Influence of rare regions on quantum phase transition in antiferromagnets with hidden degrees of freedom

12 Pages·0.12 MB·English

by  Y. N. Skryabin

Influence of rare regions on quantum phase
transition in antiferromagnets with hidden
0
0 degrees of freedom
0
2

n
a
J
Y. N. Skryabin1
5
1
Institute for Metal Physics, Russian Academy of Sciences,

] Ural Division, Kovalevskaya Str., 18,
n
n 620219, Ekaterinburg, Russian Federation
-
s
i
d
.
t A. V. Chukin and A. V. Shchanov
a
m
Ural State Technical University,
-
d Mir Str., 19, 620002, Ekaterinburg, Russian Federation
n
o
c
[

Abstract
1
v
4 The effects of rare regions on the critical properties of quantum antiferromagnets
1
with hidden degrees of freedom within the renormalization group is discussed. It
2
1 is shown that for “constrained” systems the stability range on the phase diagram
0 remains the same as in the mean-field theory while for “unconstrained” systems the
0
stability range is effectively decreased.
0
/
t
a
m Key words: Magnetically ordered materials. Point defects. Phase transitions.
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d
n
o
c
:
v
1 Introduction
i
X
r
a
The influence of quenched disorder on the critical properties of itinerant quan-
tum magnets is a very interesting problem in phase transition theory. Particu-
lar interest was given to locally ordered spatial regions (“rare regions”) which
are formed in the presence of quenched disorder even when the bulk system is
still in the paramagnetic phase [1]. The conventional theory [2] ignores these
1 Partially supported by Russian Foundation for Basic Research (97-02-17315),
Russian Federation.
Preprint submitted to Elsevier Preprint 1 February 2008
rare regions. For classical magnets, it has recently been shown that the rare
regions induce a new term in the action which breaks the replica symmetry
and that the conventional theory is unstable with respect to this perturbation
[3].
Theproblemofrareregionsforthecase ofquantum phasetransitionshasbeen
considered in Refs. [4,5]. These authors showed that the rare regions destroy
the fixed point found in the conventional theory of itinerant antiferromagnets
[6] and, in contrast, critical behavior of itinerant ferromagnets [7] is unaffected
bytherareregionsduetoaneffective long-rangeinteractionbetween theorder
parameter fluctuations.
The conventional theory of a random-T quantum antiferromagnet with the
c
hidden degrees of freedom on which various constraints are imposed has been
considered in Ref. [8] and it has been shown that for “constrained” systems
the stability range on the phase diagram remains the same as in the mean-
field theory while for “unconstrained” systems the stability range is effectively
decreased. In this paper we study the effects of rare regions on the quantum
phase transition in antiferromagnets with hidden degrees of freedom.
2 Renormalization group equations
Let us consider a disordered itinerant quantum antiferromagnet in which the
n-component vector order parameter S(r,τ) is coupled with the scalar non-
fluctuating parameter y(r,τ). We use an action
1/T
1
′ ′
S= dr dr dτ dτ [t(r −r ,τ −τ )
2 Z 1 2 Z 1 2
0
′ ′
+δ(r −r )δ(τ −τ )δt(r)]S(r ,τ)S(r ,τ )
1 2 1 2
1/T
+λ dr dτ[S(r,τ)S(r,τ)]2
sZ Z
0
1/T
+µ dr dτy(r,τ)[S(r,τ)]2
Z Z
0
1/T
1
+ β dr dτ[y(r,τ)]2. (1)
2 Z Z
0
2
Here the function t(r,τ) is the Fourier-transform of the two-point interaction
of a quantum antiferromagnet [9,6]
t(q,ω ) = t+q2 +ω , (2)
n n
where t denotes the distance from the quantum critical point, q is the wave
vector and ω is the Matsubara frequency.
n
As the given model is phenomenological, we suppose that the coupling con-
stantoftheorderparameterwiththenonfluctuatingparameterispurelyimag-
inary as it occurs in the Hubbard model [10–12]. It is easy to consider the case
with a realcoupling constant. The distinction between these two cases consists
only in the definition of the part of an appropriate phase space which corre-
sponds to a real physical model. In contrast to the classical phase transition
the order parameter depends on both the d-dimensional vector of space r and
imaginary time τ [9].
