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Numerical versus exact results for coupled magnetic spin 12 layers
•:•
Journal of Magnetism and Magnetic Materials 157/158 (1996) 336-337
~ i ~ journalof magnetism and magnetic materials
ELSEVIER
Numerical versus exact results for coupled magnetic spin 1 / 2 layers J. R i c h t e r a,*, C. G r o s b, S . E . K r i i g e r a, W . W e n z e l
b
a Institutffir Theoretische Physik, Universiti~t Magdeburg, P.O. Box 4120, D-39016 Magdeburg, Germany b lnstitutfiir Physik, Universitih Dortmund, D-44221 Dortmund, Germany
Abstract We study the phase diagram of spin 1 / 2 systems which consist of two antiferromagnetic Heisenberg layers coupled via an inter-layer coupling J ± . In dependence on the strength and the range of the inter-layer interaction J ± we discuss several magnetic phases, e.g. order from disorder as well as quantum disorder. Keywords: Antiferromagnetism; Quantum fluctuations; Bilayer; Order from disorder; Quantum disorder
Magnetic systems consisting of two interacting subsystems are realized in several experimental applications, e.g. in the Mn3Cr2Ge3Ol2 garnet [1,2] or several high-Tc superconducting compounds like YBazCu306+ x [3,4]. In these systems the competition between the inter-subsystem and intra-subsystem couplings leads to interesting magnetic phase diagrams. In the present paper we report recent calculations performed on spin systems built by two antiferromagnetic layers which interact mutually. The antiferromagnetic correlations within the layers are described by a Lieb-Mattis type Hamiltonian [5] ~
S~Sj,
second order
~0.10
E
S) s2.
(2)
iEI,jE2
Here we present the phase diagram of H = H I + H 2 + Hi~t in dependence on the strength of the interlayer coupling J± and fixed J1 = 0.2, ,/2 = 0.1 (Fig. 1). For small J ± both subsystems are antiferromagnetic at low temperatures (AF phase). Increasing the temperature the subsystem '2' undergoes at a critical temperature T~ a second order
* Corresponding author. Fax: +49-391-5592-131; emaih johannes.richter @physik.uni-magdeburg.de.
/
AFt
A / ~
AF
[,,,~Dj/II ODI
F
0.05
Fig. 1. Phase diagram of H
where N is the total number of sites, o~= 1(2) labels the layer and the antiferromagnetic interaction J~ > 0 is between spins on sublattice A and B. Recently we have shown [6] that the problem of interacting layers can be solved exactly assuming a long-ranged interaction term
N
J1=0.2 Jz=O. 1
(1)
i~A,j~B
I J± Hint=--
................................................
°°%'o ......d~ ......6"~......06
4A H,~= U
0.20
Jl
= H 1+ H 2 +
o 8 ' 'i.0 Hilt (see text).
transition to a paramagnetic phase whereas the subsystem '1' remains antiferromagnetic (AF 1 phase) till a critical temperature Tc1 > T~. For T > T] the whole system is paramagnetic. The specific heat shows two peaks at the transitions which is in accordance to the experimental observation in Ref. [1]. Physically more interesting is the region of medium strength of J ± . Here at low temperatures the magnetic structure within the subsystem '1' is canted and the subsystem '2' is ferromagnetically polarized (ODl-phase). The temperature dependence of the corresponding antiferromagnetic order parameter of system T shows interesting order from disorder phenomena [8,2], i.e. the magnetic correlations are increased by thermal fluctuations as shown in Fig. 2. This low-temperature OD 1 phase gives way to either the AF 1 phase or a ferromagnetic phase F. This ferromagnetic F phase is caused by dominating J ±
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0304-8853(95)01178-1
J. Richter et al. / Journal of Magnetism and Magnetic Materials 157/158 (1996) 336-337 1 .("
yielding a parallel (i.e. ferromagnetic) spin orientation within the subsystems. In a second part w e consider a short-range interaction b e t w e e n the layers w h i c h corresponds to the situation in bilayer cuprate superconducting c o m p o u n d s like e.g. YBa2Cu306+x. Hilnlt = J± E S1S2" iE 1,2
........ , ......... , ......... , ......... , ......
~0.4
~
"~
~
OF
Jr=0"2 3e=0-1 ....
J±=0.4
01 ............ ::(7 . . . . . . . .
0 AF
"13.00 0.02
~t.:
o15
o.~
o.to
0.,
0.05
. ....
