The effect of the three-spin interaction and the next-nearest neighbor interaction on the quenching dynamics of a transverse Ising model Uma Divakaran∗ and Amit Dutta† Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India. (Dated: February 2, 2008) We studythe zero temperature quenchingdynamics of various extensions of thetransverse Ising model (TIM) when the transverse field is linearly quenched from −∞ to +∞ (or zero) at a finite 8 anduniformrate. Therateofquenchingisdictatedbyacharacteristicscalegivenbyτ. Thedensity 0 of kinks produced in these extended models while crossing the quantum critical points during the 0 quenching process is calculated using a many body generalization of the Landau-Zener transition 2 theory. The density of kinks in the final state is found to decay as τ−1/2. In the first model n considered here, the transverse Ising Hamiltonian includes an additional ferromagnetic three spin a interaction term of strength J3. For J3 < 0.5, the kink density is found to increase monotonically J with J3 whereas it decreases with J3 for J3 > 0.5. The point J3 = 0.5 and the transverse field 7 h = −0.5 is multicritical where the density shows a slower decay given by τ−1/6. We also study 1 the effect of ferromagnetic or antiferromagnetic next nearest neighbor (NNN) interactions on the dynamicsofTIMunderthesamequenchingscheme. Inameanfieldapproximation,thetransverse ] Ising Hamiltonians with NNN interactions are identical to the three spin Hamiltonian. The NNN h interactions non-trivially modifies the dynamical behavior, for example an antiferromagnetic NNN c interactions results to a larger number of kinks in the final state in comparison to the case when e m theNNNinteraction is ferromagnetic. - PACSnumbers: t a t s I. INTRODUCTION hand, for He-4 superfluid transition, KZM could not . t be verified experimentally18. Hence more experiments a m are clearly needed to put KZM theory on a stronger The critical dynamics of classical systems have been footing. The same idea has been applied to study the - studied extensively in last three decades while the study d dynamics across a zero temperature QCP by different n ofthe dynamics ofa quantum systemwhensweptacross groups3,4,5,6,7,8,9,10. o a quantum critical point (QCP) is fairly recent and not TheextendedKZMforthezero-temperaturequantum c yet fully understood. The vanishing of energy gap be- transitions relies on the fact that during the evolution [ tween the ground state and the first excited state of the when the system is close to the static critical point, the 1 quantum Hamiltonian signals the existence of a QCP1,2. relaxation time diverges in a power-law fashion. The v AtaQCP,thecorrelationlengthaswellastherelaxation non-adiabatic effects become prominent when the time 1 timediverge,aphenomenonknownasthecriticalslowing scaleassociatedwiththe changeofthe Hamiltonianis of 2 down. This diverging timescale makes it impossible for 6 the order of the relaxation time. The loss of adiabatic- any system to cross the quantum critical point without 2 ity while crossing a quantum critical point can be quan- excitations from the ground state. The dynamics there- . tified by estimating either the density of defects (e.g., 1 fore is non-adiabatic in contrast to an adiabatic evolu- the density of oppositely oriented spins in Ising mod- 0 tionwherethesystemstickstotheinstantaneousground 8 els) in the final state3,4,5,6,7 or the fidelity of the final statethroughoutthequenchingprocess. Inrecentyears, 0 state with respect to the ground state3 or the residual therehasbeenanupsurgeinthestudyofdynamicsclose : energy19,20,21,22,23,24. The argument given above imme- v to a quantum critical point clearly indicating a growing diately leads to a (1/√τ)-dependence of the density of Xi interest in the field3,4,5,6,7,8,9,10,11,12,13. defects on the characteristic timescale τ of the quench- r OneofsuchattemptswastoextendtheKibble’stheory ing. The residual energy is defined as the