Phys. Rev. B84 (2011) 224409 arXiv:1105.5626

Resonances in a dilute gas of magnons and metamagnetism of isotropic frustrated ferromagnetic spin chains

M. Arlego,1 F. Heidrich-Meisner,2 A. Honecker,3 G. Rossini,1 and T. Vekua4 1Departamento de F´ısica, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina 2Physics Department and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universit¨at M¨unchen, 80333 M¨unchen, Germany 3Institut f¨ur Theoretische Physik, Georg-August-Universit¨at G¨ottingen, 37077 G¨ottingen, Germany 4Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, 30167 Hannover, Germany (Dated: May 27, 2011; revised November 22, 2011) We show that spinS chains with SU(2)symmetric, ferromagnetic nearestneighbor and frustrating antiferromagnetic nextnearestneighbor exchange interactions exhibit metamagnetic behavior under the influence of an external magnetic field for small S, in the form of a firstorder transition to the fully polarized state. The corresponding magnetization jump increases gradually starting from an Sdependent critical value of exchange couplings and takes a maximum in the vicinity of a ferromagnetic Lifshitz point. The metamagnetism results from resonances in the dilute magnon gas caused by an interplay between quantum fluctuations and frustration.

I. INTRODUCTION tune the effective interaction between magnons from re- pulsive to attractive by exploiting the existence of res- onances. As a main result, we demonstrate that, close Quantum spin systems at low temperatures share to resonances, where the scattering length is much larger many of the macroscopic quantum behaviors with sys- than the lattice constant, and in the case of attractive tems of bosons such as Bose-Einstein condensation, su- effective interactions, the intrinsic hard-coreness of spins perfluidity (see Ref. 1 and references therein), or macro- does not play a significant role in the limit of a dilute gas scopic quantum tunneling,2 and may indeed be viewed of magnons. A behavior resembling the collapse of attrac- as quantum simulators of interacting bosons.1 Spin sys- tively interacting bosons9,10 therefore exists in such spin tems with frustration, in particular, realize exotic phases systems close to their fully polarized state. The thermo- of strongly correlated bosons, such as spin liquids3 or dynamic instability of collapsed states causes jumps in supersolids.4 Experimentally, such systems can be stud- the magnetization curve just below saturation. This has ied both in low-dimensional quantum magnets (see, e.g., to be contrasted with the scattering length being of the Ref. 3) and in ultra-cold atomic gases.5,6 In the latter order of a few lattice sites. In that case, which is real- case, interest in the many-body physics has been excited ized for spin 1/2, the formation of mutually repulsive, by the extraordinary control over interactions between multi-magnon bound states can be observed (see, e.g., bosons via Feshbach resonances.7 These can, in particu- Refs. 11–15). lar, be used to tune interactions from repulsive to attrac- purely tive. In the attractive regime, the collapse of an ultra- Our work demonstrates that the mapping of a 1D cold gas of bosons was observed in experiments.8 In this spin system close to saturation to an effective theory work, we discuss a mechanism by which the same can be of a dilute Bose gas properly accounts for the physics of achieved in spin systems. the model. This is accomplished by connecting the scat- tering vertices of the microscopic lattice model with the In the large-S limit, spins map onto bosons with a coupling constants of the effective theory. Furthermore, finite but large Hilbert space, justifying a description in despite the purely 1D nature of our problem, a 1/S ex- terms of soft-core bosons. Yet, in many cases, similarities pansion is a valuable tool and yields the correct physics. between spins and soft-core bosons may even exist for Concretely, we study the following system: S = 1/2. In three dimensions (3D), systems of spins and bosons resemble each other the more the smaller the L z density of bosons is, whereas 1D spin-S antiferromagnets H = JS S + J ′S S hS . (1) S i i+1 i i+2 − i close to saturation behave as spinless fermions, or hard- i=1 core bosons. In spin systems, the external magnetic field h tunes the density of magnons. The limit of a dilute gas x y z Si = (Si ,Si ,Si ) is a spin-S operator acting on site i of magnons is then realized as the fully polarized state and L is the number of sites. J < 0 is the FM, nearest- (the vacuum of magnons) is approached from below. neighbor exchange interaction and J ′ = 1 is the antifer- In this work, we argue that resonances can play a cru- romagnetic (AFM), next-nearest-neighbor exchange in- cial role in frustrated quantum spin systems largely deter- teraction setting the energy scale. In the absence of an mining their low-energy behavior in an external magnetic external field h, HS has a FM ground state for J 4 field. We consider the frustrated ferromagnetic (FM) for all S (see Ref. 16). J = 4 is a ferromagnetic Lifshitz≤ − spin-S chain and show that upon changing system pa- point where the quadratic− term in the dispersion of the rameters such as the coupling constants or h, one can magnons vanishes. 2

