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Lifshitz points of higher character
Volume 61A, number 7
PHYSICS LETTERS
27 June 1977
LIFSHITZ POINTS OF HIGHER CHARACTER Walter SELKE Fachrichtung Theoretische Physik, Universität des Saarlandes, 6600 Saarbrilcken, West Germany Received 20 April 1977 Lifshitz points of higher character are found for the spherical model with multi-neighbour interactions. The Lifshitz point of character three is a point of intersection of lines of first and second order transitions.
Recently, Hornreich et al. introduced a new critical point, the Lifshitz point (LP) [1, 2], associated with the onset of helical order in magnetic systems. A more general situation has been discussed by Stanley et al. defining in a formal manner a “Lifshitz point of character L” [3—51; see also ref. [6], where this point has been called a “LP of order (L—1)”. In this note I shall discuss the critical properties of a concrete model exhibiting LP’s of higher character. I shall consider the spherical model [7] with morethan-nearest neighbour interactions along m(m ~ d) axes of ad-dimensional lattice; the Fourier transform of the exchange integrals is given by
2
~~TC
FERROMAGNETIC
1
LPOF2ND ORDER
~.
J(q)
=
2(J1
I~ cos(q,) +
~
Ji cos(iqj)),
(1)
~
j=1 ~Jl/J2
the lattice constant is taken to be one. Depending on the values ofJ. the absolute maximum of J(q) is at q = 0, at q = ir or at q ~r 0, ir leading to a ferromagnetic, antiferromagnetic or helical ground state. Choosing J~/J1 (i ~ L) such that the leading terms of the Taylor expansion of (I) are J(q) =
+ c2
~q?L + c3 ~ q? + / m+i ~
11
(2) ‘
a “rn-axial Lifshitz point of character L” will be exhibited; c~are some constants. For example, the “isotropic” (rn = d) LP can be achieved for .J2/J1 = —1/4; a LP of character three (order two) will be displayed for J1/J2 = —5/2, J3/J2 = —1/6; the LPof character 4 can be realized for J2/J1 = —29/90, J3/J1 = 1/135 and J4/J1 = 1/72 etc. In fig. 1 the phase diagram for the ground state of the spherical model with interactions up to the third neighbours is shown; see
Fig. 1. Phase diagram of the ground state of the spherical model with nearest (Ji > 0 ferromagnetic) next nearest . (~12< 0; antiferromagnetic) and third nearest neighbour (J3 > 0) interactions displaying a Lifshitz point of second order (character 3). The stability boundary between the helland ferromagnetic state is given by —J3/J2 = (4 + Ji 1.12)/9 (full line) and —J3/J2 = 1/(4(—J1/J2 — 1)) (dashed line).
also ref. [8]. The Lifshitz point of character 3 is the point of intersection of a line of “second order transitions” (full line; LP’s of character 2), where the helix decreases continuously to zero by going from the helical to the ferromagnetic state, and a line of “first order transitions” (dashed line), where the angle of the helix jumps from a finite value to zero by crossing the stability boundary to the ferromagnetic state. Thus there is a certain similarity to a tricritical point. The critical properties for the spherical model can be obtained without difficulty. The border dimension d.1. (above which mean field exponents hold) for a LP 443
Volume 61A, number 7
PHYSICS LETTERS
of character L is (4+rn—m/L d~=~ ~4L
for
the critical exponents too. For isotropic LP’s the modified scaling laws [9] are fulfilled. Some exponents rn
are
(3) for
rn=d.
Infrared divergencies, i.e. the critical temperature Tc = 0, set in for lattice dimensions d ~ d (2
+
rn
—
27 June 1977
rn/L
for
m
for
rnd.
y=2L/(2L—d), a = —d/(2L d) ;
2
—
(8)
d
If rn
12L
~j =
—
-
y2L/((2+rn—d)L—rn),
For d
a = (—rn
L (m
—L/(rn = rim = 2
=
17d
a(d—4L)/(d—2L),
+
— —
—
L(2 d.
d))/(m + rn
—
—
L(2 + rn
—
1’m
=
d));
d))
(9)
(5) 7 = 2L/(d 2L). For the rn-axial (rn
The best candidates for systems displaying LP’s of higher character should be alloys [1], where the cornposition plays the role of the exchange integrals.
