- Email: [email protected]
Quantum Monte Carlo simulation for S=1 Heisenberg model with uniaxial anisotropy
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 1360–1361 www.elsevier.com/locate/jmmm
Quantum Monte Carlo simulation for S ¼ 1 Heisenberg model with uniaxial anisotropy Mitsuaki Tsukamotoa,, Cristian Batistab, Naoki Kawashimaa a
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Condensed Matter and Statistical Physics, Los Alamos National Laboratory, University of California, Los Alamos, NM 87545, USA
b
Available online 7 November 2006
Abstract We perform quantum Monte Carlo simulations for S ¼ 1 Heisenberg model with an uniaxial anisotropy. The system exhibits a phase transition as we vary the anisotropy and a long range order appears at a finite temperature when the exchange interaction J is comparable to the uniaxial anisotropy D. We investigate quantum critical phenomena of this model and obtain the line of the phase transition which approaches a power-law with logarithmic corrections at low temperature. We derive the form of logarithmic corrections analytically and compare it to our simulation results. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.jm; 75.30.Kz; 75.40.Mg Keywords: Quantum phase transition; Quantum Monte Carlo; Heisenberg model
1. Introduction In the past years, quantum spin systems that exhibit a Bose–Einstein condensation (BEC) in applied magnetic fields have been studied extensively [1,2]. The compound for example, NiCl2 –4SCðNH2 Þ2 (DTN) is a new substance for BEC and attracts great deal of interest [2]. In this paper, we study S ¼ 1 magnetic compounds with uniaxial anisotropy such as the ones with Ni atoms. The Ni compounds are described by the S ¼ 1 Hamiltonian, X X H¼ J n S i S iþen þ D ðS zi Þ2 , (1) i;n
i
where n ¼ fx; y; zg , J n 40; D40 and en is an unit vector in the n direction. In the absence of a magnetic field, because D J n , the system has a ground state of S z ¼ 0 and a finite excitation gap between the ground state and excited states of S z ¼ 1. When the magnetic field H is applied, excited states split. At a certain H, the gap between the Corresponding author. Tel.: +81 4 7136 3262.
E-mail address: [email protected] (M. Tsukamoto). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.383
ground state and the lowest excited state closes and the system has a three-dimensional (3-D) long range order, which is regarded as BEC of excited magnons. According to the mean field theory [3], which was confirmed [4] by a scaling argument and a simulation, the phase boundary has a form T / jH H c1 j2=3 , and this is consistent with the experimental results [2]. In this paper, we study Hamiltonian (1) with isotropic interactions (J x ¼ J y ¼ J z J) for simplicity. When D J, the system has a finite excitation gap above the ground state and magnetic ordering does not occur. On the other hand, when D J, the system has a Ne´el order at a finite temperature. Tuning the ratio of D to J corresponds to controlling the applied pressure in experiments. The critical phenomenon which we consider here is a pressureinduced magnetic ordering rather than a field-induced one. We calculate the transverse staggered susceptibility, wxx by quantum Monte Carlo simulations based on the directedloop algorithm [5,6]. Critical temperatures are estimated from finite-size scaling of wxx to obtain the D vs. T phase diagram. We also discuss logarithmic corrections which appear in the present case.
