The disordered-free-moment phase: a low-field disordered state in spin-gap antiferromagnets with site dilution

Site dilution of spin-gapped antiferromagnets leads to localized free moments, which can order antiferromagnetically in two and higher dimensions. Here we show how a weak magnetic field drives this order-by-disorder state into a novel disordered-free-moment phase, characterized by the formation of local singlets between neighboring moments and by localized moments aligned antiparallel to the field. This disordered phase is characterized by the absence of a gap, as it is the case in a Bose glass. The associated field-driven quantum phase transition is consistent with the universality of a superfluid-to-Bose-glass transition. The robustness of the disordered-free-moment phase and its prominent features, in particular a series of pseudo-plateaus in the magnetization curve, makes it accessible and relevant to experiments.

Valence bond solids (VBS) in spin-gapped antiferromagnets represent some of the most fundamental examples of quantum-disordered states in condensed matter systems.The nature of such states is by now well understood theoretically and has been extensively verified experimentally.A variety of mechanisms, such as increased strength of certain bonds in the magnetic Hamiltonian [1,2], magnetic frustration [3], and the Haldane mechanism [4], can render classical Néel order unstable towards the formation of local singlets, which arrange themselves into a VBS.Evidence for such phases has been found in a large number of magnetic compounds, ranging from Haldane chains [5], to spin ladders [6], to weakly coupled dimer systems [7,8].A central focus of theoretical and experimental investigations has been the effect of doping on such states.In particular, it was soon realized theoretically, [9,10] and observed experimentally [11,12], that doping a VBS with static, non-magnetic impurities leads to the intriguing phenomenon of order-by-disorder (OBD): free S = 1/2 magnetic moments appear close to the impurity sites and interact effectively via a longrange network of unfrustrated (albeit random) couplings, which decay exponentially with the inter-moment distance.These interactions, although weak, are sufficient for the free moments (FMs) to order antiferromagnetically at experimentally relevant temperatures [11].
Given the random nature of the inter-moment couplings, the OBD state induced by doping is extremely inhomogeneous, as it contains a large variety of energy scales which depend exponentially on the spatial distribution of the impurities.In this paper we study the evolution of the OBD state upon application of a magnetic field, which represents a straightforward experimental probe of energy scales in magnetic systems.Precisely due to the large distribution of the effective couplings between the FMs, we find an amazingly rich response of the system to the applied field.The field scan reveals that the long-range order in the system is extremely ten- uous for large spin gaps, and its field-driven destruction leaves behind local singlets (or spins oppositely polarized to the field) on even-(or odd-)numbered clusters of FMs, which are coupled at energies higher than the scale characteristic for Néel order (see Fig. 1).The local polarization fields for these FM clusters cover a continuous range, so that the magnetization process with increasing field continues also after destruction of the OBD, and the resulting disordered-free-moment (DFM) phase [13] is gapless.The DFM phase persists up to the sizable field which polarizes all the FMs, and the magnetization curve within this phase shows prominent features of intermediate pseudo-plateaus, still retaining a finite albeit extremely small slope, related to the distribution of the strongly interacting clusters of FMs.
To quantitatively investigate the field response of a site-diluted spin-gapped antiferromagnet, we focus our attention on a two-dimensional S = 1/2 model of weakly coupled dimers [2,16], whose Hamiltonian reads Here i runs over the two sublattices (A and B) of a square lattice, x and ŷ are the two lattice vectors, and ǫ i is the random dilution variable taking values 0 and 1 with probability p and 1 − p respectively.h = gµ B H is the applied field.The couplings J > J ′ determine the subset of strong antiferromagnetic bonds: for J ′ /J < 0.523.. [2] the bond anisotropy stabilizes a dimer-singlet ground state against the conventional Néel ordered state of the square-lattice antiferromagnet.All the results presented here refer to the field and doping effects deep within the dimer-singlet regime at J ′ /J = 1/4.
