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Fluctuations and defects in the spiral state of magnetic superconductors
Volume 90A, number 5
PHYSICS LETTERS
12 July 1982
FLUCTUATIONS AND DEFECTS IN THE SPIRAL STATE OF MAGNETIC SUPERCONDUCTORS H. KLEINERT Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 3, 1000 Berlin 33, Germany Received 6 April 1982
We want to point out three properties of a magnetic superconductor: (i) The absence of true long-range order in the 2 and (q~)fl_2for q spiral state leads to the structure functions behaving like (q 11 — q0)~ 1 0 and q11 = 0, respectively, where q0 is the preferred momentum. The indices i~are measured via Bragg-like neutron scattering. (ii) The state is perforated by line-like defects. (iii) Above some critical temperature the defect lines proliferate, thereby destroying the spiral quasi-order.
In a recent note [11 we argued that the condensate of long-range spiral order suggested by mean-field calculations in magnetic superconductors [21 could not properly exist due to catastrophic fluctuations. The Bragg-like reflex in neutron scatteringwould then really be due to these fluctuations. Experimentally, this theorists’ subtle distinction requires high resolution of the line shape. Hoping that this will soon be available we calculate the singular behaviour of the structure factor S(q) close to q11 ~ q0, q1 = 0 and q1 ~ 0, q11 0. In addition we draw attention to the existence of line-like defects of the dislocation type (just as displayed on everyone’s fingerprints), whose configurational fluctuations eventually destroy the pseudo-order and carry the system to the normal paramagnetic superconducting state. Using the notation of ref. [lJ we begin with the free energy of the relevant order parameter M1 which is the magnetization orthogonal to the preferred momentum direction (say 1). 21 1M 2 fd3x [M~(x)]2. (I) F [r+(IqI —q0) 1(q)1 Here contains the main temperature dependence and (1) is valid for r 0 and q q 0. Our analysis will start by assuming r = r0 <0 where the energy is minimized by the spiral solution +.~-
E
i-
M1EM1 +iM2
.~/~7~eiqoz
(2)
We want to study soft long-wave this and multiply a pure space dependent phase e~~~(x). 2 ~ fluctuations (1 /4q~)(q2around q~)2and working out M1 the by lowestderivatives, F reduces to the By writing near q0: (Iqi pure gradient energy (up toq0) a constant) —
—
F = ~B [(a where X2
+ x2(a~p)2], 2 3~,)
=
(3)
4q~.Since p is a gaussian random field, the correlation function (M 1(x)M~(0)>is easily calculated
as [3] (M1(x) M~(0)>= (B/a) (exp{~i[~(x)
—
~(0)J })
=
(B/a) exp{— ~C [y,(x) ~p(o)j2>} —
where
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
259
Volume 90A, numberS
-
PHYSICS LETTERS
~([~~(x)_~(0)]2)~,(x)~(0)> (~(O)2) G~(x)= -
=
=
—
f—~.-(1
Bx(2~)3
—
-
T
_____
exp(—Xq 1IzI) cosq1x1) =
_n[log(c2x~/d2) + 27 +E1(x~/4XIzI)]
12 July 1982
-
f —i- [1
— ~
cos q .x) (q~+
—
exp(—Xq1 zI)J0(q11x11)J
.
2
Here ‘ydue ~ 0.577, ~ stands for T/87TBX, E1(x) stant to the transverse momentum cutoff. f~° Fore~dt/t Izi ~ 1xis .the exponential integral, and c 11, E1(x) -÷ —y —logx and 2’~e’l (ir2/q G~(x) c 0X~z~)h?
=
(4) q~t/q~ is a con-
(5)
.
For Ix1I~IzI,E1-~0and ~
(6)
.
This gives for the structure factor 2 for q S(q) (q11 q0)71 1 = 0, —
S(q)
(q~)712 for q11
=
0.
(7)
Such a behaviour has been detected in Bragg-like reflexes of smectic liquid crystals [4] and we hope that neutron scattering on the spiral state will soon reveal the same deviations from true long-range order. Given the simplicity of obtaining correlation functions we expect that also other fluctuation properties of the system can be understood by using this quasi-ordered state as a basis. For this it is essential to include macroscopic topological excitations [5]. Since the field p is cyclic (p and ~ + 2irn are 2indistinguishable) theissystem has singular such that b, —B ~p divergenceless, vortex lines similar to He II. Recall that there the energy isf= ~B(a~p) just as a magnetic field. Therefore one introduces a vector potential via b = a X a and finds the forces between vortex lines from the gauge invariant coupling 2irn ~ dxyl)ai (yielding the Biot-Savart law) [6]. Here the field equation —
x2 [a (a2a~p)+ a
2(a~a2~)] = 0
tells us that
2a~a
b mB(a3~p,—x
(8)
2a~a
1~p, —x
2p)
(9)
is divergenceless and may be written as ax a. In terms of b~the energy reads (2B)_1{b~+ [(xa~y’b2]b2} such that in the long wave limit [7] and the gauge a1a1 = 0 2~a~(k)l2 (k~ k~/X2k~) 1a 2]
f
~[X
+
+
[(Xa~)’b1]b1+ (10)
.
