Analytical Solution for the Steady States of the Driven Hubbard model
Joseph Tindall, Frank Schlawin, Michael A. Sentef, Dieter Jaksch
AAnalytical Solution for the Steady States of the Driven Hubbard Model
J. Tindall , F. Schlawin , , M. A. Sentef and D. Jaksch Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom ∗ The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, D-22761 Hamburg, Germany and Max Planck Institute for the Structure and Dynamics of Matter,Luruper Chaussee 149, D-22761 Hamburg, Germany (Dated: January 29, 2021)Under the action of coherent periodic driving a generic quantum system will undergo Floquetheating and continously absorb energy until it reaches a featureless thermal state. The phase-space constraints induced by certain symmetries can, however, prevent this and allow the system todynamically form robust steady states with off-diagonal long-range order. In this work, we take theHubbard model on an arbitrary lattice with arbitrary filling and, by simultaneously diagonalising thetwo possible SU(2) symmetries of the system, we analytically construct the correlated steady statesfor different symmetry classes of driving. This construction allows us to make verifiable, quantitativepredictions about the long-range particle-hole and spin-exchange correlations that these states canpossess. In the case when both SU(2) symmetries are preserved in the thermodynamic limit we showhow the driving can be used to form a unique condensate which simultaneously hosts particle-holeand spin-wave order.
I. INTRODUCTION
Coherent driving has established itself as a fundamen-tal tool for controlling and manipulating the states ofquantum systems, from implementing high fidelity gatesin few-qubit systems [1] to inducing phase transitions inmany-body optical lattices [2]. Within this paradigm, re-cent experiments have observed how intense laser pulsesin the midinfrared regime can transiently induce super-conducting features - such as the opening of a gap in thereal part of the optical conductivity and vanishing resis-tivity - when driving various solid state materials out ofequilibrium [3–11].In order to understand seminal results such as these- and more generally the role coherent driving plays inaltering the microscopic properties of many-body sys-tems - significant theoretical studies have been under-taken. Floquet theory can be used to understand howperiodic driving can modify the parameters of the sys-tem and create additional terms on top of the undrivenHamiltonian. This renormalization results in an effectiveHamiltonian which on transient scales can, for example,favour superconducting prethermal states [12–25], sup-press wavepacket spreading and induce dynamical local-ization in a many-body bosonic gas [26], control spin-charge separation in a fermionic system [27] or stabiliseexotic spin-liquid states in frustrated systems [28].It is inevitable, however, that due to Floquet heating ageneric periodically driven quantum system will continu-ously absorb energy from the driving field. This heatingcompetes with any transient order established by the ef-fective Hamiltonian, causing it to melt away and leadingto the formation of a featureless, infinite temperaturestate in the long-time limit [29, 30]. As a result, engi- ∗ [email protected] neering Floquet Hamiltonians which are stable to heat-ing on long timescales, allowing their prethermal states tobe transiently observable is a current research endeavourattracting significant attention [31–38].In contrast to these efforts to mitigate the effects ofheating in driven systems, recent theoretical work hasshown how the presence of SU(2) symmetries in thefermionic Hubbard model can prevent featureless ther-malisation and result in the formation of correlated, or-dered states as the system heats up [39, 40] - a mechanismtermed ‘heating-induced order’. The phase space con-straints induced by these symmetries mean the systemis forced to relax towards steady states with off-diagonallong-range order as it absorbs energy from an externalsource. Currently, however, this has only been demon-strated numerically for the case when a single SU(2) sym-metry is preserved in small, finite-size instances of thehalf-filled Hubbard chain [40].In this work we go beyond this, taking the drivenHubbard model on an arbitrary graph at arbitrary fill-ing and analytically constructing the correlated steadystates. We achieve this construction by simultaneouslydiagonalising the irreducible representation of the twopossible SU(2) symmetries of the system and throughit we can analytically calculate the steady state spin-exchange and particle-hole correlations, at any distance.We verify our analytical results with exact diagonalisa-tion calculations and analyse how the long-time correla-tions depend on factors such as the filling, graph size,initial state and, crucially, the symmetries the drivingpossesses. Moreover, we provide the necessary conditionsfor the steady state correlations to remain finite in thethermodynamic limit. This leads us to show how, in caseswhere both SU(2) symmetries are preserved, the drivingcan be used to merge two independent condensates andcreate a unique spin- η condensate which hosts both spin-exchange and particle-hole off-diagonal long-range order.Finally, we discuss possible experimental setups of the a r X i v : . [ c ond - m a t . s t r- e l ] J a n driven Hubbard model where the requisite symmetriesare preserved in order to observe the formation of suchunique, correlated states. II. THEORY
