Breakdown of an Electric-Field Driven System a Mapping to a Quantum Walk

a r X i v :q u a n t -p h /0407013v 3 19 M a r 2005Breakdown of an Electric-Field Driven System:a Mapping to a Quantum WalkTakashi Oka,Norio Konno †,Ryotaro Arita ∗and Hideo AokiDepartment of Physics,University of Tokyo,Tokyo 113-0033,Japan†Department of Applied Mathematics,Faculty of Engineering,Yokohama National University,Yokohama 240-8501,Japan(Dated:February 1,2008)Quantum transport properties of electron systems driven by strong electric fields are studied by map-ping the Landau-Zener transition dynamics to a quantum walk on a semi-infinite one-dimensional lattice with a reflecting boundary,where the sites correspond to energy levels and the boundary the ground state.Quantum interference induces a distribution localized around the ground state,and a delocalization transition occurs when the electric field is increased,which describes the dielectric breakdown in the original electron system.PACS numbers:05.40.Fb,05.60.Gg,72.10.BgDynamics of quantum statistical systems driven by finite external fields has attracted much attention as a typical class of non-equilibrium phenomenon.One problem is the “dissipation”arising in electron systems driven out of their ground states by strong electric fields as studied by many authors [1,2,3,4,5,6,7].In these references the electric field is expressed (via Fara-day’s electro-magnetic induction)as a time-dependent Aharonov-Bohm(AB)flux,and this induces,for strong fields,interlevel Landau-Zener transitions.There,an issue is whether the bunch of energy levels around the one having the main amplitude can play a role akin to dissipation[6].The purpose of the present study is to propose a mapping of the system onto a quantum algo-rism model for studying the problem.We start by noting that the driven quantum system and the quantum algo-rism model have in fact similarities:The former deals with quantum transitions among macroscopic number of energy levels while the latter describes the dynamics of many qubits governed by quantum logic gates,for which powerful analytical techniques are being developed.In the one-dimensional tight-bindingmodel,the Hamiltonian isH (φ(t ))=−t i,σ e −i2π2 the Landau-Zener dynamics is straightforward:A qubiton site n≥0labeled by L,R corresponds to the states atthe n-th anticrossing point,which moves down(L;with aleft-going current)or up(R;right-going)in energy afterthe tunneling event(Fig.1).One important point we notehere is that the mapped quantum walk has a reflectingboundary,since we cannot walk below the ground state inenergy.In previous quantum-walk studies[12,13,14]anabsorbing boundary was considered,for which the gen-erating functions were obtained[14].Here wefirst ob-tain the generating function for the quantum walk witha reflecting boundary.We have found the existence,andan analytic form for,the amplitude localized around theboundary(the ground state in the original problem).Thesolution exhibits an asymmetry between the L and Rstates,which represents afinite total momentum.Whenthe electricfield exceeds a critical value,a delocalizationtransition(on energy axis)is observed,which we identifyhere to describe the dielectric breakdown in the originalelectron model.Mapping—Let the wave function for the n-th en-ergy level at timeτ(measured in units of∆t/2=h/2F L)beΨ(n,τ)=t(ψL(n,τ),ψR(n,τ)),whereψL(n,τ)(ψR(n,τ))has a left-(right-)momentum.Eachenergy level is subject to a Landau-Zener tunneling(withcertain probability and phase)to neighboring levels in atime period∆t/2.While transitions among more thanthree levels exist in principle,here we restrict ourselvesto transitions among neighboring levels,for which thequantum tunneling can be described by a set of2×2unitary matrices[1,2,3,4].The diagonal elements rep-resent Landau-Zener tunneling,while the off-diagonalones the backward scattering process.Here we sim-plify the problem by assuming that the matrices are thesame U= a b c d except for the one at the bound-ary,˜U≡ ˜a˜b˜c˜d .As we shall see,an interesting struc-ture arises in the overall shape of the distribution in thebounded quantum walk even within this simplification.