One-loop functional renormalization group study for the dimensional reduction and its breakdown in the long-range random field O($N$) spin model near lower critical dimension
Yoshinori Sakamoto




yossi@phys.ge.cst.nihon-u.ac.jp

Laboratory of Physics, College of Science and Technology, 
Nihon University, 7-24-1, Narashino-dai, Funabashi-city, Chiba, 
274-8501, Japan




We consider the random-field O(N) spin model with 
long-range exchange interactions 
which decay with distance r between spins as r^-d-σ
and/or 
random fields which correlate with distance r as r^-d+ρ ,  
and reexamine the critical phenomena near the lower critical dimension 
by use of the perturbative functional renormalization group.  
We compute the analytic fixed points in the one-loop beta functions, 
and study their stability. 
We also calculate the critical exponents at the analytical fixed points. 
We show that the analytic fixed point which governs the phase transition 
in the system with the long-range correlations of random fields  
can be destabilized by the nonanalytic perturbation 
in both cases where the exchange interactions between spins are 
short ranged and long ranged. 
For the system with the long-range exchange interactions and uncorrelated random fields, 
we show that the d→ d-σ dimensional reduction at the leading order of the d-2σ expansion holds only for N>2(4+3√(3))≃ 18.3923⋯. 
Our investigation into the system 
with the long-range exchange interactions and uncorrelated random fields 
also gives the value of the boundary between critical behaviors 
in systems with long-range and short-range exchange interactions, 
which is identical to that predicted by Sak [Phys. Rev. B 8, 281 (1973)]. 
For the system with the long-range exchange interactions and the long-range correlated random fields,  
we show that the d→ d-σ-ρ dimensional reduction does not hold within the present framework, 
as far as N is finite. 



One-loop functional renormalization group study for 
the dimensional reduction and its breakdown 
in the long-range random field O(N) spin model 
near lower critical dimension

    Yoshinori Sakamoto
    December 30, 2023
=================================================================================================================================================================================








§ INTRODUCTION

The random-field O(N) spin model is the model in which 
nonrandom exchange interactions between spins are ferromagnetic 
and external magnetic fields are random. 
To clarify the critical phenomena in this model is one of the fundamental problems 
in the disordered spin system, 
and there are a lot of intensive studies on this <cit.>.  
The d→ d-θ dimensional reduction gives an important clue 
to clarify the nature of this model. 
The d→ d-θ dimensional reduction means that 
the effect of random fields reduces the spatial dimension by θ; namely, 
the critical phenomena in d-dimensional random-field system 
is equivalent to that in the (d-θ)-dimensional corresponding pure system. 
Here θ denotes the exponent describing that the flow of the renormalized temperature goes to zero 
under the renormalization-group iteration. 
If the d→ d-θ dimensional-reduction prediction is correct, 
all critical exponents in the d-dimensional random-field system 
should be the same as those in the corresponding pure system in θ dimensions less. 


In the spin system with the short-range ferromagnetic exchange interactions 
and the uncorrelated random fields (SR), 
the d→ d-2 dimensional reduction and its breakdown are one of the central issues. 
This conjecture was obtained by the perturbation theory <cit.> 
and the supersymmetry argument <cit.>. 
Rigorous proofs have shown that the d→ d-2 dimensional-reduction prediction 
is incorrect below four dimensions 
in the case of the random-field Ising model (N=1 case) <cit.>.  
The d→ d-2 dimensional reduction and its breakdown 
for the random-field O(N) spin model above four dimensions 
have been intensively studied. 
Fisher studied the critical phenomena in 4+ϵ dimensions by use of 
the O(N) nonlinear-sigma model <cit.>. 
He showed that all possible higher-rank random anisotropies 
which are all relevant operators are generated 
by the perturbative functional renormalization-group iteration 
of the O(N) nonlinear-sigma model with only the random field term. 
Then he treated the O(N) nonlinear-sigma model 
including the random-field and all the random-anisotropy terms,
and derived the one-loop beta function in 4+ϵ dimensions. 
He showed that there is no singly unstable fixed point 
corresponding to the d→ d-2 dimensional reduction at O(ϵ), 
and concluded that the d→ d-2 dimensional-reduction prediction is incorrect near four dimensions. 
The one-loop beta function obtained by Fisher 
and the two-loop beta function extended by Le Doussal and Wiese <cit.> and Tissier and Tarjus <cit.> 
have been examined carefully <cit.>. 
The breakdown of the dimensional reduction is characterized by a nonanalyticity 
which emerges in the first derivative of the function 
including the random-field and all the random-anisotropy terms. 
Namely, the nonanalyticity forms a cusp in the first derivative of the function 
including the random-field and all the random-anisotropy terms, 
which causes the breakdown of the dimensional reduction. 
The singly unstable fixed point corresponding to the d→ d-2 dimensional reduction 
exists for N>18-(49/5)ϵ, although it has the weak nonanalyticity 
which does not change the value of the fixed point. 
However, it is unstable with respect to the perturbation with nonanalyticity 
for N<2(4+3√(3))-(3(2+3√(3))/2)ϵ. 
Thus, the d→ d-2 dimensional reduction holds 
for N>2(4+3√(3))-(3(2+3√(3))/2)ϵ, 
and the critical exponents of the connected and the disconnected correlation functions 
η and η̅ satisfy η̅=η. 
Whereas in N<2(4+3√(3))-(3(2+3√(3))/2)ϵ, 
the critical phenomena is governed by the fixed point with the nonanalyticity, 
and thus the d→ d-2 dimensional reduction is broken. 
Moreover, a complete theoretical explanation of the d→ d-2 dimensional reduction 
and its breakdown has been provided 
through the nonperturbative functional renormalization group <cit.>. 

In the case where the ferromagnetic exchange interactions are short ranged 
and the random fields are correlated over the distance r as r^-d+ρ (LRF), 
the d→ d-2-ρ dimensional reduction and its breakdown are still under debate. 
The symbol ρ denotes the exponent 
which characterizes the range of the random-field correlations. 
To study the long-range effect of the random-field correlations in the system, 
we consider the case 0≤ρ<d-2d_ p, 
where d_ p is the lower critical dimension of the corresponding pure system. 
Kardar, McClain, and Taylor performed the renormalization-group calculation near the upper critical dimension d_u=6+ρ, 
and concluded that the d→ d-2-ρ dimensional reduction is broken at O(ϵ^2) in ϵ=d_u-d <cit.>. 
Bray pointed out an error in Kardar, McClain, and Taylor's result 
but their conclusion still holds <cit.>.  
Chang and Abrahams carried out the one-loop renormalization-group calculation 
for the O(N) nonlinear-sigma model 
near the lower critical dimension d_l=4+ρ, 
and showed that the d→ d-2-ρ dimensional reduction is broken 
at O(ϵ) in ϵ=d-d_l and for N>3 <cit.>. 
Fedorenko and Kühnel <cit.> examined the one-loop beta functions 
of the O(N) nonlinear-sigma model 
including not only the uncorrelated and the long-range correlated random fields 
but also all the uncorrelated and the long-range correlated random anisotropies 
which are missed in the work by Chang and Abrahams. 
They showed that the correlation length exponent ν 
and the phase diagram obtained by Chang and Abrahams are incorrect, 
and the exponents η and η̅ are correct only
in a region controlled by the singly unstable fixed point with the weaker nonanalyticity. 

In the case where the long-range ferromagnetic exchange interactions decay with distance r between spins as r^-d-σ
and random fields are uncorrelated (LRE), 
the d→ d-σ dimensional reduction and its breakdown are still under debate. 
The symbol σ denotes the exponent 
which controls the range of the exchange interactions. 
It should be positive to ensure that the energy density stays finite 
in the thermodynamic limit. 
To study the long-range character of the exchange interactions in the system, 
we consider the case 0<σ<2. 
Young performed the renormalization-group calculation near the upper critical dimension d_u=3σ, 
and concluded that the d→ d-σ dimensional reduction is broken at O(ϵ^2) 
in ϵ=d_u-d <cit.>. 
Bray pointed out an error in Young's result 
but the conclusion still holds <cit.>.  
Chang and Abrahams carried out the one-loop renormalization-group calculation 
for the O(N) nonlinear-sigma model 
near the lower critical dimension d_l=2σ, 
and showed that the d→ d-σ dimensional reduction holds at O(ϵ) in ϵ=d-d_l and for N>1 <cit.>. 
However, the O(N) nonlinear-sigma model studied by Chang and Abrahams 
does not contain an infinite number of relevant operators which should be included in the model. 
Recently, Balog, Tarjus, and Tissier studied 
the critical phenomena of a one-dimensional random-field Ising model 
with the long-range exchange interactions and uncorrelated random fields 
by use of the nonperturbative renormalization group, 
and found that there are two distinct regimes characterized by the presence or absence of the nonanalyticity 
in the region of 1/3≤σ<1/2 where the critical exponents take non-classical values  <cit.>. 

