Frustrated Ferrimagnetic Ladder in a Magnetic Field

Frustrated ferrimagnetic ladder in a magnetic ﬁeld

N. B. Ivanov1 and J. Richter2 1 Max-Planck Institut fur¨ Physik Komplexer Systeme, N¨othnitzer Strasse 38, D-39016 Dresden, Germany∗ 2Institut fur¨ Theoretische Physik, Universit¨at Magdeburg, PF 4120, D-39016 Magdeburg, Germany (Dated: October 24, 2005)

1 We study the magnetic phase diagram of two coupled mixed-spin (1, 2 ) Heisenberg chains as a function of the frustration parameter related to diagonal exchange couplings. The analysis is per- formed by using spin-wave series and exact numerical diagonalization techniques. The obtained phase diagram – containing the Luttinger liquid phase, a plateau magnetic phase with the magnetization per rung M = 1/2, and the fully polarized phase – is closely related to the generic (J/U, µ/U) phase diagram of the one-dimensional boson Hubbard model.

PACS numbers: 75.10.Jm, 75.10.Pq

Some special classes of quantum spin ladders composed are alternatively distributed on the ladder sites. The of different types of spins or/and regular mixtures of parameters J1, J⊥, and J2 are positive coupling constants ferromagnetic and antiferromagnetic exchange bonds ex- of the nearest-neighbor horizontal, vertical, and diagonal hibit a number of novel quantum spin phases and unusual exchange bonds, respectively (see Fig. 1 in Ref. 5). We thermodynamic properties.1,2 Moreover, such systems set the energy and length scales by J 1 and a = 1, 1 ≡ 0 are appropriate for studying magnetic quantum critical where a0 is the lattice spacing. For simplicity, in the points in one spatial dimension (1D), since many of them following we consider the case J⊥ = 1 and set h = gµBH. sustain magnetically-ordered ground states. Possible re- alizations of such ladder structures can be based, for in- 5 stance, on the synthesized quasi-1D bimetallic molecular M=3/2 magnets.3,4 E In a recent paper we have considered the role of ge- 4 hs ometric frustration in a system of two coupled mixed- Luttinger liquid 1 5 spin (1, 2 ) Heisenberg chains. It has been argued that 3 a relatively moderate strength of the magnetic frustra- h A B tion produced by the nearest-neighbor diagonal bonds 2 was able to destroy the ferrimagnetic phase and to sta- a b bilize the Luttinger liquid phase. The quantum phase c M=1/2 transition between these two phases is smooth and is re- 1 alized through an intermediate canted spin phase. This d D Jc latter phase is characterized by a net ferromagnetic mo- 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ment per rung, 0 < M < 0.5, and power-law transverse J spin-spin correlations. In this Brief Report we show that 2 FIG. 1: Phase diagram of the model (1) in the (J2, h) plane. the magnetic phase diagram of the discussed model as The phase boundary hs = hs(J2) of the fully polarized phase a function of the frustration parameter is closely related (M = 3/2) is exact. The dashed curve abcd marks the area to the (J/U, µ/U) phase diagram of the boson Hubbard occupied by the M = 1/2 plateau phase, as obtained in a model in periodic 1D potentials,6 where µ, J, and U are, linear spin-wave approximation. The boundary position of respectively, the chemical potential, the hopping strength this phase, as obtained from the exact numerical diagonaliza- constant, and the on-site repulsion potential. tion of periodic clusters, is traced by squares (L = 10) and The relevant spin Hamiltonian reads crosses (L = 12). The dotted curve shows the phase bound- (−) ary where ∆0 (h) = 0 [see Eq. (4)], as obtained from the L first-order spin-wave theory. The dashed curve starting from the point b and going upwards left as well as the piece bc = [J (s s + s s ) + J⊥s s H 1 1n · 2n+1 2n · 1n+1 1n · 2n represent in a linear spin-wave approximation the boundary n=1 (−) X where ∆π (h) = 0. The canted spin phase exists for h = 0 in + J (s s + s s ) gµ H(s z + s z )] , 2 1n · 1n+1 2n · 2n+1 − B 1n 2n the interval 0.342 < J2 < Jc, where Jc = 0.399 is the phase (1) transition point from the canted phase to the Luttinger liquid phase. The rest of the area is occupied by the Luttinger liquid where L, g, µB, and H are, respectively, the number phase. of rungs, the gyromagnetic ratio, the Bohr magneton, and the external magnetic field applied in z direction. s s The spin operators 1n and 2n (characterized by the The magnetic phase diagram of the model (1) in the quantum spin numbers s1 > s2 and the rung index n) (J2, h) plane is presented in Fig. 1. The diagram contains 2 the Luttinger liquid phase, the plateau magnetic phase SWT approximation, the magnon excitations above the with a quantized magnetic moment per rung M = 1/2 classical ferrimagnetic ground state are described by the and the fully polarized phase with M = 3/2. In