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The z component of the magnetization for the a-cyclic XY model in a non-equilibrium situation
PhJka
72 (1974) 123-130 0 North-Holland Publishing Co.
THE z COMPONENT
OF THE MAGNETIZATION
a-CYCLIC XY MODEL IN A NON-EQUILIBRIUM
FOR THE SITUATION
TH. J. SISKENS Instituut-Lorentz, Rijksuniversiteit te Leiden, Leiden, Nederland
Received 28 September
1973
Synopsis An exact expression is derived for the temporal behaviour of the average of the z component of the magnetization for the a-cyclic anisotropic XY model in a non-equilibrium situation, viz. the situation in which the system is initially in thermal equilibrium at temperature Tin an external, homogeneous magnetic field B along the z axis, after which the magnetic field is changed suddenly to a new value B.
1. Introduction. In two previous papers1v2) (to be referred to as I and II*, respectively) we have evaluated a number of equilibrium properties for the anisotropic c-cyclic, c-anticyclic and a-cyclic XY model in the presence of an external, homogeneous, magnetic field along the z axis. The purpose of the present paper is to calculate the average of the z component of the magnetization (instead of “z component of the magnetization” we shall shortly write “magnetization” hereafter) for the a-cyclic XY model in a nonequilibrium case. We shall study the situation in which the system is initially in thermal equilibrium at temperature Tin an external, homogeneous, magnetic field B along the z axis, after which at time t = 0 the magnetic field is changed suddenly to a new value B. This case has been studied by Niemeijer3) for the c-cyclic XY model. However, as for the physically somewhat more realistic cyclic spin XY model (i.e. the a-cyclic XY model), the results for the c-cyclic XY model give a correct description only in the limit of an infinite number of spins. Furthermore it should be stressed that, especially in this non-equilibrium case, the method we used also in I and II to compute equilibrium properties, enables one to obtain the results very quickly. In section 2 the properties of the projection operators P, 1 and P_ 1 introduced * Sections and equations of I and II to be referred to in this paper will be preceded by the prefixes I and II, respectively. 123
124
TH. J. SISKENS
in II, section 2, are summarized and it is emphasized that these properties hold for arbitrary values of the magnetic field B. In section 3 we express the average magnetization at time t for the a-cyclic XY model in the non-equilibrium situation described above, essentially in terms of the corresponding quantities for the c-cyclic and the c-anticyclic XY model, which quantities are evaluated explicitly in section 4. In section 5 the results of section 4 are combined to yield an explicit expression for the average magnetization at time t for the a-cyclic XY model. It is shown that in the thermodynamic limit the average magnetization per spin at time t reduces to the same value for the a-cyclic, c-cyclic and c-anticyclic XY model. 2. Preliminaries. The hamiltonian H(B) for the a-cyclic anisotropic XY model in the presence of an external, homogeneous, magnetic field B in the z direction can be expressed as (cf. II, section 2) H(B)
= H_,(B)
+ $h (P + l),
(1)
where H- 1(B), the hamiltonian of the c-cyclic XY model, is given by N
H_,(B)
= +NB - B t
&.I + “,gl
[(ck,+l
+ r&+J
+
h.c.1,
(2)
_i=r
and where h and P are defined by
h = - [(ch + yc$$ + h.c.]
(3)
P = exp(ia,$lcJcj).
(4)
and
The c operators are fermion operators. The hamiltonian of the c-anticyclic XY model now reads H+,(B)
= H_,(B)
(5)
+ h.
