Modelling the Berezinskii-Kosterlitz-Thouless Transition in the NiGa_2S_4
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Modelling the Berezinskii-Kosterlitz-Thouless Transition in the NiGa S Chyh-Hong Chern ∗ Institute for Solid State Physics, University of Tokyo,5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
In the two-dimensional superfluidity, the proliferation of the vortices and the anti-vortices resultsin a new class of phase transition, Berezinskii-Kosterlitz-Thouless (BKT) transition. This classof the phase transitions is also anticipated in the two-dimensional magnetic systems. However, itsexistence in the real magnetic systems still remains mysterious. Here we propose a phenomenologicalmodel to illustrate that the novel spin-freezing transition recently uncovered in the NMR experimenton the NiGa S compound is the BKT-type. The novel spin-freezing state observed in the NiGa S possesses the power-law decayed spin correlation. PACS numbers: 75.30.Gw, 75.40.Cx, 75.40.Mg
As the thermodynamic conditions in the environmentchange, for example the pressure or the temperature,matter transforms from one state to another. Steam lev-itates from the top of the hot coffee; ice melts in the softdrink. The phase transition is ubiquitous in our dailylife. Investigating the phase transition is always the cen-tral subject in physics. Most of the phase transitionscan be understood by the distinct physical properties be-tween the phases. For example, in the vapor-liquid tran-sition, the unit volumes per mole of the molecules aredifferent from the vapor to the liquid. In the ferromag-netic transition, the spins orientate randomly at the hightemperature side but start to align along the same direc-tion resulting in the net magnetization at the low tem-perature side. However, in the two-dimensional super-fluid, the Berezinskii-Kosterlitz-Thouless (BKT) transi-tion that happens when vortices and the anti-vorticesproliferate does not separate the phases with distinctthermodynamic quantities[1]. Instead, the superfluid-ity correlation changes from the power-law behavior tothe exponentially decayed one as we across the tran-sition from the lower temperature side. In the two-dimensional magnetic systems, the BKT transition is alsoanticipated in all easy-plane Heisenberg models. On theother hand, the power-law decayed spin correlation mayplay important role in the high transition temperaturesuperconductor[2]. Therefore, understanding the spindynamics in the critical phase has become tremendouslyimportant.NiGa S is originally synthesized intending to realizethe spin liquid proposed by Anderson few decades ago[3].It is the layered material that the spin-1 Ni ions formthe ideal two-dimensional triangular network. As an anti-ferromagnetic insulator, NiGa S exhibits no long-rangedmagnetic ordering down to 350mK shown in the specificheat capacity, the magnetic susceptibility, and the neu-tron scattering experiments[4]. At 1.5K, the Edwards-Anderson order parameter, Q = 1 /N P i < S i > , thattells the spin moment, is measured at 0.61, where N isthe total number of spins. This vastly reduced spin mo-ment indicates the presence of the strong quantum fluc- tuation that is highly favorable by the scenario of thespin liquid. However, in the recent experiments of theGa nuclear magnetic resonance (Ga-NMR) [5], a tem-perature T f is found around 10K below which the spindynamics is slow down and the freezing behavior is ob-served. As approaching the T f from above, both thenuclear spin-lattice relaxation rate 1/T and the nuclearspin-spin relaxation rate 1/T diverge. Moreover, spinsdo not freeze immediately at T f but persist fluctuatingdown to 2K found in the nuclear quadruple resonancemeasurement (NQR). Below 2K, the Ga-NQR spectrumbecomes very broad and featureless, which implies theformation of the static inhomogeneous internal magneticfield. First, intuitively, the static internal magnetic fieldoccurs when the spins freeze up completely. In this case,the Edwards-Anderson order parameter should be closeto the quantum number of the spin angular momentum.Second, all thermodynamic quantities change smoothlyat the transition temperature T f , but the 1/T and the1/T diverge in the NMR signals. Therefore, the laterNMR experiment apparently looks inconsistent with theprevious measurements. In this Letter, we shall providea consistent picture to compromise all the experimentalresults. Most importantly, we illustrate that the novelspin-freezing transition at T= T f is the long-sought BKTtransition in the two-dimensional magnetic systems.The way to compromise with all the experiments is toconsider the state that contains both the freezing spinsand the fluctuating ones. Then, both of them can beobserved simultaneously in the experiments. Before ex-plaining further, let us begin by reviewing the spin con-figuration, depicted in Fig.