Half-Magnetization Plateau of a Dipolar Spin Ice in a [100] Field
HHalf-Magnetization Plateau of a Dipolar Spin Ice in a [100] Field
Sheng-Ching Lin( 林 昇 慶 ) and Ying-Jer Kao( 高 英 哲 )
1, 2, ∗ Department of Physics and Center of Theoretical Sciences, National Taiwan University, Taipei 10607, Taiwan Center for Advanced Study in Theoretical Science, National Taiwan University, Taipei 10607, Taiwan (Dated: November 5, 2018)We report here numerical results of the low-temperature behavior of a dipolar spin ice in a magnetic fieldalong the [100] direction. Tuning the magnetic field, the system exhibit a half-magnetization plateau at lowtemperature. This half-polarized phase should correspond to a quantum solid phase in an effective 2D quantumboson model, and the transition from the Coulomb phase with a power-law correlation to this state can beregarded as a superfluid to a quantum solid transition. We discuss possible experimental signatures of thishalf-polarized state.
PACS numbers: 02.70.-c, 75.10.Pq, 05.10.Cc
Topological phases of matter is one of the most fascinatingphenomena in condensed matter systems. Fractional quantumHall states, spin liquids, and recently proposed topologicalinsulators are some well-known examples. Spin ice, a geo-metrically frustrated magnet with Ising spins on a pyrochlorelattice of corner-sharing tetrahedra, , at low temperatures ex-hibits a “Coulomb” phase with dipolar correlation and frac-tionalized monopole excitations. This phase emerges dueto a local constraint, the “ice rule”, with two spins pointingin and two out of each tetrahedron. The ice rule can be re-garded as a conservation law of an emergent gauge field, and it gives rise to the dipolar correlation among spins. Thislocal constraint leads to an extensive ground state degeneracy,and at low temperature, the spin ice has an extensive resid-ual entropy, S ≈ ( k B /
2) ln(3 / per spin, first estimated byPauling for the water ice. Starting from the nearest-neighbor spin ice (NNSI) andadding perturbation such as an external magnetic field orfurther-neighbor dipole-dipole interaction, the system can un-dergo a phase transition from the Coulomb phase to an or-dered state at sufficiently low temperature. This confinementtransition is highly unconventional, and can not be describedby the Landau-Ginzburg-Wilson paradigm. The transition toa non-magnetic state can be regarded as a Higgs transitioninvolving condensation of an emergent matter field coupledto the U(1) gauge field. Applying an external magnetic fieldalong the [100] direction, the transition from an ordered q = 0 fully polarized (FP) state to a Coulomb phase corresponds toa three-dimensional Kasteleyn transition, which is first-orderlike on the one side and continuous on the other. This transi-tion corresponds to a proliferation of string excitations alongthe [100] direction, which corresponds to the condensation ofbosons in an effective two-dimensional hardcore boson modelthrough a classical to quantum mapping, On the other hand, in real spin ice materials, such asDy Ti O (DTO) and Ho Ti O (HTO), the rare earth ionshave large magnetic moments ( ∼ µ B ), and the long-range dipole-dipole interaction can not be ignored. The low-temperature states of the dipolar spin ice (DSI) also satisfy theice rule, and it is shown that the NNSI and the DSI are projec-tively equivalent apart from small interactions decaying withthe separation, r , faster than /r . At very low temperature, this residual interaction selects a q = (001) long-range or-der, and the Pauling’s residual entropy is released through afirst-order phase transition. This Melko-den Hertog-Gingras(MDG) phase is non-magnetic and has so far escaped exper-imental observation, although recent specific heat measure-ment in a thermally equilibrated DTO sample shows possiblesignatures of such an ordered phase. The natural question arises: What kind of low-temperatureordered phase will emerge if one applies a [100] magneticfield in a DSI? How to characterize the phase transition fromthe disordered Coulomb phase to the ordered state? In thisLetter, using large-scale Monte Carlo (MC) simulations of theDSI in a [100] magnetic field, we find a new ordered state withhalf of the saturated magnetization stabilized by the dipolarinteraction. Starting from the MDG phase with zero magne-tization at low field, the DSI transitions to the half-polarized(HP) state upon the increase of the magnetic field, before thesystem finally reaches an FP state at high field. This shouldbe contrasted with the half-magnetization plateau found inthe Heisenberg model, which is stabilized by either the lat-tice distortion, or the quantum effects. Through a classicalto quantum mapping, the HP state should correspond to aquantum solid (QS) in a two-dimensional (2D) quantum bo- y x ++ +++ ++ +−− −−−− −− FIG. 1. (Color online). (Left) Pyrochlore lattice showing the HPstate. (Right) The HP state projected along the z -axis. The com-ponent of each spin parallel to the z axis is indicated by + and − sign. a r X i v : . [ c ond - m a t . s t r- e l ] D ec son model, and the Coulomb to the HP state transition wouldcorrespond to a superfluild (SF) to a QS transition. Signa-tures of this HP state may have been observed in the mag-netization and neutron scattering measurements of DTO andHTO. Model and Method. – The Hamiltonian of a dipolar spin icein an applied [100] magnetic field is given by H = J (cid:88) (cid:104) ( i,a ) , ( j,b ) (cid:105) σ ai σ bj + Dr nn (cid:88) i FIG. 3. (Color online). Magnetization (in units of 10 µ B ) curves at T = (a) . K, (b) . K, (c) . K, and (d) . K for L =4 (blacksquare) and 5 (red circle). M T(K) FIG. 4. (Color online) Temperature dependence of the magnetiza-tion at different magnetic fields along the [100] direction. Magneti-zation is in units of 10 µ B , and the magnetic fields are in Tesla. applied field, similar to those from the saturated FP state. Themagnetic moments from the other two tetrahedra cancel as inthe MDG state.Figure 4 shows the temperature dependence of the mag-netization at different magnetic field strength. At low fields B < . T, the systems enters from the Coulomb phaseto the MDG phase with zero magnetization at low temper-atures. For intermediate fields, the system transitions intothe HP state. At fields larger than 0.05T, a rounded Kaste-leyn transition from the disordered Coulomb phase to a FP q = (000) state is observed. This rounding might be eitherdue to the finite-size effect, or the high magnetic field. Thelow-temperature ground state is very sensitive to the delicatebalance between the magnetic field and the dipolar interac-tion. Two magnetization curves with slightly different mag-netic fields may have similar temperature dependence at high T ( K ) B(T) CoulombMDG FPHP FIG. 5. The phase diagram of the dipolar spin ice in a [100] field.Blue squares are the transition points estimated from the Binder ra-tio. The solid curves are drawn as a guide for the eye. See text fordescription of the phases. temperature; however, the states at low temperature will settleto different magnetization plateaux (Fig. 4) .The thermal phase transition from the Coulomb phase tothe ordered states show sharp specific heat anamoly due tothe strongly first-order nature of the transition, and the largeresidual entropy is released at the transition. In contrast, thetransitions between the ordered states show relatively smallspecific heat anomaly as these are ordered states with differ-ent magnetizations. Experimentally, it has been difficult toobserve the MDG phase due to the increase of the spin relax-ation time as temperature is lowered, and the system is hard toreach thermal equilibrium. On the other hand, the FP statecan be easily reached at high temperature and in large field. Here we propose another route to reach the MDG phase: First,the system is cooled down in a large [100] field from the hightemperature disordered Coulomb phase to the FP state. Then,the magnetic field is lowered at low temperature. The transi-tion from the FP state to the MDG state through an intermedi-ate HP state can be more easily reached experimentally sincethey go through transitions between ordered states.Figure 5 shows the phase diagram of the DSI for fields ap-plied along the [100] direction, using the DTO parameters.The phase boundaries are estimated using the Binder ratio(blue squares). At high temperature, the system is in thedisordered Coulomb phase with dipolar correlations. As thetemperature is lowered, the combination of the magnetic fieldand the dipolar interaction will select either the MDG or theHP state. It should be noted that there exists slight reentrantbehavior near the phase boundaries between the MDG and theHP phases, and between the HP and the MDG phases. Thethermal transitions from the Coulomb to the MDG phase, andto the HP phase are first-order, and they merge at the tip of theHP lobe as a (putative) critical point. Through the quantummapping, the MDG phase corresponds to a MI, the HP phasecorresponds to a half-filled checkerboard QS (density wave)in a 2D extended Bose-Hubbard model, and the Coulombphases corresponds to a superfluid (SF) phase. The transitionfrom the Coulomb phase to an ordered state thus correspondsto a SF-MI or SF-QS transition in the quantum model. In thismapping, the [100] field B is proportional to µ , and the mag-netic moment M would correspond to the average boson den-sity ρ . Interestingly, the magnetization (Fig. 2) and the phasediagram (Fig. 5) show great