Non-interacting Spin-1/2 Particles in Non-commuting External Magnetic Fields
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1 EJTP 6, No ) Electronc Journal of Theoretcal Physcs Non-nteractng Spn-1/ Partcles n Non-commutng External Magnetc Felds Kunle Adegoke Physcs Department, Obafem Awolowo Unversty, Ile-Ife, Ngera Receved 6 July 008, Accepted 13 January 009, Publshed 0 February 009 Abstract: We obtan, n one dmenson, all the energy levels of a system of non-nteractng spn-1/ partcles n non-commutng external magnetc felds. Examples of how to ncorporate nteractons as perturbatons are gven for the Isng model n two orthogonal felds and for the XZ model n two orthogonal felds. c Electronc Journal of Theoretcal Physcs. All rghts reserved. Keywords: Quantum Spn Systems; Isng Systems; Magnetc Felds; Nonlnear Dynamcs and Nonlnear Dynamcal Systems; Magnetc Ansotropy; Numercal Smulaton Studes PACS 008): Hk; Ds; a 1. Introducton Hamltonan spn models nvolvng two external non-commutng magnetc felds are beng ncreasngly studed these days [1,, 3]. It s almost always the case that such models cannot be solved exactly, n the presence of spn nteractons. Interacton s usually due to ether nearest neghbour exchange or next nearest neghbour exchange or both. As s common practce, approxmate calculatons can be done by frst startng wth an exactly solvable model and then ntroducng the nteractons as perturbatons. Ths s done n ths paper. We start by consderng the smple model descrbed by the Hamltonan H = h x N =1 S x h z N S z, 1) =1 where S x and S z are spn-1/ operators on the th lattce ste of a one-dmensonal chan of N partcles. h x and h z are the external transverse and longtudnal magnetc felds, respectvely, measured n unts where the splttng factor and Bohr magneton are unty. adegoke@daad-alumn.de and adegoke@rushpost.com
2 44 Electronc Journal of Theoretcal Physcs 6, No ) It wll often be convenent to wrte H = N H, =1 where H = h x S x h zs z. The model 1) s exactly solvable and n ths work we wll obtan all the energy levels and the correspondng states. The man mport of the result wll be that nteractons can be ncluded as perturbatons on the model 1), ths approach beng effectve n obtanng accurate results for models ncorporatng two external felds.. The One-partcle Model Snce the N spn-1/ partcles descrbed by 1) are non-nteractng, all results can be obtaned from the Hamltonan for a sngle partcle. We drop the ste subscrpts n 1) and wrte H ε for the one spn system and wrte H ε = h x S x h z S z, where S x and S z are smply the spn-1/ operators S x = 1 01,S z = n a bass wth S z dagonal and wth = 1. We can denote the bass states n the S z bass by the set {, } andnthes x bass by {, }, sothat S z = 1 S z = 1 S x = 1 S x = 1. S x s obtaned from S z by the smlarty transformaton S x = PS z P 1, where P s the untary matrx P = = P
3 Electronc Journal of Theoretcal Physcs 6, No ) Thus, the S x bass states and the S z bass states are related by = 1 + ) = 1 ), wth the nverse relaton = 1 + ) = 1 ). The Hamltonan of the one partcle system n the S z bass s The normalzed egenstates of H ε are wth egenenergy h x H ε = 1 h z. h x h z h x + ε 0 = h x + h z + ) h x + h z h z + ), h x + h z and wth egenenergy ε 1 = h ε 0 = x + h z, h x + h x + h z ) h x + h z h z ), h x + h z h ε 1 =+ x + h z. Now that we have obtaned the egenstates and the correspondng energes of the one-partcle Hamltonan, we return to the general Hamltonan 1). 3. Energy Levels and Degeneraces From the results of the prevous secton and the fact that the partcles are non-nteractng, t s clear that there are N + 1 energy levels. We obtan them presently.