In the spin system with the long-range ferromagnetic exchange interactions  
and the long-range correlated random fields (LREF) 
with 0<σ<2 and 0≤ρ<d-2d_ p, 
the d→ d-σ-ρ dimensional reduction and its breakdown are still under debate. 
Bray used the renormalization-group scaling theory, and showed η=2-σ, 2η-η̅=ρ, 
and θ=σ+ρ. 
Recently, we put ρ=2-σ, and studied the critical phenomena in the three-dimensional long-range random-field Ising model 
in the region of 1/2<σ<1 by using the nonperturbative functional renormalization group 
combined with the supersymmetric formalism <cit.>. 
We showed that the d→ d-2 dimensional reduction holds for 1/2<σ<σ_ DR≈0.71,  
and its breakdown is observed in the exponent ν for σ_ DR<σ<1. 

In contrast to the case 0<σ<2 
in which the long-range feature of the exchange interactions is dominant, 
the phase transition for large σ belongs to the short-range universality class. 
As the exponent σ decreases from large σ, 
the universality class of the phase transition crosses over 
from the short-range one to the long-range one at a critical value σ_*. 
In spite of theoretical and numerical studies over forty years, 
the critical behavior in the vicinity of σ=σ_* is still an ongoing problem. 
There are a lot of studies on this problem in the pure system 
<cit.>.  
In Refs. FMN,GT 
it was shown that the exponent η changes discontinuously 
from the value in the corresponding short-range system to 2-σ 
at σ=σ_*=2, 
as σ decreases from large σ. 
In Refs. Sak,HNH,LB,APR,DTC,HST,BRRZ 
it was shown that the effect of the long-range exchange interactions is relevant 
for σ<σ_*=2-η_ sr, 
where η_ sr denotes the exponent of the connected correlation function 
in the corresponding short-range system. 
Then the exponent η is continuous at σ_*=2-η_ sr, 
whose value takes η_ sr for σ≥σ_*, 
and 2-σ for σ<σ_*. 
Moreover, the presence of a logarithmic correction to the connected correlation function 
at σ_*=2-η_ sr was reported in Ref. BrePariRi. 
In Refs. Pic,BPR 
it was shown that the discontinuity of the exponent η 
at σ=2 does not occur, 
and the value of η is interpolated smoothly 
from η_ sr to 2-σ, 
as σ decreases from σ=2. 
In the random-field spin system, 
Bray showed σ_*=2-η_ sr 
by using the renormalization-group scaling theory <cit.>.  

As stated above, 
the phase transitions in this model are classified into four universality classes (SR, LRF, LRE and LREF), 
according to whether the exchange interactions and/or the random-field correlations in the system 
are short ranged or long ranged. 
However, most studies of the critical phenomena in the random-field O(N) spin model 
have been dedicated to the SR case.  
In this paper we consider all four cases. 
We study the critical phenomena near the lower critical dimension 
with the use of the O(N) nonlinear-sigma model combined with the replica formalism. 
The model treated in this paper contains 
not only the uncorrelated and the correlated random-field terms 
but also all the uncorrelated and the correlated random-anisotropy terms. 
We employ the perturbative functional renormalization group in order to obtain the one-loop beta functions. 
We examine the properties of the fixed point functions, 
and investigate the stability of the analytic fixed points 
on the basis of the argument by Baczyk, Tarjus, Tissier, and Balog <cit.>. 
Then we calculate the critical exponents η, η̅, and ν 
at each of four analytic fixed points,  
and discuss the critical properties of the system for each universality class. 
We show that the destabilization of the analytic fixed point 
controlling the critical behavior in the system with the long-range correlations 
of random fields can be caused by the perturbation with nonanalyticity 
in both cases where the exchange interactions between spins are 
short ranged and long ranged. 
In the system with LRE, 
we find that the analytic fixed point of O(ϵ) in ϵ=d-d_l 
which controls the critical behavior 
is singly unstable not for N>1 but for N>2(4+3√(3))≃ 18.3923⋯. 
We show that the validity of the d→ d-σ dimensional reduction 
at the leading order of the d-2σ expansion is confirmed 
only for N>2(4+3√(3))≃ 18.3923⋯. 
Moreover, 
by investigating the relation between the critical exponents η and η̅, 
we also obtain the critical value σ_*=2-η_ sr, 
which is the same as that obtained in Refs. Br,Sak,HNH,LB,APR,DTC,HST,BRRZ. 
In the system with LREF, 
we find that the analytic fixed point which is singly unstable exists 
under a certain condition. 
However, we show that the d→ d-ρ-σ dimensional reduction 
does not hold within the present framework, as far as N is finite. 

The organization of this paper is as follows. 
In Sec. <ref> 
we study the systems with the short-range exchange interactions, 
namely the SR and LRF cases. 
We perform the one-loop functional renormalization group analysis. 
We show that the analytic fixed point which governs the phase transition 
in the system with LRF can be destabilized by the perturbation with nonanalyticity. 
In Sec. <ref> 
we study the systems with the long-range exchange interactions, 
namely the LRE and LREF cases. 
We treat the one-loop beta functions, 
and carefully analyze the properties of the analytic fixed points and their stability. 
It is shown that 
the analytic fixed point controlling the critical behavior in the system with LRE 
becomes unstable against the perturbation with nonanalyticity 
for N<2(4+3√(3))≃18.3923⋯, 
which is the same as the case of the system with SR. 
We also show that the destabilization of the analytic fixed point 
which governs the phase transition in the system with LREF 
can occur due to the nonanalytic perturbation. 
As a result, we obtain a certain region in the plane of the parameters N and ρ  
where the analytic fixed points are singly unstable. 
In Sec. <ref>, 
we calculate the critical exponents η, η̅, and ν 
at the analytic fixed point 
which controls the critical behavior in the system with LRE. 
We reconsider the validity of the d→ d-σ dimensional reduction. 
We also present the result for the critical value σ_*. 
In Sec. <ref> 
we calculate the critical exponents η, η̅, and ν 
at the analytic fixed point which controls the critical behavior in the system with LREF. 
We show that d→ d-σ-ρ dimensional reduction 
breaks down within the present framework, as far as N is finite. 
Sec. <ref> summarizes our results. 







§ CRITICAL PHENOMENA AT ZERO TEMPERATURE OF LONG-RANGE CORRELATED RANDOM FIELD O(N) SPIN MODEL WITH SHORT-RANGE EXCHANGE INTERACTIONS IN 4+Ε DIMENSIONS

This section is intended as a reexamination of the critical phenomena 
at zero temperature of the long-range correlated random field O(N) spin model 
with short-range exchange interactions in 4+ϵ dimensions. 
We discuss the nature of analytic fixed points and their stability. 
And we calculate the critical exponents 
at the analytic fixed point which controls the critical behavior in the system with SR and with LRF. 


 §.§ Model

Let us consider an N-component vector spin system 
where an N-component vector spin S(x) 
with a fixed-length constraint S(x)^2=1 couples to a random field. 
In order to carry out the average over the random field, we use the replica method. 
The critical phenomena  
of the long-range correlated random field O(N) spin model with the short-range exchange interactions 
near lower critical dimension 
is described by the O(N) nonlinear-sigma model of the following replica partition function Z 
and effective action β H_ rep 

    Z   =   ∫∏_α=1^nD^αδ(^α(x)^2-1)
    e^-β H_ rep,
    β H_ rep   =   a^2-d/2T∫_x∑_α =1^n 
    S^α(x)·(-∂^2)S^α(x)
          -a^-d/2T^2∫_x∑_α,β^n 
    R_1
    (S^α(x)·S^β(x))
    
          -a^-d-ρ/2T^2∫_x,x'∑_α,β^n 
    g(x-x')R_2
    (S^α(x)·S^β(x')),

where a is the ultraviolet cutoff, and ∫_x:=∫ d^dx. 
The replica indices denoted 
by Greek indices take values α,β,… = 1,..., n. 
The first term in the action (<ref>) is the kinetic term 
which corresponds to the short-range exchange interactions between spins. 
The parameter T is the dimensionless temperature. 
The function 
R_i(S^α·S^β) (i=1,2)
represents the random field and all the random anisotropies, 
and is given by 

    R_i(S^α·S^β)
    =∑_r=1^∞Δ_i,r(S^α·S^β)^r.

Here Δ_i,r denotes 
the strength of the random field and the r-th rank random anisotropy 
(r=1 is the random field, 
and r=2 is the random second-rank anisotropy). 
The subscript i=1 corresponds to 
the uncorrelated random fields and random anisotropies, 
and the subscript i=2 corresponds to 
the long-range correlated random fields and random anisotropies with g(x-x')∼|x-x'|^-d+ρ. 
The lower critical dimension of this model is d_l=4+ρ. 
In the present study, we consider the case of 0≤ρ<ϵ. 



