Driven quantum spin chain in the presence of noise: Anti-Kibble-Zurek behavior
DDynamics of Driven Quantum Spin Chain in the Presence of Noise:Anti-Kibbel-Zurek Behavior of Defect Generation, Entropy and Correlation Functions
Manvendra Singh , and Suhas Gangadharaiah Department of Physics,Indian Institute of Science Education and Research, Bhopal, India (Dated: February 8, 2021)We study defect generation in a quantum XY-spin chain arising due to the linear drive of themany-body Hamiltonian in the presence of a time-dependent fast Gaussian noise. The main objectiveof this work is to quantify analytically the effects of noise on the defect density production. In theabsence of noise, it is well known that in the slow sweep regime, the defect density follows theKibble-Zurek (KZ) scaling behavior with respect to the sweep speed. We consider time-dependentfast Gaussian noise in the anisotropy of the spin-coupling term [ γ = ( J − J ) / ( J + J )] andshow via analytical calculations that the defect density exhibits anti Kibble-Zurek (AKZ) scalingbehavior in the slow sweep regime. In the limit of large chain length and long time, we calculate theentropy and magnetization density of the final decohered state and show that their scaling behavioris consistent with AKZ picture in the slow sweep regime. By considering the large n -separationasymptotes of the Toeplitz determinants, we further quantify the effect of the noise on the spin-spincorrelators in the final decohered state. We show that while the correlation length of the sub-latticecorrelator scales according to the AKZ behavior, we obtain different scaling for the magnetizationcorrelators. I. INTRODUCTION
A quantum system driven at zero temperature by somesystem dependent parameter through a quantum criti-cal point (QCP) is subjected to quantum phase transi-tion (QPT) in which the ground state of the system isfundamentally altered with completely different physicalproperties across the phase transition point. One of themain points of interest is to quantify the generation ofexcitations or the defect density generation due to thequench through the critical point. Defects are inevitablein a drive through the critical point due to the vanishingenergy gap at the critical point where the adiabaticitycriterion breaks down and non-adiabatic effects becomeimportant. In this regard, the Kibbel-Zurek mechanism(KZM) a theory originally proposed to quantify the topo-logical defect production in a cosmological phase tran-sition has been successfully applied in quantifying thedefect production in idealized condensed matter systemsundergoing QPT . The theory predicts that the de-fect density scales as n ∝ τ − β , where τ is the quench rateand the universal exponent β > .While the study of quantum systems exhibiting KZ be-havior remains an area of active interest, scenarios whichresult in deviations from this universal behavior have alsocome under increased scrutiny. Recent studies of driveprotocols in quantum systems that are coupled to ex-ternal environment, disorder or are in the presence ofnoise indicate that the defect density generated exhibitfundamentally different dynamical behavior than the onepredicted by the KZ theory . The focus of our atten- tion has been to understand experimental and numericalstudies wherein, unlike the KZ behavior, slower drivesbeyond a certain optimal quench rate/speed create moredefects . This scaling has been termed as the antiKibble-Zurek (AKZ) behavior. In all of them, the AKZscaling behavior manifested itself in the presence of thenoisy control field driving the system through the criti-cal points. In this work, we consider a time dependentquantum XY-spin chain which is driven by a transversemagnetic field with a anisotropy term that contains afluctuating Gaussian noise term. In the limit of fast noise,we perform exact analytical calculations and derive theuniversal AKZ scaling behavior of the defect density withrespect to the quench rate.Apart from the study of the defect generation, the con-sequence of the KZ picture to the entropy, magnetizationand the correlation functions have been considered before(for example in the Refs. [28, 38–42]). We furthermorequantify the effect of noise on the above physical quan-tities in the final decohered state due to the noisy drivethrough the QCPs. By considering the large n -separationasymptotes of the Toeplitz determinants, we show ana-lytically that the correlation lengths of the sub-latticecorrelators exhibits the AKZ scaling behavior in the slowsweep regime. The scaling behavior of the magnetizationdensity at the end of the protocol is also