1(k)1
The coupling of a defect loop L is obtained as follows: Let n be the vortex strength of L such that for any circuit around L, p changes by 2irn, i.e. ~ dk 1 a~p= 2irn. Then p must have a jump by 2irn on some surface spanned by L. Suppose now that this loop is inserted in a given smooth field configuration p thereby changing it to p + The field energy (3) changes by 2a~a 3x b. ~&p = d3x a ~f=Bfd3x(a3p a3~~x 1pa1~x2a~a2~pa2~)d 1(b1p) = _fdSjb1p, (11) ~.
—
—
=f
—f
where the surface integral runs over a thin ellipsoid enclosing the surface S. This integral can be evaluated via the discontinuity 1~4PIS= 2irn as ~f=2irnfdS1b1. 260
(12)
Volume 90A, number 5 Now b
&f
=
2irn
PHYSICS LETTERS
12 July 1982
a x a can be used to transform this into the gauge-invariant local coupling S6dxiai(x).
(13)
The grand-canonical ensemble of interacting lines can now be described in complete analogy with defect theories of melting [8]: We introduce a disorder field ~i(x)whose free correlation function
f dL
(~(x’)i~ti~(x)) =
e—(d/T_w)L
~
~
e~(-~’-’) [a2k2/2D
=
+
(e/T— w)]1,(14)
gives the probability for a line of any length to run from x to x’. Here D is the space dimension, w log 2D the entropy per link, a the thickiiess of the line, and e the core energy per length a. The grand-canonical ensemble of free random loops is then given by the partition function [8]
z=fcexp(_fdx[(a2I2D)Ia~I2+(e/r_w)I~I2]).
(15)
The long-range interaction due to the gauge field is incorporated by the usual minimal replacement
a a -+
—
(2iri/T)a,
(16)
where we have restricted ourselves to only the basic lines with unit winding number. Certainly, there will be short range steric interactions which we may parametrize by a term ~-qI’bI4.Thus the total partition function of the system as seen from the spiral state becomes Z
=
f7.~A(D~p cii
~
E [?c2L4
2 + (k~+ k~/X2k~)IA
exp(~
3(k)1
~
1(k)I2]) (17)
This will be the most economic tool for studying the critical behaviour at the transition temperature which is given by T~= e/w. Above Tc the disorder field destabilizes leading to (i/i) * 0 which signalizes the proliferation of defect lines and the destruction of the spiral state. It goes without saying that our partition function (17) can be used for studying the completely analogous phenomena in smectics where the importance of defect lines has been stressed before [9] and, of course, in pion condensates [7]. Li]
H. Kleinert, Phys. Lett. 83A (1981) 294. [21 E.I. Blount and C.M. Varma, Phys. Rev. Lett. 42 (1979) 1079; H. Matsumoto, H. Umeuawa and M. Tachiki, Solid State Commun. 31(1979)157; R.A. Ferrell, J.K. Bhattacharjee and A. Bagchi, Phys. Rev. Lett. 43 (1979) 154; N. Grewe and B. Schuh, Phys. Rev. B22 (1980) 3183; for a review see J. Keller and P. Fulde, preprint (Feb. 1981). [3] M.A. Caillé, C.R. Acad. Sci. Paris B274 (1972) 891. [4] J. Als-Nielsen et aL, Phys. Rev. B22 (1980) 312. [5] L. Berezinskii, Soy. Phys. JETP 32 (1971) 493; J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181. [6] See, for example, A. Fetter, in: The physics of liquid and solid He, Part I, eds. K.H. Bennemann and J.B. Ketterson (Wiley, New York, 1976) p. 207. [7] A different, more formal derivation in the context of pion condensation can be found in H. Kleinert, Lett. Nuovo Cimento, to be published. [8] For a derivation in the context of defect melting see H. Kleinert, Phys. Lett. 89A (1982) 294; Lett. Nuovo Cimento, to be published and references therein. [9] See W. Heifrich, J. de Phys. (Paris) 39 (1978) 1199 for a discussion.
