Inverted many-body mobility edge in a central qudit problem
IInverted many-body mobility edge in a central qudit problem
Saeed Rahmanian Koshkaki, Michael H. Kolodrubetz
Department of Physics, University of Texas at Dallas, Richardson, TX, USA
Many interesting experimental systems, such as cavity QED or central spin models, involve globalcoupling to a single harmonic mode. Out-of-equilibrium, it remains unclear under what conditionslocalized phases survive such global coupling. We study energy-dependent localization in the disor-dered Ising model with transverse and longitudinal fields coupled globally to a d -level system (qudit).Strikingly, we discover an inverted mobility edge, where high energy states are localized while lowenergy states are delocalized. Our results are supported by shift-and-invert eigenstate targetingand Krylov time evolution up to L = 13 and respectively. We argue for a critical energy of thelocalization phase transition which scales as E c ∝ L / , consistent with finite size numerics. Wealso show evidence for a reentrant MBL phase at even lower energies despite the presence of strongeffects of the central mode in this regime. Similar results should occur in the central spin- S problemat large S and in certain models of cavity QED. Improvements in quantum control have brought non-equilibrium quantum systems to the forefront of con-densed matter and AMO physics. Novel phases of mat-ter are possible out of equilibrium, most of which re-quire many-body localization [1, 2]. Many-body localiza-tion (MBL) results when sufficiently strong disorder pre-vents ergodicity in interacting systems, and is the onlyknown generic route to avoid thermal equilibrium in iso-lated quantum systems [3, 4]. Most numerical and ana-lytical claims of MBL rest upon the assumption of low-dimensional, locally interacting Hamiltonians, and suffi-ciently long-range non-confining interactions are gener-ally believed to destroy MBL [5–8].It was therefore surprising when we recently found thatMBL can survive coupling to a global degree of freedom[9]. Global coupling to a photon is a common occurrence
FIG. 1: Proposed phase diagram of inverted mobility edge forIsing model in the presence of global qudit or spin- S mode.For Γ < Γ c ≈ . , we predict a delocalized to MBL phasetransition as the energy is increased – an inverted mobilityedge. in many-body cavity QED, where the cavity mode is pri-marily used to create all-to-all interactions between theatoms. The key result of [9] was that the strength of thisinteraction is controlled by photon number in the cavity, N . If one takes the number of atoms L to infinity whilekeeping the ratio N/ √ L fixed, all-to-all interactions re-main sufficiently weak to allow an MBL phase.This opens the interesting possibility that, as the pho-ton number – or equivalently the energy – is lowered,all-to-all interactions will reemerge and thermalize thesystem. This implies localization at high energies andthermalization at low energies, leading to an inversionof the conventional many-body mobility edge. In thispaper, we will confirm that hypothesis using numericaland analytical tools, further uncovering a reentrant MBLphase at even lower energies. While similar phenomenaoccur in cavity QED, we argue that they are more fa-vorable in non-bosonic models such as the central spin- S and central d -level system (qudit).Model – We start from the same Hamiltonian as[9], which was motivated by a standard model of spin-1/2 particles undergoing Floquet many-body localization[10]. In the Floquet extended zone picture, the time-periodic drive is treated quantum mechanically by map-ping it to a harmonic mode. This is represented geomet-rically in the inset to Figure 1. The spins form a locallycoupled chain with periodic boundary conditions. Thesespins all couple globally to a single degree of freedom,such as a cavity photon or central spin- S . The goal ofthis work will be to study the low energy limit, wherequantization of the central degree of freedom becomesimportant.Specifically, our Hamiltonian can be written H = H + H − a + ˆ a † ) + ˆ n Ω , (1) a r X i v : . [ c ond - m a t . d i s - nn ] A ug where H ± = H z ± H x , H x = L (cid:88) i =1 g Γ σ xi ,H z = L (cid:88) i =1 σ zi σ zi +1 + L (cid:88) i =1 ( h + g (cid:112) − Γ G i ) σ zi , ˆ a † = d − (cid:88) n =1 | n (cid:105)(cid:104) n − | , ˆ n = n d − (cid:88) n =0 | n (cid:105)(cid:104) n | , (2) σ x,zi are Pauli matrices, and G i is a Gaussian randomvariable of zero mean and unit variance. The spin-1/2Hamiltonians H ± yield static models with both MBLand thermal phases. The operators ˆ a and ˆ n play therole of lowering and number operators for the