(1a), observed in the neutronscattering experiment. The correlation has the wavevector (1 / , / ,
0) with the wavelength 2 π/ a , where a names the lattice constant between two Ni ions. Thiswave vector simply means that along the a directionthe periodicity is of 6 sites and so is it along the a , andalong the b = (1 , , FIG. 1: (Color online)(a)The spin structure observed by theneutron scattering in the Ref.[4]. There are two arrows onevery site. The colored one is the spin orientation, and thethin black one is the local easy axis. (b) The ground state ofthe anti-ferromagnetic quantum Ising model on the triangularlattice. ”-” indicates the local ”-1” state and the ”+” is thelocal ”+1” state. ”0” is the linear superposition of the ”+1”and the ”-1” states. two spins point out with respect to the triangles) or the”2-in-1-out” configurations. Inspired by the observationin the NMR experiment that the spin correlation startsto develop below the Curie-Weiss temperature 80K, weassume the existence of an easy axis on every spin siteand its orientation is either parallel or anti-parallel to thecurrent spin configuration. In order to manifest the anti-ferromagnetic nature, we assign the local +z axis to bethe ”all-in” or the ”all-out” with respect to the referencetriangles alternating over the whole lattice. An exampleof the orientation of the local axes is depicted as the blackthin arrows in the Fig.(1a). In this transformed coordi-nate, spins are either +1 (parallel to the +z direction) or-1 (anti-parallel to the +z direction).Considering the quantum fluctuation explicitly, wepropose the following phenomenological Hamiltonian inthe transformed coordinate H = J X
50 100 150 200 250 300 350 400 450 0 50 100 150 200 250 300 I n v e r s e S u sc ep t i b ili t y ( χ − / m o l ) Temperature (K)
L10, K=0.5Experiment1.05*x+82.5
FIG. 2: (Color online)The system size is 10 ×
10 with 100spins in the calculation. The theoretical result has the samestructure as the experiments. At the dip of the χ − , marked T f , the spin-freezing transition occurs found in the Ga-NMRexperiments. the experimental one. We mark the temperature of thedip T c in the theoretical result and T f in the experimentalone. ξ ( T ) Temperature (J)
Quantum Monte Carlo0.75*exp(1.35/sqrt(x-0.3))
FIG. 3: (Color online)The result of ξ ( T ). The temperatureis in the unit of J , and the correlation length is in the unitof lattice constant a . At the high temperature side of T c ( ∼ . J ), the spin correlation is exponentially decayed, and thecorrelation length diverges at T c . Below T c , the phase has thepower-law spin correlation. The green line is the theoreticalfit of the 2D XY universality class. In the Ga-NMR experiments, the spin-freezing transi-tion is observed at the dip of the inverse susceptibility,and an unusual spin correlation starts to develop at theCurie-Weiss temperature 80K. To understand this, wecompute the spin-spin correlation along the a direction defined by D ( n a ) = 1 N X i < σ zi σ zi + n > (2)in the L x × M geometry with the periodic boundary con-dition, where L x is the length along the a direction and M is the one along the (1 , ,
