Propagation of local excitations through strongly correlated quantum chains
aa r X i v : . [ qu a n t - ph ] S e p Propagation of local excitations throughstrongly correlated quantum chains
Jean Richert a ∗ andTarek Khalil b,c † a Institut de Physique, Universit´e de Strasbourg,3, rue de l’Universit´e, 67084 Strasbourg Cedex,France b Department of Physics, School of Arts and Sciences,Lebanese International University, Beirut, Lebanon c Department of Physics, Faculty of Sciences(V),Lebanese University, Nabatieh, LebanonFebruary 17, 2017
Abstract
The propagation of an external transverse magnetic signal actinglocally on a 1 d chain of spins generates a disturbance which runsthrough the system. This quantum effect can be interpreted as aclassical traveling wave which contains a superposition of a large setof frequencies depending on the size of the chain. Its local amplitudefixes the size of the z-component of the spins at any location in thechain. The average and maximum value of the group velocity aredetermined and compared with the transmission velocity fixed by theLieb-Robinson upper bound inequality. ∗ E-mail address: [email protected] † E-mail address: [email protected]
Modern communication needs the development of technologies which allowfast, faithful and robust information manipulation by means of adapted trans-fer devices. These needs generate intense research in different fields of con-densed matter physics in order to develop efficient material tools on whichinformation can be safely propagated.Indeed, the last decade saw the development of new several techniques andmethods in the field of quantum information technology such as ultra-fast andflexible optical manipulations of spins in magnetically ordered materials [1,2, 3], transfer of quantum states and qubits along linear systems and differenttechniques [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], superconducting devices [15, 16],transfer of entangled states [17, 18, 19, 20, 21], investigations concerningechoes and decay times [22], transition times [23, 24], propagation velocitiesin solid state systems [26, 27, 28, 29, 30, 31, 32, 33]. The subject raisedalso more fundamental points concerning the properties related to the non-equilibrium behaviour of such systems [34, 35, 36].The present work uses a simple model, a finite XY chain submitted toeven boundary conditions in its ground state. It works here as the materialsupport for the propagation of external perturbations along the ring chain.The perturbations are chosen to be local excitations induced on a singlespin over a short interval of time and the propagation of the signal in timeis followed over the whole chain. The interest here lies essentially on theestimate of the propagation velocity of a signal through the chain. The per-turbation behaves like a traveling wave characterized by a set of frequencieswhich depend on the size and eigenenergies of a strongly correlated system.This quantity can be obtained in an analytic form and group velocities aredefined. Their order of magnitude are obtained by means of numerical ap-plications and compared with the upper velocity obtained in the applicationof the Lieb-Robinson (LR) approach to the transmission of information indiscrete quantum systems. 2ection 2 introduces the model and the time evolution of the z-componentof local spins along the chain. In section 3 different group velocities aredefined and the expression of the LR bound velocity is recalled. Numer-ical applications are presented in section 4 and the results concerning thepropagation are discussed. Section 5 is concerned with some considerationsconcerning the control of the local spin amplitudes during the propagationprocess and the implementation of sequences of local excitations in the sys-tem. Conclusions are drawn in section 6. The Hamiltonian of the finite 1 d quantum chain with N sites and asymmetryparameter γ reads H = J/ γ ) X ( i ) σ xi σ xi +1 + J/ − γ ) X ( i ) σ yi σ yi +1 − h X ( i ) σ zi (1)In the sequel even periodic boundary conditions are introduced. The inter-action strength is chosen to be J = 1 and ~ = 1. In Fourier space each site[ k = 1 , ..., N/
2] contains four single spin states. The energy spectrum can bedetermined analytically in terms of single particle energies [40, 41]. Diago-nalization of the single particle (4 ∗