consistent withthe AKZ scaling behavior in the slow sweep limit. How-ever, the correlation length of the (connected) magneti-zation correlator in the large n -separation limit continuesto follow the KZ picture.The organization of the paper is as follows. In Sec. IIwe discuss the model Hamiltonian and dynamics of theXY-spin chain with transverse magnetic field (varyinglinearly with time) in the z -direction in the absence ofnoise. In Sec. III we consider transverse protocol in the a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b presence of fast Gaussian noise and obtain analyticallythe AKZ scaling behavior of the defect density in the slowdrive regime. We also derive an expression for the opti-mal quench time with which the system must be drivenso as to minimize the defect production at the end of thedrive protocol. In Sec. IV we discuss the decoherence ofthe local observables due to the drive through the QCPsin the presence of Gaussian noise and in addition the ex-pectation values of a fermionic 2-point correlator in thefinal decohered state has been obtained. In Sec. V wederive the analytical expression for the entropy densityfor the final decohered state and show that the resultsare consistent with the AKZ scenario. Finally, in Sec. VIwe discuss in detail the analytical results for the spin cor-relators and the magnetization density at the end of thenoisy drive protocol. We have summarized our results inSec. VII. II. THE MODEL HAMILTONIAN
We consider the quantum XY-spin model driven by thetransverse external field h ( t ), H ( t ) = − N (cid:88) n =1 (cid:2) J σ n σ n +1 + J σ n σ n +1 + h ( t ) σ n (cid:3) , (1)where, J and J are respectively the spin-spin couplingsalong the x and y -spin directions. Introducing the coef-ficients J = J + J and anisotropy, γ = ( J − J ) /J ,allows us to re-express the Hamiltonian as, H ( t ) = − J N (cid:88) n =1 (cid:2) (1 + γ ) σ n σ n +1 + (1 − γ ) σ n σ n +1 (cid:3) (2) − h ( t ) N (cid:88) n =1 σ n . The γ = 0 limit represents the isotropic XY-spinchain, while γ = ± η ( t ),i.e., γ → γ ( t ) = γ + η ( t ). The Gaussian noise η ( t ) ischaracterized by, η ( t ) = 0 , η ( t ) η ( t ) = η e − Γ | t − t | , (3)where η is the noise strength and Γ is the inverse time-scale associated with the noise.In the following, we will summarize the well studiedtransverse protocol . In this protocol, the trans-verse external field h ( t ) is tuned to drive the equilibriumsystem from h → −∞ at the start of the protocol to h → ∞ at the end of the drive protocol. For a linearprotocol, h ( t ) = vt = Jt/τ Q , where v is the sweep speed of the drive, τ Q = J/v isthe ‘quench time’ and the time t runs from −∞ to ∞ .The system starts out in the paramagnetic (PM) groundstate (GS), | ↓↓↓ ... ↓(cid:105) with all the spins along the neg-ative z -axis. As t is increased the system goes throughthe QCPs at h ( t ) = ± J . The quantum phase transi-tion involves change in the nature of GS from PM toferromagnetic (FM) at h = − J and from FM to PM at h = J . The final state that emerges is not the perfect GSof the Hamiltonian, | ↑↑↑ ... ↑(cid:105) , but instead is formed outof the quantum superposition of the states of the type | .. ↑↑↓↑↑↑↑↓↑↑ .. (cid:105) . The reason for such a final state isthe inevitable violation of the adiabaticity criteria. Thiscriteria is obtained by comparing two time-scales, the re-laxation time scale, which is proportional to the inverseof minimum gap of the system and a time scale for driv-ing the system, τ Q . For a perfectly adiabatic dynamics,‘relaxation time (cid:28) τ Q ’. But in the N → ∞ limit, the en-ergy gap vanishes at the QCPs ( h ( t ) = ± J ). Thereforethe dynamics becomes non-adiabatic in close proximityto the QCPs, leading to a final state which is formedout of the quantum superposition of the states havingkinks/domain walls.It turns out that rate of production of these topolog-ical defects can be quantified with the help of KZ scal-ing theory. Qualitatively one can understand the the-ory as follows . The energy gap around the QCP de-pends on the driving field, ∆( h ) ∼ | h − h c | ν z d ∼ | vt | νz d (assuming the gap varies linearly with time) with ν be-ing the correlation length exponent and z d the dynam-ical exponent. The correlation length diverges near theQCP as ξ ∼ | h − h c | − ν , whereas the excitation energy, E ( k, h = h c ) ∼ | k − k c | z d , characterized by the dy-namical exponent z d , vanishes near the critical mode k c .The region where the adiabaticity breaks down calledthe non-adiabatic/impulse region (near the QCP) can beestimated by comparing the rate of change of the driv-ing