Probing the topological Anderson transition with quantum walks
Dmitry Bagrets, Kun Woo Kim, Sonja Barkhofen, Syamsundar De, Jan Sperling, Christine Silberhorn, Alexander Altland, Tobias Micklitz
PProbing the topological Anderson transition with quantum walks
Dmitry Bagrets, Kun Woo Kim, Sonja Barkhofen, Syamsundar De, JanSperling, Christine Silberhorn, Alexander Altland, and Tobias Micklitz Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ulpicher Straße 77, 50937 K¨oln, Germany Integrated Quantum Optics Group, Institute for Photonic Quantum Systems (PhoQS),Paderborn University, Warburger Straße 100, 33098 Paderborn, Germany Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil (Dated: February 3, 2021)We consider one-dimensional quantum walks in optical linear networks with synthetically intro-duced disorder and tunable system parameters allowing for the engineered realization of distincttopological phases. The option to directly monitor the walker’s probability distribution makes thisoptical platform ideally suited for the experimental observation of the unique signatures of the one-dimensional topological Anderson transition. We analytically calculate the probability distributiondescribing the quantum critical walk in terms of a (time staggered) spin polarization signal andpropose a concrete experimental protocol for its measurement. Numerical simulations back therealizability of our blueprint with current date experimental hardware.
I. INTRODUCTION
Low-dimensional disordered quantum systems can es-cape the common fate of Anderson localization oncetopology comes into play, as first witnessed at the in-teger quantum Hall plateau transitions . The adventof topological insulators has brought a systematic un-derstanding of topological Anderson insulators and theirphase transitions . Critical states at Anderson local-ization transitions typically show unusual spectral- andwave-function statistics , as well as anomalous diffusivedynamics. From the single parameter scaling theory oflocalization one e.g. expects a scaling (cid:104) q ( t ) (cid:105) ∼ t /d of the mean square displacement at a conventional d -dimensional localization transition . Topological local-ization transitions, on the other hand, follow a two-parameter scaling and the situation is more complex .The controlled experimental study of a critical state atthe Anderson localization transition presents an intrigu-ing challenge. It has been first accomplished within acold-atom realization of the quantum kicked rotor for thethree-dimensional Anderson localization transition in theorthogonal class . A corresponding study of a topological localization transition requires the control over additionalinternal degrees of freedom and has, to our knowledge,not been realized so far.The physics of the topological Anderson localizationtransition is particularly intriguing in one-dimensional(1 d ) systems, where disorder is exceptionally efficientin inducing quantum localization on short length scales.Topological quantum criticality then reflects a fight be-tween two powerful principles: strong localization vsthe enforced change of an integer topological invariant.Topology trumps localization and forces long range cor-relations across the system. In practical terms, this im-plies a divergent localization length, and finite conduc-tion. However, the reluctance of the system to conductshows in an extremely (logarithmically) slow spreading ofquantum states at criticality , strikingly different to q -2 -1 0 1 2 t 𝑅 𝑅𝑇 𝑇 FIG. 1. A schematic evolution of a discrete-time quantumwalk of a spin 1 / R and step ˆ T operators describethe dynamics on discrete lattice sites q ∈ Z . In step two atposition q = 0, the first interference takes place. the diffusive dynamics conventionally observed at quan-tum phase transitions between disordered phases. In thispaper, we connect the physics of one-dimensional topo-logical quantum criticality to the unique opportunitiesoffered by quantum optics experimentation. We presenta concrete and experimentally realistic blueprint for atunable 1 d quantum walk in which the unique signaturesof topological quantum criticality show via a (time stag-gered) spin polarization signal.Quantum walks have been implemented on variousexperimental platforms, such as photons , ions ,atoms and nuclear magnetic resonance . A de-tailed introduction to experimental implementations ofquantum walks can be found in Ref. . Quantum walksallow for a large tunability of the system parametersand have been used experimentally to observe Andersonlocalization , dynamical localization and topolog-ical effects . Quantum walk systems thus open the a r X i v : . [ qu a n t - ph ] F e b perspective of a low-dimensional system in which disorderand nontrivial topology can be introduced in controlledmanners. Direct experimental access to the probabilitydistribution allows, moreover, for a full characterizationof the walker’s dynamics.A prototypical quantum walk is depicted in Fig. 1. Itis generated by the single time-step evolution ˆ U = ˆ R ˆ T ,iteratively acting on a walker with a two dimensionalinternal degree of freedom, refered to as ‘spin’ in thefollowing. Here ˆ T translates the particle on a discreteone-dimensional lattice. Depending on its internal spin-state, the walker propagates to the left or right, and ˆ R is a rotation in spin-space. Using linear optical elements,discrete time quantum walks have been used to measureprobability distributions of walkers exposed to tunabledisorder and decoherence . Specifically, conditionsfor ballistic, Anderson-localized and classically diffusivedynamics were prepared, and the corresponding walkersprobability distributions [see also Eq. (9)] were observed.That is, the following scenarios apply: (i) P σ (cid:48) σ ( t, q ) ∼ δ ( t − σq ) δ σσ (cid:48) for a translational invariant quantum sys-tem; (ii) P σ (cid:48) σ ( t, q ) ∼ exp( −| q | /ξ loc ) for a disorderedquantum system; and (iii) P σ (cid:48) σ ( t, q ) ∼ exp( − q /Dt ) fora disordered classical system, where ξ loc and D are lo-calization length and diffusion constant. In the photonicimplementation, the internal ‘spin’-states correspond tohorizontal, | H (cid:105) , and vertical, | V (cid:105) , polarization direc-tions, and disorder is controlled by local variations ofpolarization plates . Rotations that do not explore allSU(2)-angles independently leave symmetries, which canplace the walk into one of the five nonstandard symmetryclasses hosting topologically interesting phases .In this paper, we explore a quantum walk operating ata topological Anderson localization transition. We derivethe walker’s critical probability distribution at the topo-logical transition and find a (time-staggered) spin polar-ization as a smoking-gun evidence for the critical dynam-ics. We indicate a protocol which allows for an observa-tion of the discussed features within existing experimen-tal platforms, and compare results of our effective fieldtheory approach to numerical simulations. The remain-der of the paper is organized as follows. In Sec. II, weintroduce a quantum walk with chiral symmetry that canbe tuned to a quantum critical point separating two topo-logically different Anderson insulators. In Sec. III, weanalyze the probability distribution of the critical walkerand propose, in Sec. IV, an experimental protocol thatallows one to study the predicted effect. We conclude inSec. V with a discussion and an outlook. Several techni-cal discussions are relegated to appendices. II. CHIRAL QUANTUM WALK