The projection operators P, (o = - 1, + 1) defined as P, = 3 (1 + UP),
1 = unit operator
(a = -1,
+1),
(6)
have the following properties (cf. II, section 2) P,1P_1
= P_,P,l
= 0,
(7)
U’,, H-,(B)1 = [P,, hl = 0 J’,H (B) = P,H, (8
(c = -1,
(0 = -1,
+l),
+1),
(8) (9)
FOR a-CYCLIC
and more in general, iffis
Furthermore trary spin
XY MODEL
IN NON-EQUILIBRIUM
an analytic function,
the P,, (o = - 1, + 1) commute with the z component
[PO,qfl = 0
125
(0 = -1,
+1;j
= I, . ..) N).
of an arbi-
(11)
We want to stress that the properties summarized above, are valid for arbitrary values of the magnetic field, and this enables one to compute quantities involving various field dependences very easily. 3. A formal expression for the average magnetization for the a-cyclic XY model in a non-equilibrium situation. We suppose the system to be initially in thermal equilibrium at temperature Tin a magnetic field B. At time t = 0 the magnetic field is changed suddenly to the value 8. The average magnetization at time t is given by
where e(B) is defined as
e(B) =
expi -BH (B))
(13)
and where MZ
(t; E) z exp {itH (B)} M’ exp { - itH (8)) -
(14)
For t < 0 the average magnetization is given by the equilibrium value at field B, which has been calculated in II [c$ (11.52)]:
+ CJ(e,(B)
P> Me WI + id),
(15)
where Z(B) is given by (11.40), Z,(B) (CT= - 1, + 1) by (11.38), (e,(B) P) (a = - 1, + 1) by (11.39), the functionals iU, [@I (CS= - 1, + 1) by (11.66), and the explicit form of the functionals (l/N) iU, [@I + iz] (a = - 1, + 1) by (11.51). All these quantities refer to the value B of the magnetic field.
TH. J. SISKENS
126 For t > 0 the average
magnetization
can be written
as
w=(o) =
b= _I
C
= 0=-l. +1
+1 (exp
{ -BH
(B)1 PO>
@)I M’ (t; B; 4 Pm> (16)
where M= (t;
B; o)
exp {it&, (B)} M’ exp { - itH, (B)}
E
It follows immediately
(0 = - 1, +l).
(17)
that
1
=-
c
Z(B)
+z, (B) {
R
4)
a=-1,+1
(18) We shall first, in the next section, calculate the “c-cyclic” and “c-anticyclic” quantities (eb(B) M’ (t; B; a)) and (e,(B) M’ (t; B; a) P) (cr = - 1, + I), and subsequently, in section 5, evaluate (M”(t)) for the a-cyclic XY model. 4. Evaluation of the “c-cyclic” and “c-anticyclic” quantities (Q,(B) W (t; B; a)) and (e,(B) M’ (t; B; a) P) (a = - 1, + 1). First a remark about the notation. The matrices
M,,
S,,
S, and
the eigenvalues
A,, p, A,, y, a,, ,, (a = - 1, + 1;
p = 1, . ..) N) (c$ I, section 3) are dependent on the magnetic field. For simplicity of notation M,, S,, s,, A,, p, A,#p and 1,. p are understood to refer to the magnetic field B, and tiO, .$,, 3,) An, p, I,, p, I,, p to the magnetic field B. (The A bra and A=,. will be chosen positive.) In terms of the OLand #? operators (CA I, section 3) the quantities (e,(B) M’ (t; B; a)) read
j=l
(e,(B)ar,(t;B;a)BI(t;B;a))
(a = -1,
+l).
(1%
(&P(t))
FOR a-CYCLIC
XY MODEL
127
IN NON-EQUILIBRIUM
Inserting (1.24) and (1.25) into (19) we obtain
- i j *i= l (cos , 9 + i
5 j,k,
A7$t)jk ($JW
sin /i&$t),r (Q,(B) LY&
(S,/G,J+ I=1
The quantum-mechanical Substitution yields
traces on the r.h.s. of (20) are given by (1.36) and (1.37).
The second and third terms on the r.h.s. of (20) cancel, as they should. Performing the summations on the r.h.s. of (21) we obtain = -+ Tr [cos2 (@t)
- 3 Tr [&Vi’
S,M~’
tanh &?A&]
sin2 (@t) S,Mif
tanh +/?A@]
= - 4 Tr (SJM,~ tanh &!?A& + + Tr [(l - s”,Mi’) sin2 (/i&)
S,M~’ tanh @M,f] (0 = -1,
+ 1).