The presence of lattice vacancies induces FMs which are localized in the vicinity of the unpaired spins which have lost their J-neighbor.Perturbation theory [10,13,17] provides an effective coupling J ij ≈ (−1) i−j−1 (J 1 /r) exp(−r/ξ 0 ) between these FMs, where r = |i − j|, ξ 0 is the correlation length of the undoped system.We choose J 1 ≈ J ′ exp(1/ξ 0 ) in order for J ij to correctly reproduce the limit J ′ for neighboring unpaired spins.For a deeper understanding of the Hamiltonian Eq. ( 1), it is illuminating [18] to study an effective model for the network of FMs, consisting of randomly distributed S = 1/2 spins with effective couplings J ij We investigate the original and the effective Hamiltonian, given by Eq. ( 1) and (2), using Stochastic Series Expansion (SSE) Quantum Monte Carlo simulations based on the directed-loop algorithm [19].For the original Hamiltonian Eq. ( 1) we study L × L lattices up to L=40 with dilution p = 1/8, whereas for the effective model Eq. ( 2) we randomly distribute spins on the same lattice sizes with a density p equal to that of the vacancies in the original model.Disorder averaging is typically performed over ≈ 300 realizations.The ground-state properties are systematically obtained using a β-doubling approach [20].Inverse temperatures up to βJ = 2 15  are necessary to observe the physical T → 0 behavior.In the following, we focus our attention on the uniform magnetization per spin m u = 1/N s i S z i , where N s is the total number of spins in the system considered; on the uniform susceptibility χ u /J = ∂m u /∂h; on the staggered magnetization m s = S ⊥ (π, π)/L 2 , where S ⊥ (π, π) = 1/(2L 2 ) ij S x i S x j + S y i S y j is the transverse static structure factor; on the correlation length ξ, extracted from the q-dependent structure factor; and on the superfluid density ρ s = 1/(2βJ) W 2 x + W 2 y , where W x(y) are the winding numbers of the SSE worldlines.In the absence of a magnetic field, the effective couplings J ij give rise to long-range magnetic order of the network of coupled free moments.In particular, for J ′ /J=1/4 and p=1/8 we extrapolate a staggered magnetization m s = 0.032(3) in the thermodynamic limit.Although the couplings J ij range between 0 and J ′ , the average coupling strength responsible for the long-range order turns out to be much smaller than J ′ [13].Fig. 2 shows the evolution of the ordered moment under application of a field.It reveals that the antiferromagnetic order is already destroyed at a field h ≪ J ′ , namely at h = h DFM = 0.007 (1).Yet, a striking feature of this disordered phase is that the destruction of long-range order is not accompanied by the full polarization of the FMs, as it would ordinarily happen in a homogeneous antiferromagnet.At the critical field h DFM the uniform magnetization is found to be m u = 0.0208 (6), much less than the value m u = pS = 1/16 corresponding to fully polarized FMs, which is attained at a much larger field h plateau /J ≈ 0.5.Consequently the DFM phase, appearing between h DFM and h plateau , is highly unconventional, retaining a finite uniform susceptibility and a gapless spectrum.These unconventional properties have also been observed in the magnetic Bose-glass phase of triplet quasiparticles living on intact dimers [13,14,15,21].Later we will argue that these two phases bear indeed strong analogies, although involving different degrees of freedom.