2
[1] R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev.
—
—
a = rnvm + (d
—
m)vd
(6)
.
Some exponents for the spherical model are a = (rn
—
4
—
7 = 2L/((d —2
—
rn)L
+
L(d
rn))/(rn +
+ L(d
rn),
—
2
—
rn)) (7)
0; Vd =y/2, vm =y/(2L). Eqs. (3)—(7) agree with the results in refs. [1,2, 5] for rn = d and rn = 1. The critical temperature Tc decreases with increasing L For example, for d = 3 and rn = 1 one gets Tc i/L for large L. For d ~ d the phase transitions for the models exhibiting LP’s occur at zero temperature. Following the notation of Baker and Bonner [9] we can calculate ~d ~m
-
444
I wish to thank I. Peschel for the useful discussions.
References
Lett. 35 (1975) 1678. [21 R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Lett. 55A (1975) 269. [3] J.F. Nicoll, T.S. Chang and H.E. Stanley, Phys. Rev. A13 (1976) 1251. [4J J.F. Nicoll, G.F. Tuthill, T.S. Chang and H.E. Stanley, Phys. Lett. 58A (1976)1. [5] J.F. Nicoll, G.F. Tuthill, T.S. Chang and H.E. Stanley, Physica 86-88B (1977) 618. [61 W. Selke, Z. Physik B (1977) in press.
171
G.S. Joyce, in: Phase transitions and critical phenomena, eds. C. Domb and M.S. Green (Academic Press, New York, 1972) Vol. 2, p. 375. [8] T. Nagamiya, K. Nagata Phys. 27 (1962) 1253. and Y. Kitano, Prog. Theor. [9] G.A. Baker and J.C. Bonner, Phys. Rev. B12 (1975) 3741.
PHYSICS LETTERS
27 June 1977
LIFSHITZ POINTS OF HIGHER CHARACTER Walter SELKE Fachrichtung Theoretische Physik, Universität des Saarlandes, 6600 Saarbrilcken, West Germany Received 20 April 1977 Lifshitz points of higher character are found for the spherical model with multi-neighbour interactions. The Lifshitz point of character three is a point of intersection of lines of first and second order transitions.
Recently, Hornreich et al. introduced a new critical point, the Lifshitz point (LP) [1, 2], associated with the onset of helical order in magnetic systems. A more general situation has been discussed by Stanley et al. defining in a formal manner a “Lifshitz point of character L” [3—51; see also ref. [6], where this point has been called a “LP of order (L—1)”. In this note I shall discuss the critical properties of a concrete model exhibiting LP’s of higher character. I shall consider the spherical model [7] with morethan-nearest neighbour interactions along m(m ~ d) axes of ad-dimensional lattice; the Fourier transform of the exchange integrals is given by
2
~~TC
FERROMAGNETIC
1
LPOF2ND ORDER
~.
J(q)
=
2(J1
I~ cos(q,) +
~
Ji cos(iqj)),
(1)
~
j=1 ~Jl/J2
the lattice constant is taken to be one. Depending on the values ofJ. the absolute maximum of J(q) is at q = 0, at q = ir or at q ~r 0, ir leading to a ferromagnetic, antiferromagnetic or helical ground state. Choosing J~/J1 (i ~ L) such that the leading terms of the Taylor expansion of (I) are J(q) =
+ c2
~q?L + c3 ~ q? + / m+i ~
11
(2) ‘
a “rn-axial Lifshitz point of character L” will be exhibited; c~are some constants. For example, the “isotropic” (rn = d) LP can be achieved for .J2/J1 = —1/4; a LP of character three (order two) will be displayed for J1/J2 = —5/2, J3/J2 = —1/6; the LPof character 4 can be realized for J2/J1 = —29/90, J3/J1 = 1/135 and J4/J1 = 1/72 etc. In fig. 1 the phase diagram for the ground state of the spherical model with interactions up to the third neighbours is shown; see
Fig. 1. Phase diagram of the ground state of the spherical model with nearest (Ji > 0 ferromagnetic) next nearest . (~12< 0; antiferromagnetic) and third nearest neighbour (J3 > 0) interactions displaying a Lifshitz point of second order (character 3). The stability boundary between the helland ferromagnetic state is given by —J3/J2 = (4 + Ji 1.12)/9 (full line) and —J3/J2 = 1/(4(—J1/J2 — 1)) (dashed line).