Systems with competing FM and AFM interactions are This leads to the recurrence relations of timely interest,17 in particular, the spin-1/2 version of 11–14,18–20 S Eq. (1), motivated by the experimental real- Ω0C0 = (ζ1C1 + ζ2C2) S(2S 1) izations in, e.g., LiCuVO4 (Refs. 21,22) and Li2ZrCuO4 − (Ref. 23). The 1D case of J < 0 and S > 1/2 is largely (2S 1)3/2 (Ω J)C = − ζ C + ζ C unexplored (for J > 0 and S > 1/2, see Refs. 19 and 24). 0 − 1 S3/2 1 0 1 2 We are mainly interested in the region 4 0. The remaining ± unknown quantities C0, v and Eb are determined from A. Solution of the two-magnon problem the remaining relations listed in Eq. (5).

We now solve the interacting two-magnon problem, starting with the thermodynamic limit (for S =1/2, see B. Scattering length in the lattice problem Ref. 29). On a chain of finite length L with periodic boundary conditions the total momentum K is a good For S > 1/2, bound states with energies below the quantum number due to the translational invariance of minimum of the two-magnon scattering continuum ex- the Hamiltonian HS in Eq. (1). Thus, it is convenient to ist only for K 2kcl and only in a finite window of use a basis separating momentum subspaces couplings ≃ ±

L 4 1/2 which this window disappears completely. and L even, r =0, 1, .., L/2 1, (L/2) for q odd (even). − We define the scattering length of bound states, in the We expand a general two-magnon state with momen- thermodynamic limit L , from their spatial extent tum K into the (unnormalized) basis of Eq. (2) as → ∞

S 1 3/2 2 5/2 Ψ2M = Cr K, r (3) | | −Jcr(S) 2.11 (2.95) 3.31 (3.42) 3.68 (3.66) 3.84 (3.80) r cr and determine Cr analytically by solving the two-magnon TABLE I: Critical exchange couplings J (S) for the existence Schr¨odinger equation of metamagnetism in Eq. (1) derived from solving the two magnon problem (values in parenthesis: Results from the 1/S H Ψ = E Ψ . (4) expansion of Sec. III). S| 2M 2M | 2M 3