resemblance to their counterpartsin a 2D extended hardcore Bose-Hubbard model on a squarelattice, H = − t (cid:88) (cid:104) ij (cid:105) (cid:16) b † i b j + h. c. (cid:17) + V (cid:88) (cid:104) ij (cid:105) n i n j − µ (cid:88) i n i , (2)where b i and b † i are the annihilation and creation operators ofhardcore bosons on site i , n i is the boson density, µ is thechemical potential, and V is the nearest-neighbor repulsiveinteraction. The HP phase can be identified as a half-filledcheckerboard QS (density wave) in this model. The SF-QStransition in the bosonic model is generally first-order as thetwo phases break different symmetries, and similarly for thephase transition from the Coulomb to HP phase. Further anal-ysis based on symmetries is required to construct an effectivequantum model. What are the possible experimental signatures of the HPstate? One should observe a clear half-magnetization plateauat low temperatures in magnetization measurements. The sys-tems should also exhibit magnetic susceptibility anomaliesnear the plateau transitions. Since the HP phase is a mixtureof the q = (000) and (001) states, one should observe in neu-tron scattering two peaks at (000) and (001), and the spectralweights for each peak should be a fraction of that observed inthe MDG and the FP phase, respectively. The available mag-netization data show no signs of the magnetization plateau attemperature as low as 0.6K and the system remains paramag-netic up to saturation. On the other hand, the integratedintensity at (000) obtained from neutron scattering on DTO at T =0.05K shows steps with hysteresis at B ∼ . T . Moreinterestingly, the ordered moment in HTO at T =0.05K showsa plateau of 6 µ B near B ∼ . T. Also, the diffuse scatteringat (001) persists in finite field and shows a steplike decrease incommensuration with the steplike increase in the (000) scat-tering between B ∼ . to 0.5T (Fig. 8 in Ref. 18 ). Thesefeatures have been previously attributed to the lack of relax-ation of the magnetic moments at low temperatures, and thesystem is trapped in a metastable state. On the other hand,these features resemble the signatures of a HP state, althoughthe field range observed experimentally is one order of mag-nitude larger than our prediction, and the signatures disappearupon cycling through the field. Further experiments are nec-essary to determine whether these indicate the existence of theHP phase. Conclusion. – Using large-scale Monte Carlo simulations,we construct the full phase diagram of a DSI in a magneticfield along the [100] direction. We find a new intermedi-ate HP phase with half of the saturated magnetization, whichshould correspond to a QS phase in a 2D quantum bosonmodel. The transition from the Coulomb phase to this phasethus corresponds to a SF-QS transition in the bosonic model. This points to a new direction which can be further exploredthrough the classical to quantum mapping. More theoreticalanalysis based on symmetry considerations has to be done todetermine the general form for the quantum Hamiltonian, andthe corresponding continuum quantum field theory that de-scribes the characteristics of the phase transition, in particularnear the putative critical point at the tip of the HP phase. Inother words, studying the purely classical DSI in a magneticfield may provide useful information regarding the quantumphase transitions in a 2D quantum boson model, and exoticphases such as the supersolid phase in the lattice boson modelmay also be realized in the DSI. On the experimental side,we believe the data presented in Ref. 18 indicates possible sig-natures of this phase in both DTO and HTO, and our resultswill stimulate further experimental efforts to resolve this is-sue. It will also be interesting to see what are the quantumeffects on the magnetic plateau in the quantum spin ice. ACKNOWLEDGMENTS We thank L. D. C. Jaubert for a discussion on the improvedworm algorithm, and S. Powell for very useful comments. Wealso thank J. T. Chalker and R. Moessner for helpful discus-sions. Y. J. K. thanks the hospitality of the Aspen Center forPhysics, where part of this work was done. This work is par-tially supported by the NSF under Grant No. PHY-1066293,the NSC in Taiwan under Grants No. 100-2112-M-002-013-MY3, 100-2923-M-004-002 -MY3 and 102-2112-M-002-003-MY3, and by NTU Grant No. 101R891004. Travel sup-port from NCTS in Taiwan is also acknowledged. Appendix A: Comparison of loop and worm updates Two types of cluster updates were used in our simulation:the loop and worm algorithms . In the loop algorithm, aloop is randomly generated and is flipped with a Metropolismove. It depends strongly on the energy difference involvedbetween the initial and proposed states of the the loop moveand can