ARTICLE IN PRESS M. Tsukamoto et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 1360–1361
2. Finite size scaling
0.103
3. Logarithmic corrections We are interested in the quantum critical phenomena at T ¼ 0 where D has a critical value Dc , above which the long range order does not exist. The universality class of the quantum critical point (QCP) is a that of ð3 þ zÞdimensional classical XY model, where z denotes the dynamic exponent. According to the spin wave theory, the system described by the Hamiltonian (1) has a linear dispersion relation which means z ¼ 1. The 4-D classical XY model has a marginal operator that gives rise to logarithmic corrections [8]. In the present case, the power low behavior of the phase boundary is affected by logarithmic corrections near the QCP. The cross-over from the 4-D XY universality class to the 3-D XY universality class can be observed in the vicinity of the QCP [9]. The form of logarithmic corrections can be calculated by renormalization group equations for the classical Oð2Þ model [8]. Taking the logarithmic corrections into account, the phase boundary has the following form: L=8 L = 12 L = 16
0
(T-Tc) χ1/γ
-5
0.1025
0.15
0.102 J/D
In the present case, the Hamiltonian has the U(1) symmetry. At the critical point, the magnetic ordering occurs in the xy-plane by breaking the U(1) symmetry. This means that the universality class of critical points where the magnetic ordering occurs at finite temperature is that of the classical 3-D XY model. We calculate wxx with the system size of L ¼ 8; 12 and 16, and perform the finite size scaling of wxx to estimate critical temperatures. Fig. 1 shows an example of the finite-size scaling of wxx at D=J ¼ 1:25, using critical exponents of the 3-D XY model, g ¼ 1:3177 and n ¼ 0:67155 [7]. The estimated critical temperature is T c ¼ 2:394. We calculate T c at various D to obtain the phase diagram, which is shown in the inset of Fig. 2.
5
1361
0.1
0.1015
0
0.2
0.4
0.6
0.101 0.1005
Simulation Fitting with log correction Fitting without log correction
0.1 0
0.02
0.04
0.06
0.08
0.1
0.12
Tc/J Fig. 2. The phase diagram of J=D vs. T c =J. The marks are simulation results. Simulation results are fitted by using Eq. (2) (solid line) and the form without the logarithmic correction (dashed line). The inset is a global view.
T=J / jD=J ðD=JÞc j1=2 j ln jD=J ðD=JÞc jj1=5 .
(2)
In Fig. 2 (main panel) the numerical estimates of T c and fitting curves near the QCP are shown. In addition to Eq. (2), the form without logarithmic corrections, T / jD=J ðD=JÞc j1=2 , are also examined. Here, six points at lowest temperatures are used for fitting. Although it is difficult to distinguish between two fitting curves in a global scale (inset), in the vicinity of the QCP, it seems that the form Eq. (2) fits better than the form without the logarithmic correction. 4. Summary We have performed quantum Monte Carlo simulations to investigate the quantum critical phenomena for S ¼ 1 Heisenberg Model with uniaxial anisotropy. The existence of a logarithmic correction has been confirmed by our numerical results. The calculation has been done using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. References
-10 -15 -20 -25 -10
-5
0
5
10
15
(T-Tc) L1/ν Fig. 1. Finite size scaling plots for wxx at D=J ¼ 1:25 with system sizes of L ¼ 8; 12 and 16 by using the critical exponents of 3-D classical XY model, g ¼ 1:3177, n ¼ 0:67155. The estimated critical temperature is T c ¼ 2:394.
[1] Z. Honda, K. Katsumata, H.A. Katori, K. Yamada, T. Ohishi, T. Manabe, M. Yamashita, J. Phys.: Condens. Matter 9 (1997) L83. [2] V.S. Zapf, D. Zocco, B.R. Hansen, M. Jaime, N. Harrison, C.D. Batista, M. Kenzelmann, C. Niedermayer, A. Lacerda, A. PaduanFilho, Phys. Rev. Lett. 96 (2006) 077204. [3] T. Nikuni, M. Oshikawa, A. Oosawa, H. Tanaka, Phys. Rev. Lett. 84 (2000) 5868. [4] N. Kawashima, J. Phys. Soc. Japan 73 (2004) 3219. [5] O.F. Sljuasen, A.W. Sandvik, Phys. Rev. E 66 (2002) 046701. [6] N. Kawashima, K. Harada, J. Phys. Soc. Japan 73 (2004) 1379. [7] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, E. Vicari, Phys. Rev. B 63 (2001) 214503. [8] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, Cambridge, New York, 1997. [9] S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, New York, 1999.
