof h are 12 23 31 current along the link opposite to the well. The pumped very strongly renormalized with respect to the given v. charge is related to the quasienergy according to Therenormalizationofthehoppingandon-siteelements hij canbeunderstoodalongthelinesoftherenormaliza- Q ∂ΩT tion in the Gutzwiller approximation. = , (12) e − ∂Φ That h must contain magnetic fields is not obvious. If all the reduced system had to do was pump charge, which is similar to a formula for Cooper pair pumping electric fields would be sufficient. However, h must also insuperconductingcircuits[40]. Equation(12)isrelated reproduce the dynamics of all ψk , including their indi- tothestationarityofthequasienergyΩ[JT][31]. There- | (cid:105) (cid:82)t+T vidual dynamical and geometric phases, and that would duced geometric phases i ψ ∂ ψ ds contribute to t (cid:104) k| s k(cid:105) 5 the pumped charge since their sum gives the geometric Porto, and I. B. Spielman, Nature 462, 628 (2009). contributiontothequasienergy. Apartfromthecoupling [26] J. Dalibard, F. Gerbier, G. Juzeliunas, and P. O¨hberg, to external electric fields, our model is a closed system. Rev. Mod. Phys. 83, 1523 (2011). [27] R. Requist and O. Pankratov, Phys. Rev. A 83, 052510 The effect of dissipation on the pumped charge in a non- (2011). interacting three-site ring coupled to a bath of harmonic [28] R. Requist, Phys. Rev. A 86, 022117 (2012). oscillators has been studied [41]. [29] K. G. Wilson, Phys. Rev. D 10, 2445 (1974). In summary, we identified two types of induced gauge [30] P. B. Allen, A. G. Abanov, and R. Requist, Phys. Rev. geometry resulting from the conjunction of interactions A 71, 043203 (2005). andnonadiabaticityinmanybodysystems. Theimplica- [31] See Supplemental Material. tionsoftheassociatedeffectivemagneticfieldsforcharge [32] K. C. Wright, R. B. Blakestad, C. J. Lobb, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 110, andenergytransferwereillustratedinadriventhree-site 025302 (2013). Hubbard ring; the predicted phenomena are potentially [33] G. Arwas, A. Vardi, and D. Cohen, arxiv:1308.5860v1 observable in triple quantum dots or ultracold atoms. (2013). Early stages of the research were supported by the [34] C.-Y.Hsieh,Y.-P.Shim,M.Korkusinski,andP.Hawry- Deutsche Forschungsgemeinshaft (Grant No.PA516/7-1) lak, Rep. Prog. Phys. 75 (2012). andlaterstagesbyPRIN/COFIN2010LLKJBX004and [35] P. W. Langhoff, S. T. Epstein, and M. Karplus, Rev. Mod. Phys. 44, 602 (1972). 2010LLKJBX 007, Sinergia CRSII2136287/1 as well as [36] R. Requist and O. Pankratov, Phys. Rev. A 79, 032502 ERC Advanced Grant 320796 – MODPHYSFRICT. (2009). [37] K.J.H.Giesbertz,O.V.Gritsenko,andE.J.Baerends, Phys. Rev. Lett. 105, 013002 (2010). [38] S. Ghosh and A. Dhara, Phys. Rev. A 38, 1149 (1988). [39] G. Vignale, Phys. Rev. B 70, 201102(R) (2004). ∗ Electronic address: [email protected] [40] A. Russomanno, S. Pugnetti, V. Brosco, and R. Fazio, [1] Y.AharonovandD.Bohm,Phys.Rev.115,485(1959). Phys. Rev. B 83, 214508 (2011). [2] T.T.WuandC.N.Yang,Phys.Rev.D12,3845(1975). [41] F. Pellegrini, C. Negri, F. Pistolesi, N. Manini, G. E. [3] V. Fock, Zeit. fu¨r Phys. 39, 226 (1926). Santoro, and E. Tosatti, Phys. Rev. Lett. 107, 060401 [4] H. Weyl, Proc. Nat. Acad. Sci. 