Long range interactions in antiferromagnetic quantum spin chains
B. Bravo, D. C. Cabra, F. A. Gómez Albarracín, G. L. Rossini
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Long range interactions in antiferromagnetic quantum spin chains
B. Bravo, D.C. Cabra, F.A. G´omez Albarrac´ın, and G.L. Rossini IFLP-CONICET and Departamento de F´ısica, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina (Dated: August 30, 2017)We study the role of long range dipolar interactions on antiferromagnetic spin chains, from theclassical S → ∞ limit to the deep quantum case S = 1 /
2, including a transverse magnetic field. Tothis end, we combine different techniques such as classical energy minima, classical Monte Carlo,linear spin waves, bosonization and DMRG. We find a phase transition from the already reporteddipolar ferromagnetic region to an antiferromagnetic region for high enough antiferromagnetic ex-change. Thermal and quantum fluctuations destabilize the classical order before reaching magneticsaturation in both phases, and also close to zero field in the antiferromagnetic phase. In the ex-treme quantum limit S = 1 /
2, extensive DMRG computations show that the main phases remainpresent with transition lines to saturation significatively shifted to lower fields, in agreement withthe bosonization analysis. The overall picture keeps close analogy with the phase diagram of theanisotropic XXZ spin chain in a transverse field.
PACS numbers: 75.10.Pq,75.10.Jm, 75.30.Kz
I. INTRODUCTION
Long range interactions in quantum systems have re-cently attracted much attention. While short-range in-teractions are naturally present in quantum gases, longer-range interactions are much harder to control. To in-vestigate them, ultracold gases of particles with largemagnetic or electric dipole moments, atoms in Ryd-berg states, or cavity-mediated interactions have beenstudied. These experiments also open the possibility ofsimulating dipole-dipole interactions in one dimensionalspin chains. . Since then, theoretical and numerical in-vestigation of dipolar spin chains has been revitalized. On the other hand the inclusion of dipolar, and moregenerally long-ranged interactions, in classical and quan-tum models has proven to modify in different degrees theoutcoming physics .Motivated by these studies, we consider the competi-tion between short range antiferromagnetic exchange andlong range dipolar interactions in spin chains in order toexplore the presence of novel phases, either ordered ordisordered. We also include an external magnetic field,transverse to the dipole-dipole induced anisotropy, whichcompetes with both antiferromagnetic and dipolar clas-sical ordering.The present analysis follows different approaches thatallow to explore from the classical S → ∞ limit to thedeep quantum case S = 1 /
2, as well as parameter regionswhere antiferromagnetic exchange dominates over dipo-lar interactions and the other way around. For large S weperform classical energy analysis, classical Monte Carlosimulations and linear spin wave fluctuations. On theother extreme, for spin S = 1 / S = 1 / S = 1 / S = 1 / II. MODEL AND CLASSICAL DESCRIPTION
We consider a spin chain in the x direction with longrange dipolar interactions and nearest neighbours antifer-romagnetic exchange coupling J >
0. The Hamiltonianreads H = J X i ~S i · ~S i +1 + µ X i In this section we review the classical ground configu-rations of the Hamiltonian in Eq. (1). Dipolar interac-tions tend to align classical spins along the x axis whilethe transverse magnetic field induces a tilting towardsthe z axis. Ferromagnetic exchange couplings J < , but the presentantiferromagnetic couplings J > xy plane. One can then search for theclassical ground state configurations within the manifoldparameterized by ~S i = S (sin θ cos φ, ( − i sin θ sin φ, cos θ ) (2)where θ is the tilting angle w.r.t. the magnetic field and φ describes a staggered deviation from the chain directioninto the xy plane. The energy per site of such configura-tions reads E S = J (cid:0) cos θ + sin θ cos(2 φ ) (cid:1) + ζ (3) D (cid:18) cos θ − θ cos φ − 34 sin