As a starting point we consider the long-time states ofa quantum system, with Hamiltonian H , subject to con-tinued periodic driving under the modified Hamiltonian H + H D ( t ), where H D ( t ) is the time-dependent periodicdriving term H D ( t + T ) = H D ( t ). We then assume thatwe can find a set of X operators C = { C , C , ..., C X } ,which form a linearly independent, irreducible repre-sentation of the symmetries of the system satisfying[ C i , C j ] = 0 ∀ i, j . We note that whilst this is not al-ways possible in a general quantum system, for the caseswe consider in this work we are able to identify the com-pletely commuting set C . Being a representation of thesymmetries of the system we also clearly have that eachmember of C satisfies[ H + H D ( t ) , C i ] = 0 , i = 1 , ..., X. (1)Under the action of H + H D ( t ) the system will con-tinuously absorb energy until, in the long-time limit, itreaches a state of maximum entropy [29]. The system is,however, under the constraint that its probability distri-bution over the eigenspace of the operators in C mustalways be conserved. Consequently, the long-time stateof the system will effectively have the form lim t →∞ ρ ( t ) = ρ ∞ = (cid:88) α =( α ,α ,...α N ) P α D α (cid:88) β =1 | α, β (cid:105) (cid:104) α, β | , (2)with (cid:80) α P α D α = 1 and the multi-index/quantum num-ber α running over the combinations of possible eigenval-ues of the operators in C . For a given α , the D α vectors | α, β (cid:105) form the basis which simultaneously diagonalisesthe operators C , C , ..., C X and P α is the probability offinding the state in this subspace. This probability mustbe preserved throughout the dynamics and thus P α = D α (cid:88) β =1 Tr( ρ (0) | α, β (cid:105) (cid:104) α, β | ) , (3)where ρ (0) is the initial state of the system. If the com-plete basis {| α, β (cid:105)} can be constructed and the corre-sponding probabilities P α calculated then the long-timestate of the system is known. Whilst clearly in a closed system a pure state will always remainpure, this mixed state ansatz is reasonable in a many-body sys-tem as the energy of the driving will have scrambled the phases ofthe wavefunction sufficiently to destroy any coherences betweenthe | α, β (cid:105) eigenstates [30]. The structure and properties of the set C has a signif-icant influence on the properties of the steady state ρ ∞ .For example, consider the case of a many-body latticewith a single U(1) symmetry such as the total particlenumber. The corresponding total number operator is di-agonal in the Fock basis and thus, by Eq. (2), so is ρ ∞ .All the states in the Fock basis can be written as a prod-uct state over the different lattice sites, making ρ ∞ afeatureless (outside of the well-defined particle number),unentangled thermal state.The same cannot be said, howevever, for more compli-cated symmetries. In the case of an SU(n) symmetry fora many-body lattice, one can form the set C by using thegenerators to construct the n − ρ ∞ is diagonal. These basis statescannot all be written as product states over the differentlattice sites and often contain correlations in the formof excitations which are spread between sites. ThroughEq. (2) the long-time state will inherit these properties,along with the complete translational invariance of theCasimir operators, and thus possess correlations or ex-citations which are independent of the lattice geometry- i.e. they are completely uniform with distance. Thisinduction of uniform, long-range correlations by heatingan SU(n) symmetric system up to its steady state hasbeen termed ‘heating-induced order’ and has been stud-ied numerically on small, finite-sized half-filled Hubbardlattices with a single preserved SU(2) symmetry [39, 40].We note that the formation of these off-diagonal long-range ordered states as t → ∞ does not exclude thepossibility that, due to the driving, one could also ob-serve the emergence of some transient, dynamical or-der on an intermediate timescale. This would then befollowed by the melting of this order and the onset ofour robust, steady state off-diagonal long-range order(ODLRO) once sufficient heating has occurred. The for-mation of transient, dynamical order is, however, usuallyreliant on a careful of choice of driving terms and param-eters [14, 16, 18, 19, 22–24]. Meanwhile, if the requisitesymmetries are satisfied, the emergence of steady stateODLRO is guaranteed due to the inevitability of Floquetheating under periodic driving - giving us significant free-dom in the driving fields which can be used to observeheating-induced order. A. The Hubbard model on an arbitrary graph
Here, we consider the Hubbard model on an arbitrarygraph with arbitrary filling and, by simultaneously di-agonalising the dual SU(2) symmetries, analytically con-struct the long-time states for the different symmetryclasses of driving. We define this graph as G = ( V, E )where V are the vertices (or sites) and E the edges. The FIG. 1. Fermionic Hubbard model on a graph G = G ( V, E )where V are the vertices and E are the edges. The M ver-tices form the lattice sites on which the fermions reside andinteract with strength U whilst the edges form the nearestneighbour bonds over which the fermions can hop with con-stant anplitude τ . The Hamiltonian is defined in Eq. (4) and G , G and G are the 3 different M = 10 vertex graphs we usefor our numerics. The grey vs blue sites represent a bi-partitesplitting on the graphs G and G . Hamitonian can then be written as H = − τ (cid:88) V,V (cid:48) (cid:88) σ = ↑ , ↓ ( c † σ,V c σ,V (cid:48) +h . c)+ U (cid:88) V n ↑ ,V n ↓ ,V , (4)where c † σ,V and its adjoint are the usual creation and an-nihilation operators for a fermion of spin σ on vertex V .The first summation runs over all the edges in the graph,kinetically coupling together the two sites V, V (cid:48) residingat the ends of each edge with strength τ . The secondsummation runs over all the vertices in the graph andcreates an energy penalty U for vertices simultaneouslyoccupied by both spin species. Additionally, n σ,V is thenumber operator for a particle of spin σ on site V andwe use the integer M to denote the number of verticeson the graph. We also depict this Hamiltonian in Fig.1, showing the 3 finite-size graphs we use for our nu-merical results, which serve to benchmark our analyticalpredictions. These analytical predictions, however, canbe immediately applied to any graph of any number ofvertices.The Hamiltonian in Eq. (4) has a rich symmetry struc-ture comprised of either one or two SU(2) symmetries[41], which are fundamental to our results. The first,permanent, SU(2) symmetry can be introduced throughthe spin-raising operator S + = (cid:80) V c † V, ↑ c V, ↓ , its conju-gate S − and the total magnetisation S z = (cid:80) V n ↑ ,V − n ↓ ,V . These operators are conserved over all graphs, i.e.[ H, S ± ,z ] = 0 , ∀G and we refer to this symmetry as the‘spin’ symmetry with the corresponding operators only acting non-trivially on the sites of the lattice occupiedby a single fermion (singlons).The second of the SU(2) symmetries is introducedthrough the η -raising operator η + = (cid:80) V f ( V ) c † V, ↑ c † V, ↓ ,its conjugate η − and the modified total number operator η z = (cid:80) V ( n ↑ ,V + n ↓ ,V − G is bi-partite, i.e.the vertices can be split into two sets with the edges ofthe graph only forming connections between the two sets,then [ H, η + η − ] = 0 if we set the function f ( V ) to take thevalues ± H, η z ] = 0 ∀G and thus the η -operators form an SU(2)or U(1) symmetry depending on whether the graph isbi-partite or not. The corresponding operators only actnon-trivially on the empty and full sites within the lat-tice and these sites are often referred to as occupied bya ‘hole’ or a ‘doublon’ quasiparticle respectively.In addition to this we note that there may be addi-tional symmetries that H posseses which will reflect the polygon symmetries of the graph G . For example on anopen boundary 1D chain the Hubbard model has a re-flection symmetry about the central site, whilst on morecomplex geometries there may be mutiple reflection andtranslational shift symmetries [41].Our goal is to determine the long-time states reachedunder continued periodic driving on top of the Hamilto-nian in Eq. (4). For simplicity, we will assume that thedriving either breaks any polygon symmetries in the sys-tem, or that they can be ignored due to their sufficientlysmall effect on the long-time properties of the system. Wewill see this is a reasonable assumption for the graphs weuse and therefore the relevant symmetries to consider inour system are the two possible SU(2) symmetries of thelattice.From here on we will also, without loss of generality, fixthe quantities N ↑ and N ↓ which correspond to the totalnumber of fermions of spin ↑ and spin ↓ respectively.Our results in the full Hilbert space can be recovered byperforming a direct sum over all possible values of N ↑ and N ↓ . To keep the equations we derive more concise, we willrestrict ourselves to lattices with an even number of sites M and an even total number of particles N = N ↑ + N ↓ .For brevity we will also introduce the integer quantities α = ( M − N ) / β = ( N ↑ − N ↓ ) / B. Simultaneously diagonalising the HubbardSU(2) Casimir operators
In order to construct the long-time state for arbi-trary driving we must be able to diagonalise the ir-reducible representation of the dual SU(2) symmetries.This representation, i.e. our set C , consists of the twoSU(2) Casimir operators η = η + η − + η − η + + ( η z ) and S = S + S − + S − S + + ( S z ) and their corresponding ‘z’operators η z and S z . By fixing the particle numbers wehave already removed the dependency on η z and S z andthis restriction also means that the eigenvectors of theoperator O + O − are also those of O − O + , where O is ei-ther η or S . Hence, our problem is immediately simplifiedto simultaneously diagonalising S + S − and η + η − .We can make progress with this problem by noticingthat both operators commute with the total doublon op-erator N D = (cid:80) V n ↑ n ↓ . We can therefore simultaneouslyreduce them into block matrices with the blocks indexedby i , the number of doublons on the graph, which rangesfrom Max(0 , − α ) to Min( N ↑ , N ↓ ). For a given value of i , we observe that there must be M + i − N holes inthe graph and so the remaining vertices, or sites, will beoccupied by N ↑ − i and N ↓ − i singlons of spin ↑ and ↓ respectively.We can use this knowledge to take any given block andarrange the sites of the lattice into two sets, with the firstset ( A ) containing M +2 i − N sites and the second set ( B )containing the remaining N − i sites. There are (cid:0) MN − i (cid:1) different ways in which the sites can be arranged in thismanner and in A we place all of the doublons and holeswhilst in B we place all of the singlons. If the operator S + V S − V (cid:48) ( η + V η − V (cid:48) ) acts on a vertex which is in set A ( B )then it will immediately annihilate any given basis state,and thus we can let η + η − → ( η + η − ) (cid:48) = (cid:88) V,V (cid:48) ∈ A η + V η − V (cid:48) ,S + S − → ( S + S − ) (cid:48) = (cid:88) V,V (cid:48) ∈ B S + V S − V (cid:48) , (5)and ignore the other terms in these two operators. Ifwe can now construct a state | η (cid:105) , within A , which is aneigenvector of ( η + η − ) (cid:48) with eigenvalue λ A and a state | S (cid:105) within B which is an eigenvector of ( S + S − ) (cid:48) witheigenvalue λ B then their ‘tensor product’ | η (cid:105) ˜ ⊗ | S (cid:105) will si-multaneously be an eigenvector of both η + η − and S + S − on the full graph; with eigenvalues λ A and λ B respec-tively. The tilde on the tensor product means we willtake into account the way the sets A and B were formedand re-order the vertices of the graph back to their orig-inal order.We now need to determine the eigenspectrum of( η + η − ) (cid:48) and ( S + S − ) (cid:48) . Crucially, we know that ( η + η − ) (cid:48) and ( S + S − ) (cid:48) correspond to the SU(2) Casimir operators in a, respectively, M +2 i − N and N − i -fold tensor prod-uct representation of the fundamental representation ofSU(2) [42, 43]. By exploiting the ‘ladder’ structure ofSU(2) representations we can determine the eigenvaluesof ( η + η − ) (cid:48) and ( S + S − ) (cid:48) as λ η ( k ) = k ( k + 1) − α ( α + 1) , k = | α | , ...α + iλ s ( m ) = m ( m + 1) − β ( β + 1) , m = | β | , ..., N/ − i. (6)These eigenvalues have the following degeneracies D η ( k ) = C ( α + i + k, α + i − k − δ k ) ,D S ( m ) = C ( N/ − i + m, N/ − i − m − δ m ) , (7)where δ a is the Kronecker delta function and C ( x, y ) ≡ C xy is the Catalan Triangle Number [44] C xy = (cid:40) ( x + y )!( x − y +1) y !( x +1)! x, y > , | η k,l (cid:105) and | S m,n (cid:105) asthe eigenvectors of ( η + η − ) (cid:48) and ( S + S − ) (cid:48) where k and m index the respective eigenvalues and l and n run throughthe degenerate eigenvectors for a given k and m . Thestate | ψ (cid:105) = | η k,l (cid:105) ˜ ⊗ | S m,n (cid:105) is then an eigenvector of η + η − and S + S − on the full graph G . The relevant indices tospecify the state | ψ (cid:105) are then | ψ i,j,k,l,m,n (cid:105) where i is thenumber of doublons, j indexes the (cid:0) LN − i (cid:1) ways in whichthe lattice can be split into the two aforementioned sets, k and l index the degenerate eigenvectors of ( η + η − ) (cid:48) whilst m and n do the same for ( S + S − ) (cid:48) . The states | ψ (cid:105) form a complete, orthonormal basis which diagonalise η + η − and S + S − for the given filling. C. Steady states of the driven Hubbard model
We can now combine the basis we have constructedwith the general result of Eq. (2) and write down thesteady state ρ ∞ of the periodically driven Hubbard modelon an arbitrary graph with arbitrary filling as ρ ∞ = 1 Z Min( N ↑ ,N ↓ ) (cid:88) i =Max(0 , − α ) α + i (cid:88) k = | α | N/ − i (cid:88) m = | β | P k,m C α + i + kα + i − k − δk (cid:88) l =1 C N/ − i + mN/ − i − m − δm (cid:88) n =1 ( LN − i ) (cid:88) j =1 | ψ i,j,k,l,m,n (cid:105) (cid:104) ψ i,j,k,l,m,n | , (9)where Z is the partition function and the values P k,m area series of probabilities (analogous to the P α in Eq. (2))which are are dependent on both the initial state as wellas which, if any, of the SU(2) symmetries are present inthe system. This dependency is encapsulated by Table I, where we have also introduced the following projectors P ηk = (cid:88) i,j,m,n,l | ψ i,j,k,l,m,n (cid:105) (cid:104) ψ i,j,k,l,m,n | , P Sm = (cid:88) i,j,k,n,l | ψ i,j,k,l,m,n (cid:105) (cid:104) ψ i,j,k,l,m,n | , P S,ηm,k = (cid:88) i,j,n,l | ψ i,j,k,l,m,n (cid:105) (cid:104) ψ i,j,k,l,m,n | , (10) Spin Symmetry η Symmetry No YesNo P m,k = const . P m,k = P m P k , P k = Tr( ρ (0) P ηk ) P m = const . Yes P m,k = P m P k , P k = const ., P m = Tr( ρ (0) P Sm ) P m,k = Tr( ρ (0) P S,ηm,k )TABLE I. Form of the probabilities P k,m which characterise the steady state of the driven Hubbard model (see Eq. (9)). Theprobabilities are defined by their relationship to the initial state ρ (0) and the projectors P ηk , P Sm and P S,ηm,k , which are defined inEq. (10). This relationship changes depending on which of the two SU(2) symmetries are present during the systems evolutionto the steady state. with the indices used all retaining their original meaningand ranges from Eq. (9).Table I, alongside Eq. (9), allows us to classify andwrite down the steady states of the driven Hubbardmodel on an arbitrary graph. This classification is basedon which of the SU(2) symmetries are present during thetime evolution and - due to the excited, entangled na-ture of the eigenstates of the SU(2) Casimir operators -the long-time state will contain long-range correlationsin the channels corresponding to the preserved symme-tries. Meanwhile, in the channels where the underlyingsymmetry was not present the constant nature of theprobabilities means all excitations are equally likely andthey will destructively interfere with each other to ensurethere is no long-range order in that channel.Given the probabilities P k,m we can calculate a numberof properties of the state in Eq. (9). For example we canimmediately deduce the moments of the doublon number N d (cid:104) N αd (cid:105) = 1 Z (cid:88) i i α f ( i ) ,f ( i ) = (cid:88) k,m P k,m C α + i + kα + i − k − δ k C N/ − i + mN/ − i − m − δ m (cid:18) MN − i (cid:19) . (11)These equations are useful because we can take advan-tage of the distance-invariance of correlations in thelong-time state and use the first moment of the dou-blon