Spin-1/2 XX chain in a transverse field with regularly alternating g -factors: Static and dynamic properties
Taras Krokhmalskii, Taras Verkholyak, Ostap Baran, Vadim Ohanyan, Oleg Derzhko
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Spin-1/2 XX chain in a transverse field with regularly alternating g -factors:Static and dynamic properties Taras Krokhmalskii,
Taras Verkholyak, Ostap Baran, Vadim Ohanyan,
3, 4, 5 and Oleg Derzhko
1, 6 Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Svientsitskii Str. 1, 79011 L’viv, Ukraine Department for Theoretical Physics, Ivan Franko National University of L’viv, Drahomanov Str. 12, 79005 L’viv, Ukraine Laboratory of Theoretical Physics, Yerevan State University,Alex Manoogian Str. 1, 0025 Yerevan, Armenia Joint Laboratory of Theoretical Physics – ICTP Affiliated Centre in Armenia CANDLE Synchrotron Research Institute, Acharyan Str. 31, 0040 Yerevan, Armenia Department of Metal Physics, Ivan Franko National University of L’viv,Kyrylo & Mephodiy Str. 8, 79005 L’viv, Ukraine (Dated: November 5, 2020)We consider the spin-1/2 isotropic XY chain in an external magnetic field directed along z axiswith periodically varying g -factors. To reveal the effects of regularly alternating g -factors, we calcu-late various static and dynamic equilibrium quantities in the ground state and at finite temperatures.We demonstrate that because of the regularly alternating g -factors the saturation field may disap-pear and the field dependence of the susceptibility in the ground state has additional logarithmicsingularity at zero field. Moreover, the zero-field susceptibility has a logarithmic singularity as T → . Furthermore, the dynamic structure factors exhibit much more structure in the “wave vec-tor – frequency” plane that can be traced out to modifications of the two-fermion excitation continuawhich exclusively determine S zz ( κ, ω ) and dominate the properties of S xx ( κ, ω ) . We discuss whatchanges can be observed in dynamic experiments on the corresponding substances. PACS numbers: 71.10.-w, 75.10.Lp, 75.10.JmKeywords: spin-1/2 XY chain, nonuniform g -factors, dynamic properties I. INTRODUCTION
The magnetic moment of an electron is related to itsangular momentum by the g -factor. The magnetic mo-ment of a free electron is associated with its spin an-gular moment only and the magnitude of the electron g -factor (or more precisely the electron spin g -factor) is ≈ .
002 319 [1]. In atoms, both orbital angular momen-tum and spin angular momentum of electron contributeto the magnetic moment of an atomic electron and thespin g -factor has to be replaced by the Landé g -factor.Furthermore, in crystalline solids, the Landé g -factor (orin what follows simply g -factor) may be, in principle, sitedependent.From the solid-state-physics side, one can mention anumber of spin-chain compounds with regularly alter-nating g -factor values [2–10]. Thus, one-dimensionalcopper-iridium oxide Sr CuIrO which contains both 3 d (Cu ) and 5 d (Ir ) magnetic ions can be well de-scribed by an effective spin-1/2 ferromagnetic Heisen-berg model with an Ising-like exchange anisotropy ( ∆ ≈ . ) [2, 3]. Moreover, the Cu sites carry the Cu spin s = 1 / with g -factor ≈ and the Ir sites carry theIr isospin s = 1 / with g -factor ≈ − [3, 4]. An-other instance is a one-dimensional molecular magnet[{Co II (∆) Co II (Λ) }(ox) (phen) ] n [5]. Magnetic proper-ties of this compound can be explained using a one-dimensional Ising-chain model with two different ex-change couplings and two different g -factors, . and . .Next example of single-chain molecular magnet is a coor-dination polymer compound [ { ( CuL ) Dy }{ Mo ( CN ) } ] · CN · H O, in which L − is N,N-propylenebis(3-methoxysalicylideneiminato). The magnetic unit cellin this compound contains four magnetic ions withthree different values of the g -factors. The presenceof highly anisotropic Dy ion makes possible an ex-act solution for the corresponding spin-chain model [6].One more example is the spin-1/2 chain antiferromag-net CuCl · ) SO) [7]. There are results of veryrecent studies of another heterotrimetallic coordination-polymer single-chain magnet with large difference be-tween the g -factors of the magnetic ions in the mag-netic unit cell, [ Cu II Mn II ( L )][ Fe III ( bpb )( CN ) ] · ClO · H O [8]. In this system, a staggered g -tensor and/orDzyaloshinskii-Moriya interactions lead to a staggeredfield along x direction upon application of a uniformfield along z direction. As a result, a spin-1/2 anti-ferromagnetic Heisenberg chain with an alternating g -factor emerges (see also Ref. [9] discussing the quasi-one-dimensional spin-1/2 antiferromagnet Cu benzoate).Finally, one may also mention a two-sublattice one-dimensional system Ni (EDTA)(H O) · O, the mag-netic behavior of which was discussed in terms of a spin-1 g − g antiferromagnetic Heisenberg (or Ising) chain with g /g about . [10].From the theoretical side, since the g -factor entersmany standard lattice models of crystalline solids, it isquite natural to address a question about the conse-quences of a regular non-uniformity of the g -factor forthe observable magnetic properties. There are severalexact calculations for the spin-chain systems aimed onexploring the essential effects of nonuniform g -factors.Spin-1/2 XY chains provide an excellent playground forsuch analysis because they correspond to noninteractingfermions [11, 12]. Prior work, which is closely related toour study, concerns the two-sublattice [13, 14] and theinhomogeneous periodic (i.e., with several sites in a cellwhich periodically repeats) [15] spin-1/2 XX chain in a z -aligned field with various interaction constant and g -factor values. The reported results refer to the magneti-zation, susceptibility and equal-time two-spin zz correla-tion functions [13, 14], as well as to some dynamic quan-tities related to correlations of the average cell operators[15]. The continued-fraction method was also used tofigure out the magneto-thermal properties of the generalinhomogeneous isotropic XX chain including the caseof random Lorentzian transverse field [16]. The sameprogram has been performed also for the quantum Isingchain [17]. In the most recent papers, the detailed anal-ysis of the ground-state properties for general boundaryconditions for the quantum Ising chain with the period-2modulated transverse field have been done [18]. Free-fermion models in which the period-2 alternation of thenearest-neighbor interactions is accompanied by multi-ple spin exchange were considered in Refs. [19–21]. XX chains is the extreme limit of the Heisenberg chains withan XY -like exchange anisotropy. The opposite limitingcase is the Ising chains. Recently, a spin-1/2 Ising chainwith period-2 regularly alternating g -factors has beenstudied in context of unusual properties of Sr CuIrO [3, 4]. Moreover, this material, as was mentioned above,features not only alternating g -factors of magnetic ionalong the chain, but also the negative sign of the one ofthem. Negative g -factors (for the pseudospin operators)are interesting by themselves as they are the result ofstrong interplay between the ligand field and spin-orbitinteraction [22–24]. Very recently it has been shown thateven in the simplest case of ferromagnetic Ising modelwith g -factors of different sign on bipartite lattice, thefrustration takes place and there are configurations con-taining ordered and disordered sublattices at the sametime [4, 25]. Rigorous results for finite quantum spinclusters and an Ising-Heisenberg chain with different g -factors have been obtained recently in Ref. [26].In the present paper we report results of the systematicstudy of the spin-1/2 XX chain in a transverse field withregularly alternating g -factors including the case when g -factors have different sings. We pay special attentionto manifestation of regularly alternating g -factors in thetransverse magnetization, the static zz susceptibility, aswell as in the two dynamic structure factors S zz ( κ, ω ) and S xx ( κ, ω ) . S yy ( κ, ω ) behaves identically to S xx ( κ, ω ) due to the symmetry of the model. Dynamic quantitiesare accessible experimentally and therefore understand-ing of the effects generated by nonuniform g -factors maybe useful for interpreting experimental data. The recentdevelopment of the exact and numerical calculations ofthe spin dynamic structure factors for the integrable one-dimensional quantum spin systems are really impressive[27]. However, the examined in what follows spin-chain model, although corresponds to noninteracting fermions,may be of interest for the full Heisenberg exchange inter-action case too: Since the seminal papers by G. Mülleret al. [28] we know that many dynamic features of thespin-1/2 Heisenberg chain can be analyzed starting fromthe free-fermion limit.It might be worth it to list here the main findings ofthe present paper.