centralmode. In this work, we mainly study the case of a cen-tral d -level system – a qudit – for which ˆ a lowers theexcitation number by one with unit matrix element; thiswill be compared to photons and central spin- S later inthe paper. The qudit levels are split by a bare energy Ω (with (cid:126) = 1 ) and can be excited through coupling to thespin Hamiltonian H − . The most important parameterfor localization is Γ . The limits Γ = 0 and representtrivially localized and thermalizing phases, respectively.Other parameters are chosen as g = 0 . , h = 0 . and Ω = 3 . . We expect that our results will be inde-pendent of these particular parameters, though we notethat MBL is generally favored by large Ω . Furthermore,we will use d = 12 throughout to approximate d = ∞ ,such that only the lower cutoff on qudit number, n ≥ ,plays a role.Mobility edge – In this model, [9] found evidence for aninfinite temperature phase transition ( E ≈ tr[ H ] / tr[ ] )between MBL at small Γ and thermalization at large Γ upon taking L → ∞ at finite d/ √ L . The transition oc-curs at Γ c ≈ . for d/ √ L (cid:29) , which corresponds tothe Floquet limit. In this paper, we will study the energydependence of this transition. In order to obtain initialinsight into energy dependence, we utilize the results ofthe high-frequency expansion [9], rederived in the Sup-plementary Information for clarity [11]. Physically, thehigh-frequency expansion (HFE) involves perturbativelyeliminating fluctuations of the central mode via a canon-ical transformation, similar to the Floquet-Magnus ex-pansion [12, 13] or Schrieffer-Wolff transformation [14].For d = ∞ , this gives an effective Hamiltonian, H eff = H + − ( H − ) | (cid:105)(cid:104) | + ˆ n Ω + O (Ω − ) The first term in H eff consists of the undriven Hamil-tonian H + . The second comes from commutators of thequdit operators, which lead to infinite-range interactions.The leading term is ∼ ( H − ) / Ω but, importantly, it isonly active when the qudit is at its extreme value of | (cid:105) .For our model, this gives infinite range interactions near the zero energy state, which compete with local interac-tions in H + to thermalize the system. Higher order termswill give long-range interactions mediated by states | (cid:105) , | (cid:105) , etc., but suppressed by powers of Ω − .The HFE suggests the existence of an inverted mo-bility edge. For large energy, E/ Ω (cid:29) , for whichthe qudit number is n (cid:29) , no infinite-range interac-tions are produced, and the MBL-delocalized transition isgiven by that of the locally dressed H + Hamiltonian with Γ c ≈ . . For E ≈ ( n ≈ ), infinite range interactionscompete with H + , generically leading to thermalization.Numerics – To distinguish the MBL and thermalphases numerically, we first study energy eigenstates ofthe Hamiltonian (1) using shift-and-invert methods [15]to target eigenstates near a given energy, up to amaximum system size of L = 13 . Thermal systems areexpected to follow the rules of random matrix theory,while MBL phases do not. Looking at their energy lev-els, this implies that thermal eigenstates undergo levelrepulsion, following Wigner-Dyson level statistics, whileMBL eigenstates follow Poisson level statistics with nolevel repulsion. This is captured by the level spacingstatistic [16]: r n = min( δE n , δE n +1 )max( δE n , δE n +1 ) , (3)where δE n = E n − E n − is the gap between orderedeigenenergies E n . For Poisson statistics, (cid:104) r n (cid:105) = 0 . ≡ r P ois , while for the Gaussian orthogonal ensemble, (cid:104) r n (cid:105) =0 . ≡ r GOE . Figure 2(g-i) shows the numerically cal-culated level statistics. At the largest L , states near E = 0 , corresponding to the bare energy of the quditground state | (cid:105) , converge toward r GOE . At both lowerand higher energies, the level statistics appear Poisso-nian, suggesting that the system is localized. An ap-proximate window for thermalization is sketched in theplots based on where the level statistics start to drift to-wards the GOE value. Interestingly, a reemergent MBLphase appears at low energies