 §.§ One-loop beta functions and the zero-temperature fixed points

To perform the renormalization group transformation, 
we put each replicated vector spin S^α(x) 
as a combination of a slow field n_0^α(x) of the unit length 
and fast fields φ_i^α(x), i=1,…,N-1 such that 

    S^α(x)
       =   n_0^α(x)
    √(1-φ^α(x)^2)
    +φ^α(x)
       ≃   n_0^α(x)
    -1/2(φ^α(x)^2)n_0^α(x)
    +φ^α(x),
    
    φ^α(x)   =   ∑_i=1^N-1φ_i^α(x)e_i^α(x),

where the unit vectors _i^α(x) are perpendicular 
to each other and also to the vector 
_0^α(x). 
Integrating out the fast fields φ_i^α(x), 
and calculating the new replicated action 
β H_ rep' up to the second order of the perturbation expansion, 
we get the one-loop beta functions for T, R_1, and R_2, 
which have been obtained by Fedorenko and Kühnel <cit.>. 
The one-loop beta function for the temperature T is 


    ∂_tT
    =
    -(d-2)T+(N-2)T(T+R_1'(1)+R_2'(1)),

where ∂_t denotes a derivative with respect to t=log l 
with l being the length-scale parameter which increases toward the infrared direction. 
Here we have rescaled T, R_1, and R_2 by 2/((4π)^d/2Γ(d/2)). 
We find that T=0 is the fixed point, at which the parameter T is irrelevant for d>2. 
The one-loop beta functions at T=0 for R_1 and R_2 are 

    ∂_tR_1(z)
       =   -ϵ R_1(z)+2(N-2)(R_1'(1)+R_2'(1))R_1(z)-(N-1)z(R_1'(1)+R_2'(1))R_1'(z)
          +(1-z^2)(R_1'(1)+R_2'(1))R_1”(z)+1/2(N-2+z^2)(R_1'(z)+R_2'(z))^2
          -z(1-z^2)(R_1'(z)+R_2'(z))(R_1”(z)+R_2”(z))+1/2(1-z^2)^2(R_1”(z)+R_2”(z))^2,
    ∂_tR_2(ζ)
       =   -(ϵ-ρ)R_2(ζ)+2(N-2)(R_1'(1)+R_2'(1))R_2(ζ)
          -(N-1)ζ(R_1'(1)+R_2'(1))R_2'(ζ)+(1-ζ^2)(R_1'(1)+R_2'(1))R_2”(ζ),

where z=n_0^α(x)·n_0^β(x), 
ζ=n_0^α(x)·n_0^β(x'). 
Here we have put d=4+ϵ. 
Practically, the beta functions for the first and second derivatives of R_1 and R_2 play a central role 
in the critical phenomena at zero temperature near the lower critical dimension. 
The beta functions at T=0 for R_1', R_1”, R_2', and R_2” in d=4+ϵ are 

    ∂_tR_1'(z)
       =   -ϵ R_1'(z)+(N-3)(R_1'(1)+R_2'(1))R_1'(z)-(N+1)z(R_1'(1)+R_2'(1))R_1”(z)
          +(1-z^2)(R_1'(1)+R_2'(1))R_1”'(z)+z(R_1'(z)+R_2'(z))^2
          +(N-3+4z^2)(R_1'(z)+R_2'(z))(R_1”(z)+R_2”(z))-z(1-z^2)(R_1'(z)+R_2'(z))(R_1”'(z)+R_2”'(z))
          -3z(1-z^2)(R_1”(z)+R_2”(z))^2+(1-z^2)^2(R_1”(z)+R_2”(z))(R_1”'(z)+R_2”'(z)),
    
    ∂_tR_1”(z)
       =   -ϵ R_1”(z)-4(R_1'(1)+R_2'(1))R_1”(z)-(N+3)z(R_1'(1)+R_2'(1))R_1”'(z)
          +(1-z^2)(R_1'(1)+R_2'(1))R_1^(IV)(z)+(R_1'(z)+R_2'(z))^2
          +10z(R_1'(z)+R_2'(z))(R_1”(z)+R_2”(z))+(N-4+7z^2)(R_1'(z)+R_2'(z))(R_1”'(z)+R_2”'(z))
          -z(1-z^2)(R_1'(z)+R_2'(z))(R_1^(IV)(z)+R_2^(IV)(z))+(N-6+13z^2)(R_1”(z)+R_2”(z))^2
          -11z(1-z^2)(R_1”(z)+R_2”(z))(R_1”'(z)+R_2”'(z))+(1-z^2)^2(R_1”(z)+R_2”(z))(R_1^(IV)(z)+R_2^(IV)(z))
          +(1-z^2)^2(R_1”'(z)+R_2”'(z))^2,


    ∂_tR_2'(ζ)
       =   -(ϵ-ρ)R_2'(ζ)+(N-3)(R_1'(1)+R_2'(1))R_2'(ζ)-(N+1)ζ(R_1'(1)+R_2'(1))R_2”(ζ)
          +(1-ζ^2)(R_1'(1)+R_2'(1))R_2”'(ζ),
    
    ∂_tR_2”(ζ)
       =   -(ϵ-ρ)R_2”(ζ)-4(R_1'(1)+R_2'(1))R_2”(ζ)-(N+3)ζ(R_1'(1)+R_2'(1))R_2”'(ζ)
          +(1-ζ^2)(R_1'(1)+R_2'(1))R_1^(IV)(ζ).


The properties of the fixed point solution (R_1'(z)^*, R_2'(ζ)^*) are determined 
under the condition that |R_1'(z)| and |R_2'(ζ)| remain finite during the renormalization group flows.  
We discuss the properties of the fixed point solution (R_1'(z)^*,R_2'(ζ)^*). 
Eq.(<ref>) is linear in the function R_2'(ζ), which can be solved analytically. 
Solving the fixed point equation ∂_tR_2'(ζ)^*=0, 
we can find that the fixed point solution R_2'(ζ)^* is analytic on ζ. 
Next, we assume that the functions R_1'(z) and R_2'(ζ) take the following form: 

    R_1'(z)   =   R_1'(1)-R_1”(1)(1-z)+⋯+a_l(1-z)^α+⋯,
    
    R_2'(ζ)   =   R_2'(1)-R_2”(1)(1-ζ)+R_2”'(1)/2(1-ζ)^2+⋯,

with α>0. 
To keep |R_1'(z)| and |R_2'(ζ)| finite, 
the following condition on the function (<ref>) is required; 

    .

Thus, the fixed point function R_1(z)^* also has the same behavior of (1-z)^α^* 
with α^*=1/2 or α^*≥1. 
Only in the case of α^*=1/2, R_1”(1)^* diverges. 
We use the term “cuspy” on a function with (1-z)^1/2 
and “cuspless” if the first and the second derivatives of a function are finite. 




 §.§ Stability of fixed points and critical exponents η, η̅ and ν

The critical exponents η and η̅ 
of the connected and disconnected correlation functions 
are expressed by use of R_1'(1)^* and R_2'(1)^* which are the values of R_1'(1) and R_2'(1) 
at the fixed point: 

    η=R_1'(1)^*+R_2'(1)^*,
          η̅=(N-1)(R_1'(1)^*+R_2'(1)^*)-ϵ.

The critical exponent ν of the correlation length 
is given by the inverse of the maximal eigenvalue of the scaling matrix at the fixed point. 
Then, we find the fixed points 
by solving ∂_tR_1'(1)^*=0, ∂_tR_1”(1)^*=0, ∂_tR_2'(1)^*=0, and ∂_tR_2”(1)^*=0, 
study their stability, 
and calculate the critical exponents η, η̅, and ν in the following. 

The fixed points are 

    (R_1'(1)^*,R_2'(1)^*,R_1+”(1)^*,R_2”(1)^*)
       =   (
    ϵ/N-2,0,
    ϵ[N-8+√((N-2)(N-18))]/2(N+7)(N-2),0
    ),
    
    (R_1'(1)^*,R_2'(1)^*,R_1-”(1)^*,R_2”(1)^*)
       =   (
    ϵ/N-2,0,
    ϵ[N-8-√((N-2)(N-18))]/2(N+7)(N-2),0
    ),
    
    (R_1'(1)^*,R_2'(1)^*,R_1+”(1)^*,R_2”(1)^*)
       =   (
    (ϵ-ρ)^2/(N-3)^2ρ,
    (ϵ-ρ)^2/(N-3)^2ρ{(N-3)ϵ̂-(N-2)},
    
          (ϵ-ρ)
    [(N-3)ϵ̂-6
    +√({(N-3)ϵ̂-6}^2-4(N+7)
    )]
    /2(N+7)(N-3),
    0
    ),
    
    (R_1'(1)^*,R_2'(1)^*,R_1-”(1)^*,R_2”(1)^*)
       =   (
    (ϵ-ρ)^2/(N-3)^2ρ,
    (ϵ-ρ)^2/(N-3)^2ρ{(N-3)ϵ̂-(N-2)},
    
          (ϵ-ρ)
    [(N-3)ϵ̂-6
    -√({(N-3)ϵ̂-6}^2-4(N+7)
    )]
    /2(N+7)(N-3),
    0
    ).


Here we have introduced the reduced variable ϵ̂: 

    ϵ̂=ϵ/ϵ-ρ.

The stability of the cuspless fixed points with respect to the cuspless perturbation 
can be investigated by calculating eigenvalues of the 4× 4 scaling matrix whose elements are 
the first derivatives of the beta functions 
∂_tR_1'(1), ∂_tR_1”(1), ∂_tR_2'(1), and ∂_tR_2”(1)
at the cuspless fixed points. 