4) Hamiltonian leads to the stationaryorthonormalized eigenstates | ψ k i = α k | i k + α k a + k a + − k | i k | ψ k i = β k | i k + β k a + k a + − k | i k | ψ k i = a + k | i k | ψ k i = a + − k | i k (2)where | i k is the spin vacuum, a + k creates a spin at site k , α k and β k arecomplex amplitudes which are shown in Appendix A.The corresponding eigenvalues read3 (1) k = cos φ k − [(cos φ k − h ) + γ sin φ k ] / ǫ (2) k = cos φ k + [(cos φ k − h ) + γ sin φ k ] / ǫ (3) k = cos φ k = ǫ (4) k (3)where φ k = 2 πk/N .The N th spin of the system experiences an external magnetic field h which induces a perturbation H ( N )1 ( t ) = h exp( − t/τ H ) S zN (4)with S zN = σ zN /
2. The external field acts on the z component of the spinat site N over a finite time interval fixed by τ H and starting at t = 0. Thegenerated signal is transmitted to the whole chain of connected interactingspins. The localized action of an external magnetic field may be an experi-mental challenge. There exists however experimental work which shows thatthis might be possible in the not too far future [37, 38, 39]. The z component of each spin site n evolves in time as S zn ( t ) S zn ( t ) = exp( i Z t H ( t ′ ) dt ′ ) S zn (0) exp( − i Z t H ( t ′ ) dt ′ ) (5)where H ( t ) = H + H ( N )1 ( t ).The physical situation considered here corresponds to an excitation in-duced by H ( N )1 ( t ) which relaxes over a time τ H . This time is chosen assmall compared to the time unit. In this regime a perturbative treatment ofthe expectation value of S zn ( t ) makes sense. The perturbation is essentiallygoverned by the parameter τ H and h τ H which can be small in practical ap-plications if τ H is a very short relaxation time even for sizable magnetic fields h . Using a perturbative treatment up to second order in the perturbationdevelopment provides an analytic expression of the time evolution of thespins along the chain. The ground state | Ψ i of the system is the product ofthe single particle states. A second order perturbation expansion gives the4xpectation value h S zn ( t ) i of the z-component of the spin with respect to theground state delivers three contributions h S zn ( t ) i = h S zn i (0) + h S zn ( t ) i (1) + h S zn ( t ) i (2) (6)where h S zn i (0) = 2 /N N/ X k =1 | α k | − / h S zn ( t ) i (1) = 1 / N/ X k,l =1 v ( k,l )2 [ A kl [sin( ψ + n ; kl ) + sin( ψ − n ; kl )]+ B kl [cos( ψ + n ; kl ) + cos( ψ − n ; kl )]] + C ( t, τ H ) (8)where v ( k,l )2 are transition matrix elements between different single spin states( k, l ), A kl and B kl amplitudes, C ( t, τ H ) a time-dependent and state indepen-dent contribution which relaxes to a constant over a time interval t ≥ τ H .The phases read ψ + n ; kl = ω ( kl ) t + nK ( k, l ) ψ − n ; kl = ω ( kl ) t − nK ( k, l ) (9)with ω ( kl ) = ǫ (3) k + ǫ (4) l − ǫ (1) k − ǫ (1) l and K ( k, l ) = ( φ k − φ l ). The expressionsof the matrix elements v ( k,l )2 and the amplitudes A kl and B kl are shown inAppendix B.The expression of the second order contribution h S zn ( t ) i (2) is given by alarge number of terms. Present applications show that their contributionsare sizably smaller than those of the first order when t ≫ τ H . In AppendixC we estimate their order of magnitude. The time-dependent contribution of h S zn ( t ) i (1) given by equation (8) can beinterpreted as a sum of classical traveling waves characterized by frequencies5 ( kl ) and phases ( φ k − φ l ). Considering the analytic expressions of the ω ( kl ) and the energies ǫ ( i ) k given above one sees that these quantities are incom-mensurate. Hence the oscillations will not be periodic in time. The groupvelocity for a fixed wave number K ( k, l ) characterizes the propagation of thewaves. This quantity is defined as v gr = dω ( K ) /dK (10)For a fixed value of K ( k, l ) more than one value of ω ( k, l ) is possible since K ( k, l ) depends on the difference k − l . In the sequel we consider ω max ( K )which correponds to maximum values of ω s for a given set of couples ( k, l )corresponding to a fixed K and the averages < ω > over all possible couples( k, l ) generating the same K .The group velocity is a measure of the speed at which the external per-turbation propagates through the medium, here in the case of a spin chain.By means of Eq.