number ( α = 1), along with the initial values (cid:104) η + η − (cid:105) and (cid:104) S + S − (cid:105) , to directly extract values for the off-diagonal spin-exchange and particle-hole order parame-ters (cid:104) S + V S − V (cid:48) (cid:105) and (cid:104) η + V η − V (cid:48) (cid:105) . Specifically, we know that (cid:104) η + η − (cid:105) = (cid:104) N d (cid:105) + M ( M − (cid:104) η + V η − V (cid:48) (cid:105) , (12)and (cid:104) S + S − (cid:105) = N ↑ − (cid:104) N d (cid:105) + M ( M − (cid:104) S + V S − V (cid:48) (cid:105) , (13)where V (cid:54) = V (cid:48) . Moreover, higher moments of the doublonnumber, α > η and spin symmetry sectors (for example we canshow that (cid:104) n ↑ ,V n ↓ ,V n ↑ ,V (cid:48) n ↓ ,V (cid:48) (cid:105) ∝ (cid:104) N d (cid:105) − (cid:104) N d (cid:105) ).In principle, however, in order to calculate the mo-ments of the doublon number we need to know the exactvalues of the P k,m s - which, in some cases, could be quite complicated and would involve taking a number of pro-jective measurements on the initial state. We find fromour equations, however, that the first moment of the dou-blon number is only dependent on the probabilities P k,m through its relationship to (cid:104) η + η − (cid:105) and (cid:104) S + S − (cid:105) and thusknowledge of these two values, and the graph size andfilling, is enough to calculate (cid:104) N d (cid:105) . The steady stateoff-diagonal order parameters (cid:104) η + V η − V (cid:48) (cid:105) and (cid:104) S + V S − V (cid:48) (cid:105) thenfollow immediately from Eq. (12) and the corresponding‘spin’ version.These quantities are particularly important becausewhen finite in the thermodynamic limit and completelyuniform with distance, the latter of which is automat-ically satisfied by ρ ∞ , they describe the existence of aspin-wave or η condensate. These condensates are un-derpinned by excitations which are completely spreadout in space and in an η condensate the long-range fi-nite value of (cid:104) η + V η − V (cid:48) (cid:105) directly implies superconductivityas the Meissner effect and flux quantisation can be ob-served [45, 46]. III. RESULTS - HUBBARD MODEL UNDERGENERIC DRIVING
We now demonstrate the results we have derived ex-plicitly, first focussing on the case where a single SU(2)symmetry is preserved and then moving on to the casewhere both SU(2) symmetries are preserved. We notethat the scenario where both symmetries are not pre-served needs no attention as it is trivial and the long-time state will simply be a featureless thermal state witha fixed particle number.
A. Single Symmetry Preservation
We consider a setup in which the spin SU(2) symmetryis present whilst the η SU(2) symmetry is not. These re-sults are immediately analagous to the case where thespin-symmetry is not present and the η symmetry is,which was studied in Ref. [40] but only for a small, half-filled chain. We introduce the graph measure d ( V, V (cid:48) )which is the minimum number of edges that must be tra-versed to move between the vertices V and V (cid:48) . With this FIG. 2. (a) Spin-exchange correlations versus distance δ , seeEq. (14), for the periodically driven graph G pictured inFig. 1, with the site index V running from 1 to 10, startingat V = 1 for the bottom left site and increasing in an anti-clockwise manner. The driving term we use is of the form H D = A cos(Ω t ) (cid:80) V V n V and the grey vs blue markers/linescorrespond to the fillings N ↑ = N ↓ = 5 and N ↑ = N ↓ =3 respectively. At time tτ = 0 the system is initialised inthe ground state of the undriven Hamiltionian H with U =0 . τ and then evolved under the Hamiltonian H + H D with U = τ , A = 6 . τ and Ω = 1 . τ . The solid lines vs markersindicate the correlations at tτ = 0 and tτ = 100 respectively.The black-dotted lines give the exact results in the long-timelimit. b) Off-diagonal spin-exchange order as a function of thenumbers of ↑ and ↓ fermions in the steady state of a driven,arbitrary, M = 100 vertex Hubbard graph where the spinSU(2) symmetry is preserved whilst the η SU(2) symmetryisn’t. The system is initialised with (cid:104) S + S − (cid:105) = 0. measure we can then define the correlation function O ( δ ) = 1 N (cid:88) (cid:104) V,V (cid:48) (cid:105) d ( V,V (cid:48) )= δ (cid:104) O + V O − V (cid:48) (cid:105) , (14)where O is either S or η , the summation is over all pairsof vertices where d ( V, V (cid:48) ) = δ and N is the number ofpairs of vertices which satisfy d ( V, V (cid:48) ) = δ . Hence, O ( δ )measures the average of the spin-exchange or particle-hole correlations at a distance δ for any given graph andwe can introduce | O ( δ ) | δ>l as the average magnitude ofthese correlations at distances greater than l .In Fig. 2a we demonstrate agreement, for two differ-ent particle fillings, between the equations in the previoussection and exact diagonalisation code which reaches thelong-time limit of the graph G from Fig. 1. As the driv-ing explicitly breaks η symmetry the long-time predictionfrom these equations is independent of the lattice struc-ture and whether it is bi-partite or not. Figure 2 showsthat when driving the ground state out of equilibrium,the preservation of (cid:104) S + S − (cid:105) under driving causes the es-tablishment of completely uniform, long-range spin-waveorder in the long-time limit, with a significant enhance-ment of the long-range correlations. This long-range or-der is largest at the higher filling and, more generally, ourequations show that the steady state order will always bemaximised when the system is closest to 0 total magneti-sation and half-filling, where the largest number of spin- FIG. 3. Scaling of the steady state off-diagonal spin-exchangecorrelations with the number of vertices and initial value of (cid:104) S + S − (cid:105) = Tr( ρ (0) S + S − ) for the half-filled Hubbard modelon an arbitrary graph. The system is time evolved underdriving which preserves the spin SU(2) symmetry and breaksthe η symmetry. a) Scaling versus both system size and m where m = ( − (cid:112) (cid:104) S + S − (cid:105) ) /