"
0.04 0.06 kBT
0.08
o
o.~
)o.o0 J±
Fig. 3. Excitation gap (dashed lines) and antiferromagnetic order parameter ](~A~B + ~A~)I1/2/N (solid line) for H = H a + H 2 q - HinUt .
scaling analysis yields a critical J ± = J1 where the order breaks down. At the same point a gap opens to the first triplet excitation, w h i c h is in accordance with the above m e n t i o n e d spin-gap scenario in bilayer cuprates. Acknowledgement: The paper was supported by the country S a c h s e n - A n h a l t . References
Jz=0.63
~
0.3
0.20
(3)
In this case no c o m p l e t e solution o f the m o d e l is available, h o w e v e r because o f the high s y m m e t r y o f the p r o b l e m we could p e r f o r m exact diagonalization up to N = 240 sites [7]. (Notice that at present the exact diagonalization e.g. for the square lattice antiferromagnet is limited to N < 36). As it has been discussed recently [4] the short-range inter-layer interaction might b e responsible for the experimentally observed spin gap in t h e s e materials. According to the situation in the cuprate superconductors w e consider identical layers, i.e. J1 =-/2. Our results for the antiferromagnetic order parameter and the excitation gap are presented in Fig. 3. The inter-layer coupling J . acts against the ordering within the layers. A finite size 0.5
337
0.10
Fig. 2. Magnetic order parameters O ~ ( a = 1(2)) and O F versus i The order parameters are temperature for H = H 1 + / ~ + Him. ce 2 ~ ~ 2 defined as O~F = (4/N)<(S~, - SB ) } (sublattice magnetization of subsystem 'oe') and O F =(4/N-)((S~ 2) +(S~ 2) ) (magnetization of subsystems '1' and '2').
[1] I.V. Golosovskii et al., JETP Lett. 24 (1976) 423; T.A. Valyanskaya and V.I. Sokolov, Soy. Phys. JETP 48 (1978) 161. [2] E.F. Shender, Soy. Phys. JETP 56 (1982) 178. [3] J.M. Tranquada et al., Phys. Rev. B 46 (1992) 5561;R. Stem et al., Phys. Rev. B 50 (1994) 426. [4] A.J. Millis and H. Monien, Phys. Rev. Lett. 70 (1993) 2810; A.W. Sandvik and D.J. Scalapino, Phys. Rev. Lett. 72 (1994) 2777. [5] E. Lieb and D.C. Mattis, J. Math. Phys. 3 (1962) 749; J. Richter, Phys. Rev. B 47 (1993) 5794. [6] J. Richter, S.E. Kri.iger, A. Voigt and C. Gros, Europhys. Lett. 28 (1994) 363. [7] C. Gros, W. Wenzel and J. Richter, Europhys. Lett. 32 (1995) 747. [8] J. Villain, et al., J. Phys. (Paris) 41 (1980) 1263.
Journal of Magnetism and Magnetic Materials 157/158 (1996) 336-337
~ i ~ journalof magnetism and magnetic materials
ELSEVIER
Numerical versus exact results for coupled magnetic spin 1 / 2 layers J. R i c h t e r a,*, C. G r o s b, S . E . K r i i g e r a, W . W e n z e l
b
a Institutffir Theoretische Physik, Universiti~t Magdeburg, P.O. Box 4120, D-39016 Magdeburg, Germany b lnstitutfiir Physik, Universitih Dortmund, D-44221 Dortmund, Germany
Abstract We study the phase diagram of spin 1 / 2 systems which consist of two antiferromagnetic Heisenberg layers coupled via an inter-layer coupling J ± . In dependence on the strength and the range of the inter-layer interaction J ± we discuss several magnetic phases, e.g. order from disorder as well as quantum disorder. Keywords: Antiferromagnetism; Quantum fluctuations; Bilayer; Order from disorder; Quantum disorder
Magnetic systems consisting of two interacting subsystems are realized in several experimental applications, e.g. in the Mn3Cr2Ge3Ol2 garnet [1,2] or several high-Tc superconducting compounds like YBazCu306+ x [3,4]. In these systems the competition between the inter-subsystem and intra-subsystem couplings leads to interesting magnetic phase diagrams. In the present paper we report recent calculations performed on spin systems built by two antiferromagnetic layers which interact mutually. The antiferromagnetic correlations within the layers are described by a Lieb-Mattis type Hamiltonian [5] ~
S~Sj,
second order
~0.10
E
S) s2.
(2)
iEI,jE2
Here we present the phase diagram of H = H I + H 2 + Hi~t in dependence on the strength of the interlayer coupling J± and fixed J1 = 0.2, ,/2 = 0.1 (Fig. 1). For small J ± both subsystems are antiferromagnetic at low temperatures (AF phase). Increasing the temperature the subsystem '2' undergoes at a critical temperature T~ a second order
* Corresponding author. Fax: +49-391-5592-131; emaih johannes.richter @physik.uni-magdeburg.de.
/
AFt
A / ~
AF
[,,,~Dj/II ODI
F
0.05
Fig. 1. Phase diagram of H
where N is the total number of sites, o~= 1(2) labels the layer and the antiferromagnetic interaction J~ > 0 is between spins on sublattice A and B. Recently we have shown [6] that the problem of interacting layers can be solved exactly assuming a long-ranged interaction term
N
J1=0.2 Jz=O. 1
(1)
i~A,j~B
I J± Hint=--
................................................