• We have performed the detailed study of the dy-namic properties. We calculated the dynamicstructure factors S zz ( κ, ω ) and S xx ( κ, ω ) and in-spected how they change in the external magneticfield for different period-2 alternations of g -factors.• In the case when both g -factors are of the samesign, the correspondence between the boundariesof the zz and xx structure factors is still present.• On the contrary, if g g ≤ , a large enough mag-netic field leads to the highly intense modes in the xx structure factor.• Analyzing the absorption intensity I α ( ω, h ) , wefound that in the Voigt configuration ( α = z ),the model with uniform g -factors does not haveany response. In the case when g differs from g , we obtain the nonzero contribution to the ab-sorption intensity. For sufficiently large frequencies ω > | J | (where J denotes the exchange coupling)the van Hove singularity arises at the magnetic field h = √ ω − J / | g − g | .• In the Faraday configuration ( α = x ), the situationis a bit different. The absorption spectra can beobserved in the uniform case. It shows a broadmaximum at some resonance field. The alternationof g -factor leads to the doubling of this resonanceline.• Although in our study we focus on the exactly solv-able XX chain, we know that such analysis of dy-namics is useful for understanding a more realisticcase of the Heisenberg chains. Many qualitativefeatures (e.g., doubling of the resonance line) ofthe absorption profiles can be found also in caseof Heisenberg of XXZ -model with alternating g -factors.The rest of the paper is organized as follows. We be-gin with introducing the model to be studied and thefree-fermion representation of the model which emergesafter applying the Jordan-Wigner transformation, Sec. II.After that we discuss the magnetization and the suscep-tibility in the ground state (Sec. III) and some finite-temperature quantities (Sec. IV). In Sec. V we examinethe dynamic structure factors of the model. We reportthe results for S zz ( κ, ω ) obtained mainly analytically andfor S xx ( κ, ω ) obtained mainly numerically. We concludethe paper with a summary, Sec. VI. II. THE MODEL AND ITS FREE-FERMIONREPRESENTATION
In the present study, we consider the spin-1/2 isotropic XY chain in a transverse (i.e., aligned along z axis) mag-netic field. The peculiarity of the model is the regularlyalternating g -factor which acquires periodically two val-ues, g and g . The Hamiltonian of the model reads H = N X l =1 (cid:2) J (cid:0) s x l − s x l + s y l − s y l + s x l s x l +1 + s y l s y l +1 (cid:1) − g µ B H s z l − − g µ B H s z l (cid:3) . (2.1)Here J is the exchange interaction (we may put | J | = 1 without loss of generality), µ B is the Bohr magneton, H is the value of the magnetic field measured, e.g., inTeslas (then with µ B ≈ . K/T the field h = µ B H ismeasured in Kelvins), and g µ B H = g h , g µ B H = g h .Furthermore, N is the number of lattice sites which isassumed to be even, and periodic boundary conditionsare imposed for convenience. After introducing g ± = g ± g , (2.2)we can rewrite Eq. (2.1) in a more compact form H = N X l =1 (cid:2) J (cid:0) s xl s xl +1 + s yl s yl +1 (cid:1) − h l s zl (cid:3) ,h l = [ g + − ( − l g − ] h. (2.3)This is the Hamiltonian of the spin-1/2 isotropic XY chain in a regularly alternating (with period 2) transversemagnetic field.The defined model is exactly solvable by making useof the famous Jordan-Wigner fermionization [11, 12] (seealso Refs. [29, 30]). In terms of the Jordan-Wignerfermions the spin Hamiltonian (2.3) becomes H = N X l =1 (cid:20) J (cid:16) c † l c l +1 + c † l +1 c l (cid:17) − h l (cid:18) c † l c l − (cid:19)(cid:21) . (2.4)Again periodic boundary conditions are implied inEq. (2.4) [31]. After the Fourier transformation c l = 1 √ N X κ e − iκl c κ ,κ = 2 πjN , j = − N , − N , . . . , N − , (2.5)Eq. (2.4) can be cast into H = X − π ≤ κ<π (cid:2) ( J cos κ − g + h ) c † κ c κ + g − hc † κ c κ ± π (cid:3) + g + h N. (2.6) Next, we perform the Bogolyubov transformation, c κ = u κ α κ − v κ α κ + π , (2.7) c κ + π = v κ α κ + u κ α κ + π ( − π/ ≤ κ < π/ ,u κ = 1 √ vuut | J cos κ | q J cos κ + g − h ,v κ = sgn( g − hJ cos κ ) √ vuut − | J cos κ | q J cos κ + g − h , leading to H = X − π ≤ κ<π Λ κ (cid:18) α † κ α κ − (cid:19) , (2.8) Λ κ = − g + h + sgn( J cos κ ) q J cos κ + g − h . Hence, we have arrived at the free-fermion representation(2.8) of the initial spin model (2.1). Within this repre-sentation many calculations for the thermodynamicallylarge system can be performed rigorously analytically orwith very high accuracy numerically. From Eq. (2.8) itis immediately evident that nonzero magnetic field de-velops a gap in the excitation spectrum splitting it intotwo branches. In the limiting case of large g -factors (orfield h ) the system becomes close to the two-level modelwith only two possible eigenenergies on each site − g h and − g h . The position of the Fermi level is importantfor the understanding of the ground state and thermody-namics of the model given in the next section.Although the isotropic XY interactions may occur insome spin-1/2 chain compounds (see, e.g., Ref. [34]), theycan be viewed as a limiting case of more common XXZ interactions. Consider the spin-1/2
XXZ chain in a z -directed magnetic field. The Hamiltonian of such modelcontains in addition to the one given in Eqs. (2.1) or(2.2) the interaction of the z components of neighboringspins with the strength J ∆ , where ∆ is the anisotropyparameter. As a result, in terms of the Jordan-Wignerfermions the spin Hamiltonian becomes H = N X l =1 (cid:20) J (cid:16) c † l c l +1 + c † l +1 c l (cid:17) + J ∆ c † l c l c † l +1 c l +1 − ( h l + J ∆) c † l c l + h l J ∆4 (cid:21) . (2.9)One way to proceed is to apply a mean-field like approx-imation for the four-fermion term [35, 36]: c † l c l c † l +1 c l +1 → (cid:18)
12 + m (cid:19) (cid:16) c † l c l + c † l +1 c l +1 (cid:17) − (cid:18)
12 + m (cid:19) − t (cid:16) c † l c l +1 + c † l +1 c l (cid:17) + t − sc † l c † l +1 − s ∗ c l c l +1 + | s | , (2.10)where the parameters m ≡ h c † l c l i − / , t ≡ h c † l c l +1 i , and s ≡ h c l c l +1 i have to be determined self-consistently. Itshould be noted that the Jordan-Wigner fermionizationapproach was successfully used for examining the staticand dynamic properties away from the free-fermion point[37–41]. III. ZERO-TEMPERATURE PROPERTIES
Let us first present the ground-state ( T = 0 ) prop-erties of the system. Although some particular resultshave been already obtained in Refs. [13–15], we pro-vide here the ground-state analysis for consistency. Par-ticularly, we focus on calculating the ground-state en-ergy e = h H i /N , the transverse magnetization m = − ∂e /∂h , the sublattice average z -component of spin, h s z i = − ∂e /∂ ( g h ) , h s z i = − ∂e /∂ ( g h ) , and thestatic zz susceptibility χ zz = ∂m/∂h . For the model athand, one has to differ the magnetization and the av-erage of the z -component of the spin operator, i.e., themagnetic moment and the angular moment at site. It isobvious that m = 12 ( g h s z i + g h s z i ) . (3.1)In what follows we distinguish two cases: g g > and g g < .The case g g > . There are two values of the Fermimomenta κ F defined as the solutions of the equation Λ κ = 0 : κ F = ± κ , if 0 < Jg + h < | Jg + | h s , (3.2) κ F = ± ( π − κ ) , if − | Jg + | h s < Jg + h < ,κ = arccos | h/h s | (0 < κ < π/ , where the saturation field h s is given by h s = | J | / √ g g > . It is worth to note that the saturationfield exists if the fully polarized state | ↑ . . . ↑i , whichis obviously the eigenstate of the Hamiltonian (2.1), be-comes the ground state as