parameter/energy-gap with the energy which is pro-portional to the square of the energy i.e., d ∆ /dt ≈ ∆ . One finds that the energy scale at which the adiabatic-ity breaks down is given by ∆ ∗ ∼ | v | ν z d / ( ν z d +1) . Cor-responding to this energy scale a length scale ξ ∗ ∼| v | − ν/ ( z d ν +1) can be associated beyond which the fluctua-tions of the order parameter can not follow the adiabaticdynamics resulting in the creation of defects/excitations. n ∼ | ξ ∗ | − d ∼ | v | νd/ ( νz d +1) ∝ | τ Q | − νd/ ( νz d +1) (4)where d is the dimension of the system. For a 1-D XY-spin chain under going a transverse protocol (linearlywith time) and ν = z d = 1, the KZ scaling theory pre-dicts the defect density to scale as n ∼ √ v ∼ / √ τ Q .These predictions were confirmed by the exact solu-tion of the defect density generation in a linearly drivenXY-spin chain Hamiltonian by mapping it to a set of in-dependent two-level Landau-Zener (LZ) problems. Thefirst step involves the use of Jordan-Wigner (JW) trans-formation, σ ± n = ( e ± iπ (cid:80) n − l =1 c † l c l ) c n and σ zl = 2 c † l c l − σ ± l = σ x ± iσ y ), to map the spin-1/2 Hamiltonianto the spinless free fermion Hamiltonian H ( t ) = − J N (cid:88) n =1 [( c † n c n +1 + γ c n +1 c n + h.c.) − h ( t )(2 c † n c n − . (5)Restricting to the even parity subspace with c N +1 = − c and using the Fourier transformation, c n = e − i π/ √ N (cid:80) k c k e i k n yields H ( t ) = (cid:88) k (cid:104) ( h ( t ) + J cos k ) c † k c k + J γ sin kc − k c k (cid:105) + h.c. . (6)The above Hamiltonian can be written as a sum of in-dependent terms, H ( t ) = (cid:80) k> H k ( t ), where for eachvalue of k, H k ( t ) acts on the 4-dimensional Hilbert spacespanned by the basis vectors: | (cid:105) , | k (cid:105) = c † k | (cid:105) , | − k (cid:105) = c †− k | (cid:105) and | k, − k (cid:105) = c † k c †− k | (cid:105) . We note that the Hamil-tonian with or without the noise term proportional to η ( t ) leaves the parity unchanged. Although the 1-particlestates | k (cid:105) and | − k (cid:105) evolve in time with a global phasefactor only, the states | (cid:105) and | k, − k (cid:105) couple to each otherand exhibit LZ dynamics. The projected Hamiltonian inthe subspace of | (cid:105) and | k, − k (cid:105) has the structure of LZtype Hamiltonian and is given by H k ( t ) = 2 (cid:20) h ( t ) + J cos( k ) J γ sin( k ) J γ sin( k ) − h ( t ) − J cos( k ) (cid:21) , (7)where the off-diagonal term, ∆ k = 2 J γ sin( k ), is k -dependent. Applying the LZ transition theory, yieldsthe k -dependent transition/excitation probability (in thelarge time limit), p k ( z ) = e − πz sin k , (8)where z is the dimensionless quench parameter given by z = J γ /v = J γ τ Q (see Fig. 1).Integrating it over the k modes one obtains the exactresult for the total defect density n in the absence ofnoise, n ( z ) = 1 π (cid:90) π dk e − πz sin k = e − π z I [ π z ] , (9)where I ( π z ) is the modified Bessel function of the firstkind. The limit of large z (or the small sweep speedregime) reveals that the defect density scales as n ∼ / √ z , which indeed matches with the prediction of theKZ scaling behavior (see Fig. 2).Interestingly, recent numerical studies have shown thatthe defect density production exhibits completely differ-ent scaling when in addition to the usual linear protocola small Gaussian noise is present in the control field .In particular, the defect density scales as n ≈ c τ − / Q + d η τ Q , (10)where c and d are system dependent parameters. Thefirst term in the above equation Eq. (10) accounts for FIG. 1. p k Vs k in the absence of noise. In the large z or the small sweep speed limit, most of the contribution tothe defect generation comes from the regions near the criticalpoints k = 0 , ± π . It is to be noted that z = log 2 / π is aspecial value of z , where p k = 1 / k = ± π/ n ( z ) Vs z in the absence of noise. The approximateresult supports KZ mechanism i.e., n ∼ / √ z , and matcheswith the exact result in the large- z regime. the usual KZ scaling behavior. The second term is pro-portional to the square of noise amplitude and representsincreased defect-density production with the increase of τ Q (or is inversely proportional to the sweep speed). Thisis converse to that of the KZ scaling and is termed as theAKZ scaling behavior.In the next section we will consider linear protocol withtransverse noise and obtain analytical results for the de-fect density. We will show that for large- z values AKZscaling behavior dominates. In the subsequent sections,we will calculate the entropy and correlation functions ofa driven quantum XY-spin chain in the presence of noise. III. EFFECT OF NOISE ON THE DEFECTGENERATION