261
PHYSICS LETTERS
12 July 1982
FLUCTUATIONS AND DEFECTS IN THE SPIRAL STATE OF MAGNETIC SUPERCONDUCTORS H. KLEINERT Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 3, 1000 Berlin 33, Germany Received 6 April 1982
We want to point out three properties of a magnetic superconductor: (i) The absence of true long-range order in the 2 and (q~)fl_2for q spiral state leads to the structure functions behaving like (q 11 — q0)~ 1 0 and q11 = 0, respectively, where q0 is the preferred momentum. The indices i~are measured via Bragg-like neutron scattering. (ii) The state is perforated by line-like defects. (iii) Above some critical temperature the defect lines proliferate, thereby destroying the spiral quasi-order.
In a recent note [11 we argued that the condensate of long-range spiral order suggested by mean-field calculations in magnetic superconductors [21 could not properly exist due to catastrophic fluctuations. The Bragg-like reflex in neutron scatteringwould then really be due to these fluctuations. Experimentally, this theorists’ subtle distinction requires high resolution of the line shape. Hoping that this will soon be available we calculate the singular behaviour of the structure factor S(q) close to q11 ~ q0, q1 = 0 and q1 ~ 0, q11 0. In addition we draw attention to the existence of line-like defects of the dislocation type (just as displayed on everyone’s fingerprints), whose configurational fluctuations eventually destroy the pseudo-order and carry the system to the normal paramagnetic superconducting state. Using the notation of ref. [lJ we begin with the free energy of the relevant order parameter M1 which is the magnetization orthogonal to the preferred momentum direction (say 1). 21 1M 2 fd3x [M~(x)]2. (I) F [r+(IqI —q0) 1(q)1 Here contains the main temperature dependence and (1) is valid for r 0 and q q 0. Our analysis will start by assuming r = r0 <0 where the energy is minimized by the spiral solution +.~-
E
i-
M1EM1 +iM2
.~/~7~eiqoz
(2)
We want to study soft long-wave this and multiply a pure space dependent phase e~~~(x). 2 ~ fluctuations (1 /4q~)(q2around q~)2and working out M1 the by lowestderivatives, F reduces to the By writing near q0: (Iqi pure gradient energy (up toq0) a constant) —
—
F = ~B [(a where X2
+ x2(a~p)2], 2 3~,)
=
(3)
4q~.Since p is a gaussian random field, the correlation function (M 1(x)M~(0)>is easily calculated
as [3] (M1(x) M~(0)>= (B/a) (exp{~i[~(x)
—
~(0)J })
=
(B/a) exp{— ~C [y,(x) ~p(o)j2>} —
where
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
259
Volume 90A, numberS
-
PHYSICS LETTERS
~([~~(x)_~(0)]2)~,(x)~(0)> (~(O)2) G~(x)= -
=
=
—
f—~.-(1
Bx(2~)3
—
-
T
_____
exp(—Xq 1IzI) cosq1x1) =
_n[log(c2x~/d2) + 27 +E1(x~/4XIzI)]
12 July 1982
-
f —i- [1
— ~
cos q .x) (q~+
—
exp(—Xq1 zI)J0(q11x11)J
.
2
Here ‘ydue ~ 0.577, ~ stands for T/87TBX, E1(x) stant to the transverse momentum cutoff. f~° Fore~dt/t Izi ~ 1xis .the exponential integral, and c 11, E1(x) -÷ —y —logx and 2’~e’l (ir2/q G~(x) c 0X~z~)h?
=
(4) q~t/q~ is a con-
(5)
.
For Ix1I~IzI,E1-~0and ~
(6)
.
This gives for the structure factor 2 for q S(q) (q11 q0)71 1 = 0, —
S(q)
(q~)712 for q11
=
0.
(7)
Such a behaviour has been detected in Bragg-like reflexes of smectic liquid crystals [4] and we hope that neutron scattering on the spiral state will soon reveal the same deviations from true long-range order. Given the simplicity of obtaining correlation functions we expect that also other fluctuation properties of the system can be understood by using this quasi-ordered state as a basis. For this it is essential to include macroscopic topological excitations [5]. Since the field p is cyclic (p and ~ + 2irn are 2indistinguishable) theissystem has singular such that b, —B ~p divergenceless, vortex lines similar to He II. Recall that there the energy isf= ~B(a~p) just as a magnetic field. Therefore one introduces a vector potential via b = a X a and finds the forces between vortex lines from the gauge invariant coupling 2irn ~ dxyl)ai (yielding the Biot-Savart law) [6]. Here the field equation —
x2 [a (a2a~p)+ a
2(a~a2~)] = 0
tells us that
2a~a
b mB(a3~p,—x
(8)
2a~a
1~p, —x
2p)
(9)
is divergenceless and may be written as ax a. In terms of b~the energy reads (2B)_1{b~+ [(xa~y’b2]b2} such that in the long wave limit [7] and the gauge a1a1 = 0 2~a~(k)l2 (k~ k~/X2k~) 1a 2]
f
~[X
+
+
[(Xa~)’b1]b1+ (10)
.