(22)
Inserting (1.38) into (22) we get GM)
M’ 0; B; 4)
= (Q,(B) M’)
+ 3 Tr [& (s, - 5,) fli’
sin’ (@t)
S,,MY* tanh =@M$]
(CT= -1,
+1).
(23)
128
TH. 1. SISKENS
In terms of the eigenvalues we obtain the following expression
=
+ i 5
(cx 1
A LB Im CL d sin2 (JO, ,t) a.p R2
p=l
tanh 3/U,, I
A
0, P
-1,
+l).
(24)
Since Im A#,, = ImA,,,
(0 = -1,
= y sinpl,,.
+l;p
= 1, . ..) N),
(25)
it follows that
=
il
I#,, -
[(Re
x sin2 (A#,,t) tanh !&A, p A 0. P
1
ReL,,.)
(‘m$‘*‘)* a. P
(0 = -1,
(26)
+I),
or
= (Q,(B) M’)
(CT = 1
p”*psin2 ((i,, J) 5 ” “j;2;
+ (B - B)
p=l
tanh 38&, p A
b* p
-1,
+“‘1”,,
(27)
where, in terms of the eigenvalues, <&#) M’) = _ 4 5 p=l
(‘OS 2,” r
- @ tanh 3&%, p
(a = -1, +1)
b,P WY
(1.38)]. Let us define the functionals M&g; _; t [PA] (o = - 1, + 1) as
[cf:
%;E;,;,
[WI =
(0 = -1,
+l),
(29)
the r.h.s. being given explicitly by (27) and (28). Following the procedure of II, section 5, we obtain for the quantities (e,(B) M” (t; i?; a) P) (o = - 1, + 1) the following expression = M& 6; O;t W
+ ix1
(0 = -1, +l). (30)
FOR a-CYCLIC
XY MODEL
129
IN NON-EQUILIBRIUM
The quantities (e,(B) P) (a = - 1, + 1) are given by (11.39) [c$ also (11.37)] and the functionals M~;s;,;~ [@A + in] read explicitly
sin’ (Cr,, &
1 (a =
coth384, p A be D
-1,
+l).
(31)
5. The average magnetization at time t (>O) for the a-cyclic XY model. Substituting (29) and (30) into (18), 0) can be expressed as follows
Substituting the explicit forms (27), (28) and (31) into (32), we obtain
+
(B -
7’ sin* s, p sin*
-p
(?r,,
tanh
3bL
p
,t)
A 0. P
a. 1
>
+ (B - B) 0 (@,(B) P> ;
p=l
x sin*
(33)
,
where [c$ (11.39)] (a = -1,
+1)
(34)
130
TH. J. SISKENS
with PO; ,,(W = + I or - 1, depending on the values of o and B and on N being even or odd [c$ (11.37)]. From (11.52) it follows that instead of (33) we may also write
(5 - B) -!-
=
Z(B)
A a,a
+
G
(p,(B)
V’b7psin? (/r,,,t)
y2 Fl
P> 5
p=l
cot~“A”*p
0. P
0. D
3
II
.
(35)
From (34) it follows that for finite /3 (CT= -1,
lim (e,(B) P} = 0
+l).
(36)
N-+W
Following the argument of II, section 5 we obtain in the thermodynamic limit for the average magnetization per spin at time t (> 0) the following expression 2x
lim L (W(t)) N-rm
= :t
N
-t
i
(p(B) ill”) + (B 0
(37) where 2x
f; -+
$
(Q(B) M’)
dp,
= - $b
s
(cos ‘p - B)
tanh +%I (v) .
(38)
4v)
0
It is obvious that in the thermodynamic limit the results for the c-cyclic, the c-anticyclic and the a-cyclic XY model are the same.
REFERENCES
1) Mazur, P. and Siskens, Th. J., Physica 69 (1973) 259. 2) Siskens, Th.J. and Mazur, P., Physica 71 (1974) 560. 3) Niemeijer,
Th., Physica
36 (1967) 377.