The time dynamics of states spanned by | (cid:105) and | k, − k (cid:105) is governed by the Hamiltonian H ηk ( t ), H ηk ( t ) = 2 (cid:20) h ( t ) + J cos( k ) J γ sin( k ) J γ sin( k ) − h ( t ) − J cos( k ) (cid:21) + 2 η ( t ) (cid:20) J sin( k ) J sin( k ) 0 (cid:21) , (11)where the first term in the above Hamiltonian is the usualdeterministic part. The second term couples the ho-mogeneous time-dependent Gaussian noise η ( t ) to eachof the k -mode within the restricted subspace. In thissubspace, a general state at any given time t for asingle realization of the noise η ( t ) can be written as, | Ψ ηk ( t ) (cid:105) = u ηk ( t ) | (cid:105) + v ηk ( t ) | k, − k (cid:105) , where u ηk ( t ) and v ηk ( t )are the time-dependent amplitudes. The system startsout in the perfect PM state defined by the initial condi-tions u ηk ( −∞ ) = 1 and v ηk ( −∞ ) = 0. The time evolutionof the general state | Ψ ηk ( t ) (cid:105) for a single noise realizationis governed by the stochastic Schr¨odinger equation, i ddt | Ψ ηk ( t ) (cid:105) = H ηk ( t ) | Ψ ηk ( t ) (cid:105) , (12)where the solution has to be averaged over all possiblerealizations (ensemble averaging due to the noise) of the2-level system corresponding to each k -mode .The projected Hamiltonian Eq. (11) is equivalent tothe noisy LZ problem with the noise present only in thetransverse part of the Hamiltonian. We consider the den-sity matrix, ˆ ρ ηk ( t ) = | Ψ ηk ( t ) (cid:105)(cid:104) Ψ ηk ( t ) | , and set up the timeevolution equation for the population inversion ρ ηk (dif-ference between the unexcited and the excited density fora given mode k ): ddτ ρ ηk ( τ ) = − (cid:90) τ −∞ d τ e i (cid:82) ττ dτ ( v LZ τ ) / ρ ηk ( τ ) − γ (cid:90) τ −∞ d τ e i (cid:82) ττ dτ ( v LZ τ ) / × η ( τ ) η ( τ ) ρ ηk ( τ ) + h.c. , (13)where τ = 2∆ k ( t + cos k/v ), v LZ = v/ ∆ k , and ∆ k =2 Jγ sin k . Taking the noise average one obtains ddτ ρ k ( τ ) = − (cid:90) τ −∞ d τ cos[ v LZ τ − τ )] ρ k ( τ ) − γ (cid:90) τ −∞ d τ cos[ v LZ τ − τ )] η ( τ ) η ( τ ) ρ k ( τ ) , (14)where ρ k ( τ ) is obtained by performing noise averageover ρ ηk ( τ ). The fast noise criteria allows us to decouple η ( τ ) η ( τ ) ρ ηk ( τ ) into a separate product of the noise termsand the density matrix term, η ( τ ) η ( τ ) ρ k ( τ ) . The so-lution of the reduced master equation in the t → ∞ limit FIG. 3. p k Vs k in the presence of the fast noise: Interest-ingly, in the the large- z regime ( z (cid:29) z ) or slower sweepsthe fast noise begins to affect the system. New critical region(where the excitation probability becomes non-zero) opensup symmetrically around k = ± π/ is obtained by following the approach of Ref 47 and isgiven by ρ k = e − π η / (2 v LZ γ ) (2 e − π/ (2 v LZ − p k ( z ) = 12 (cid:104) e − πz η sin k (2 e − πz sin k − (cid:105) , (15)where η = η /γ . It is interesting to note that the exci-tation probability [Eq. (15)] which is non-zero around the k = 0 , ± π points in the absence of noise, also opens uparound k = ± π/ z -limit or the small sweep speedscenario. Although the fast sweep regime ( z (cid:28) z ) isunaffected by the fast noise (see Fig. 3).We next evaluate the defect density by integrating n ( z ) = (cid:82) π dk p k /π and obtain n ( z ) = 12 + e − π ( z +¯ z ) I [ π ( z + ¯ z )] − e − π ¯ z I [ π ¯ z ] , (16)where ¯ z = 2 zη . In the limit, η (cid:28)
1, the defect densityis approximated as n ( z ) ≈ √ π √ z + πη z, (17) FIG. 4. Effect of the fast noise on the defect density: Thedefect density has been plotted with respect to z for differ-ent noise strength, which is consistent with the AKZ picturefor large- z regime i.e., enhanced defect generation for slowersweeps beyond the optimal quench rate which depends on thenoise strength.FIG. 5. Anti-Kibbel Zurek scaling behavior of the defectdensity: The difference, δn = n ( z ) − n ( z ), scales linearlywith z for different noise strengths for η (cid:28) zη sim z ) regime. which gives the AKZ scaling behavior (see Figs. 4 and 5).From the above expression it can be deduced that thedefect density is minimized for a optimal quench rategiven by, z O ≈ √ π η − / ∝ η − / . (18)We note that the optimal quench time has a universaluniversal power-law dependence on the noise strength. IV. DECOHERENCE OF LOCALOBSERVABLES