The cuspless fixed points (<ref>) and (<ref>) exist for N≥18. 
The eigenvalues λ_1,…,λ_4 of the scaling matrix at the cuspless fixed points 
(<ref>) and (<ref>) are given by 

    λ_1   =   ϵ,
    λ_2   =   (ϵ-ρ)(N-3)ϵ̂-(N-2)/N-2,
    
    λ_3^±   =   ±ϵ√(N-18/N-2),
    λ_4   =   -(ϵ-ρ+4ϵ/N-2),

Thus, the cuspless fixed point (<ref>) is multiply unstable. 
If 1<ϵ̂<(N-2)/(N-3), namely λ_2<0, 
the cuspless fixed point (<ref>) is singly unstable. 
Due to R_2'(1)^*=0, 
the long-range correlations of random fields and random anisotropies 
are irrelevant, 
and thus the cuspless fixed point (<ref>) governs the phase transition 
in the system with SR. 
The critical exponents η_ SR of the connected correlation function 
and η̅_ SR of the disconnected correlation function 
at the cuspless fixed point (<ref>) are 

    η_ SR   =   ϵ/N-2,
    η̅_ SR   =   ϵ/N-2.

And the critical exponent ν_ SR 
which characterizes the divergence of the correlation length 
in the vicinity of transition is 

    ν_ SR=1/ϵ.

Whereas in N≤18 
the cuspless fixed points (<ref>) and (<ref>) merge 
and annihilate, and thus 
the beta functions have no cuspless fixed point of O(ϵ). 

The cuspless fixed points (<ref>) and (<ref>) exist 
for N>3 and 

    ϵ̂≥ϵ̂_ cuspless
    =6+2√(N+7)/N-3.

The eigenvalues λ_1,…,λ_4 of the scaling matrix at the fixed points 
(<ref>) and (<ref>) are given by 

    λ_1   =   (ϵ-ρ)N-2/N-3-ϵ/2
          +ϵ/2√(1+4[N-2-ϵ̂(N-3)]/ϵ̂^2(N-3)^2),
    λ_2   =   (ϵ-ρ)N-2/N-3-ϵ/2
          -ϵ/2√(1+4[N-2-ϵ̂(N-3)]/ϵ̂^2(N-3)^2),
    
    λ_3^±   =   ±(ϵ-ρ)√({(N-3)ϵ̂-6}^2-4(N+7))/N-3,
    
    λ_4   =   -(ϵ-ρ)N+1/N-3,

Thus, the cuspless fixed point (<ref>) is multiply unstable. 
If ϵ̂>(N-2)/(N-3), namely λ_2<0, 
the cuspless fixed point (<ref>) is singly unstable. 
Due to R_2'(1)^*>0, 
the effect of the long-range correlation of random fields and random anisotropies appears, 
and then the cuspless fixed point (<ref>) governs the phase transition 
in the system with LRF. 
Thus, the critical exponents η_ LRF and η̅_ LRF 
at the cuspless fixed point (<ref>) are 

    η_ LRF   =   ϵ-ρ/N-3,
    η̅_ LRF   =   2ϵ-(N-1)ρ/N-3.

These exponents satisfy the Schwartz-Soffer inequality 
η̅_ LRF≤ 2η_ LRF <cit.>,  
and saturate the generalized Schwartz-Soffer inequality 
η̅_ LRF≤ 2η_ LRF-ρ <cit.>.  
And the inverse of the exponent ν_ LRF is 

    ν_ LRF^-1   =   (N-2)(ϵ-ρ)/N-3[
    1-(N-3)ϵ̂/2(N-2)
          +(N-3)ϵ̂/2(N-2)√(1+4[N-2-ϵ̂(N-3)]/ϵ̂^2(N-3)^2)].

Whereas in ϵ̂≤(6+2√(N+7))/(N-3) 
the cuspless fixed points (<ref>) and (<ref>) merge and annihilate, 
and thus the beta functions have no cuspless fixed point of O(ϵ). 

As Tissier and Tarjus (TT) and co-workers argued in Ref. TT1,TT2,TT3,TT4,BTTB, 
the cuspless fixed points (<ref>) and (<ref>) have weaker nonanalyticities 
(1-z)^α^* with a noninteger α^*≥1. 
The weaker nonanalyticity is called “subcusp”. 
We refer to the cuspless fixed points (<ref>) and (<ref>) 
as “SR TT FP” and “LRF TT FP”, respectively. 
The weaker nonanalyticity does not alter the flow equations 
for R_1'(1) and R_2'(1). 
The power α^* is obtained as follows. 
Calculating the flow of a_l in Eq. (<ref>), 
we have 

    ∂_ta_l=a_lΛ_α+1(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*),
          Λ_α+1(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)
          =2[R_1'(1)^*+R_2'(1)^*+3(R_1”(1)^*+R_2”(1)^*)]α^2
          -[(N-5)(R_1'(1)^*+R_2'(1)^*)
            -(N+7)(R_1”(1)^*+R_2”(1)^*)]α
          +(N-1)(R_1'(1)^*+R_2'(1)^*)
          +(N+1)(R_1”(1)^*+R_2”(1)^*)-ϵ.

The power α^* is determined from 

    Λ_α^*+1(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)=0.

Substituting the SR TT FP (<ref>) 
and the LRF TT FP (<ref>) 
into the above equation, we have explicit expressions for α^*, respectively. 
Here we treat only the LRF case (see Ref. TT3 for the SR case). 
From Eqs.(<ref>) and (<ref>), 
we obtain the following quadratic equation for α^*: 

    (
    2+3/N+7
    [(N-3)ϵ̂-6
          -√({(N-3)ϵ̂-6}^2-4(N+7))]
    )α^*^2
          -(
    N-5-1/2
    [(N-3)ϵ̂-6
          -√({(N-3)ϵ̂-6}^2-4(N+7))]
    )α^*
          +N-1-(N-3)ϵ̂
          +N+1/2(N+7)
    [(N-3)ϵ̂-6
          -√({(N-3)ϵ̂-6}^2-4(N+7))]=0.

Solving the above quadratic equation, we obtain the solution α^* 
as a function of N and ϵ̂. 
It goes to N/2+ O(1) at large N. 
The graphs of α^*=α(N,ϵ̂)^* for some values of ϵ̂ 
are depicted in Fig. <ref>. 




We proceed to investigate the stability of the cuspless fixed points 
with respect to the cuspy perturbation, 
following the work by Baczyk, Tarjus, Tissier, and Balog <cit.>. 
The eigenvalue λ relating to the cuspy deformation 
from the cuspless fixed points is given by 

    λ
    =Λ_3/2(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)

Substituting the SR TT FP (<ref>) 
and the LRF TT FP (<ref>) 
into the above equation, we have explicit expressions for λ, respectively. 
The eigenvalues λ_ SR for the system with SR 
and λ_ LRF for the system with LRF are as follows: 

    λ_ SR   =   -ϵ/4(N+7)[3(N+4)√(N-18/N-2)-N+8],
    λ_ LRF   =   -ϵ-ρ/4(N-3)(N+7)
          ×[3(N+4)√({(N-3)ϵ̂-6}^2-4(N+7))
          +(N+16){(N-3)ϵ̂-6}-2(N+7)(N-8)].


In the case of the system with SR, we find that, 
below N=N_ cusp=2(4+3√(3))≃18.3923…, 
the eigenvalue λ_ SR takes a positive value. 
Thus the cuspy perturbation becomes relevant for N<N_ cusp, 
where the SR TT FP (<ref>) is multiply unstable with respect to the cuspy perturbation. 
Whereas it remains singly unstable with respect to the perturbation with and without the cuspy behavior 
for N>N_ cusp. 
As shown in Ref. BTTB, 
there exists a singly unstable cuspy fixed point below N=N_ cusp. 
As N decreases from sufficiently large N, 
the fixed point which governs the phase transition in the system continuously changes 
from the SR TT FP to the singly unstable cuspy fixed point at N=N_ cusp 
before N reaches to N=18. 
Accordingly, the values of the critical exponents η_ SR and η̅_ SR  
deviate from the dimensional-reduction results (<ref>) and (<ref>) 
below N=N_ cusp. 

In the case of the system with LRF, the eigenvalue λ_ LRF takes a positive value 
below 

    ϵ̂=ϵ̂_ cusp
          =3(N+4)√(N^2-8N+48)-(N+4)(N-20)/4(N-2)(N-3).

Since ϵ̂_ cusp≥ϵ̂_ cuspless 
for 4(1+√(7))≤ N ≤ 2(4+3√(3)), 
the LRF TT FP (<ref>) is destabilized by the cuspy perturbation 
for 4(1+√(7))≤ N ≤ 2(4+3√(3)) and ϵ̂<ϵ̂_ cusp. 
Even in this case, a singly unstable cuspy fixed point which governs the phase transition in the system 
is considered to exist 
for 4(1+√(7))≤ N ≤ 2(4+3√(3)) and ϵ̂<ϵ̂_ cusp. 