72 (1974) 123-130 0 North-Holland Publishing Co.
THE z COMPONENT
OF THE MAGNETIZATION
a-CYCLIC XY MODEL IN A NON-EQUILIBRIUM
FOR THE SITUATION
TH. J. SISKENS Instituut-Lorentz, Rijksuniversiteit te Leiden, Leiden, Nederland
Received 28 September
1973
Synopsis An exact expression is derived for the temporal behaviour of the average of the z component of the magnetization for the a-cyclic anisotropic XY model in a non-equilibrium situation, viz. the situation in which the system is initially in thermal equilibrium at temperature Tin an external, homogeneous magnetic field B along the z axis, after which the magnetic field is changed suddenly to a new value B.
1. Introduction. In two previous papers1v2) (to be referred to as I and II*, respectively) we have evaluated a number of equilibrium properties for the anisotropic c-cyclic, c-anticyclic and a-cyclic XY model in the presence of an external, homogeneous, magnetic field along the z axis. The purpose of the present paper is to calculate the average of the z component of the magnetization (instead of “z component of the magnetization” we shall shortly write “magnetization” hereafter) for the a-cyclic XY model in a nonequilibrium case. We shall study the situation in which the system is initially in thermal equilibrium at temperature Tin an external, homogeneous, magnetic field B along the z axis, after which at time t = 0 the magnetic field is changed suddenly to a new value B. This case has been studied by Niemeijer3) for the c-cyclic XY model. However, as for the physically somewhat more realistic cyclic spin XY model (i.e. the a-cyclic XY model), the results for the c-cyclic XY model give a correct description only in the limit of an infinite number of spins. Furthermore it should be stressed that, especially in this non-equilibrium case, the method we used also in I and II to compute equilibrium properties, enables one to obtain the results very quickly. In section 2 the properties of the projection operators P, 1 and P_ 1 introduced * Sections and equations of I and II to be referred to in this paper will be preceded by the prefixes I and II, respectively. 123
124
TH. J. SISKENS
in II, section 2, are summarized and it is emphasized that these properties hold for arbitrary values of the magnetic field B. In section 3 we express the average magnetization at time t for the a-cyclic XY model in the non-equilibrium situation described above, essentially in terms of the corresponding quantities for the c-cyclic and the c-anticyclic XY model, which quantities are evaluated explicitly in section 4. In section 5 the results of section 4 are combined to yield an explicit expression for the average magnetization at time t for the a-cyclic XY model. It is shown that in the thermodynamic limit the average magnetization per spin at time t reduces to the same value for the a-cyclic, c-cyclic and c-anticyclic XY model. 2. Preliminaries. The hamiltonian H(B) for the a-cyclic anisotropic XY model in the presence of an external, homogeneous, magnetic field B in the z direction can be expressed as (cf. II, section 2) H(B)
= H_,(B)
+ $h (P + l),
(1)
where H- 1(B), the hamiltonian of the c-cyclic XY model, is given by N
H_,(B)
= +NB - B t
&.I + “,gl
[(ck,+l
+ r&+J
+
h.c.1,
(2)
_i=r
and where h and P are defined by
h = - [(ch + yc$$ + h.c.]
(3)
P = exp(ia,$lcJcj).
(4)
and
The c operators are fermion operators. The hamiltonian of the c-anticyclic XY model now reads H+,(B)
= H_,(B)
(5)
+ h.