In both the noiseless and noisy drive protocols, theXY-spin chain (with N spins) is prepared in a PM state at the initial time t in = − T . This initial state is a purestate i.e., the full system density matrix can be writ-ten as ρ ( t = − T ) = | N (cid:105)(cid:104) N | . Subsequently the systemis driven by the transverse magnetic field h ( t ) = t/τ Q through the quantum critical points ( h = ∓ J ) up to thefinal time t f = T . In the noiseless drive scenario the fulldensity matrix of the evolved N -spin chain remains in thepure state due to the unitary time evolution. However,for large system size ( N → ∞ ) and in the long time limiti.e., T → ∞ , the coherences of the density matrix de-velop highly fluctuating phases (dependent on k and T )which vanishes when integrated over k for all local ob-servables . This decohered density matrix correspondsto the nonequilibrium steady state (NESS) which is fun-damentally different from the decoherence process dueto any external or internal noise . In addition to theinternal decoherence the dephasing is further enhancedby the noise in the drive protocol which results in theexponential suppression of the fluctuating coherences.For η J (cid:28) ∆ k the noise has negligible effect on thesystem. The crossover region is around η J ∼ ∆ k atwhich the noise begins to play a role in the dynamicsof the system. In the limit η J (cid:29) ∆ k , the fast noise ef-fects the system the most specifically in the non-adiabaticregions (when the gap ∆ k →
0) around the quantumcritical points ( k = 0 , ± π ), in addition new critical re-gions around k = ± π/ ρ k and ¯ ρ k vanish. Therefore the noise averaged decohered densitymatrix, ¯ ρ D = ⊗ k> ¯ ρ D,k (with ¯ ρ D,k a diagonal 2 × | (cid:105) , | k, − k (cid:105) ) in the limit N → ∞ and T → ∞ , with respect to the final decohered state can bewritten as, ¯ ρ D,k = (cid:18) p k
00 1 − p k (cid:19) . (19) Similar to Ref. 38, one can define a 2-point correlatorin terms of the fermionic operators as, d ( x − x (cid:48) ) = (cid:104) c x c † x (cid:48) (cid:105) = 12 π (cid:90) π − π dk e − i k ( x − x (cid:48) ) p k . (20)This correlator is non-zero for x − x (cid:48) = 2 n , where n is aninteger. In the absence of noise the 2-point correlator is d n ( z ) = 12 π (cid:90) π − π dk e − i k n p k = e − π z I n ( π z ) , (21)where I n ( π z ) is the modified Bessel function of the firstkind. The large- n expansion of d n at fixed z yields d n ( z ) ≈ π (cid:90) π − π dk e − π z k / e i n k = e − n / πz √ π z . (22)Thus consistent with the KZ-picture the correlationlength of the density correlator is proportional to √ z andthe magnitude of the correlator is inversely proportionalto √ z .A similar calculation for the 2-point correlator in thepresence of noise, p k yields d n ( z ) = 12 π (cid:90) π − π dk e − i k n p k = 12 δ n, − e − π ¯ z I n [ π ¯ z ] + e − π ( z +¯ z ) I n [ π ( z + ¯ z )] . (23)In the limit of η (cid:28) π ¯ z < n ex-pansion ( n > (cid:112) πzη ln 1 /η ) the above expression isapproximated as, d n ( z ) ≈ (cid:112) π ( z + ¯ z ) e − n / [2 π ( z +¯ z )] . (24)Thus the correlation length in the presence of noise l n = (cid:112) π ( z + ¯ z ) remains proportional to √ z and in-creases with the strength of the noise. V. ENTROPY DENSITY IN THE FINALDECOHERED STATE
The decohered state has a finite entropy density whichis a clear indication that the final state is a mixed state.To quantify the amount of information lost in the deco-herence process (at the end of the drive protocol) we cal-culate the Von-Neumann entropy ( S = − N tr ρ D ln ρ D )in terms of the excitation probability as follow, S = − N π (cid:90) π − π dk [ p k ln p k + (1 − p k ) ln(1 − p k )] , (25)where p k is given in Eq [15]. The above integration isperformed by expanding both the log terms in terms of e − πzη sin k (2 e − πz sin k −
1) and integrating each of theterms individually. The final result of the entropy density(
S/N ) can be expressed in the following series form,
S/N = ln 2 − ∞ (cid:88) m =1 2 m (cid:88) r =0 m − r ( − r (cid:0) mr (cid:1) m (2 m − e − πzY rm I [ πzY rm ] , (26)where Y rm = (2 m − r + 4 mη ). In Fig. 6 we plot Eq. (26)for different noise strengths. One can observe from thefigure that the finite entropy density depends on thesweep speed and also that for each noise strength theirexists an optimal quench rate either side of which the en-tropy density increases. In particular for z > z O (where z O is the optimal quench rate) the entropy increases FIG. 6. Entropy density Vs. z plotted for different noisestrengths. The entropy density is consistent with the AKZpicture. The entropy density increases after the optimalquench time which is the signature of increased defect gen-eration due to the fast noise in slower sweep regime. For veryslow sweeps, the noise (with η (cid:28)
1) can randomize the sys-tem to the maximally mixed state (i.e.