In order to write an action of the conventional theory the replica trick should
be used and then an integration over random variables should be performed.
For simplicity we consider the case when the coefficient of the quadratic term
in the order parameter in the action, δt(r), is the fluctuating Gaussian vari-
able with zero mean and variance ∆. Finally we obtain an action which has
homogeneous saddle points only [8].
To incorporate rare regions into the theory we need a different approach. In
analogy to Refs. [3–5], we consider inhomogeneous saddle-point solutions for a
fixed realization of disorder. The partition function can be written as the sum
of all contributions obtained from the vicinity of each saddle-point [5]. Then
we can average over disorder by means of the replica trick. We finally obtain
the following effective action
1/T
1
S = dr dr dτ dτ′t(r −r ,τ −τ′) Sα(r ,τ)Sα(r ,τ′)
eff 2 Z 1 2 Z 1 2 1 2
Xα
0
1/T
+λ dr dτ [Sα(r,τ)Sα(r,τ)]2
sZ Z
Xα
0
1/T
− (∆+xδ ) dr dτ dτ′[Sα(r,τ)Sα(r,τ)]×
α,β Z Z
X
α,β 0
[Sβ(r,τ′)Sβ(r,τ′)]
1/T
+µ dr dτ yα(r,τ)[Sα(r,τ)]2
Z Z
Xα
0
3
1/T
1
+ β dr dτ [yα(r,τ)]2. (3)
2 Z Z
Xα
0
Here α and β are replica indices. The x-term is generated by taking into
account the inhomogeneous saddle points. The conventional theory misses
this term.
Let us assume the temperature T to be equal zero and use double ǫ expansion
[6] according to which the space dimensionality is equal 4−ǫ and the dimen-
sionality of imaginary time is ǫ . Of course, for the real physical case we have
τ
ǫ = ǫ = 1.
τ
Defining x¯ = xT−ǫτ, and putting T = 0, we obtain the following renormaliza-
tion group flow equations in the one-loop approximation
du
=(ǫ−2ǫ )u−4(n+8)u2 +6u∆, (4)
τ
dl
d∆
=ǫ∆−8(n+2)u∆+4∆2 +8n∆x¯, (5)
dl
dx¯
=(ǫ−2ǫ )x¯−8(n+2)ux¯+4nx¯2 +6∆x¯, (6)
τ
dl
dz
=(ǫ−2ǫ )z −8(n+2)uz −2nz2 +2z∆, (7)
τ
dl
dw
=(ǫ−2ǫ )w−8(n+2)uw−2nw2 −4nzw +2w∆, (8)
τ
dl
where l = lnb with b the scale parameter, and we have scaled K u →
4
u,K ∆ → ∆,K x¯ → x¯,K z → z,K w → w with K = 1/8π2. We also
4 4 4 4 4
denote here
µ2
u=λ − , (9)
s
2β
µ2 µ2
z= − 0, (10)
β β
0
µ2
w= 0. (11)
β
0
Inequations(4)-(8)weseparatethecoefficientofthenonfluctuatingparameter
y(q = 0) from y(q 6= 0) because of its possible role in constraining systems
[10,13].
4
3 Fixed points
Before discussing the renormalization group analysis we consider the mean-
field theory result. After integrating over the nonfluctuating parameter in the
equation for the partition function we can obtain a new effective action in
terms of the order parameter S(r). In the mean-field approximation for this
effective action the boundaries of a stability range can be easily found. It
is convenient to introduce new notations for the coupling constants of the
effective action as
λ(0) = µ2/2β , λ(1) = µ2/2β. (12)
c 0 0 c
For the unconstrained system λ(0) = λ(1) ≡ λ (or z = 0) while the boundaries
c c c
of the stability range correspond to the equations of lines
λ −λ(0) = 0, λ(0) = 0. (13)
s c c
For the constrained system λ(0) = 0 (or w = 0) while the boundaries of the
c
stability range can be written as
λ = 0, λ(1) = 0. (14)
s c
The stability ranges are represented on planes (λ −λ ) (Fig. 1) and (λ(1)−λ )
c s c s
(Fig. 2).
