QMC simulations of Heisenberg ferromagnet

Transcription

1 and Wolfard Janke Institut für Teoretisce Pysik, Universität Leipzig, Augustusplatz /, D-49 Leipzig, Germany We use te Quantum Stocastic Series Expansion (SSE) algoritm to investigate te properties of te one-dimensional Heisenberg ferromagnet. We consider te model for two spin values s = 2 and s = and study te dependence of te specific eat, magnetisation, magnetic susceptibility and correlation functions on temperature and magnetic field. Obtained results are compared to tose derived by analytical metods (Bete ansatz, Green s function, Random Pase Approximation). In particular, te surprising prediction of a double-peak structure in te specific eat at low temperatures and small magnetic fields can be confirmed. PoS(LAT25)24 XXIIIrd International Symposium on Lattice Field Teory 25-3 July 25 Trinity College, Dublin, Ireland Work supported by te EU Marie Curie Host Development Fellowsip IHP-HPMD-CT-2-8. Speaker. c Copyrigt owned by te autor(s) under te terms of te Creative Commons Attribution-NonCommercial-SareAlike Licence. ttp://pos.sissa.it/

2 . Introduction Te study of low-dimensional quantum ferromagnets is mainly motivated by progress in te syntesis of new materials wic can be described by effective one-dimensional (D) [, 2] or two-dimensional (2D) [3, 4, 5, 6] spin s = 2 Heisenberg Hamiltonians. In a recent paper [7] Green s function analytical metods were applied to suc systems. It was claimed tat, witin te employed approximation, te eat-capacity curve for te (isotropic) s = 2 Heisenberg cain sows an unexpected double peak at very low temperatures. Additionally it was sown tat te field dependent position of te maxima and te eigts of te specific eat and magnetic susceptibility fit to power laws. Te simulations we carried out confirm tese predictions for s = 2. Additionally we ave investigated te model for spin s =. Our simulations sow tat a similar scaling beaviour can be expected also in tat case. Te double peak in te specific eat is visible as well. However, te magnetic field range were it appears is sifted in comparison to te s = 2 case. Te outline of te paper is as follows. A brief definition of te model is given in section 2. Te section 3 contains an outline of te SSE algoritm, wic we employed in te simulations. Example tests of te metod s correctness are sown in section 4. Te results of te simulations of te s = 2 system are collected in section 5, and te results of te s = system in section 6. Finally, section 7 contains our summary. 2. Te model We consider te standard D Heisenberg ferromagnet (FHM), wic can be described by te following Hamiltonian: H = J Ŝ i Ŝ j Ŝi z, (2.) i j i were i j denotes a pair of spin operators Ŝ i = {Ŝi x,ŝy i,ŝz i } placed on te nodes of a D cain. Te coupling constant J is negative, J <, and te magnetic field is directed along te z-axis. We assume periodic boundary conditions, i.e., a ring of spins. PoS(LAT25)24 3. Te algoritm Te numerical simulations were carried out by means of te directed loop algoritm, a particular version of te stocastic series expansion (SSE) [8] metod. Te detailed description of tis algoritm can be found elsewere (see for example Ref. [9]), but for completeness we briefly present its key points. Te SSE algoritm is based on a power-series expansion of te partition function: Z = Tre βh = α n= ( β) n α H n α. (3.) n! 24 / 2

3 Magnetic susceptibility s=/2, =, L=6 Exact diagonalization Specific eat s=, =, L= Exact diagonalization Figure : Comparison of and direct diagonalization results for: magnetic susceptibility of te s = 2, L = 6 system in magnetic field = (left), and specific eat for s =, L =, = (rigt). It is convenient to use te eigenvectors of te operator Ŝi z as a base α, and express te Hamiltonian as a sum of bond operators: H = J L b= H b = J L b= (H,b + H 2,b ), (3.2) were L is te number of nodes. Eac bond operator is a sum of diagonal part: and off-diagonal part: H,b = C + Ŝ z i(b)ŝz j(b) 2J [Ŝz i(b) + Ŝz j(b) ], (3.3) H 2,b = 2 [Ŝ+ i(b)ŝ j(b) + Ŝ i(b)ŝ+ j(b) ], (3.4) Ŝ ± i = Ŝ x i ± Ŝ y i. To ensure tat all H,b matrix elements are positive, te additional C constant as been inserted. Finally te partition function is expressed as PoS(LAT25)24 were Z = α n= S n ( Jβ)n n! α n i= H ai,b i α, (3.5) S n = [a,b ],[a 2,b 2 ],...,[a n,b n ] (3.6) denotes te sequence of n diagonal (a i = ) and off-diagonal (a i = 2) operators living on bonds b i. After introducing an additional cut-off for te maximal number of operators, n = n max, tis expression can be used to simulate te system on a finite L n max lattice. 4. Tests One of te tests for correctness of our simulation program is te comparison of te data wit results obtained by oter metods. For very small systems (up to L = 6 spins for s = /2, 24 / 3

