《Field theory of selfavoiding walks in random media.pdf》由会员分享，可在线阅读，更多相关《Field theory of selfavoiding walks in random media.pdf（9页珍藏版）》请在三一文库上搜索。
1、arXiv:cond-mat/9909407v1 cond-mat.stat-mech 28 Sep 1999 Field theory of self-avoiding walks in random media A V Izyumov and K V Samokhin Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, UK (February 1, 2008) Abstract Based on the analogy with the quantum mechanics of a particle propagating
2、in a complex potential, we develop a fi eld-theoretical description of the sta- tistical properties of a self-avoiding polymer chain in a random environment. We show that the account of the non-Hermiticity of the quantum Hamiltonian results in a qualitatively diff erent structure of the eff ective a
3、ction, compared to previous studies. Applying the renormalisation group analysis, we fi nd a transition between the weak-disorder regime, where the quenched random- ness is irrelevant, and the strong-disorder regime, where the polymer chain collapses. However, the fact that the renormalised interact
4、ion constants and the chiral symmetry breaking regularisation parameter fl ow towards strong coupling raises questions about the applicability of the perturbative analysis. 1 I. INTRODUCTION The problem of a polymer in a random environment is among the most interesting in statistical physics.It has
5、been known for a long time that the mean square end-to-end distance of a pure self-avoiding walk (SAW) of length L obeys the scaling law hr2i L2, where 0.59 in three dimensions 14 (for a Gaussian random walk, one has the classical exact result = 0.5 in all dimensions).The question of how this scalin
6、g behaviour is aff ected by external impurities has attracted considerable research eff ort for more than a decade 517.In his pioneering work, Harris 6 argued that, treated perturbatively, quenched disorder is irrelevant, and, therefore, no modifi cation of the critical exponent should be expected (
7、see also Ref. 7). This conclusion found support in the Monte-Carlo simulations on weakly diluted lattices 5. The opposite case of strong disorder has also been studied, both numerically 8 and analytically 9,10. Edwards and Muthukumar 9 and Cates and Ball 10 predicted that a Gaussian chain placed in
8、the fi eld of impurities would collapse to a localised state, in which hr2i const at L . The localisation breaks down if one introduces a weak repulsive interaction between the monomers in the chain. In this case, it was found 11 that the polymer behaves as a free random walk with = 0.5. Later, a cr
9、ossover between the regimes of weak ( 0.59) and strong ( = 0.5) disorder was predicted to occur at some critical concentration of impurities 12. However, these results seem to contradict the conclusions of a number of other authors 14,15, who argued that, at any concentration of impurities, the scal
10、ing of a long chain with excluded volume interactions is controlled by the critical exponent of a directed random walk, = 2/3. Overall, a comprehensive theory that would describe the eff ects of disorder on self-avoiding polymers is still missing. Previous studies of the interplay of disorder and ex
11、cluded volume interactions have been largely based on qualitative arguments, although a number of mathematical techniques (including the variational methods 11,12, replica fi eld theories 7,13,14 and the real-space renormalisation group 16) have also been used. A more general framework would therefo
12、re be benefi cial to a better understanding of these eff ects. We believe that such a framework could be designed by employing a fi eld-theoretical approach, which draws on the connection between the statistical mechanics of a polymer and the quantum mechanics of a particle in a complex random poten
13、tial. Despite its apparent simplicity, the use of this method has been hampered by the fact that, as the Hamiltonian of the particle in a complex potential is non-Hermitian, it is impossible to represent the Green function in the form of a convergent functional integral (see below). A similar proble
14、m has been encountered, and successfully resolved, in the spectral theory of non-Hermitian operators. The latter has attracted great interest in recent years. A variety of applications have been identifi ed including the study of anomalous diff usion in random media 18, scattering in open quantum sy
15、stems 19, neural networks 20, and the statistical mechanics of fl ux lines in superconductors 21. The problem of a self-avoiding walk without impurities, which can be mapped onto the quantum mechanics of a particle propagating in a random imaginary potential, has also been analysed in this context 2
16、2. These studies have led to the development of a new technique, based on the representation of the spectral properties of non-Hermitian operators through an auxiliary Hermitian operator of twice the dimension 23, which serves as a good starting point for a fi eld-theoretical approach. 2 The main pu
17、rpose of the present paper is to derive a consistent fi eld-theoretical formu- lation of the problem of a self-avoiding polymer chain in a random white-noise potential, using the methods of non-Hermitian quantum mechanics, which is done in Section II. The large-scale behaviour of this model is studi