(10) it is possible to define and obtain an order of magnitudeof this quantity and an upper value of the propagation of a signal in this typeof quantum devices. Explicit calculations are presented and discussed below.On the other hand the Lieb-Robinson procedure [26] allows the determi-nation of an upper limit of the velocity v with which a signal is transmittedin a discrete system from an initial point at a fixed initial time to a finalpoint at a fixed later time [27, 28, 29]. This velocity depends on metricfactors which define distances between sites of the system and the norm k H k of the Hamiltonian matrix. The limit velocity is obtained by means of avariational procedure depending on a real positive parameter a . Workingout the expression and determining the upper bound by varying a leads tothe expression v LR = eN k H k / e is the exponential constant, N the number of sites of the system.Some details of the derivation are given in Appendix D. In section 4 . v LR and compare it with v gr . The time evolution of a fixed spin and the behaviour of the spins in the chainare worked out for a spin chain of finite length. The physical conditions under6 n < S z ( t = ) > Figure 1: z-components of the spins n for t=1.which the perturbative treatment discussed above is valid are realized in thepresent applications. The zeroth order contribution to the z-component ofthe spin h S zn i (0) is time-independent and the second order term h S zn i (2) issmaller than h S zn i (1) by two orders of magnitude. Due to the symmetriesinduced by the boundary conditions imposed on the chain the oscillationfrequencies ω ( k, l ) are symmetric with respect to k and l belonging to theintervals [1 , N/ , [ − , − N/ , N/ N = 100and a set of parameters ( h = 0 . , h = 1 . , γ = 0 . h = h c = − . τ H = 10 − . The time evolution of h S zn ( t ) i (12) = h S zn ( t ) i (1) + h S zn ( t ) i (2) (12)for fixed time and a set of sites n = 1 to 50 is shown in Figs. 1 and 2.For short times the essential part of the excitation resides in the sites withlow n which are close to N = 100 due to the periodic boundary conditions.For large times the longitudinal part of all the spins is affected and showsa somewhat irregular and non repetitive pattern due to the incommensura-bility of the values of ω and K ( k, l ) discussed above. The calculations havebeen repeated for an excitation h > h = h c = − / n -0.03-0.02-0.0100.010.02 < S z ( t = ) > Figure 2: z-components of the spins n for t=200.to a crossing of the critical point when t goes to infinity. In both cases nospecific behaviour of the spin component is observed. Individual spins whichare affected by local excitations are qualitatively insensitive to the globaleffects induced phase transitions which affect the whole system.Figs. 3 and 4 show the behaviour of h S zn =1 ( t ) i (12) and h S zn =40 ( t ) i (12) overan interval of time t = [1 , Figs. (5) and (6) show the values of the maximal and average group velocities v maxgr ( K ) and < v gr ( K ) > for fixed K = [1 , N/ − K ( k, l ) = | k − l | . Theparameters of the model are those given in section 4 .
1. One observes sizablevariations of these quantities over the range of K values which shows a largedispersive behaviour of the system due to the fact that the frequencies show astrongly non-linear dependence on K ( k, l ). This quantity is is itself a mixtureof many single particle energies. The absolute value of the maxima are 1.5 for v maxgr ( K ) and 0.55 for < v gr ( K ) > which correspond to a maximum transittime between two sites of 0.66 time units for v maxgr ( K ) and 1.8 for < v gr ( K ) > .These quantities are independent of the amplitude of h , the velocities dependon the characteristics of the system only.Using the expression of the Lieb-Robinson velocity for a signal which8volves over the whole system one finds v LR = 1 . ∗ ( N = 100 and k H k = 75) in the same physical units as those used for v gr . It appears thatthere is a large quantitative difference between the two velocities. One canmake the following comments.The group velocity v gr generated by an external disturbance is a quantitywhich is local in K space and depends on single particle energy differences,see section 2.2. On the other hand v LR is the maximum speed with which asignal emitted at some time at some place is transmitted through the systemand observed at some other place at a later time. It depends explicitly on theextensive properties of the system, the maximum of its total energy ( k H k )and its size ( N ). The transmission between two neighbouring sites ( N = 2)one would lead to v LR = 4 . v LR depends on the geometric stucture of the systemonly through its total energy ( k H k ) which although different should be of thesame order of magnitude for open and closed systems. This effect is certainlyweaker than the difference between v gr in a closed and an open system dueto the wave nature of the propagation. Consider the transmission of the amplitude of the z-component h S zN ( t ) i (1) located at site N at time t to some other site in the chain. This may berealized in the following way.At a given initial