S/N → ln 2). Apartfrom that, the entropy density maximizes (locally) at z = z which is the signature of the crossover behavior of the spincorrelation functions from monotonically decreasing behaviorfor z < z to the oscillatory behavior for z > z as discussedin Ref. 38. which is the signature of AKZ behavior of defect pro-duction. For η (cid:54) = 0, the entropy density asymptoticallyapproaches ln 2 for large- z value i.e., a fully mixed stateis formed or in other words, the system approaches anasymptotic infinite temperature steady state. However,for the fast sweep speeds ( z < z ), the driven system isnot affected by the noise. In the intermediate region thesystem has some finite non-zero entropy density whichsignifies a partially mixed state. VI. MAGNETIZATION AND SPIN-SPINCORRELATIONS
The expectation value of the spin-spin correlators withrespect to the decohered state is conveniently obtainedin terms of the pair products of Majorana fermion opera-tors. Consider the Majorana fermion operators A x = c † x + c x , B x = c † x − c x . (27)The pair product of spins σ αx σ αx + n and that of the Jordan-Wigner string variable τ x τ x + n (with τ x = Π x The average magnetization density (cid:104) σ x (cid:105) at the end ofthe quench protocol is given by, m z = (cid:104) A x B x (cid:105) = 1 − n ( z ) , (35)where n ( z ) is the noise dependent defect density[Eq. (16)] and in terms of which, m z = e − π ¯ z I [ π ¯ z ] − e − π ( z +¯ z ) I [ π ( z + ¯ z )] . (36)Thus the large- z limit of the magnetization density givenby, m z − ≈ − √ π √ z − πzη , (37)is consistent with the AKZ picture or in other words, itdecreases after the optimal quench time, Eq [18], whenthe defect production starts to increase due to the noise(see Fig. 7).Consider next the magnetization correlator in the z -direction, (cid:104) σ x σ x +2 n (cid:105) , given by, (cid:104) σ x σ x +2 n (cid:105) = (cid:104) σ x (cid:105) − (cid:18) (cid:90) π − π dk π e − ikn p k (cid:19) . (38) FIG. 7. Magnetization density as a function of z reduces afterthe optimal quench time/rate in agreement with the AKZpicture. For very large- z case the noise can randomize thesystem completely resulting in a zero magnetization density. The connected correlator C n ( z ) is obtained by subtract-ing the position independent part, (cid:104) σ x (cid:105) from (cid:104) σ x σ x +2 n (cid:105) and is given by C n ( z ) = − d n ( z ), where d n ( z ) is given bythe equation Eq. (23). Therefore, the magnetization cor-relation for large- n retains the KZ scaling relation withthe correlation length given by l noisy = (cid:112) π ( z + ¯ z ) / , (39)where we note that the magnetization correlation lengthincreases as compared to the noiseless scenario. It isinteresting to note that the presence of noise in theanisotropy decreases the magnetization density due toincreased defect production, the correlation length ofthe magnetization correlator, however, increases with thestrength of the noise. The amplitude of the magnetiza-tion correlator nevertheless decreases with the noise. B. Spin correlators: (cid:104) σ , x σ , x +2 n (cid:105) As shown in Eq. (31) the spin correlators can be ex-pressed as product of sublattice correlators. One can rep-resent the sublattice correlators at n -separation in termsof the determinants of the Toeplitz matrices: (cid:104)(cid:104) σ x σ x + n (cid:105)(cid:105) = D n [ g +1 ,z ] , (40) (cid:104)(cid:104) σ x σ x + n (cid:105)(cid:105) = D n [ g − ,z ] , (41) (cid:104)(cid:104) τ x τ x + n (cid:105)(cid:105) = D n [ g ,z ] , (42)where g m,z are the generating functions defined as, g m,z ( ξ ) = − ( − ξ ) m (1 − p k ) , (43)where ξ = e ik and D n [ g m,z ] are the correspondingToeplitz matrix determinants for different sublattice spincorrelators. Given the generating function g m,z the de-terminant D n [ g m,z ] are defined as , D n [ g m,z ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( m )0 f ( m ) − .... f ( m ) − ( n − f ( m )1 f ( m )0 .... f ( m ) − ( n − . . .... .. . .... .. . .... .f ( m ) n − f ( m ) n − .... f ( m )0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (44)where the elements of the determinant f ( m ) l is the l th cummulant of the generating function g m,z ( ξ ) = (cid:80) l f ( m ) l ξ l and are obtained by performing the followingcontour integration, f ( m ) l = (cid:73) C dξ πiξ ξ − l g m,z ( ξ ) , (45)where C is a unit circle contour with | ξ | = 1. The aboveintegral in terms of the k -variable acquires the form, f ( m ) l = (cid:90) π/ − π/ dkπ e − i kl g m,z ( e i k ) . (46)The integral is evaluated by taking the integral represen-tation of the modified Bessel function of the first kind, I ν ( z ) = 12 π (cid:90) π − π dθe z cos θ − iνθ (47)where ν is an integer and Re( z ) > 0. Using the aboveintegral identity yields f ( m ) l = e − π ¯ z π I l − m [ π ¯ z ] − e − π ( z +¯ z ) π I l − m [ π ( z + ¯ z )] , (48)with m = 0 , ± 1. In the large- z limit (and with zη (cid:54) = 0),the expression reduces to f ( m ) l = ( − m π (cid:34) − e − ( l − m ) / π ¯ z √ z + √ e − ( l − m ) / π ( z +¯ z ) √ z + ¯ z (cid:35) . (49) C. Spin Correlations at Large Separation In this section, we investigate the behavior of spin cor-relators at large separation in the presence of the fastnoise. In the asymptotic limit, following Szeg¨o’s limittheorem, the Toeplitz determinant acquires the form, D n [ g ] ≈ exp (cid:18) n (cid:90) π dθ π ln g ( e iθ ) (cid:19) , (50)here g ( ξ ) is the generating function which should havezero winding number and no singularities on the unit cir-cle for the results of the Szeg¨o limit theorem to hold. Fur-thermore the asymptotic behavior given by Eq. (50) can be modified when the generating function has non-zerowinding number/singularities on the unit circle .It is clear from Eq. (50) that the zeroes of the gen-erating function g m,z ( ξ ) play an important role in theanalyticity of the asymptotic behavior of the spin corre-lators i.e., when z < z the exponent (cid:82) π dθ π ln g m,z ( e iθ )in Eq. (50) is analytical, it has a singularity at z = z and it becomes ill-defined for z > z . The asymptoticbehavior of the spin correlators near z = z is betterunderstood by analyzing the effects of the zeroes of thegenerating function. In the following, we will summarizethe discussion of Cherng and Levitov (Ref. 38) whichhave important implications for our work. The zeroes ofthe generating function, g m,z ( ξ ) = − ( − ξ ) m λ − (1 − λ ξ )(1 − λ ξ − ) e h ( ξ ) , (51) ξ = λ p , λ − p (where p is an integer) are infinite in numberwith multiplicity 1 and are given by, λ p = exp (cid:20) − arccosh (cid:18) − z z − ipz (cid:19)(cid:21) , (52)where the branch of arccosh has been chosen with pos-itive real part so that | λ p | ≤ | λ p | > | λ p (cid:48) | for | p | < | p (cid:48) | . The closest zeroes to the unit circle are λ and λ − which satisfy, λ + λ − − z z . (53)Furthermore, the pair of zeroes, λ and λ − , lie on theunit circle of the complex plane and play a central role inthe behavior of spin correlator’s asymptotes near z = z .Isolating the effect of these roots, the generating functionEq. (51) can also be written as, g m,z ( ξ ) = − ( − ξ ) m λ − (1 − λ ξ )(1 − λ ξ − ) e h ( ξ ) , (54)where, e h ( ξ ) = e π z ( ξ + ξ − ) / π z √ e π z Π p (cid:54) =0 z (1 − λ p ξ )(1 − λ p ξ − )4 | p | λ p . (55)The function h ( ξ ) can be simply written as, h ( ξ ) = ln (cid:20) − e − π z (1 − z /z − x ) − z /z − x ) (cid:21) , (56)where x = ( ξ + ξ − ) / 2. In the noisy drive case, we noticethat the generating function, g m,z fn ( ξ ) = − ( − ξ ) m e − πz η (1 − x ) (1 − e − π z (1 − z /z − x ) ) , (57)(the subscript ’fn’ represents fast noise) has same set ofzeroes as for the noiseless drive given by Eq. (52). There-fore the effect of zeroes of the generating function on theanalyticity of the spin correlators with respect to z willremain the same as for the noiseless drive case. The dif-ference is that the generating function with noisy drive ismultiplied by an extra exponential factor, e − π z η (1 − x ) ,which depends on the noise. We are mainly interestedin the role of this extra term on the correlation lengthsof the sublattice and the full lattice spin correlators. Inthe following discussion we show analytically that thisextra term is responsible for the AKZ scaling behavior ofthe correlation lengths. The generating function Eq. (57)for the noisy drive can be written as (after isolating theeffect of the pair λ and λ − ), g m,z fn ( ξ ) = − ( − ξ ) m λ − (1 − λ ξ )(1 − λ ξ − ) e H ( ξ ) , (58)with, H ( ξ ) = ln (cid:18) − e − π z (1 − z /z − x ) − z /z − x )) (cid:19) − π z η (1 − x ) . (59)Both h ( ξ ) and H ( ξ ) can be expanded as a function of x = ( ξ + ξ − ) / h ( x ) = (cid:88) n ≥ h n x n , H ( x ) = (cid:88) n ≥ H n x n , (60)where we notice that H = h − π z η and H = h + 2 π z η , while for n (cid:54) = 0 , 1, all other coefficientssatisfy H n = h n . One can write the sublattice corre-lator asymptotes obtained from the generating functionsEq. (57) as , (cid:104)(cid:104) σ αn σ αx + n (cid:105)(cid:105) ≈ E ( − G ) n , α = 1 , (cid:104)(cid:104) τ n τ x + n (cid:105)(cid:105) ≈ E ( − G ) n λ n +10 E − λ n +10 E − λ − λ − , (61)where for the noisy drive we have,ln G = (cid:90) − dx √ − x H ( x ) = (cid:90) − dx √ − x h ( x ) − π ¯ z. (62)The sublattice correlators can also be written in the fol-lowing canonical form, (cid:104)(cid:104) σ αn σ αx + n (cid:105)(cid:105) ≈ A σ e − n/l σ cos πn (cid:104)(cid:104) τ n τ x + n (cid:105)(cid:105) ≈ A τ e − n/l τ cos ( ω τ n − φ τ ) , (63)and using Eq. (31), the physically relevant quantity, i.e.,the full lattice