600 states are not formed below the minimum of the scat- S=1/2 tering continuum). In the attractive regime aS > 0, the /a S S=1 i.e. S=3/2 scattering lengths obtained from both approaches [ , 300 Eq. (9) and Eq. (13)] are in excellent agreement with each other. As a side note on terminology, we call at- tractive (repulsive) regime the one in which the effective 0 interaction between magnons is attractive (repulsive). We can now generalize the procedure31 of mapping the scattering length a Jcr(S=3/2) Jcr(S=1) antiferromagnetic (unfrustrated) spin-S chain close to -300 saturation onto the low-density limit of the Lieb-Liniger -4 -3.5 -3 -2.5 -2 J model with a coupling constant 2 S FIG. 1: (Color online) 1D scattering length a /a (a: lattice g0 = . (14) spacing) for S = 1/2, 1, 3/2 (solid, dashed, dotdashed line). −maS However, in our model the single-magnon dispersion (in analogy to the continuum problem of particles inter- has two minima. The effective theory thus will be a two- acting via a short-range, attractive potential): component (two species) Lieb-Liniger model. There are two types of low-energy scattering processes, first when 1 momenta of two magnons are in the vicinity of the same a = . (9) S min Re[κ ] dispersion minimum that we have presented above (in- { ± } traspecies scattering), and second, when the momenta The binding energy takes its lowest value (i.e., the largest k1 and k2 of two magnons are in the vicinity of different absolute value) for K = K∗ 2kcl = 2arccos( J/4) minima of dispersion, i.e., k = k +k and k = k k ≃± ± − 1 cl 2 cl and this quantity, with extremely high accuracy, is re- (interspecies scattering). For the latter case we− can− re- lated to the scattering length by peat all steps presented above and extract another cou- pling constantg ˜ from the interspecies scattering length, 1 0 a˜ in analogy with Eq. (14). We obtain thatg ˜ > 0 (im- Eb(K∗) 2 , (10) S 0 ≃−maS plying that bound states with total momentum K = 0 where m is the one-magnon mass, are never formed below the scattering continuum), and g˜0 > g0 for any S in the region 4 g always holds,− the relevant scatter- m = . (11) 0 0 S(4 J)(4 + J) ing length at low energies is the intraspecies scattering − length aS. The relation Eq. (10) between the binding energy and The scattering length, shown in Fig. 1, can be well the scattering length that holds for our microscopic lat- described, for small S > 1/2, by a sum of two terms + tice model is typical for a 1D Bose gas in the continuum (resonances): a λ−/(4 + J)+ λ /(J (S) J) [where S ≃ S S cr − interacting via an attractive contact potential, the Lieb- λ± are numerical prefactors]. We emphasize that, for 30 S Liniger model. S 1, the scattering length is in general much larger The scattering length aS can also be determined from than≥ the lattice spacing, as is evident from Fig. 1: For the scattering problem of two magnons for general S in S = 1, aS takes a minimum at J 3.3 with aS 80a. the thermodynamic limit. We have solved this problem, Additionally, the emergence of bound≃− states manifests≃ with the momenta of the two magnons (which partici- itself by a diverging scattering length at Jcr(S), where aS pate in scattering) taken in the vicinity of the same dis- changes its sign jumping from to + . Thus, bound persion minimum, k = k + k and k = k k. From −∞ ∞ 1 cl 2 cl − states are typically shallow, with a binding energy given the asymptotic form of the two magnon scattering state by Eq. (10). In addition, the minima in their dispersion wavefunction we extract the scattering phase shift δS(k) occur at incommensurate momenta K∗. for any S, For S =1/2, any J < 0 induces a two-magnon bound 18,20 state with total momentum K = π, i.e., aS > 0 lim Cr cos(rk + δS(k)). (12) r ∼ and there is no resonance at 4 1/2 case To extract the scattering length from the scattering phase where bound states with K = π are never below the two- shift we use the same relation as in a 1D continuum model magnon scattering continuum, and, as discussed above, of particles interacting via a short-range potential a resonance exists for 4

L 1000 2000 4000 ∞ vides equivalent results) −Jcr(S = 1) 2.25 2.16 2.13 2.11 z + S = S a†a , S = √2Sa , i − i i i i TABLE II: Finitesize dependence of critical values Jcr(S) S− = √2Sa†(1 a†a /2S) , (15) for the emergence of bound states below the minimum of the i i − i i twomagnon continuum of scattering states, for S = 1. we map Eq. (1) onto a bosonic problem:

Γ0(q; k, k′) ′ H = (2Sǫk )ak† ak + ak†+qak† ′ qakak , − ′ 2L − k k,k,q bound/antibound states if present, for selected values (16) of J [ 4, 0] in systems with up to L = 4000 sites where and several∈ − values of S. Table II shows the numerical ǫk = J cos k + cos2k (J cos kcl + cos2kcl) 0 determination of Jcr(S) for S = 1 and different sys- − ≥ tem sizes. Although finite-size effects are apparent, a is the single-magnon dispersion and k = arccos( J/4). cl − quadratic fit in 1/L to numerical data for Jcr(S) extrap- Note that in our normalization, the minima of the single- olates to Jcr 2.11 in agreement with the result de- particle dispersion are at zero energy: ǫ k = 0. The ≃ − ± cl termined directly in the thermodynamic limit (see the bare interaction vertex Γ0 is given by Γ0(q; k, k′) = 1 ′ preceding discussion and Table I). Vq 2 (Vk + Vk ) with Vk = 2J cos k + 2cos2k. The − cl cl chemical potential is = hs h, where hs is the clas- cl− 2 sical saturation field value hs = S(J + 4) /4. We are interested in the dilute regime 0. III. MAPPING OF THE SPIN HAMILTONIAN Concentrating on the low-energy→ behavior we arrive at, TO A DILUTE GAS OF BOSONS via a Bogoliubov procedure,25,26 a two-component Bose gas interacting via a δ-potential with Hamiltonian den- In this section, we describe our effective theory in the sity thermodynamic limit, for the case of a finite (though 2 ψα g0(S) 2 2 vanishingly small) density of magnons. The mapping eff = |∇ | + (n +n )+˜g0(S) n1n2 . (17) H − 2m 2 1 2 α to a dilute gas of bosons is motivated by the following observation: For S > 1/2, we have shown that aS is Here ψα, α = 1, 2 describe bosonic modes with mo- large. Hence in the dilute limit, we can safely neglect menta close to kcl (i.e., the Fourier transforms of ψα are the hard-core constraint and take the continuum limit. ± ψ1(k 0) akcl+k, ψ2(k 0) a kcl+k) and nα = ψα† ψα → ≈ → ≈ − For S =1/2, on the contrary, the scattering length aS is are the corresponding densities. The bare coupling con- typically of the order of a few lattice constants and only stants of the effective 1D model of the two-component for 4

Next, we apply a 1/S expansion to calculate the inter- action vertices and extract the coupling constants g0(S) A. Effective Hamiltonian andg ˜0(S). Using a standard ladder approximation the Bethe-Salpeter equation for the vertices Γ reads: Using the Dyson-Maleev transformation34 (the Dyson- Γ(q; k, k′) = Γ0(q; k, k′) (19) Maleev representation is used here for convenience. We 1 Γ0(q p; k + p, k′ p) have checked that the explicitly hermitean Holstein- − − Γ(p; k, k′) . 35 − 2SL ǫ + ǫ ′ Primakoff representation, to leading order 1/S, pro- p k+p k p − 5

-3 Setting the transferred momentum q = 0 in Γ and the 2 10 0 S=3/2 incoming momenta to k = k′ = k , we get (see Appendix

cl (S)