1(k)1
The coupling of a defect loop L is obtained as follows: Let n be the vortex strength of L such that for any circuit around L, p changes by 2irn, i.e. ~ dk 1 a~p= 2irn. Then p must have a jump by 2irn on some surface spanned by L. Suppose now that this loop is inserted in a given smooth field configuration p thereby changing it to p + The field energy (3) changes by 2a~a 3x b. ~&p = d3x a ~f=Bfd3x(a3p a3~~x 1pa1~x2a~a2~pa2~)d 1(b1p) = _fdSjb1p, (11) ~.
—
—
=f
—f
where the surface integral runs over a thin ellipsoid enclosing the surface S. This integral can be evaluated via the discontinuity 1~4PIS= 2irn as ~f=2irnfdS1b1. 260
(12)
Volume 90A, number 5 Now b
&f
=
2irn
PHYSICS LETTERS
12 July 1982
a x a can be used to transform this into the gauge-invariant local coupling S6dxiai(x).
(13)
The grand-canonical ensemble of interacting lines can now be described in complete analogy with defect theories of melting [8]: We introduce a disorder field ~i(x)whose free correlation function
f dL
(~(x’)i~ti~(x)) =
e—(d/T_w)L
~
~
e~(-~’-’) [a2k2/2D
=
+
(e/T— w)]1,(14)
gives the probability for a line of any length to run from x to x’. Here D is the space dimension, w log 2D the entropy per link, a the thickiiess of the line, and e the core energy per length a. The grand-canonical ensemble of free random loops is then given by the partition function [8]
z=fcexp(_fdx[(a2I2D)Ia~I2+(e/r_w)I~I2]).
(15)
The long-range interaction due to the gauge field is incorporated by the usual minimal replacement
a a -+
—
(2iri/T)a,
(16)
where we have restricted ourselves to only the basic lines with unit winding number. Certainly, there will be short range steric interactions which we may parametrize by a term ~-qI’bI4.Thus the total partition function of the system as seen from the spiral state becomes Z
=
f7.~A(D~p cii
~
E [?c2L4
2 + (k~+ k~/X2k~)IA
exp(~
3(k)1
~
1(k)I2]) (17)
This will be the most economic tool for studying the critical behaviour at the transition temperature which is given by T~= e/w. Above Tc the disorder field destabilizes leading to (i/i) * 0 which signalizes the proliferation of defect lines and the destruction of the spiral state. It goes without saying that our partition function (17) can be used for studying the completely analogous phenomena in smectics where the importance of defect lines has been stressed before [9] and, of course, in pion condensates [7]. Li]
H. Kleinert, Phys. Lett. 83A (1981) 294. [21 E.I. Blount and C.M. Varma, Phys. Rev. Lett. 42 (1979) 1079; H. Matsumoto, H. Umeuawa and M. Tachiki, Solid State Commun. 31(1979)157; R.A. Ferrell, J.K. Bhattacharjee and A. Bagchi, Phys. Rev. Lett. 43 (1979) 154; N. Grewe and B. Schuh, Phys. Rev. B22 (1980) 3183; for a review see J. Keller and P. Fulde, preprint (Feb. 1981). [3] M.A. Caillé, C.R. Acad. Sci. Paris B274 (1972) 891. [4] J. Als-Nielsen et aL, Phys. Rev. B22 (1980) 312. [5] L. Berezinskii, Soy. Phys. JETP 32 (1971) 493; J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181. [6] See, for example, A. Fetter, in: The physics of liquid and solid He, Part I, eds. K.H. Bennemann and J.B. Ketterson (Wiley, New York, 1976) p. 207. [7] A different, more formal derivation in the context of pion condensation can be found in H. Kleinert, Lett. Nuovo Cimento, to be published. [8] For a derivation in the context of defect melting see H. Kleinert, Phys. Lett. 89A (1982) 294; Lett. Nuovo Cimento, to be published and references therein. [9] See W. Heifrich, J. de Phys. (Paris) 39 (1978) 1199 for a discussion.
261