0) direction, and L x ∼ M istaken. In this case, the correlation length ξ ( T, M ) scaleswith M . We define ξ ( T ) = lim M →∞ ξ ( T, M ) , (3)and it is shown in Fig.(3). We find that the spin correla-tion starts to develop at 1 . J and it diverges at T c ∼ . J .At T c , the BKT-type transition occurs. Above T c , forexample T=0.4 J , ξ ( T, M ) saturates exponentially shownin the Fig.(4). Below T c , for example T=0.2 J , ξ ( T, M ) islinear to M . This linearity has only two possibilities: 1)the correlation length is so large that our system sizes aretoo small to reach the saturation. 2) it is in the criticalphase. Because of the conformal invariance of the criticalphase in the two dimensions, the spin correlation decaysexponentially in the torus geometry. In this case, ξ ( T, M )is linear to M. Furthermore, the scaling dimension ∆( T )can be obtained by ξ ( T ) = 12 π ∆( T ) M (4)where ∆ is defined by < σ z (0) σ z ( r ) > ∼ r (5)The first possibility can be ruled out as the following.In Fig.(5), we plot the initial slope of ξ ( T, M ) definedby ∂ξ ( T, M ) /∂M | M =0 . It shows a clear phase transi-tion at T= T c . If the first possibility were true, the ini-tial slope would have been monotonic and grew expo-nentially as temperature decreases. However, the slopehas the discontinuity at T c , indicating a phase transition.Moreover, it is not a second-order phase transition, be-cause both the magnetic susceptibility and the specificheat are smooth functions at T c . If it is a classical Isingtransition, ξ ( T ) should be symmetric with respect to T c as | T − T c | <<
1. Here, although the initial slopes atT=0.2 J and T=0.4 J are the same within the error bar,their asymptotic behaviors are entirely different as shownin Fig.(4). Therefore, the phase below T c should be criti-cal with the power-law spin correlation. Additionally, thescaling dimension ∆( T ) has the monotonic temperaturedependence, which is the typical behavior of the scalingdimension in the critical phase. Due to the restriction ofthe technique, we are not able to compute the free energyto find the central charge in the critical phase. Classifica-tion by the conformal field theory may be an interestingdirections for the future research. ξ ( M ) M T=0.21.15*xT=0.449*(1-exp(-0.0255*x))
FIG. 4: (Color online) ξ ( T, M ) at T=0.2 J and 0 . J . The ver-tical axis is the ξ ( T, M ), and M is in the unit of the latticeconstant a . At T=0.4 J , the correlation length saturates ex-ponentially. At T=0.2 J , the linear scaling implies the criticalphase as explained in the text. Two lines are the functionalfit to the data d ξ ( M ) / d M | M = Temperature(J)
FIG. 5: (Color online)The temperature is in the unit of J .The vertical axis is the initial slope of ξ ( T, M ) defined in thetext. If there is no phase transition, the initial slope should beexponentially and monotonically increasing. However, thereis a discontinuity at T = 0.3 J , which indicates a phase tran-sition. The existence of the BKT transition in the quan-tum Ising model on the triangular lattice was previouslypointed out by R. Moessner, et al.[7]. Here we summa-rize their argument and further construct the topologicalobject in this model. The quantum 2D model of Eq.(1)at finite temperature can be mapped to a 3D classicalIsing model with the ferromagnetic exchange along theimaginary time direction with finite dimension . Usingthe Landau-Ginzburg-Wilson analysis, the 3D model canbe mapped to an XY model with a sixth-order symmet-ric breaking term which has the sixfold clock symmetry.At zero temperature, there is a quantum phase transi-tion described by the 3D XY universality class at finite K , because the clock term is dangerously irrelevant in3D. However, at finite temperature and in the thermo-dynamic limit, the 3D model crosses over to the 2D modeland it results in two transitions: one is BKT transition athigher temperature and the other at lower temperaturecorresponds to the sixfold clock symmetry breaking, forexample, to the state in Fig.(1b). Therefore, the BKTtransition in NiGa S actually belongs to the 2D XY uni-versality class! In Fig.(3), we fit the correlation resultwell with the 2D XY model[10]. These two transitionsare both seen in the Ga-NMR experiment[5]. The spin-freezing transition at 10K is the BKT transition, and thetransition at 2K where spins completely freeze up corre-sponds to the second transition.It is not coincident that the phase transition occurs atthe dip of the χ − . Once the power-law spin correlationsurvives in the phase rather than a critical point, theresponse to the external field could be weaker. Due to theslow spin dynamics, the magnetic susceptibility begins todrop in the critical phase. C v / N Temperature (J)
FIG. 6: (Color online) The temperature dependence of thespecific heat capacity. The temperature is in Kelvin.