The characteristic feature of flow equations is the closed system of three equa-
tions (4)-(6). Within the notations this set of equations coincides with the
appropriate set of equations of Ref. [4] where however the nonfluctuating de-
grees of freedom were not taken into account. It is easy to find all eight fixed
∗
points (4)-(6). Four of the fixed points have a zero fixed-point value of x¯ ,
∗ ∗
x¯ = 0. The other four fixed points have x¯ 6= 0.
Using these fixed points we can find other fixed points from the flow equations
∗
(7),(8).There arethesixteen fixedpointswithx¯ = 0(Table1)studied before
∗
inRef.[8].Theothersixteenfixedpointshavex¯ 6= 0(Table2). Ascanbeseen
from Table 1 we can separate all fixed points on groups consisting of two fixed
points, for example group of Gaussian points (G) and group of renormalized
Gaussian points (RG). Each group has its proper set of critical exponents.
Such classification of fixed points was used for description of phase transition
in classical systems [10,14,15] where for renormalized fixed points the criti-
cal exponents can be found according to Fisher [16] via critical exponents of
5
Table 1
∗
Fixed points with x¯ = 0
∗ ∗ ∗ ∗ ∗
u ∆ x¯ z w
G 0 0 0 0 ǫ−2ǫτ
1 2n
G 0 0 0 0 0
2
RG 0 0 0 ǫ−2ǫτ 0
1 2n
RG 0 0 0 ǫ−2ǫτ −ǫ−2ǫτ
2 2n 2n
H ǫ−2ǫτ 0 0 0 (ǫ−2ǫτ)(4−n)
1 4(n+8) 2n(n+8)
H ǫ−2ǫτ 0 0 0 0
2 4(n+8)
RH ǫ−2ǫτ 0 0 (ǫ−2ǫτ)(4−n) 0
1 4(n+8) 2n(n+8)
RH ǫ−2ǫτ 0 0 (ǫ−2ǫτ)(4−n) −(ǫ−2ǫτ)(4−n)
2 4(n+8) 2n(n+8) 2n(n+8)
U 0 −ǫ 0 0 ǫ−4ǫτ
1 4 4n
U 0 −ǫ 0 0 0
2 4
RU 0 −ǫ 0 ǫ−4ǫτ 0
1 4 4n
RU 0 −ǫ 0 ǫ−4ǫτ −ǫ−4ǫτ
2 4 4n 4n
R ǫ+4ǫτ (4−n)ǫ+4(n+2)ǫτ 0 0 (n−4)ǫ−12nǫτ
1 16(n−1) 8(n−1) 8n(n−1)
R ǫ+4ǫτ (4−n)ǫ+4(n+2)ǫτ 0 0 0
2 16(n−1) 8(n−1)
RR ǫ+4ǫτ (4−n)ǫ+4(n+2)ǫτ 0 (n−4)ǫ−12nǫτ 0
1 16(n−1) 8(n−1) 8n(n−1)
RR ǫ+4ǫτ (4−n)ǫ+4(n+2)ǫτ 0 (n−4)ǫ−12nǫτ −(n−4)ǫ−12nǫτ
2 16(n−1) 8(n−1) 8n(n−1) 8n(n−1)
appropriate non-renormalized values
α ν
α = − , ν = . (15)
renorm renorm
1−α 1−α
The eigenvalues λ (i = 1,...,5) of flow equations linearized about the fixed
i
point (4)-(8) are indicated in Table 3 and Table 4.
Itshouldbenotedthateigenvaluesλ andλ forrandom(R)andrenormalized
1 2
random (RR) fixed points for n > 1 are complex.