°°%'o ......d~ ......6"~......06
4A H,~= U
0.20
Jl
= H 1+ H 2 +
o 8 ' 'i.0 Hilt (see text).
transition to a paramagnetic phase whereas the subsystem '1' remains antiferromagnetic (AF 1 phase) till a critical temperature Tc1 > T~. For T > T] the whole system is paramagnetic. The specific heat shows two peaks at the transitions which is in accordance to the experimental observation in Ref. [1]. Physically more interesting is the region of medium strength of J ± . Here at low temperatures the magnetic structure within the subsystem '1' is canted and the subsystem '2' is ferromagnetically polarized (ODl-phase). The temperature dependence of the corresponding antiferromagnetic order parameter of system T shows interesting order from disorder phenomena [8,2], i.e. the magnetic correlations are increased by thermal fluctuations as shown in Fig. 2. This low-temperature OD 1 phase gives way to either the AF 1 phase or a ferromagnetic phase F. This ferromagnetic F phase is caused by dominating J ±
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0304-8853(95)01178-1
J. Richter et al. / Journal of Magnetism and Magnetic Materials 157/158 (1996) 336-337 1 .("
yielding a parallel (i.e. ferromagnetic) spin orientation within the subsystems. In a second part w e consider a short-range interaction b e t w e e n the layers w h i c h corresponds to the situation in bilayer cuprate superconducting c o m p o u n d s like e.g. YBa2Cu306+x. Hilnlt = J± E S1S2" iE 1,2
........ , ......... , ......... , ......... , ......
~0.4
~
"~
~
OF
Jr=0"2 3e=0-1 ....
J±=0.4
01 ............ ::(7 . . . . . . . .
0 AF
"13.00 0.02
~t.:
o15
o.~
o.to
0.,
0.05
. ....
"
0.04 0.06 kBT
0.08
o
o.~
)o.o0 J±
Fig. 3. Excitation gap (dashed lines) and antiferromagnetic order parameter ](~A~B + ~A~)I1/2/N (solid line) for H = H a + H 2 q - HinUt .
scaling analysis yields a critical J ± = J1 where the order breaks down. At the same point a gap opens to the first triplet excitation, w h i c h is in accordance with the above m e n t i o n e d spin-gap scenario in bilayer cuprates. Acknowledgement: The paper was supported by the country S a c h s e n - A n h a l t . References
Jz=0.63
~
0.3
0.20
(3)
In this case no c o m p l e t e solution o f the m o d e l is available, h o w e v e r because o f the high s y m m e t r y o f the p r o b l e m we could p e r f o r m exact diagonalization up to N = 240 sites [7]. (Notice that at present the exact diagonalization e.g. for the square lattice antiferromagnet is limited to N < 36). As it has been discussed recently [4] the short-range inter-layer interaction might b e responsible for the experimentally observed spin gap in t h e s e materials. According to the situation in the cuprate superconductors w e consider identical layers, i.e. J1 =-/2. Our results for the antiferromagnetic order parameter and the excitation gap are presented in Fig. 3. The inter-layer coupling J . acts against the ordering within the layers. A finite size 0.5
337
0.10
Fig. 2. Magnetic order parameters O ~ ( a = 1(2)) and O F versus i The order parameters are temperature for H = H 1 + / ~ + Him. ce 2 ~ ~ 2 defined as O~F = (4/N)<(S~, - SB ) } (sublattice magnetization of subsystem 'oe') and O F =(4/N-)((S~ 2) +(S~ 2) ) (magnetization of subsystems '1' and '2').
[1] I.V. Golosovskii et al., JETP Lett. 24 (1976) 423; T.A. Valyanskaya and V.I. Sokolov, Soy. Phys. JETP 48 (1978) 161. [2] E.F. Shender, Soy. Phys. JETP 56 (1982) 178. [3] J.M. Tranquada et al., Phys. Rev. B 46 (1992) 5561;R. Stem et al., Phys. Rev. B 50 (1994) 426. [4] A.J. Millis and H. Monien, Phys. Rev. Lett. 70 (1993) 2810; A.W. Sandvik and D.J. Scalapino, Phys. Rev. Lett. 72 (1994) 2777. [5] E. Lieb and D.C. Mattis, J. Math. Phys. 3 (1962) 749; J. Richter, Phys. Rev. B 47 (1993) 5794. [6] J. Richter, S.E. Kri.iger, A. Voigt and C. Gros, Europhys. Lett. 28 (1994) 363. [7] C. Gros, W. Wenzel and J. Richter, Europhys. Lett. 32 (1995) 747. [8] J. Villain, et al., J. Phys. (Paris) 41 (1980) 1263.