[8]The wave function evolves deterministically following arecursion formula,Ψ(n,τ+1)=PΨ(n+1,τ)+QΨ(n−1,τ)(1)for excited(n≥2)states,while we putΨ(1,τ+1)=PΨ(2,τ)+˜QΨ(0,τ),Ψ(0,τ+1)=PΨ(1,τ)(2)between the ground and thefirst excited levels.Herewe have decomposed U into P≡(ab00),Q≡(00cd)and˜Q≡(00˜c˜d ).Equations(1),(2)define the mapping to a one-dimensional quantum walk on a semi-infinite space with a reflecting boundary at n=0.We take the initial wavefunction|Ψ0 to be the ground state,i.e.,Ψ(0,0)=t(1,0) andΨ(n,0)=0for n≥1.Generating function for the quantum walk—Now we state our main result for the quantum walk:We can ob-tain the generating function for the wave function in the reflecting boundary condition,where the generating func-tion is defined asΨL,R(0→n;z)= ∞τ=0ψL,R(n,τ)zτ, with the summation taken over all the paths startingfrom0and ending at n.The obtained expression is ΨL(0→n;z)= dλ+(z)acλ+(z)−azan−1˜c z∆2z4+2∆(1−2|a|2)z2+13˜Ξb (0→0;τ)and Ξb (0→n −1;τ−1),we obtain ˜B ˜q (0→n ;z )= 1+˜c ˜B ˜r (0→0;z )dzB q (0→n −1;z ),(5)˜B ˜r (0→n ;z )= 1+˜c ˜B ˜r (0→0;z )dzB r (0→n −1;z ).With a repeated application of the no-tilde ver-sion of Eq.(5)we obtain B q (0→n ;z )=[dλ+(z )/a ]n /d,B r (0→n ;z )=[dλ+(z )/a ]n [(λ+(z )−az )/(acz )].Again using (5)with a relation between ˜B ˜r (0→0;z )and A r (0→0;z ),we arrive at eqs.(3)after a bit of algebra.Edge states in energy space —One asset of the gen-erating function is that we can show that there exist “edge states”in the bounded quantum walk.To actu-ally show this let us take the Landau-Zener tunneling probability,p =exp(−πF ∝∆2depends on the size of the gap∆[16,17].We plug this in the transfer matrix,U = √1−pe iγ−√pe −iβ ,˜U = 0e i ˜γ−e −i ˜γ0 ,(6)where the diagonal elements of ˜Uare set to zero to rep-resent a complete reflection.The phases β,γ,˜γconsist of the dynamical phase and the geometric phase (nonadia-batic counterpart to Berry’s phase).In physical systems (such as the Hubbard model)these phases at the level anti-crossings are generally nonzero (and depend on the driving force F ).Expanding the generating function (3)into a power series in z yields the time evolution of the wave func-tion (Fig.2).We can see that Ψ(distribution on energy axis)breaks up,after a transient period,into two parts,Ψ(n,τ)∼Ψedge (n,τ)+Ψbulk (n,τ),where Ψedge is a com-ponent localized around the boundary (i.e.,the ground state)while Ψbulk is a component traveling into the bulk (excited states)with a nearly constant velocity.Interest-ingly,the edge state only appears when the phase differ-ence,θ≡γ−˜γ,between the bulk and boundary transfer matrices isnonzero (as is the case with physical systems as men-tioned).What is the nature of the edge state?The two generat-ing functions ΨL,R (0→n )(z )(eqs.(3))have a common,first-order pole in z 2at z 2pole=1−e −iθ√1−e iθ√F /F )involves F .We note that ǫcan beexpressed as ǫ=−i lim t →∞11−p ),(7)with |φL (0)|2=(1−r )2.Thus Φ(n )is an edge state ex-ponentially localized (on energy axis)around the bound-ary n =0,whose weight is |Φedge |2= n (|φL (n )|2+|φR (n )|2)=1−r .The size (on energy axis)of the edge state is ξ=1/|ln r |,which behaves as ξ∝F/F |F −F c (θ)|−1in the vicinity of the threshold,p c (θ)=sin 2θ,F c (θ)=−π4FIG.3:Total momentum J edge and energy E edge contributedby the edge component (eq.