E < . This is consistentwith the high-frequency expansion, which at low enoughenergies will be dominated by the term − ( H − ) / (16Ω) .While this term has been argued to give infinite-rangeinteractions that compete with short-range interactions,in isolation it shares eigenstates with the local Hamilto-nian H − . Therefore, MBL for E < apparently comesfrom the static MBL phase of H − .Full convergence to r GOE is difficult to see, particu-larly for small values of Γ . Therefore, we turn to thehalf-system mutual information and Kullback-Leibler di-vergence. The Kullback-Leibler divergence (KL) mea-sures similarity between eigenstates [17]. For each eigen-state | n (cid:105) , a probability distribution is defined by p n ( i ) = |(cid:104) i | n (cid:105)| , where | i (cid:105) is an element of the σ z ⊗ ˆ n basis. Fortwo neighboring energy eigenstates | n (cid:105) and | n + 1 (cid:105) , theKL is defined by KL = (cid:80) dim ( H ) i p n ( i ) ln p n ( i ) p n +1 ( i ) . In the FIG. 2: (a-c) Half-system mutual information (I ( L/ ), (d-f) Kullback-Leibler divergence, and (g-i) level statistic (cid:104) r n (cid:105) asfunction of energy. The shaded red color indicates the approximate region of delocalization. MBL phase, this quantity increases linearly with systemsize ∝ ln (dim( H )) because nearby eigenstates are com-pletely uncorrelated. For the thermal phase, one expects KL = 2 in the thermodynamic limit from random matrixtheory [17]. The energy-dependent KL is shown in Fig-ure 2(d-f). The KL of the thermal phase is notably lowerthan MBL phase and, for small Γ , shows an inversion ofthe finite size dependence; KL increases with system sizein the MBL phase and decreases with system size in thedelocalized phase. A finite size crossing of the KL givesan approximate location of the delocalized phase, whichis seen to increase for increasing Γ .Similar behavior is seen in the half-system mutual in-formation (MI) of the energy eigenstates, defined as I(L / ≡ I( A, B ) = S ( A ) + S ( B ) − S ( A ∪ B ) (4)where S ( A ) = − tr[ ρ A ln ( ρ A )] is the von Neumann entan-glement entropy of subsystem A . The system is split intothree pieces, as shown in the inset to Figure 1, where A and B correspond to dividing the spin system intohalves and S ( A ∪ B ) = S q is the entanglement entropy ofthe qudit. Mutual information is chosen to best captureentanglement between the subsystems A and B, whichshould be area law in the MBL phase and volume lawin the delocalized phase [18]. As seen in Figure 2(a-c), mutual information is indeed higher in the delocal-ized phase, though the apparent super-volume-law scal-ing is a finite size effect which is expected to go awayat larger system sizes [9, 19]. Note that the mutual in-formation remains well below its maximal (Page) valueof I ( L/ → L ln 2 ≈ . L , further demonstrating thelarge finite size effects.To approach larger system sizes up to L = 18 , we useKrylov time evolution [15], which is limited to shortertimes. For the localized phase, we expect the system to FIG. 3: (a) Dynamics of I( L/ ) obtained using Krylov timeevolution starting from a product state. Dashed curves cor-respond to the MBL regime ( E/ √ L = − . ) and solid linescorrespond to the delocalized regime ( E/ √ L = 0 . ). (b) En-ergy dependence of I ( L/ at late time, t = 3900 . All dataare for Γ = 0 . . retain memory of its initial state to exponentially longtime, resulting in a quick plateau of the mutual informa-tion, followed by slow – potentially logarithmic – growth[20, 21]. By contrast, ergodic phases should quick reachthermal equilibrium with much larger entanglement. Thecrossover behavior is more complicated, but physics deepin these phases should be well-approximated by this sim-ple picture.We studied time evolution by preparing initial productstates in the σ z ⊗ ˆ n basis and evolving the wave functionusing the Krylov method [22]. Beginning these stateswithin a given energy window of width ∆ E = 0 . , Figure3 shows energy-resolved mutual information. We are notable to obtain data for sufficiently long times to clearlyidentify a late-time plateau, but points within the MBLand delocalized regions show different trends. For delo-calized values of E/ √ L = 0 . , the mutual informationapproaches a plateau value near the theoretical maxi-mum, I( L/ ) = L × ln (2) . On the other hand, for thestates near the ground state and in the middle of thespectrum, data consistent with a logarithmic growth ofmutual information is detected, which is suggestive of lo-calization in thermodynamic limit [20, 21]. Taking theinstantaneous mutual information at late time, t = 3900 ,we observe same trend as the data obtained using energyeigenstates (top panel of Figure 3).Discussion – Our data are consistent with the picturefrom the high-frequency expansion suggesting an invertedmobility edge for Γ < Γ c . A concern for this analysis isthe fact that the HFE is asymptotic rather than conver-gent for L = ∞ and finite Ω [23]. Therefore, we nu-merically compare the results from the exact numericsto those from the HFE. The HFE matches very well inthe low energy re-emergent MBL phase and appears toapproach the correct answer in the high energy phase,where convergence is expected when the system is local-ized by standard