0 -0.04 S=1 A for details): -3 g 10 -0.08 -4 -3.5 -3 -2.5 -2 V V 1 (S) 0 kcl 0 0 g J Γ(0; kcl, kcl) 1+ − = S=3 2SL ǫk +p + ǫk p p cl cl− S=10 -3 S=∞ 1 Vp V0 -10 V0 Vkcl + 1 − Γ(p; kcl, kcl) − 2SL − ǫk +p + ǫk p p cl cl -3 − -2 10 1 (V V ) [Γ(p; k , k ) Γ(0; k , k )] -4 -3.95 -3.9 0 − kcl cl cl − cl cl . J −2SL ǫ + ǫ p kcl+p kcl p − FIG. 2: (Color online) Effective bare intraspecies interaction (20) g0(S), for S = 3 (solid line), representative of the generic behavior for SScr [dotdashed line: g0(S = ∞)]. Inset: g0 for S = 1 renormalized vertex with the bare vertex on the right and 3/2. hand side of Eq. (20), Γ(p; kcl, kcl) 2ǫp, which is possi- ble since there are no infrared divergences.→ Regularizing the left hand side of Eq. (20) as in Refs. 32,36, and using To corroborate this, using ED for Eq. (1) and S = 1 Eq. (18) we extract the coupling constants of the effective with periodic boundary conditions, we have calculated model. The analytical expression for g0(S) is the ground-state momentum of the states with a small, F but finite number of magnons, which is incommensurate, g0(S)= F (J2 8) (21) supporting the picture of a uniform chiral state in the 1 − J S(16 J2)3/2 repulsive case J >Jcr, and a collapsed state in the at- − | | − tractive case J 1/2) can be trusted only for J 4, (the derivation is presented in detail in Appendix A; see where the scattering length becomes much larger→ than − Eq. (A25) for the expression for F˜). the lattice constant. In that case we can easily incorpo- 26 From Eqs. (21) and (23), we notice that g0(S) < g˜0(S) rate the exact hard-core constraint into our formalism for 4 4 in the spin-1/2 when approaching the ferromagnetic Lifshitz point,→ frustrated ferromagnetic Heisenberg chain.− 12–14,20 ∞ g0 m 0. Going back to S > 1/2, at lower M, corresponding to | The| → effective bare intraspecies interaction is depicted higher densities of magnons, the hard-core nature of spins in Fig. 2 and behaves as g (S) [J J (S)] for J eventually prevails as well, resulting in a uniform ground 0 ∼ − cr → Jcr(S). The scattering length is related to the effective state at a nonzero momentum. However, as already men- coupling constant by Eq. (14), signaling a resonance at tioned, from the finite-size analysis of the two-magnon Jcr(S). Thus, we see that for S

z (a) S=1, L=128 S=1, L=128 J=-3 (jump at S =97) 1 0.5 J=-1 (no jump) ∆ Mjump 0.8 0.4 > i 0.6 z 0.3 M(h) 1- 97 for J = 3), (c) M(h) for S = 3/2 at J = −3.5 (all for L = 128). magnons collapse into the center of the system, whereas− for actual ground states below the metamagnetic transi- tion, the magnetization profiles become flat. By contrast, repulsive upon increasing the magnon densities. Thus, in the case of J = 1 where the transition to the fully the state below the jump (i.e., 0 Jcr, i.e., it is a translationally uniform chiral state. B. Central charge

IV. DMRG RESULTS Our effective theory developed in Section III suggests that the gapless phase in the region hc

Spin S=1, PBC (b) J=-2, M=1/2 (c) J=-3, M=5/16 4 4 L=64 0.6 Spin S=1 (a) J=-1, M=1/2 L=128

0.4 jump 3 3 M 0.2 ∆ hsat (l) h

vN 0 c h

S 2 2 -4 -3 -2 J gapless (chiral) 1 DMRG Fully polarized Fit with c~1 1

0 (Double-)Haldane 0 10 20 30 40 50 60 10 20 30 40 50 60 10 20 30 40 50 60 0 l l l -4 -3 -2 -1 0 J [t!] FIG. 5: (Color online) DMRG results for the vonNeumann entropy SvN (l) in the gapless phase hc 54 from the fit). In this figure, we display as well as finitesize scaling (not shown here) supports that results for periodic boundary conditions and L = 64 sites. both Mjump and hc are finite in extended regions of J.