Finally, we calculate the eigenfunction 
which belongs to the eigenvalue (<ref>). 
Solving the eigenvalue equation, 
we obtain two solutions. 
One takes the form of (1-z)^α_-(λ) with α_-(λ)=1/2 when z→ 1, 
and the other takes the form of (1-z)^α_+(λ). 
Both solutions individually diverge in z=-1. 
The physical eigenfunction is represented as a linear combination 
of two solutions of the eigenvalue equation, 
in which the coefficients should be chosen to eliminate the singularities at z=-1. 
The power α_+(λ) of the function (1-z)^α_+(λ) 
can be obtained by imposing 

    Λ_α_++1(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)
          =Λ_3/2(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*).


In the case of the system with SR, 
substituting the SR TT FP (<ref>) into Eq. (<ref>), 
we have 

    α_+(λ_ SR)=1/4(N-10+√((N-2)(N-18))).

For N≥ 18, α_+(λ_ SR) takes α_+(λ_ SR)≥ 2. 

In the case of the system with LRF, 
substituting the LRF TT FP (<ref>) into Eq. (<ref>), 
we have 

    α_+(λ_ LRF)
       =   (N-14)(N-3)ϵ̂+(N-2)(N-6)/2[3(N-3)ϵ̂+N-2]
          +√({(N-3)ϵ̂-6}^2-4(N+7))/3(N-3)ϵ̂+N-2.
 
The power α_+(λ_ LRF) takes 
α_+(λ_ LRF)≥ 1+√(3) 
for N≥ N_ cusp and ϵ̂≥(N-2)/(N-3), 
and α_+(λ_ LRF)≥ 1 
for 4(1+√(7))≤ N<N_ cusp and ϵ̂≥ϵ̂_ cuspless. 
However, we should note that, 
for N<4(1+√(7)) and in the region of 

    ϵ̂_ cuspless≤ϵ̂<(N-2)(N^2+32)/(N-3)(N-8)(N+16),

α_+(λ_ LRF)<1, 
which is in contradiction with the condition (<ref>). 
Thus, for N<4(1+√(7)) and in the region (<ref>), 
the cuspy deformation from the LRF TT FP (<ref>) is unphysical. 
Then, the destabilization of the LRF TT FP (<ref>) by the cuspy perturbation does not occur 
for ϵ̂>2(23+19√(7))/111. 

The regions where the various fixed points are singly unstable are depicted 
in Fig. <ref>. 
Outside the areas where the SR TT and the LRF TT FPs are singly unstable, 
the cuspy fixed point is considered to control the critical behavior in the system. 
Particularly, in the region of 1≤ϵ̂<2(23+19√(7))/111≃1.32, 
the destabilization of the SR TT and the LRF TT FPs by the cuspy perturbation is caused 
at N_ cusp for the SR TT FP, and at ϵ̂_ cusp for the LRF TT FP, 
respectively. 










§ FIXED POINTS AND THEIR STABILITY IN THE RENORMALIZATION GROUP 
OF LONG-RANGE CORRELATED RANDOM FIELD O(N) SPIN MODEL 
WITH LONG-RANGE EXCHANGE INTERACTIONS IN 2Σ+Ρ+Ε DIMENSIONS

We now study the critical phenomena at zero temperature 
of the long-range correlated random field O(N) spin model 
with the long-range exchange interactions in 2σ+ρ+ϵ dimensions 
by use of the renormalization group. 
The critical phenomena at zero temperature 
of the long-range correlated random field O(N) spin model 
with the long-range exchange interactions near lower critical dimension 
is described by the O(N) nonlinear-sigma model. 
In this section 
we investigate the fixed points and their stability of the one-loop beta functions 
in the O(N) nonlinear-sigma model. 
The critical phenomena are carefully discussed in the subsequent sections. 


 §.§ Model

We start from the O(N) nonlinear-sigma model with the replica effective action  

    β H_ rep   =   a^σ-d/2T∫_x∑_α =1^n 
    S^α(x)·(-∂^2)^σ/2S^α(x)
          -a^-d/2T^2∫_x∑_α,β^n 
    R_1
    (S^α(x)·S^β(x))
    
          -a^-d-ρ/2T^2∫_x,x'∑_α,β^n 
    g(x-x')R_2
    (S^α(x)·S^β(x')).

The first term in the action (<ref>) is the kinetic term 
which corresponds to the long-range exchange interactions between spins. 
The operator (-∂^2)^σ/2 denotes the fractional Laplacian in the Euclidean space.  
In the present study we consider the case of 0<σ<2. 
The parameter T denotes the dimensionless temperature. 
The function 
R_i(z) (i=1,2)
represents the random field and all the random anisotropies, 
which is defined by Eq.(<ref>).  
The lower critical dimension of this model is d_l=2σ+ρ. 
In the present study, we consider the case of ρ≥0. 





 §.§ One-loop beta functions

To carry out the renormalization group transformation, 
it is convenient to use the momentum representation. 
The fractional Laplacian (-∂^2)^σ/2 is written by its Fourier transformation: 

    (-∂^2)^σ/2f(x)=∫_kk^σf̃(k)e^ikx,


where 
kx=k^(1)x^(1)+⋯+k^(d)x^(d), 
k^σ=(k^(1)^2+⋯+k^(d)^2)^σ/2, 
and ∫_k≡∫d^dk/(2π)^d. 
The correlation of the random fields g(x-x') is written as 

    g(x-x')∼1/|x-x'|^d-ρ
    =∫_kk^-ρe^ik(x-x'),

in the momentum representation. 
The N-component replicated vector spin ^α(x) 
of the magnetization (<ref>) 
is rewritten in the momentum representation as follows: 

    ^α(x)
       ≃   _0^α(x)
    -1/2(^α(x)^2)
    _0^α(x)
    +^α(x)
       =   ∫_k_0^α(k)e^ikx
    -1/2∫_k,k_1,⋯,k_4(∑_i,j^N-1φ̃_i^α(k_1)φ̃_j^α(k_2)
    _i^α(k_3)·_j^α(k_4)
    )_0^α(k)
    e^i(k_1+⋯+k_4+k)x
          +∫_k_1,k_2(∑_i=1^N-1φ̃_i^α(k_1)
    _i^α(k_2)
    )
    e^i(k_1+k_2)x.

We integrate out the fast fields φ̃_i^α(k), 
and calculate the new replicated action 
β H_ rep' up to the second order of the perturbation expansion. 
After rewriting β H_ rep' in the coordinate representation again, 
we can then obtain the one-loop beta functions for T, R_1, and R_2. 
The one-loop beta function for the temperature T is 

    ∂_tT=
    -(d-σ)T+(N-1)T(T+R_1'(1)+R_2'(1)).

Here we have rescaled T, R_1, and R_2 by 2/((4π)^d/2Γ(d/2)). 
For d>σ, 
we find that T=0 is the fixed point, 
at which the parameter T is irrelevant. 
The one-loop beta functions at T=0 for R_1 and R_2 are

    ∂_tR_1(z)
       =   -(ϵ+ρ)R_1(z)+2(N-1)(R_1'(1)+R_2'(1))R_1(z)-(N-1)z(R_1'(1)+R_2'(1))R_1'(z)
          +(1-z^2)(R_1'(1)+R_2'(1))R_1”(z)+1/2(N-2+z^2)(R_1'(z)+R_2'(z))^2
          -z(1-z^2)(R_1'(z)+R_2'(z))(R_1”(z)+R_2”(z))+1/2(1-z^2)^2(R_1”(z)+R_2”(z))^2,
    
    ∂_tR_2(ζ)
       =   -ϵ R_2(ζ)+2(N-1)(R_1'(1)+R_2'(1))R_2(ζ)-(N-1)ζ(R_1'(1)+R_2'(1))R_2'(ζ)
          +(1-ζ^2)(R_1'(1)+R_2'(1))R_2”(ζ).

Here we have put d=2σ+ρ+ϵ. 
To study the fixed points and their stability, 
we consider the beta functions for their derivative. 
Differentiating Eqs.(<ref>) and (<ref>) 
with respect to z and ζ, respectively, 
we obtain the one-loop beta functions for R_1'(z), R_1”(z), R_2'(ζ), 
and R_2”(ζ); 

    ∂_tR_1'(z)
       =   -(ϵ+ρ) R_1'(z)+(N-1)(R_1'(1)+R_2'(1))R_1'(z)-(N+1)z(R_1'(1)+R_2'(1))R_1”(z)
          +(1-z^2)(R_1'(1)+R_2'(1))R_1”'(z)+z(R_1'(z)+R_2'(z))^2
          +(N-3+4z^2)(R_1'(z)+R_2'(z))(R_1”(z)+R_2”(z))-z(1-z^2)(R_1'(z)+R_2'(z))(R_1”'(z)+R_2”'(z))
          -3z(1-z^2)(R_1”(z)+R_2”(z))^2+(1-z^2)^2(R_1”(z)+R_2”(z))(R_1”'(z)+R_2”'(z)),
    