correlations are given by, (cid:104) σ αx σ αx +2 n (cid:105) ≈ A σ A τ e − n/l cos [( π − ω τ ) n + φ τ ] , α = 1 , . (64)Thus the corresponding correlation length is given by, l − = l − σ + l − τ . (65)By comparing Eqs. (61) and (63), we obtain the followingcorrelation length of the sub-lattice spin correlators l − σ = l − τ = − ln G = − (cid:90) − dx √ − x h ( x ) + π ¯ z. (66) FIG. 8. The sub-lattice inverse correlation lengths have beenplotted as a function of z for the different noise strengths,both the l − σ and l − τ show the anti-Kibbel Zurek scaling be-havior i.e., the sub-lattice correlation lengths l σ,τ decreasesafter the optimal quench rate which depends on the noisestrength. Furthermore, the l − τ shows non-analytical behav-ior at z = z and beyond z > z both have same values.Therefore, it is understood that the increased noise strengthof the fast noise further decreases l , making the spin correla-tors relatively short ranged in the presence of the fast noise.The l − = l − σ + l − τ also shows the non-analytical behavior at z = z as a result of the non-analytical behavior of l − τ , thisis the indication of the crossover of different behaviors of thespin correlators i.e. non-oscillatory monotonically decreasingbehavior for z < z regime and exponentially suppressed os-cillatory behavior of spin correlators for z > z regime withrespect to the spatial separation. The result of the above integration yields l − σ = l − τ ≈ ∞ (cid:88) m =0 Γ( m + 1 / Re Li m +3 / (2) π / Γ( m + 1)(2 π z ) m +1 / + π ¯ z, (67)where in the absence of noise and for slow sweep speedsthe correlation length or the domain size is proportionto √ z thus satisfying the KZ scenario . On the otherhand, for the fast noise and large z scenario the scaling0of the correlation lengths is consistent with the AKZ pic-ture suggesting that the domain size reduces, this resultquantifies how the fast noise randomizes the driven sys-tem spatially at the end of the protocol. The sub-latticeand full lattice correlation lengths have been plotted inthe Fig. [8] with respect to z for different noise strengths.They clearly show the AKZ behavior in the large- z case.i.e., beyond the optimal quench rate z O the inverse cor-relation length starts to increase. For small- z values thefast noise has negligible effect. VII. CONCLUDING REMARKS In this work the effects of the fast Gaussian noise inthe driven quantum XY-spin chain have been quantifiedanalytically. We have considered transverse protocol inwhich the external transverse magnetic field drives thesystem linearly with respect to time, starting from theparamagnetic phase to a region with the ferromagneticphase and finally back to the paramagnetic phase in thepresence of the time dependent Gaussian noise in theanisotropy term. In this protocol, the system passesthrough two quantum critical points at h = ∓ J wherethe energy gap vanishes resulting in non-adiabatic ef-fects. We map the problem to the noisy LZ problemand in terms of the density matrix formalism we obtaina reduced master equation for the population inversion.The solution of the equation in the T → ∞ limit hasbeen utilized to obtain the final excitation probability.The implications for the correlators due to the non-equilibrium dynamics of the noisy transverse drive pro-tocol are as follows: first, the fast fluctuating coherencesvanishes in the large time limit (even without the noise)due to the internal decoherence arising from the large system size ( N → ∞ ) and this allows the course grain-ing in momentum space and transforms the pure stateinto an entropic state with finite non-zero entropy. Thetime dependent Gaussian fast noise further exponentiallysuppresses the highly fluctuating coherences and in par-ticular affects the system most when the system passesthrough the quantum critical points. Finally, the noisecan heat up the population to the asymptotic infinitetemperature state for the slower sweeps or large- z case,maximizing the entropy density to log 2 and at the sametime minimizing the average magnetization density tozero. The effects of the noise are minimized when drivenat an optimal sweep rate which turns out to scale univer-sally with the strength of the fast noise.We have analyzed the spin-spin correlation functions(in the presence of noise) at the end of the protocol forlarge separation using the Toeplitz determinant asymp-totes at large- n . For slow sweep speeds the effect ofthe fast gaussian noise on the correlation lengths of thespin correlators at large-separation reveals behavior con-sistent with the Anti-Kibbel-Zurek picture. The sub-lattice correlation lengths for σ , and τ -spin correla-tors decreases with the strength of the noise accordingto the Anti-Kibbel-Zurek scaling behavior. 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