4 and up to L = spins for s = ) we were able to diagonalize te Hamiltonian matrix. Tis allowed us to calculate all interesting quantities exactly. An example comparison of te data and te direct diagonalization results is sown in figure, demonstrating perfect agreement (witin te statistical error bars of te metod). 5. Results for spin s = /2 We compare te results of te simulations of te spin s = 2 system wit te Beteansatz results for te infinite cain []. As an example we sow in figure 2 te magnetisation and te specific eat as a function of temperature, obtained by tese two metods. As it can be seen, already for te size L = 28, te finite-size effects are so small tat, in te sown temperature range, te and Bete-ansatz results are in perfect agreement. Te double maximum in te specific-eat curve can be seen for small magnetic fields <., as it was claimed in Ref. [7] using te Green s function approximation. We cecked te dependence of te temperature for wic te specific eat maximum appears, Tmax, C and te maximum eigt C max on te magnetic field. If in te range of small fields <., were te double maximum occurs, te left (lower temperature) maximum is taken into account, Magnetization s=/2, L=28 Bete ansatz =. =.5 =.5 = =.4 =.6 =. Specific eat s=/2, L=28 Bete ansatz =. =.5 =. =.5 = =.4 =.6 =. =2. Figure 2: Te dependence of te magnetisation (left) and te specific eat (rigt) on te temperature, for te s = 2 and L = 28 system, in various magnetic fields. Te Bete ansatz results are plotted by solid lines. PoS(LAT25)24 s=/2, L=28.4 s=/2, L=28 T C max Cmax.2 Tmax C.. Figure 3: Te dependence of te position of te specific-eat maximum (left) and te maximal value (rigt) on te magnetic field for s = 2, L = 28. Te fitted power-law ansatz is sown by a dased line. 24 / 4

5 Susceptibility s=, L=64 Green function RPA =.5 = =.2 =.4 =.6 =. Specific eat s=, L=64 =.5 =. =.3 =.5 = Figure 4: Magnetic susceptibility (left) and specific eat (rigt) of te s =, L = 64 system as a function of temperature. On te left plot, te analytical results (Green s function, random pase approximation (RPA)) are sown by solid lines. T χ max s=, L=64 Green function RPA. T χ max χmax. s=, L=64 Green function RPA Figure 5: Te dependence of te position of te susceptibility maximum (left) and te maximal value (rigt) on te magnetic field for spin s =, calculated by tree different metods (Green s function, random pase approximation (RPA), and simulations). te scaling wit field obeys a power-law beaviour as demonstrated in te log-log plots of figure 3. By fitting a power-law ansatz in te range < <., we obtain te following dependence: PoS(LAT25)24 T C max =.7().6(), C max =.5().22(). (5.) Tis can be compared wit results obtained by te Green s function metod [7]: T C max = , C max = (5.2) 6. Results for spin s = We compare te results of te spin s = simulations wit te results obtained by Green s function [7] and random pase approximation (RPA) [] metods. Te figure 4 sows te dependence of te magnetic susceptibility and specific eat on te temperature. In te plot sowing te susceptibility, Green s function and RPA results are also depicted. Te agreement is quite good; small differences are only visible for small magnetic fields. 24 / 5

6 Te double maximum in te specific eat curve is visible, but in contrast to te s = 2 case it does not appear for very small fields. Te range were it is visible starts at. (for spin s = 2 it starts already at = ). In te log-log plots of figure 5 te scaling beaviours of te position and eigt of te magnetic susceptibility maximum as a function of te magnetic field are sown. Again, all tree metods sow power-law scaling. 7. Summary Te results of our simulations of te D Heisenberg ferromagnet were briefly presented. A comparison wit oter analytical metods sows good agreement. Terefore te existence of te double peak in te specific-eat curve and te power-law scaling of te positions and eigts of te specific eat and magnetic susceptibility maxima ave been confirmed. We would like to tank Dieter Ile and Iren Junger for many useful discussions. References [] M. Takaasi, P. Turek, Y. Nakazawa, M. Tamura, K. Nozawa, D. Siomi, M. Isikawa, and M. Kinosita, Pys. Rev. Lett. 67 (99) 746. [2] C.P. Landee and R.D. Willett, Pys. Rev. Lett. 43 (979) 463. [3] N. Read and S. Sacdev, Pys. Rev. Lett. 75 (995) 359. [4] C. Timm, S.M. Grivin, P. Henelius, and A.W. Sandvik, Pys. Rev. B 58 (998) 64. [5] S. Feldkemper, W. Weber, J. Sculenburg, and J. Ricter, Pys. Rev. B 52 (995) 33. [6] H. Manaka, T. Koide, T. Sidara, and I. Yamada, Pys. Rev. B 68 (23) [7] I. Junger, D. Ile, J. Ricter, and A. Klümper, Pys. Rev. B 7 (24) 449. [8] A.W. Sandvik and J. Kurkijärvi, Pys. Rev. B 43 (99) 595. [9] O.F. Syljuasen and A.W. Sandvik, Pys. Rev. E 66 (22) 467. [] A. Klümper, Eur. Pys. J. B 5 (998) 667. [] S.V. Tjablikov, in Metods in te quantum teory of magnetism, Plenum Press, New York, 967. PoS(LAT25)24 24 / 6