    ∂_tR_1”(z)
       =   -(ϵ+ρ) R_1”(z)-2(R_1'(1)+R_2'(1))R_1”(z)-(N+3)z(R_1'(1)+R_2'(1))R_1”'(z)
          +(1-z^2)(R_1'(1)+R_2'(1))R_1^(IV)(z)+(R_1'(z)+R_2'(z))^2
          +10z(R_1'(z)+R_2'(z))(R_1”(z)+R_2”(z))+(N-4+7z^2)(R_1'(z)+R_2'(z))(R_1”'(z)+R_2”'(z))
          -z(1-z^2)(R_1'(z)+R_2'(z))(R_1^(IV)(z)+R_2^(IV)(z))+(N-6+13z^2)(R_1”(z)+R_2”(z))^2
          -11z(1-z^2)(R_1”(z)+R_2”(z))(R_1”'(z)+R_2”'(z))+(1-z^2)^2(R_1”(z)+R_2”(z))(R_1^(IV)(z)+R_2^(IV)(z))
          +(1-z^2)^2(R_1”'(z)+R_2”'(z))^2,
    
    ∂_tR_2'(ζ)
       =   -ϵ R_2'(ζ)+(N-1)(R_1'(1)+R_2'(1))R_2'(ζ)-(N+1)ζ(R_1'(1)+R_2'(1))R_2”(ζ)
          +(1-ζ^2)(R_1'(1)+R_2'(1))R_2”'(ζ),
    
    ∂_tR_2”(ζ)
       =   -ϵ R_2”(z)-2(R_1'(1)+R_2'(1))R_2”(ζ)-(N+3)ζ(R_1'(1)+R_2'(1))R_2”'(ζ)
          +(1-ζ^2)(R_1'(1)+R_2'(1))R_1^(IV)(ζ).


We discuss the properties of the fixed point solution (R_1'(z)^*,R_2'(ζ)^*). 
First, we investigate the fixed point solution R_2'(ζ)^* 
for Eq.(<ref>). 
Since Eq.(<ref>) is linear in the function R_2'(ζ), 
the fixed point equation ∂_tR_2'(ζ)^*=0 can be solved analytically. 
The fixed point equation ∂_tR_2'(ζ)^*=0 
takes the form 

    (1-ζ^2)(R_1'(1)^*+R_2'(1)^*)R_2”'(ζ)^*
    -(N+1)ζ(R_1'(1)^*+R_2'(1)^*)R_2”(ζ)^*
    +[(N-1)(R_1'(1)^*+R_2'(1)^*)-ϵ]R_2'(ζ)^*=0.


The solutions of this equation have regular singular points at ζ=±1 
for the interval -1≤ζ≤1. 
Under the condition of |R_2'(ζ)^*|<∞ on the interval -1≤ζ≤1, 
the solutions of Eq.(<ref>) can be expressed 
in terms of the Gaussian hypergeometric function: 

    R_2'(ζ)^*
          ={[  C_2F_1(x_1,x_2;y;(1-ζ)/2)                           ; C'_2F_1(x_1,x_2;y;(1+ζ)/2)                            ].,

where C and C' are constants 
fulfilling the condition |R_2'(ζ)^*|<∞. 
Here, the generalized hypergeometric function is defined by the following series expansion: 

    _mF_n(x_1,x_2,…,x_m;y_1,y_2,…,y_n;z)
          ≡∑_k=0^∞(x_1)_k(x_2)_k⋯(x_m)_k/(y_1)_k(y_2)_k⋯(y_n)_kz^k/k!,
          (x)_k=Γ(x+k)/Γ(x).

And, x_1, x_2 and y are 

    x_1,x_2
       =   1/2[
    N±√(N^2+4{N-1-ϵ/R_1'(1)^*+R_2'(1)^*})],
    
    
    y   =   N+1/2.

Thus the fixed point solution R_2'(ζ)^* is an analytic function on ζ. 
Next we examine the renormalization group flow of R_1'(z). 
We assume that the functions R_1'(z) and R_2'(ζ) take the forms 
given by Eqs.(<ref>) and (<ref>) with α>0. 
To keep |R_1'(z)| and |R_2'(ζ)| finite, 
the following condition on the function (<ref>) is required; 

    .

The fixed point solution R_1'(z)^* also has the same singularity. 
Only in the case of α=1/2, R”(1)^* diverges. 





 §.§ Stability of cuspless fixed points

The critical exponents η and η̅ are expressed 
by use of R_1'(1)^* and R_2'(1)^*: 

    η   =   2-σ,
    η̅   =   (N-1)(R_1'(1)^*+R_2'(1)^*)
          -(2σ+ρ+ϵ-4).

The critical exponent ν is determined from the inverse of the maximum eigenvalue 
of the 4×4 scaling matrix at the fixed point. 
Then, we find the fixed points by solving 
∂_tR_1'(1)^*=0, ∂_tR_1”(1)^*=0, 
∂_tR_2'(1)^*=0, and ∂_tR_2”(1)^*=0, 
and study their stability. 

The cuspless fixed points are 

    (R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)
          =(
    ϵ+ρ/N,0,
    (ϵ+ρ)[N-8+√((N-2)(N-18))]/2N(N+7),0
    ),


    (R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)
          =(
    ϵ+ρ/N,0,
    (ϵ+ρ)[N-8-√((N-2)(N-18))]/2N(N+7),0
    ),


    (R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)
          =(
    ϵ^2/(N-1)^2ρ,
    ϵ^2[(N-1)ϵ̃-N]/(N-1)^2ρ,
    
          ϵ
    [(N-1)ϵ̃-8
    +√({(N-1)ϵ̃-8}^2-4(N+7))]
    /2(N+7)(N-1),
    0
    ),


    (R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)
          =(
    ϵ^2/(N-1)^2ρ,
    ϵ^2[(N-1)ϵ̃-N]/(N-1)^2ρ,
    
          ϵ
    [(N-1)ϵ̃-8
    -√({(N-1)ϵ̃-8}^2-4(N+7))]
    /2(N+7)(N-1),
    0
    ).

Here we have introduced the reduced variable ϵ̃: 

    ϵ̃=ϵ+ρ/ϵ.

The stability of the cuspless fixed points with respect to the cuspless perturbation 
can be investigated by calculating eigenvalues of the 4× 4 scaling matrix 
whose elements are the first derivatives of the beta functions 
∂_tR_1'(1), ∂_tR_1”(1), 
∂_tR_2'(1), and ∂_tR_2”(1) 
at the cuspless fixed points. 

The cuspless fixed points (<ref>) and (<ref>) exist for N≥18. 
The eigenvalues λ_1,…,λ_4 of the scaling matrix 
at the cuspless fixed points (<ref>) and (<ref>)
are given by 

    λ_1   =   ϵ+ρ,
    λ_2   =   -ϵ(1-N-1/Nϵ̃),
    
    λ_3^±   =   ±(ϵ+ρ)√((N-2)(N-18))/N,
    
    λ_4   =   -ϵ(1+2/Nϵ̃),

Thus the fixed point (<ref>) is multiply unstable. 
If 1≤ϵ̃<N/(N-1) or 0≤ρ<ϵ/(N-1), 
the fixed point (<ref>) is singly unstable.  
Due to R_2'(1)^*=0, the long-range correlations of random fields and random anisotropies 
are irrelevant, 
and thus the fixed point (<ref>) governs the phase transition 
in the system with LRE. 
Whereas, in N≤ 18, 
the cuspless fixed points (<ref>) and (<ref>) merge and annihilate, 
and thus the beta functions have no cuspless fixed point of O(ϵ). 

The cuspless fixed points (<ref>) and (<ref>) exist for 
N>1 and 

    ϵ̃≥ϵ̃_ cuspless=8+2√(N+7)/N-1.

The eigenvalues λ_1,…,λ_4 of the scaling matrix 
at the cuspless fixed points (<ref>) and (<ref>)
are given by 

    λ_1
       =   ϵ[
    N/N-1
    -ϵ̃/2
    +ϵ̃/2√(1+4[N-ϵ̃(N-1)]/ϵ̃^2(N-1)^2)],
    λ_2
       =   ϵ[
    N/N-1
    -ϵ̃/2
    -ϵ̃/2√(1+4[N-ϵ̃(N-1)]/ϵ̃^2(N-1)^2)],
    λ_3^±   =   ±(ϵ+ρ)
    √(1-4[N-9+4ϵ̃(N-1)]/ϵ̃^2(N-1)^2),
    
    λ_4
       =   -ϵN+1/N-1,

Thus the fixed point (<ref>) is multiply unstable. 
If ϵ̃>N/(N-1) or ρ>ϵ/(N-1), 
the fixed point (<ref>) is singly unstable. 
Due to R_2'(1)^*>0, 
the effect of the long-range correlation of random fields and random anisotropies 
appears, 
and then the fixed point (<ref>) governs the phase transition 
in system with LREF. 

The cuspless fixed points (<ref>) and (<ref>) 
have subcuspy singularities (1-z)^α_* with a noninteger α_*≥1. 
Then we call the singly unstable fixed points (<ref>) and (<ref>) 
as the “LRE TT FP” and the “LREF TT FP” respectively. 
The power α_* is obtained as follows. 
Calculating the flow of a_l in Eq.(<ref>), we have  

    ∂_ta_l=a_lΛ_α+1(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*),
          Λ_α+1(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)
          =2[R_1'(1)^*+R_2'(1)^*+3(R_1”(1)^*+R_2”(1)^*)]α^2
          -[(N-5)(R_1'(1)^*+R_2'(1)^*)
            -(N+7)(R_1”(1)^*+R_2”(1)^*)]α
          +(N+1)(R_1'(1)^*+R_2'(1)^*+R_1”(1)^*+R_2”(1)^*)
            -(ϵ+ρ).