2. b) Scaling with systemsize for various initial states. The solid lines correspond tothe initial values (cid:104) S + S − (cid:105) = M α /
20 for varying α whilst thedashed line is for (cid:104) S + S − (cid:105) = 0 exchange excitations are available. In Fig. 2b we showthis result explicitly, observing significant order aroundthis half-filled non-magnetic point which then decays to0 as either of the filling numbers, N ↑ and N ↓ , approachtheir maximal or minimal values (where it is not possibleto perform a spin-exchange S + V S − V (cid:48) without annihilatingthe state of the system).We then concentrate on the non-magnetic half-filledpoint, where the order is maximised, and in Fig. 3plot the magnitude of the long-time spin-exchange or-der for a large range of system sizes and initial valuesTr( ρ (0) S + S − ). These results apply to any driven Hub-bard graph where the spin SU(2) symmetry is preservedand the η symmetry isn’t - which could be a result of thedriving or the underlying lattice structure.Notably, as system size increases there is a growingspace of initial values where there is significant spin-waveorder in the long-time limit. In fact we can easily arguethat the spin-wave order will be finite in the thermody-namic limit of any graph if, and only if, the initial statesatisfies (cid:104) S + S − (cid:105) ∝ M , with Fig. 3b showing this ex-plicitly. Such states are already likely to have finite long-range spin correlations to satisfy this requirement, how-ever the driving will still act to renormalize these correla-tions and make them completely uniform with distance,stabilizing the spin-wave order. Meanwhile, for simula-tions which start in the ground state (where (cid:104) S + S − (cid:105) ≡ /M but re-mains finite for any finite-size system - with the dynamicsunderpinned by a drastic amplification of the long-rangecorrelations at the expense of the short-range ones.Interestingly, in Ref. [12], a type of spin preservingperiodic driving was studied for the 1D Hubbard chainin the thermodynamic limit. There it was shown howdriving the ground state can renormalize the exchange FIG. 4. a-b) Off-Diagonal Spin and η Correlations versus time and distance for 3 different half-filled Hubbard lattices withdriving of the form H D = δU cos( ωt ) (cid:80) i n ↑ ,i n ↓ ,i and the bare Hamiltonian H as defined in Eq. (4) with U = 4 . τ . The systemis initialised, at time tτ = 0, in the thermal state ρ ∝ exp( − βH ) with βτ = 5 and then time-evolved under H with U = 4 . τ , δU = 1 . τ and ω = 1 . τ . Black-Dotted lines represent the long-time analytical predictions for the 3 respective graphs. Inset)Spin-correlations versus distance at times tτ = 0 and tτ = 100 for the lattice G . c) Map of the doublon/ spin order oflong-time states of the Hubbard model on a 100 vertex bi-partite graph with driving which preserves both SU(2) symmetries.The indices m and k are dependent on the initial values of the spin and η symmetry via m = ( − (cid:112) (cid:104) S + S − (cid:105) ) / k = ( − (cid:112) (cid:104) η + η − (cid:105) ) /
2. The two maps are related via a reflection over the black-dotted line. Insets: Doublon (SolidLine) and Spin (Dashed Line) order for the long-time state at the circled point on the map. parameters in the system and transiently enhance long-range singlet pairing. Even though the long-time spin-order will be 0, our results here suggest this transientresponse could be a result of the preservation of (cid:104) S + S − (cid:105) .Under driving, this preservation forces a drastic reorgan-isation of the spin degrees of freedom, which will involvea transient enhancement of the long-range correlationsat the expense of the shorter ones, before they mutuallydecay away to 0 in the long-time limit.Despite the ground state of hypercubic Hubbard lat-tices possessing the smallest possible value of (cid:104) S + S − (cid:105) ,our equations show that the magnitude of the inducedoff-diagonal spin order |(cid:104) S + V S − V (cid:48) (cid:105)| under driving whichpreserves the SU(2) spin symmetry is larger than that forany initial states which have finite (cid:104) S + S − (cid:105) < ( M − / (cid:104) S + V S − V (cid:48) (cid:105) isa monotonic function of (cid:104) S + S − (cid:105) but is negative for fi-nite M and (cid:104) S + S − (cid:105) < M/
4, at which point it changessign. Hence, the ground state spin order is the most neg-ative and it can be shown from our equations that themagnitude of this order is larger than that of any otherstates in the range 0 < (cid:104) S + S − (cid:105) < ( M − /