Journal of Magnetism and Magnetic Materials 310 (2007) 1360–1361 www.elsevier.com/locate/jmmm
Quantum Monte Carlo simulation for S ¼ 1 Heisenberg model with uniaxial anisotropy Mitsuaki Tsukamotoa,, Cristian Batistab, Naoki Kawashimaa a
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Condensed Matter and Statistical Physics, Los Alamos National Laboratory, University of California, Los Alamos, NM 87545, USA
b
Available online 7 November 2006
Abstract We perform quantum Monte Carlo simulations for S ¼ 1 Heisenberg model with an uniaxial anisotropy. The system exhibits a phase transition as we vary the anisotropy and a long range order appears at a finite temperature when the exchange interaction J is comparable to the uniaxial anisotropy D. We investigate quantum critical phenomena of this model and obtain the line of the phase transition which approaches a power-law with logarithmic corrections at low temperature. We derive the form of logarithmic corrections analytically and compare it to our simulation results. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.jm; 75.30.Kz; 75.40.Mg Keywords: Quantum phase transition; Quantum Monte Carlo; Heisenberg model
1. Introduction In the past years, quantum spin systems that exhibit a Bose–Einstein condensation (BEC) in applied magnetic fields have been studied extensively [1,2]. The compound for example, NiCl2 –4SCðNH2 Þ2 (DTN) is a new substance for BEC and attracts great deal of interest [2]. In this paper, we study S ¼ 1 magnetic compounds with uniaxial anisotropy such as the ones with Ni atoms. The Ni compounds are described by the S ¼ 1 Hamiltonian, X X H¼ J n S i S iþen þ D ðS zi Þ2 , (1) i;n
i
where n ¼ fx; y; zg , J n 40; D40 and en is an unit vector in the n direction. In the absence of a magnetic field, because D J n , the system has a ground state of S z ¼ 0 and a finite excitation gap between the ground state and excited states of S z ¼ 1. When the magnetic field H is applied, excited states split. At a certain H, the gap between the Corresponding author. Tel.: +81 4 7136 3262.
E-mail address: [email protected] (M. Tsukamoto). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.383
ground state and the lowest excited state closes and the system has a three-dimensional (3-D) long range order, which is regarded as BEC of excited magnons. According to the mean field theory [3], which was confirmed [4] by a scaling argument and a simulation, the phase boundary has a form T / jH H c1 j2=3 , and this is consistent with the experimental results [2]. In this paper, we study Hamiltonian (1) with isotropic interactions (J x ¼ J y ¼ J z J) for simplicity. When D J, the system has a finite excitation gap above the ground state and magnetic ordering does not occur. On the other hand, when D J, the system has a Ne´el order at a finite temperature. Tuning the ratio of D to J corresponds to controlling the applied pressure in experiments. The critical phenomenon which we consider here is a pressureinduced magnetic ordering rather than a field-induced one. We calculate the transverse staggered susceptibility, wxx by quantum Monte Carlo simulations based on the directedloop algorithm [5,6]. Critical temperatures are estimated from finite-size scaling of wxx to obtain the D vs. T phase diagram. We also discuss logarithmic corrections which appear in the present case.