Due to the relation
u = λ −λ(1), (16)
s c
∗ ∗
the fixed points with ∆ = 0 and x¯ = 0 align on parallel lines:
(1) λ = λ for the unconstrained system (z = 0);
s c
(2) λ = λ(1) for the constrained system (w = 0);
s c
6
Table 2
∗
Fixed points with x¯ 6= 0
∗ ∗ ∗ ∗ ∗
u ∆ x¯ z w
1 0 0 −ǫ−2ǫτ 0 ǫ−2ǫτ
4n 2n
2 0 0 −ǫ−2ǫτ 0 0
4n
3 0 0 −ǫ−2ǫτ ǫ−2ǫτ 0
4n 2n
4 0 0 −ǫ−2ǫτ ǫ−2ǫτ −ǫ−2ǫτ
4n 2n 2n
5 ǫ−2ǫτ 0 (n−4)(ǫ−2ǫτ) 0 (ǫ−2ǫτ)(4−n)
4(n+8) 4n(n+8) 2n(n+8)
6 ǫ−2ǫτ 0 (n−4)(ǫ−2ǫτ) 0 0
4(n+8) 4n(n+8)
7 ǫ−2ǫτ 0 (n−4)(ǫ−2ǫτ) (ǫ−2ǫτ)(4−n) 0
4(n+8) 4n(n+8) 2n(n+8)
8 ǫ−2ǫτ 0 (n−4)(ǫ−2ǫτ) (ǫ−2ǫτ)(4−n) −(ǫ−2ǫτ)(4−n)
4(n+8) 4n(n+8) 2n(n+8) 2n(n+8)
9 0 −ǫ+4ǫτ −ǫ+4ǫτ 0 3ǫ−4ǫτ
8 16n 8n
10 0 −ǫ+4ǫτ −ǫ+4ǫτ 0 0
8 16n
11 0 −ǫ+4ǫτ −ǫ+4ǫτ 3ǫ−4ǫτ 0
8 16n 8n
12 0 −ǫ+4ǫτ −ǫ+4ǫτ 3ǫ−4ǫτ −3ǫ−4ǫτ
8 16n 8n 8n
13 ǫ+4ǫτ (n−4)ǫ+24ǫτ (n−4)(ǫ+4ǫτ) 0 (4−n)3ǫ−(n+8)4ǫτ
8(10−n) 4(10−n) 8n(10−n) 4n(10−n)
14 ǫ+4ǫτ (n−4)ǫ+24ǫτ (n−4)(ǫ+4ǫτ) 0 0
8(10−n) 4(10−n) 8n(10−n)
15 ǫ+4ǫτ (n−4)ǫ+24ǫτ (n−4)(ǫ+4ǫτ) (4−n)3ǫ−(n+8)4ǫτ 0
8(10−n) 4(10−n) 8n(10−n) 4n(10−n)
16 ǫ+4ǫτ (n−4)ǫ+24ǫτ (n−4)(ǫ+4ǫτ) (4−n)3ǫ−(n+8)4ǫτ −(4−n)3ǫ−(n+8)4ǫτ
8(10−n) 4(10−n) 8n(10−n) 4n(10−n) 4n(10−n)
(3) λ −λ = (ǫ−2ǫ )/4K (n+8) for the unconstrained system (z = 0);
s c τ 4
(4) λ −λ(1) = (ǫ−2ǫ )/4K (n+8) for the constrained system (w = 0).
s c τ 4
It is easy to see that for the quantum phase transition (ǫ 6= 0) in the case
τ
when the inequality ǫ < 2ǫ is satisfied the fixed points lying on the two last
τ
lines are not situated in the stability range of the effective action (Fig. 1 and
Fig. 2). Thus the stability range for the quantum phase transition in the con-
strained system remains such as in the mean-field theory in contrast with the
classical phase transition where these fixed points are situated in the stability
range and cause an effective reduction of this range [11,12]. However for the
unconstrained system there is the unstable fixed point G on the boundary of
1
the stability range. Its critical exponents coincide with critical exponents of
the point G . Analogous to the classical phase transition a separatrix going
2
out this point divides the stability range in two regions. One is the region of
accessibility of the fixed point G and the other is the region of phase space
2
where flow trajectories approach to the “invariant” line G −G .
