Materals Scence-Poland, Vol. 25, No. 2, 2007 Quantum spn system wth on-ste exchange n a magnetc feld G. PAWŁOWSKI * Insttute of Physcs, Adam Mckewcz Unversty, 61-614 Poznań, ul. Umultowska 85, Poland We present the results of a full dagonalsaton appled to a 1 D quantum spn system wth on-ste exchange ansotropy. The model consdered s a quantum generalzaton of the 1 D classcal Blume Capel model. Thermodynamc propertes of the system n the presence of magnetc feld are examned takng nto account a quantum spn ladder wth a perodc boundary condton, where each rung of the ladder contans two nteractng spns 1/2. Ths effectve spn 1 system exhbts very rch thermodynamc behavour. We present the ground state results, showng many varous types of magnetc confguratons and compare t wth the classcal case. The calculatons are performed n the unfed smulaton envronment ALPS, whch offers the fulldag applcaton. Key words: quantzed spn model; spn ladder 1. Introducton We have studued the magnetc orderngs n the frustrated quantum spn 1/2 ladder wth the nteracton ansotropy (the AF Hesenberg generalzed ladder wth J II = J dag = J, J = Δ [1]). The scheme of the consdered structure s shown n Fg. 1. * E-mal: gpawlo@amu.edu.pl Fg. 1. The scheme of the frustrated quantum spn 1/2 ladder wth the nteracton ansotropy
566 G. PAWŁOWSKI The exchange nteractons between spns on the rung and along the man drecton of the ladder are dfferent. We examne the thermodynamc propertes of the system as a functon of Δ/J. The Hamltonan of the system consdered takes the form: (1) Hˆ = J ˆ σ ˆ ˆ ˆ ασ β + Δ σ σ, αβ, + 1 A B where σ s the quantum spn 1/2 operator on the th rung and α-leg of the ladder. ˆα We take nto account the nfluence of external magnetc feld h on the z component of the spn and we add the Zeeman term ˆ ˆ z z Hh ( ) = H h ( ˆ σ + ˆ σ ) (2) A B Such a model can be treated as the effectve nteractng (J) chan of spn 1 on the th ste wth the on-ste exchange (Δ) [2] n the presence of a magnetc feld (h). We can show that the Hamltonan (1) has a smlar form as the well known Blume Capel (BC) model [3] (we use the classcal sgn conventon) H = J SS + D S 2 BC j, j (3) where S s the classcal spn 1 Isng operator whch takes the values { 1,0,1}, J s the exchange nteracton and D s the sngle-on ansotropy. Formally, the Hamltonan (3) can be rewrtten nto the equvalent form n terms of spn 1/2. Let us express each spn S over the sum S = σ A + σ B of two classcal spns σ α =± 1/2 on the th ste. Ths transformaton s non-one-to-one and changes the number of egenstates of effectve spn S = { 1,0,0,1}. In our recent paper [4], we show that such a spn model wth zero -state degeneracy can be transformed nto the BC model wth a temperature-dependent sngle on ansotropy (at T=0 the models are dentcal). We rewrte the Hamltonan (3) for S ~ n the new varable σ α H = J σ σ + 2D σ σ + C (4) α jβ A B, j, α, β where α, β = {A,B}, J = J, D = D k ln 2 BT and C = ND/2, whch yelds a smlar form as n Eq. (1). In the analyss of the system, we have mplemented the unfed smulaton envronment ALPS [5], whch offers the fulldag applcaton [6]. Usng the XML language, we have frst defned a new dagonally connected ladder graph structure, where = 1..., x (Fg. 2) where x s the number of rungs, 2x number of vertces, 1A, 1B, xα, xβ, are the proper symbols of the vertex number.
Quantum spn system wth on-ste exchange n a magnetc feld 567 Fg. 2. Dagonally connected ladder graph structure On the last defned EDGE we mpose perodc boundary condtons. In the dagonalzaton procedure we use also the model-dspn.xml fle n whch the quantum nteractons between local spns 1/2 are defned. After preparng the parameter.xml fle, where we specfy the type of the system, temperature, the value of the nteractons etc., we start wth the fulldag procedure to get the egenvalues of the system. Then we use the fulldag-evaluate program to obtan thermodynamc propertes of the system. 2. Results We present the exact results of 12 quantum spns on the ladder wth antferromagnetc (AF) exchange nteracton (Δ > 0, J = 1). For the famly of Δ/J values we obtaned the followng thermodynamc characterstcs: magnetzaton, specfc heat, susceptblty, entropy and energy as functons of the magnetc feld. In Fgure 4, exemplary results for Δ = 1.8 are presented. The dagrams are symmetrcal due to the spn degeneracy n the absence of magnetc feld. Fgure 4 presents possble magnetc orderngs n the ground state n the plane hδ for the quantum case (Fg. 4a) and n hd for the classcal case (Fg. 4b) (the results have been extrapolated from very low temperatures). In the quantum case, we observe some states wth magnetzaton analogous as n the classcal case. The classcal dsordered phase (M = 0) corresponds to the dmer phase D2 of snglets along the rungs, whle the Isng spn 1 AF1 phase ( 1,1) s replaced by the dmer phase D1 (equvalent to the Haldane phase of the S = 1 chan) [3, 7]. For M = 0.5 we observe equvalent magnetc order n the classcal, as well as n the quantum case. For h > h c there s paramagnetc state (the saturated ferromagnetc) n both stuatons.
568 G. PAWŁOWSKI Fg. 3. The thermodynamc characterstcs of the system as a functon of external magnetc feld h for Δ = 1.8 and T = 0.05 h Fg. 4. The magnetc orderngs n the ground state n the presence of magnetc feld for the antferromagnetc ladder descrbed by the quantum model (a) and by the classcal one (b)
Quantum spn system wth on-ste exchange n a magnetc feld 569 Though the detaled structure of phase dagrams of the classcal BC model (Fg. 4b) must be dfferent from that presented for the quantum system (Fg. 4a), the multphase composton for large Δ and h of both models s smlar. Acknowledgement The author would lke to thank S. Robaszkewcz for useful dscussons. Ths work was supported n part by the Polsh State Commttee for Scentfc Research, Grant No. 1 P03B 08426. References [1] CHATTOPADHYAY E., BOSE I., Physca A, 318 (2003), 14. [2] WHITE S., Phys. Rev. B, 53 (1996), 52; HIDA K., Phys. Rev. B, 45 (1992), 2207. [3] BLUME M., Phys. Rev., 141 (1966), 517; Capel H.W., Physca (Amsterdam), 32 (1966), 966. [4] PAWŁOWSKI G., Eur. Phys. J. B, 53 (2006), 471. [5] ALBUQUERQUE A.F., ALET F., CORBOZ P., DAYAL P., FEIGUIN A., FUCHS S., GAMPER L., GULL E., GÜRTLER S., HONECKER A., IGARASHI R., KÖRNER M., KOZHENIKOV A., LÄUCHLI A., MANMANA S.R., MATSUMOTO M., MCCULLOCH I.P., MICHEL F., NOACK R.M., PAWŁOWSKI G., POLLET L., PRUSCHKE T. SCHOLLWÖCK U., TODO S., TREBST S., TROYER M., WERNER P., WESSEL S., J. Magn. Magn. Mater., 310 (2007), 1187; http://alps.comp-phys.org. [6] TROYER M., Lect. Notes Comp. Sc., 1732 (1999), 164. [7] XIAN Y., Phys. Rev. B, 52 (1995), 12485. Do Autora: proszę uaktualnć Ref. 4. Receved 7 May 2006 Revsed 1 September 2006