the field h exceeds a certainfinite value. This is the case for g g > but not for g g < . Here we may consider two separate ranges ofthe magnetic field h . The first one, when | h | > h s , cor-responds to the saturated phase with all spins aligned inthe field direction. There is no solution for κ F and, thus,the ground state energy as well as the averages of spinshave simple expressions: e = − | g + h | , m = sgn( h ) g + , (3.3) h s z i = h s z i = sgn( h )2 , χ zz = 0 . More interesting is the second range, − h s < h < h s , when e = −| g + h | (cid:18) − κ π (cid:19) − π q J + g − h E( κ , κ ) , (3.4) m = g + sgn( h ) (cid:18) − κ π (cid:19) + g − hπ q J + g − h F( κ , κ ) , h s z i = sgn( h ) (cid:18) − κ π (cid:19) + g − hπ q J + g − h F( κ , κ ) , h s z i = sgn( h ) (cid:18) − κ π (cid:19) − g − hπ q J + g − h F( κ , κ ) ,χ zz = g + κ π p h s − h + g − π q J + g − h (F( κ , κ ) − E( κ , κ )) . Here κ = | J | / q J + g − h and we have also introducedthe elliptic integrals of the first and second kind given bythe following standard expressions [42]: F( κ , κ ) = Z κ d θ p − κ sin θ , (3.5) K( κ ) = F (cid:16) π , κ (cid:17) , E( κ , κ ) = Z κ d θ p − κ sin θ, E( κ ) = E (cid:16) π , κ (cid:17) . As can be seen from the reported formulas, the sus-ceptibility diverges at h = ± h s showing the square-rootsingularity χ zz ≈ g − g − πg + p h s − h , h → | h s | . (3.6)If g = g an additional weak divergence of χ zz occurs at h = 0 : χ zz ≈ g + πh s + g − π (cid:18) ln 2 h s h − (cid:19) , | h | → . (3.7)It was noticed for the first time apparently in Ref. [13].The case g g < . In this case the equation for theFermi momenta Λ κ = 0 does not have real solutions,which means that the Fermi level lays in the forbiddenband between two branches of the spectrum. Since theodd and even spins are directed oppositely in a field,there is also no saturation field, i.e., the magnetizationnever attains its saturation value corresponding to h s z i = −h s z i = ± / . The ground-state energy is given by thefollowing formula: e = − π q J + g − h E( κ ) . (3.8)After straightforward differentiation we get m = g − hπ q J + g − h K( κ ) , (3.9) h s z i = −h s z i = g − hπ q J + g − h K( κ ) ,χ zz = g − π q J + g − h (K( κ ) − E( κ )) for the magnetization, the sublattice average z -component of spin, and the susceptibility, respectively.These formulas can be simplified in the strong-field andweak-field limits. We obtain m ≈ g − h q J + g − h , (3.10) χ zz ≈ g − J (cid:0) J + g − h (cid:1) , as | h | → ∞ and m ≈ g − hπ q J + g − h ln 4 q J + g − h | g − h | , (3.11) χ zz ≈ g − π q J + g − h ln 4 q J + g − h | g − h | − , as | h | → . While Eq. (3.10) demonstrates explicitlythat the saturation is never achieved for any finite h ,Eq. (3.11) demonstrates a non-analyticity of the ground-state energy which manifests itself as a logarithmic pe-culiarity of the magnetization and the susceptibility invanishing field.In Fig. 1 we show the ground-state magnetization andsusceptibility. In all numerical investigations, withoutloss of generality, we assume first that g = g = 1 andthen g starts to decrease. These plots illustrate the re-ported above analytical results including the asymptoticbehavior of the susceptibility. It is worthwhile to stressthat the logarithmic singularity of the susceptibility χ zz can be detected not only in the case g g < , when it isquite natural to expect it, but also in the opposite case g g > , see Eq. (3.7). It is the consequence of anotherpeculiar property shown in Fig. 2 where the total magne-tization and spin moment is confronted with the averagespin moments of each sublattices. We can see that evenfor positive g (see Fig. 2 for g = 0 . ) the average spinmoment at small fields started to evolve in the oppo-site to the field direction feeling the competition betweenthe applied magnetic field and quantum interaction withstronger magnetized neighboring spins.Let us denote by h ( h > ) the value of the fieldat which h s z i = 0 if | g | < | g | (or h s z i = 0 if | g | < -0.4-0.2 0 0.2 0.4 -2 -1 0 1 2 3 4 m h g =1 g =0.5 g =0 g =-0.5 g =-1 0 0.2 0.4 0.6 0.8 1 1.2 -2 -1 0 1 2 3 4 χ zz h g =1 g =0.5 g =0 g =-0.5 g =-1 FIG. 1: (Color online) Ground-state magnetization (upperpanel) and susceptibility (lower panel) vs field h . | J | = 1 , g = 1 , g = 1 (solid), g = 0 . (long-dashed), g = 0 (short-dashed), g = − . (dashed-dotted), g = − (dotted). | g | ); h exists in the case g g > only. After usingapproximate formulas for the elliptic integrals one canshow that h ≈ h s e − α , where α = √ g g / | g − g | . If g (or g ) approaches zero we can again use approximateformulas for the elliptic integrals to conclude that h ≈ h s / √ . Both limiting cases can be combined into thefollowing approximate expression h ≈ e − α √ − e − α h s , (3.12)which yields the correct value of h for the whole region g g > with the accuracy of less than 1.5%. IV. FINITE-TEMPERATURE PROPERTIES
Finite-temperature quantities can be easily calculatedfrom the free energy per site f ( T, h ) = − T π Z π − π d κ ln (cid:18) κ T (cid:19) (4.1)with Λ κ given in Eq. (2.8). For example, for the specificheat one finds c ( T, h ) = 12 π Z π − π d κ (cid:18) Λ κ T (cid:19) cosh − Λ κ T . (4.2) -0.4-0.2 0 0.2 0.4 -3 -2 -1 0 1 2 3 h FIG. 2: (Color online) Ground-state values of h s z i (dotted), h s z i (dashed), ( h s z i + h s z i ) / (dot-dashed), and m (solid) vsfield h . | J | = 1 , g = 1 , g = 0 . . Furthermore, for the finite-temperature magnetizationand susceptibility one finds m ( T, h ) = 14 π Z π − π d κ ∂ Λ κ ∂h tanh Λ κ T (4.3)and χ zz ( T, h ) = (4.4) π Z π − π d κ " ∂ Λ κ ∂h tanh Λ κ T + 12 T (cid:18) ∂ Λ κ ∂h (cid:19) cosh − Λ κ T , respectively. Here, the derivatives ∂ Λ κ /∂h and ∂ Λ κ /∂h are given by the following formulas: ∂ Λ κ ∂h = − g + + sgn( J cos κ ) g − h q J cos κ + g − h , (4.5) ∂ Λ κ ∂h = sgn( J cos κ ) g − J cos κ (cid:0) J cos κ + g − h (cid:1) / . In Fig. 3 we demonstrate the temperature behavior ofthe specific heat (4.2) for several regimes: 1) gapless zero-field and finite-field regimes ( < | h | < h s ) (solid blackand dashed brown), 2) two cases when | h | = h s or g = 0 (dashed-dotted blue), and 3) two gapped regimes when | h | > h s , g g > or when g g < at h = 0 (dottedgreen).The gapless regime features the universal linear-temperature dependence of the specific heat: c ( T ) ≃ π c v F T, T → . (4.6)Here, in our case the central charge c = 1 and the Fermivelocity for the case of zero field coincides with the thosefor the XX -chain, v F = | J | , whereas for the case of thegapless finite-field regime ( < | h | < h s , g g > ) itis v F = J p − h /h s / ( h s | g + | ) . When the magneticfield reaches the saturation value | h | = h s ( g g > ) C T FIG. 3: (Color online) Temperature dependence of the spe-cific heat for | J | = 1 at h = 0 (solid black); h = 0 . , g =1 , g = 0 . (dashed brown); h = 0 . , g = 1 , g = 0 (dashed-dotted blue); and h = 0 . , g = 1 , g = − . (dotted green).The inset shows the same plots in log − log scale. The linear,square-root and exponential behavior of the specific heat areclearly visible here. Thin red lines represent the asymptoticforms from Eqs. (4.6), (4.7), and (4.8). the Fermi level touches the bottom points of the upperpart of the spectrum (van Hove singularity). The low-temperature behavior of the specific heat in this case isgiven by the square-root temperature dependence, c ( T ) ≃ (cid:0) √ − (cid:1) ζ (cid:0) (cid:1) p | g + h | √ π | J | √ T , (4.7)where ζ ( x ) is the standard zeta-function. The same ex-pression is valid for the case g = 0 for arbitrary nonzerovalues of the magnetic field. Finally, two gapped regimesare possible: i) | h | > h s , g g > and ii) g g < atany h = 0 . The specific heat has universal exponentiallow-temperature behavior, given by c ( T ) ≃ ∆ √ πr e − ∆ T T , (4.8)where for the | h | > h s regime r = | J | κ / , ∆ = | g + h | − q J + g − h , whereas for the g g < regime r = J / (2 | g − h | ) , ∆ = | g h | ( ∆ = | g h | ) if | g | < | g | ( | g | > | g | ).Let us also consider the low-temperature behavior ofthe magnetic susceptibility at zero field. We have theuniversal formula with logarithmic singularity given by χ zz ( T ) ≃ π | J | (cid:20) g − g − (cid:18) ln πT | J | − C (cid:19)(cid:21) , (4.9)where C ≃ .