2. On ahalf-filled hypercubic lattice, this range includes all ini-tial states in thermal equilibrium ρ (0) ∝ e − βH as thevalue of (cid:104) S + S − (cid:105) for these states monotonically increasesfrom 0 to M/ β decreasesfrom ∞ to 0. Hence, for these thermal initial states ona finite sized lattice, the magnitude of the steady statespin order is maximised for the ground state and remainsfinite for any finite temperature initial state - asymptot-ically tending to 0 as the temperature of the initial stateapproaches ∞ where (cid:104) S + S − (cid:105) = M/ β is likely to be morecomplex. This could, however, lead to the exciting pos-sibility of heating ρ (0) ∝ e − βH under certain symmetriesand forming a state with uniform, finite, off-diagonal or-der even in the thermodynamic limit – dynamically trans-forming a system in thermodynamic equilibrium into a‘hot’ condensate.It is worth emphasizing that in this paper we havetaken the hopping strength τ to be homogeneous acrossall edges of the lattice. The spin SU(2) symmetry of theHubbard model is, however, preserved even in the caseof an inhomogeneous hopping strength and so our resultsin this section immediately apply to this more generalscenario. B. Both SU(2) Symmetries Preserved
We now move to the case where the driving preservesboth SU(2) symmetries and in order for this preserva-tion to also be true for the full Hamiltonian H + H D ( t )we require the underlying graph to be bi-partite, whichwe will assume in our analytical calculations. For ournumerics, the graphs G and G are bi-partite whilst G isn’t and so we shall see that this leads to distinctly dif-ferent dynamics when they are driven. We fix ourselvesto symmetric half-filling ( N ↑ = N ↓ = M/
2) where thelong-time order will be maximal due to the maximumavailability of both particle-hole and spin-exchange ex-citations. In Figure 4a-b. we start in a thermal stateand apply driving which preserves both SU(2) symme-tries to the three different lattice structures G , G and G , calculating the average amplitude of the correlationsbetween all pairs of sites not connected by an edge. Inthe particle-hole channel, we observe a pronounced in- FIG. 5. Merging of two condensates under long-time driving. The Hubbard model on a 1D chain is split into two independenthalves which contain an η and spin-wave condensate respectively. The system is then time-evolved under generic drivingwhich respects both SU(2) symmetries of the model, causing the condensates to merge into a single hybrid spin- η condensate.Left) Pictorial depiction of the process. Right) Matrices of η and spin- exchange correlations for the whole lattice in thethermodynamic limit ( M → ∞ ) at times t = 0 and t = ∞ , i and j index the different sites of the chain. crease in this average, with the large value of U havingsuppressed them in the initial thermal state. The aver-age in the spin-exchange sector instead remains relativelyconstant, but the inset in Fig. 4c shows that the driv-ing has still reordered these correlations to be completelyuniform with distance. We find our predictions from Eq.(11) are in perfect quantitative agreement with the long-time order observed in these states and, as we expect,the non-bipartite graph G cannot support particle-holeorder as it lacks the requisite symmetry.In Fig. 4c we present maps of the long-time off-diagonal particle-hole and spin-exchange order as a func-tion of the η and spin eigenvalues of the initial state fora bi-partite graph with M = 100 sites. There is a wholemanifold of states with significant, co-existing, particle-hole and spin-exchange order, and we present an examplestate in the inset of each map. Here, we can also showthat, similar to the single symmetry case, when startingin the ground state the off-diagonal spin and η order al-ways remains finite for finite systems but asymptoticallydecays to 0 as 1 /M .In fact, in the thermodynamic limit, the condition (cid:104) η + η − (cid:105) ∝ M or (cid:104) S + S − (cid:105) ∝ M is necessary to observe fi-nite ODLRO in the η and spin sectors respectively. Thesetwo conditions are not mutually exclusive, allowing us toexploit the driving to form a unique spin- η condensate.Specifically, consider a bi-partite M site lattice whichhosts the initial state ( S + ) M/ | χ (cid:105)⊗ ( η + ) M/ | χ (cid:105) , where | χ (cid:105) = |↓ , ↓ , ..., ↓(cid:105) on M/ | χ (cid:105) is the vac-uum state on the other M/ H over the full lattice, thedynamics will involve the system heating up whilst con-serving the values of (cid:104) S + S − (cid:105) and (cid:104) η + η − (cid:105) - forcing the condensates to merge and phase-lock into a larger, sin-gle condensate which, remarkably, hosts ODLRO in theparticle-hole and spin-exchange sectors simultaneously.Specifically, our equations tell us lim M →∞ | η ( δ ) | δ> =lim M →∞ | S ( δ ) | δ> = 0 . S + ) M/ | χ (cid:105) ⊗ ( S + ) M/ | χ (cid:105) or ( η + ) M/ | χ (cid:105)⊗ ( η + ) M/ | χ (cid:105) . In this casethe driving only needs to preserve the relevant SU(2)symmetry in order to phase-lock and merge the two con-densates into a larger one. C. Experimental Implementation of SU(2)Symmetry Preservation