arguments for Floquet MBL [24, 25].Unsurprisingly, the HFE does not converge in the ther-mal regime, which is a signature of resonant delocaliza-tion [11].The HFE also shows why inverted mobility edges aremore apparent with central qudits or spins than withbosonic modes. In the photonic HFE, the leading long-range interactions become − ( H − ) / Ω independent ofphoton number [11]. Higher order corrections will pickup photon number dependence, but are more difficult tosee due the Ω − r suppression at r th order. For a centralspin S , the relevant commutator is [ S − , S + ] = − S z ,which becomes large at the edges of the spin spectrum(large | S z | ), similar to the qudit, and thus will also showan inverted mobility edge as seen in the supplement [11].In general, the energy-dependence of localization will de-pend on the manner in which the photon couples to themany-body system, and similar HFEs should enable anal-ysis of the energy dependence.We can also get some insight from the HFE aboutthe energy at which the MBL-delocalized transition oc-curs. Within the HFE, the density of states for eachqudit level | n (cid:105) is approximately Gaussian with mean n Ω and width ∼ J √ L . From the HFE, only states from the n = 0 branch contribute to the infinite-range thermaliz-ing interactions at leading order. Since the energy win-dow corresponding to n = 0 extends up to E ≈ J √ L ,we postulate that the critical energy will scale similarly: E c ∝ L / . We are unable to definitively confirm thisscaling given our small, finite-size numerics. However,plotting data as a function of E/ √ L – as is done through-out the paper – appears to give better data collapse thanplotting as a function of E (see Supplemental material[11]).Experimentally, a few systems exist in which a non-bosonic central mode is globally coupled to an interact- ing spin or electron system as required for this physics. Anotable example is the recent realization of a cavity QED-like architecture with superconducting qubits playing therole of mirrors [26, 27]. The cavity mode is replaced bythe dark state manifold of a qubit chain, whose raisingand lowering operators satisfy the commutation relationsof large spin- S ) [11]. The size of this spin is controllableby the number of qubits in the chain, hence can be scaledto large values as we use here. Currently, experimentshave shown coupling of the dark mode to a single atom-like qubit to simulate cavity QED, but we expect thatcoupling to a disordered interacting spin chain is prac-tical through conventional superconducting qubit archi-tectures [28]. Similar large-spin algebra results for cou-pling between polaritons in a semiconductor microcavityand spin impurities in the semiconductor, since in certainregimes the polaritons “inheret” the non-bosonic commu-tation relations of their matter component. [29, 30].Finally, we note that, for generic cavity-atom couplingin conventional cavity QED we also expect an inversionof the mobility edge in certain regimes, as will be detailedin an upcoming paper [31].In summary, we have shown that in centrally coupledspin chains, such as those with a central qudit or spin- S in a magnetic field, an inversion of the mobility edge ispossible. We postulate that this will be a generic featureof many such models, since long-range thermalizing inter-actions are most strongly induced at the edge of the spec-trum where the compactness of the central mode becomesapparent. This phenomology opens up further intriguingquestions about localization in such systems with compe-tition between local and global interactions, such as theexistence and character of localized bits ( (cid:96) -bits [32, 33]).Furthermore, as the energy-dependent phase transitioncomes from global interactions, it should be in a differentclass than recent avalanche pictures of the MBL transi-tion [34–37].Acknowledgments – We would like to acknowledgeuseful discussions with R. Nandkishore, N. Ng, andA. Polkovnikov. This work was performed with sup-port from the National Science Foundation throughaward number DMR-1945529 and the Welch Foundationthrough award number AT-2036-20200401. We used thecomputational resources of the Lonestar 5 cluster oper-ated by the Texas Advanced Computing Center at theUniversity of Texas at Austin and the Ganymede andTopo clusters operated by the University of Texas at Dal-las’ Cyberinfrastructure & Research Services Department [1] J. R. Wootton and J. K. 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Supplementary InformationA. More details regarding the high-frequency expansion
1. Non-Floquet derivation of high-frequency expansion