577 215 6 is the Euler-Mascheroni constant.As it is seen from this expression, the logarithmic diver-gence at T → is the consequence of the non-uniformityof the g -factors and it disappears when g − = 0 . This isillustrated in Fig. 4. χ zz T FIG. 4: (Color online) Low-temperature behavior of the zero-field susceptibility for | J | = 1 , g = 1 and g = 1 (solid black), g = 0 . (dashed brown), g = 0 (dashed-dotted blue), and g = − . (dotted green). The inset shows the same plots in log − log scale. Thin red lines represent the asymptotic formfrom Eq. (4.9). V. DYNAMIC PROPERTIES
In this section, we study dynamic quantities of themodel. Dynamic properties of quantum spin-chain com-pounds are observable in the neutron scattering [43] andelectron spin resonance (ESR) [44] experiments.We start with the dynamic structure factor related tothe inelastic neutron scattering cross section [43, 45]: S αα ( κ, ω ) = (5.1) N N X j =1 N X n =1 exp (i κn ) ∞ Z −∞ d t exp (i ωt ) g j g j + n h s αj ( t ) s αj + n i c , where h s αj ( t ) s αj + n i c = h s αj ( t ) s αj + n i − h s αj ih s αj + n i and s αj ( t ) = exp( iHt ) s αj exp( − iHt ) . The inclusion of the g -factors in Eq. (5.1) here implies that we have the dynamicstructure factors of the magnetic moments. In general, g -factors may also depend on the probing field direction α . But if we imply that the ratio between g and g ispreserved for any direction α , Eq. (5.1) will acquire ascaling factor. In the case of site-independent g -factorsEq. (5.1) coincides with the definition of Refs. [46–48].For the chain with site-dependent g -factors with periodtwo the dynamic structure factor has the following gen-eral structure: S αα ( κ, ω ) = g S αα ( κ, ω ) + g − S αα ( κ + π, ω ) (5.2) − g − g + (cid:16) S αα ( κ, ω ) + S αα ( κ + π, ω ) (cid:17) , where the uniform spin structure factor S αα ( κ, ω ) andthe staggered spin structure factor S αα ( κ, ω ) are defined in the standard way: S αα ( κ, ω ) = (5.3) N N X j =1 N X n =1 exp (i κn ) ∞ Z −∞ d t exp (i ωt ) h s αj ( t ) s αj + n i c ,S αα ( κ, ω ) =1 N N X j =1 N X n =1 exp (i κn ) ∞ Z −∞ d t exp (i ωt ) ( − j h s αj ( t ) s αj + n i c . Furthermore, we consider S zz ( κ, ω ) and S xx ( κ, ω ) structure factors separately. In the former case one facesa problem of two-fermion excitations only and all cal-culations can be performed analytically. The latter casecorresponds to many-fermion excitations problem and re-quires, in general, the calculation of Pfaffians. We per-form these calculations numerically [30, 46–48] carefullycontrolling the accuracy of computations. As in previousstudies on the dynamics of spin-1/2 XY chains, bothstructure factors exhibit some similarities. In what fol-lows, we discuss the changes in these quantities causedby regular alternation of g -factors.The dynamic structure factors allow us to calculate theenergy absorption intensities I α ( ω, h ) , α = z, x observedin the ESR experiments. Following the procedure givenin Appendix A of Ref. [49], we can get for the linearlypolarized electromagnetic wave: I α ( ω, h ) ∝ ωχ ′′ αα (0 , ω ) , (5.4) χ ′′ αα (0 , ω ) = 1 − exp( − βω )2 S αα (0 , ω ) , where χ ′′ αα (0 , ω ) is the imaginary part of the αα dy-namic susceptibility and S αα (0 , ω ) is the correspondingdynamic structure factor at κ = 0 defined in Eq. (5.1). Inthe ESR experiment two configurations are distinguished[44]: i) the Voigt configuration, when the magnetic polar-ization of the electromagnetic wave is collinear with theconstant field, and ii) the Faraday configuration, whenthe magnetic polarization of the electromagnetic wave isperpendicular to the constant field. In our model, the z [ x ] polarized electromagnetic wave corresponds to theVoigt [Faraday] configuration, i.e., the absorption inten-sity is I z ( ω, h ) [ I x ( ω, h ) ]. Again, as discussed in whatfollows, the regularly alternating g -factors change dra-matically the ESR absorption intensity. A. zz dynamics One can work out the closed-form expression for thedynamic structure factor S zz ( κ, ω ) . It is given by thefollowing expression: S zz ( κ, ω )= π Z − π d κ B + ( κ ; κ ) C ( κ ; κ ) δ ( ω − D ( κ ; κ ))+ π Z − π d κ B − ( κ ; κ ) C ( κ + π ; κ ) δ ( ω − D ( κ + π ; κ )) ,B ± ( κ ; κ ) = [ g ± ( u κ u κ + κ ± v κ v κ + κ ) ∓ g ∓ ( u κ v κ + κ ± v κ u κ + κ )] ,C ( κ ; κ ) = n κ (1 − n κ + κ ) ,D ( κ ; κ ) = Λ κ + κ − Λ κ , (5.5)where n κ = 1 / (cid:0) e Λ κ /T + 1 (cid:1) is the Fermi-Dirac functionfor the spinless fermions (2.8). Hence, S zz ( κ, ω ) is gov-erned exclusively by two-fermion excitation continua.Let us discuss this two-fermion quantity in more detail.For fixed κ and ω , one has to solve the equations ω − D ( κ ; κ r ) = 0 , ω − D ( κ + π ; κ ′ r ) = 0 , (5.6)i.e., to find all roots κ r , κ ′ r . Then Eq. (5.5) can be writtendown as follows: S zz ( κ, ω ) = X κ r B + ( κ ; κ r ) C ( κ ; κ r ) A ( κ ; κ r )+ X κ ′ r B − ( κ ; κ ′ r ) C ( κ + π ; κ ′ r ) A ( κ + π ; κ ′ r ) , (5.7)where A ( κ ; κ ) = (cid:12)(cid:12)(cid:12)(cid:12) ∂D ( κ ; κ ) ∂κ (cid:12)(cid:12)(cid:12)(cid:12) (5.8) = J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | cos( κ + κ ) | sin( κ + κ ) q J cos ( κ + κ )+ g − h − | cos κ | sin κ q J cos κ + g − h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . In Fig. 5(a) we show (for a representative set of pa-rameters) the regions in the κ – ω plane where equa-tions (5.6) have four roots (black), two roots (gray) orno roots (white). In other words, we plot S zz ( κ, ω ) (5.7) assuming A ( κ ; κ ) = A ( κ + π ; κ ) = 1 as well as B ± ( κ ; κ ) = 1 and C ( κ ; κ ) = C ( κ + π ; κ ) = 1 . Clearly,the dynamic structure factor S zz ( κ, ω ) is identically zerowithin the white regions in the κ – ω plane [equations (5.6)have no roots]. Furthermore, any two-fermion quantityhave some structure coming from the factors /A ( κ ; κ ) and /A ( κ + π ; κ ) . It is nicely seen in the infinite-temperature limit when C ( κ ; κ ) = C ( κ + π ; κ ) = 1 / shown in Fig. 5(b). Next, deviating from the infinite-temperature limit we have to examine the effect of theFermi-Dirac functions in Eq. (5.7) which may suppressthe dynamic structure factor S zz ( κ, ω ) even in the grayor black regions, especially at T = 0 . In Fig. 5(c) we showthe effect of the ground state Fermi-Dirac distributionsfor the same set of parameters [we plot S zz ( κ, ω ) (5.7) as-suming A ( κ ; κ ) = A ( κ + π ; κ ) = 1 and B ± ( κ ; κ ) = 1 ]. ω κ ω κ ω κ ω κ FIG. 5: (Color online) Towards the dynamic structure factor S zz ( κ, ω ) . | J | = 1 , g = 1 , g = 0 . , h = 0 . . (a) Number ofroots of two equations (5.6). (b) S zz ( κ, ω ) at T = ∞ . (c) Thesame as in panel (a) but taking into accounting the Fermi-Dirac functions at T = 0 . (d) S zz ( κ, ω ) at T = 0 . Green andred lines are the boundaries (A1) and (A2) correspondingly. In addition to the two- and four-roots regions, the regionswith one and three roots, surviving after the thermody-namic averaging, come into play [compare Figs. 5(c) and5(a)]. Moreover, some allowed previously regions becomewhite at T = 0 signalizing the action of the Fermi-Diracfunctions in the ground state. The final gray-scale plot