18、ed perturbatively in Section III by means of the momentum-shell renormalisation group (RG) in D = 4 dimensions. Section IV concludes with a discussion of the results obtained and of possible limitations on the validity of the perturbative approach. II. DERIVATION OF THE FIELD THEORY Let us consider
19、a continuum self-avoiding chain of length L, with one end fi xed at the origin. Then the probability to fi nd the other end at a point r can be expressed as a path integral (Edwards model 24,25): P(r,L) = Z x(L)=r x(0)=0 Dx(s) exp ? 1 2a Z L 0 ds dx(s) ds !2 Z L 0 ds V1(x(s) 2 2 Z L 0 ds1 Z L 0 ds2(
20、x(s1) x(s2) ? .(1) The fi rst term in the exponent corresponds to the entropic contribution (the length a, called the Kuhn length, is a microscopic parameter with the physical meaning of the monomer size, which provides a natural ultraviolet cutoff scale). The second term is the potential energy of
21、the chain in an external potential, which is assumed to be a Gaussian distributed random function with the correlator V1(r)V1(r) = 1(r r). The last term takes into account the excluded volume eff ects (2 0), the limit 2 describing the situation where the intersections of diff erent fragments of the
22、chain are penalized by an infi nite energy barrier and thus completely forbidden. This interaction term can be decoupled by introducing an auxiliary white-noise potential. Using the identity exp ( 2 2 Z L 0 ds1 Z L 0 ds2(x(s1) x(s2) ) = * exp ( i Z L 0 ds V2(x(s) )+ , where the angular brackets deno
23、te averaging over a Gaussian distributed random fi eld V2(r) with correlator hV2(r)V2(r)i = 2(rr), we can represent Eq. (1) as an averaged Feynman propagator of a fi ctitious quantum particle moving in a complex random potential: P(r,L) = hU(r,L)i, where U(L) = exp(LH) with the Hamiltonian H = 2+ V1
24、(r) + iV2(r)(2) (V1,2 (r) are independent Gaussian random fi elds). Note that H is non-Hermitian and may therefore have complex eigenvalues. It is convenient to use the “energy representation”, which is achieved by introducing the Green operator as a function of z = x + iy, g(z) 1 z H = X k |Rki 1 z
25、 zk hLk|,(3) 3 where |Rki and hLk| are the right and left eigenfunctions of H, and zkdenote the complex eigenvalues. Using the identity z1/z= 2(z) (x)(y), one can relate U(L) to g(z), and express the end-to-end probability distribution of a self-avoiding chain in a given distribution of impurities i
26、n the form P(r,L|V1) = 1 Z d2z exp(zL) z hg(r,z)i,(4) where the integration runs over the entire complex plane.It should be noted that the standard fi eld-theoretical methods, based on the representation of an inverse matrix in the form of a Gaussian functional integral, cannot be directly used for
27、the calculation of g(z), the reason being that, as H may have eigenvalues anywhere in the complex plane, it is impossible to guarantee the convergence of the functional integral. As pointed out in Ref. 26, a naive attempt to calculate the functional integral by the analytical continuation from the r
28、egion where it converges to the whole complex plane fails. The problems are revealed by representing the density of complex eigenvalues through the identity (z) X k 2(z zk) = 1 z Sp g(z),(5) wherein the Green function is shown to be non-analytic everywhere in which the density of states is non-vanis
29、hing. To circumvent these diffi culties, a representation has been introduced 23,27,28, in which the complex Green function g(z) is expressed through an auxiliary Hermitian operator, which in our case has the form G1(z) ? 0z H z H0 ? = ? x + 2 V1(r) ? 1 (y V2(r)2,(6) where iare the Pauli matrices. A
30、 relationship between g and G is straightforward: g(z) = G21(z).(7) Using the replica trick, the matrix Green function G(z) can be written as a functional integral over 2n-component complex Bose fi elds (in the limit n 0): Gij(r,z) = i lim +0 Z D2aexp ? i Z dDr a( G1(z) + i0)a ? i 1(r) j, 1 (0)(8) (
31、0is the unit matrix, summation over repeated replica indices is assumed), where a(r) = 1 a(r) 2 a(r) ! ,D2a= n Y a=1 Y i=1,2 D(Rei a)D(Im i a) . Due to the Hermiticity of G1(z) and the presence of the term with (“regulator”), the functional integral is well defi ned and convergent. Note that, althou
32、gh the matrix Green function (6) possesses a chiral symmetry, 3G1(z)3= G1(z), this symmetry is broken in Eq. (8) if 6= 0. 4 Eqs. (7) and (8) allow one to average g(z) over V2and calculate the polymer partition function in a given confi guration of the external disorder. In order to obtain physically
33、 observable quantities, one has to average the free energy F(r,L) = lnP(r,L) over the quenched random fi eld V1, which can be done using the replica trick once again: F(r,L) = lim m0 Pm(r,L) 1 m = lim m0 1 m (Z m Y =1 d2z exp L m X =1 z ! m Y =1 z m Y =1 hg(r,z)i 1 ) .(9) It is thus necessary to int
34、roduce a second set of replica indices and to integrate over 2nm- component fi elds i a,(r) (i = 1,2; a = 1,.,n; = 1,.,m). Averaging over V1, we obtain from Eqs. (7) and (8): Y hg(r,z)i = Z D2a,eiS a,a, Y 2 1,(r) 1, 1,(0) (n,m 0),(10) where the eff ective action has the following form: iS = Z dDr ?
