Critical Phenomena and Critical Dimensions in Anisotropic Nonlinear Systems

CRITICAL PHENOMENA AND CRITICAL DIMENSIONS IN ANISOTROPIC NONLINEAR SYSTEMS

A.V. Babich,∗ S.V. Berezovsky, L.N. Kitcenko, V.F. Klepikov Institute of Electrophysics and Radiation Technologies NAS of Ukraine, 61108, Kharkov, Ukraine (Received October 30, 2011)

The model that allows one to generalize the notions of the multicritical and Lifshitz points is considered. The model under consideration includes the higher powers and derivatives of order parameters. Critical phenomena in such systems were studied. We assess the lower and upper critical dimensions of these systems. These calculation enable us to find the fluctuation region where the mean field theory description does not work. PACS: 64.60.Kw, 64.60-i

1. INTRODUCTION by various methods based on the renormalization group. In condensed matter theory a consideration Second order phase transitions (PT) are ones of the of systems with d>3 is just a trick that allows one most intensively investigated phenomena in theoret- to calculate the critical indexes by the renormgroup ical physics. A lot of different kinds of PT have methods. But in such fields as particle physics, gen- been observed and investigated in various physical eral relativity and cosmology it is necessary to use systems [1–3]. Initially the basics objects of PT the- models with higher space dimensionalities [7,8]. ory application were various condensed matter systems. However such fundamentals of PT theory 1 dl 2 du 3 as the spontaneous symmetry breaking, the group- d theoretical approach allow one to use predictions of Fig. 1. 1. No ordering. 2. Fluctuation region. PT the theory of PT in the fields of physics which from are possible, but the mean field approximation is the first view have nothing in common with above invalid. 3. The fluctuations are damped. The mean mentioned condensed matter systems. field approximation is valid One of the most important properties of the systems in vicinity of critical point is a strong increase of fluctuations influence. The influence of the fluc- 2. MULTICRITICAL AND LIFSHITZ tuations strongly depends on spatial dimension. In POINTS modern theory of critical phenomena the space dimensionality “d” is usually considered as continuous In the simplest model of PT an expansion of thermo- value [4,5]. It appears in thermodynamic relations as dynamic potential (TP) looks as follows: one of the parameters of the system. As mentioned 2 4 Φ=Φ0 + aϕ + bϕ . (1) above a critical behavior of the system strongly depends on it. Here lower and upper critical dimensions equal 2 and One of the effects of this dependence is an ex- 4 correspondingly. In more complicated models crit- istence of the 2 critical (or borderline) dimensions. ical dimensions depend on the model parameters. Lower critical dimension determines the range of the There are a few ways to generalize model (1). existence ordering states: there are no PT at nonzero The first one is to take into account terms with temperature if the space dimensionality is less than higher powers of the order parameters in the ex- lower critical dimension, or in other words at lower pansion of the TP. It leads one to the notion critical dimension Goldstone bosons start to interact of the multicritical point. Multicritical points of strongly [6]. Upper critical dimension determines a various types have been observed in many physi- range of the mean field based theories applicability. cal systems. The most known example of a sys- So if we plot an axis (Fig. 1) of dimensionality then tem with the tricritical point (TCP) is the mixture 3 4 the critical dimensions divide it into 3 regions (dl and of helium isotopes He -He . There are TCPs on du are the upper and lower critical dimensions). phase diagrams of some ferroelectrics for example The critical dimensions are important not only in KH2PO4 (Fig. 2), antiferromagnetics. Upper CD as “borders”. They are necessary elements for cal- of systems with TCP is equal to 3, so correspond- culations of critical indexes in the fluctuation region ing models are renormalazed in usual physical space. ∗Corresponding author E-mail address: [email protected]

268 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 268-272. dients of OP and the highest power of OP in such model may take arbitrary values.

Fig. 2. Schematic phase diagram for systam with TCP. β>0 – second order PT, β<0 –ﬁrstorder PT