straightforward to measure SvN (l) with this numerical corresponding spin gap hc is a bit subtle. Namely, a Hal- method. dane chain with open boundaries gives rise to spin-1/2 47 Some typical DMRG results (squares) for systems of excitations at the open ends. Since we have two chains (for small J ), we have a total of four spin-1/2 end spins. L = 64 and periodic boundary conditions are presented | | in Fig. 5. We have fitted the expression Eq. (25) to our Hence, the spin gap in Fig. 6 is determined from numerical data (shown as solid lines in the figure) and ob- z z tain c = 1.0 0.1 in all examples. Therefore, we expect hc = E(S = 3) E(S = 2) , (26) ± − the gapless phase to be a (chiral) one-component liquid. where E(Sz) is the ground-state energy in a sector with Note, though, that at both small M and J , where the z | | a given total S . convergence of DMRG is notoriously difficult, we can- It is worth emphasizing several differences with the not completely rule out the presence of a c = 2 region, phase diagram of the spin-1/2 version of Eq. (1). First, which, however, is irrelevant for the main conclusions of for S = 1, there are no multipolar phases, which oc- our work. cupy a large portion of the corresponding spin-1/2 phase diagram.11–14,20 Second, the spin-1/2 system features an instability towards nematic order,11,12,18 which can be C. Phase diagram for S = 1 excluded on general grounds for integer spin-S chains,19 even for 0 0, see Ref. 46), (ii) a gapless (chiral) finite-field phase for hc 4) and quantum fluctuations − − small rendering it difficult to resolve it numerically), con- (1/2 g(S = ) . (A8) ∞ 2kcl 0 − kcl ∞ Acknowledgments Thus, classically (S ,m 0), we haveg ˜0 > g0 for 4 g0, Γ(p; kcl, kcl) 2ǫp (which is possible because there are no infrared divergences→ anymore), and use that V V = p − 0 = g(S)n (A3) 2(ǫp ǫ0). Equation (A10), after passing to infinite sys- tem− size, becomes and forg ˜0

(J + 4)2 g0(S) g(S = )= V V = (A7) g(S)= , (A13) ∞ 0 − kcl 4 1+ g0(S)√2m/(π√) 9 where we have introduced the intraspecies Lieb-Liniger and coupling constant g0(S) as in Eq. (21). Note that all integrals presented in this section are evaluated analyti- cally, which is a nice feature of the 1/S treatment of our π k − cl problem. 1 Vp + V 2kcl p 2Vkcl I2 = dp − − − (Γ˜p Γ˜0) Now we outline the calculation ofg ˜(S). We denote −4πS 2ǫkcl+p − kcl Γ(p; kcl, kcl)= Γ˜p and obtain the Bethe-Salpeter equa- − − π kcl tion for − Γ˜0 Vp + V 2k p 2Vk π π dp − cl− − cl . (A20) 1 1 V V −4πS 2ǫ + C/2S ˜ ˜ p kcl ˜ kcl+p Γ0 = V0 Vk + dpΓp dp − − Γp kcl cl − − 4πS − 4πS 2ǫkcl+p π π − − and π The last terms in I1 can be written as 1 Γ˜ 2k = V 2k Vk + dpΓ˜p − cl − cl − cl 4πS π − k π ˜ − cl Γ 2kcl Vp + V 2kcl p 2Vkcl 1 V 2kcl p Vkcl − dp − − − = dp − − − Γ˜p. (A14) − 4πS 2ǫkcl+p + C/2S −4πS 2ǫkcl+p πk π − − cl − k ˜ ˜ ˜ − cl Adding these two equations givesg ˜(S)= Γ0 + Γ 2k , Γ 2kcl Vp + V 2kcl p (V0 + V 2kcl ) − cl − dp − − − − π − 4πS − 2ǫkcl+p 2 π kcl g˜(S) = V + V 2V + dpΓ˜ − − 0 2kcl kcl p k − − 4πS − cl π Γ˜ 2k V0 + V 2k 2Vk − − cl − cl cl π dp − . (A21) − 4πS 2ǫkcl+p + C/2S 1 Vp + V 2kcl p 2Vkcl πkcl dp − − − Γ˜p.(A15) − − −4πS 2ǫkcl+p π − We divide the last term in Eq. (A15) into two pieces, Similarly, for the last terms in I2 we get, π