2 1
∗ ∗
Correspondingly, the fixed points with ∆ 6= 0 and x¯ = 0 are situated along
parallel lines:
7
Table 3
Eigenvalue of fixed points with x¯∗ = 0 (λR = 1 {−3nǫ+4(n−4)ǫ ±[(5n−
1,2 8(n−1) τ
8)2ǫ2+24(16−12n−n2)ǫǫ +48(16−8n−5n2)ǫ2]1/2})
τ τ
λ λ λ λ λ
1 2 3 4 5
G ǫ−2ǫ ǫ ǫ−2ǫ ǫ−2ǫ −ǫ+2ǫ
1 τ τ τ τ
G ǫ−2ǫ ǫ ǫ−2ǫ ǫ−2ǫ ǫ−2ǫ
2 τ τ τ τ
RG ǫ−2ǫ ǫ ǫ−2ǫ −ǫ+2ǫ −ǫ+2ǫ
1 τ τ τ τ
RG ǫ−2ǫ ǫ ǫ−2ǫ −ǫ+2ǫ ǫ−2ǫ
2 τ τ τ τ
H −ǫ+2ǫ (4−n)ǫ+4(n+2)ǫτ (4−n)(ǫ−2ǫτ) (4−n)(ǫ−2ǫτ) −(4−n)(ǫ−2ǫτ)
1 τ n+8 n+8 n+8 n+8
H −ǫ+2ǫ (4−n)ǫ+4(n+2)ǫτ (4−n)(ǫ−2ǫτ) (4−n)(ǫ−2ǫτ) (4−n)(ǫ−2ǫτ)
2 τ n+8 n+8 n+8 n+8
RH −ǫ+2ǫ (4−n)ǫ+4(n+2)ǫτ (4−n)(ǫ−2ǫτ) −(4−n)(ǫ−2ǫτ) −(4−n)(ǫ−2ǫτ)
1 τ n+8 n+8 n+8 n+8
RH −ǫ+2ǫ (4−n)ǫ+4(n+2)ǫτ (4−n)(ǫ−2ǫτ) −(4−n)(ǫ−2ǫτ) (4−n)(ǫ−2ǫτ)
2 τ n+8 n+8 n+8 n+8
U −1ǫ−2ǫ −ǫ −1ǫ−2ǫ 1ǫ−2ǫ −1ǫ+2ǫ
1 2 τ 2 τ 2 τ 2 τ
U −1ǫ−2ǫ −ǫ −1ǫ−2ǫ 1ǫ−2ǫ 1ǫ−2ǫ
2 2 τ 2 τ 2 τ 2 τ
RU −1ǫ−2ǫ −ǫ −1ǫ−2ǫ −1ǫ+2ǫ −1ǫ+2ǫ
1 2 τ 2 τ 2 τ 2 τ
RU −1ǫ−2ǫ −ǫ −1ǫ−2ǫ −1ǫ+2ǫ 1ǫ−2ǫ
2 2 τ 2 τ 2 τ 2 τ
R λR λR (4−n)(ǫ+4ǫτ) (n−4)ǫ−12nǫτ −(n−4)ǫ−12nǫτ
1 1 2 4(n−1) 4(n−1) 4(n−1)
R λR λR (4−n)(ǫ+4ǫτ) (n−4)ǫ−12nǫτ (n−4)ǫ−12nǫτ
2 1 2 4(n−1) 4(n−1) 4(n−1)
RR λR λR (4−n)(ǫ+4ǫτ) −(n−4)ǫ−12nǫτ −(n−4)ǫ−12nǫτ
1 1 2 4(n−1) 4(n−1) 4(n−1)
RR λR λR (4−n)(ǫ+4ǫτ) −(n−4)ǫ−12nǫτ (n−4)ǫ−12nǫτ
2 1 2 4(n−1) 4(n−1) 4(n−1)
λ
c
H G
2 2 λ
s
H
1
G
1
∗ ∗
Fig. 1. Phase diagram for the unconstrained system (∆ = 0, x¯ = 0)
8
Table 4
Eigenvalue of fixedpoints withx¯∗ 6= 0 (λ(9) = 1{4ǫ −3ǫ±[25ǫ2−24ǫǫ −240ǫ2]1/2}
2,3 8 τ τ τ
and λ(13) = − 1 {(16−n)ǫ+4(n−4)ǫ ±[9(8−n)2ǫ2+24(n2−4n−48)ǫǫ +
1,2 4(10−n) τ τ
16(n2 +40n−464)ǫ2]1/2})
τ
λ λ λ λ λ
1 2 3 4 5
1 ǫ−2ǫ −ǫ+4ǫ −ǫ+2ǫ ǫ−2ǫ −ǫ+2ǫ
τ τ τ τ τ
2 ǫ−2ǫ −ǫ+4ǫ −ǫ+2ǫ ǫ−2ǫ ǫ−2ǫ
τ τ τ τ τ
3 ǫ−2ǫ −ǫ+4ǫ −ǫ+2ǫ −ǫ+2ǫ −ǫ+2ǫ
τ τ τ τ τ
4 ǫ−2ǫ −ǫ+4ǫ −ǫ+2ǫ −ǫ+2ǫ ǫ−2ǫ
τ τ τ τ τ
5 −ǫ+2ǫ (n−4)ǫ+24ǫτ (n−4)(ǫ−2ǫτ) (4−n)(ǫ−2ǫτ) −(4−n)(ǫ−2ǫτ)
τ n+8 n+8 n+8 n+8
6 −ǫ+2ǫ (n−4)ǫ+24ǫτ (n−4)(ǫ−2ǫτ) (4−n)(ǫ−2ǫτ) (4−n)(ǫ−2ǫτ)
τ n+8 n+8 n+8 n+8
7 −ǫ+2ǫ (n−4)ǫ+24ǫτ (n−4)(ǫ−2ǫτ) −(4−n)(ǫ−2ǫτ) −(4−n)(ǫ−2ǫτ)