0 1 2 3 0 0.5 1 1.5 2 0 0.5 1 1.5 2 S zz ( κ , ω )(a) κ ω
0 1 2 3 0 0.5 1 1.5 2 0 0.5 1 1.5 2 S zz ( κ , ω )(b) κ ω FIG. 6: (Color online) S zz ( κ, ω ) vs ω at κ = 0 , κ = π/ , κ = π/ , κ = 3 π/ , and κ = π . | J | = 1 , g = 1 , g = 0 . , h = 0 . , T = 0 (left), cf. Fig. 5(d), and T → ∞ (right), cf.Fig. 5(b). Green and red lines are the boundaries (A1) and(A2) correspondingly. of the zz dynamic structure factor (5.7) at T = 0 is pre-sented in Fig. 5(d). The frequency profiles for the cho-sen set of parameters are also plotted in Fig. 6 comple-menting the gray-scale plot in Figs. 5(b,d). It is clearlyseen that the zz dynamic structure factor at T → ∞ shows the van Hove divergence at the edges of the two-fermion continua which is typical for the XX chains (seeRefs. [30, 47, 48] for a review). S zz ( κ, ω ) in the groundstate [Fig. 5(d)] demonstrates even richer behavior dueto the step-like form of the Fermi-Dirac functions [seeFig. 5(c,d) and Fig. 6(a)]. The analytical formulas forthe boundaries of the two-fermion continua are given inAppendix.We can understand the reported findings taking intoaccount that the dynamic structure factor S zz ( κ, ω ) isgoverned by two-fermion continua. The general effect ofalternating g -factors can be understood from Figs. 7–9,where some results for S zz ( κ, ω ) for different fields h andvalues of g at T = 0 are collected. The decreasing of g from 1 to − at fixed value of magnetic field h and g = 1 leads to redistribution of the intensity of the zz dynamicstructure factor from the boundary to the center of theBrillouin zone. For g ∈ (0 , , there are two regions with S zz ( κ, ω ) = 0 (top and bottom) which are disconnected,see Figs. 7(b), 8(b) and 9(b). The distances betweenthese top and bottom regions increase with decreasing g and with increasing h . For g ∈ [ − , , the increas-ing of the magnetic field h leads to redistribution of theintensity of the zz dynamic structure factor to higherfrequencies.Let us consider the effect of changes g -factors and h in more detail. At zero field, the zz structure factoris extremely simple [see Eqs. (5.2) and (5.3)] and canbe presented as a sum of two contributions for the uni-form model shifted by π along the wave-vector axis [i.e.,Eq. (5.2) in the case of zero staggered spin structure fac-tor S zz ( κ, ω ) ]. It is definitely also the case of a small field(see Fig. 7 for h = 0 . ). It is clearly seen that at small h , the deviation of g from g = 1 induces a tiny strip ofnew two-fermion continuum at lower frequencies. The in-tensity of this low-energy two-fermion continuum waneswith decreasing g . Surprisingly, S zz ( κ, ω ) for g ≤ does not show any trace of the low-energy continuumanymore [see Figs. 7(c,d)]: The zz structure factor showsone two-fermion continuum only. In contrast to h = 0 ,at small fields, two opposite cases g = 1 and g = − are not identical [compare Fig. 7(a) and Fig. 7(d)].At higher fields, the magnetic structure factor cannotbe approximated by the sum of uniform spin structurefactors S zz ( κ, ω ) anymore. Even for a moderate alter-nation of g -factors [ g = 1 , g = 0 . in Fig. 8(b)] weobserve the appearance of another two-fermion contin-uum at lower frequencies. It can be treated as a splittingof the initial continuum inherent in the uniform model[see Fig. 8(a)] in two parts, which is a signal of the two-band structure of the fermion excitation spectrum (2.8).It should be noted that the two-fermion continuum atlower frequencies induced by small deviation of g (from g = 1 ) is not a tiny strip anymore as it was at small fields( h = 0 . ). At higher fields as well as at small ones, the zz structure factor for g ≤ shows just one two-fermioncontinuum only [Figs. 8(c,d)]. This picture keeps the ten-dency with increasing field as it is shown in Fig. 9. Intwo top panels we present results at magnetic fields closeto h s whereas for g g ≤ we put h = 1 [Figs. 9(c,d)],because at g g ≤ the saturation field does not exist.The fact, that in Fig. 9(b) both the low-energy and hight-energy two-fermion continua are tiny strips, is caused bythat the field is very close to h s .We also examine the temperature effect on the zz structure factor for non-positive g ≤ . The resultsfor T → ∞ in Fig. 10 show an additional two-fermioncontinuum for low frequencies. In case of zero tempera-ture this continuum was hidden owing to the Fermi-Diracfunctions, compare Fig. 10 to Fig. 8.In the case κ = 0 , Eq. (5.5) can be transformed to thefollowing form: S zz (0 , ω )= δ ( ω ) π Z − π d κ ( g + − g − u κ v κ ) n κ (1 − n κ )+ g − q ω − g − h ω q J +4 g − h − ω X κ r n κ r (1 − n κ r + π ) , (5.9)where κ r are solutions of the equation ω = Λ κ r + π − Λ κ r .The latter equation has solutions only in the restrictedregion | g − h | ≤ ω < q J + g − h . (5.10)We can use Eqs. (5.9) and (5.4) to get explicit expres-sions for the absorption intensity I z ( ω, h ) : I z ( ω, h ) ∝ g − q ω − g − h q J + 4 g − h − ω (5.11) × − exp( − βω )(1 + exp[ β (cid:0) g + h − ω (cid:1) ])(1 + exp[ − β (cid:0) g + h + ω (cid:1) ]) . ω κ ω κ ω κ ω κ FIG. 7: (Color online) The density plot of the dynamic struc-ture factor S zz ( κ, ω ) at T = 0 : | J | = 1 , g = 1 , g = 1 (a), g = 0 . (b), g = 0 (c), g = − (d), h = 0 . . In the ground state we arrive at the following formula: I z ( ω, h ) ∝ g − q ω − g − h q J + 4 g − h − ω , (5.12)where in case g g > the Fermi-Dirac functions shrinkfurther the condition of allowed ω [see Eq. (5.10)] to thefollowing one: | g + h | < ω < q J + g − h . ω κ ω κ ω κ ω κ FIG. 8: (Color online) The density plot of the dynamic struc-ture factor S zz ( κ, ω ) at T = 0 : | J | = 1 , g = 1 , g = 1 (a), g = 0 . (b), g = 0 (c), g = − (d), h = 0 . . It is evident from Eq. (5.11) that there is no energy ab-sorption in case of the uniform g -factors ( g = g = 1 ),since the total magnetization commutes with the Hamil-tonian. The alternation of g -factors destroys this prop-erty and leads immediately to nonzero absorption inten-sity I z ( ω, h ) . From Eqs. (5.12) and (5.11) one can deducethe shape of the absorption line. The field profiles of theabsorption intensity for alternating g -factors are shownin Fig. 11. The absorption intensity curve I z ( ω, h ) for1 ω κ ω κ ω κ ω κ FIG. 9: (Color online) The density plot of the dynamic struc-ture factor S zz ( κ, ω ) at T = 0 : | J | = 1 , g = 1 , g = 1 , h = 0 . (a), g = 0 . , h = 1 . (b), g = 0 , h = 1 (c), g = − , h = 1 (d). any frequency ends continuously at h = ω/ (2 | g − | ) forboth T = 0 , g g < and T > cases. It is clearlyseen in Figs. 11(a,b); short-dashed blue line. If the fre-quency exceeds | J | , we observe also a van Hove sin-gularity at h = √ ω − J / (2 | g − | ) [see Figs. 11(a,b);solid blue line]. In the ground state for g g > this singularity disappears at ω = 2 | J | / p − ( g − /g + ) .If ω < | J | / p − ( g − /g + ) for zero temperature and ω κ ω κ FIG. 10: (Color online) The density plot of the dynamic struc-ture factor S zz ( κ, ω ) at T → ∞ : | J | = 1 , g = 1 , h = 0 . , g = 0 (a), g = − (b).