15, 323 (1929). (2011). [5] C.A.MeadandD.G.Truhlar,J.Chem.Phys.70,2284 (1979). [6] H. C. Longuet-Higgins, U. O¨pik, M. H. L. Pryce, and R. A. Sack, Proc. R. Soc. London, Ser. A 244, 1 (1959). [7] M. V. Berry, Proc. Roy. Soc. Lond. A 392, 45 (1984). [8] B. Simon, Phys. Rev. Lett. 51, 2167 (1983). [9] Y.AharonovandJ.Anandan,Phys.Rev.Lett.58,1593 (1987). [10] M.V.Berry,Thequantumphase,fiveyearsafter (World Scientific, Singapore, 1989), pp. 7–28. [11] R. Jackiw, Commun. At. Molec. Phys. 21, 71 (1988). [12] M. V. Berry and J. M. Robbins, Proc. R. Soc. Lond. A 442, 641 (1993). [13] J. P. Provost and G. Vallee, Commun. Math. Phys. 76, 289 (1980). [14] A. S. Goldhaber, Phys. Rev. A 71, 062102 (2005). [15] J.E.Avron,A.Raveh,andB.Zur,Rev.Mod.Phys.60, 873 (1988). [16] R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993). [17] M. C. Chang and Q. Niu, Phys. Rev. Lett. 75, 1348 (1995). [18] D. Culcer and Q. Niu, Phys. Rev. B 74, 035209 (2006). [19] R. Shindou and L. Balents, Phys. Rev. Lett. 97, 216601 (2006). [20] J. Zak, Phys. Rev. B 40, 3156 (1989). [21] R. B. Laughlin, Phys. Rev. B 23, 5632 (1981). [22] J. Fro¨hlich and U. M. Struder, Rev. Mod. Phys. 65, 733 (1993). [23] C. H. Wong and Y. Tserkovnyak, Phys. Rev. B 84, 115209 (2011). [24] R. Dum and M. Olshanii, Phys. Rev. Lett. 76, 1788 (1996). [25] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V. 1 Supplemental Material S1. Lie algebra parameterization, angular parameterization and a generalized Bloch equation Theanalysisofthethree-siteHubbardmodelinvolvesHermitian3 3matrices, whicharenaturallyparameterized × (cid:80) using the su(3) Lie algebra. For example, the spin-summed one-body rdm ρ = ρ can be expanded as 1,ij σ 1,iσjσ ρ =ρ(cid:126) (cid:126)ν, (S1) 1 1 · where (cid:126)ν is a nine-component vector whose first eight elements are the Gell-Mann matrices and whose ninth element is the identity matrix. Similarly, the one-body terms of the Hamiltonian in Eq. (5) can be expressed as 1 v = V(cid:126) (cid:126)νˆ, (S2) 2 · where (cid:126)νˆ is the vector of operators νˆ =(cid:80) c† ν c and the elements of V(cid:126) =Tr(Hˆ(cid:126)νˆ) are k µνσ µσ k,µν νσ V = 2Ret = 2 V = 2Ret = 2 V = 2Ret = 2 1 12 4 31 6 23 − − − − − − V = 2Imt =0 V = 2Imt =0 V = 2Imt =0 2 12 5 31 7 23 − − − (cid:114) 1 2 V =(cid:15) (cid:15) V = ((cid:15) +(cid:15) 2(cid:15) ) V = ((cid:15) +(cid:15) +(cid:15) ). (S3) 3 1 2 8 1 2 3 9 1 2 3 − √3 − 3 The on-site energies (cid:15) depend only on the variables (V ,V ,V ). In the main text, the driving is chosen to be i 3 8 9 (cid:104) (cid:105) V = √3(cid:15) sin φ(t)+φ 3 0 − (cid:104) (cid:105) V =+√3(cid:15) cos φ(t)+φ (S4) 8 0 and, without loss of generality, the spatial constant V is set to zero. The factor √3 is introduced to normalize the (cid:15) , 9 i so for example ((cid:15) ,(cid:15) ,(cid:15) ) equals (cid:15)( 1,1,1) if φ+φ = 2π and (cid:15)( 1, 1,1) if φ+φ =π. 1 2 3 − 2 2 0 3 −2 −2 0 Since the number of electrons is conserved, the occupation numbers n (eigenvalues of ρ ) can be parameterized as k 1 2 A n = + +B a 3 3 2 A n = + B b 3 3 − 2 2A n = , (S5) c 3 − 3 where A=(n +n 2n )/2 and B =(n n )/2 satisfy the inequality constraints 0 B A 1 due to the Pauli a b c a b − − ≤ ≤ ≤ principle and because we choose n n n . c b a ≤ ≤ The natural orbitals φ are parameterized in terms of 6 angle variables (θ ,θ ,θ ,ϕ ,ϕ ,ϕ ) as follows [S1] k 1 2 3 1 2 3 | (cid:105) cosθ1cosθ2e−i(ϕ1−ϕ2)/2  φa = cosθ1sinθ2e−i(ϕ1+ϕ2)/2  sinθ1e+i(ϕ1+ϕ2)/2  sinθ1cosθ2sinθ3e−i(ϕ1+ϕ3)/2 sinθ2cosθ3e+i(ϕ1+ϕ3)/2e+iϕ2  − − φb =+cosθ2cosθ3e+i(ϕ1+ϕ3)/2 sinθ1sinθ2sinθ3e−i(ϕ1+ϕ3)/2e−iϕ2  − +cosθ1sinθ3e+i(ϕ1−ϕ3)/2  sinθ1cosθ2cosθ3e−i(ϕ1+ϕ3)/2+sinθ2sinθ3e+i(ϕ1+ϕ3)/2e+iϕ2  − φc = cosθ2sinθ3e+i(ϕ1+ϕ3)/2 sinθ1sinθ2cosθ3e−i(ϕ1+ϕ3)/2e−iϕ2 . (S6) − − +cosθ1cosθ3e+i(ϕ1−ϕ3)/2 2 Thevariables(θ ,ϕ )describebeyondmeanfielddynamicsbecausetheyarenotpresentinthemoststronglyoccupied 3 3 orbital φ , which is the only occupied orbital in a mean field-like theory. To investigate the structure of the (θ ,ϕ ) a 3 3 | (cid:105) subspace, consider the unitary transformation from the site basis of Eq. (S6) to the basis (φ ,φ ,φ ), where a u v  sinθ2e+iϕ2/2   sinθ1cosθ2e−i(ϕ1−ϕ2)/2  − − φu =+cosθ2e−iϕ2/2 , φv = sinθ1sinθ2e−i(ϕ1+ϕ2)/2  (S7) − 0 cosθ1e+i(ϕ1+ϕ2)/2 are two states orthogonal to φ . In this basis ρ is block diagonal a 1     n 0 0 0 0 0 a 1 ρ1 = 0 21(nb+nc) 0 + 2(nb−nc)0 cos2θ3 sin2θ3 e−i2ϕ3 , 0 0 12(nb+nc) 0 sin2θ3 e+i2ϕ3 −cos2θ3 and we see that (θ ,ϕ ) parameterize the orbit of an SU(2) subgroup of SU(3) acting on ρ . 3 3 1 Thewavefunctioncanbeexpressedintermsofthefullsetof10independentoccupationnumberandanglevariables (A,B θ ,θ ,θ ,ϕ ,ϕ ,ϕ ζ,η,µ) as follows 1 2 3 1 2 3 | | 1 (cid:88) Ψ = e−i2ζk√n c† c† 0 , (S8) | (cid:105) √2 k k↑ k↓| (cid:105) k where ζ =(µ+ζ +η)/2, ζ =(µ+ζ η)/2 and ζ =(µ 2ζ)/2. To obtain the results reported in the Letter, the a b c − − Schrödinger equation was solved in two ways: (i) directly in the complete eigenbasis of many body singlet states, see Eqs. (S12), and (ii) via the explicit equations of motion for the 10 occupation number and angle variables. Exactly the same results were obtained in both cases. The latter equations of motion were derived from the stationary action principle and will be reported elsewhere [S2]. Having solved for the 10 occupation number and angle variables, we can construct the dynamics of the states ψk = e−iζk√nk φk or, alternatively, the phase-including natural orbitals | (cid:105) | (cid:105) χk =e−iζk φk . The χk and nk are plotted in Figs. S1 and S2 for the same parameters as Figs. 1, 2, and 4. | (cid:105) | (cid:105) | (cid:105) Inanalogywiththetwo-siteHubbardmodel[cf.Eq.(31)ofRef.S3],theequationofmotionforρ canbeexpressed 1 as a generalized Bloch equation ∂ ρ(cid:126) =V(cid:126) ρ(cid:126) +U(cid:126), (S9) t 1 1 ∧ wherethewedgeproductrepresents 1(cid:80) C V ρ andC arethestructureconstantsofthesu(3)Liealgebra,and 2 ij ijk i 1,j ijk V(cid:126) andU(cid:126) aredefinedaccordingtov = 1V(cid:126) (cid:126)ν andu= 1U(cid:126) (cid:126)ν. InFig.S3,weplotthesiteoccupationsandtheelements 2 · 2 · ofρ(cid:126) correspondingtooff-diagonalGell-Mannmatrices,i.e.