We start our discussion with a general one-dimensionalquantum walk of a spin-1 / 𝑝 ε ( 𝑝 ) - 𝜋/2 - 𝜋𝜋 - 𝜋 𝜋 𝜋 - FIG. 2. Dispersion-relation (cid:15) ( p ) of Floquet bands shown forangles ϕ = 0 and θ = 0 (black line), π/ π/ time-step evolution ˆ U ( φ, ϕ, θ ) = ˆ R ( φ, ϕ, θ ) ˆ T . (1)The spin-dependent ‘shift’ operator ˆ T here is diagonal inthe ˆ s -eigenbasis,ˆ T = (cid:88) q ( | q + 1 , ↑(cid:105)(cid:104)↑ , q | + | q − , ↓(cid:105)(cid:104)↓ , q | ) , (2)where q ∈ Z are the lattice sites with unit spacing, and‘spin’ states {| ↑(cid:105) , | ↓(cid:105)} parametrize the walker’s two-dimensional internal degrees of freedom, see also Fig. 1.Local ‘coin’ rotations,ˆ R = (cid:88) q,σσ (cid:48) | q, σ (cid:105) R σσ (cid:48) q (cid:104) q, σ (cid:48) | , (3)are conveniently parametrized by (site-dependent) Eulerangles R q ( φ, ϕ, θ ) = exp( i φ q σ ) exp( iϕ q σ ) exp( i θ q σ ),where Pauli matrices σ i operate in spin-space. Employ-ing a symmetrized time-splitting, we can writeˆ U ( φ, ϕ, θ ) = R z ( ˆ ϕ ) R x ( ˆ θ ) ˆ T R x ( ˆ φ ) R z ( ˆ ϕ ) , (4)with ˆ φ , ˆ ϕ , ˆ θ being site-diagonal matrices of angles and R i (ˆ α ) = exp( i ˆ ασ i ) defines a rotation along i -th direc-tion. From Eq. (4), one readily verifies that quantumwalks subject to the constraint ˆ θ = ˆ φ exhibit a chiralsymmetry , σ ˆ U σ = ˆ U † . (5)That is, the latter are members of the chiral symmetryclass AIII, which may host Z × Z topological insulatingphases for one-dimensional quantum walks . In the fol-lowing, we focus on walks with the chiral symmetry, asdenoted in Eq. (5).Chiral symmetry of the time-evolution operator re-flects in a spectrum which is symmetric around zero inthe 2 π -periodic ‘Brillouin zone’ of quasi-energies (cid:15) ± p = ± (cid:15) p . For spatially constant rotations, Floquet eigen-states are plane-waves and the two energy bands are de-fined by the relationcos( (cid:15) p ( θ, ϕ )) = cos( ϕ + p ) cos ( θ ) − cos( ϕ − p ) sin ( θ ) , (6)with p being the momentum of the plane wave. Finiteangles ϕ and θ shift the momentum and tune the band-width of Bloch-bands; see Fig. 2. Specifically, linearlydispersing bands that extend over the entire Brillouinezone exist at values θ = 0 , π , with (cid:15) p ( θ, ϕ ) = θ + p + e iθ ϕ, (7)and ϕ = ± π , with (cid:15) p ( θ, ϕ ) = π ± sgn( ϕ ) p. (8)For any other values of the angles, the spectrum is gappedaround (cid:15) = 0 and π . Disorder can be introduced in a con-trolled manner by randomizing angles. Assuming shortrange site-to-site correlations, rotations are then charac-terized by average angles ¯ θ , ¯ ϕ and their deviations γ θ , γ ϕ ,which we assume to be identical for all lattice sites. In onedimension, even weak disorder γ θ,ϕ (cid:28) , which is set by the spatial scaleon which the random rotation angles fluctuate. The pres-ence of the chiral symmetry, Eq. (5), on the other hand,allows the quantum walker to escape the common fate ofAnderson localization. This happens when the system isfine tuned to the quantum critical point, separating twotopologically different Anderson insulating phases, as wediscuss next. III. CRITICAL DISTRIBUTION
To elaborate on the last mentioned point, we considerthe probability distribution, P σ (cid:48) σ ( t, q ) = (cid:104)|(cid:104) q, σ (cid:48) | ˆ U t | , σ (cid:105)| (cid:105) θ,ϕ , (9)for a walker who is initially in eigenstate | σ (cid:105) = | ←(cid:105) , | →(cid:105) of the chiral operator ˆ σ . This distribution yields theprobability of the walker to be found after t time-stepsat a distance q in eigenstate | σ (cid:48) (cid:105) . Here and in the follow-ing, (cid:104) . . . (cid:105) θ,ϕ denotes averages over distributions of an-gles. Since particles conserve their (quasi-)energies, itis convenient to Fourier transform Eq. (9) to a spectralrepresentation P σ (cid:48) σ ( ω, q ) = (cid:90) d(cid:15) (cid:104)(cid:104) q, σ (cid:48) | ˆ G R(cid:15) + ω | , σ (cid:105)(cid:104) , σ | ˆ G A(cid:15) − ω | q, σ (cid:48) (cid:105)(cid:105) θ,ϕ . (10)Here we introduced the retarded (particle) and advanced(hole) propagators ˆ G R(cid:15) = [1 − e i(cid:15) − ˆ U ] − and ˆ G A(cid:15) = [ ˆ G R(cid:15) ] † ,respectively.The chiral symmetry relates particle and hole dynam-ics for states in the vicinity of particle-hole symmetricpoints (cid:15) (cid:39) , π in the 2 π -periodic spectrum. More specif-ically, the chiral symmetry Eq. (5) translates into therelation, ˆ G A − (cid:15) = σ ˆ G R(cid:15) σ , (11) FIG. 3. Two-parameter flow of conductance g ( L ) and aver-age topological index χ ( L ) for class AIII nonlinear σ -modelwith bare values g (1) ∼ χ (1) = ¯ χ . Inset: the phasediagram of the quantum walk. Assuming ¯ ϕ (cid:54) = 0 the systemis at criticality provided ¯ θ = 0 or π . Away from criticality,the pair of topological indices ( χ , χ π ) flow to either (0 ,
1) or(1 ,
0) which defines two distinct Anderson localized topologi-cal phases. indicating that the dynamics of particles and holes at afixed energy is only related for (cid:15) (cid:39) − (cid:15) . That is, for statesin the vicinity of particle-hole symmetric points (cid:15) (cid:39) , π .To account for the (breaking of) symmetry betweenparticle and hole propagators in different ranges of thequasi-energy spectrum, we change to an energy represen-tation of Eq. (9), and separate Fourier components intotwo contributions , P σ (cid:48) σ ( ω, q ) (cid:39) P reg ( ω, q ) + P chiral σ (cid:48) σ ( ω, q ) . (12)Herein, the first (spin-independent) contribution resultsfrom single-particle states, with | (cid:15) | , | (cid:15) − π | (cid:38) ω , for whichenergy detuning is large enough to break the chiral sym-metry between propagators that compose the probabilitydistribution as P ∼ G R G A . Consequently, the probabil-ity distribution for states breaking the chiral symmetryis (on long time and length scales) identical to that ofconventional Anderson insulators. That is, upon Fouriertransform the first contribution is given by a static prob-ability distribution, P reg ( t, q ) ∼ θ ( t ) e −| q | /ξ loc . (13)By contrast, the second contribution results from stateswith quasi-energies | (cid:15) | , | (cid:15) − π | (cid:46) | ω | , for which both prop-agators are related by chiral symmetry. That is, P chiral σ (cid:48) σ ( ω, q ) (cid:39) | ω |(cid:104)(cid:104) q, σ | ˆ G R ω | , σ (cid:105)(cid:104) , σ | ˆ G A − ω | σ, q (cid:105)(cid:105) θ,ϕ (14)encodes the critical dynamics of the walker, and a non-trivial time resolved behavior can be expected. We nextapply field-theory methods to identify quantum criticalpoints of the chiral walk and calculate the critical distri-bution in Eq. (14). A. Sinai diffusion
Following standard approach to disordered sys-tems we describe the physics of the critical statesaround (cid:15) (cid:39) , π by a Ginzburg Landau type effectivetheory. More specifically, we derive in Appendix A anonlinear σ -model action which encodes the full quan-tum dynamics of soft diffusion modes and interferenceprocesses which eventually drive strong Anderson local-ization. What changes this conventional behavior in ourcase is a topological contribution to the effective action.The σ -model is parametrized by the frequency ω (which however does not ‘flow’ in an renormalizationgroup sense), and two coupling constants, viz. the con-ductance g and topological angle χ , see Eq. (A25). In aconventional disordered system g follows a single param-eter scaling , which in 1 d predicts a single, Andersoninsulating phase. The angle χ , on the other hand, allowsfor a characterization of topologically different Andersoninsulating phases. This opens the possibility to escapeAnderson localization when fined tuned to specific, criti-cal values, separating two topologically distinct Andersoninsulators.The bare topological angle for the chiral quantum walkreads (see Appendix A for details)¯ χ (cid:15) = 12 (cid:0) − e i(cid:15) (cid:104) sin( θ ) cos( ϕ ) (cid:105) θ,ϕ (cid:1) , (15)with (cid:15) = 0 , π indicating the critical states describedby the effective action. The presence of two couplingconstants in the σ -model action — the ‘conductance’and ‘topological angle’ — leads to a two-parameterflow and resulting phase-diagram shown in Fig. 3 . Forgeneric bare values, the average topological angle flowsto the closest integer value χ = 0 and 1, characterizingthe two Anderson insulating phases realized by the chiralquantum walk, while χ = 1 / ϕ , a critical line corresponds to¯ θ = 0 , ± π . The same configuration of angles in the cleanlimit gives rise to gapless linearly dispersing bands, asexpected from analogy to the corresponding Hamiltoniansystem. For generic ¯ θ , on the other hand, criticality isachieved at ¯ ϕ = ± π/