0 1 2 3 4 1.8 1.9 2 2.1 2.2 2.3 0 0.1 0.2 0.3 0.4 I z ( ω , h )(a) h ω
0 0.5 1 1.5 1.9 2 2.1 2.2 2.3 0 1 2 3 4 5 I z ( ω , h )(b) h ω FIG. 11: (Color online) Field profiles of the absorption inten-sity I z ( ω, h ) at different frequencies ω for | J | = 1 , g = 1 , g = 0 . (a), g = − . (b), and temperatures T = 1 (solidblack curves) and T = 0 (dashed red curves). The dashed-dot-dot violet (dashed green) curve indicates the intensity at h = 0 and T = 0 ( T = 1 ). The solid and short-dashedblue curves show the boundaries given in Eq. (5.10) while thedashed-dot green curve in panel (a), given by h = ω/ (2 | g + | ) ,denotes the upper boundary of I z ( ω, h ) at T = 0 (see thediscussion in the text). g g > , the absorption intensity curve I z ( ω, h ) endsabruptly at h = ω/ (2 | g + | ) [see Fig. 11(a); dashed-dotgreen line], and at ω > | J | / p − ( g − /g + ) this ground-state absorption intensity vanishes, I z ( ω, h ) = 0 . B. xx dynamics We pass to another dynamic structure factor, namely,the xx structure factor S xx ( κ, ω ) . We perform the com-2putation of the xx time correlation functions numericallyusing the previously elaborated method [30, 46, 47]. Inwhat follows, we consider the finite chain of N = 400 spins with open boundary conditions. To avoid theboundary effect, we have to adapt Eq. (5.1). Thus,we choose a “central” spin at the site j = 61 , (de-pending on the adopted parameters) and then calcu-late the time correlation functions h s xj ( t ) s xj + n i as well as h s xj +1 ( t ) s xj + n +1 i for n ≥ . Finally, we present the Fouriertransform in Eq. (5.1) in the following symmetrized form: S xx ( κ, ω ) = 12 Re Z ∞ d te − ǫt e iωt (5.13) × g h s xj ( t ) s xj i + 2 N X n =1 cos(2 nκ ) h s xj ( t ) s xj +2 n i +2 g g N X n =1 cos((2 n − κ ) h s xj ( t ) s xj +2 n − i + g h s xj +1 ( t ) s xj +1 i +2 N X n =1 cos(2 nκ ) h s xj +1 ( t ) s xj +1+2 n i +2 g g N X n =1 cos((2 n − κ ) h s xj +1 ( t ) s xj +1+2 n − i . In numerical calculations we restrict the sum over n upto . . . depending on the correlation length.The results of the numerical calculation for S xx ( κ, ω ) at sufficiently low temperature T = 0 . are shown inFigs. 12–15. In contrast to the zz structure factor, S xx ( κ, ω ) is not governed exclusively by the continuum oftwo-fermion excitations. However, the deeper inspectionof Figs. 12–15 reveals some resemblance between the zz and xx structure factors. Although there is no singularparts visible in S xx ( κ, ω ) as well as abrupt boundariesfor the regions with nonzero values, the dominating con-tribution in the case of positive g is circumscribed bythe boundaries of the two-fermion continua outlined inAppendix. The same feature was demonstrated earlierfor the uniform and dimerized XX chains [47, 48]. Wecan deduce from relation (5.2) and Fig. 12 that the stag-gered spin structure factor (5.3) is minor at small fields.Thus, one can observe how the intensity of the structurefactor S xx ( κ, ω ) is redistributed between two basic con-tinua of the uniform chain [see Fig. 12(a)] shifted by π with respect to each other when g decreases from 1 up tonegative values. One can still recognize the similar fea-ture even at intermediate field h = 0 . in case of g > in Fig. 13(b) where the combination of two continua of S xx ( κ, ω ) and S xx ( κ + π, ω ) creates an intricate intensitypicture.Interestingly, the structure factor S xx ( κ, ω ) for non-positive g ≤ is concentrated mainly along the lines λ ± κ = q J sin κ + g − h ± g + h. (5.14) Although the exact xx correlation functions and the ex-act xx structure factor are not known for g g < , onecan adapt the procedure of Refs. [48, 50] for the case ofthe uniform and dimerized chains above the saturationfield. We need to make the crucial assumption that theaction of the Jordan-Wigner phase factors on the groundstate is equivalent to its action on the ideal antiferromag-netic state. Then, the problem is reduced to calculationof the pair correlation functions for spinless fermions withthe final result S xx ( κ, ω ) ≈ (5.15) π (cid:8)(cid:0) g + g − +4 g + g − sgn( h ) u κ + π/ | v κ + π/ | (cid:1) δ ( ω − λ + κ )+ (cid:0) g + g − − g + g − sgn( h ) u κ + π/ | v κ + π/ | (cid:1) δ ( ω − λ − κ ) (cid:9) . Equation (5.15) although approximate, agrees with nu-merics shown in Figs. 13, 14 for negative g (dashed anddashed-dot lines).If g ∈ (0 , for magnetic fields close to h s , the many-fermion continua shrink [see Fig. 14(a,b)] and above thesaturation fields they reduce to the one-fermion excita-tion spectrum shifted by π along the κ axis with thereversed sign [i.e., − Λ κ + π , dashed line in Fig. 14(a,b)]and if g ∈ (0 , , also by the one-fermion excitationspectrum multiplied by − [i.e., − Λ κ , dashed-dot linein Fig. 14(b)], S xx ( κ, ω ) = π (cid:2) ( g + u κ − g − v κ ) δ ( ω − Λ κ ) (5.16) +( g + v κ + g − u κ ) δ ( ω − Λ κ + π ) (cid:3) , if h < − h s ,S xx ( κ, ω ) = π (cid:2) ( g + v κ − g − u κ ) δ ( ω +Λ κ )+( g + u κ + g − v κ ) δ ( ω +Λ κ + π ) (cid:3) , if h > h s . In case of g ≤ , in Fig. 14(c,d) we observe for higherfield even more pronounced mode along the lines givenin Eq. (5.14).In Fig. 15 we show the frequency profiles of the struc-ture factor for several values of κ = 0 , π/ , π/ , π/ , π .It is clearly seen there that the non-uniform g -factor leadsto many-peak structure in the frequency dependences of S xx ( κ, ω ) at the low temperature T = 0 . , see Fig. 15(a).In contrast, the infinite temperature smears out the finestructure of S xx ( κ, ω ) transforming the frequency profilesinto κ -independent Gaussian ridges, see Fig. 15(b). Sucha form can be obtained using the exact results for thetime correlation functions of dimerized chain [51]. Thosecorrelation functions vanish if the sites are different thatleads to a κ -independent structure factor S xx ( κ, ω ) . Uti-lizing the result of Ref. [51], we get the following explicit3 ω κ ω κ ω κ ω κ FIG. 12: (Color online) The density plot of the dynamic struc-ture factor S xx ( κ, ω ) . J = − , g = 1 , g = 1 (a), g = 0 . (b), g = 0 (c), g = − (d), h = 0 . at low temperature T = 0 . . ω κ ω κ ω κ ω κ FIG. 13: (Color online) The density plot of the dynamic struc-ture factor S xx ( κ, ω ) . J = − , g = 1 , g = 1 (a), g = 0 . (b), g = 0 (c), g = − (d), h = 0 . at low temperature T = 0 . . Dashed and dashed-dot curves follow Eq. (5.14). ω κ ω κ ω κ ω κ FIG. 14: (Color online) The density plot of the dynamic struc-ture factor S xx ( κ, ω ) . J = − , g = 1 , g = 1 , h = 0 . (a), g = 0 . , h = 1 . (b), g = 0 , h = 1 (c), g = − , h = 1 (d)at low temperature T = 0 . . Dashed and dashed-dot curvesin panels (a) and (b) correspond to − Λ κ + π and − Λ κ . Dashedand dashed-dot curves in panels (c), (d) follow Eq. (5.14). formula for S xx ( κ, ω ) at T → ∞ : S xx ( κ, ω ) = 18 ∞ Z −∞ d te iωt Re (cid:8) g Z o ( t ) + g Z e ( t ) (cid:9) , (5.17) Z e ( t )= θ ( z, q ) θ ( z , q ) θ ( z ′ , q ) θ ( z ′ , q ) exp (cid:20) ig + ht − (cid:18) − E( e κ )K( e κ ) (cid:19) J t (cid:21) ,Z o ( t ) = exp ( i g + ht ) Z ∗ e ( t ) ,J ± = 12 (cid:18)q J + g − h ± | g − h | (cid:19) , e κ = J − J + = J (cid:16)q J + g − h + | g − h | (cid:17) ,q = exp − π K( √ − e κ )K( e κ ) ! , where θ ( z ′ , q ) , θ ( z, q ) are the Jacobi theta-functions(see [51] and references therein) with z = π ( J + t + iv )2K( e κ ) , z ′ = π ( J + t − iv )2K( e κ ) z = iπv e κ ) , z ′ = − iπv e κ ) , (5.18)and the parameter v is defined by the following relation: dc( iv , e κ ) = J J + , (5.19)where dc( iv , e κ ) = dn( v , − e κ ) is the elliptic deltaamplitude function for imaginary argument.In case of strong magnetic field h and non-uniform g -factors g − = 0 we have ˜ κ ≪ . Expanding the correlationfunctions for small ˜ κ , we get the xx structure factor inthe explicit Gaussian form: S xx ( κ, ω ) ≈ √ π | J | " A − e − ( ω + ω − )22 J − + e − ( ω − ω − )22 J − ! + A + e − ( ω + ω +)22 J − + e − ( ω − ω +)22 J − ! ,ω ± = J + ± g + h,A ± = ( g + g − ) J + | J | ± g + g − r J J − . (5.20)From Eq. (5.20) it is clear that the intensity of the xx structure factor in the infinite-temperature limit is con-centrated near two Gaussian peaks at ω = ω ± .In Fig. 16 we present the absorption intensity I x ( ω, h ) as a function of the magnetic field. In contrast to the I z ( ω, h ) case, here the field profiles do not exhibit anysingularities. A prominent feature of the absorption pro-files I x ( ω, h ) is a two-peak structure for the case of differ-ent nonzero g -factors. The cases g = 0 . and g = − . demonstrate additional satellite peak [Figs. 16(b,d)]. For5
0 0.5 1 1.5 2 3 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 S xx ( κ , ω )(a) κ ω
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 S xx ( κ , ω )(b) κ ω FIG. 15: (Color online) S xx ( κ, ω ) vs ω at κ = 0 , κ = π/ , κ = π/ , κ = 3 π/ , and κ = π . J = − , g = 1 , g = 0 . , h = 0 . , T = 0 . (left panel), cf. Fig. 13, and T → ∞ (rightpanel).
0 1 2 3 4 5 0 1 2 3 0 1 2 I x ( ω , h )(a) h ω
0 1 2 3 4 5 0 1 2 3 0 1 2 I x ( ω , h )(b) h ω
0 1 2 3 4 5 0 1 2 3 0 1 2 I x ( ω , h )(c) h ω
0 1 2 3 4 5 0 1 2 3 0 1 2 I x ( ω , h )(d) h ω FIG. 16: Field profiles of the absorption intensity I x ( ω, h ) at different frequencies ω for J = − , g = 1 , g = 1 (a), g = 0 . (b), g = 0 (c), and g = − . (d) at T = 1 . the uniform chain ( g = g ) we can see one peak whichmoves with increasing of frequency to a higher value ofmagnetic field [Fig. 16(a)]. Qualitatively the same pic-ture is seen for g = 0 in Fig. 16(c), where the peak isless steeper in comparison to the case in Fig. 16(a). VI. SUMMARY
To summarize, we have studied the effect of the alter-nation of g -factors on the static and dynamic propertiesof the spin-1/2 XX chain in a transverse field. The cru-cial point is that the conservation of the total magnetiza-tion is lost in this case. This evokes non-trivial changes inthe thermodynamic and dynamic behavior of the model.While the logarithmic peculiarities of the magnetiza-tion and the susceptibility at T = 0 were obtained ear-lier [13], we found peculiarities in the low-temperaturethermodynamics. In particular, we have shown that thespecific heat can change its behavior from the linear de-pendence in the spin-liquid phase to the √ T dependenceat the saturation field, and finally transformed to the ex-ponential law (4.8). The susceptibility at zero magneticfield displays the logarithmic divergence with tempera-ture as it follows in Eq. (4.9). We have performed the detailed study of the dynamicproperties. We calculated the dynamic structure factors S zz ( κ, ω ) and S xx ( κ, ω ) and inspected how they change inthe external magnetic field for different period-2 alterna-tions of g -factors. In the case when both g -factors are ofthe same sign, the correspondence between the bound-aries of the zz and xx structure factors is still presentlike it was observed previously [47, 48]. On the contrary,if g g ≤ , a large enough magnetic field leads to thehighly intense modes in the xx structure factor. In ad-dition, we calculated the absorption intensity I α ( ω, h ) for the different configuration of ESR experiments. Inthe Voigt configuration ( α = z ), the model with uni-form g -factors does not have any response. In the casewhen g differs from g , we obtain the nonzero contri-bution to the absorption intensity. For sufficiently largefrequencies ω > | J | the van Hove singularity arises at h = √ ω − J / (2 | g − | ) . In the Faraday configuration( α = x ), the situation is a bit different. The absorptionspectra can be observed in the uniform case. It shows abroad maximum at some resonance field. The alternationof g -factor leads to the doubling of this resonance line.Although in our study we focus on the exactly solvable XX chain, from Ref. [28] we know that such analysis ofdynamics is useful for understanding a more realistic caseof the Heisenberg chains. Acknowledgments
The present study was supported by the ICTP (OEA,network-68 and NT-04): V. O. acknowledges the kindhospitality of the ICMP during his visits in 2015–2019;T. V. and O. B. acknowledge the kind hospitality of theYerevan University in 2016, 2017, and 2018. The work ofT. K. and O. D. was partially supported by Project FF-30F (No. 0116U001539) from the Ministry of Educationand Science of Ukraine. V. O. acknowledges the par-tial support from the ANSEF project condmatth-5212,as well as the support from the HORIZON 2020 RISE"CoExAN" project (GA644076).