ARTICLE IN PRESS M. Tsukamoto et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 1360–1361
2. Finite size scaling
0.103
3. Logarithmic corrections We are interested in the quantum critical phenomena at T ¼ 0 where D has a critical value Dc , above which the long range order does not exist. The universality class of the quantum critical point (QCP) is a that of ð3 þ zÞdimensional classical XY model, where z denotes the dynamic exponent. According to the spin wave theory, the system described by the Hamiltonian (1) has a linear dispersion relation which means z ¼ 1. The 4-D classical XY model has a marginal operator that gives rise to logarithmic corrections [8]. In the present case, the power low behavior of the phase boundary is affected by logarithmic corrections near the QCP. The cross-over from the 4-D XY universality class to the 3-D XY universality class can be observed in the vicinity of the QCP [9]. The form of logarithmic corrections can be calculated by renormalization group equations for the classical Oð2Þ model [8]. Taking the logarithmic corrections into account, the phase boundary has the following form: L=8 L = 12 L = 16
0
(T-Tc) χ1/γ
-5
0.1025
0.15
0.102 J/D
In the present case, the Hamiltonian has the U(1) symmetry. At the critical point, the magnetic ordering occurs in the xy-plane by breaking the U(1) symmetry. This means that the universality class of critical points where the magnetic ordering occurs at finite temperature is that of the classical 3-D XY model. We calculate wxx with the system size of L ¼ 8; 12 and 16, and perform the finite size scaling of wxx to estimate critical temperatures. Fig. 1 shows an example of the finite-size scaling of wxx at D=J ¼ 1:25, using critical exponents of the 3-D XY model, g ¼ 1:3177 and n ¼ 0:67155 [7]. The estimated critical temperature is T c ¼ 2:394. We calculate T c at various D to obtain the phase diagram, which is shown in the inset of Fig. 2.
5
1361
0.1
0.1015
0
0.2
0.4
0.6
0.101 0.1005
Simulation Fitting with log correction Fitting without log correction
0.1 0
0.02
0.04
0.06
0.08
0.1
0.12
Tc/J Fig. 2. The phase diagram of J=D vs. T c =J. The marks are simulation results. Simulation results are fitted by using Eq. (2) (solid line) and the form without the logarithmic correction (dashed line). The inset is a global view.
T=J / jD=J ðD=JÞc j1=2 j ln jD=J ðD=JÞc jj1=5 .
(2)
In Fig. 2 (main panel) the numerical estimates of T c and fitting curves near the QCP are shown. In addition to Eq. (2), the form without logarithmic corrections, T / jD=J ðD=JÞc j1=2 , are also examined. Here, six points at lowest temperatures are used for fitting. Although it is difficult to distinguish between two fitting curves in a global scale (inset), in the vicinity of the QCP, it seems that the form Eq. (2) fits better than the form without the logarithmic correction. 4. Summary We have performed quantum Monte Carlo simulations to investigate the quantum critical phenomena for S ¼ 1 Heisenberg Model with uniaxial anisotropy. The existence of a logarithmic correction has been confirmed by our numerical results. The calculation has been done using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. References
-10 -15 -20 -25 -10
-5
0
5
10
15
(T-Tc) L1/ν Fig. 1. Finite size scaling plots for wxx at D=J ¼ 1:25 with system sizes of L ¼ 8; 12 and 16 by using the critical exponents of 3-D classical XY model, g ¼ 1:3177, n ¼ 0:67155. The estimated critical temperature is T c ¼ 2:394.
[1] Z. Honda, K. Katsumata, H.A. Katori, K. Yamada, T. Ohishi, T. Manabe, M. Yamashita, J. Phys.: Condens. Matter 9 (1997) L83. [2] V.S. Zapf, D. Zocco, B.R. Hansen, M. Jaime, N. Harrison, C.D. Batista, M. Kenzelmann, C. Niedermayer, A. Lacerda, A. PaduanFilho, Phys. Rev. Lett. 96 (2006) 077204. [3] T. Nikuni, M. Oshikawa, A. Oosawa, H. Tanaka, Phys. Rev. Lett. 84 (2000) 5868. [4] N. Kawashima, J. Phys. Soc. Japan 73 (2004) 3219. [5] O.F. Sljuasen, A.W. Sandvik, Phys. Rev. E 66 (2002) 046701. [6] N. Kawashima, K. Harada, J. Phys. Soc. Japan 73 (2004) 1379. [7] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, E. Vicari, Phys. Rev. B 63 (2001) 214503. [8] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, Cambridge, New York, 1997. [9] S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, New York, 1999.