2. Finally, we remark that, in thestrong disorder limit where angels are randomly drawnfrom the entire unit circle, γ θ,ϕ = 2 π , the quantum walkis always critical. The same effective action also describesdisordered quantum wires with chiral symmetry and barelocalization length ξ loc = 1 / . Criticality in the strongdisorder limit is, however, a peculiarity of the Floquetsystem.Concentrating then on the vicinity of a criticalpoint, we can calculate the walker’s critical distributionEq. (14). The rather technical calculation is detailed inAppendix B and indicates the scaling form P chiral σ (cid:48) σ = N ( t ) F σ (cid:48) σ ( | q | ξ − t ) , ξ t ≡ π ln t, (16) with a time-dependent normalization factor N ( t ) ∝ ln − t and the explicit form of F ( x ) to be given given below. The scaling of ξ t implies anomalously retarded ‘Sinai dif-fusion’ of critical states, characterized by a mean dis-placement (cid:104)| q |(cid:105) ∼ log t. (17)Another feature of the critical distribution is the depen-dence on spin orientations σσ (cid:48) = ± σ . This can be seen from the scalingfunctions in the long time and distance limits, t, q (cid:29) P chiral σ (cid:48) σ ( t, q ) ∝ t ∞ (cid:88) n =1 ( σσ (cid:48) ) ( n +1) n e − n | q | /ξ t . (18)Focusing on the tails | q | (cid:29) ξ t , one finds from Eq. (18)the exponential profiles F ± σ,σ ( x ) = e − x ± e − x + . . . , (19)where the leading spin-independent contribution remindsus of conventional Anderson insulators. The directly fol-lowing terms indicate, however, dependence of the criti-cal distribution on the spin orientation of the final state ± σ , with interesting implications, being discussed in thenext subsection. The full distributions [cf. Eq. (18)] areshown in Fig. 4, and we refer the interested reader toAppendix B for more detailed analytical expression. Wenext discuss how the characteristic features of the criticaldistribution, viz. (i) slowly increasing width ξ t in timeand (ii) dependence on spin-orientation with respect tothe basis of the chiral symmetry operator, can be ob-served in experiments. B. “Time-staggered” spin polarization
Sinai diffusion has previously been predicted for dis-ordered one-dimensional systems with particle-hole sym-metry , and our above result for a system with chi-ral symmetry indicates that it is a universal dynamicalfeature at one-dimensional topological Anderson localiza-tion transitions. Arguably, observation of the weak loga-rithmic time-dependence presents an experimentally in-triguing challenge . Recalling, moreover, the contribu-tion from non-critical states to the total probability dis-tribution implies that Sinai diffusion is generally maskedby conventionally Anderson localization. Complicatingthis matter even further, the number of critical statesresolved in time t reduces as | ω | ∼ /t . This gener-ates additional time dependencies in the distributions ofnon-critical and critical states, as summarized in the nor-malization N ( t ) of Eq. (16). The optical linear networkrealization of a quantum walk discussed in the introduc-tion, on the other hand, allows for a direct observation ofspin-resolved probability distributions. This opens an in-teresting opportunity to observe the second feature, i.e.,
20 10 0 10 20 q P ( t , q )
20 0 200.00.20.40.6
FIG. 4. Walker’s critical probability distribution P chiral σ (cid:48) σ ( t, q ),Eq. (18), for t = 10. Spin-configurations ( σ (cid:48) , σ ) of final andinitial states are aligned ( → , → ) (light red) and anti-aligned( ← , → ) (dark red), and distributions are normalized by theaverage return probability per spin. Inset: Spin polarization∆ P ( t, q ) (peak at origin is not fully shown). the peculiar spin-dependence of the critical walk. Specif-ically, this suggests to measure the difference∆ P ( t, q ) ≡ P chiral →→ ( t, q ) − P chiral ←→ ( t, q ) , (20)which only depends on the critical contribution to thetotal probability. A finite spin polarization of the criticalwalker may be viewed as a precursor of spin polarizedboundary states emerging in the topologically non trivialphase .The q -dependence of the difference ∆ P ( t, q ) is shownin the inset of Fig. 4. The corresponding long time prob-ability distributions for non-critical states and conven-tional Anderson insulators (with spin orbit interaction)do not keep the memory of the initial spin-configuration.The observation of ∆ P ( t, q ) would thus provide clear ev-idence for the critical walk at an Anderson localizationtransition.Further smoking-gun evidence for the critical distribu-tion is then obtained from an additional symmetry of theFloquet operator, not discussed so far. The discrete lat-tice structure motivates the introduction of the sublatticeoperator ˆ S ≡ (cid:88) q | q (cid:105) ( − q (cid:104) q | , (21)anti-commuting with the Floquet operator ˆ U . FromEq. (21), one can construct a chiral-sublattice operatorˆ C sl ≡ σ ⊗ ˆ S, (22)satisfying ˆ C sl i ˆ U ˆ C sl = ( i ˆ U ) † , and consequently resultingin ˆ G A(cid:15) − (cid:15) = ˆ C sl ˆ G R(cid:15) + (cid:15) ˆ C sl , (23)whenever (cid:15) = ± π . That is, ˆ C sl is an additional chiralsymmetry that applies to critical states (cid:15) (cid:39) ± π . This symmmetry is also visible in the density of states as weshow in Appendix C. Interestingly, ˆ C sl has different im-plications for time evolution when extending over an evenor odd number of time steps t . We show in Appendix Dthat, for critical states induced by the chiral-sublatticesymmetry ˆ C sl , the spin-polarization alternates in betweentime-steps; that is∆ P ( t, q ) = ( − t | ∆ P ( t, q ) | (24)holds true. The two main obervations leading to Eq. (24)are the following (for a more rigorous explanation, seeAppendix D). (i) The Floquet operator induces transi-tions between subspaces of opposite site-parity. Thatis, walkers positioned at an even site propagate in thefollowing time step to an odd site, and vice versa. (ii)Eigenstates of ˆ C sl have alternating spin-polarization oneven and odd sites; e.g., | q, →(cid:105) are eigenstates of ˆ C sl witheigenvalues ( − q , and analogously for | q, ←(cid:105) . Combin-ing both observation, we notice that walkers propagatingfor an even number of time steps have spins aligned iftheir initial and final states are eigenstates of ˆ C sl to thesame eigenvalue. (The same applies for the chiral oper-ator σ .) In contrast, for an odd number of time steps,the walker has spins anti-aligned if initial and final statesare eigenstates of ˆ C sl to the same eigenvalue. This dif-ference simply follows from the observation that, for anodd number of time steps, the walk starts and ends inopposite parity sectors. Finally, we also remark that theprobability P σ (cid:48) σ has to be read as the transition within( σ (cid:48) = σ ) or between ( σ (cid:48) = − σ ) eigenspaces of the chiraloperator. Combining the above, it follows that for oddnumbers of time steps spin polarization reverses its sign.We thus conclude that, for critical states induced by thechiral-sublattice symmetry ˆ C sl , the spin-polarization al-ternates in between time steps, as indicated in Eq. (24). IV. EXPERIMENTAL PROPOSAL
We now devise an experimental protocol which allowsus to observe the discussed characteristic features of thequantum critical walk.
A. Experimental protocols
Quantum walks are typically initialised at a localisedposition and thus involve eigenmodes from the entirequasi-energy domain. The experimental challenge thenis to restrict the dynamics to states that approximatelypreserve chiral symmetry. We suggest to prepare a singlephoton in a coherent superposition described by | ψ p M (cid:105) = 1 √ M + 1 (cid:88) | q |≤ M/ | (2 q ) (cid:105) ⊗ | →(cid:105) e iqp , (25)occupying ( M +1) even lattice sites, where M (cid:29) (cid:15) p . Alternatively, one can make use of the equivalence ofcoherent light and a single quantum particle when prop-agating in a linear optical network and directly apply atrain of coherent laser pulses. With | ψ p M (cid:105) as the delo-calised initial state, time dependence of the localizationlength ξ t cannot be captured, but the spatially integratedspin-polarization ∆ P ( t ) can serve as a key measure forcritical phases. Specifically, we define the latter as∆ P ( t ) ≡ (cid:88) q ∆ P ψ ( t, q ) , (26)where ∆ P ψ ( t, q ) ≡ P → ψ ( t, q ) − P ← ψ ( t, q ) and the spin-dependent local probabilities are P σψ ( t, q ) ≡ (cid:104)|(cid:104) q, σ | ˆ U t | ψ p M (cid:105)| (cid:105) θ,ϕ . (27)Figures 5 and 6 show our numerical simulations forthe spin polarization ∆ P ( t ) using the initial state fromEq. (25) for M = 10 . We here did not assume peri-odic boundary conditions, i.e. the signal could propa-gate without restrictions to the left and right. In theseplots, light red histograms simulate the critical walk atthe topological Anderson localization transition, corre-sponding to critical energies (cid:15) (cid:39) (cid:15) (cid:39) π/ t = 40 time-steps, which is in the reach of current ex-periments, indeed . By contrast, the dark red curveis a simulation of the quantum walk in a conventionalAnderson insulating phase. In this case, the spin polar-ization of the initial state | ψ p M (cid:105) quickly scrambles anddecays. In all simulations static uncorrelated angles θ q and ϕ q were randomly drawn from intervals (¯ θ − δ, ¯ θ + δ )and ( ¯ ϕ − δ, ¯ ϕ + δ ) of size 2 δ = π/ θ and ¯ ϕ referringto their mean values, and we performed ensemble averageover 5 · realizations. Overall, we find clear evidenceof the discussed features in different variants of the sug-gested protocol, starting at a number of t ∼ O (10) timesteps. As we have also checked, the results demonstratedin Figs. 5 and 6 remain qualitatively unchanged providedonly one angle, ϕ q , is random but θ q = ¯ θ doesn’t fluc-tuate, which potentially is easier to realize in practice asdiscussed in the next subsection. B. Experimental setup