Consider the following generic cavity/qubit many-body Hamiltonian: H = Ωˆ n + H + H (ˆ a + ˆ a † ) , (5)where we can either have ˆ a represent qudit lowering operator, ˆ a q | n (cid:105) = (1 − δ n ) | n − (cid:105) , bosonic annihiliation operator, ˆ a b | n (cid:105) = √ n | n − (cid:105) , or spin lowering operator, ˆ a s | n (cid:105) = (cid:112) − n ( n − / [ s ( s + 1)] | n − (cid:105) , where for spins n = − s, − s +1 , . . . , s and otherwise n = 0 , , . . . . We’ve picked the simplest case where there is only a single harmonic and H isreal so that it only couples to the real quadrature of the cavity, but this expansion can be easily amended to treatother terms. The idea is to do a high-frequency expansion, i.e., a perturbative expansion around Ω = ∞ . Thereforewe only want to diagonalize the ˆ n operator and don’t care about diagonalizing the spin/electron operators. We willdo so by canonical transformation, similar to Schrieffer-Wolff: H eff = e iS He − iS , S = S Ω + S + S + · · · , where each S j is Hermitian. At second order, we can expand the exponentials and collect terms: H eff ≈ (cid:18) iS Ω + iS − S (cid:19) (cid:0) Ωˆ n + H + H (ˆ a + ˆ a † ) (cid:1) (cid:18) − iS Ω − iS − S (cid:19) ≈ ˆ n Ω + (cid:0) iS ˆ n − i ˆ nS + H + H (ˆ a + ˆ a † ) (cid:1) +1Ω (cid:18) ˆ n (cid:18) − iS − S (cid:19) + (cid:0) H + H (ˆ a + ˆ a † ) (cid:1) ( − iS ) + ( iS ) (ˆ n ) ( − iS ) + ( iS ) (cid:0) H + H (ˆ a + ˆ a † ) (cid:1) + (cid:18) iS − S (cid:19) ˆ n (cid:19) ≈ ˆ n Ω + (cid:0) i [ S , ˆ n ] + H + H (ˆ a + ˆ a † ) (cid:1) +1Ω (cid:18) i [ S , ˆ n ] − (cid:0) ˆ nS − S ˆ nS + S ˆ n (cid:1) + i (cid:2) S , H + H (ˆ a + ˆ a † ) (cid:3)(cid:19) . The goal is to select S j order-by-order to cancel off-diagonal terms of ˆ n . The first order term, H (1)eff = i [ S , ˆ n ] + H + H (ˆ a + ˆ a † ) has two off-diagonal terms, H ˆ a and H ˆ a † . We want to pick S to cancel these, so the natural ansatz is S = A ˆ a + A ˆ a † ,where A , are matrices acting on the spins/electrons. For any of the above choices for ˆ a , we see that [ˆ a, ˆ n ] | n (cid:105) = nc n | n − (cid:105) − c n ( n − | n − (cid:105) = c n | n − (cid:105) = ˆ a | n (cid:105) , where we define ˆ a | n (cid:105) = c n | n − (cid:105) . Thus [ˆ a, ˆ n ] = ˆ a . Similarly, (cid:2) ˆ a † , ˆ n (cid:3) = − ˆ a † . Plugging in the ansatz for S , we have i [ S , ˆ n ] = iA ˆ a − iA ˆ a † . Thus, to cancel out the off-diagonal terms, we must choose A = iH and A = − iH , i.e., S = iH (ˆ a − ˆ a † ) .One can follow a similar strategy for S , S , etc. with the natural ansatz S j = M ˆ a + M † ˆ a † + M ˆ a + M † (ˆ a † ) + . . . + M j ˆ a j + M † j (ˆ a † ) j . To obtain the effective Hamiltonian at 2nd order, we simply plug in the results from 1st orderand drop all off-diagonal terms (which will be accomplished by S ). H eff ≈ ˆ n Ω + H + 1Ω (cid:18) H (cid:16) ˆ n (cid:0) ˆ a − ˆ a † (cid:1) − (cid:0) ˆ a − ˆ a † (cid:1) ˆ n (cid:0) ˆ a − ˆ a † (cid:1) + (cid:0) ˆ a − ˆ a † (cid:1) ˆ n (cid:17) − (cid:2) H (cid:0) ˆ a − ˆ a † (cid:1) , H (ˆ a + ˆ a † ) (cid:3)(cid:19) diag = ˆ n Ω + H + H Ω (cid:18) (cid:0) − ˆ n (cid:0) ˆ a ˆ a † + ˆ a † ˆ a (cid:1) + 2ˆ a ˆ n ˆ a † + 2ˆ a † ˆ n ˆ a − (cid:0) ˆ a ˆ a † + ˆ a † ˆ a (cid:1) ˆ n (cid:1) − (cid:2) ˆ a, ˆ a † (cid:3)(cid:19) = ˆ n Ω + H + H Ω (cid:18) (cid:0) − (cid:2) (cid:26)(cid:26) ˆ a ˆ n − ˆ a (cid:3) ˆ a † − (cid:2) (cid:8)(cid:8) ˆ a † ˆ n + ˆ a † (cid:3) ˆ a + (cid:24)(cid:24)(cid:24) a ˆ n ˆ a † + (cid:24)(cid:24)(cid:24) a † ˆ n ˆ a − ˆ a (cid:2) (cid:8)(cid:8) ˆ n ˆ a † − ˆ a † (cid:3) − ˆ a † (cid:2) (cid:26)(cid:26) ˆ n ˆ a + ˆ a (cid:3)(cid:1) − (cid:2) ˆ a, ˆ a † (cid:3)(cid:19) = ˆ n Ω + H − H Ω (cid:2) ˆ a, ˆ a † (cid:3) . This is the same expression derived in [9] with the notable exception that the harmonic level spacing ˆ n Ω remainsexplicitly present. In that paper it was instead argued to derive by adiabatic continuation from the rotating frameto the lab frame. The commutator (cid:2) ˆ a, ˆ a † (cid:3) is where the difference between qudit, boson, and spin- S arises. It is,respectively, (cid:104) ˆ a b , ˆ a † b (cid:105) = 1 (cid:2) ˆ a q , ˆ a † q (cid:3) = (cid:32) d − (cid:88) n =1 | n − (cid:105)(cid:104) n | (cid:33) (cid:32) d − (cid:88) n =1 | n (cid:105)(cid:104) n − | (cid:33) − (cid:32) d − (cid:88) n =1 | n (cid:105)(cid:104) n − | (cid:33) (cid:32) d − (cid:88) n =1 | n − (cid:105)(cid:104) n | (cid:33) = | (cid:105)(cid:104) | − | d − (cid:105)(cid:104) d − | (cid:2) ˆ a s , ˆ a † s (cid:3) = 2ˆ ns ( s + 1) (6)While higher order terms can be recovered by a similar procedure, the Floquet high-frequency expansion in generalmakes this simpler by replacing direct solution of canonical perturbation theory by a solved Floquet problem [13, 38].