35、i( a,1 2 a,) + ix( a,1a,) iy( a,2a,) ( a,0a,) 1 2 ( a,1a,)( b,1b,) 2 2 ( a,2a,)( b,2b,) 1 2 ( a,1a,)( b,1b,) ? .(11) This action is diff erent from the replica n-vector model, used previously in the RG analysis of the problem of a SAW in a random medium 7,13,14, because of the double dimensionality
36、of the fi elds involved and a larger number of the coupling constants (three instead of two). The additional term proportional to 1 cannot be obtained in a formal derivation of Eq. (11), but should be added to the long-wavelength eff ective action for consistency, as we shall see below. Note also th
37、at we have kept the term with in Eq. (11) (the importance of this term will become clear shortly). The asymmetry between the ways the Latin and Greek replica indices appear in Eq. (11) refl ects the diff erences in the nature of the random potentials V1,2: V1describes the external quenched disorder,
38、 while V2 is a fi ctitious annealed random fi eld. III. RENORMALISATION GROUP ANALYSIS The long-distance properties of the fi eld theory (11) can be investigated using the mo- mentum shell RG approach and the -expansion near the upper critical dimension Dc= 4. Using the standard procedure, consistin
39、g of the separation of “slow” and “fast” degrees of freedom followed by a rescaling of lengths and fi elds 29, we obtain the fl ow equations in one-loop order: 5 dlnx d = 2 + 1 82 (1 2+ 1), dlny d = 2 + 1 82 (1 2+ 1), dln d = 2 + 1 82 (1+ 2+ 1), d1 d = 1+ 1 42 (22 1 12+ 1 1), (12) d2 d = 2+ 1 42 (31
40、2 2 2 + 32 1), d 1 d = 1+ 1 42 (31 1+ 2 2 2 1+ 2 2 1), where = ln(L/a) is the logarithmic RG parameter. The real and imaginary components of the complex “energy” z = x+iy are renormalised in the same way. It also follows from Eqs. (12) that, even though initially 1 = 0, it acquires a non-zero value
41、under renormalisation. Introducing a diff erent set of independent variables, u = 1 2+ 1, v = 2,w = 1, we rewrite Eqs. (12) as dlnx d = 2 + 1 82 u, dlny d = 2 + 1 82 u,(13) dln d = 2 + 1 82 (u + 2v),(14) du d = u + 1 22 u2,(15) dv d = v + 1 42 (3uv + 2v2),(16) dw d = w + 1 42 (3uw + v2+ 2vw w2).(17)
42、 Note that the fl ow equations for x,y and u are decoupled from those for v and w. Let us neglect the renormalisation of , i.e. put 0= 0 (the consequences of allowing for 06= 0 will be discussed below). As seen from Eq. (9), the main contribution to the average free energy at L comes from x,y 1/L, w
43、hich provide the initial conditions for the equations (13). The scale Rc, at which the renormalised values of x and y become of the order of unity, represents the correlation length of our fi eld theory and should be identifi ed with the average size of a polymer of length L. Integrating Eqs. (13) w
44、ith respect to from = 0 to c= ln(Rc/a), we obtain the equation which relates the polymer size to L: ln L a = Z c 0 d 2 + u() 82 ! .(18) The behaviour of u() essentially depends on the bare values of the coupling constants 1,2 (the bare value of 1 is zero).One should distinguish between the two possi
45、bilities: In 6 the weak disorder regime (1 2, i.e. u0 2, i.e. u0 0), the solution of Eq. (15) u() = u0 e 1 u0 22(e 1) (19) has a pole at = , where = 1 ln 1 + 22 u0 ! .(20) Substituting u() ( )1 in Eq. (18), we fi nd that, at L , Rc(L) ae . Thus, the polymer size tends to some constant independent of
46、 its length. Although this conclusion coincides with the results of Refs. 9,16,17, the applicability of our one-loop RG equations in this strong-coupling regime is limited. IV. CONCLUSIONS AND DISCUSSION Our results show that a proper account of the non-Hermiticity leads to signifi cant changes in t
47、he structure of the eff ective fi eld theory for self-avoiding walks in random media. The application of the one-loop renormalisation group analysis to the action with the doubled number of the degrees of freedom allowed us to predict the existence of a phase transition between the weak disorder reg
48、ime (in which the scaling behaviour of the polymer is not aff ected by the external randomness), and the strong disorder regime (in which the polymer collapses into a localised state). This conclusion seems to be in agreement with one of the scenarios considered previously, notably in Refs. 7,16. Ho
49、wever, our RG calculations also indicate that there might be some hidden fl aws within the perturbative approach, one of which is revealed by taking into consideration the infi nites- imal regulator , which explicitly breaks the chiral symmetry of Eq. (6). It follows from Eq. (16) that, in the weak disorder case, v() has a pole at = v, where v = 1 ln 1 + 22 4v0 |u0| (2v0 |u0|)2 ! .(21) After substitution in Eq. (14) and integration over , we fi nd that () grows