Finally, it is important to discuss how driving whichpreserves the SU(2) symmetries of the Hubbard modelcan be achieved experimentally. If we consider an opticallattice implementation of the Hubbard model [47–49] wecan take advantage of the fact the Hubbard interactionand hopping strengths have a well defined relationshipwith the depth and separation of the potential minimawhich form the optical lattice sites. These quantities canbe directly controlled, and made to oscillate, by modulat-ing the standing-wave interference pattern which gener-ates the potential landscape - a process which has alreadyled to the experimental realisation of a Hubbard Hamil-tonian with time-dependent parameters [50] and could beused to realise the unique states we have observed here,including the exotic η -spin condensate in Fig. 5.Moreover, in a quantum materials setting, a recentexperiment has shown how laser excitation of the vi-brational modes of the organic charge transfer salt κ − (BEDT − TTF) Cu[N(CN) ]Br, whose conductinglayers can be described by a triangular Hubbard model,leads to the formation of transient superconducting fea-tures [11]. Density functional theory modelling of thematerial has shown that the laser excitation induces aperiodic time-dependence in the parameters of the trian-gular Hubbard Hamiltonian which, in combination withthe irregular geometry of the system, leads to the systemtransiently establishing particle-hole order as it absorbsenergy from the driving field [51]. The numerical cal-culations in Ref. [51] thus provide a possible connectionbetween the mechanism of heating-induced order and thisrecent experiment.It is also worth mentioning that, in this work, we haveconsidered the situation where the relevant SU(2) sym-metries are completely preserved and so the correlatedstates we observe will form in the long-time limit andpersist indefinitely - i.e they are not prethermal statesbut are exact steady states of the system. In a realis-tic experimental setup, however, these SU(2) symmetrieswill never be perfectly preserved due to the presence ofthermal effects and lattice imperfections. In this case, as-suming these unwanted mechanisms are sufficiently smallin magnitude compared to the driving strength, then weexpect the correlated states to instead form transientlyand be observable on some intermediate timescale, priorto the system eventually heating up to a featureless infi-nite temperature states. IV. CONCLUSION
In this paper we have simultaneously diagonalised thedual SU(2) symmetries of the Hubbard model on an ar-bitrary graph. This diagonalisation has allowed us toconstruct the long-time states of the driven model andclassify and predict their properties under various sym-metry classes of driving. The preservation of either, orboth, of the SU(2) symmetries leads to a significant dy-namical re-ordering of the long-range correlations of thesystem, resulting in states with off-diagonal long-rangecorrelations in the corresponding symmetry sectors. Wehave analysed how these correlations scale with the rele-vant initial state properties, lattice filling and the graphsize. This analysis led us to identify a mechanism bywhich a unique condensate in the thermodynamic limit- hosting both spin-exchange and particle-hole order si- multaneously - can be formed.Here, we have focused directly on the case of peri-odic driving, where a many-body system will genericallyundergo the desired heating for most non-trivial driv-ing terms and experimental implementations are possiblewith current technologies. We emphasize, however, thatthis mechanism of ’heating-induced order’ can occur inany quantum system which continuously absorbs energyfrom an external source whilst the requisite symmetriesare preserved. For example, coupling the system to anenergetic Markovian external source which introduces de-coherence through local, hermitian jump operators hasbeen shown to cause the desired heating [40, 52] and driv-ing in the form of strong kicking or non-monochromaticpulses [51, 53, 54] are likely to also provide a route toinducing these correlated states.Alongside this, our work opens up several further ques-tions. Firstly, we have not quantified the effect of the polygon symmetries of the Hubbard lattice on the long-time states reached under periodic driving. Whilst thereis a minimal effect on the geometries we consider, morecomplicated structures could significantly alter the pair-ing landscape of the system, revealing exotic states whichhave preferential directions for the flow of supercurrentswithin the lattice structure.Furthermore we emphasize that this intuition ofsymmetry-constrained relaxation to ordered states is notlimited to the single-band Hubbard model. For example,the n species Hubbard model is SU(n) symmetric [55] anddriving terms which preserve this symmetry would leadto a unique state which possesses off-diagonal long-rangespin-exchange correlations between each distinct pair offermionic species. Moreover, a number of other systems,such as the SU(n) Heisenberg model [56] or multi-band/multi-orbital Hubbard models [57, 58], possess specialunitary/ orthogonal symmetries which could be exploitedto realise similar exotic, correlated states under heating. ACKNOWLEDGMENTS
We would like to thank Martin Claassen, Yao Wang,Andrea Cavalleri, Michelle Buzzi and Daniele Nicolettifor helpful comments. This work has been supported byEPSRC grants No. EP/P009565/1 and EP/K038311/1and is partially funded by the European Research Coun-cil under the European Union’s Seventh Framework Pro-gramme (FP7/2007-2013)/ERC Grant Agreement No.319286 Q-MAC. JT is also supported by funding from Si-mon Harrison. MAS acknowledges support by the DFGthrough the Emmy Noether programme (SE 2558/2-1)and F. S. acknowledges support from the Cluster of Ex-cellence ‘Advanced Imaging of Matter’ of the DeutscheForschungsgemeinschaft (DFG) - EXC 2056 - project ID390715994.0 [1] H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T.Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuleti´c,H. Pichler, and M. D. Lukin, Parallel implementation ofhigh-fidelity multiqubit gates with neutral atoms, Phys.Rev. Lett. , 170503 (2019).[2] A. Zenesini, H. Lignier, D. Ciampini, O. Morsch, andE. 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