The projection operators P, (o = - 1, + 1) defined as P, = 3 (1 + UP),
1 = unit operator
(a = -1,
+1),
(6)
have the following properties (cf. II, section 2) P,1P_1
= P_,P,l
= 0,
(7)
U’,, H-,(B)1 = [P,, hl = 0 J’,H (B) = P,H, (8
(c = -1,
(0 = -1,
+l),
+1),
(8) (9)
FOR a-CYCLIC
and more in general, iffis
Furthermore trary spin
XY MODEL
IN NON-EQUILIBRIUM
an analytic function,
the P,, (o = - 1, + 1) commute with the z component
[PO,qfl = 0
125
(0 = -1,
+1;j
= I, . ..) N).
of an arbi-
(11)
We want to stress that the properties summarized above, are valid for arbitrary values of the magnetic field, and this enables one to compute quantities involving various field dependences very easily. 3. A formal expression for the average magnetization for the a-cyclic XY model in a non-equilibrium situation. We suppose the system to be initially in thermal equilibrium at temperature Tin a magnetic field B. At time t = 0 the magnetic field is changed suddenly to the value 8. The average magnetization at time t is given by
where e(B) is defined as
e(B) =
expi -BH (B))
(13)
and where MZ
(t; E) z exp {itH (B)} M’ exp { - itH (8)) -
(14)
For t < 0 the average magnetization is given by the equilibrium value at field B, which has been calculated in II [c$ (11.52)]:
+ CJ(e,(B)
P> Me WI + id),
(15)
where Z(B) is given by (11.40), Z,(B) (CT= - 1, + 1) by (11.38), (e,(B) P) (a = - 1, + 1) by (11.39), the functionals iU, [@I (CS= - 1, + 1) by (11.66), and the explicit form of the functionals (l/N) iU, [@I + iz] (a = - 1, + 1) by (11.51). All these quantities refer to the value B of the magnetic field.
TH. J. SISKENS
126 For t > 0 the average
magnetization
can be written
as
w=(o) =
b= _I
C
= 0=-l. +1
+1 (exp
{ -BH
(B)1 PO>
@)I M’ (t; B; 4 Pm> (16)
where M= (t;
B; o)
exp {it&, (B)} M’ exp { - itH, (B)}
E
It follows immediately
(0 = - 1, +l).
(17)
that
1
=-
c
Z(B)
+z, (B) {
R
4)
a=-1,+1
(18) We shall first, in the next section, calculate the “c-cyclic” and “c-anticyclic” quantities (eb(B) M’ (t; B; a)) and (e,(B) M’ (t; B; a) P) (cr = - 1, + I), and subsequently, in section 5, evaluate (M”(t)) for the a-cyclic XY model. 4. Evaluation of the “c-cyclic” and “c-anticyclic” quantities (Q,(B) W (t; B; a)) and (e,(B) M’ (t; B; a) P) (a = - 1, + 1). First a remark about the notation. The matrices
M,,
S,,
S, and
the eigenvalues
A,, p, A,, y, a,, ,, (a = - 1, + 1;
p = 1, . ..) N) (c$ I, section 3) are dependent on the magnetic field. For simplicity of notation M,, S,, s,, A,, p, A,#p and 1,. p are understood to refer to the magnetic field B, and tiO, .$,, 3,) An, p, I,, p, I,, p to the magnetic field B. (The A bra and A=,. will be chosen positive.) In terms of the OLand #? operators (CA I, section 3) the quantities (e,(B) M’ (t; B; a)) read
j=l
(e,(B)ar,(t;B;a)BI(t;B;a))
(a = -1,
+l).
(1%
(&P(t))
FOR a-CYCLIC
XY MODEL
127
IN NON-EQUILIBRIUM
Inserting (1.24) and (1.25) into (19) we obtain
- i j *i= l (cos , 9 + i
5 j,k,
A7$t)jk ($JW
sin /i&$t),r (Q,(B) LY&
(S,/G,J+ I=1
The quantum-mechanical Substitution yields
traces on the r.h.s. of (20) are given by (1.36) and (1.37).
The second and third terms on the r.h.s. of (20) cancel, as they should. Performing the summations on the r.h.s. of (21) we obtain
- 3 Tr [&Vi’
S,M~’
tanh &?A&]
sin2 (@t) S,Mif
tanh +/?A@]
= - 4 Tr (SJM,~ tanh &!?A& + + Tr [(l - s”,Mi’) sin2 (/i&)
S,M~’ tanh @M,f] (0 = -1,
+ 1).