4 46 Electronc Journal of Theoretcal Physcs 6, No ) Ground State The ground state E 0 of the system of non-nteractng N spn-1/ partcles s the drect product state E 0 = ε 0 ε 0 ε 0 ε 0 = ε 0 ) N. ) The ground state energy of the system s found as follows H E 0 = H E 0 =H 1 ε 0 ) ε 0 ε 0 ε 0 + ε 0 H ε 0 ) ε 0 ε 0 + ε 0 ε 0 H 3 ε 0 ) ε 0 ε ε 0 ε 0 ε 0 ε 0 H N ε 0 ). Snce H ε 0 = ε 0 ε 0 and there are N terms n the above sum, we thus fnd that H E 0 = Nε 0 E 0. The ground state energy of the model s therefore E 0 = Nε 0 = N h x + h z. We shall have more to say about the ground state, n the next secton, but presently we obtan the other energy levels. 3. Excted States A frst excted state FES) of the model s E 1 1 = ε 0 ε 0 ε 0 ε 0 ε 1. We see at once that the FES s N fold degenerate, snce ε 1 can occur anywhere n the drect product state. The superscrpt 1 was affxed n antcpaton. The remanng N 1 states that are degenerate wth E 1 1 are E 1 = ε 0 ε 0 ε 0 ε 1 ε 0 E 1 3 = ε 0 ε 0 ε 1 ε 0 ε 0 E 1 N = ε 1 ε 0 ε 0 ε 0 ε 0.
5 Electronc Journal of Theoretcal Physcs 6, No ) The energy of the frst excted state s found from H E 1 1 = H E 1 1 =H 1 ε 0 ) ε 0 ε 0 ε 0 ε 1 + ε 0 H ε 0 ) ε 0 ε 0 ε ε 0 ε 0 ε 0 H N ε 1 ) From whch t follows that =N 1)ε 0 E ε 1 E 1 1 =[N 1)ε 0 + ε 1 ] E 1 1. E 1 =N 1)ε 0 + ε 1 =N )ε 0. In general, a kth excted state has k ε 1 factors n the drect product state. Snce there are N,k) ways of arrangng the k ε 1 factors, ths means that the kth excted state has degeneracy g E k ), gven by g E k )= N k = N! k!n k)!. The degenerate energy E k s easly found by the above scheme to be E k =N k)ε 0 + kε 1 =N k)ε 0. We see that only the ground state E 0 and the Nth excted state E N are nondegenerate. 4. An Explct Expresson for the Ground State We recall from equaton ) that the ground state s the drect product state where ε 0 = E 0 = ε 0 ) N, 3) h x + h z + ) h x + h z h x h + z + ). h x + h z
6 48 Electronc Journal of Theoretcal Physcs 6, No ) Performng the bnomal expanson suggested by equaton 3), we have that E 0 = N h N m x h z + m h x z) + h Sm [ h x + h z + ) ] N/, 4) h x + h z m=0 where S m s the lnear combnaton of the N,m) states wth m spns up n the total S z bass), that s, the lnear combnaton of all states wth total S z = m N/. We can check the trval lmts of 4): 1) h z =0 H = h x ) h x =0 that s Puttng h z = 0 n 4) gves E 0 = N m=0 = 1 ) N S x h N m x h m x S m h x )N/ N S m m=0 = 1 + )N ) N = ) N = H = h z Here the ground state s trvally the all spns up, non-degenerate ferromagnetc state E 0 = S z wth energy E 0 = Nh z / Puttng h x = 0 n n 4), we see that the only non-vanshng term n the sum occurs when m = N and so, E 0 = h z) N h z ) S N N =.