2. Floquet derivation of high-frequency expansion
To obtain higher order terms, we follow the same approach as in [9, 13]. Specifically, we go to a rotating framewhere the harmonic degrees of freedom in the static Hamiltonian in Eq. 5 are replaced by Floquet time-dependenceon the a and a † terms. Then we have H rot = H + H (ˆ ae − i Ω t + ˆ a † e − i Ω t ) , (7)Then we calculate higher-order terms of the effective Hamiltonian H eff using van Vleck expansion [13]: H roteff = ∞ (cid:88) n = − n H ( n )eff H ( − =ˆ n Ω H (0)eff = H (0) = H H (1)eff =[ H (1) , H ( − ] = H [ a † , a ] H (2)eff =[[ H (1) , H (0) ] , H ( − ] + h.c. = [[ H , H ] a + , H a ] + h.c. =[[ H , H ] , H ] (cid:16) a † a + aa † (cid:17) H (3)eff = 12 [[[ H (1) , H (0) ] , H (0) ] , H ( − ] + 14 [[ H (1) , [ H (1) , H ( − ]] , H ( − ] + h.c. = 12 [[[ H , H ] , H ] , H ] (cid:16) a † a + aa † (cid:17) + 12 H [[ a † , [ a † , a ]] , a ] (8)In this picture the global mode is decoupled from the spin chain, meaning [ H eff , ˆ n ] = 0 and thus we have an n -dependent spin chain Hamiltonian (cid:104) n | H eff | n (cid:105) .Using this effective Hamiltonian, one can derive the time evolution in the rotating frame U rot ( t ) using a time-dependent kick operator iK eff ( t ) , U rot ( t ) = e − iK roteff ( t ) e − iH roteff t e iK roteff (0) (9)where the first few terms of the kick operator are, iK roteff ( t ) = ∞ (cid:88) n = − n iK ( n )eff ( t ) iK (0)eff ( t ) =0 iK (1)eff ( t ) =( e i Ω t H a † − h.c. ) iK (2)eff ( t ) = e i Ω t [ H a † , H ] − h.c.iK (3)eff ( t ) = e i Ω t [[ H , H ] , H ] a † + e i Ω t H , H ] , H ]( a † ) + 2 e i Ω t H [ a † , [ a † , a ]] − h.c. (10)Finally, using the rotation operator e − i ˆ n Ω t , we can go back to the lab frame: e iK labeff ( t ) = e − i ˆ n Ω t e iK roteff ( t ) , H roteff = H labeff (11)For comparison with numerical simulations of the full Hamiltonian, Eq. 5, we use the qudit algebra as in the maintext: ˆ a q = n − (cid:88) n =1 | n − (cid:105) (cid:104) n | , ˆ n = d − (cid:88) n =1 n | n (cid:105) (cid:104) n | , (cid:2) ˆ a q , ˆ a † q (cid:3) = | (cid:105)(cid:104) | − | d − (cid:105)(cid:104) d − | . (12)Substituting these relations into Eq. 8, we can derived H eff for d → ∞ : H eff =ˆ n Ω + H + H − ) (cid:16) | (cid:105)(cid:104) | (cid:17) + 116Ω [[ H − , H + ] , H − ] (cid:16) − | (cid:105)(cid:104) | (cid:17) +132Ω (cid:16)
12 [[[ H − , H + ] , H + ] , H − ](2 − | (cid:105)(cid:104) | ) + 1512 H − ( | (cid:105) (cid:104) | − | (cid:105) (cid:104) | ) (cid:17) + O (Ω − ) (13)In this expansion, the zeroth order term and all terms with commutators are short-range-interacting with standardMBL-thermal phase transitions. But terms proportional to H n +1 − / Ω n for odd n are non-local and introduce all-to-allcoupling. In expansion terms we derived here, these non-local terms only show up for the states at extreme of quditlevels, here | (cid:105) and | (cid:105) , but it also will be present in higher-order terms for other qudit levels. These higher orderterms are suppressed by the Ω − n factors, and thus it can be ignored for large Ω .The presence of these all-to-all terms is responsible for delocalizing the states near the edge of the spectrum at Γ < Γ c . In our study, we use a moderate frequency ( Ω = 5 π/ ), for which the high-frequency expansion is, at best,asymptotic. For even higher frequencies, the dominant terms at our system size would simply be a time average of theoriginal Hamiltonian, which cannot give rise to new physics. These effect of coupling between number sectors wouldthen become apparent at much larger system sizes, beyond our numerical reach. Lower frequencies, meanwhile, willdestroy convergence of the high-frequency expansion.