(22)
Inserting (1.38) into (22) we get GM)
M’ 0; B; 4)
= (Q,(B) M’)
+ 3 Tr [& (s, - 5,) fli’
sin’ (@t)
S,,MY* tanh =@M$]
(CT= -1,
+1).
(23)
128
TH. 1. SISKENS
In terms of the eigenvalues we obtain the following expression
=
+ i 5
(cx 1
A LB Im CL d sin2 (JO, ,t) a.p R2
p=l
tanh 3/U,, I
A
0, P
-1,
+l).
(24)
Since Im A#,, = ImA,,,
(0 = -1,
= y sinpl,,.
+l;p
= 1, . ..) N),
(25)
it follows that
=
il
I#,, -
[(Re
x sin2 (A#,,t) tanh !&A, p A 0. P
1
ReL,,.)
(‘m$‘*‘)* a. P
(0 = -1,
(26)
+I),
or
= (Q,(B) M’)
(CT = 1
p”*psin2 ((i,, J) 5 ” “j;2;
+ (B - B)
p=l
tanh 38&, p A
b* p
-1,
+“‘1”,,
(27)
where, in terms of the eigenvalues, <&#) M’) = _ 4 5 p=l
(‘OS 2,” r
- @ tanh 3&%, p
(a = -1, +1)
b,P WY
(1.38)]. Let us define the functionals M&g; _; t [PA] (o = - 1, + 1) as
[cf:
%;E;,;,
[WI =
(0 = -1,
+l),
(29)
the r.h.s. being given explicitly by (27) and (28). Following the procedure of II, section 5, we obtain for the quantities (e,(B) M” (t; i?; a) P) (o = - 1, + 1) the following expression
+ ix1
(0 = -1, +l). (30)
FOR a-CYCLIC
XY MODEL
129
IN NON-EQUILIBRIUM
The quantities (e,(B) P) (a = - 1, + 1) are given by (11.39) [c$ also (11.37)] and the functionals M~;s;,;~ [@A + in] read explicitly
sin’ (Cr,, &
1 (a =
coth384, p A be D
-1,
+l).
(31)
5. The average magnetization at time t (>O) for the a-cyclic XY model. Substituting (29) and (30) into (18),
Substituting the explicit forms (27), (28) and (31) into (32), we obtain
+
(B -
7’ sin* s, p sin*
-p
(?r,,
tanh
3bL
p
,t)
A 0. P
a. 1
>
+ (B - B) 0 (@,(B) P> ;
p=l
x sin*
(33)
,
where [c$ (11.39)] (a = -1,
+1)
(34)
130
TH. J. SISKENS
with PO; ,,(W = + I or - 1, depending on the values of o and B and on N being even or odd [c$ (11.37)]. From (11.52) it follows that instead of (33) we may also write
(5 - B) -!-
=
Z(B)
A a,a
+
G
(p,(B)
V’b7psin? (/r,,,t)
y2 Fl
P> 5
p=l
cot~“A”*p
0. P
0. D
3
II
.
(35)
From (34) it follows that for finite /3 (CT= -1,
lim (e,(B) P} = 0
+l).
(36)
N-+W
Following the argument of II, section 5 we obtain in the thermodynamic limit for the average magnetization per spin at time t (> 0) the following expression 2x
lim L (W(t)) N-rm
= :t
N
-t
i
(p(B) ill”) + (B 0
(37) where 2x
f; -+
$
(Q(B) M’)
dp,
= - $b
s
(cos ‘p - B)
tanh +%I (v) .
(38)
4v)
0
It is obvious that in the thermodynamic limit the results for the c-cyclic, the c-anticyclic and the a-cyclic XY model are the same.
REFERENCES
1) Mazur, P. and Siskens, Th. J., Physica 69 (1973) 259. 2) Siskens, Th.J. and Mazur, P., Physica 71 (1974) 560. 3) Niemeijer,
Th., Physica
36 (1967) 377.