7 Electronc Journal of Theoretcal Physcs 6, No ) Example Applcatons In ths secton, we wll consder two models wth nteracton, the nteractons wll be treated as perturbatons. Specfcally we wll obtan the ground state energy of the Isng model n two felds and also the XZ model n non-commutng felds to second order n nearest neghbour exchange nteractons. We wll also compute frst order correctons to the energy of the frst excted state for each model. We frst note that for the one-partcle system and S z ε 0 = 1 h x + h z + ) h x + h z h x h + z + ) h x + h z = α S z ε 1 = 1 h x + h x + = β, so that we have the matrx elements h z ) h x + h z h z ) h x + h z and Furthermore, ε 0 S z ε 0 = ε 0 α = 1 h z h x+h z ε 1 S z ε 0 = ε 1 α = 1 h x = ε 0 β h x+h z ε 1 S z ε 1 = ε 1 β = 1 h z. h x+h z S x ε 0 = 1 h x + h z + ) h x + h z h x h + z + ) h x + h z S x ε 1 = 1 h x + h z ) h x + h z h x h + z ), h x + h z 5) so that we have also the followng marx elements: ε 0 S x ε 0 = ε 0 λ = 1 h x = ε 1 α h x +h z ε 1 S x ε 0 = ε 1 λ = 1 h z = ε 0 S x ε 1 = ε 0 δ h x +h z ε 1 S x ε 1 = ε 1 δ = 1 h x = ε 1 α = ε 0 λ h x +h z
8 50 Electronc Journal of Theoretcal Physcs 6, No ) We are now ready to compute the energy correctons. 5.1 Perturbaton Expanson of the Ground State Energy of the Isng Model n Non-Commutng Felds The Isng model n non-commutng magnetc felds s descrbed by the Hamltonan where H I = j S z S+1 z h x S x h z S z = V + H, 6) V = j S z S z +1 and H s as gven n 1). The nearest neghbour exchange nteracton j s assumed postve, so that we have a ferromagnetc model. Furthermore, we assume a perodc boundary condton, so that SN+ z = Sz. For weak nteracton, V can be treated as a perturbaton Frst Order Energy Correcton The frst order correcton to the ground state energy of the Hamltonan 6) s gven by ΔE 1) 0 I = E 0 V E 0 = j E 0 S z Sz +1 E 0. Now, S z S+1 z E 0 = S1S z z E 0 + SS z 3 z E SN z Sz 1 E 0 = α α ε 0 ε 0 ε 0 + ε 0 α α ε 0 ε 0 + ε 0 ε 0 α α ε 0 7) + + α ε 0 ε 0 ε 0 α. Multplyng from the left by E 0,wehave E 0 S z Sz +1 E 0 = N ε 0 α ) = N 4 h z h x +. h z
9 Electronc Journal of Theoretcal Physcs 6, No ) We therefore have that the frst order correcton to the ground state energy of the Isng model n non-commutng magnetc felds s 5.1. Second Order Correcton ΔE 1) 0 I = jn 4 h z h x +. h z The second order correcton to the ground state energy of the Isng model n noncommutng felds has the form ΔE ) 0 I = E E V E 0 E 0 E. 8) From the form of 7) and the fact that ε 0 ε 1 = 0 t s clear that only states wth E = E 1 and those wth E = E can contrbute to the sum n 8). We consder them n turns. When E = E 1,weobservethat E 1 1 = ε 0 ε 0 ε 0 ε 1 gves 1 E 1 S z Sz +1 E 0 = ε 1 ε 0 ε 0 ε 0 ε 0 ε 0 ε 0 ε 0 α α + ε 1 ε 0 ε 0 ε 0 α ε 0 ε 0 ε 0 ε 0 α = ε 1 α ε 0 α. We have smlar results from the remanng N 1 states that are degenerate wth E 1 1.TheE = E 1 states therefore contrbute N ε 1 α ε 0 α ) j) ε 0 = Nj 8ε 0 h x h z h x + h z) to the sum n 8). We have used 5) to substtute for ε 1 α and ε 0 α and we have also used the fact that E 0 E k =kε 0. As for the states wth E = E,onlyN those wth two consecutve ε 1 factors) of the N,) states contrbute to the sum. Ther contrbuton s N ε 1 α ) j) 4ε 0 = Nj h 4 x 64ε 0 h x + h z). The second order correcton to the ground state energy of the Isng model n noncommutng felds s therefore ΔE ) 0 I = Nj 8ε 0 h x h z h x + h z) + Nj 64ε 0 h 4 x h x + h z).