1 Vp + V 2kcl p 2Vkcl dp − − − Γ˜p = I1 + I2 , (A16) −4πS 2ǫkcl+p π π kcl − ˜ − Γ0 Vp + V 2kcl p 2Vkcl where dp − − − = − 4πS 2ǫkcl+p + C/2S kcl k − − cl 1 Vp + V 2kcl p 2Vkcl π k I1 = dp − − − ˜ − cl −4πS 2ǫk +p Γ0 Vp + V 2kcl p (V0 + V 2kcl ) cl − − − πkcl dp − − − −4πS − 2ǫkcl+p ˜ ˜ ˜ kcl (Γp Γ 2kcl + Γ 2kcl ) (A17) − × − − − π k − cl and Γ˜0 V0 + V 2k 2Vk dp − cl − cl . (A22) π kcl −4πS 2ǫ + C/2S 1 − V + V 2V kcl+p p 2kcl p kcl kcl I2 = dp − − − − −4πS 2ǫkcl+p k − cl (Γ˜ Γ˜ + Γ˜ ) . (A18) × p − 0 0 Noting that Note that, for convenience, we have shifted the first Bril- louin zone, ( π, π) ( π k , π k ). Shifting the − → − − cl − cl T-matrix off-shell, we have (the integral with a dash de- π k − cl notes the principal value) 1 Vp + V 2kcl p (V0 + V 2kcl ) dp − − − − − 4πS − 2ǫ kcl kcl+p − kcl 1 Vp + V 2kcl p 2Vkcl − I1 = dp − − − (Γ˜p Γ˜ 2k ) k − cl − cl −4πS − 2ǫkcl+p − 1 Vp + V 2kcl p (V0 + V 2kcl ) π kcl = dp − − − − − − −4πS − 2ǫ kcl kcl+p ˜ − π kcl Γ 2kcl Vp + V 2kcl p 2Vkcl − − − dp − − − (A19) 2 4πS 2ǫ + C/2S J 8 − kcl+p = − , (A23) πk − − cl 16S 10 and gathering all contributions, Eq. (A15) takes the form S=1, L=48, PBC 1 (a) J=-3.5 (b) J=-1 1 π 2 ∆M J 8 V0 +V 2k 2Vk dp 0.8 jump g˜(S) 1+ − + − cl − cl  16S 16πS ǫp + C/4S  0.6