A schematic drawing of such quantum critical walk isshown in Fig. 7. Pulses with a fixed phase relation areentering neighbouring input modes of the network. Overthe course of the evolution, they undergo polarisation ro-tations with particular angles (indicated by the differentcolours of vertices) and a polarisation dependent rout-ing. Finally, the detectors resolve internal (i.e., polariza-tion) as well as external degree of freedom for extract-ing spin-resolved probability distributions. Ensemble av-erages over a few thousand realisations of disorder arenecessary since a single realisation only shows very little
FIG. 5. Spin-polarization ∆ P ( t ) as a function of time steps t for the initial states | ψ M (cid:105) with M = 10 and different choicesof mean angles (¯ θ, ¯ ϕ ). ∆ P ( t ) remains finite for a large num-ber, t ∼ O (10 ), of time steps (light red) if the walker probescritical states (cid:15) (cid:39) θ, ¯ ϕ ) = (0 ,
0) (cf. aspectral decomposition of the initial state as shown in the in-set, with a n = |(cid:104) (cid:15) n | ψ M (cid:105)| ). On other hand, ∆ P ( t ) is scram-bled (dark red) if non-critical states at (¯ θ, ¯ ϕ ) = ( π/ ,
0) areprobed.FIG. 6. The spin-polarization ∆ P ( t ) shows the predictedtime-staggered behavior with a long-lived finite amplitude forthe initial state | ψ π/ M (cid:105) with M = 10 which is chosen to filtercritical energies (cid:15) (cid:39) ± π related to the chiral-sublattice sym-metry ˆ C sl . The criticality implies (¯ θ, ¯ ϕ ) = (0 ,
0) (light red).Similar to Fig. 5, ∆ P ( t ) is scrambled when non-critical statesat (¯ θ, ¯ ϕ ) = ( π/ ,
0) are probed (dark red). signatures of the critical states because of the impact ofall the non-critical states. Only through this averagingprocedure, the staggering behaviour of the critical statesbecomes visible and can be reliably extracted.In addition to the precise control of all local coin rota-tions and the easy reconfigurability of the experiment toprogramme the high number of realisations, the gener-ation of the delocalized input state is one of the mainexperimental challenges. When considering a spatialimplementation of the quantum walk network , aspatial-multiplexing techniques, as in , can be adoptedto produce the input state to be fed into the networkports. Analogously, time-multiplexing networks canbe adapted in the (temporal) position spacing to be di-rectly compatible with the pulse train produced by a co-herent cavity laser source. For the advanced control ofthe phase between the pulses, we envision the usage of di-rectly modulated light source . Alternatively, an exter-nal time-multiplexing loop, as suggested for driven quan-tum walks in and which controls timings and phases ofthe initial pulse train, can be connected to the setup.Standard optical waveplates take care of the desired cir-cular input polarization resulting in the state | ψ p M (cid:105) —i.e.,a state in the form of Eq. (25).The non-random rotations R x (¯ θ/
2) and the randomrotation R z ( ¯ ϕ q / ϕ q in a controlled fash-ion. Since the evolution in the network typically takesplace in horizontal and vertical polarisation, the mea-surement basis has to be rotated again to circular states,such as by using QWPs at 45 ◦ angle in front of the detec-tors. Crucially, to extract the spin-polarization, ∆ P ( t ),both polarization modes must be measured separatelyfor every step. In Appendix E, we provide estimates fortime and spacial scales which validate a feasibility of ourproposal within existing experimental techniques. V. DISCUSSION
We have studied the one-dimensional quantum walk ofa spin-1/2 particle with chiral symmetry and tunable dis-order. The quantum walk allows to realize topologicallydifferent Anderson insulating phases, and can be tunedto a quantum critical point separating two such phases.Building on a low energy effective field theory, we havederived the walker’s phase-diagram as a function of theaverage values of the coin operators rotation angles andtheir variances. We found that in the Floquet systemfully unitary disorder always realizes the critical point.The critical dynamics reflects the fight between stronglocalization in a 1 d system on the scale of the mean freepath, and the formation of a nontrivial topological in-variant which forces long range correlation through thesystem. We calculated the critical probability distribu-tion of the walker, and verified that the powerful, op-posing strong localization in 1 d manifests in extremelyslow critical dynamics. That is, in Sinai diffusion as pre-viously also found for quasi one-dimensional disorderedtopological superconductors . We identified a (time-staggered) spin polarization as a promissing observablesignature witnessing the quantum critical correlations.More specifically, we noticed that the walker’s criticaldistribution keeps memory of the initial spin configura- FIG. 7. A prototype of a linear optical network to real-ize a quantum critical walk discussed in details in the maintext. A train of M + 1 coherent pulses with a fixed relativephase difference, 2 p , between adjacent pulses enters neigh-bouring ports of the network. Each vertex illustrated thecoupling of two spatial modes and is implemented by a se-ries of waveplates and a polarizing beam splitters, realisingthe operators according to Eq. (4). The different colors ofthe vertices denote the (static) randomness in the phase φ q .Eventually, the distribution for each disorder realization ismeasured polarization-resolved in the circular basis. tion, when prepared in an eigenstate of the chiral oper-ator. Moreover, we noted that the combination of chiralhopping on a discrete lattice and the chiral symmetryleads to a second, ‘chiral lattice symmetry’. The spin-polarization then becomes staggered in time, when criti-cal states protected by this second symmetry are probed.The underlying mechanism suggests that, quite generally,in systems with chiral hopping on a discrete lattice onemay expect time-staggering of observables which are sen-stitive to the eigenvalues of the chiral symmetry operator.Taking advantage of the versatile opportunities offeredby optics, we have proposed a protocol that should allowfor the observation of the spin polarization within ex-isting experimental set up. One experimental challengeis to minimize contributions from uncritical states thatsuffer from conventional Anderson localization and whichmay mask the spin polarization. We proposed to filtercritical states in the quasi-energy spectrum by preparingthe walker initially in a spatially extended state. We con-firmed the viability of our proposal by numerically simu-lations of the protocol for experimentally realistic systemparameters, and verfied a strong suppression of the spinpolarization by either activating uncritical states, break-ing chiral symmetry or introducing dephasing. For anexperimental platform we e.g. indicate an optical linearnetwork similar to that used in Ref. . The preparationof an extended initial state with a stable fixed phase rela-tion may still be challenging. We are, however, optimisticthat some variant of the protocol is in experimental reachin one of the discussed platforms. The experimental ob-servation of the time staggered spin polarization wouldprovide intriguing evidence for the quantum critical dy-namics manifesting as a competition of strong localiza-tion and nontrivial topology in disordered quantum sys-tems. ACKNOWLEDGMENTS
We wish to thank Benjamin Brecht for discussions.TM acknowledges financial support by Brazilian agenciesCNPq and FAPERJ. AA, DB, and KWK were funded bythe Deutsche Forschungsgemeinschaft (DFG) Project No.277101999 TRR 183 (project A01/A03). The IntegratedQuantum Optics group acknowledges financial supportthrough the Gottfried Wilhelm Leibniz-Preis (Grant No.SI1115/3-1) and the European Commission through theERC project QuPoPCoRN (Grant No. 725366).