θ sin φ (cid:19) − hS cos θ (3)where ζ (3) = P ∞ n =1 1 n ≈ . J ≤ J c ≡ ζ (3) D and below a saturation field h DF sat ( D ) =6 ζ (3) SD , exhibits a minimum at φ = 0 (parallel spins inthe xz plane) and tilting angle θ DF ( h, D ) = arccos (cid:18) h ζ (3) SD (cid:19) (4)defining a ”dipolar-ferromagnetic” (DF) phase. Thisphase has a Z mirror degeneracy under exchange θ →− θ . Instead, for J ≥ J c and below a saturation field h AF sat ( J, D ) = 4 S ( J + ζ (3) D ), the minimum appears at φ = ± π/ θ AF ( h, J, D ) = arccos (cid:18) h S ( J + ζ (3) D ) (cid:19) (5) defining an antiferromagnetic (AF) phase where spins liein staggered tilted directions in the yz plane. This phasehas a Z discrete translation degeneracy along the chaindirection. The classical phase diagram in the h vs J planeis shown in Fig. 2. FMDF AF FIG. 2: (color online) Classical phases are described in the h vs. J plane: tilted dipolar-ferromagnetic (DF), tilted an-tiferromagnetic (AF) and magnetic saturation (FM). Shadedregions indicate predominance of both thermal and spin wavequantum fluctuations. We show below that the boundariesto saturation are significantly shifted down in the quantum S = 1 / At the specific value J = J c > φ that can continuously interpolate between thedipolar-ferromagnetic and the antiferromagnetic phases.Also notice that at zero field the classical AF phase pos-sesses an extra U (1) degeneracy associated to rotationsaround the chain direction. Such degeneracies enhancethe role of classical and quantum fluctuations, as we dis-cuss in the next Section. III. CLASSICAL AND QUANTUMFLUCTUATIONSA. Monte Carlo simulations We study here the effects of classical thermal fluctu-ations on the model presented in Section II, in order tocheck the stability of the phase boundaries of the zerotemperature classical phase diagram shown in Fig. 2. Tothis end we have run Monte Carlo simulations using thestandard Metropolis algorithm combined with overrelax-ation (microcanonical) updates for chains of L = 300sites. Finite size systems with periodic boundary con-ditions are considered, so that long range interactionsare taken up to L/ Monte Carlo steps (mcs) were dedicated to thermaliza-tion, lowering the temperature with the annealing tech-nique at a rate T n +1 = 0 . × T n . Measurements are thentaken during the 2 × subsequent mcs. The resultspresented for each data point describe the average over100 independent simulations.In order to describe the effects of thermal fluctuationsin both DF phase, J < J c and AF phase, J > J c , wedefine two different order parameters: a mean magneti-zation m defined as m = q m x + m y + m z (6)where m α = L P i S αi ( α = x, y, z ) and an antiferromg-netic magnetization m AF defined as m AF = q m x + ( m stagy ) + m z (7)where m stagy = L P i ( − i S yi picks the staggered contri-bution of the y components.We show m and m AF as functions of the external mag-netic field for different values of J and T /D in Fig. 3.The top panel shows m for two values of J < J c at T /D = 0 . 02. The external magnetic field is normalizedby the corresponding T = 0 saturation value h DF sat (seesection II). There is a clear dip at h/h DF sat < J , due to temperature effects. This is illustratedin the inset of the top panel, which shows m vs h/h DF sat for J/D = 0 . m AF versus h/h AF sat for three valuesof J > J c and T /D = 0 . 02, while the inset shows theeffects of temperature for J/D = 1. m h/h satDF J/D=0.5J/D=0.2 m A F h/h satAF J/D=0.8J/D=1J/D=1.2 FIG. 3: (color online) Order parameters m (top) and m AF (bottom) as functions of the external magnetic field scaledwith the corresponding T = 0 saturation values. Temperatureis set at T /D = 0 . 