(ρ ,ρ ,ρ ,ρ ,ρ ,ρ ),forU =8,ω =1/4,(cid:15)=4andφ = 1 1 2 4 5 6 7 0 0 2π/3. ThesystemreachesadefiniteFloquetstatewhereallvariablesareperiodic; evidencefortheadiabaticityofthe ramping wrt the basis of instantaneous Floquet states is given in Sec. S3. The function ρ(cid:126) (A,B,θ ,θ ,θ ,ϕ ,ϕ ,ϕ ) 1 1 2 3 1 2 3 can be inverted analytically, and the reduced geometric phases of the ψ can be evaluated analytically [S2]. k | (cid:105) Induced gauge geometries and reduced geometric phases are related to the fiber bundles associated with the orbits of ρ in case I or ψ in case II. The manifold on which the ρ dynamics takes place can be identified with the 1 k 1 {| (cid:105)} coadjointorbitofaLiegroupGactingonthedualg∗ ofaLiealgebrag. Coadjointorbitshaveanaturalfiberbundle structure [S4]. What is interesting about such bundles in the framework of the BBGKY hierarchy is that the fibers are degrees of freedom of higher-order rdms. For example, a holonomy generated by the dynamics of ρ in the base 1 space influences higher-order rdms and hence two-body correlation functions. Similar arguments apply to ψ and k {| (cid:105)} analogous sets of variables associated with higher-order rdms [S5]. S2. Derivation of an effective many body Hamiltonian with dynamically induced magnetic fields Theinducedmagneticgaugepotentialsstudiedinthemaintextappearedintheexactreducedone-bodyequations of motion, and since those equations have a single-particle form, they can be interpreted as the equations of motion ofaneffectivenoninteracting system, seeSec.(S4). Itisinteresting, asitgivesanalternativeperspective, toexamine induced gauge potentials at the fully interacting level, i.e. within a many body approach that retains all interactions. Floquet theory is one method for doing so, although it appears difficult to obtain analytic results for the present case except within perturbation theory [S6–S8], e.g. in the limit U ω . Applying Floquet theory to the present model 0 (cid:28) is an interesting problem for future work. Here we shall instead consider the adiabatic limit ω 0, where we can 0 → use standard adiabatic analysis, which has the additional advantage of not being limited to time-periodic dynamics. 3 We shall find not only generic dynamically-induced gauge potentials but also new types of complex interactions with their own gauge structure. Intheadiabaticregime,thewavefunctionofasystemthatstartsinthegroundstatestaysclosetotheinstantaneous ground state η (t) throughout the dynamics. Consider a unitary transformation U(t) to the adiabatic basis η(t) = 0 | (cid:105) | (cid:105) U†(t)ψ(t) , where U(t) diagonalizes H(t), i.e. U†HU =diag(E ,E ...). The dynamical equation for η(t) is 0 1 | (cid:105) | (cid:105) i∂ η(t) =H (t)η(t) (S10) t 1 | (cid:105) | (cid:105) with H = U†HU iU†∂ U. The nonadiabatic term iU†∂ U couples the instantaneous eigenstates of H , thereby 1 t t 1 − − inducinganonvanishingcurrentintheinstantaneousgroundstateofH (seeSec.S5)[S3]. Thepresenceofpersistent 1 currents in the ground state suggests that the effective Hamiltonian H contains induced magnetic gauge potentials. 1 Toprovethatitdoes,itsufficestoshowthatthegaugeinvariantquantityt1 t1 t1 = 1(V1 iV1)(V1 iV1)(V1 iV1) 12 23 31 8 1 − 2 6 − 7 4 − 5 hasanonvanishingargumentΦ1 =Argt1 t1 t1 . Here,t1 aretheeffectivehoppingelementsofH andV1 =Tr(Hˆ νˆ) 12 23 31 ij 1 i 1 i is a vector V(cid:126)1, analogous to V(cid:126) in Eq. (S3). If Φ1 = 0, then H contains an induced magnetic flux. Plots of the real 1 (cid:54) and imaginary parts of t1 t1 t1 in Fig. S4 show this is indeed the case. The procedure leading to H can