The power α^* is determined from 

    Λ_α^*+1(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)=0.

Substituting the LRE TT FP (<ref>) 
and the LREF TT FP (<ref>) 
into the above equation, we have explicit expressions for α^*, respectively. 
Firstly, substituting the LRE TT FP (<ref>) 
into Eq.(<ref>), 
we obtain the following quadratic equation for α^*: 

    (2+3/N+7[N-8-√((N-2)(N-18))])α^*^2
          -(N-5-1/2[N-8-√((N-2)(N-18))])α^*
          +1+N+1/2(N+7)[N-8-√((N-2)(N-18))]=0.

Solving the above equation, we have the solution α^* as a function of N. 
It goes to N/2+ O(1) at large N. 
The solution is the same as that of the system with SR <cit.>. 
The graph of α^*=α^*(N) is depicted in Fig. <ref>. 



Next, substituting the LREF TT FP (<ref>) 
into Eq.(<ref>), 
we obtain the following quadratic equation for α^*: 

    (
    2+3/N+7[(N-1)ϵ̃-8
            -√({(N-1)ϵ̃-8}^2-4(N+7))]
    )α^*^2
          -(
    N-5-1/2[(N-1)ϵ̃-8
            -√({(N-1)ϵ̃-8}^2-4(N+7))]
    )α^*
          +N+1-(N-1)ϵ̃
    +N+1/2(N+7)[(N-1)ϵ̃-8
            -√({(N-1)ϵ̃-8}^2-4(N+7))]=0.

Solving the above equation, we obtain the solution α^* as a function of N and ϵ̃. 
It goes to N/2+ O(1) at large N. 
The graphs of α^*=α(N,ϵ̃)^* for some values of ϵ̃ 
are depicted in Fig. <ref>. 




We investigate the stability of the TT fixed points with respect to the cuspy perturbation.  
It can be done by calculating the eigenvalue λ 
relating to the cuspy deformation from the TT fixed point, 
which is given by 

    λ
    =Λ_3/2(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*).

Substituting the LRE TT FP (<ref>) 
and the LREF TT FP (<ref>) 
into the above equation, we obtain explicit expressions for λ, respectively. 
The eigenvalues λ_ LRE for the system with LRE
and λ_ LREF for the system with LREF 
are as follows: 

    λ_ LRE   =   -(ϵ+ρ)(N-2)/4N(N+7)
          ×[3(N+4)√(N-18/N-2)-N+8].
    λ_ LREF   =   -ϵ/4(N-1)(N+7)
          ×[
    3(N+4)√([(N-1)ϵ̃-8]^2-4(N+7))
          -2(N+7)(N-8)+(N+16)[(N-1)ϵ̃-8]].


In the case of the system with LRE, we find that, 
below N=N_ cusp=2(4+3√(3))≃18.3923…, 
the eigenvalue λ_ LRE takes a positive value. 
Thus the cuspy perturbation becomes relevant for N<N_ cusp, 
where the LRE TT FP (<ref>) is multiply unstable 
with respect to the cuspy perturbation. 
Whereas it remains singly stable 
with respect to the perturbation with and without the cuspy behavior for N>N_ cusp. 

In the case of the system with LREF, 
the eigenvalue λ_ LREF takes a positive value below 

    ϵ̃   =   ϵ̃_ cusp
       =   3(N+4)√(N^2-8N+48)-(N^2-24N-64)/4(N-1)(N-2).

Since ϵ̃_ cusp≥ϵ̃_ cuspless 
for 4(1+√(7))≤ N ≤ 2(4+3√(3)), 
the LREF TT FP (<ref>) is destabilized by the cuspy perturbation 
for 4(1+√(7))≤ N ≤ 2(4+3√(3)) and ϵ̃<ϵ̃_ cusp. 
Even in this case, a singly unstable cuspy fixed point which governs the phase transition in the system 
is considered to exist 
for 4(1+√(7))≤ N ≤ 2(4+3√(3)) and ϵ̃<ϵ̃_ cusp. 

Finally, we calculate the eigenfunction 
which belongs to the eigenvalue (<ref>). 
Solving the eigenvalue equation, 
we obtain two solutions. 
One takes the form of (1-z)^α_-(λ) with α_-(λ)=1/2 when z→ 1, 
and the other takes the form of (1-z)^α_+(λ). 
Both solutions individually diverge in z=-1. 
The physical eigenfunction is represented as a linear combination 
of two solutions of the eigenvalue equation, 
in which the coefficients should be chosen to eliminate the singularities at z=-1. 
The power α_+(λ) of the function (1-z)^α_+(λ) 
can be obtained by imposing 

    Λ_α_++1(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*)
          =Λ_3/2(R_1'(1)^*,R_2'(1)^*,R_1”(1)^*,R_2”(1)^*).


In the case of the system with LRE, 
substituting the LRE TT FP (<ref>) into Eq. (<ref>), 
we have 

    α_+(λ_ LRE)=1/4(N-10+√((N-2)(N-18))).

For N≥18, α_+(λ_ LRE) takes α_+(λ_ LRE)≥2. 

In the case of the system with LREF, 
substituting the LREF TT FP (<ref>) into Eq. (<ref>), 
we have 

    α_+(λ_ LREF)
       =   (N-14)(N-1)ϵ̃+N^2-10N+40/2[3(N-1)ϵ̃+N-8]
          +(N-2)√({(N-1)ϵ̃-8}^2-4(N+7))/3(N-1)ϵ̃+N-8.

The power α_+(λ_ LREF) takes 
α_+(λ_ LREF)≥ 1+√(3) 
for N≥ N_ cusp and ϵ̃≥ N/(N-1), 
and α_+(λ_ LREF)≥ 1 
for 4(1+√(7))≤ N<N_ cusp and ϵ̃≥ϵ̃_ cuspless. 
However, we should note that, 
for N<4(1+√(7)) and in the region of 

    ϵ̃_ cuspless≤ϵ̃<N^3+48N-320/(N-1)(N-8)(N+16),

α_+(λ_ LREF)<1, 
which is in contradiction with the condition (<ref>). 
Thus, for N<4(1+√(7)) and in the region (<ref>), 
the cuspy deformation from the LREF TT FP (<ref>) is unphysical. 
Then, the destabilization of the LREF TT FP (<ref>) by the cuspy perturbation 
does not occur for ϵ̃>2(10+21√(7))/103. 

The regions where the various fixed points are singly unstable are depicted 
in Fig. <ref>. 
Outside the areas where the LRE TT and the LREF TT FPs are singly unstable, 
the cuspy fixed point is considered to control the critical behavior in the system. 
Particularly, in the region of 1≤ϵ̃<2(10+21√(7))/103≃1.27, 
the destabilization of the LRE TT and the LREF TT FPs by the cuspy perturbation is caused 
at N_ cusp for LRE TT FP, and at ϵ̃_ cusp for the LREF TT FP, respectively. 





In the following sections, we carefully examine the critical phenomena governed by 
the LRE TT FP (<ref>) and the LREF TT FP (<ref>). 







§ CRITICAL PHENOMENA IN THE SYSTEM WITH LRE IN 2Σ+Ρ+Ε DIMENSIONS

In this section we study the critical phenomena controlled by the LRE TT FP (<ref>). 
We calculate the critical exponents η, η̅ and ν at O(ϵ). 
We put ρ=0, and investigate the d→ d-σ dimensional reduction. 
We also discuss the relations between η and η̅, 
and present the result for the critical value σ_*. 

Substituting the LRE TT FP (<ref>) into Eqs.(<ref>) and (<ref>), 
we obtain the critical exponents η_ LRE and η̅_ LRE 
at the LRE TT fixed point (<ref>): 

    η_ LRE   =   2-σ,
    η̅_ LRE   =   4-2σ-ϵ+ρ/N.

These exponents satisfy η_ LRE≥(4-d)/2, 
η̅_ LRE≥4-d, and the Schwartz-Soffer inequality 
η̅_ LRE≤ 2η_ LRE. 
In the large N limit, the relation 
between η_ LRE and η̅_ LRE 
satisfies η̅_ LRE=2η_ LRE, 
which is identical to the result of the previous study for the critical properties 
of the random field spherical model by Vojta and Schreiber <cit.>. 
For finite N but N>N_ cusp, the relation 
between η_ LRE and η̅_ LRE 
satisfies 2η_ LRE-η̅_ LRE
=(ϵ+ρ)/N for ϵ=d-2σ-ρ. 
Our result is consistent with the result of 1/N expansion study by Bray <cit.>.  
He showed 2η_ LRE-η̅_ LRE=ϵ/N
for ϵ=d-2σ by the use of the 1/N expansion. 
Thus, the relation 2η_ LRE-η̅_ LRE=(d-2σ)/N 
holds in the region where the scaling behavior in the system is controlled by the LRE TT fixed point. 