For h > h plateau the saturation of the magnetization of FMs leads to the full restoration of a gapped disordered phase due to the field [17].Once the field reaches the value corresponding to the gap of the clean system, h = ∆ = 0.60(1), a further Bose-glass phase is established in which rare clean regions develop a local magnetization without the appearence of spontaneous order, corresponding to localized triplet quasiparticles [13,15].A delocalization transition of the triplet bosons into a superfluid condensate corresponds to a further onset of long-range transverse order (m s > 0) at even higher fields [13,15].As shown in Fig. 2, the main features of the field dependence m u and m s for h < h plateau in the doped coupled-dimer model of Eq. ( 1) are very well reproduced by the effective model Eq. ( 2), for which we take ξ 0 = 1 as found by simulations at h = 0 and p = 0.In particular, the fundamental appearence of a DFM phase with m s = 0 and χ u > 0 is confirmed in the effective model.This reveals that the FMs are essentially the only degrees of freedom responding to a field h < h plateau in the doped coupled-dimer system.The novel quantum phase transition (QPT) between the OBD and the DFM phase is studied using finite-size scaling analysis of the correlation length, and of the staggered magnetization where δh = h−h DFM , as shown in Fig. 3.This allows us to extract the critical exponents z, ν and β.For the original Hamiltonian Eq. ( 1) we find z = 2.0(1), ν = 1.0(1) and β = 0.9 (1).These estimates are also confirmed in the effective FM model.The above exponents are fully consistent with those of the 2d superfluid-to-Bose-glass (SF-BG) QPT previously studied in diluted bilayers [13], and with the exponents found at the BG-SF transition for higher fields for the model Eq. ( 1) [22].In particular z is in agreement with the general theoretical prediction z = d [14], and ν satisfies the fundamental Harris criterion ν ≥ 2/d.Altogether the present results and those of Refs.13, 22 point towards a general SF-BG universality in d = 2 for order-disorder transitions at which the uniform susceptibility χ u remains finite, corresponding to the absence of a gap.
In the DFM phase, the magnetization curves of the original Hamiltonian Eq. ( 1) and the effective model Eq.
(2) show a dramatic feature: beside the large pleateau appearing at h ≥ h plateau , one observes the presence of apparent intermediate plateaus at around 3/4 and 95% of the saturation magnetization.A detailed study of the temperature-dependent susceptibility in this field region reveals that these features are actually pseudoplateaus (PPs), which retain an extremely small slope (Fig. 2).For both H and H FM the first PP extends up to h ≈ 0.7J ′ ; a second PP markedly appears around h ≈ 1.2J ′ for H (it is rounded off for H FM [22]); the true saturation plateau is only attained at h ≈ 2J ′ .These fundamental features can be understood within the picture of strongly interacting clusters of FMs in the DFM phase.As shown in Fig. 1, the zero-field OBD phase is essentially inhomogeneous due to the random nature of the couplings.A majority of FMs are spaced from each other by an average distance r = p −1/d , large in the small dilution limit, and interact via weak average couplings J eff ∼ pJ ′ [13].However, fluctuations in the spatial distribution of the impurities also lead to small clusters of free moments located on neighboring sites, and thus interacting with much stronger couplings J ′ .If antiferromagnetically coupled in even-numbered clusters, the strongly interacting FMs participate only marginally in the OBD state of the system, and have a significant singlet component in their ground state wave function.This is directly revealed in a histogram of the bond energies (Fig. 4 Applying a magnetic field h J eff has clearly the effect of destroying the long-range order of the FMs, but at the same time the FM singlets are left intact while oddnumbered clusters are not fully polarized (Fig. 1), with the fundamental consequence that the antiferromagnetic order disappears but the FMs are far from saturation.This is clearly seen in the histogram of the local magnetic moments S z i of the unpaired spins only (Fig. 4(b)): for small fields, a double-peak structure appears with a peak at S z i = S corresponding to fully polarized FMs, and a strong quantum peak at S z i = 0 corresponding to FM singlets.The large tails for 0 < S z i < S and for S z i < 0 come instead from FMs in odd-numbered clusters.In fact, we can resolve two more peaks at S z i = 1/3 and S z i = −1/6 at larger fields, which correspond to partially polarized spins in FM trimers.Local FM clusters have widely different local gaps to full polarization, both due to their geometric structure (dimers, trimers, quadrumers, etc.), and to the local field they experience from the other FMs.Yet the distribution of rare FM clusters clearly assigns dominant statistical weight to the dimers, and this simple geometric fact is the reason for the appearence of the first PP: the magnetization process nearly stops until the local gap of the dominant FM dimers is overcome at h J ′ .Nonetheless, the slope remains finite because the FM dimers have a distribution of local gaps.The magnetization value and the field location of the PP can be quantitatively related to the statistics of FMs clustered in dimers [22].Analogously, one can quantitatively associate the second plateau with the statistics of the FM trimers [22].Higher-order plateaus associated with larger local polarization fields should be expected, but they cannot be resolved within the given numerical accuracy.