3. Numerical tests of the high-frequency expansion
FIG. 4: Mutual information I( L/ ) calculated for each global mode level separately in H eff . For E < , the major contributionis from n = 0 , which strongly supports our argument that localization of these states is due to the ( H ) term. For states nearthe middle of the spectrum, the sum of all separate levels is given in a dashed black line. The match between actual Hamiltonianand H eff depends on the expansion order. As we include higher-order terms, the localized phase seems to be converging, whilethe delocalized phase does not. Slow convergence even in the localized phase is expected, but should eventually occur at higherenergies were the full Hamiltonian is equivalent to Floquet MBL. Finally, we numerically verified whether the effective Hamiltonian, Eq. 13, fits the actual data up to order O (Ω − ) .In the high-frequency expansion picture, the Hamiltonian is diagonalized within the qudit Hilbert space, giving ablock diagonal matrix of size d × d . In this picture, the first block refers to the n = 0 qudit level, second block tothe n = 1 qudit level, and so on. Each block is individually calculated and then the results are summed over qubitsectors, taking care to apply the kick operator to rotate back to the lab frame. The result is shown in Figure 4. Inall cases, the low energy states of the full Hamiltonian are consistent with the high-frequency expansion, since the ( H − ) term dominates. The inclusion of higher-order terms alters blocks at higher n , closer to the middle of thespectrum. Since the frequency we have chosen is not particularly large, we find the expected results that a large0number of H eff expansion terms are required to get an acceptable numerical consistency between the full Hamiltonianand expansion terms. We note that convergence is much cleaner in the high-energy MBL phase than the E ≈ thermal phase – where the HFE is expected to break down – but that the MBL has not fully converged by fourthorder. We nevertheless expect that the HFE will converge within the MBL phase at sufficiently high order, as this isequivalent to the well-established Floquet MBL phase in the extended zone picture. FIG. 5: Mutual information and KL divergence as a function of either unscaled energy E or scaled energy E/ √ L . All plotsare for Γ = 0 . . Scaling E/ √ L appears to give better collapse. Dashed lines show the maximum values (minimum values) formutual information (KL divergence), which seem to converge better for E/ √ L scaling. Having argued for the consistency of the HFE, we can now use it to predict properties such as the scaling of thetransition energy with system size. We use the fact that density of states (DOS) for the short-range interacting spinchain has Gaussian distribution of the form [39], D n ( E ) ≈ L √ πLJ eff exp (cid:32) − (cid:18) E − n Ω J eff √ L (cid:19) (cid:33) , (14)where we assumed that this distribution is centered at n Ω for qudit level n . This form of DOS should be correct fora finite value of L , though as we go to the thermodynamic limit, the non-local terms such as ( H − ) can modify thedensity of states at low n [9]. Setting aside this potential issue, the standard deviation of the DOS for n = 0 shouldset the approximate energy scale of the MBL-delocalized phase transition, since this sets the scale over which thesenon-local terms compete with local Hamiltonians at higher n . This predicts a transition energy E c ∝ L . To testthis numerically, we plotted I( L/ ) and KL as a function of either E and E/ √ L . Figure 5 shows that E/ √ L scalinggives a better convergence for the maximum (minimum) value of I( L/ ) (KL), though we are unable to state anythingconclusively for such small system size. B. Localization with large- s central spin While central qudits provide a numerically tractable test of localization with a central mode, it is not experimentallyclear how to realize such a system. Two main types of central mode occur naturally: central bosonic modes, as incavity QED, or central spins. In this section, we discuss and present data for localization in the presence of a large- s central spin.For the central spin mode, where ˆ a s ∼ S − and ˆ a ∼ S + , the algebra reads ˆ a † s = 1 (cid:112) s ( s + 1) s (cid:88) n = − s +1 (cid:112) s ( s + 1) − n ( n + 1) | n (cid:105) (cid:104) n − | , ˆ a s = 1 (cid:112) s ( s + 1) s (cid:88) n = − s +1 (cid:112) s ( s + 1) − n ( n − | n − (cid:105) (cid:104) n | , ˆ n = s (cid:88) n = − s n | n (cid:105) (cid:104) n | ∝ S z , (cid:2) ˆ a s , ˆ a † s (cid:3) = 2ˆ ns ( s + 1) (15)1 FIG. 6: Mutual information