be iterated 12 23 31 1 to give an nth-order adiabatic Hamiltonian H =U†H U iU†∂ U [S9] and further approximations to Φ1. n n n−1 n− n t n Wenowinvestigatewhetherthetwo-bodyinteractionsarealsomodifiedbythenonadiabaticcoupling. Wewillagain use Lie algebras, this time su(6) instead of su(3). Since the space of two-electron spin singlet states is 6-dimensional, themostgeneralHamiltoniancanbenon-redundantlyparameterizedintermsofthegeneratorsofsu(6). However,the standardgeneratorsofsu(6),i.e.the6 6matriceswithonlytwononzeroelements,turnouttobelinearcombinations × of one-body and two-body operators. To see this, first note that e.g. νˆ can be expressed as 2  0 0 0 i√2 i√2 0 −  0 0 i 0 0 0  −  νˆ2 =(cid:88)(cid:0)−ic†1σc2σ+ic†2σc1σ(cid:1)= i0√2 0i 00 00 00 00 (S11) σ −   i√2 0 0 0 0 0 0 0 0 0 0 0 in the following basis of two-electron states: 1 0 1 (cid:0)c† c† c† c† (cid:1) 0 | (cid:105)≡|↑↓ (cid:105)≡ √2 1↑ 2↓− 1↓ 2↑ | (cid:105) 2 0 1 (cid:0)c† c† c† c† (cid:1) 0 | (cid:105)≡|↑ ↓(cid:105)≡ √2 1↑ 3↓− 1↓ 3↑ | (cid:105) 3 0 1 (cid:0)c† c† c† c† (cid:1) 0 | (cid:105)≡| ↑↓(cid:105)≡ √2 2↑ 3↓− 2↓ 3↑ | (cid:105) 4 00 c† c† 0 | (cid:105)≡|↑↓ (cid:105)≡ 1↑ 1↓| (cid:105) 5 0 0 c† c† 0 | (cid:105)≡| ↑↓ (cid:105)≡ 2↑ 2↓| (cid:105) 6 00 c† c† 0 . (S12) | (cid:105)≡| ↑↓(cid:105)≡ 3↑ 3↓| (cid:105) Since there is no linear transformation among the set of 6 off-diagonal νˆ that makes them coincide with 6 of the n standardgeneratorsofsu(6),weconcludethatatleastsomeofthosegeneratorsmustcorrespondtolinearcombinations of one-body and two-body operators. Therefore, to build up a complete set of two-body operators that are linearly independent of all one-body operators νˆ , we shall have to find appropriate linear combinations of the standard n generators. BythelinearindependenceoftwooperatorsAândBˆ wemeanthatTr(AˆBˆ)=0,wherethetraceistaken wrt the basis in Eq. (S12), and we normalize all operators such that Tr(AÂˆ)=10. First, consider the on-site Hubbard interactions U nˆ n .They correspond to the diagonal elements H , H , iiii i↑ i↓ 44 55 H of the Hamiltonian in the basis (S12). Although they are not orthogonal to νˆ , νˆ and νˆ under the trace, we 66 3 8 9 4 can define the following operators that are: (cid:114) 5 µˆ = diag( 1, 1,+1,+1,0,0) 1 2 − − (cid:114) 5 µˆ = diag( 1,+1, 1,0,+1,0) 2 2 − − (cid:114) 5 µˆ = diag(+1, 1, 1,0,0,+1). (S13) 3 2 − − Second, note that the double hopping interactions such as W c† c† c c +W∗ c† c† c c (S14) 1122 1↑ 1↓ 2↓ 2↑ 1122 2↑ 2↓ 1↓ 1↑ arealreadyorthogonaltoallone-bodyoperatorsbecausetheyareonlynonzerointhelowerrightblockwhenexpressed inthe6-dimensionalbasis(S12),i.e.theyonlyactinthesectorofdoublyoccupiedstates. Clearly,thedoublehopping terms can be put in a one-to-one correspondence with the off-diagonal elements of a set of Gell-Mann matrices for the 4 , 5 , 6 sector, for example, {| (cid:105) | (cid:105) | (cid:105)}     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 ωˆ1 =0 0 0 0 1 0 and ωˆ2 =0 0 0 0 i 0.    −  0 0 0 1 0 0 0 0 0 i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The double hopping amplitudes define a nontrivial gauge invariant loop quantity W W W like the hopping 1122 2233 3311 amplitudes. In the two-site Hubbard model, the complex phase of the double hopping amplitude W was found to 1122 couple strongly to the dynamics of the occupation numbers n and relative phases ζ [S10]. k k Third, consider correlated hopping terms such as 1 1 4 + 4 1 =S (c† c† c† c† )c c +H.c. (S15) | (cid:105)(cid:104) | | (cid:105)(cid:104) | 1211√2 1↑ 2↓− 1↓ 2↑ 1↓ 1↑ We can decompose these terms into three types of interactions. The first, which we denote by σˆ , are nonzero only n in the upper right and lower left blocks, e.g.  (cid:113) (cid:113)   (cid:113) (cid:113)  0 0 0 5 5 0 0 0 0 i 5 i 5 0  2 − 2   2 2   0 0 0 0 0 0  0 0 0 0 0 0      0 0 0 0 0 0  0 0 0 0 0 0 σˆ = (cid:113) , σˆ = (cid:113) , 1  5 0 0 0 0 0 2  i 5 0 0 0 0 0  (cid:113)2  − (cid:113)2   5 0 0 0 0 0  i 5 0 0 0 0 0 − 2  − 2  0 0 0 0 0 0 0 0 0 0 0 0 or, as an operator in second quantization, (cid:114) (cid:114) 5 5 σˆ = (n n )(c† c +c† c )+ (n n )(c† c +c† c ). (S16) 1 2 1↑− 2↑ 2↓ 1↓ 1↓ 2↓ 2 1↓− 2↓ 2↑ 1↑ 1↑ 2↑ The second type of interaction τˆ has a form such as n  0 0 0 √1 √1 0  0 0 0 i√1 i√1 0 − 2 − 2 − 2 2  0 0 2 0 0 0  0 0 i2 0 0 0    −   0 2 0 0 0 0  0 i2 0 0 0 0 τˆ = , τˆ = , 1  √1 0 0 0 0 0 2  √1 0 0 0 0 0 − 2  − 2   √1 0 0 0 0 0  √1 0 0 0 0 0 − 2 − 2 0 0 0 0 0 0 0 0 0 0 0 0 5 or 1 1 τˆ = ( n n +n )(c† c +c† c )+ ( n n +n )(c† c +c† c ). (S17) 1 2 − 1↑− 2↑ 3↑ 2↓ 1↓ 1↓ 2↓ 2 − 1↓− 2↓ 3↓ 2↑ 1↑ 1↑ 2↑ The third type of interaction χˆ corresponds to the elements H (i=1...6) and leads to terms such as n i,7−i n (c† c +c† c )+n (c† c +c† c ), (S18) 3↑ 2↓ 1↓ 1↓ 2↓ 3↓ 2↑ 1↑ 1↑ 2↑ All of the correlated hopping operators σˆ , τˆ and χˆ are orthogonal to νˆ under the trace. Numerical calculations n n n 1 confirmthatallofthedynamicallyinducedinteractionoperatorswithimaginarymatrixelements(e.g.ωˆ , σˆ andτˆ ) 2 2 2 are generically present in H . Figure S5 shows the amplitudes of double hopping terms ωˆ and correlated hopping 1 n σˆ , τˆ and χˆ obtained from the Hamiltonian H . Correlated hopping terms similar to these were studied for solids n n n 1 with intermediate valency [S11] and, recently, for ultracold atoms in optical lattices [S12–S14]. The operators (νˆ µˆ ,σˆ ,τˆ ,ωˆ ,χˆ ) form a complete set of generators for the su(6) algebra. As expected, there n n n n n n | are a total of 36=62 independent operators. One advantage of these operators is that they can be used to explicitly separate one-body and two-body degrees of freedom. The most general two-body rdm ρˆ can be expanded uniquely 2 in terms of the one-body and two-body operators. The above approach to orthogonalizing one-body and two-body operatorscanbeextendedtomorecomplexsystemsandmightbeusefulinthestudyofgenerallatticemodels. Itmight also be useful in identifying appropriate gauge invariant quantities for density functional-type theories, e.g. current density functional theory [S15], where the basic independent variables should be gauge invariant. There are likely connections with the geometry of entanglement (see Ref. S16 and references therein) in quantum information theory. S3. Verifying adiabatic ramping and tracking the adiabatically continued quasienergies Starting from a given stationary state, perhaps the simplest way to bring a system to a Floquet state is to turn on the periodic driving slowly enough that the system has a chance to adjust