In Fig.(6), we show the calculation of the specific heatcapacity. It illustrates the double-peak structure and thereason is similar to the one given in the Ref.[11]. Al-though the valley between the peaks is not as deep as theexperimental one shown in Ref.[4]. The magnetic specificheat in Ref.[4] is obtained by subtracting the specific heatof ZnIn S , which is non-magnetic and iso-structural toNiGa S , from the one of the NiGa S . We remark thatthere is 29.68% difference in the total atomic mass be-tween these two compounds. How reliable their magneticspecific result is suspicious to us. However, their low-temperature result may be correct. Here we also findno evidence of the existence of the energy gap, which isconsistent with their experiment.In summary, we have shown that the novel spin-freezing transition seen in the Ga-NMR experiment onthe NiGa S compound is the BKT-type transition whichbelongs to the 2D XY universality class. The divergenceof the spin correlation leads to the divergence of the 1/T and 1/T . Below T f , the power-law spin correlation de-velops in the state in the Fig.(1). The truly long-rangedspin correlation happens at the zero temperature, dubbedby the ”order by disorder”[6, 7]. Through our analysis,NiGa S should be removed from the candidate list forthe spin-liquid ground state. Finally, the long-soughtBKT transition in the quantum spin system is unexpect-edly found. The new phase accompanying with the novelphase transition will refresh our understanding of the spindynamics in the critical phase.CHC particularly acknowledges S. Nakatsuji and Y.Nambu for providing the experimental data and the stim-ulated discussion. He deeply appreciates the discussionwith M. Oshikawa, with whom the argument for the BKTtransition is formulated. He is also grateful for the fruit-ful discussions with J. Moore and D. Agterberg. Most ofthe calculation is done in the Supercomputer Center inthe Institute for Solid State Physics. ∗ Electronic address: [email protected] [1] J. Kosterlitz and D. Thouless, J. Phys. C , 1181 (1973).[2] W. Rantner and X.-G. Wen, Phys. Rev. Lett. , 3871(2001).[3] P. Anderson, Mater. Res. Bull. , 153 (1973).[4] S. Nakatsuji, Y. Nambu, H. Tonomura, O. Sakai,S. Jonas, C. Broholm, H. Tsunetsugu, Y. Qiu, andY. Maeno, Science , 1697 (2005).[5] H. Takeya, K. Ishida, K. Kitagawa, Y. Ihara, K. Onuma,Y. Maeno, Y. Nambu, S. Nakatsuji, D. E. MacLaughlin,A. Koda, et al., arXiv:0801.0190, to appear in Phys. Rev.B. (2008).[6] R. Moessner, S. Sondhi, and P. Chandra, Phys. Rev. Lett. , 4457 (2000).[7] R. Moessner and S. Sondhi, Phys. Rev. B , 224401(2001).[8] D. Blankschtein, M. Ma, A. N. Berker, G. S. Grest, andC. M. Soukoulis, Phys. Rev. B , 5250 (1984).[9] K. Takubo, T. Mizokawa, J.-Y. Son, Y. Nambu, S. Nakat-suji, and Y. Maeno, Phys. Rev. Lett. , 037203 (2007).[10] J. V. Jose, L. P. Kadanoff, S. Kirkpatrick, and D. R.Nelson, Phys. Rev. B , 1217 (1977).[11] C.-H. Chern and M. Tsukamoto, Phys. Rev. B77