difference be- a ofdefectproductionintroducedtoexplainearlyuniverse tween the energy of the evolved ground state and the behavior14 to the second order quantum phase transi- true ground state. This residual energy for the inte- tions. This method of calculating the density of defects grable disorder free systems is trivially proportional to is knownas the Kibble-Zurekmechanism (KZM)15. The the density of kinks with the proportionality constant theory of KZM for a classical second order phase tran- being equal to the strength of interaction. In an op- sition is based on the universality of the critical slowing timization approach popularly known as the “quantum down and leads to the prediction that the linear dimen- annealing”19,20,21,22,23,24, the strength of the quantum sion of the ordered domains scales with the transition fluctuations is slowly reduced to zero so that a disor- time τ as τw where w is some combination of critical dered and frustrated system of finite size is expected to exponents. KZMhasbeenconfirmedbynumericalsimu- reach adiabatically its true classical ground state. In lationsoftimedependentGinzburg-Landaumodel16 and the present literature, the expressions “annealing” and also for various experimental systems17. On the other “quenching” are used synonymously. The residual en- 2 ergyturns outto be a moreappropriatemeasureofnon- in the x and y components of the spin27. Interestingly, adiabaticityfortheannealingapproach. Inarecentwork even in the presence of the three spin interaction term, by Caneva et. al.25, it has been shown that for a disor- theHamiltoniangivenbyEq.(1)isexactlysolvedbythe dered quantum Ising spin chain, the residual energy and Jordan-Wigner (JW) transformation29,30,31 which maps the density of kinks show different scaling behavior with this interacting spin system to a system of noninteract- τ. Recently a general analysis has been carried out of ing spinless fermions. Moreover, this three spin term is the breakdown of the adiabatic limit in low-dimensional found to be irrelevant in determining the quantum crit- gapless systems26. ical behavior of the system. The critical exponents are In this paper, we will concentrateonthe estimationof the same as that of Ising model in a transverse field ex- densityofdefectsproducedduringthedynamicsofthree cept for the case J = 0 ,and J = h. For the sake of x 3 different types of model Hamiltonians, all of them being completeness,letusnowprovideabriefdiscussiononthe exactly solved,at least in a mean field level, via the Jor- diagonalizationof the Hamiltonian. dan Wigner transformation. These three Hamiltonians In the JW-transformation, the Pauli matrices are are extensions of the TIM with an additional interac- transformed to a set of fermionic operators (c ) defined i tion term in each and our aim is to study the effect of as suchinteractionsonthe densityofdefects produceddur- i−1 ing the quenching. The additional terms are i) a ferro- c = σ−exp( iπ σ†σ−) magneticthreespininteraction27ii)anantiferromagnetic i i − j j Xj=1 nextnearestneighborinteractionandiii)aferromagnetic next nearest neighbor interaction, respectively. We con- σiz = 2c†ici−1 (2) siderthe unitaryevolutionofthe systempreparedinthe with σ† = (σx + iσy)/2 and σ− = (σx iσy)/2, and ground state of the initial Hamiltonian which crosses its − satisfy the standard anticommutation relations equilibrium critical line as the system evolves. As de- scribed later, in all the cases, the fermionization of the c†,c =δ , c†,c† = c ,c =0. Hamiltonian reduces it to a quadratic form and hence { i j} ij { i j} { i j} one can reduce the dynamics of a many-body Hamilto- Weshallworkinthebasisinwhichσz isdiagonalsothat nianeffectivelytoa2 2LandauZenerproblem28 inthe thepresenceofafermionataparticularsiteicorresponds × fourier representation. to an up spin (i.e., eigenvalue +1 of the operator σz) i The paper is organized as follows. Section II includes at that site. Using a periodic boundary