(8))against the electric field F for θ=π/4.The upper figure depicts the weight of the edge state.the electric field and becomes zero at F =F c (θ)(toppanel of Fig.3).Following [1,2,3,4],we choose j R,Ln =±j 0n,E R,L n =E 0n 2,n =0,2,...as the momentum and energy of the states,respectively,where j 0,E 0are units of momentum and energy.We note that the mo-mentum measures the chiral asymmetry (between L and R )of the distribution.By plugging in eq.(7)we ob-tain J edge =j 0p (2−p −2cos θ√2√1−p cos θ−1)2,and a similar ex-pression for E edge ,valid for F <F c (θ)(Fig.3).In the weak-field regime,the J edge is suppressed until the Landau-Zener tunneling is activated for F >F /F )starts to rise.When F is furtherincreased to reach F =F c (θ),the electric-field induced breakdown occurs,which corresponds,in the present pic-ture,to the edge-to-propagating transition in the quan-tum walk.At the breakdown point the energy expecta-tion value diverges as E edge ∼|F −F c (θ)|−2.The J edge ,however,does not diverge but shows a smooth increase.Discussions —The idea of mapping the non-equilibrium problem to a quantum walk may apply to wider range of systems having many energy gaps.Here we have concentrated on the “edge state”when there is randomness (i.e.,θ=0)only at the edge.Randomness in the transfer matrices in the excited state,ignored here,is expected to enhance the localization to bring |Φedge |2closer to unity.Blatter et.al [4]have in fact obtained the expectation value of the momentum in such a situation,where the result resembles J edge obtained here.However,Blatter et.al did not obtain,in the range of electric field F they have studied,a delocalization transition.In fact,Cohen et.al ,in a random matrix model,found a disap-pearance of localized states in strong external fields[6].If such transitions also occur in electron systems,the quantum-walk picture may be used in understanding the transition.Indeed,some of the quantum-walk models now under study exhibit localization (e.g.,[20]),and the present mapping may shed lights on the non-equilibrium properties of driven quantum systems.It is a pleasure to acknowledge V.Kagalovsky,Y.Mat-suo,S.Miyashita,N.Nagaosa,and Y.Tokura for fruitful discussions in the early stage of the present work.We also wish to thank M.Katori and T.Sasamoto who suggested a link between the Landau-Zener transition and quantum walks.One of us (TO)thanks the Yukawa Institute for Theoretical Physics at Kyoto University,where this work was initiated during the YITP-W-03-18on “Stochastic models in statistical mechanics”.Part of this work was supported by a JSPS fellowship for Young Scientists.∗Present address:Max-Planck-Institut f¨u rFestk¨o rperforschung,Stuttgart,D-70569Germany[1]D.Lenstra and W.van Haeringen,Phys.Rev.Lett.57,1623(1986).[2]ndauer,Phys.Rev.B 33,6497(1985).[3]Y.Gefen and D.J.Thouless,Phys.Rev.Lett.59,1752(1987).[4]G.Blatter and D.A.Browne,Phys.Rev.B 37,3856(1988).[5]T.Oka,R.Arita,and H.Aoki,Phys.Rev.Lett.91,66406(2003).[6]D.Cohen and T.Kottos,Phys.Rev.Lett.85,4839(2000).[7]S.Fishman,D.R.Grempel,and R.E.Prange,Phys.Rev.Lett.49,509(1982).[8]When the evolution matrix U (including the phase γ)differs from one level crossing to another,we can still use eq.(5)to recursively obtain the generating functions,which will be discussed in a separate publication.[9]J.Kempe,Contemporary Physics 44,307(2003).[10]B.Tregenna,et al.,New J.Physics 5,83(2003).[11]A similar model has been employed to analyze the crit-ical behavior of the integer quantum Hall system by J.T.Chalker and 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