Dynamics of dimer and z spin component fluctuations in spin-1 / 2 XY chain

j(t)Bj+n(0)i for the local spin operators {Am, Bm} = � s z , Dmwhere Dm = s x s x+1 + s y s y+1 is the dimer operator. The results for the dynamic transverse structure factor Szz(�, !) and for the dynamic dimer structure factor SDD(�, !) are known, whereas the analysis of the dynamic struc- ture factor SzD(�, !) = (SDz(�, !)) ? has not been reported so far. We

One-dimensional quantum spin-1/2 XY models admit the rigorous analysis not only of their static properties (i.e. the thermodynamic quantities and the equal-time spin correlation functions) but also of their dynamic properties (i.e. the different-time spin correlation functions, the dynamic susceptibilities, the dynamic structure factors).This becomes possible after exploiting the Jordan-Wigner transformation which reduces the spin model to a model of spinless noninteracting fermions.A number of dynamic quantities (e.g.related to transverse spin operator or dimer operator fluctuations) are entirely determined by two-fermion excitations and can be examined in much detail.We consider the spin-1/2 XY chain in a transverse ( z) magnetic field with the Hamiltonian is the dimer operator.The results for the dynamic transverse structure factor S zz (κ, ω) and for the dynamic dimer structure factor S DD (κ, ω) are known, whereas the analysis of the dynamic structure factor S zD (κ, ω) = (S Dz (κ, ω)) has not been reported so far.We compare different two-fermion dynamic quantities contrasting their generic and specific features.
hand, quite often the relevant quantities can be examined rigorously, especially if the space dimension is equal to one.This is important even if the models in question are simplified since conventional approximations usually fail after being applied to low-dimensional quantum spin systems.On the other hand, material science provides a number of magnetic materials which can be modelled using the spin-1/2 XXZ Heisenberg chains.Therefore, to interpret the experimental data obtained in neutron scattering experiments or resonance experiments for such compounds one needs a corresponding theory of the dynamic properties.A particular case of the spin-1/2 XXZ Heisenberg chain, the XY chain, owing to the Jordan-Wigner fermionization trick can be investigated analytically, thus shedding light on the physical effects that may be observed in less tractable cases.
In what follows, we consider the one-dimensional spin-1/2 XY model in a transverse ( z) external field [1] defined by the Hamiltonian Here J is the exchange interaction constant (we will set further J = −1), Ω is the external magnetic field, s α = 1/2•σ α , σ α with α = x, y, z are the Pauli matrices, and N → ∞ is the number of sites.We imposed periodic boundary conditions in (1).
After exploiting the Jordan-Wigner transformation the considered spin model can be presented in terms of noninteracting spinless fermions with the Hamiltonian Here periodic boundary conditions are implied (the so-called boundary term is not important for the dynamic quantities examined below) and κ is the quasi-momentum which takes N values in the region from −π to π. Important information about the behavior of the system under small perturbations follows from the dynamic susceptibilities [2] Here A m , B m are the local operators attached to the site m (for example, the spin operators s x m , s y m , s z m , the dimer operator D m = s x m s x m+1 + s y m s y m+1 or the trimer operator T m = s x m s x m+2 +s y m s y m+2 [3][4][5][6][7][8]).Another quantities which reflect the dynamic properties of the system are the dynamic structure factors S AB (κ, ω) is connected to the imaginary part of χ AB (κ, ω) through the fluctuationdissipation theorem.The imaginary and real parts of χ AB (κ, ω) are connected by the Kramers-Kronig transformation.In what follows we focus on the dynamic structure factors (4).
Introducing the fermionic representation (2) and exploiting the Wick-Bloch-de Dominicis theorem we easily calculate the two-spin correlation functions entering equations ( 3), ( 4) where n κ = (1 + exp (βΛ κ )) −1 is the Fermi function and The associated dynamic structure factors are obtained from equations ( 5)-( 7) by Fourier transform.The resulting expressions for N → ∞ can be brought into the form where −π κ π are the solutions of the equation We can combine formulas ( 8)- (11) rewriting them in the form (2) with The dynamic transverse structure factor S zz (κ, ω) was obtained by Th.Niemeijer (see [9]), the dynamic dimer structure factor S DD (κ, ω) was examined in [3][4][5].To the best of our knowledge, the dynamic structure factor S zD (κ, ω) = (S Dz (κ, ω)) has not been discussed so far.