condition, the a detailed discussiononthe analyticaldiagonalizationof Fourier transform of the Hamiltonian can be cast in the the transverse Ising Hamiltonian with a three spin in- form teraction term. In section III, we have described the transverse quenching scheme along with the results for H = [ (h+cosk J cos2k)(c†c +c† c ) − − 3 k k −k −k the above model. We have presented a comparison be- kX>0 tween the three spin Hamiltonian and the Hamiltonians +i(sink J sin2k)(c†c† +c c )]. (3) with next nearest neighbor interactions when treated at − 3 k −k k −k a mean field level in section IV. A brief summary of the Clearly, in the momentum representation of c-fermions, work is presented in the concluding section with a brief theHamiltonianisquadraticandistranslationallyinvari- discussionbasedontherecentdevelopmentsinthisfield. ant. Using the Bogoliubov transformation, the Hamilto- niancanbe diagonalizedto the form ǫ η†η where − k k k k ηk are the Bogoliubov quasiparticles anPd ǫk is the exci- II. MODEL AND THE PHASE DIAGRAM tation energy or gap given by27,29 ǫ =(h2+1+J2+2hcosk 2hJ cos2k 2J cosk)1/2(4) The Hamiltonian of a one-dimensional three spin in- k 3 − 3 − 3 teracting transverse Ising system is given by27 with J set equal to unity. x It can be easily shown that the gap of the spectrum H = 1 σz[h+J σx σx ] −2{ i 3 i−1 i+1 vanishes at h = J3 + 1 and also at h = J3 1 with Xi ordering (or mode-softening) wave vectors π a−nd 0 re- + J σxσx , (1) spectively. Thesetwolinescorrespondtoquantumphase x i i+1} X transitions from a ferromagnetically ordered phase to a whereσz andσx arenon-commutingPaulispinmatrices, quantum paramagnetic phase with the associated expo- J is the strength of the nearest neighbor ferromagnetic nents being the same as the transverse Ising model30. x interaction while J denotes the strength of the three The wave vector at which the minima of ǫ (Eq. 4) oc- 3 k spin interaction. In the limit J 0, the model reduces curs, gets shifted from k =0 to k =π wave vector when 3 → to the celebrated transverse Ising model studied exten- one crosses the line h = J . Moreover, there is an ad- 3 sively in recent years1,2. By a duality transformation31, ditional phase transition at h = J . This transition 3 − the above Hamiltonian can be mapped to a transverse belongs to the universality class of the anisotropic tran- XY model with competing (ferro-antiferro) interactions sition observed in the transverse XY-model dual to the 3 Hamiltonian (1)32 and the phase boundary is flanked by H(t) = [ H (t)] where each H (t) operates on a k>0 k k the incommensurate phases on either side with ordering fourdimePnsionalHilbertspacespannedbythe basisvec- wave vector given by tors 0 , k, k , k and k . Thevacuumstatewhere | i | − i | i |− i no c-particle is present, is denoted by 0 which corre- h J3 | i cosk = − . (5) sponds to a spin configuration with all spins pointing in 4hJ 3 the z-direction. The form of the Hamiltonian readily − This incommensurate wave vector picks up a value k suggeststhatthe parity(evenorodd)oftotalnumberof o such that cosk0 = 1/2J3 at the phase boundary. Obvi- fermions (given by nk = c†kck +c†−kc−k) for each mode ously, for J3 < 0.5, the anisotropic phase transition can is conserved. Therefore, to study the quenching dynam- not occur. The equilibrium phase diagram of the model ics, it is convenient to project the Hamiltonian H (t) in k is shown in Fig. 1. the subspacespannedby 0 and k, k . The projected | i | − i Hamiltonian has a form 4 h(t)+cosk J cos2k i(sink J sin2k) 3 3 − − Para h=1+J3 (cid:20) i(sink J3sin2k) (h(t)+cosk J3cos2k)(cid:21) − − − − 2 In the reduced Hilbert space, any generalstate can be Ferro h=J3−1 represented as a superposition of 0 and k, k with h 0 Commensurate time dependent amplitudes u (t) |anid v (t|) s−uchi that k k 3−spin dominated Incommensurate ψk(t) = uk(t)0 +vk(t)k, k . The time evolution of | i | − i Com−2mensPuraartea IncPoamrmaensurate h=−J3 the state is given by the Schroedinger