addition, following dispersion relations: there is a canted magnetic phase which exists only at zero magnetic field (in the interval 0.342 < J < J = 0.399) 3 k 2 c ω( )(k) = h + J sin2 and for h > 0 merges into the Luttinger liquid phase. ∓ 4 2 2 Some properties of the latter phase at h = 0 have al- 2 2 ready been studied in Ref. 5, by using a conformal-field- 1 3 2 k 2 k + 9 2J2 sin 2 3 4 sin . theory analysis of the numerical exact-diagonalization 2 2 − 2 − − 2 s (ED) data for the ground-state energy and the energies of (3) some low-lying excited states in finite periodic systems. As is well-known, the external magnetic field varies the In the last equation, ω(+)(k) is a FM branch, whereas − parameters of the Luttinger liquid phase without chang- ω( )(k) describes antiferromagnetic (AFM) one-magnon ing its basic properties up to a saturation field (hs), where excitations. In the whole region occupied by the M = 1/2 the system becomes fully polarized.7 phase, the dispersion relations (3) exhibit minima at the In this connection, let us firstly consider the critical center (k = 0) as well as at the boundary (k = π) of the field hs defining the phase boundary of the fully polarized Brillouin zone. Close to these wave vectors the magnon M = 3/2 phase. The boundary hs = hs(J2) can be ob- energies take the generic form tained by examining the instabilities of the one-magnon 2 excitations above the ferromagnetic vacuum. We assume, () () (k k0) ωk0 (k) = ∆k0 (h) + − () , k0 = 0, π . (4) as generally accepted, that the multimagnon excitations 2mk0 become unstable at a weaker magnetic field as compared () () to the one-magnon excitations. As a matter of fact, the Here ∆k0 (h) and mk0 are, respectively, the magnon latter assumption is confirmed by our ED data, which (+) gaps and effective masses. Since ω0 (k) is a Goldstone exactly reproduce the analytical result for hs presented (+) below. A straightforward calculation results in the fol- mode, we have ∆0 (0) = 0. lowing exact expression for the one-magnon dispersion The magnon modes in Eq. (4) become gapless on relations in the fully polarized phase different parts of the phase boundary marked by the a b c d (−) (−) (+) points , , , and ; ω0 (k), ωπ (k), and ωπ (k) 9 k are gapless, respectively, on the pieces ab, bc, and cd. ω(1,2)(k) = h 3J sin2 − 4 − 2 2 At the special point b (analogous to the point E), both (−) (−) AFM modes ω (k) and ωπ (k) are gapless. Similar 1 3 k 2 k 2 0 2J sin2 + 2 3 4 sin2 , special points have been found in other 1D spin mod- 2 2 − 2 2 − 2 8 s els with folding excitation spectra. In the case when (2) the related magnon masses in (4) diverge, such points produce cusps in the magnetization curves. Otherwise, where the wave vector k runs in the lattice Brillouin zone as in our case, these are crossover points marking the π k < π. Equation (2) describes two folding ferro- boundary between regions with different low-lying exci- −magnetic≤ (FM) branches of one-magnon excitations. The tations which belong to a single excitation branch. On phase boundary h = h (J ) is defined by the expressions s s 2 the other hand, at the special point c with coordinates ∗ ∗ 9 (J2 , h ) = (0.75 √2/6, 1.5 √2/6), two different types hs = , 0 J2 0.5 − −(+) 2 ≤ ≤ of modes – the FM mode ωπ (k) and the AFM mode 1 − 2 2 ( ) c 9 1 3 ωπ (k) – reach the ground state. The point belongs = + 3J2 + 2J2 + 2 , J2 0.5, to the special line h = 0.75 + J2 on which the energies 4 2 " 2 − # ≥ (+) (−) of both magnon modes coincide: ωπ (k) = ωπ (k). Us- where the first and second lines correspond, respectively, ing the picture of ”particle-hole” excitations defined by (2) (+) (−) to instabilities of ω (k) at the wave vectors k = 0 and ωph(k) = ω (k) + ω (k), we find that close enough ∗ ∗ π. For h < hs(J2), the modes around k = 0 and π to (J2 , h ) the dispersion relation ωph(k) takes the rela- become unstable and destroy the fully polarized state. tivistic form As discussed in the following text, the point E in Fig. (2) 2 2 2 1 is a crossover point at which both minima in ω (k) ωph(k) = ∆ + v (k π) . (5) ph ph − reach the ground state. q ∗ ∗ Now we address the M = 1/2 plateau phase. The pres- While the velocity vph remains finite at (J2 , h ), the ex- ∗ 1/2 ence of quantum fluctuations in the ferrimagnetic state citation gap vanishes as ∆ph (J2 J2) . Clearly, does not allow exact considerations, so that we shall rely the discussed structure of the lo∝w-lying− excitations in a ∗ ∗ on a qualitative analysis based on the spin-wave theory vicinity of (J2 , h ) remembers the known structure of (SWT) supplemented by numerical ED data for finite pe- low-lying excitations close to the commensurate Mott- riodic systems containing up to L = 12 rungs. In a linear insulator-superfluid transition.6 3