possibly be rejected . For simulations in a field, thisalgorithm becomes less efficient as the rejection rate can bevery high. On the other hand, in the worm algorithm, the de-tailed balance condition is imposed along the construction ofthe loop, and it allows to flip a long loop of spins in a fieldwithout rejection for the NN spin ice model . In our imple-mentation of the worm algorithm, the loop is generated usingthe transition probabilities specified in the nearest-neighborspin ice (NNSI) model, and the loop is flipped with theMetropolis probability P = min(1 , exp( − β ∆ E dip )) , where ∆ E dip is the energy change due to the dipolar interaction.Figure 6 shows the acceptance rates of the loop and the wormupdates at a field of B = 6 . × − T. The worm update ismore efficient at high temperatures since the loop can be gen-erated without any backtracks . However, the worm updatebecome inefficient when temperature is lower than . K.This temperature roughly coincides with the Kasteleyn tran-sition temperature T k = h √ for the NNSI, below which the A cce p t an ce R a t e ( % ) T h=6.696e-02, worm h=6.696e-02, loop FIG. 6. (Color online) Comparison of the conventional loop and di-rected loop updates at B = 6 . × − T. Worm updates becomesless efficient below T = 0 . K. E / G ( J ) B(T) MDG HP FP 0.18K 0.2K 0.22K FIG. 7. (Color online). Field dependence of the energy ( E ) per spinfor the MDG phase (black square), the HP phase (red circle) andthe FP (blue upper triangle). The level crossings correspond to theplateau transitions at zero temperature. Also plotted is the free en-ergy ( G ) per spin at T =0.22K (purple left triangle), . K (greendiamond), and . K (pink down triangle). backtracking must be included to ensure a physically mean-ingful solution for the transition probability, and the loop mayterminate prematurely. On the other hand, the loop in the loopupdate is generated by random walks through tetrahedra ,and there still exists finite probability to update the system. Appendix B: Level crossings To better understand the temperature evolution of the fielddependence of the magnetization, we examine the energy ofthe possible ordered states and the free energy density of thesystem at different temperatures in Fig. 7. First, we considerthe field dependence of the energy for the ordered states: the MDG state (black square), the HP state (red circle), and the . . . . . . . . . . . . . . . . . . T ( K ) C v B ( T ) FIG. 8. (Color online.) The specific heat data showing very sharpanomalies at the thermal transitions from the Coulomb to the orderedstates. In contrast, relatively small specific heat anomaly is observedfor the low-temperature transitions between the ordered states. FP state (blue upper triangle). There exist two level cross-ings at B = 0 . , and 0.052T respectively, which corre-sponds to the plateau transitions at low temperature. We alsocompute the the free energy G = E − T S at T = 0 . K,0.20K and 0.22K from the simulation data. The entropy isobtained by numerically integrating the specific heat data. At T = 0 . K (purple left triangle), none of the ordered stateshas the lowest free energy and the system remains disordered.At . K (green diamond), the free energy curve touches theHS state near B = 0 . T, indicating the appearance of thehalf-magnetization plateau. At . K (pink down triangle),for B < . T, the MDG state has the lowest free energy,for B =0.025 to 0.045T, the HP state is the ground state andfor B > . T, the system enters the fully polarized state.Between these plateaus, the system remains in the disorderedCoulomb phase. These results are consistent with what is ob-served in the field dependence of the magnetization (Fig. 3 inthe main text). Appendix C: Specific heat and magnetic susceptibility Figure 8 shows the specific heat data for L = 5 . The ther-mal transitions from the spin ice state to the ordered statesshow sharp peaks due to the release of the large residual en-tropy at the transition. In contrast, the transitions between theordered states show relatively small specific heat anomaly asthese are ordered states with different magnetizations. Fig-ure 9 shows the magnetic susceptibility versus temperatureand magnetic field for L = 5 . Strong magnetic susceptibilityanomalies can be observed near the plateau transitions. Thisshould be contrasted with the specific heat plot where it showssmall signature of anomalies near the plateau transitions sincethe states are all ordered states with different magnetization. FIG. 9. (Color online) Magnetic susceptibility versus temperatureand magnetic field. Strong magnetic susceptibility anomalies can beobserved near the plateau transitions. ∗ [email protected] X.-G. 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