equation i∂ ψ (t)=H (t)ψ (t). (6) t k k k −4 h= J3 1−4J3 We shall here use the initial conditions u ( ) = 1 k 0 0.5 1 1.5 2 and v ( ) = 0 which in the spin langu−ag∞e corre- J k −∞ 3 sponds to the state with all spins down. The off diag- onal term ∆ = sink J sin2k represents the interac- FIG. 1: Equilibrium phase diagram of the three spin inter- − 3 tionbetweenthetwotimedependentlevelswithenergies acting Ising model. Solid lines show phase boundaries and E = [h(t)+cos(k) J cos2k]. The zeroes of the dottedlinemarkstheboundarybetweentheincommensurate 1,2 ± − 3 off-diagonal term yield the mode softening wave vectors and the commensurate phase. k =0,π and cos−11/(2J ) (provided J >0.5) at which 3 3 thesystembecomesquantumcriticalforappropriatepa- rameter values. At these parameter values and wave III. TRANSVERSE QUENCHING AND vectors, the system undergoes a nonadiabatic transition RESULTS fromits instantaneousgroundstate. Ameasureofnona- diabaticity can be obtained by comparing the two level problem to the corresponding Landau-Zener transition The dynamics of the three spin interacting TIM is equations5,7. For a completely adiabatic transition, we found to be very interesting due to the fact that the expect the final state to be described by the probability system crosses various quantum critical lines during the amplitudes u (+ )=0 and v (+ )=1, i.e., the com- process of dynamics. As mentioned already, the sys- k ∞ k ∞ pletespin-flipfromdowntoupoccurs. Thenonadiabatic tem deviates from the adiabatic evolution in the neigh- transition probability p is directly given by u (+ )2 borhood of a quantum critical point where nonadia- k | k ∞ | wheretheprobabilityamplitudesu (t)andv (t)arenor- baticity dominates due to the divergence of relaxation k k malized at each instant of time. Equivalently, p also time. We shall introduce the time dependence in the k measures the probability that the system remains in its Hamiltonianthroughthetransversefieldwhichislinearly initial state 0 even at the final time. Using the results quenched from −∞ to +∞ at a steady finite rate given of Landau-Z|enier transitions21,28, p is found to be by h(t) t/τ, where the quenching time τ determines k the rate∼of quenching3,4,5. At time t = the trans- verse field h = and hence all the spin−s∞are pointing p = u (+ )2 =exp( 2πγ) where γ = ∆2 .(7) −∞ k | k ∞ | − d(E E ) in the negative z direction. By virtue of the duality dt 1− 2 − transformation, the transverse quenching of the 3 spin Therefore, in this model Hamiltoniancorrespondstothe anisotropicquenchingof the transverse XY model where the interaction term of p =exp[ πτ(sink J sin2k)2]. (8) k 3 the later Hamiltonian is adiabatically changedfrom − − to 10. −∞ The variationof p as a function of k for different values k ∞ Let us recall the Hamiltonian given in Eq. 3 with of quenching time τ is shown in Fig. 2. It is to be noted a time dependent transverse field h(t). This Hamil- that for J < 0.5, there are peaks at π,0 and π in the 3 − tonian can be split into a sum of independent terms, whole range of wave vectors from π to π whereas for − 4 J3=0.1 J3=1 The density of kinks monotonically increases with in- 1 00..15 00..15 creasing J3 provided J3 < 0.5 because of the decrease 10 10 in the off-diagonal term making the probability of exci- 0.8 tations higher. On the other hand, for J > 0.5, the 3 off-diagonal term monotonically increases with with in- 0.6 creasingJ resultingtoanoveralldecreaseinthedensity p 3 k ofkinks,seefigure3. Theseresultscanalsobeseenfrom 0.4 theapproximateanalyticalexpressionofthekinkdensity given in Eq. (10 a and b) for both the cases. 0.2 We shall now focus our attention to the case J =0.5. 