Another way is to take into account an inhomo- geneity of the order parameters. It leads one to the notion of the Lifshitz point (LP). The theoretical in- Fig. 4. Phase diagram for Sn2P2(SexS1−x)6 in the vestigations of PT in systems with anisotropic mod- vicinity of TCLP ulations of the order parameters near the Lifshitz points began in the 1970th and have continued up In the vicinity of the critical point under consid- to date. Initially the main fields of application of eration the effective Hamiltonian can be written as these theories were systems with anisotropic mag- follows: 1 2 netic ordering. But recently systems with Lifshitz m d−m r 2 γ 2 H = d xid xc η + Δ η + points have increasingly arisen interest in such fields 2 2 i of physics as particle physics, quantum gravity and N+1 1 2 p 2 cosmology [7,8]. Lifshitz points arise in systems with δ 2 β 2 + Δc η + Δ η + uη , (3) modulated structures of OP. There are 3 phases co- 2 2 i exist in LP: initial ordering phase, homogeneous dis- where η is a one-component order parameter, d is the ordered phase and modulated phase (Fig. 3). Modu- space dimension and r, γ, δ, β, u are the model para- lated structures of different types have been observed meters. We assume that the space can be divided into in a lot of magnetic and ferroelectric crystals (MnP, two subspaces of dimensions m and d−m denoted by NaNO2,K2SeO4 etc.). i and c respectively. There are wave modulation vec- tors in the first subspace and none in the second one. Let us assume d and m to be continuous variables and d>m.Δc and Δi are the Laplacian operators available in the corresponding subspaces. In this case the operators Δl are defined as Δl =Δ Δl−1 . For non-integer values of l and m the corresponding operators are determined using the inverse Fourier transformation. In the critical point r = γ =0,and the other parameters in (3) are positive quantities. Using the Fourier transformation : η (x)= ddq · η (q) eiqx, (4) Fig. 3. Schematic phase diagram for system with we rewrite the Hamiltonian (3) in momentum space: Lifshitz point. 1 – ordered state, 2 – state with modulated structures, disordered state 1 H = ddqν (q) η (q) η (−q)+ 2 This paper is about CDs of systems with mutual d d multicritical and Lifshitz behavior. The simplest ex- + u d q1 · ... · d qN+1 (5) ample of such point is tricritical Lifshitz point (Fig.4). Point of that kind has been studied in ferroelectrics × δ (q1 + ... + qN+1)(η (q1) ...η (qN+1)) . of type Sn2P2(SexS1−x)6. Corresponding TP: 2 2p 2 Here ν (q)=r+γq +βq +δq , qi and qc are the ab- i i c−→ α β γ δ g solute values of the wave vector q being in the sectors 2 4 6 2 2 m d Φ=Φ0 + ϕ + ϕ + ϕ + ϕ + ϕ . (2) 2 2 2 2 2 4 6 2 4 i and c respectively, and qi = qα, q c = qα. α=1 α=m+1 We introduce a model that allows one to study a The main parameters which determine the criti- system with a new type of critical point. The gra- cal behavior of the system with the Hamiltonian (3)

269 are the order of higher gradients p, the dimension of here σ(d)=2− (m/2p +(d − m)/2) and the func- the subspace of modulation, and the order of nonlin- tion I (κi,κc) does not depend on t. And ﬁnally for earity of the model. σ(d): m d − m σ(d)= + − 1, (14) 2p 2 3. THE CRITICAL DIMENSIONS OF THE From (8) and (14): ANISOTROPIC SYSTEMS 1 dl = m 1 − +2. (15) The lower critical dimension dl is deﬁned by the fol- p lowing condition: there are no ordering states in space with d

entropy is a dl - comparing the fluctuation contribution to a heat divergent function of temperature. The fluctuation capcity: contribution to entropy looks as follows: 2 ∂ Φ 2(N +1) −α CV = T =Φ=B · t l , (16) σ(d) 2 − 2 Sfl = sτ , (6) ∂T (N 1) N+1 where αl =2− . here τ =(T − Tc)/Tc is the reduced temperature, N−1 σ(d) is a function of space dimensionality and does With fluctuation correction to the heat capacity: not depend on T. We are interested in the critical −α C1,fl = t fl · I (κi,κc) , (17) behavior of Sfl,so: m d−m where αfl =2− 2p + 2 and the function 0 σ(d) < 0, lim Sfl = (7) I (κi,κc) does not depend on t. τ→0 ∞ σ(d) > 0. Second is to find du from the stability condition of the fixed point of corresponding renomgroup trans- One can see that under condition Tc = 0 the fluctu- formation: ation contribution to entropy is a divergent function if σ(d) > 0 otherwise it goes to zero. Thus one can ν(q)=z2a−mb−(d−m) find the lower critical dimension from the following 2 2 2p condition: × qc qi qi r + δ 2 + γ 2 + β 2p + ... , (18) σ(dl)=0. (8) b a a N+1 −N −N(d−m) · In order to calculate σ(dl) let us find how the fluc- u (q)=z a b (u+...) . (19) tuation contribution to entropy depends on the tem- Here we take into account that the scale parameters perature. By definition: changing the scale for the momenta are independent ∂Gfl in the sectors i and c.Wedenotethema and b re- Sfl(τ)= , (9) ∂τ spectively. The scale parameter for u is denoted by z. Both of those ways lead us to the following expression here Gfl is a fluctuation contribution to the thermo- for du [9]: dynamic potential, in model (3) it looks as: 1 N +1 2p 2 du = m 1 − +2 . (20) m d−m βqi + δqc + ατ p N − 1 Gfl = A d qi · d qc ln , (10) πT Also one can find du from the condition of scale varia- where A is a parameter that does not depend on tem- tion invariance of the model (3) under transformation perature. From (9) and (10): with generator: −1 N − 1 ∂ N − 1 ∂ ∂ m d−m 2p 2 X = + − . (21) Sfl = αA d qi · d qc βqi + δqc + ατ , 2 ∂xc 2p ∂xi ∂ϕ (11) after some manipulations: 4. DISCUSSION 2p −1 m d−m 2 − 1 · · · 2p Sfl = τ αA d qi d qc β qi τ + The spatial distribution of an order parameter is defined by the solution of the variational equation for