also ref. [8]. The Lifshitz point of character 3 is the point of intersection of a line of “second order transitions” (full line; LP’s of character 2), where the helix decreases continuously to zero by going from the helical to the ferromagnetic state, and a line of “first order transitions” (dashed line), where the angle of the helix jumps from a finite value to zero by crossing the stability boundary to the ferromagnetic state. Thus there is a certain similarity to a tricritical point. The critical properties for the spherical model can be obtained without difficulty. The border dimension d.1. (above which mean field exponents hold) for a LP 443
Volume 61A, number 7
PHYSICS LETTERS
of character L is (4+rn—m/L d~=~ ~4L
for
the critical exponents too. For isotropic LP’s the modified scaling laws [9] are fulfilled. Some exponents rn
are
(3) for
rn=d.
Infrared divergencies, i.e. the critical temperature Tc = 0, set in for lattice dimensions d ~ d (2
+
rn
—
27 June 1977
rn/L
for
m
for
rnd.
y=2L/(2L—d), a = —d/(2L d) ;
2
—
(8)
d
If rn
12L
~j =
—
-
y2L/((2+rn—d)L—rn),
For d
a = (—rn
L (m
—L/(rn = rim = 2
=
17d
a(d—4L)/(d—2L),
+
— —
—
L(2 d.
d))/(m + rn
—
—
L(2 + rn
—
1’m
=
d));
d))
(9)
(5) 7 = 2L/(d 2L). For the rn-axial (rn
The best candidates for systems displaying LP’s of higher character should be alloys [1], where the cornposition plays the role of the exchange integrals.
2
[1] R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev.
—
—
a = rnvm + (d
—
m)vd
(6)
.
Some exponents for the spherical model are a = (rn
—
4
—
7 = 2L/((d —2
—
rn)L
+
L(d
rn))/(rn +
+ L(d
rn),
—
2
—
rn)) (7)
0; Vd =y/2, vm =y/(2L). Eqs. (3)—(7) agree with the results in refs. [1,2, 5] for rn = d and rn = 1. The critical temperature Tc decreases with increasing L For example, for d = 3 and rn = 1 one gets Tc i/L for large L. For d ~ d the phase transitions for the models exhibiting LP’s occur at zero temperature. Following the notation of Baker and Bonner [9] we can calculate ~d ~m
-
444
I wish to thank I. Peschel for the useful discussions.
References
Lett. 35 (1975) 1678. [21 R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Lett. 55A (1975) 269. [3] J.F. Nicoll, T.S. Chang and H.E. Stanley, Phys. Rev. A13 (1976) 1251. [4J J.F. Nicoll, G.F. Tuthill, T.S. Chang and H.E. Stanley, Phys. Lett. 58A (1976)1. [5] J.F. Nicoll, G.F. Tuthill, T.S. Chang and H.E. Stanley, Physica 86-88B (1977) 618. [61 W. Selke, Z. Physik B (1977) in press.
171
G.S. Joyce, in: Phase transitions and critical phenomena, eds. C. Domb and M.S. Green (Academic Press, New York, 1972) Vol. 2, p. 375. [8] T. Nagamiya, K. Nagata Phys. 27 (1962) 1253. and Y. Kitano, Prog. Theor. [9] G.A. Baker and J.C. Bonner, Phys. Rev. B12 (1975) 3741.