02 for different values of J/D for J < J c (top) and J > J c (bottom). A dip close to saturation is observed, as expected, inboth DF and AF phases which increases with tempera- ture. In the AF phase, on the other hand, one observesa much pronounced dip in the staggered magnetizationclose to zero field due to the U (1) symmetry of the AFclassical solution in absence of magnetic field. B. Spin Waves spectrum Linear spin waves (LSW) fluctuations around the clas-sical S → ∞ solutions can be analyzed in a standardway. One first introduces local axes such that at eachsite i a new z ′ axis coincides with the classical solutionspin orientation. Within the LSW approximation, thespin operator components in the local axes can be repre-sented by bosonic Holstein-Primakoff local operators a i , a † i as S z ′ i = S − a † i a i (8) S x ′ i = r S a † i + a i ) S y ′ i = i r S a † i − a i )After a Fourier transformation in a periodic chain with L sites, a i = 1 √ L X k e ikx i a k (9)where k = p πLa , p = − L/ , · · · , L/ x i = i a , andignoring cubic and higher order terms in a k , a † k , one canget the form H = S X k n A k ( a † k a k + a †− k a − k ) + B k ( a † k a †− k + a k a − k ) o (10)with real A k ≥ | B k | . Then the Hamiltonian can be diag-onalized by a Bogoliubov transformation a k = u k α k + v k α †− k a †− k = v ∗ k α k + u ∗ k α †− k (11)with | u k | − | v k | = 1 ensuring that α k , α † k are bosonicmodes. The Hamiltonian finally reads H = N E + S X k ( ε k − A k ) + S X k ε k α † k α k (12)where E is the classical energy per site given in Eq. (3)and ε k = p A k − | B k | are the Bogoliubov mode ener-gies.At the dipolar-ferromagnetic phase an appropriateglobal orthogonal coordinate system is set by rotatingthe original axes an angle θ DF ( h, D ) around the y direc-tion. The Hamiltonian coefficients A DF k , B DF k in Eq. (10)are given by A DF k = J (cos( ka ) − 1) ++ D (cid:0) θ DF − (cid:1) (cid:18) ζ (3) + 12 F ( ka ) (cid:19) + h S cos θ DF B DF k = − D cos θ DF F ( ka ) (13)where F ( ka ) ≡ Re [ Li ( e ika )] with Li ( z ) thepolylogarithm series Li ( z ) ≡ P ∞ n =1 1 n z n .At the antiferromagnetic phase the appropriate axesare local ones, obtained by rotating the original axes ateach site i with an angle ( − i θ AF ( h, J, D ) about the x direction. The Hamiltonian coefficients A AF k , B AF k inEq. (10) are then given by A AF k = J cos θ AF cos( ka ) − J cos(2 θ AF ) ++ D ( 12 cos θ AF − F ( ka ) + 12 D sin θ AF G ( ka ) + − Dζ (3)(cos θ AF − 34 sin θ AF ) + h S cos θ AF B AF k = J sin θ AF cos( ka ) − D (cid:18) 12 cos θ AF + 1 (cid:19) F ( ka ) + − D sin θ AF G ( ka ) (14)where G ( ka ) ≡ F (2 ka ) − F ( ka ).In either case, the semiclassical ground state is theBogoliubov vacuum | i annihilated by the operators α k .The ground state energy is then given by H = N E + S X k ( ε k − A k ) . (15)Within the LSW framework the sensible order parame-ter to compute is the average of the local magnetizationsalong the classical directions z ′ , defined as m z ′ = 1 N X i h | S z ′ i | i = S + 12 − X k A k p A k − | B k | . (16)A large value of the summation in the last term signalsthe breakdown of the LSW approximation, meaning thatquantum fluctuations destroy the classical order. Thisoccurs in the shaded regions of the classical phase dia-gram in Fig. 2.We show in Fig. 4 m z ′ vs. h for coupling ratios withinthe DF phase, J/D < ζ (3), while in Fig. 5 we showthe corresponding results for ratios within the AF phase, J/D > ζ (3). These results extend those obtained inRef. 3, at J = 0, to the whole DF region and show upa new phase with antiferromagnetic characteristics. Onecan observe that the LSW corrections to the order pa-rameter diverge as h approaches h sat , both in the DFand the AF phases. From the slopes in these figures thesensitivity to quantum fluctuations shows to be higherwhen J approaches J c , i.e. when the competition be-tween dipolar and exchange interactions is stronger. Inthe AF phase quantum fluctuations are crucial not onlyclose to saturation but also close to zero field, as expectedfor antiferromagnetic systems. Here the LSW corrections m z ’ / S h / h DF sat J/D=0.0J/D=0.5J/D=0.6 FIG. 4: (color online) Mean local magnetization vs. h fordifferent values of antiferromagnetic J < J c , in the LSW ap-proximation. m z ’ / S h / h AF sat J/D=0.8J/D=1.2J/D=2.0 FIG. 5: (color online) Mean local magnetization vs. h fordifferent values of antiferromagnetic J > J c , in the LSW ap-proximation. are more important for larger J . Indeed, the crossings inthe curves