We turn to compute the exponent ν_ LRE of the correlation length. 
The critical exponent ν_ LRE is determined from the inverse of 
the maximal eigenvalue given by Eq.(<ref>). 
Thus, we obtain the inverse of the critical exponent ν_ LRE as 

    ν_ LRE^-1=ϵ+ρ.


If we put ρ=0, the spatial dimension in the present system becomes d=2σ+ϵ, 
and then ν_ LRE^-1 is 

    ν_ LRE^-1=ϵ,

which is in agreement with that in the pure long-range system in σ dimensions less <cit.>.  
Therefore, the d→ d-σ dimensional reduction holds at O(ϵ), 
and its validity is recognized only for N>N_ cusp. 

The relation between η_ LRE and η̅_ LRE is classified on the basis of the value of σ as follows: 

    1.  σ<2-ϵ+ρ/N: η̅_ LRE>η_ LRE,
          2.  σ=2-ϵ+ρ/N: η̅_ LRE=η_ LRE,
          3.  σ>2-ϵ+ρ/N: η̅_ LRE<η_ LRE,

for N>N_ cusp. 
Since η_ LRE≤η̅_ LRE≤ 2η_ LRE, 
the case 3 is unphysical. 
Thus, the critical value σ=σ_* 
which separates between the long-range and the short-range exchange regimes of the theory is

    σ_*=2-ϵ+ρ/N.

Here we comment on the critical value σ_*. 
If σ>2-(ϵ+ρ)/2, 
the spatial dimension in the present system is above four. 
Then we put d=2σ+ρ+ϵ=4+ϵ' (0<ϵ'≪1). 
The critical value σ_* is rewritten in terms of ϵ' as follows: 

    σ_*=2-ϵ'/N-2.

Since the exponent η of the random field O(N) spin model 
with SR in 4+ϵ' dimensions is η=η_ SR=ϵ'/(N-2) 
at O(ϵ') and for N>N_ cusp, 
our result confirms that the critical value σ_* 
which separates between the long-range and the short-range exchange regimes of the theory is 

    σ_*=2-η_ SR.








§ CRITICAL PHENOMENA IN SYSTEM WITH LREF IN 2Σ+Ρ+Ε DIMENSIONS

In this section we study the critical phenomena controlled by the LREF TT fixed point (<ref>). 
We calculate the critical exponents η, η̅, and ν, 
and investigate the d→ d-σ-ρ dimensional reduction and the d→ d-2 dimensional reduction.  

Substituting the LREF TT FP (<ref>) into Eqs. (<ref>) and (<ref>),  
we obtain the critical exponents η_ LREF and η̅_ LREF 
at LREF TT fixed point (<ref>):  

    η_ LREF   =   2-σ,
    η̅_ LREF   =   4-2σ-ρ.

These exponents satisfy the Schwartz-Soffer inequality 
η̅_ LREF≤ 2η_ LREF <cit.>,  
and saturate the generalized Schwartz-Soffer inequality 
η̅_ LREF≤ 2η_ LREF-ρ <cit.>.  
And the inverse of the critical exponent ν_ LREF is 

    ν_ LREF^-1
    =ϵ[
    N/N-1-ϵ̃/2
    +ϵ̃/2√(1+4[N-ϵ̃(N-1)]/ϵ̃^2(N-1)^2)].

Since lim_N→∞ν_ LREF^-1=ϵ, in the large N limit, the exponent ν_ LREF agrees with 
that of the pure long-range system in σ+ρ dimensions less <cit.>. 
However, as long as N is finite,  ν_ LREF^-1≠ϵ. 
Thus, the d→ d-σ-ρ dimensional reduction is broken for finite N. 
Hence, the d→ d-2 dimensional reduction in the case of ρ=2-σ is also broken for finite N,  
although the exponents η_ LREF and η̅_ LREF satisfy 
η̅_ LREF=η_ LREF=2-σ for N>1 and σ>2-ϵ/(N-1). 
The graphs of (ϵν_ LREF)^-1 for some values of N 
are depicted in Fig. <ref>. 
It shows that (ϵν_ LREF)^-1 tends to draw to 1 
as the value of the parameter σ decreases. 
Then one expects that (ϵν_ LREF)^-1 reaches 1 
if the value of the parameter σ decreases even further. 
However, it is impossible to study within the present framework, 
since the nontrivial fixed point of O(ϵ) disappears. 











§ SUMMARY

In this paper we have reexamined the critical phenomena 
of the long-range random field O(N) spin model near the lower critical dimension 
by using the O(N) nonlinear-sigma model 
with the random fields and all possible higher-rank random anisotropies. 
By the use of the perturbative functional renormalization group, 
we have investigated the stability of the analytic fixed points in the one-loop beta functions. 
Also, we have calculated the critical exponents η, η̅, and ν 
at the analytic fixed point controlling the critical behavior in the system. 

We have shown that the analytic fixed point controlling the critical behavior 
in the system with the long-range correlations of the random fields has the subcusp, 
and that it can be destabilized by the cuspy perturbation 
in both cases where the exchange interactions between spins 
are short ranged and long ranged. 

We have studied the critical phenomena in the spin system with LRE. 
We have investigated the d→ d-σ dimensional reduction. 
We have found that there exists the once-unstable analytic fixed point 
corresponding to the d→ d-σ dimensional reduction 
for N>N_ cusp=2(4+3√(3))≃18.3923⋯. 
Although it has the subcusp, the weaker nonanalyticity does not change the value of the fixed point. 
Then the critical exponents η_ LRE and η̅_ LRE 
evaluated at the once-unstable analytic fixed point 
are 
η_ LRE=2-σ and η̅_ LRE=4-2σ-(d-2σ)/N, respectively, 
and satisfy the relation 2η_ LRE-η̅_ LRE=(d-2σ)/N. 
The inverse of the exponent ν_ LRE takes ν_ LRE^-1=ϵ 
at O(ϵ) in ϵ=d-2σ. 
Therefore, we conclude that the d→ d-σ dimensional reduction  
at the leading order of the d-2σ expansion 
holds only for N>N_ cusp. 
For N<N_ cusp, the nonanalyticity occurring by the appearance of the linear cusp  
breaks down the d→ d-σ dimensional reduction. 
This is considered to violate the simple relation between the exponents. 
Thus, one expects that 
the critical scaling behavior in the spin system with LRE 
is described by three independent exponents <cit.>. 
Moreover, we have also obtained the critical value σ_*=2-η_ SR 
on the basis of the condition η≤η̅≤2η. 
Thus, our result supports the prediction that the crossover 
between the long-range and the short-range exchange regimes of the theory 
occurs at σ_*=2-η_ sr <cit.>. 

We have studied the critical phenomena in the spin system with LREF. 
We have investigated the d→ d-σ-ρ dimensional reduction 
and the d→ d-2 dimensional reduction. 
We have found the once-unstable analytic fixed point  
controlling the critical behavior. 
Although it has the subcusp, 
the weaker nonanalyticity does not change the value of the fixed point. 
Then the critical exponents η_ LREF and η̅_ LREF 
evaluated at the once-unstable analytic fixed point 
are 
η_ LREF=2-σ and η̅_ LREF=4-2σ-ρ, respectively, 
and satisfy 2η̅_ LREF-η_ LREF=ρ. 
However, we have shown that the d→ d-σ-ρ dimensional reduction 
does not holds within the present analysis, as far as N is finite;  
the exponent ν_ LREF does not coincide with 
that of the pure long-range system in σ+ρ dimensions less. 
Thus, the d→ d-2 dimensional reduction in the case of ρ=2-σ 
is also broken for finite N. 
The result does not contradict that in our previous study 
for the three-dimensional long-range random field Ising model <cit.>. 
Since our present study by the use of the perturbative renormalization group 
has been restricted to ϵ+ρ=ϵ+2-σ∼ O(ϵ), 
only the breakdown of the d→ d-2 dimensional reduction has been observed. 
Then, to study the d→ d-2 dimensional reduction and its breakdown 
in the (σ+2+ϵ)-dimensional long-range 
random field O(N) spin model, 
the nonperturbative analysis are needed.  

Finally, we comment on the validity of the d→ d-σ dimensional reduction 
in the system with LRE 
near the lower critical dimension and for N>N_ cusp. 
As shown in the previous works by Young <cit.> and Bray <cit.>, 
the value of ν_ LRE^-1 coincides with 
that of the pure long-range system in σ dimensions less 
at the leading order in ϵ=d_u-d near the upper critical dimension d_u=3σ. 
However, it fails at O(ϵ^2). 
Thus, although we have shown that the d→ d-σ dimensional reduction holds 
at the leading order in ϵ=d-d_l near the lower critical dimension d_l=2σ
and for N>N_ cusp in the present work, 
there is room for doubt whether it holds beyond one loop, even if N>N_ cusp. 
Further studies by using the higher-loop calculation should shed light on this problem. 






The author would like to thank to Matthieu Tissier and Gilles Tarjus 
for discussions in early stage of this work. 








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