Appendix A: Effective field theory
In this section, we discuss how the evaluation ofthe probability distributions P σ (cid:48) σ ( t, q ) at large distancescales q (cid:29) G ψ = N − (cid:88) q =0 | q, σ (cid:105) e iψ q (cid:104) q, σ | , (A1)with ψ q = qφ + ψ . Here φ and ψ are some constants;the phase ψ q is a linear ramp; and, to comply with the pe-riodic boundary conditions, we require that φ = 2 πn/N with n ∈ Z . Then, by definition, ˆ G ψ | q, σ (cid:105) = | q, σ (cid:105) e iψ q holds true, and this enables one to represent Eq. (4) inthe equivalent form P σ (cid:48) σ ( t, q ) = (cid:104)|(cid:104) q, σ (cid:48) | G † ψ ˆ U t G ψ | , σ (cid:105)| (cid:105) θ,ϕ . (A2)This is only nominally ψ -dependent, and the usefulnessof such artificial representation is going to be evidentmomentarily. As one can see, the operator ˆ G ψ commuteswith the (local) coin operator ˆ R . On the other hand,ˆ G † ψ ˆ T ˆ G ψ = e iφ σ ˆ T applies. With this observation, wecan write the probability distribution as P σ (cid:48) σ ( t, q ) = (cid:104)|(cid:104) q, σ (cid:48) | [ ˆ U φ ] t | , σ (cid:105)| (cid:105) θ,ϕ , (A3)where ˆ U φ = e i ˆ ϕ σ e i ˆ θ σ e iφ σ ˆ T e ˆ θ σ e i ˆ ϕ σ , (A4)and average the latter over the auxiliary angle φ . For that, let us consider the (disorder specific) SUSYaction S [ ¯ ψ, ψ ] = (cid:90) dq (cid:48) ¯ ψ q (cid:48) (1 − e i ω − ˆ U φ ) ψ q (cid:48) . (A5)Here the fields ¯ ψ q = { ¯ ψ α,σq } are four-component su-pervectors, consisting of (anti-)commuting components α = (f)b and carrying spin index σ = ± . The latter de-note the eigenvalue of a spin operator ˆ s i , which we herekeep rather general, i.e. i = 1 , ,
3, although our focuswas on the chiral operator i = 2 in the main text. Thenotation (cid:82) dq above is symbolic in the sense that opera-tor ˆ U φ in fact maps the spinor ψ q onto ψ q ± . On takinginto account the chiral symmetry, ˆ G A − (cid:15) = σ G R(cid:15) σ , theprobability in Eq. (14) can be then obtained via a Gaus-sian functional average P chiral σ (cid:48) σ ( ω, q ) (A6)= | ω | (cid:10)(cid:90) D ( ¯ ψ, ψ ) ψ b σ (cid:48) q ¯ ψ b σ [ σ ψ f0 ] σ [ ¯ ψ f q σ ] σ (cid:48) e − S [ ¯ ψ,ψ ] (cid:11) θ,ϕ (here we used (1 − e i ω − ˆ U φ ) ≡ [ G R ω ] − ). To facilitatethe subsequent derivation, it is advantageous to augmentthe action by a source term, S J [ ¯ ψ, ψ ] = S [ ¯ ψ, ψ ] − (cid:90) dq (cid:48) ¯ ψ q (cid:48) ( j q (cid:48) σ ) ψ q (cid:48) . (A7)Here the current j q (cid:48) = αδ q (cid:48) π bf ⊗ π σi + βδ q (cid:48) q π fb ⊗ π σ (cid:48) i involves projection matrices in spin- and graded-space, π σi = (1 + σσ i ), π bf = ( ), π fb = ( ), and affords acalculation of the probability distribution according to P chiral σ (cid:48) σ ( ω, q ) = | ω |(cid:104) ∂ αβ Z J (cid:105) φ,ϕ | α = β =0 , (A8)where Z J is a partition sum of the action S J [ ¯ ψ, ψ ]. Thisidentity can be checked by a straightforward computationwhich invokes the Wick’s theorem for the Gaussian action S .The probability in Eq. (A6) is φ –independent by con-struction. Thus, one can average the generating func-tion over all equivalent gauge configurations, (cid:104)Z J (cid:105) φ = (cid:82) π dφ π Z J . This integral can be done via the color-flavortransformation by trading the ‘microscopic’ degrees offreedom ¯ ψ q , ψ q for bi-local matrix fields ¯ Z qq (cid:48) , Z qq (cid:48) , repre-senting the Goldstone, viz. diffusion modes, of the disor-dered single-particle system. To see its working principle,let us decompose the Floquet evolution operator as U φ = V ( θ, ϕ ) T + e iφ σ T − V ( θ, ϕ ) , (A9)where partial ‘coin’ rotations are V ( ϕ, θ ) = e i ϕq σ e i θq σ and V ( ϕ, θ ) = e i θq σ e i ϕq σ , while T σ = T π σ + π − σ are‘shift’ operators describing individual hopping of the spinup and down particle to the left and right, respectively.If one further introduces auxiliary spinors( ψ T , ψ T ) = ¯ ψe i ω V ( θ, ϕ ) T + , ( ψ (cid:48) , ψ (cid:48) ) T = T − V ( θ, ϕ ) ψ, (A10)where the two component structure refers to the spinsubspace, then the free action S can be cast into theequivalent form S [ ψ, ¯ ψ ]= (cid:90) dq (cid:0) ¯ ψ q ψ q − ψ T ,q e iφ ψ (cid:48) ,q − ψ T ,q e − iφ ψ (cid:48) ,q (cid:1) . (A11)At the heart of the color-flavor transformation lies theidentity (cid:90) π dφ π e (cid:80) q ( ψ T ,q e iφ ψ (cid:48) ,q + ψ T ,q e − iφ ψ (cid:48) ,q ) (A12)= (cid:90) dZd ¯ Z sdet(1 − ¯ ZZ ) e (cid:80) qq (cid:48) ( ψ T ,q Z qq (cid:48) ψ (cid:48) ,q (cid:48) + ψ T ,q ¯ Z qq (cid:48) ψ (cid:48) ,q (cid:48) ) , and ‘sdet’ referes to the graded determinant. Here ¯ Z = { ¯ Z αα (cid:48) } and Z = { Z αα (cid:48) } are the (graded) matrix-fieldsmentioned above, with components α, α (cid:48) ∈ { b , f } , ¯ Z bb = − [ Z bb ] † and ¯ Z ff = [ Z ff ] † to guarantee convergence ofEq. (A12), and the additional matrix structure of Z is inspin-space. The anti-commuting blocks, Z αα (cid:48) and ¯ Z αα (cid:48) with α (cid:54) = α (cid:48) are independent varibales. On applying thisidentity to the partition sum (cid:104)Z J (cid:105) φ and then integratingover fields ( ¯ ψ q , ψ q ), we can reduce the former to the pathintegral over collective matrix fields ( ¯ Z, Z ) with an action S [ ¯ Z, Z ] = − str ln(1 − ¯ ZZ ) (A13)+ str ln (cid:0) − e i ω V T + (cid:0) Z ¯ Z (cid:1) T − V − jσ (cid:1) . Recalling the chiral symmetry, σ T − V σ = ( T + ) † V † ,one identifies two Goldstone modes of this action, Z = − ¯ Z , whenever ω → π are satisfied (this correspondsto particle/hole energies ± (cid:15) being close to 0 or ± π , re-spectively). Indeed, if Z = − ¯ Z are constant in space and j = 0, then action (A13) vanishes identically. Physically,the field Z αα (cid:48) qq (cid:48) ∼ ψ αq ¯ ψ α (cid:48) q (cid:48) describes a pairwise propagationof a retarded and an advanced single-particle amplitudeat a slight (mod 2 π ) difference in frequency ω . At longspatial scales, off-diagonal components (with q (cid:54) = q (cid:48) ) re-lax quickly due to accumulation of random phases, andGoldstone modes assume the form Z qq (cid:48) = Z q δ qq (cid:48) . Assum-ing Z q to vary slowly in space, we further expand (A13)in small spatial gradients and frequency. a. Topological and 2nd order gradient terms Let us first discuss terms with spatial derivatives andset j = ω = 0 — this corresponds to the 1st Goldstonemode — and we comment on the 2nd one (with ω → π )in the end of this subsection. By defining Z = (1 − iZσ ),one can rewrite the action (A13) as S [ Z ] = − str ln( Z ) + str ln( Z + δ Z ) (A14)with δ Z = − V T + [ Z, T † + ] V † iσ . (A15) To second order, S [ Z ] (cid:39) Str (cid:0) Z − δ Z (cid:1) − Str (cid:0) Z − δ ZZ − δ Z (cid:1) + . . . = S (1) [ Z ] + S (2) [ Z ] + . . . , (A16)while (cid:104) q | [ T + [ Z, T † + ] | q (cid:105) = ( Z (cid:48) q + Z (cid:48)(cid:48) q ) P + . . . . The topologi-cal and so-called Gade terms originate from the 1st-orderterms in these series. Using that Z − = (1 + iσ Z ) / (1 + Z ) and evaluating traces in the spin subspace, one ar-rives at (cid:104) S (1)1 (cid:105) θ,φ = ¯ χ (cid:90) dq str (cid:0) g − ∂ q g (cid:1) − (cid:90) dq ∂ q str ln(1 + g ) ≡ S top + S r . (A17)Here ¯ χ = 12 (1 − (cid:104) sin θ q cos ϕ q (cid:105) θ,φ ) , (A18)and we introduced g = (1 + iZ ) / (1 − iZ ). Geometri-cally, the unconstrained pair ( ¯ Z, Z ) defines a set of stere-ographic coordinates parametrizing a two-dimensionalsphere in the ‘fermionic’ ff-sector, respectively, hyper-boloid in the ‘bosonic’ bb-sector. This is readily verifiedrecalling that ¯ Z ff / bb = ± [ Z ff / bb ] ∗ and stereographic co-ordinates( x , x , x ) = 11 ± ¯ zz ( ± z, ∓ z ) , ∓ ¯ zz ) (A19)for the two-sphere/hyperboloid, respectively. TheGoldstone-mode restriction ¯ Z = − Z defines one-dimensional submanifolds which result from their inter-section with two-dimensional planes, viz. a circle, respec-tively, hyperbola. The latter identifies g ∈ Gl(1 |
1) as asupersymmetric group manifold.For a system with periodic boundary conditions we canomit the 2nd (residual) term S r and keep only the 1st(topological) one. In fact, both terms are full derivativessince str( g − ∂ q g ) = ∂ q ln det( g ). However, S top is non-trivial. Consider a configuration g = (cid:0) e x
00 1 (cid:1) bf , where x isa compact fermion angle. Assuming periodic boundaryconditions, mappings x q : S → S may have windings,i.e. x L = x + 2 πW , where W ∈ Z . Then action S top onsuch configuration becomes non-zero, S top = 2 iπnχ . Forthe residual term one finds S r = (cid:90) πn ie ix e ix dx = n (cid:73) | w | =1 dw (1 + w ) . (A20)If one regularizes this integral by slightly shifting the pole w = 1 outside the unit circle | w | = 1, then S r vanishes.The Gade term is obtained if one keeps the 2nd ordercumulant expansion when averaging over disorder, S G [ g ] = − c (cid:90) dq str ( g − ∂ q g ) , (A21)where c = (cid:104)(cid:104) χ ( θ, ϕ ) (cid:105)(cid:105) = (cid:104) χ ( θ, ϕ ) (cid:105) θ,φ − ¯ χ . This termis exactly zero at criticality (where χ = and does not0fluctuate), and is known to give inessential modificationsaway from it .Since in this paper we are interested in critical quan-tum walks only, we can derive the diffusive action S [ g ]by setting V = V = θ =0 , π ). The latter simplifies the variation, δ Z = − ( Z (cid:48) q + Z (cid:48)(cid:48) q ) P + iσ . . . , and action S [ g ] originates from twopieces. The 1st piece is S (2) [ Z ] = −
12 str (cid:0) (1 + Z ) − ZZ (cid:48) (cid:1) , (A22)while the 2nd piece stems from the 2nd order gradientterm ( ∝ Z (cid:48)(cid:48) ) in S (1) [ Z ]. It evaluates to S (1)2 [ Z ] = 12 str (cid:0) (1 + Z ) − ZZ (cid:48)(cid:48) (cid:1) . (A23)By adding these two contributions and integrating byparts one finds the diffusive action of the class AIII σ -model S [ g ] = −
12 str (cid:18)
11 + Z Z (cid:48) (cid:19) = − (cid:90) dq str( ∂ q g − ∂ q g ) . (A24)Let us now comment on the 2nd Goldstone mode with ω → π . If we change Z → − Z in the prototype ac-tion (A13), it is reduced to the one with the 1st Gold-stone ( ω →
0) and at the same time g → g − . The latterdoes not change S [ g ], but transforms the topological an-gle, ¯ χ → ¯ χ π = 1 − ¯ χ , in the action (A17). At criticalityboth Goldstone modes are described by the same actionwith ¯ χ = 1 / and the colour-flavour transforma-tion , we arrive at the effective action, S (cid:15) = S + S (cid:15) top ,where S = 12 (cid:90) dq (cid:2) − g str (cid:0) ∂ q g − ∂ q g (cid:1) + iω str( g + g − ) (cid:3) ,S (cid:15) top = ¯ χ (cid:15) (cid:90) dq str( g − ∂ q g ) . (A25)Here g denotes a group-valued matrix field that describesthe critical fluctuations in the system, g = 1 / χ (cid:15) = 12 (cid:0) − e i(cid:15) (cid:104) sin( θ ) cos( ϕ ) (cid:105) θ,ϕ (cid:1) (A26)is the bare topological angle with (cid:15) = 0 , π indicating thecritical states described by the effective action. Its is alsoworth mentioning here that the action S (cid:15) is identical tothe one describing disordered quantum wires of a sym-metry class AIII .