π M(h) M(h) − 0.5  8+ J 2  0.4 = V0 + V2kcl 2Vkcl + − 4S 0.2 h 0 c 1 V + V 2V 0 0 p kcl p+kcl kcl ˜ ˜ 0 0.05 0.1 0 0.5 1 1.5 2 2.5 3 dp − − (Γp kcl Γ 2kcl ) −4πS − 2ǫp − − − h h π − π FIG. 7: (Color online) Magnetization curves M(h) for S = 1 1 V + V 2V p kcl p+kcl kcl ˜ ˜ (a) J = −3.5 and (b) J = −1 (L = 48), calculated with dp − − (Γp kcl Γ0).(A24) −4πS 2ǫp − − DMRG on systems with periodic boundary conditions. In 0 panel (a), we clearly see the metamagnetic jump of height Now we can plug the zeroth-order vertices into the Mjump and in panel (b), the Haldane gap that defines hc right hand side of Eq. (A24), in the spirit of the 1/S shows up as a zerofield magnetization plateau. ˜ expansion: Γp 2ǫp. Noting that V0 + V 2kcl 2Vkcl = ˜ → − − F + O(1/S), where here demonstrate that all main features can be seen with 2 both OBC and PBC, namely the existence of the meta- ˜ 8+ J F = V0 + V2kcl 2Vkcl + (A25) magnetic jump and the Haldane gap. Moreover, the − 4S quantitative results are comparable, except for the ex- 0 1 Vp k + Vp+k 2Vk pected finite-size effects due to the incommensurability. − cl cl cl dp − (ǫp kcl ǫ 2kcl ) − 2πS − 2ǫp − − − π − π 1. Magnetization curves from periodic boundary 1 V + V 2V p kcl p+kcl kcl conditions for S = 1 dp − − (ǫp kcl ǫ0), − 2πS 2ǫp − − 0 First, we discuss some examples of magnetization in the same way as for g(S) we obtain curves obtained for PBC. We kept up to m = 1600 states for the PBC DMRG computations presented here. In g˜ (S) g˜(S)= 0 (A26) addition, we used exact diagonalization for those sectors 1+˜g0(S)√2m/(π√) which are sufficiently small. Therefore, in particular the data close to the saturation field are free of truncation withg ˜ (S) given in Eq. (23). 0 errors. We see that in order 1/S, quantum fluctuations do not Figure 7 shows the magnetization curves for J = 3.5 modify the relationg ˜ (S) >g (S) for J < 0. Therefore, 0 0 [panel (a)] and J = 1 [panel (b)] for L = 48 and PBC.− the interspecies interaction is always positive (repulsive) As in the results for OBC,− we observe the presence of the for any J and S > 1/2, and behaves asg ˜ (S) (J + 4)2 0 metamagnetic jump (here in the case of J = 3.5) and for J 4 for all S. A hard-core constraint,∼ stemming the Haldane gap (see the J = 1 curve), which− manifests from the→− mapping of spins to bosons, is not included in itself as a plateau in the magnetization− curve at M = 0. our theory. While it is well-known how to treat the hard- The latter defines the critical field h that separates the core constraint in an exact way for S =1/2,26 this is not c gapped Double-Haldane phase from the gapless finite-M the case for S > 1/2. phase (compare Fig. 5). Note that at intermediate M, the PBC data show spu- z Appendix B: DMRG results for periodic boundary rious small steps with ∆S > 1. We have checked that conditions these features disappear as one goes to larger system sizes. The DMRG results presented in the main text were computed for open boundary conditions (OBC), except for those in Fig. 5 where we used periodic boundary con- 2. Comparison of OBC vs PBC for S = 1 ditions (PBC). Alternatively, one may compute all the data, e.g., the magnetization curves M(h) using PBC. Figure 8 contains the results for the Haldane gap that This approach is generally expected to suffer from (i) defines the critical field hc separating the gapped zero- slower convergence with respect to the number of states field phase from the gapless phase at finite magnetiza- 28 kept in the DMRG runs and (ii) large finite-size ef- tions, the saturation field hsat, and the jump height (in- fects due to the incommensurability in the problem. We set), comparing data for OBC (open symbols) with data 11

4 PBC 0.6 Spin S=1, L=64 of L = 64 for both PBC and OBC. OBC h (PBC) For PBC we can determine the spin gap from 0.4 jump 3 c M h (PBC)

∆ sat 0.2 PBC z z h hc = E(S = 1) E(S = 0) , (B1) 0 sat − h 2 -4 -3 -2 hc J z gapless (chiral) where, as in Eq. (26), E(S ) is the ground-state energy in Fully polarized z 1 a sector with a given total S . The good agreement be- tween the OBC and PBC results for hc in Fig. 8 confirms (Double-)Haldane that it is indeed appropriate to use Eq. (26) for OBC. 0 -4 -3 -2 -1 0 The PBC data for ∆Mjump in Fig. 8 suffer from the J presence of several peaks, resulting in a non-monotonic FIG. 8: (Color online) Phase diagram of the frustrated ferro dependence on J. This is due to the incommensurabil- magnetic S = 1 chain, comparing DMRG results from systems ity in the finite-magnetization region, which is incom- with OBC (open symbols) to results from systems with PBC patible with the lattice vectors of a system with peri- (solid symbols). L = 64 in all cases. Lines are guides to the odic boundary conditions that is translationally invari- eye. ant. Therefore, in the main text we focus on the discus- sion of DMRG data from systems with OBC. The results from OBC and PBC data for the saturation field hsat, from PBC (solid symbols). Here we choose a chain length however, agree very well with each other.

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