10 5 Electronc Journal of Theoretcal Physcs 6, No ) Perturbaton expanson of the ground state energy of the XZ model n non-commutng felds. The XZ model n one dmenson, n the presence of non-commutng external felds s descrbed by the Hamltonan H xz = j 1 N =1 S x S x +1 j N =1 S z S z +1 h x N =1 S x h z where j 1 > 0andj > 0 are the nearest neghbour exchange nteracton energy. Agan we assume a perodc boundary condton, so that SN+ x = Sx and SN+ z = Sz. For weak nearest neghbour nteractons, after calculatons smlar to that n the prevous secton, we obtan the followng correctons to the ground state energy: ΔE 1) 0 XZ = N 1 4 h x + j1 h h x + j hz) z and ΔE ) 0 XZ = N h x h ) z j 8ε 0 h x j h z ) + N 1 j 64ε 0 h x + h z) 1 h 4 x + ) j h4 z. The ground state energy of the XZ model n non-commutng felds, to second order n nteractons j 1 and j s thus E 0XZ = N h x + h z N 1 j1 h 4 h x + h x + j hz) z + N 8ε 0 h x h z h x + h z) j 1 + j + N 64ε 0 1 h x + h z ) j 1 h 4 x + j h4 z 5.3 Frst Order Correcton to the Energy of the Frst Excted State FES) of the Isng Model n Non-Commutng Felds The frst excted state, n the absence of nteractons, s N fold degenerate, wth the N states as gven n 3.). We therefore use degenerate perturbaton theory to fnd the frst order correcton to the frst excted state energy of the Isng model n non-commutng felds. S z S+1 z E 1 1 = α α ε 0 ε 0 ε 1 + ε 0 α α ε 0 ε 0 ε ε 0 ε 0 ε 0 ε 0 α β + α ε 0 ε 0 ε 0 ε 0 β. ) ). N =1 S z,
11 Electronc Journal of Theoretcal Physcs 6, No ) We therefore have that 1 E 1 S z Sz +1 E 1 1 =N ) ε 0 α ) + ε 0 α ε 1 β ) = h z N h x + h z 4 1. A smlar calculaton for the remanng N 1 states shows that the dagonal elements of the N N perturbaton matrx V, are equal, beng gven by: V = E 1 j ) S z Sz +1 E 1 ) = jh z N h x + h z 4 1 = f. 9) We can also compute V 1 thus V 1 = E 1 j ) S z Sz +1 E 1 1 = j ε 0 ε 1 ε 0 ε 0 ε 0 α β = j ε 1 α ε 0 β = j h x 4 h x +. h z In fact, for m n, wehave V mn = m E 1 j ) S z Sz +1 E 1 n = j h x δ 4 h x + h m,n±1 z = V nm = g. 10) The N N perturbaton matrx V s therefore fg g gfg gfg gf g 0 0 V = 000g f g g f g g g f
12 54 Electronc Journal of Theoretcal Physcs 6, No ) where f and g are as gven n 9) and 10). Clearly f and g are negatve for N>4. The smallest egenvalue of V s g + f and s non-degenerate, so that the correcton to the frst excted state energy, to frst order n nteracton j s ΔE 1) 1 I =g + f = j h x h x + h z jh z h x + h z ) N 4 1. The nearest neghbour nteractons thus lft the degeneracy n the frst excted state of the model. The energy of the frst excted state of the Isng model n non-commutng felds, to frst order n j s therefore E 1I = 1 N ) h x + h z j h x h x + h z jh z h x + h z ) N Frst Order Correcton to the Energy of the Frst Excted State FES) of the XZ Model n Non-Commutng Felds A completely analogous treatment to the above gves the energy of the frst excted state of the XZ model n non-commutng felds, to frst order n the nteractons j 1 and j as E 1XY ) N h = 1 x + h z 1 j h x + j 1 h z h x + h z N 4 1 ) j1 h x + j h z h x + h z. Concluson We have obtaned all the energy levels of a system of N non-nteractng spn-1/ partcles n non-commutng external magnetc felds, n one dmenson. The energy of the kth level s E k =N k)ε 0,whereε 0 = h x + h z )/ andk =0, 1,,...,N. The kth energy level was found to be N,k) = N!/[k!N k)!]-fold degenerate. An expct expresson for the ground state was also derved equaton 4)). It s not dffcult to gve smlar explct expressons for the excted states. A scheme for obtanng the energes of the excted states was hghlghted n secton 3.. Snce most real systems do nteract, examples of how the nteracton terms of an Hamltonan can be ncluded by Raylegh-Schrödnger perturbaton was demonstrated by fndng energy correctons to the ground state energy of the Isng model n non-commutng felds, as well as to the ground state energy of the XZ model n non-commutng felds. Correctons to the frst excted state energy were also calculated.
13 Electronc Journal of Theoretcal Physcs 6, No ) Acknowledgments The author s grateful to the DAAD for a scholarshp and thanks the Physcs Insttute, Unverstät Bayreuth for hosptalty. References [1] D.V.DmtrevandV.Y.Krvnov,Phys.Rev.B70, ). [] M. Kenzelmann, R. Coldea, D. A. Tennant, D. Vsser, M. Hofmann, P. Smebdl, and Z. Tylczynsk, Phys. Rev B. 65, ). [3] A. Ovchnnkov, D. V. Dmtrev, V. Y. Krvnov, and V. O. Cheranovsk, Phys. Rev. B 68, ).
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