time t and fixed h and τ H one applies an excitation atsite N which generates a perturbation h S zN ( t ) i (1) . At time t + ∆ t , h S zN − ( t +∆ t, h ) i (1) has a well defined value. If one applies a further excitation h ′ atsite N − t + ∆ t such that h S zN ( t ) i (1) = h S zN − ( t + ∆ t, h ) i (1) + h S zN − ( t + ∆ t, h ′ ) i (1) (13)the amplitude of the spin component at site N − t + ∆ t will be thesame as the amplitude at site N at time t . The field h ′ needed is easily fixedsince any z-component of a spin h S z (1) n ( t + ∆ t, h ) i depends linearly on the9 t -0.2-0.100.10.2 < S z ( n = ) > Figure 3: Time dependence of the z-component h S zn ( t ) i (12) of the spin at siten=1.field h under the physical conditions described above. The operation can beiterated over a finite interval of sites of a given length.The fidelity with which these operations can be realized as presented hereare of course subject to limitations. Indeed the excitation must be such thathigher than first order corrections are effectively negligible. Furthermore thephysical realization of a transmission device of this type is also subject to theprecision with which the operations described above can be realized. Thisconcerns essentially the mechanical and thermal fluctuations of the externalfield applied at site N . In practical applications the excitation to be transmitted from site N to somesite m through the system might be repeated a finite number of times startingfrom an initial time t = 0. Consider the case where a set of n excitations aresent from site N over regular intervals of time t ≫ τ H . For times t > nt there are two contributions to h S zm ( t ) i (1) , h S zm ( t ) i (1) = h S zm ( t ) i (1 a ) + h S zm ( t ) i (1 b ) which read 10 t -0.00500.0050.01 < S z ( n = ) > Figure 4: Time dependence of the z-component of the spin h S zn ( t ) i (12) at siten=40. h S z ) m ( t ) i (1 a ) = n X p =0 X k,l D ( k, l ) [ ω ( kl ) (1 − cos( ω ( kl ) ( t − pt )) exp( − ( t − pt ))) − τ H sin( ω ( kl ) ( t − pt )) exp( − ( t − pt ))] (14)where D ( k, l ) = [1 /τ H + ω kl ) ] and h S zm ( t ) i (1 b ) = 12 X k,l v ( k,l )2 S n ( k, l )[ A kl [(sin( ϕ + n,m ; kl ) + sin( ϕ − n,m ; kl )] ++ C kl [(cos( ϕ + n,m ; kl ) + cos( ϕ − n,m ; kl )]] (15)with S n ( k, l ) = sin[ ω ( kl ) ( n + 1) / t ]sin[ ω ( kl ) / t ] (16)and 11 + n,m ; kl = ω ( kl ) ( t − nt /
2) + mK ( k, l ) ϕ − n,m ; kl = ω ( kl ) ( t − nt / − mK ( k, l ) (17)The observable h S zm ( t ) i (1) is known at each time t and can in principlebe renormalized to a fixed value as discussed above in the case of a singleexcitation. The propagation of a local perturbation generated on a quantum spin chainat temperature T = 0 has been investigated. The perturbation generated byan external magnetic field starts at an initial time t = 0 at some local spinsite of a 1 d chain and affects the transverse component of the other spins.The conditions imposed on the relaxation time of the external field allow aperturbative treatment.The action of the external field induces time-dependent oscillations ofthe transverse components along the spin sites of the chain which behaveslike a ring due to the imposed boundary conditions. The oscillations of thespin components show the complex behaviour of classical traveling waveswith a large number of harmonic modes due to the spectral properties of thephysical system on which the perturbation propagates. The case where finitesequences of perturbations are locally emitted at fixed time intervals of equallength has been worked out.In order to characterize the propagation of the perturbation differentgroup velocities have been introduced. These velocities show a strong dis-persive behaviour far from sharp wave packets. They reflect the complexnature of the strongly coupled spins in the chain. It comes out that thesevelocities are sizably smaller than the Lieb-Robinson propagation velocitybound defined in [28, 29]. Reasons for this have been presented and dis-cussed in section 4.The chain undergoes a phase transition for a specific value h c of theinternal magnetic field which acts on the transverse components of the spins.The propagating wave is not qualitatively influenced by this property. Itis due to the fact that the propagation concerns individual spins and theexternal field does not affect the global behaviour of the system. This pointis analysed in section 4. 