3 0 In the process of the transverse quenching, the system −3 −2 −1 0 1 2 k 3 −3 −2 −1 0 1 2 3 crosses the multicritical point at h = 0.5,J3 = 0.5 as − shown in the Fig. 1 and a special power-law behavior FIG. 2: Non-adiabatic transition probability pk for thethree of the kink density is observed at these parameter val- spin interacting Hamiltonian with J3 = 0.1 in Fig. 2(a) and ues. The transition probability p maximizes at k = 0 k J3 =1 in Fig. 2(b) for various τ. It should be noted that for as shown above. The argument of the exponential in p k J3 = 1, the system undergoes a non-adiabatic transition at is expanded about k=0 at J = 0.5, leading to a form anincommensuratewavevectork=π/3andtherefore,there 3 p = exp[ πτk6/4]. The contribution to the the kink- is an additional peak at this wave vector . For large τ, pk is dkensity sca−les as 1/τ1/6 which can be obtained by sim- nonzeroonlyforwavevectorsveryclosetothecriticalmodes. ply integrating this p , see figure 3. This relatively slow On the other hand, for small values of τ, levels cross quickly k resulting to a non-zero valueof p for all values of k. decay of density is a special characterisitc of quenching k through a multicritical point. A similar behavior is also seen in the anisotropic quenching of the transverse XY J3 > 0.5 there are additional peaks at the incommensu- model10. rate values cos−1(1/2J ). 3 As menti±oned already, the degree of nonadiabaticity −0.4 (a) (b) can be quantified through the density of kinks n gener- −0.6 J3=0.5 ated at t = + which is obtained by integrating the ∞ probability pk over the entire range of wave vector. −0.8 1 π n n= p = dk p . (9) −1 k k 2π Z Xk −π −1.2 J3=0.2 A close inspection of Eq. 8 (see also Fig. 2) shows that J3=0.9 for sufficiently slow quenching (i.e., largeτ), only modes −1.4 J3=1.0 closetothecriticalmodesareexcited. Onecantherefore, J3=0.1 J3=1.2 −1.6 to the lowest order in k, replace sink by k in the expo- 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 τ nential of Eq. 8 and arrive at an approximate analytical expressionfor density of kinks in the largeτ limit, given FIG. 3: The variation of kink density with τ for different by: J3 (< 0.5) is shown in Fig. 3a. Kink density increases with J3. On the other hand for J3 > 0.5, this variation decreases n= 1 + 1 (for J <0.5) with J3 as shown in Fig. 3b. The thick line has slope −0.5 2π(1 2J )√τ 2π(1+2J )√τ 3 andtheslopeof thedotted lineis−1/6, abehaviorobserved 3 3 − (10a) at J3 =0.5 2 2 n= + (for J3 >0.5). Onecanalsostudytheeffectoftheanisotropicquench- 2π(2J 1)√τ 2π(1+2J )√τ 3− 3 ing which involves quenching of the nearest neighbor (10b) Ising interaction term Jx(t)( t/τ) from to + In Eq. 10(a), the first term corresponds to the contri- ∼ −∞ ∞ instead of the transverse field with the three spin inter- bution from modes close to k = 0 whereas the second actiontermsettounity. Att ,thegroundstateof term is due to the peaks at k = π, π. For the case →−∞ − the systemisantiferromagneticalongx. Theprobability J > 0.5, the contribution from peaks at k = 0,π and 3 of the non-adiabatic transition is similarly given by π happens to be the same as Eq. 10(a) whereas the c−ontribution n from the modes close to k = k is also p =exp[ πτ (h+1)sink 2]. (12) 1 0 k − { } equal to 10(a) in the following way: It is interesting to note that for h = 1, p is unity for k n = 1 πexp[−πτ{(cosk0−2J3cos2k0)(k−k0)}2] all values of k. The density of kinks −for the anisotropic 1 π Z quenching is given as 0 1 1 1 1 = [ + ]. (11) n= dkp =exp(( πτ(h+1)sink)2) (13) 2π√τ 2J3+1 2J3 1 2π Z k − − 5 with an approximate analytical form given as analyticalsolutionispossibleatleastinthelimitofsmall J . Deep in the paramagnetic phase all the spins are 1 2 n= (14) oriented in the direction of the transverse field so that π(h+1)√τ <σz >=1 or in the fermionic language 1 2c† c = i − i+1 i+1 which shows that the density of kinks decreases mono- 1. We shall approximate < σz >= 1 for all pos- − i tonically with h. This can be attributed to an increase itive values of h including h 0. This approxima- ∼ in the off-diagonal term of the Hamiltonian. tion, though crude, transforms the four fermion term (c† c )(1 2c† c )(c† +c ) to a quadraticform. i − i − i+1 i+1 i+2 i+2 The Hamiltonian becomes