τ n+8 n+8 n+8 n+8
8 −ǫ+2ǫ (n−4)ǫ+24ǫτ (n−4)(ǫ−2ǫτ) −(4−n)(ǫ−2ǫτ) (4−n)(ǫ−2ǫτ)
τ n+8 n+8 n+8 n+8
9 ǫ+4ǫτ λ(9) λ(9) 3ǫ−4ǫτ −3ǫ+4ǫτ
4 2 3 4 4
10 ǫ+4ǫτ λ(9) λ(9) 3ǫ−4ǫτ 3ǫ−4ǫτ
4 2 3 4 4
11 ǫ+4ǫτ λ(9) λ(9) −3ǫ+4ǫτ −3ǫ+4ǫτ
4 2 3 4 4
12 ǫ+4ǫτ λ(9) λ(9) −3ǫ+4ǫτ 3ǫ−4ǫτ
4 2 3 4 4
13 λ(13) λ(13) (n−4)(ǫ+4ǫτ) (4−n)3ǫ−(n+8)4ǫτ −(4−n)3ǫ−(n+8)4ǫτ
1 2 2(10−n) 2(10−n) 2(10−n)
14 λ(13) λ(13) (n−4)(ǫ+4ǫτ) (4−n)3ǫ−(n+8)4ǫτ (4−n)3ǫ−(n+8)4ǫτ
1 2 2(10−n) 2(10−n) 2(10−n)
15 λ(13) λ(13) (n−4)(ǫ+4ǫτ) −(4−n)3ǫ−(n+8)4ǫτ −(4−n)3ǫ−(n+8)4ǫτ
1 2 2(10−n) 2(10−n) 2(10−n)
16 λ(13) λ(13) (n−4)(ǫ+4ǫτ) −(4−n)3ǫ−(n+8)4ǫτ (4−n)3ǫ−(n+8)4ǫτ
1 2 2(10−n) 2(10−n) 2(10−n)
λ(c1)
H G
2 2 λ
s
RH
1
RG
1
∗ ∗
Fig. 2. Phase diagram for the constrained system (∆ = 0, x¯ = 0)
9
λ
c
R
2 λ
s
R
1
∗ ∗
Fig. 3. Phase diagram for the unconstrained system (∆ 6= 0, x¯ = 0)
λ(c1)
R
2 λ
s
RR
1
∗ ∗
Fig. 4. Phase diagram for the constrained system (∆ 6=0, x¯ = 0)
(1) λ −λ = (ǫ+4ǫ )/16K (n−1) for the unconstrained system (z = 0);
s c τ 4
(2) λ −λ(1) = (ǫ+4ǫ )/16K (n−1) for the constrained system (w = 0).
s c τ 4
For the unconstrained system the fixed points lying on these lines for both
the classical (ǫ = 0) and the quantum phase transition are situated in the
τ
stability range (Fig. 3). Thus the accessibility range of fixed points is also
reduced in comparison with the mean-field result. For the constrained system
not all points are in the range of stability (Fig. 4).
The analysis of the renormalization group equations (4), (6) and (5) shows
that as well as in the case of systems without hidden degrees of freedom [4,5]
the Gaussian fixed point G is stable for d > 4 and unstable to disorder for
2
d < 4. The random fixed point R is stable for 3nǫ > 4(n−4)ǫ , n > 4 and
2 τ
12nǫ > (n− 4)ǫ. It is easy to see that this point is always stable for d < 4
τ
and 4 < n < n where n = 16 for the particular case ǫ = ǫ. It should be
c c τ
10