12 K -2-1.5-1-0.50 V g r m a x Figure 5: Maximum group velocity as a function of K(k,l) for a chain withN=100 and the parameters given in the text.The way in which the value of the transverse component of a spin atone site can be retrieved at another site has been investigated under idealnoiseless conditions. A more realistic approach should include the presenceof noise which can be induced by initial thermal effects and fluctuations ofthe external magnetic field acting on the system. It would also be of interestto consider the case of a chain coupled to a heat bath at temperature T = 0at time t = 0 which would introduce the action of excited states of the chain.The present study concerns systems which are influenced by a field whichacts over a small amount of time and cannot produce a reversal of the spins.The extension of the present investigation to chains excited by strong fieldsable to affect sensibly or even to reverse the transverse spin direction wouldrequest a non-perturbative treatment. Such a case needs further develop-ments.Finally the choice of even boundary conditions on the end sites of thechain closes it in practice. The behaviour of the propagation in an openchain may lead to a different behaviour of the perturbation. In the prospectof practical applications it would be particularly interesting to investigatethis point.Thanks are due to J.-Y. Fortin for a careful reading of the manuscript. Appendix A: explicit expressions of the sin-gle particle amplitudes K -0.200.20.40.6 < V g r > Figure 6: Average group velocity as a function of K(k,l) for a chain withN=100 and the parameters given in the text.The energy spectrum of the 1 d XY chain of length N reduces to a productof single particle energies with 4 states for a given momentum ( k = [1 , N/ k can be analytically diago-nalized leading to the eigenvalues given by the expressions shown in Eq.(3).The amplitudes of the eigenvectors corresponding to the states 1 and 2 aregiven by α k = ( ǫ (1) k − (2 cos( φ k − h )) /N (1) α k = ( − iδ p / /N (1) β k = ( iδ p / /N (2) β k = ( ǫ (2) k − h ) /N (2) (18)with δ p = − γ sin( φ k ) and the normalization factors N (1) = [1 / δ p + ( ǫ (1) k − (2 cos φ k − h ) ] / N (2) = [1 / δ p + ( ǫ (2) k − h ) ] / (19) Appendix B: matrix elements and amplitudesentering h S z (1) j ( t ) i v ( k,l )2 of the first order contributions to h S z (1) J ( t ) i are given by v ( k,k )2 = h Ψ ( k ) | ˆ V | Ψ i ( k ) i v ( k,l )2 = h Ψ ( k ) | ˆ V | Ψ ij ( kl ) i (20)where | Ψ ( k ) i , | Ψ i ( k ) i , | Ψ ij ( kl ) i are the single state eigenfunctions at sites( k, l ) in Fourier space in the ground state | > and excited states | i > and | j > , ˆ V the time-independent part of the diagonal and non-diagonal elementsof the perturbation interaction ˆ V ( t ) = h exp( − t/τ H ) c + N c N expressed in thesecond quantization formalism in ordinary space [40].The amplitudes A kl and B kl depend on the structure of the chain and theapplied magnetic field at site N . They read: A kl = 1 /τ H /τ H + (∆ ( i,j )( k,l ) ǫ ) B kl = − ∆ ( i,j )( k,l ) ǫ /τ H + (∆ ( i,j )( k,l ) ǫ ) (21)where ∆ ( i,j )( k,l ) ǫ = ǫ ( i ) k + ǫ ( j ) l − ǫ (1) k − ǫ (1) l , i and j corresponding to excited states. Appendix C: second order contributions tothe tranverse component of spin n
The order of magnitude of the matrix elements given by the coefficientslinked to the time-dependent part of the different contributions of secondorder are given by the strength factors S tr S = h τ H S = h (1 /D ) S = h / ( τ H D ) D ∼ (1 /τ H + ∆ ǫ ) (22)where ∆ ǫ ∼ ǫ ( i ) − ǫ ( j ) , ( i, j ) corresponding to single particle ground or excitedstates. The single particle state energies are of the order of unity. In thenumerical applications h is chosen to be of the same order of magnitude and15 H is four order of magnitude smaller. As a consequence the second ordercontributions to h S zj i are two to three orders of magnitude smaller than thefirst order ones. This justifies their neglect in the numerical estimates. Appendix D: estimate of the Lieb-Robinsonbound for the chain.
The expression of the velocity reads v a = inf a C a k Φ a k /a (23)Here a is a real positive number, C a depends on the distances between discretesites ( x, y, z ) of the system and k Φ a k = k H k /D a ( x, y ) (24)where k H k is the norm of the Hamiltonian matrix and D a ( x, y ) a measureof the distance between sites x and y multiplied by an exponential factor exp ( − a | x − y | ) [27, 28].Working out this expression leads to the expression v a = inf a N/ aN )Φ( a = 1) /aN (25)which shows an extremum for a = 1 /N . This extremum is a minimum. Thefinal result is given by v LR in section 3. References [1] M. Ban, S. Kitajima, F. Shibata, Phys. Rev.
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