2 −2 the functional of the free energy. This equation is a − 1 +γ · qc · τ 2 + α . (12) nonlinear differential equation (NDE). In our model the order of corresponding equation is 2p.Atpresent − 1 After the change of variables κi = qi · τ 2p ,κc = there are no universal methods for solving the NDE, − 1 qc · τ 2 the final expression for Sfl takes the form: therefore of fundamental importance are the methods enabling to simplify the NDE analysis. One of σ(d) Sfl = τ · I (κi,κc) , (13) the most effective ways for the NDE analysis is the

270 analysis of its symmetries. Investigation in the group Let us consider what types of anisotropic system NDE structure allows one to determine the behavior in 3 and 4 dimensional spaces are possible accord- of solutions without defining their explicit form, to ing to the condition d>dl.Thereare2typesof find different NDE invariants, in particular, to con- anisotropic systems in three-dimensional space: struct the conservation laws. Knowing the symmetry 1) usual critical point (without anisotropy) dl = groups of the assumed NDE enables also to construct 2, and important classes of particular solutions. Every addi- 2) 1-axial Lifshitz point dl =2, 5. In other cases tional NDE-admitted Lie group allows reducing the dl ≥ 3. equation order by one. The variational symmetries In the case of 4-dimensional space: play here a very important role. These symmetries 1) usual critical point (without anisotropy) dl =2, belong to the variational equation, as well as, to the and functional, for which the given equation is variational. 2) 1-, 2-, 3-axial Lifshitz point of order 2 dl = The presence of the variational symmetry in the NDE 2.5, 3, 3.5. makes it possible to reduce its order by 2. The im- 3) 1-, 2-axial Lifshits point of order p>2, dl = portance of the variational symmetries is due to the 3 − 1/p,4− 1/p. In other cases dl ≥ 4. conservation laws related to them. In paper [9] was In general cases: for some initial order p of the shown that model (3) is invariant under scale varia- Lifshitz point, the ordering in the space with dimen- tional transformations with generators: sionality d>2p with any kind of anisotropy is possible. In case of d ≤ 2p, there are various situations N − 1 ∂ N − 1 ∂ ∂ Xˆ = + − . (22) and this case demands additional investigation. 2 ∂xc 2p ∂xi ∂ϕ The obtained results are correct for classical PTs. As we know, in the theory of quantum PTs the ef- Let us find the range of the fluctuation region: fective dimension of a system in the vicinity of the 4 quantum critical point is higher than a dimension of Δd ≡ du − dl = , and lim (du − dl)=0. (23) N − 1 N→∞ space. So it is apparent that there are more possible types of PTs in a quantum case. In particular, So, the fluctuation region decreases as a function our results do not contradict a possibility of quantum of power of nonlinearity (see Fig. 5). This fact is PTs in 2-dimensional systems. physically reasonable. Strong coupling supresses the fluctuations. As it is expected, the lower critical dimension of any systems is not less then 2. References

1. J.C. Toledano and P. Toledano. The Landau Theory of Phase Transitions. Application to Structural, Incommensurate, Magnetic and Liq- uid Crystal Systems. World Scientiﬁc Publishing Company, 1987, 461 p. 3 2. A.D. Bruce and R.A. Cowley. Structural Phase Transitions. 1981, London: Taylor & Francis 2 Ltd., 302 p. 3. A.I. Olemskoi, V.F. Klepikov. The theory of 1 spatiotemporal patterns in nonequilibrium systems // Physics Reports. 2000, v. 338, p. 571-677. N 4. A.Z. Patashinskii and V.L. Pokrovskii. Flucta- Fig. 5. Dependence of Δd on the order of nonlin- tion Theory of Phase Transition. 1979, New earity of the model York: “Pergamon”, p. 524. 5. Shang-keng Ma. Modern Theory of Critical Phe- Let us compare the obtained results with those nomena. 1976, Perseus Books, U.S., p. 299. known previously. As mentioned above, in the case of the simple critical point (N =3,p =1,m =0) 6. A.M. Polyakov. Gauge Fields and Strings. the lower and upper CDs are equal to 2 and 4 respec- Switzerland: “Harwood Academic Publishers”, tively. The CDs of the system with m-axial Lifshitz 1987, p. 327. point (N =3,p = 2) depend on m linearly (some 7. B.E. Meierovich. Vector order parameter in gen- special cases in 3 and 4 dimensional spaces will be eral relativity: Covariant equations // Phys. Rev. discussed later) . An interesting case is the model 2010, v. D82, 024004. with an isotropic Lifshitz-like point with derivatives of order up to p (Lifshitz point of order p). The varia- 8. K.A. Bronnikov, S.G. Rubin, and tional equations in this model coincide with Multidi- I.V. Svadkovsky. Multidimensional world, mensional Polywave Equation that is invariant under inﬂation, and modern acceleration // Phys. Rev. conformal transformations. 2010, v. D81, 084010.

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Critical dimensions of systems with joint multi- drecht: Kluwer Academic Publishers, 306 p.

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