Appendix: Boundaries of the two-fermion excitationcontinua
Let us present the expressions for the lines in the ( κ, ω ) plane, which restrict the regions for different number ofsolutions of Eqs. (5.6) as it is shown in Fig. 5(a); greenlines. We have ω , ( κ ) = q (cid:0) J + 2 g − h ± J cos κ (cid:1) , (A1) ω , ( κ ) = | sin κ | (cid:18)q J + g − h ± | g − h | (cid:19) ,ω , ( κ ) = q J sin κ + g − h ± | g − h | . Let us also present the expressions in the case | h | < h s , g g > for the characteristic lines, which boundednonzero values of the Fermi-Dirac functions at T = 0 [seealso Fig. 5(c); red lines]. We have ω , ( κ ) = (cid:12)(cid:12)(cid:12)(cid:12) g + h + q J cos ( κ ± κ ) + g − h (cid:12)(cid:12)(cid:12)(cid:12) , (A2) ω , ( κ ) = (cid:12)(cid:12)(cid:12)(cid:12) g + h − q J cos ( κ ± κ ) + g − h (cid:12)(cid:12)(cid:12)(cid:12) . Here κ is defined in Eq. (3.2). [1] B. Odom, D. Hanneke, B. D’Urso, and G. Gabrielse,Phys. Rev. Lett. , 030801 (2006); G. Gabrielse,D. Hanneke, T. Kinoshita, M. Nio, and B. Odom, ,039902E (2007).[2] A. Niazi, P. L. Paulose, and E. V. Sampathkumaran,Phys. Rev. Lett. , 107202 (2002); P. D. Battle,G. R. Blake, J. Darriet, J. G. Gore, and F. Weill, J.Mater. Chem. , 1559 (1997); P. D. Battle, G. R. Blake,J. Sloan, and J. F. Vente, J. Solid State Chem. , 103(1998); G. R. Blake, J. Sloan, J. F. Vente, and P. D. Bat-tle, Chem. Mater. , 3536 (1998).[3] W.-G. Yin, X. Liu, A. M. Tsvelik, M. P. M. Dean,M. H. Upton, J. Kim, D. Casa, A. Said, T. Gog, T. F. Qi,G. Cao, and J. P. Hill, Phys. Rev. Lett. , 057202(2013).[4] W.-G. Yin, C. R. Roth, and A. M. Tsvelik,arXiv:1510.00030 (2015).[5] P. Bhatt, N. Thakur, M. D. Mukadam, S. S. Meena, andS. M. Yusuf, J. Phys. Chem. C , 1864 (2014).[6] D. Visinescu, A. M. Madalan, M. Andruh, C. Duhayon,J.-P. Sutter, L. Ungur, W. Van den Heuvel, andL. F. Chibotaru, Chem. Eur. J. , 11808 (2009);W. Van den Heuvel and L. F. Chibotaru, Phys. Rev. B , 174436 (2010); S. Bellucci, V. Ohanyan, and O. Ro-jas, Europhys. Lett. , 47012 (2014).[7] M. Kenzelmann, C. D. Batista, Y. Chen, C. Broholm,D. H. Reich, S. Park, and Y. Qiu, Phys. Rev. B ,094411 (2005).[8] F. Souza, M. L. Lyra, J. Strečka, and M. S. S. Pereira,J. Magn. Magn. Mater. , 423 (2019).[9] M. Oshikawa and I. Affleck, Phys. Rev. Lett. , 2883(1997); I. Affleck and M. Oshikawa, Phys. Rev. B ,1038 (1999); , 9200(E) (2000).[10] E. Coronado, M. Drillon, A. Fuertes, D. Beltran, A. Mos-set, and J. Galy, J. Am. Chem. Soc. , 900 (1986).[11] E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. (N.Y.) , 407 (1961).[12] S. Katsura, Phys. Rev. , 1508 (1962); , 2835(1963).[13] V. M. Kontorovich and V. M. Tsukernik, Sov. Phys.JETP , 687 (1968).[14] J. H. H. Perk, H. W. Capel, M. J. Zuilhof, andTh. J. Siskens, Physica A , 319 (1975).[15] J. P. de Lima, T. F. A. Alves, and L. L. Gonçalves,J. Magn. Magn. Mater. , 95 (2006); J. P. de Lima,L. L. Gonçalves, and T. F. A. Alves, Phys. Rev. B ,214406 (2007); J. P. de Lima and L. L. Gonçalves, Phys.Rev. B , 214424 (2008).[16] O. Derzhko, J. Richter, and O. Zaburannyi, Physica A , 495 (2000); J. Magn. Magn. Mater. , 207 (2000).[17] O. Derzhko, J. Richter, T. Krokhmalskii, and O. Zabu- rannyi, Phys. Rev. E , 066112 (2004).[18] A. D. Varazi and R. C. Drumond, Phys. Rev. E ,022104 (2019).[19] A. A. Zvyagin and G. A. Skorobagat’ko, Phys. Rev. B , 024427 (2006).[20] A. A. Zvyagin, Quantum Theory of One-DimensionalSpin Systems (Cambridge Scientific Publishers, Cam-bridge, 2010).[21] A. A. Zvyagin, Fiz. Nizk. Temp. , 1240 (2016) [LowTemp. Phys. , 971 (2016)].[22] M. Atanasov, P. Comba, and C. A. Daul, Inogr. Chem. , 2449 (2008).[23] L. F. Chibotaru and L. Ungur, Phys. Rev. Lett. ,246403 (2012).[24] L. F. Chibotaru, in Advances in Chemical Physics , editedby S. A. Rice and A. R. Dinner, Vol. 153 (Wiley, NewJersey, 2013), pp. 397-519.[25] J. Torrico, V. Ohanyan, and O. Rojas, J. Magn. Magn.Mater. , 85 (2018).[26] V. Ohanyan, O. Rojas, J. Strečka, and S. Bellucci, Phys.Rev. B , 214423 (2015).[27] J.-S. Caux and J. Maillet, Phys. Rev. Lett. , 077201(2005); R. G. Pereira, J. Sirker, J.-S. Caux, R. Hagemans,J. M. Maillet, S. R. White, and I. Affleck, J. Stat. Mech.:Theory Exp. P08022 (2007); J.-S. Caux, J. Mossel, andI. P. Castillo, J. Stat. Mech.: Theory Exp. P08006 (2008);J.-S. Caux, J. Math. Phys. , 095214 (2009); R. Vlijm,I. S. Eliëns, and J.-S. Caux, SciPost Phys. , 008 (2016).[28] G. Müller, H. Thomas, H. Beck, and J. Bonner, Phys.Rev. B , 1429 (1981); G. Müller, H. Thomas,M. W. Puga, and H. Beck, J. Phys. C , 3399 (1981).[29] O. Derzhko, Journal of Physical Studies (L’viv) , 49(2001).[30] O. Derzhko, in Condensed Matter Physics in the Prime ofthe 21st Century: Phenomena, Materials, Ideas, Meth-ods , edited by J. J¸edrzejewski (World Scientific, Singa-pore, 2008), pp. 35-87.[31] The periodic spin chain ( a -cyclic model) is represented bythe fermionic spin chains with periodic and antiperiodicboundary conditions ( c -cyclic and c -anticyclic model),see Refs. [11, 32, 33]. However, as it was shown inRef. [33], in the thermodynamic limit, for the thermo-dynamic quantities as well as for the time correlationfunctions of the z components of two spins one arrives atthe same result for the a -cyclic, c -cyclic, and c -anticyclicmodel, i.e., the imposed boundary conditions are irrele-vant. Note that the open spin chain (used in numerics tofind the time correlation functions of the x componentsof two spins) is represented by the open fermionic chain.[32] B. M. McCoy, E. Barouch. and D. B. Abraham, Phys.Rev. A , 2331 (1971). [33] P. Mazur and Th. J. Siskens, Physica , 259 (1973);Th. J. Siskens and P. Mazur, Physica , 560 (1974).[34] M. Kenzelmann, R. Coldea, D. A. Tennant, D. Visser,M. Hofmann, P. Smeibidl, and Z. Tylczynski, Phys. Rev.B , 144432 (2002).[35] L. N. Bulaevskii, Zh. Eksp. Teor. Phys. , 1008 (1963)[Sov. Phys. JETP 17, 684 (1963)]; see also Ref. [29].[36] A. A. Zvyagin, Phys. Rev. B , 174408 (2020).[37] J.-S. Caux, E. H. L. Essler, and U. Löw, Phys. Rev. B , 134431 (2003).[38] D. V. Dmitriev, V. Ya. Krivnov, and A. A. Ovchinnikov,Phys. Rev. B , 172409 (2002).[39] R. Hagemans, J.-S. Caux, and U. Löw, Phys. Rev. B ,014437 (2005).[40] T. S. Nunner and Th. Kopp, Phys. Rev. B , 104419(2004).[41] B. Bruognolo, A. Weichselbaum, J. von Delft, andM. Garst, Phys. Rev. B , 085136 (2016).[42] E. Jahnke and F. Emde, Tables of Functions (Dover Pub-lishers, New York, 1945). [43] I. A. Zaliznyak and J. M. Tranquada, in
Strongly Cor-related Systems , edited by A. Avella and F. Mancini,Springer Series in Solid-State Sciences Vol. 180 (Springer,Berlin, 2015).[44] Y. Ajiro, J. Phys. Soc. Jpn. , 12 (2003).[45] J. Jensen and A. R. Mackintosh, Rare Earth Magnetism:Structures and Excitations (Clarendon Press, Oxford,1991).[46] O. Derzhko and T. Krokhmalskii, Phys. Rev. B , 11659(1997); Phys. Status Solidi (b) , 221 (1998).[47] O. Derzhko, T. Krokhmalskii, and J. Stolze, J. Phys. A , 3063 (2000).[48] O. Derzhko, T. Krokhmalskii, and J. Stolze, J. Phys. A , 3573 (2002).[49] M. Brockmann, F. Göhmann, M. Karbach, A. Klümper,and A. Weiße, Phys. Rev. B , 134438 (2012).[50] H. B. Cruz and L. L. Gonçalves, J. Phys. C , 2785(1981).[51] J. H. H. Perk and H. W. Capel, Physica A100