From the above data a clear picture of the DFM phase emerges: in this phase, a majority of the FMs are polarized, but antiparallel spins exist, corresponding to rare FM clusters.Upon a spin-to-hardcore-boson transformation, these antiparalell spins take the nature of bosonic spin-down quasiparticles (↓-QPs) localized on rare regions of the lattice [22].This aspect connects with the ordinary picture of a Bose glass [14], and it further endows the OBD-to-DFM quantum phase transition with the nature of a localization transition: in the OBD phase the ↓-QPs form a superfluid condensate, which is progressively depleted by the applied field (acting as a negative chemical potential), up to the point where the ↓-QPs undergo localization into a Bose-glass state, losing super-fluidity but retaining compressibility, which corresponds to a finite uniform susceptibility.
It is evident from the above results that the DFM phase is relevant for experiments on site-diluted spin-gapped antiferromagnets which display an OBD phase in zero field.The fundamental condition for the observation of the DFM phase is that the spin gap of the pure system be much larger than the maximum energy scale of the FM interaction (average inter-dimer coupling in weakly coupled dimer systems, inter-chain coupling in Haldane chains).This condition is necessary to ensure that the physics of the field response of the FMs is well separated in energy from that of the field-induced ordered state (for a detailed discussion, see Ref. 13).Joint magnetometry and neutron scattering measurements at relatively low fields should be sufficient to fully pinpoint this phase by demonstrating the absence of spontaneous order and the finite susceptibility down to zero temperature.Furthermore, NMR measurements can show the rich structure of the distribution of local magnetic moments, similar to the histogram of Fig. 4. The low-field location of the DFM phase and its strong physical signatures in the magnetic observables make it the most accessible novel disordered phase in quantum magnets with lattice randomness.

FIG. 1 :
FIG.1:(Color online) Quantum phase transition between the order-by-disorder (OBD) phase and the novel disorderedfree-moment (DFM) phase in a diluted coupled-dimer system.Left panel : At zero applied field, spins on intact dimers form singlets (solid ellipses), dimers of free moments (FMs) have a strong singlet component (dashed ellipse), whereas the other FMs (blue arrows) participate in the OBD state.Right panel : Upon applying a field, the OBD-FMs are mostly polarized, but local singlets and localized down-spins can survive on clustered FMs, leading to the DFM phase.

FIG. 2 :
FIG.2:(Color online) Field scan of uniform magnetization, uniform susceptibility, and staggered magnetization in the diluted system (a) and in the effective model (b).The lower right panel in (a) shows the temperature dependence of the uniform susceptibility at h = 0.175J, indicating a saturation to a small but finite value as T → 0. PL=plateau, BG=Bose glass, SF=superfluid.

FIG. 3 :
FIG. 3: (Color online) Finite-size scaling of the correlation length, staggered magnetization and spin stiffness in the vicinity of the quantum critical point hDF M .
(b)) E b = J b S 1,b • S 2,b where J b = J, J ′ and (1, b), (2, b) are the two neighboring lattice sites participating in the bond b.Beside the peak at E b ≈ −3J/4, corresponding to singlets on intact dimers, a further peak at E b ≈ −3J ′ /4 is observed, corresponding to FM dimer singlets, as well as a peak at E b ≈ −J ′ /2 corresponding to FM trimers.
Useful discussions with T. Giamarchi, W. Li, O. Nohadani, P. Sengupta, M. Sigrist are gratefully acknowledged.It is a pleasure to thank M. Vojta for a remark which has sparked the present investigation.T.R. is supported by the E.U. through the SCALA integrated project.R.Y. and S.H. are supported by DOE under grant No. DE-FG02-05ER46240.