in Fig. 5 signal a crossover from a dipolarreminiscent behavior, close to J C , towards an exchangedominated behavior for J ≫ D .These features agree with the classical thermal fluctua-tions picture and are confirmed by extensive DMRG com-putations and a bosonization analysis in the next Section. IV. EXTREME QUANTUM LIMIT: SPIN S = 1 / CASE In this Section we discuss specific features of the spin S = 1 / A. Bosonization approach For the present discussion we find it convenient torewrite the Hamiltonian in Eq. (1) by separating thenearest neighbors interactions ( H NN ) from the longer dis-tance dipolar terms ( H int ), which we then treat pertur-batively. For the sake of clarity, we consider first the zerofield case. The Hamiltonian reads H = H NN + H int (17)where H = ( J + D ) X i (cid:20) S yi S yi +1 + S zi S zi +1 + J − DJ + D S xi S xi +1 (cid:21) (18)and H int = D X j>i +1 − S xi S xj + S yi S yj + S zi S zj ( j − i ) ! . (19)Notice that H NN corresponds to the well knownanisotropic XXZ Heisenberg chain with ∆ = J − DJ + D (seefor instance Ref. 7). In this sense, the short range effectof dipolar interactions is the onset of an exchange-likeanisotropy along the chain direction. Without furtherinteractions, a Luttinger liquid phase is present, extend-ing from a ferromagnetic transition point at J = 0 . D (∆ = − 1) up to an isotropic Heisenberg point (∆ = 1)to be reached at J/D → ∞ , with a free fermion point(∆ = 0) located at J = 2 D . Bosonization of the spin S = 1 / /r decaying long range dipolar interactions in H int do not alter the Luttinger liquid behavior, but only renor-malize the Luttinger parameters. In accordance withthe spin wave indications, the system enters a disorderedphase for J > J qc , a quantum critical point which is even-tually shifted from 0 . D by dipolar interactions.In the region J < . D the truncated Hamiltonian H NN enters a gapped ferromagnetic phase , with a two-fold degenerate ground state characterized by the orderparameter h S xi i 6 = 0. Though conformal perturbationscannot be applied in this region, notice that the fullHamiltonian H NN + H int classically exhibits the sameordering, which is robust against thermal and quantumfluctuations. For the quantum case S = 1 / y direction ( i.e. transverse to both the anisotropy and the magnetic field).On the gapped ferromagnetic phase, the transverse fielddiminishes the order parameter, until a quantum phasetransition into a paramagnetic phase is reached. Dis-cussing how the longer range interactions in H int maymodify this picture is beyond the scope of the presentpaper. These predictions, namely the renormalization of thequantum critical point and the behavior in a transversemagnetic field, are confirmed below by a DMRG analysis. B. DMRG calculations With the aim of characterizing the spin S = 1 / .Including long range interactions is a non-trivial task,comparable with current studies of two-dimensional spinsystems . Based on our experience in such investiga-tions, we use here open boundary conditions and long-range interactions involving all available neighbors inchains of finite size. We have considered chains of length L up to 64 sites, keeping m = 350 states and achievingtruncation errors in the density matrix of the order of10 − .As a first step to identify the configurations of the sys-tem we compute the total magnetic moment m definedin Eq. (6), where now m α = L P i h S αi i stands for the av-erage of quantum expectation values. In Fig. 6 we showDMRG results for the total magnetic moment m as afunction of the magnetic field for two representative val-ues J = 0 . D and J = 1 . D of the antiferromagneticexchange interaction, chosen to be compared with thefigures shown in Section III. As in the classical chain,we observe two different configurations depending on thevalue of the exchange interaction J being below or abovea critical value. In more detail (not shown in the fig-ure) we have been able to estimate a quantum criticalcoupling J qc = 0 . J c = ζ (3) D ≈ . J < J qc , the magnetization exhibits aminimum, a similar behavior to that of the pure dipolarcase in Ref. 