exactly solvable but the rich IV. CONNECTION TO THE TRANSVERSE phase diagram of the model is not captured in this QUENCHING OF THE MODELS WITH NEXT approximation35,36. Within this approximation,we shall NEAREST NEIGHBOR INTERACTIONS explorethe roleofsmallNNN antiferromagneticinterac- tion on the density of kinks produced during the trans- We shall now use the results of the previous section versequenching. Asdescribedbelow,thisapproximation to study the transverse quenching of a quantum Ising at least shows a decrease of critical field h with J for c 2 model with uniform ferromagnetic nearest neighbor in- J <0.5. 2 teraction and also an additional NNN interaction which The mean field Hamiltonian in the momentum space is either antiferromagnetic or ferromagnetic. The model is withNNNantiferromagneticinteractionhasregularfrus- trations and is popularly known as Axial Next Nearest H = [ (h+cosk+J cos2k)(c†c +c† c ) − 2 k k −k −k Neighbor Ising (ANNNI) model33 in a transverse field. kX>0 We shall show below that within a mean field approxi- +i(sink+J sin2k)(c†c† +c c )] (17) mation, the three spin model has a close resemblance to 2 k −k k −k the one-dimensional NNN interacting TIMs. Comparing Eq. (17) with Eq. (3), one finds that the The Hamiltonian of the transverse ANNNI model is transverse ANNNI chain Hamiltonian in the mean field given by approximationis identicalto the three spin Hamiltonian iftheantiferromagneticinteractionJ oftheformerisre- N 2 1 placedby the negative of the three spin interactionterm H = [hσz +J σxσx J σxσx ] (15) −2{Xi i 1 i i+1− 2 i i+2 } (J3) in the latter. Using the results of the previous sec- tion, the phase diagramof the meanfield ANNNI model where J1,J2 > 0. Henceforth, without loss of any gen- canbefoundoutforh>0(seeFig.4). Thephasebound- erality, we shall set J1 = 1. At h = 0, the ground state ary between the ferromagnetic phase and the paramag- is ferromagnetically ordered for J2 < 0.5, whereas the neticisgivenbyh=1 J2 withanorderingwavevector − system shows an “anti-phase” ordering (where two up π (this corresponds to the Ising transition at h=J +1 3 spins are followed by two down spins) for J2 >0.5. The of Fig. 1). For J3 > 0.5, i.e., the transition between the twophasesmeetataninfinitelydegeneratemulti-critical antiphase and the paramagnetic phase, is given by the point J2 = 0.5 and h = 0. The quantum fluctuations corresponding anisotropic transition of three spin model introduced by the transverse field h competes with the with the phase boundary given by the equation h = J 2 ferromagnetic (or the antiphase) order and eventually and the ordering wave vector has an incommensurate the system undergoes a quantum phase transition to a value as given in Eq. 5. paramagnetic phase at a critical value of the transverse Theapproximation1 2c†c = 1isvalidforpositiveh field givenby hc whichis a function of the NNN interac- only,wechooseaquenc−hingisichem−ewherethetransverse tionJ2. One-dimensionalquantumANNNI modelshows field has a functional dependence h(t) t/τ with t a rich phase diagram which is not fully understood till goingfrom to0sothath(t) remains∼po−sitiveforthe date2,33,34. entire quen−ch∞ing period and vanishes at the end of the When mapped to the corresponding fermionic Hamil- quenching. ThereforethesystemdoesnotcrosstheIsing tonian via a JW transformation, the NNN interaction critical line h=J +1. 