1. Sources
Finally let us turn to source contributions S J to theaction. Relevant contributions result from an expansion of the action (A13) to linear order in j , S J = − str (cid:0) jσ Z − (cid:1) = − α (cid:20) iZ + σδ i Z Z (cid:21) fb − β (cid:20) iZ q + σ (cid:48) δ i Z q Z q (cid:21) bf , (A27)or S J [ g ] = − (cid:40) α (cid:2) g − g − (cid:3) fb + β (cid:2) g q − g − q (cid:3) bf , i = 1 , σα [ g σ ] fb + σ (cid:48) β [ g σ (cid:48) q ] bf , i = 2 , (A28)resulting in P chiral σ (cid:48) σ = (cid:88) σ,σ (cid:48) = ± σσ (cid:48) | ω | (cid:104)(cid:104) [ g σ ] fb [ g σ (cid:48) q ] bf (cid:105)(cid:105) , i = 1 , , (A29) P chiral σ (cid:48) σ = σσ (cid:48) | ω |(cid:104)(cid:104) [ g σ ] fb [ g σ (cid:48) q ] bf (cid:105)(cid:105) , i = 2 , (A30)We evaluate these propagators in the next section. Appendix B: Transfer matrix method
When evaluating the probability distributions (A29)and (A30), the non-perturbative nature of Anderson lo-calization requires the functional integration over the en-tire group-manifold, which usually is a highly compli-cated task. We are here, however, in a better situationsince powerful alternative non-perturbative methods areavailable for the one-dimensional σ -model . The lat-ter is based on the interpretation of the action S [ g ] asthe action of a quantum mechanical particle with coor-dinate g moving in the potential V ( g ) = η str( g + g − )where η = − iω . Changing then from the path-integral-to the Schr¨odinger-description, one expresses the prob-ability distribution in a spectral decomposition with re-spect to the corresponding Hamilton-operatorˆ H = ∆ g + V ( g ) (B1)where ∆ g = − J − ∂ i G ij J∂ j is the Beltrami-Laplace op-erator on the AIII-manifold, with metric tensor G ij andJacobian J = √ sdet G .In what follows we sketch the details of such programat criticality when ¯ χ = 1 / g in terms of 4 coordinates z = ( x, y, ¯ ξ, ξ ) suchthat g = U (cid:18) e x e iy (cid:19) bf U − , U = exp (cid:18) ξ ¯ ξ (cid:19) bf (B2)with x, y ∈ R being commutative while ¯ ξ, ξ being Grass-mann anti-commutative fields, which results in the fol-lowing metric dl = − str( dgdg − ) = G ij dz i dz j (B3)= dx + dy + 8 sinh ( x − iy ) d ¯ ξdξ (B4)1on the GL(1 |
1) manifold. The Eq. (B4) above definesnon-zero elements of the tensor G ij . With the Jaco-bian J ( z ) = sinh − ( x − iy ) and the vector potential A = ¯ χ ( i, , ,
0) this metric defines the transfer matrixHamiltonian H = − J − ( z )( ∂ µ − iA µ ) G µν J ( z )( ∂ ν − iA ν ) + V ( z ) , (B5)where V ( x, y ) = η (cosh x − cos y ) is the potential energydue to frequency term in the action and η = − iω . Thenthe Sutherland transformation, H = e ¯ χ ( x − iy ) J / H J − / e − ¯ χ ( x − iy ) , (B6)complemented by the ’gauge’ transform eliminating thevector potential brings the Hamiltonian to a simpler formˆ H = − ∂ x − ∂ y −
12 sinh − (cid:0) x − iy (cid:1) ∂ ¯ ξ ∂ ξ + V ( x, y ) . (B7)The ground state | (cid:105) ≡ Φ ( x, y ) of ˆ H — it obeys ˆ H | (cid:105) =0 due to supersymmetry — depends only on bosonic an-gles ( x, y ) and can be approximated byΦ ( x, y ) = − coth (cid:18) x − iy (cid:19) K (cid:16)(cid:112) ηe | x | / (cid:17) / ln η. (B8)If η (cid:28) | (cid:105) in the limit x ∼ | x | (cid:29)
1, resp. . Theexcited states | k (cid:105) ≡ Φ k ( z ) of ˆ H with energies E k > k = ( n, l, ¯ λ, λ ),where n and l are integers and ¯ λ, λ are Grassmanns.Specifically, Φ k ( z ) = R k ( x, y ) × e ¯ ξλ + ξ ¯ λ (B9)can be split into radial and angular parts where R k ( x, y )satisfies to the radial Schr¨odinger equation (cid:18) − ∂ x − ∂ y + V ( x, y ) −
12 sinh − (cid:0) x − iy (cid:1) ¯ λλ (cid:19) R k Φ k = E k R k . (B10)Since ¯ λλ is the nilpotent of the Grassmann algebra, thespectrum and eigenstates of the above radial equationshould have the following form: E k = (cid:15) n,l + ¯ λλ (cid:15) (cid:48) n,l and R k ( x, y ) = R n,l ( x, y ) + ¯ λλR (cid:48) n,l ( x, y ) , (B11)where n = 1 , , . . . , and l ∈ Z are radial quantum num-bers. It turns out (see Sec.B 0 b below) that only the 0thorder terms in bilinear ¯ λλ are required to evaluate thepropagator P chiral σ (cid:48) σ ( η, q ) of the quantum Sinai diffusion.We proceed by constructing an asymptotic form of theradial wave function R n,l ( x, y ) at η (cid:28) t (cid:29)
1) in thenext section and then find P σ (cid:48) σ in Sec.B 0 b. a. Radial wave function We now concentrate on the spectrum (cid:15) n,l and eigen-states R n,l ( x, y ) of the 0th order Hamiltonianˆ H = − ∂ x − ∂ y + η (cosh x − cos y ) . (B12)It will be seen in Sec.B 0 b that in the limit η (cid:28) x ’s satisfy ηe | x | ∼ y term in ˆ H can be neglected. We thusapproximate R n,l ( x, y ) ≈ R n ( x ) e ily , which leads to (cid:15) n,l = (cid:15) n + l together with a simple radial equation[ − ∂ x + η cosh x ] R n ( x ) = (cid:15) n R n ( x ) . (B13)To solve it we introduce momenta k n = √ (cid:15) n and dividethe x -axis in three intervals: (I) ’small’ angles with | x | <
1; (II) ’intermediate’ ones, such that 1 < | x | < ln(1 /η )and (III) ’large’ angles, where | x | > ln(1 /η ). In the fol-lowing, it will be sufficient to consider the domain x > x is symmetric. In the intervalsII & III one can approximate (B13) by (cid:2) − ∂ x + ηe x (cid:3) R n ( x ) = k n R n ( x ) . (B14)Up to a normalization factor which is found below, thesolution of this equation is a modified Bessel function R n ( x ) ∝ K ik n ( √ ηe x/ ). Taking a limit of K ν ( z ) atsmall argument, the wave function R n ( x ) in the intervalII is reduced to the plane wave R n ( x ) ∝ A ( k n ) e ik n x + A ∗ ( k n ) e − ik n x ,A ( k ) = Γ( − ik ) ( η/ ik . (B15)As to interval I, one can neglect η -dependent poten-tial whatsoever, and therefore by continuity the planewave (B15) is also a solution in the interval I. We canthus introduce a scattering matrix and a phase shiftfrom the right potential barrier, S ( k ) = A ( − k ) /A ( k ) = e − iφ ( k ) , which finally gives us a quantization condition φ ( k n ) = πn . Here n = 0 , , , . . . with even/odd n cor-responding to even/odd wave functions R n ( x ), resp., i.e. R n ( − x ) = ( − n R n ( x ). For small momenta, k n (cid:28)
1, weget with log-accuracy η ik (cid:39) e iπ ( n +1) which leads to thespectrum (cid:15) n,l = π η ( n + 1) + l , n = 0 , , , . . . , l ∈ Z . (B16)We now proceed to find a normalization factor for theradial wave function. For that let’s note that the maincontribution to its norm (cid:82) + ∞−∞ R n ( x ) dx = 1 comes fromthe intervals I and II. The wave function in these regionsis a plane wave, R n ( x ) ∝ | A ( k n ) | cos (cid:0) k n x + πn (cid:1) . (B17)It can be matched to the one found within the semiclas-sical approximation, R n ( x ) = ( C n / (cid:112) k n ) cos (cid:0) k n x + πn (cid:1) , (B18)2where the normalization constant is fixed by C n =(2 k n /π )( ∂k n /∂n ). The comparison of these two repre-sentations leads to the following normalized radial wavefunction R n,l ( x, y ) = e ily (cid:18) π ∂k n ∂n (cid:19) / | A ( k n ) | − K ik n ( (cid:112) ηe x/ ) , | A ( k ) | − = (cid:18) k sinh 2 πkπ (cid:19) / , x > . (B19)We use this intermediate result in the next subsection toevaluate the series expansion of the propagator P chiral σ (cid:48) σ . b. Propagator of Sinai diffusion Employing a spectral decomposition, the propaga-tor (7) can be written as a sum over excited eigenstates | k (cid:105) , P chiral σ (cid:48) σ ( η, q ) = η (cid:88) n,l (cid:90) dλd ¯ λ Γ σ (cid:48) k ¯Γ σk e − E k | q | , (B20)whereΓ σ (cid:48) k = (cid:104) | [ g σ (cid:48) ] bf | k (cid:105) (B21)= σ (cid:48) π + ∞ (cid:90) −∞ dx π (cid:90) dy (cid:90) d ¯ ξdξ Φ ( x, y )[ g σ (cid:48) ] bf Φ k ( z ) . is a matrix element of the field [ g σ (cid:48) ] bf = ( e iσ (cid:48) y − e σ (cid:48) x ) ξ between the ground and excited states and a similar ex-pression is valid for a conjugated matrix element ¯Γ k ofthe field [ g σ ] fb = ([ g σ ] bf ) ∗ . Using the explicit form of theexcited state, Φ k ( z ) = R k ( x, y ) × e ¯ ξλ + ξ ¯ λ , (B22)one can first perform the integral over Grassmanns( ¯ ξ, ξ ) in Eq. (B21) and verify that the nilpotent part ∼ R (cid:48) n,l ( x, y ) of the radial wave function does not con-tribute to the matrix elements. The latter are then sim-plified to Γ σ (cid:48) k = − λ Γ σ (cid:48) n,l and ¯Γ σk = ¯ λ Γ σn,l withΓ σn,l = σ π + ∞ (cid:90) −∞ dx π (cid:90) dy Φ ( x, y )( e σx − e iσy ) R n,l ( x, y ) . (B23)Separating here y -dependent parts of the wave functions,the integration over the compact angle y yields σ π (cid:90) π − π dy coth (cid:18) x − iy (cid:19) ( e σx − e iσy ) e ily = e σx δ l + δ l + σ . (B24)Here the l = ± (cid:15) n, ± = (cid:15) n + 1) thus we keep l = 0 contribution only, thelatter readsΓ σn, = − (cid:90) e σx K ( (cid:112) ηe | x | / ) R n, ( x ) dx/ ln η. (B25) It is worth mentioning that R n, ( x ) is either even or odddepending on a parity of n , thus Γ + n, = ( − n Γ − n, . Onchanging the integration variable to z = √ ηe x/ , theremaining integral for Γ σn, is reduced to a table one, (cid:90) + ∞ zK ( z ) K ik ( z ) dz = ( k π /
2) sinh − ( πk ) . (B26)Finally, taking into account proper normalization factorsgiven in Eq. (B19), one finds the following matrix ele-ments M σ (cid:48) σn = Γ σ (cid:48) n, Γ σn, = ( σ (cid:48) σ ) n π k n η ln η (cid:18) ∂k n ∂n (cid:19) × k n cosh( πk n )sinh ( πk n ) k n (cid:28) −→ ( σ (cid:48) σ ) n ( n + 1) η ln (1 /η ) . (B27)From here the propagator of Sinai diffusion is constructedas P chiral σ (cid:48) σ ( η, q ) = η (cid:88) n,l =0 (cid:90) dλd ¯ λ Γ σ (cid:48) k ¯Γ σk e − | q | E k = η + ∞ (cid:88) n =0 M σ (cid:48) σn e − | q | (cid:15) n, . (B28)When evaluating the above integral over Grassmann vari-ables one may notice that the nilpotent correction to thespectrum, ¯ λλ(cid:15) (cid:48) n,l , does not contribute to the net result.At large distances, q (cid:29)
1, essential momenta are small, k n (cid:28)
1, and the Laplace transform of (B28) from η tothe time domain yields the result Eq. (18) in the maintext.As a final remark let us evaluate the integrated prob-ability P chiral ( η ) = (cid:88) σ (cid:48) σ (cid:90) dq P chiral σ (cid:48) σ ( η, q ) = 4 η + ∞ (cid:88) k =0 M ++2 k (cid:15) k, . (B29)This series is convergent owing to the exponential decayof M σ (cid:48) σn at large momenta k n >
1. In the limit η (cid:28) P chiral ( η ) = π η ln η (cid:90) + ∞ dk k cosh πk sinh πk = 14 η ln η . (B30)Hence the overall contribution of the critical states tothe walker’s probability decreases in time as P chiral ( t ) =1 / (4 ln t ). Appendix C: Density of states
In the main text we focused on the walker’s criticaldynamics at the topological Anderson localization tran-sition. As discussed there, the critical dynamics describes3
FIG. 8. Density of states (DoS) of the quantum walk model with averaged angles (¯ θ, ¯ ϕ ) = (0 ,
0) and disorder strengths γ θ = γ φ = π/ (cid:15) ∈ [ − π, π ] (upper panel) and (cid:15) ∈ [ − . , .