3. Instead, in the AF phase, J > J qc , we findthat the total magnetization smoothly increases with theapplied magnetic field.As a further step we analyze the components of themagnetic moments along the three directions x , y and z separately. In Fig. 7 we show the values of m x and m z as functions of the applied magnetic field, for exchange J = 0 . D < J qc and J = 1 . D > J qc . In both phases m z increases smoothly with the applied field, approach-ing to saturation at about the classical saturation field.Instead, the m x component makes apparent the differ-ence between the DF and AF regimes. The component m y vanishes in both cases, and for the sake of clarity it isnot shown in the figures; we argue below that the reasonwhy this happens is very different for each phase.In the DF phase ( J < J qc ) we found a two-fold de-generate ground state, as dictated by parity Z symme-try under x → − x reflections. A parity resolved basisfor this ground state subspace is given by states with m x < m x > 0. A tiny magnetic field h x = 10 − D acting just on the end sites of the chain is enough to m / S h/h sat J/D=0.6, L=36J/D=0.6, L=48J/D=0.6, L=64J/D=1.0, L=36J/D=1.0, L=48J/D=1.0, L=64 FIG. 6: (color online) Total magnetic moment m/S as a func-tion of the applied magnetic field h for two representative val-ues of the antiferromagnetic exchange J/D = 0 . < J c and J/D = 1 . > J c and lattice sizes L =36,48,64 (each curveis normalized by its corresponding saturation magnetic field h DFsat and h AFsat ). m α / S h/h satDF m x m z m α / S h/h satAF m x m z FIG. 7: (color online) Magnetic moments m x and m z as afunction of the applied magnetic field for a chain of L = 48sites. Left panel: Results for J/D = 0 . J < J c ). Rightpanel: Results for J/D = 1 . J > J c ). explicitly break parity Z symmetry, selecting the statewith m x > 0. This response supports the interpretationthat the finite size ground state is a simple superposi-tion of disentangled ferromagnetic product states. Fol-lowing this strategy we produced the states shown in theleft panel of Fig. 7: at low fields the most importantcontribution to the magnetization is provided by the m x component, followed by a sudden drop well before m z ap-proaches saturation. This explains the pit in Fig. 6. Asthe magnetic field increases, we found a magnetic field h c ≈ . h DFsat above which all the magnetization weightis already in the z -direction. The value of h c is obtainedby extrapolation of the h c ( L ) for different chain lengths,and signals that all spins already align with the magneticfield at less than half the classical saturation field. In thisphase, as in the pure dipolar case, there is no symmetryreason for the system to choose an orientation in the y direction; we accordingly found that h S yi i = 0 at eachsite.In the AF phase ( J > J qc ) we observed distinct groundstates in the absence or presence of the external mag-netic field. Without external field, in agreement with the Luttinger regime found in the bosonization analy-sis, the ground state shows no order; local expectationvalues vanish for any spin component as shown in theright panel of Fig. 7 at h = 0. Under a magnetic field,also in agreement with the gapped Ising N´eel phase pre-dicted by bosonization, we found a two-fold degenerateground state, related to one-site Z translation invari-ance. While we expect a spontaneous symmetry break-ing in the thermodynamic limit, leading to a staggeredmagnetization along the y direction, one should noticethat translation symmetry is not broken in the availablefinite size systems ; in consequence, one point operatorscan only show homogeneous expectation values and astaggered magnetization is compatible with the observedlocal result h S yi i = 0. In contrast to the DF phase, inthis case a tiny staggered magnetic field h y acting on theend sites of the chain is not enough to explicitly break Z translation invariance. In order to obtain clear sig-nals of the N´eel order classically observed and quantummechanically predicted in the AF phase, we resorted tothe computation of S y two-point spin correlations h S