2 leads to a four-fermion term in the fermionic version of Inthe fina−lstate att=0,allthe spinsareexpectedto the Hamiltonian. In the limit J2 0, this term van- orientinthex-directionwithaferromagneticorder. The → ishes so that the model is exactly solvable in terms of densityofoppositelyorientedspinsatt 0is relatedto non-interacting fermions. For non-zero J2, the fermionic J2 as → Hamiltonian is written as 1 H =−21{[Xi h(2c†ici−1)+(c†i −ci)(c†i+1+ci+1) n= 2π(1−2J2)√τ (18) J (c† c )(1 2c† c )(c† +c )] . (16) whichshowsthatthedensityofkinksincreasesmonoton- − 2 i − i − i+1 i+1 i+2 i+2 } ically with J2. The occurrence of the four-fermion term renders the Itshouldbenotedthatifwefollowaquenchingscheme model analytically intractable though an approximate inwhichthetransversefieldischangedfrom tozero, −∞ 6 Quenching direction frommodeltomodel. Thefirstofthevariantsincludesa three spin interaction with strength J . Here, the phase 3 1 diagram indicates the existence of an anisotropic phase h=1−J h=J 2 2 transition at an incommensurate value of wave vector in addition to the normal Ising transition for J > 0.5. In- 3 h Paramagnet terestingly,weobservethatthedensityofkinksincreases monotonicallywithJ forJ <0.5whereasdecreasesfor 3 3 0.5 J3 > 0.5. On the other hand, at J3 = 0.5, the con- tribution to the kink density scales as τ−1/6 due to the Ferromagnet Antiphase existence of a multicritical point at J =0.5. The other 3 set of Hamiltonians include a ferromagnetic or an anti- ferromagnetic next nearest neighbor interactions. The presence of the four fermion term makes such a Hamil- 0 0.5 J 1 tonian analytically intractable. We have used a mean 2 field approximation to reduce the four fermion term in FIG. 4: Mean field phase diagram of the ANNNI model in the fermionized representationto a quadratic term. The the h−J2 plane. We study the quenching dynamics across quenching scheme is chosen carefully so that the regions thephase boundary close to J2 →0. where the approximation is not valid are avoided in the process of dynamics. Using the similarity between the fermionized next nearest neighbor interacting Hamilto- we must approximate the term 1 2c† c with +1 − i+1 i+1 niansunder the meanfield approximation,andthe three rather than 1 for above calculations to be viable. In − spin interacting model, the density of kink in the final theprocessofdynamics,thesystemcrossesthequantum state is estimated. It is observed that the ferromagnetic critical line h = J +1 with the modes close to k = 0 − 2 next nearest neighbor interactions reduces the density getting excited. This approach as well leads to identical of kinks produced as opposed to the case of antiferro- result for kink density (as given in Eq. 18). Therefore, magnetic next nearestneighbor interactionbecause such the presence of a small antiferromagnetic NNN interac- a ferromagnetic interaction enhances the ferro-ordering tion adds to the kink-production in comparison to the discouragingtheproductionofkinks. Ontheotherhand, ferromagnetic transverse Ising model (J = 0) with the 2 frustration leads an enhanced non-adiabatic transitions. same quenching scheme. We should mention in conclusion that it is in principle One can also study, in the similar spirit, a model with possible to construct a better mean field theory for the a small ferromagnetic NNN interaction J . We use FM ANNNI model36, however, no qualitative change in the thesamemeanfieldapproximationforh 0sothatthis modelisidenticaltothethreespinmodelw≥ithJ3 =JFM. dynamical behaviour in the region J2 →0 is expected. We conclude with the comment that the models stud- A similar calculation leads once again to a 1/√τ fall of ied in the present work are integrable (at least in the the density of kinks given by mean field limit) which leads to a 1/√τ scaling behav- 1 ior of the defect density. However, in a random or n= . (19) 2π(1+2J )√τ a non-integrable system such a behavior need not be FM expected25. The quenching and annealing dynamics of This is expected because the NNN ferromagnetic inter- several non-integrable systems along with the depen- action enhances the strength of the ferromagnetic order- dence of the defect density on the integrability of the ing andhence the probabilityofexcitationsor densityof modelareyettobecompletelyunderstood. Wehavealso kinks is lowered. observed a much slower decay of the form 1/τ1/6 when quenchedthroughthemulticriticalpointofthethreespin model as in the anisotropic quenching of the transverse V. CONCLUSIONS XY chain10. In this paper, we have studied the effect of various Acknowledgments additionalinteractionsonthedynamicsofthetransverse Isingmodelwhensweptacrossthequantumcriticallines. The defect density scale with the timescale τ as τ−1/2, We acknowledge Victor Mukherjee and Diptiman Sen likeintransverseIsingcase,withaprefactorwhichvaries for collaboration in related works. ∗ Electronic address: [email protected] 1 Sachdev S, 1999 Quantum Phase Transitions, Cambridge † Electronic address: [email protected] University Press, Cambridge. 