In fact, the outlined peculiarities of the (J2, h) phase excitations, and zero compressibility, κ = ∂ρ/∂µ=0. Ex- diagram – basically deduced from the linear SWT ap- amples of magnetization curves m(h) for different param- proximation – share a number of common features with eters J2 are presented in Fig. 2. These are analogues of the (J/U, µ/U) phase diagram of the boson Hubbard the ρ(µ) curves for the 1D boson Hubbard model.11 Note model on 1D periodic lattices. Before discussing this that in the case J2 = 1 the curve m(h) describes the 3 issue, let us present our quantitative estimates for the reduced magnetization of the spin- 2 antiferromagnetic boundary positions based on higher-order spin-wave se- Heisenberg chain.12 The magnetization curves also indi- ries and numerical ED data. Unfortunately, already the cate a divergence of the zero-field magnetic susceptibility first-order corrections in the SWT series exhibit diver- χ = χ(J2) as J2 approaches the isolated critical point −1/2 gences of the form δ , δ being the distance in the Jc (J2 Jc 0), in accord with the generic Luttinger ∗ ∗ → 7 parametric (J2, h) space from the point (J2 , h ). Thus, liquid relation χ(h, J2) = K(h, J2)/[πvs(h, J2)] and the reliable quantitative results can be achieved only for large observation that the spin-wave velocity vs(0, J2) 0 as enough δ. As an example, we show in Fig. 1 the first- J J 0.5 Here K(h, J ) is the Luttinger liquid→ pa- 2 → c 2 order SWT result for the critical field hc defined through rameter at finite h and J2. Up to now, only a few studies (−) the relation ∆0 (hc) = 0. In spite of the relatively large analyzing such ferromagnetic quantum critical points in correction of the boundary position, we observe a good 1D have been published.13–15 In particular, as recently agreement with the ED data. In fact, one may expect fur- shown in Ref. 15, the Berry term plays an important role ther improvement of the theoretical result in a second- in the critical theory, so that in the isotropic limit one can 9 order SWT approximation. Relatively large finite-size expect different critical behavior near Jc, as compared to effects in the ED data can be indicated only in the vicin- the behavior established in the Ising limit.14 The mag- ity of the tip of the region occupied by the M = 1/2 netization curves also indicate that the canted magnetic phase. It is remarkable that the peculiarities of the phase phase is characterized by a finite susceptibility χ(0, J2), boundary – found in the framework of the linear SWT as can be expected since this state may be viewed as a approximation – are reproduced by the ED data. In par- kind of ferromagnetic Luttinger liquid phase.16 ticular, the cusp at the point B – signaling a change in Discussed instabilities at hc and hs share the same the type of low-lying excitations – can easily be indicated physics as those at the lower (hc1) and upper (hc2) crit- by the ED data. This is important because the ED data ical fields in Haldane-gap chains in a magnetic field.17 (due to finite-size effects) do not fix the position of the In particular, the instability at hc may be viewed as tip. a condensation of dilute gas of magnon excitations, as in the case of Haldane-gap chains slightly above the 18–20 m=1 lower critical field hc1. At the critical field hc, the − b ( ) 1 gap of the AFM mode ω0 (k) vanishes and, as a result, a macroscopic amount of AFM magnons condense 0.5 0.8 onto the ground state. The collapse of quasiparticles is 1.0 prevented by the repulsive on-site quasiparticle interac-