yi S yj i which allow for staggered non-vanishing results, invari-ant under one-site translations. For J > J qc we indeedfound staggered correlations, which include an exponen-tially decaying connected part and a non-vanishing longrange order disconnected part, h S yi S yj i = h S yi S yj i + h S yi ih S yj i (20)with h S yi ih S yj i ∝ ( − j − i ( m stagy ) . We thus can ex-tract the antiferromagnetic order parameter from thetwo-point spin correlations. The long distance behav-ior can be better analyzed by considering the end-to-endcorrelations C ,L = h S y S yL i . (21)As pointed out in Ref. 14 these correlations betweenthe spins at opposite ends of the chain provide a moretractable distance scaling than bulk correlations. Be-cause of the even-odd sign of correlators, we rather plot | C ,L | to identify the response of end-to-end correlationsto the applied magnetic field. We plot in Fig. 8 the re-sults for the considered finite size chains at J/D = 1 . J > J qc . As a secondone, the staggered correlations drop down and vanish atabout 60% of the classical saturation field. This coin-cides with the high slope rise of m z in the right panel ofFig. 7. Finally, end-to-end correlations approach to zeroas the magnetic field vanishes (this is more clear deepinside the AF phase, for instance at J/D = 1 . | C , L | h/h satAF J/D=1.0, L=36J/D=1.0, L=48J/D=1.0, L=64J/D=1.0, L →∞ FIG. 8: (color online) End-to-end correlations for latticelengths L = 36 , , 64 and the L → ∞ extrapolation as afunction of the applied magnetic field for J/D = 1 . Luttinger phase of the anisotropic XXZ spin chain in thepresence of a transverse field (see for instance Fig. 5 inRef. 10). V. CONCLUSIONS In the present work we have described the phase di-agram for an antiferromagnetic nearest neighbors spinchain (with exchange strength J ) including the effects oflong range dipole-dipole interactions (of strength D ) anda transverse magnetic field (of strength h ).On the one hand, we have characterized the presenceof ordered phases for classical spins and their stabil-ity under the influence of classical and quantum fluc-tuations. The main feature here is the presence of aphase transition at J/D = ζ (3) from a dipolar domi-nated phase to an antiferromagnetic phase. In the for-mer one classical spins are aligned in a ferromagneticpattern forming an angle θ DF ( h, D ) = arccos (cid:16) h ζ (3) SD (cid:17) with the external field, according to the competition be-tween the dipolar tendency to align them parallel to thechain and the Zeeman energy, while in the latter phasespins order antiferromagnetically, transverse to the dipo-lar anisotropy and canted towards the external field atan angle θ AF ( h, J, D ) = arccos (cid:16) h S ( J + ζ (3) D ) (cid:17) . Classi-cal and quantum fluctuations destroy both these classicalorders before reaching the saturation field, and also theantiferromagnetic order close to zero field.On the other hand, in the extreme quantum case S = 1 / 2, DMRG computations indicate that classical order disappears well before reaching the classical satu-ration fields. This is in agreement with previous studieson the purely dipolar S = 1 / where a quantumcritical point belonging to the 2 d Ising universality classwas identified and the effect of quantum fluctuations wasproved to reduce the value of the critical field to magneticsaturation. Regarding the competition between dipo-lar ferromagnetic and antiferromagnetic orders, we havefound that the quantum critical point separating thesephases is slightly shifted (about 3%) to a lower value.This behavior is fully compatible with the bosonizationanalysis discussed in the text. The observed quantumphases are similar to those present in the well knownXXZ anisotropic spin chain. Also the quantum criticaltransition reported in Ref. 3 for the dipolar S = 1 / Acknowledgments BB thanks C.J. Gazza for useful help in DMRG imple-mentation details. FGA acknowledges H.D. Rosales andR. Borzi for discussions on Monte Carlo implementation.DCC and GLR acknowledge M.D. Grynberg, A. Iucci andR. 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