7 2 ForareviewonphasetransitionsinTIMssee: Chakrabarti Ed. by A. Das and B. K. Chakrabarti (Springer-Verlag, BK,DuttaAandSenP,1996Quantum IsingPhases and Berlin, 2005). Transitions in Transverse Ising Models vol m41 (Heidel- 21 Suzuki S and Okada M, in Quantum Annealing and Re- berg: Springer) . lated Optimization Methods, Ed. by A. Das and B. K. 3 ZurekWH,DornerUandZollerP,2005 Phys.Rev.Lett. Chakrabarti (Springer-Verlag, Berlin, 2005). 95, 105701. 22 Brooke J, Bitko D, Rosenbaum T F, and Aeppli G, 1999 4 Dziarmaga J, 2005 Phys. Rev.Lett. 95, 245701 . Science 284, 779. 5 Damski B, 2005 Phys. Rev. Lett. 95, 035701; Damski B 23 Santoro G E, Martonak R, Tosatti E, and Car R, 2002 and ZurekW H, 2006 Phys.Rev.A 73, 063405. Science 295, 2427. 6 Polkovnikov A, 2005 Phys.Rev.B 72, 161201. 24 Santoro G E and Tosatti E, 2006 J. Phys. A: Math. Gen. 7 Cherng R W and Levitov L S, 2006 Phys. Rev. A 73, 39, R393. 043614. 25 Caneva T, Fazio R and Santoro G E, arXiv:0706.1832. 8 Cincio L, Dziarmaga J, Rams Marek M and Zurek W H, 26 Polkovnikov A and Gritsev V, arxiv:cond-mat/0706.0212. 2007 Phys. Rev.A, 75 052321. 27 Kopp A and ChakravartyS, 2005 Nat.Phys., 1 53. 9 Cramer M, Dawson Christopher M, Eisert J, Osborne, T 28 Zener C, 1932 Proc. Roy. Soc. London Ser A 137, 696; J, arxiv:cond-mat/0703314. Landau L D and Lifshitz E M, Quantum Mechanics: non 10 Mukherjee V, Divakaran U, Dutta A and Sen D, relativistic theory, 2nded.(PergamonPress,Oxford1965) arxiv:cond-mat/0708.0278 29 Lieb E, Schultz E and Mattis D, 1961 Annals of Physics, 11 K.Sengupta,S.Powell, andS.Sachdev,Phys.Rev.A69, 61 407. 053616 (2004). 30 Pfeuty P, 1970 Annals of Phys.(NY) 57, 79 . 12 P. Calabrese and J. Cardy, J. Stat. Mech: Theory Expt 31 John B Kogut, 1979 Rev.Mod. Phys.,51 659. P04010 (2005). 32 Bunder J E and McKenzie R H, 1999 Phys. Rev. B, 60, 13 A.Das,K.Sengupta,D.Sen,andB.K.Chakrabarti,Phys. 344. Rev.B 74, 144423 (2006). 33 Selke W, 1988 Phys. Rep. 170, 213; Villain J and Bak 14 Kibble T B W, 1976 J. Phys. A,9, 1387. P, 1981 J. Physique (Paris) 42, 657; Nijs M den, 1988 15 ZurekAH,1985 Nature(London),317,505; Phys.Rep., inPhase Transitions and Critical Phenomena, Vol.12,ed. 1996 276, 177. DombCandLebowitzJL(Academic,NewYork);SelkeW 16 P. Laguna and W. H. Zurek, 1997 Phys. Rev. Lett. 78, 1992, in Phase Transitions and Critical Phenomena, Vol. 2519; 1998 Phys. Rev. D 58, 5021 ; Yates A and Zurek 15,ed.DombCandLebowitzJL(Academic,NewYork); W H, 1998 Phys. Rev.Lett 80, 5477; StephensG J et. al, Yeomans J,i 1987 Solid State Physics, Vol. 41, ed. Ehren- 1999 Phys.Rev.D59, 045009; N.D.Antuneset. al, 1999 reich H and TurnbullJ L (New York,Academic, 1987); Phys.Rev.Lett.82,2824;DziarmagaJ,P.LagunaandW. 34 DuttaAandSenD,2003Phys.Rev.B67,094435; Shira- H.Zurek,1999Phys.Rev.Lett.82,4749;HindmarshMB hata T and Nakamura T, 2001 Phys. Rev. B 65, 024402; andRajantieA,2000Phys.Rev.Lett.85,4660; Stephens Chandra A K and Dasgupta S, 2007 J. Phys. A. Math. GJ,Bettencourt,andW.H.Zurek,2002Phys.Rev.Lett. Theor. 40 6251; Beccaria M, Camostrini M and Feo A, 88, 137004; 2007 arxiv/cond-mat:0702676 17 Ruutu V M H et. al., 1996 Nature 382, 334; Bau¨rle C et. 35 Sen P and Chakrabarti and B K, 1989 Phys. Rev. B 40, al., ibid,1996 382, 332. 760 . 18 P. C. Hendry et.al., 1994 Nature 368, 315; M. E. Dodd 36 Sen P and Chakrabarti B K,1991 Phys. Rev.B 43, 13559 et.al., 1998 Phys. Rev.Lett. 81, 3703 . (1991). 19 KadowakiTandNishimoriH,1998Phys.Rev.E.58,5355. 20 Quantum Annealing and Related Optimization Methods,