0.6 tion. For h > h , the magnetization M continuously m c m=1/3 exceeds the quantized value 1/2. The instabilities re- 0.38 0.4 0.42 a lated to the FM modes, as in the case of a Haldane chain at the upper critical field hc2, are expected to share the same critical properties.7,21 In fact, discussed tran- 0.2 h s 22 hc sitions are of a commensurate-incommensurate type, 0 like the insulator-superfluid transition in the 1D bo- 0 1 2 3 4 5 6 7 h son Hubbard model for non-integer ρ. Since magnons FIG. 2: Magnetization curves of the model (1) for different behave as a dilute gas of hard-core bosons, in a first values of the frustration parameter J2 (= 0.38, 0.42, 0.5, and approximation the specific form of the interaction is 1). m = M/Ms is the reduced magnetization, M being the irrelevant.20,23 This results, in particular, in the uni- magnetic moment per rung, and Ms = 3/2. The symbols versal behavior of the Haldane chain magnetization19,24 mark the midpoints of the magnetization steps, as obtained M = [2∆ (h h )]1/2 /(πv ), where ∆ is the Haldane from the ED data for L = 12. For clarity, we do not show the H − c1 s H gap and vs ( 2.49) is the spin-wave velocity. Similar data points for J2 = 0.38 and 0.42 in the interval hc < h < hs ≈ since they closely follow the presented data for J2 = 0.5. universal expressions can be obtained close to the phase To compare with the phase diagram of the boson Hub- boundaries of the (J2, h) diagram of our model. Using bard model, recall that the magnetic field h plays the the equivalence of 1D hard-core bosons and free spin- role of a chemical potential and the magnetization M is less fermions, this implies, in particular, the Luttinger an analogue of the conserved density of particles ρ. As is liquid parameter K = 1 on the entire boundary of the well-known, the Luttinger liquid phase is an analogue of M = 1/2 phase as well as on the hs(J2) boundary, ex- the superfluid phase in the boson model, and the plateau cluding the special point at the tip of the region occupied phases can be viewed as Mott-insulator phases,10 char- by the M = 1/2 phase. Respectively, the spin-spin cor- 1 acterized by an integer ρ, a finite gap for particle-hole relation exponents take the values ηz = 2 and ηx = 2 , 4 and the magnetization exponent β = 1/2. Based on the mixed-spin ladder exhibits a magnetic phase diagram established picture of the low-lying excitations close to which is closely related to the one of the boson Hub- ∗ ∗ the point (J2 , h ) and using the analogy with the boson bard model in one spatial dimension. Similar magnetic phase diagram, it is plausible to assume that the latter phase diagrams may be expected for arbitrary quantum critical point describes a Berezinskii-Kosterlitz-Thouless spin numbers s and s , provided that s = s and 1 2 1 6 2 (BKT) type transition. Although our ED data do not fix s1 + s2 =half-integer. the position of the tip, it is easy to indicate the typical pointed-type shape of the boundary25 resulting from the specific BKT particle-hole excitation gap ∆ph close to ∗ Acknowledgments this transition: ∆ph exp const/ J2 J2 . As might be expected, the SWT∼ does not reproduce− the correct p behavior of the gap ∆ph close to this special point. This work was partially supported by the Bulgarian In conclusion, we have demonstrated that a generic Science Foundation, Grant No. 1414/2004.

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