16] (lower pannel) using different energyresolution. In the calculations we used averages over 10 disorder configurations. Peaks of the DoS at chiral symmetric energy (cid:15) = 0 , ± π/ , π are clearly visible. quasi-energy states centered around the chiral symmetricenergies (cid:15) = 0 , ± π/ , π . To substantiate this statementwe provide here the numerical results for the density ofstate (DoS) of the quantum walker with periodic bound-ary conditions. Fig. 8 (left) shows the disorder averagedDoS in the entire quasi-energy domain for a system of N x = 400 sites. As expected, sharp peaks are visibleat the chiral symmetric energies (cid:15) = 0 , ± π/ , π . Fig. 8(right) shows a magnified view of the region colored inred of Fig. 8. With a smaller energy scale for the his-togram, we can see further structures of the DoS, whichis known to diverge as ∼ / ( (cid:15) ln (cid:15) ) .For our discussion it is important to notice that thenumber of eigenstates within these energy domains is notthe dominant contribution to the total density of states.This indicates that a walker initially localized on a singlesite is not the optimal choice for a protocol aiming to testthe walker’s critical dynamics, as it involves quasi energystates from the entire energy band approximately withequal weight. That is why in the main text we proposeto use the plane wave with momentum p = 0 , π/ Appendix D: Time-staggered spin polarization
In this Appendix we discuss the time-staggered spinpolarization, observable in a quantum critical walk at atopological Anderson localization transition. As statedin the main text, the time-staggered spin polarization in-volves critical states (cid:15) (cid:39) ± π/
2, related to the chiral sub- lattice symmetry ˆ C sl ≡ σ ⊗ ˆ S , where ˆ S ≡ (cid:80) q | q (cid:105) ( − q (cid:104) q | the sublattice operator. Before discussing the relation be-tween ˆ C sl and a time-staggered signal, it is instructive toreformulate our discussion in the main text on the chiralsymmetry ˆ C ≡ ˆ s , and related spin polarization ∆ P , ina more formal way which readily allows for an extensionto the chiral sublattice symmetry of interest. Chiral symmetry:—
In the main text we introduced theprobability distribution, P σ (cid:48) σ ( t, q ) = (cid:104)|(cid:104) q, σ (cid:48) | ˆ U t | , σ (cid:105)| (cid:105) θ,ϕ , (D1)for a walker initially prepared in eigenstate | σ (cid:105) = | ←(cid:105) , | →(cid:105) of the chiral operator ˆ C to be found after t time-steps at a distance q in eigenstate | σ (cid:48) (cid:105) . More formally,we can separate the walker’s Hilbert space into the directsum of subspaces characterized by the quantum numbers s = ± of the chiral operator ˆ C , H = H ⊕ H − , andspanned by H = span {| q, ←(cid:105)} , (D2) H − = span {| q, →(cid:105)} . (D3)The statement on the positive ‘spin polarization’ dis-cussed in the main text, can then be restated as follows:for critical states related to the chiral symmetry ˆ C theprobability distributions P s (cid:48) s : H s ˆ U t −→ H s (cid:48) for initial andfinal states belonging to the same and different subspaces, s (cid:48) = s respectively s (cid:48) = − s , differ and their difference isstrictly positive P ss ( t, q ) − P − ss ( t, q ) > . (D4)Formulated in terms of quantum numbers of the chi-ral operator, the statement on the positivity (D4) holds4for critical states related to the chiral symmetry, in-dependently of its specific form. For the specific chi-ral symmetry ˆ C = ˆ s quantum numbers are simplyspin-orientations, and the positive difference is indeedequivalent to the positive ‘spin polarization distribution’, P chiral →→ ( t, q ) − P chiral ←→ ( t, q ) ≡ ∆ P ( t, q ) >
0, discussed inthe main text. Statement (D4) can now be applied tothe chiral sublattice symmetry ˆ C sl , where it shows moreinteresting consequences. Chiral sublattice symmetry:—
Separating the walker’sHilbert space into the direct sum of subspaces, H = H sl+ ⊕H sl − , characterized by the quantum numbers s = ± of thechiral sublattice symmetry ˆ C sl ≡ σ ⊗ ˆ S , we notice thatquantum numbers differ from the spin orientations, andsubspaces are now spanned by H sl+ = span {| q, ←(cid:105) , | q − , →(cid:105)} , (D5) H sl − = span {| q, →(cid:105) , | q − , ←(cid:105)} . (D6)Positivity (D4) holds for critical states related to the chi-ral operator independently of its specific form, and wenext have to relate this statement to the spin polariza-tion. The relation is more involved for ˆ C sl than for ˆ C ,since the spin structure of eigenstates of the former al-ternates between even and odd sites. More specifically,this implies that the spin structure of P s (cid:48) s depends onthe (parity of the) propagated distance q , i.e. P ss ( t, q ) = (cid:40) P σσ ( t, q ) , q even ,P − σσ ( t, q ) , q odd ,P − ss ( t, q ) = (cid:40) P − σσ ( t, q ) , q even ,P σσ ( t, q ) , q odd , (D7)where s, s (cid:48) are the eigenvalues of ˆ C sl and σ, σ (cid:48) those of σ .To structure then above probabilities (D7) according tothe parity of propagated time steps t , we notice that thesingle time-step evolution ˆ U propagates states by exactlyone lattice site. Starting e.g. from the even site q = 0and propagating for an even number of time steps t one,therefore, ends again on an even site. For an odd numberof time steps t , on the other hand, one ends on an oddsite. That is,span {| q, σ (cid:105)(cid:105)} ˆ U t −→ span {| q, σ (cid:105)(cid:105)} , (D8)span {| q, σ (cid:105)(cid:105)} ˆ U t +1 −→ span {| q + 1 , σ (cid:105)(cid:105)} . (D9)and we can relate probabilities Eqs. (D7) to the parityof propagated steps t as follows. For even numbers oftime steps probabilities P s (cid:48) s : H sl s ˆ U t −→ H sl s (cid:48) , conserving(changing) the quantum number of the chiral sublatticeoperator coincides with probabilities preserving (chang-ing) spin orientation, P s (cid:48) s = P σ (cid:48) σ . The difference (D4)is again the spin polarization, P ss ( t, q ) − P − ss ( t, q ) = P chiral →→ ( t, q ) − P chiral ←→ ( t, q ). For odd numbers of time steps,on the other hand, probabilities P s (cid:48) s : H sl s ˆ U t +1 −→ H sl s (cid:48) QWP,HWP,
X-rotation, R 𝑥 ( q/2 ) a=q/4 QWP QWP R 𝑥 ( q/2 ) | y 𝑀𝑝 > PBS PBSq-1q+1 R 𝑥 ( q/2 ) D t … M+1 pulses 𝑒 Input state preparation:
EOM R 𝑧 ( /2 ) R 𝑧 ( /2 )EOMDetectionPBS SPD P −, y (q) P +, y (q) D t QWPSPD BS
95% 5% PC PCPBS a = - 𝜋 a = 𝜋4 BS FIG. 9. A prototype of a linear optical network to realizea quantum walk discussed in details in the main text (alongthe lines of Refs. ). A phase modulated laser source (notshown) generates a train of pulses with a fixed time inter-val, ∆ t , and relative phase difference, 2 p , between adjacentpulses. HWP: half-wave plate rotated by angle α . QWP:quarter wave plate; PBS: polarizing beam splitter; BS: beamsampler; EOM: fast switching electro-optic modulator; SPD:single-photon detector; PC: polarization controller. Fibers ofdifferent lengths ensure a time delay 2∆ t between | H (cid:105) and | V (cid:105) states thereby realizing ’shift’ operator T . conserving (changing) the quantum number of the chi-ral sublattice operator correspond to probabilities chang-ing (preserving) spin orientation. In this case P ss ( t, q ) − P − ss ( t, q ) = P chiral ←→ ( t, q ) − P chiral →→ ( t, q ) is the negative spinpolarization distribution. Summarizing, we find that forcritical states (cid:15) (cid:39) ± π/ C sl positivity (D4) translates into a time-staggered spinpolarization distribution, P chiral →→ ( t, q ) − P chiral ←→ ( t, q ) = ( − t | ∆ P ( t, q ) | , (D10)as stated in the main text. Appendix E: Details of experimental proposal
Here we discuss few technical details related to thetime-multiplexing experimental proposal mentioned inthe main text, see Fig. 9. One envisions a train of equidis-tant pulses with controlled phase relation to be producedby a coherent laser source. The half-wave (HWP) andquarter-wave (QWP) plates are used for the initializa-tion of input state in the form (25), implementation ofthe rotation R x ( θ ) as well as in the detection. Witha fast axis aligned horizontally, the plates in the basisof linearly polarized states, {| H (cid:105) , | V (cid:105)} , are characterizedby the diagonal Jones matrices M / = diag(1 , −
1) and M / = diag(1 , i ). Then, for instance, the Jones matrixof the HWP rotated at α degrees becomes M / ( α ) = (cid:18) cos 2 α sin 2 α sin 2 α − cos 2 α (cid:19) , (E1)and, on other hand, M / ( − π/ ∼ R x ( π/
4) where thelast equality holds up to inessential phase factor.5Consider now left/right circular polarized states, | L/R (cid:105) = √ ( | H (cid:105) ± i | V (cid:105) ), which are eigenstates of theoperator ˆ σ and thus can be identified with spin states | →(cid:105) and | ←(cid:105) discussed in the main text. Assum-ing that a light from a laser source is linearly polar-ized along | H (cid:105) , one checks that M / ( − π/ | H (cid:105) = | L (cid:105) ,which generates incoming state | ψ p M (cid:105) , cf. Eq. (25) inSec. IV A. The same is true for the detection. Owing topolarizing beam splitters (PBS), two single-photon detec-tors (SPDs) detect linearly polarized states. Because ofidentity (cid:104) R ( L ) | = (cid:104) H ( V ) | M / ( π/
4) the later are trans-formed into circular polarized ones and thereby the mea-surement of spin-dependent probabilities P σψ ( t, q ) de-fined by Eq. (27) in Sec. IV A can be achieved. Finally,the identity R x ( θ/
2) = M / · M / ( θ/ · M / (E2)is a key to implement a (half)-rotation along x -axis usingthree plates as shown in Fig. 9.Few remarks are now in order with regard to possi-ble time and spatial scales of the experiment. FollowingRefs. we assume that a laser emits a train of pulseswith interval ∆ t ∼
106 ns at the telecom wavelength λ ∼ (cid:104) n (cid:105) in ∼
1, to eliminate many photon contributions in the click detectors. The inter-val ∆ t requires a fibre length mismatch ∆ L ∼
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