as a function of energy using central spin- s with s = 12 . Note that the normalization of ˆ a s is chosen such that matrix elements are equal to near the center of the spectrum(state | (cid:105) ), so that this part of the spectrum reproduces the Floquet extended zone picture. The high-frequencyexpansion for large central spin is H effrot = ˆ n Ω + H + ( H ) Ω [ˆ a † s , ˆ a s ] + 12Ω (cid:16) [[ H , H ]ˆ a † s , H ˆˆ a s ] + h.c. (cid:17) + . . . = ˆ n Ω + H + ( H ) s ( s + 1)Ω 2ˆ n + 1 s ( s + 1)Ω [[ H , H ] , H ] (cid:16) ˆ a † s ˆ a s + ˆ a s ˆ a † s (cid:17) + . . . (16)To observe the mobility edge numerically for large central spin, we need to find an appropriate regime where themagnitude of the third term in Eq. 16 is comparable to that of the second term. Near the low energy states, ˆ n ≈ − s ,we can write H + ( H ) s ( s + 1)Ω 2ˆ n ≈ H − H ) ( s + 1)Ω ≈ H − H (cid:20) ( H )( s + 1)Ω (cid:21) . (17)Since H is a local Hamiltonian with extensive energy variance, the relevant magnitude is the standard deviationover energy eigenstates, σ ( (cid:104) H (cid:105) ) , which is proportional to √ L . Therefore, to make the term in square brackets inEq. 17 of order unity, we need s (cid:63) √ L . We therefore used moderate system sizes of L ∈ [8 , , , with s = 12 .The Hamiltonian is identical to that used for the central qudit in the main text with ladder operators ˆ a s and ˆ a † s from Eq. 15. The results are shown in Figure 6. Clearly, the mutual information behaves similarly to the mutualinformation obtained with the central qudit (main text, Figure 2). The mutual information near E = − s Ω (minimal ˆ n ) is much larger than near the middle or edges of the spectrum. There is a region near zero energy which showsan unexpected decrease in mutual information, suggesting stronger localization. As of yet, we are unable to explainthis result, though we note that it occurs near the energy E = 0 = ⇒ n = 0 where the long-range term in the HFE, ( H − ) ˆ n/ Ω , vanishes.These results suggest that an inverted mobility edge is possible for the experimentally relevant case of large centralspin. We note briefly that similar results can be obtained as a side effect of coupling bosonic modes in a different waythan in this paper, such as via the vector potential: H int = (cid:88)
In [26], the authors demonstrate a cavity QED-like architecture in which multiple “cavity” qubits are coupledtogether to form a collective dark state manifold, which in turn couples to a single “probe” qubit. While the dark2state plays a role similar to the bosonic cavity in cavity QED, here we show that its commutation relations are actuallythose of a large spin- s , with the number of levels s + 1 set by the number of cavity qubits.The collective dark states defined in [26] has lowering and raising operators of the form, ˆ S D = 1 / √ N (cid:88) m> (ˆ σ mge + ˆ σ − mge )( − m , (18)where N , an even number, is the total number of qubits evenly installed about the probe qudit, and m runs up to N/ . Here ˆ σ mge = | g m (cid:105) (cid:104) e m | where g and e are referring to the ground and excited states of the m th qubit respectively.Using Eq. 18, the commutation relation of raising and lowering operators becomes, (cid:104) ˆ S D , ˆ S † D (cid:105) = 1 /N (cid:88) m,n> (cid:2) ˆ σ mge + ˆ σ − mge , ˆ σ † nge + ˆ σ †− nge (cid:3) ( − m + n = 1 /N (cid:88) m,n> [ | g m (cid:105) (cid:104) e m | + | g − m (cid:105) (cid:104) e − m | , | e n (cid:105) (cid:104) g n | + | e − n (cid:105) (cid:104) g − n | ] ( − m + n (19)Since (cid:104) g m | g − n (cid:105) = (cid:104) e m | e − n (cid:105) = 0 for any m and n , we can write (cid:104) ˆ S D , ˆ S † D (cid:105) = 1 /N (cid:88) m,n> (cid:16) [ | g m (cid:105) (cid:104) e m | , | e n (cid:105) (cid:104) g n | ] + [ | g − m (cid:105) (cid:104) e − m | , | e − n (cid:105) (cid:104) g − n | ] (cid:17) ( − m + n (20)We also have [ | g m (cid:105) (cid:104) e m | , | e n (cid:105) (cid:104) g n | ] = | g m (cid:105) (cid:104) e m | | e n (cid:105) (cid:104) g n | − | e n (cid:105) (cid:104) g n | | g m (cid:105) (cid:104) e m | = | g m (cid:105) (cid:104) g n | δ m,n − | e n (cid:105) (cid:104) e m | δ m,n (21)Finally, we can write Eq. 20 as (cid:104) ˆ S D , ˆ S † D (cid:105) = 1 /N (cid:88) m (cid:18) | g m (cid:105) (cid:104) g m | − | e m (cid:105) (cid:104) e m | + | g − m (cid:105) (cid:104) g − n | − | e − n (cid:105) (cid:104) e − m | (cid:19) ( − m (22) ( − m = 1 . Changing the summation range, (cid:104) ˆ S D , ˆ S † D (cid:105) = 1 /N (cid:88) − N 2) + 1 as for a single spins. For spin we have [ ˆ S + , ˆ S − ] = 2 ˆ S zz