In figures 1-3 we plot the dynamic structure factor S zD (κ, ω) at different temperatures.These plots demonstrate not only the specific features of the dynamic structure factor S zD (κ, ω) but also the generic features of all two-fermion dynamic structure factors (13).From equation (13) it is clearly seen that the dynamic structure factors ( 8), ( 9), ( 10), (11) are governed entirely by the two-fermion (particle-hole) excitation continuum the properties of which were examined in [9,10].Hereinafter we briefly account for these results.At zero temperature β → ∞ the two-fermion excitation continuum exists only if |Ω| < |J| and has the following lower, middle and upper boundaries in the plane wave-vector κ -frequency ω respectively; here we have introduced the parameter α = arccos(Ω/|J|) which varies from π when Ω = −|J| to 0 when Ω = |J|.The region ω l ω ω m corresponds to the lower two-fermion excitation continuum, whereas the region ω m ω ω u corresponds to the upper two-fermion excitation continuum.The ω-profiles at fixed κ of the two-fermion dynamic structure factors (13) may exhibit Van Hove's singularities (as the density of one-particle states in one dimension) when ω → ω s − 0, Figure 4 shows expressions (18)-( 21) plotted for |J| = 1 and various Ω.As temperature increases the lower boundary of the two-fermion excitation continuum smears out, i.e. ω l = 0, and the upper boundary becomes ω u = 2|J| sin(|κ|/2).The specific features of different dynamic structure factors are connected with the explicit form of the function C (2) AB (κ, κ ) ( 14)-( 17).Thus, the dynamic dimer structure factor vanishes (and hence does not diverge) along ω s (21) since C (2) DD (κ, κ ) (17) cancels the zero in the denominator in equation ( 13).This is contrary to the case of the dynamic transverse structure factor with C (2) zz (κ, κ ) (14).Moreover, the zerotemperature dynamic structure factor S zD (κ, ω) is zero in the upper two-fermion excitation continuum, i.e. for ω m ω ω u .For this region of the κ-ω plane the two roots κ 1 and κ 2 of equation ( 12) yield in equation (13) two contributions of the same value but with opposite signs.At nonzero temperatures the values of these contributions become different and S zD (κ, ω) deviates from zero in the upper twofermion excitation continuum.In the lower two-fermion excitation continuum one of the roots does not contribute at zero temperature due to the Fermi functions in equation (13).However, when temperature tends to infinity β → 0 the restrictions owing to the Fermi functions in equation (13) disappear and two roots κ 1 and κ 2 come into play for all 0 ω ω u .They again have the same value but opposite signs and as a result, S zD (κ, ω) disappears within the whole two-fermion excitation continuum in the κ-ω plane.This behavior is illustrated in figures 1-3.
Thus, all the considered dynamic quantities exhibit the same generic properties controlled by the δ-functions and the Fermi functions in equation (13) (lower, middle and upper boundaries, soft modes, Van Hove's singularities) and some specific properties controlled by the function C (2) AB (κ, κ ) (zero values in the accessible region of the κ-ω plane, disappearance of Van Hove's singularities).Our results may be important from the theoretical point of view helping to understand the dynamic properties of quantum spin chains.Thus, equation (13) gives a hint to the form of multi-fermion excitation continua contributions to dynamic quantities (see [7,8]).On the other hand, we note that spin-1/2 XY chains are realized in some quasi-onedimensional magnetic insulators [11,12], and hence our findings may have a relation to the experimental data obtained in dynamic experiments.
dynamic structure factorsS AB (κ, ω) = n exp (−iκn) ∞ −∞ dt exp (iωt) A j (t)B j+n (0) for the local spin operators {A m , B m } = s z m , D m where D m = s x m s x m+1 + s y m s y m+1

Figure 2 .
Figure 2. The same as in figure 1 at β Note the different vertical scales in figure 1 and figure 2.

Figure 3 .
Figure 3.The same as in figure 1 at β = 1.Note the different vertical scales in figure 1 and figure 3.