and adiabatically build up periodically oscillating components. A convenient way to formulate this mathematically is to send the initial time back to −∞ and employ an adiabatic ramping function such as f(t)=(1/2)(1+tanhαt), which turns the perturbation on over a slow time scale τ =α−1 and approaches a constant value of 1 as t . In this way the Hamiltonian, although not →∞ perfectly periodic during the ramping, approaches a periodic function in the limit t . →∞ If the ramping is successful, the system will have evolved adiabatically from a given stationary state to a given Floquetstate. Byvaryingthedetailsofthetime-periodicHamiltonianandtherampingfunction,onecanmapoutthe set of Floquet states that are reachable from the initial stationary state. A version of the adiabatic theorem has been proved for adiabatically varied time-periodic Hamiltonians [S17]. The key point is that the adiabatic eigenenergies which enter in the adiabatic theorem get replaced by the instantaneous quasienergies, which are the quasienergies one would obtain by solving Eq. (S21) below for the time-periodic Hamiltonian with a “frozen” value of the ramping function. One can then adiabatically continue these instantaneous quasienergies by varying the parameters of the ramping. Indoingso,theFloquetstatecorrespondingtoagivenquasienergyisadiabaticallytransportedinthespace of Floquet states. Any two states that can be connected in this way will be called adiabatically connnected. Two states might not be adiabatically connected if the quasienergy in question undergoes any avoided crossings with other quasienergies during the ramping. In analogy with Landau-Majorana-Zener transitions between adiabatic eigenstates, there may be appreciable nonadiabatic transitions at such avoided crossings (see e.g. [S18]). Here we demonstrate numerically that our system with the ramping given by Eqs. (S4) does indeed evolve adiabatically to a Floquet state to high accuracy. The figure of merit is the periodicity of the factor ξ(t) . Deviations from periodicity | (cid:105) are measured by δ =limsup ξ(t+T) ξ(t) . The quantity ξ(t+T) ξ(t) is plotted in Fig. S6 for the same t→∞|| − || || − || parameters that were used in Figs. 1, 2 and 4, namely α=0.11, (cid:15)=4, ω =0.2, U =7 and φ =0, and in the limit 0 0 t it approaches δ .00045. The error in the quasienergy is (δ2). To convey a sense of the effectiveness of → ∞ ≈ O adiabaticrampinggloballyinparameterspace,inFig.S7weplotδasafunctionof(U,ω )for(cid:15)=4,α=min(ω ,1/8), 0 0 φ =0 and t = 3T/2. 0 0 − Time dependent quasienergies Ω (t) can be defined by propogating from the nth stationary state of the initial n HamiltonianH( ). Inordertoinvestigatewhetherthereareavoidedcrossingsofthequasienergiesduringadiabatic −∞ ramping,therunningtimeaveragesΩ (t)areplottedinFig.S8intheadiabaticregime. Alsoshownaretheadiabatic nT eigenenergies E (t) and their running time averages E (t). All running time averages approach constants in the n nT limit t . There is apparently a strong level attraction between the quasienergies of the highest sector of states, → ∞ but the quasienergies of the Floquet states obtained from the lowest three states remain separated from each other by an energy gap uniformly throughout the ramping.