Spectral Properties of Weakly Coupled Landau-GinzburgStochastic ModelsRicardo Schor1, Jo~ao C. A. Barata2, Paulo A. Faria da Veiga3, Emmanuel Pereira11Dep. F��sica-ICEx, UFMG, CP 702, 30.161-970 Belo Horizonte MG, Brazil2IFUSP-USP, CP 66318, 05315-970 S~ao Paulo SP, Brazil3ICMC-USP, CP 668, 13560-970 S~ao Carlos SP, BrazilAugust 19, 1998AbstractWe study existence of bound states in the generator of the stochastic dynamics associatedto weakly coupled lattice Landau-Ginzburg models. By analyzing the Bethe-Salpeter kernelin the ladder approximation, these states are shown to exist if the polynomial interaction hasa negative quartic term and the lattice dimension is smaller than three. Asymptotic valuesfor the masses are also obtained, giving precise relaxation rates for even correlations.PACS number(s): 2.50.-r, 05.40.+j, 05.50+q1 IntroductionIn this paper, we consider some aspects for the stochastic dynamics of lattice systems describedby an action of the formS(') = X~x2ZZd(12 " dXi=1('(~x + ~ei)� '(~x))2 +m2'(~x)2#+ �P('(~x))) (1)where '(~x) is a real continuous spin variable at ~x 2 ZZd, the unit d-dimensional lattice, ~ei is theunit vector along the ith coordinate, P is an even polynomial bounded from below, m > 0 and1

� � 0.For t 2 lR denoting the time variable, the dynamics is introduced by the Langevin equation@@t'(~x; t) = �12 ��'(~x; t)S + �(~x; t) (2)where f�(~x; t)g is a family of Gaussian white noise processes with the expectations E(�(~x; t)) =0 and E(�(~x; t)�(~x0; t0)) = �~x;~x0�(t � t0). Such models can be used to describe the (purelyrelaxational) evolution of an order parameter in statistical mechanical systems [1,2].The dynamics induced by (2) is associated with a Markov semigroup and leaves invariant theGibbs distribution d� = e�S(')d'=normalization de�ned by the action (1). More speci�cally,if f is any function of the spin con�guration ' = f'(~x)g, we de�ne its time evolution ft byft( ) = E(f('(t))) (3)where '(0) = is the initial condition in eq.(2). It follows then that ft is determined by theMarkov semigroup e�tH whose generator H, given byHf = �12 X~x2ZZd " @2@'(~x)2 f � @S@'(~x) @f@'(~x)# (4)is positive and Hermitian on L2(d�). Clearly, the constant function f = 1 is an eigenfunctionof H with zero eigenvalue. As m is nonzero, for small �, there is a gap in the spectrum of H,implying an exponentially fast approach to equilibrium in the system.It is possible and indeed desirable to associate a quasiparticle structure to the operatorH, which can then be viewed as a Hamiltonian, since this structure provides information oncorrections to the exponential law of approach to equilibrium. Momentum operators ~P arenaturally de�ned by space translations and commute with H. >From this point of view, thenatural question to ask is about the nature of the spectrum of (H; ~P ).This problem has been recently considered by Kondratiev and Minlos [3], in the contextof the stochastic XY model at high temperatures. They constructed one-particle states (fortwo di�erent species of quasiparticles) and showed that they are isolated from the rest of thespectrum. 2

The existence of isolated one-particle states for the model de�ned by (1) and (2) followsalmost automatically from standard techniques of constructive �eld theory [4], taking as inputthe convergent cluster expansion established by Dimock in ref. [5]. This paper intends to furtherour knowledge about the spectrum of (H; ~P ). More precisely, we analyze the existence of boundstates of two quasi-particles. We remark that the mass of such a bound state shows up directlyin the exponential approach to equilibrium for even observables. Hence this question is of directphysical relevance.To attack this problem, we use a functional integral representation for the associated correla-tion functions and look at the dynamical system as a quantum �eld theory in discrete coordinatespace and continuous time. This �eld theory turns out to present non-local interactions. Thepart of the spectrum above the one-particle states is studied through a Bethe-Salpeter (B-S)equation in a way which is similar to the methods employed previously in local relativistic �eldtheory. Here, however, the discreteness of space and the non-locality of the interactions repre-sent additional complications in the analysis of the B-S kernel. In order to simplify them, wedo restrict ourselves to the spectral analysis of translationally invariant states. Also, the B-Skernel is computed only in the ladder approximation. This procedure has been justi�ed in caseswhere a rigorous analysis was possible [6].To state our results, we write the polynomial interaction P in (1) as an expansion in anHermite basis (see Section 3), starting with the fourth power:P(x) = NXn=2 an(2n)! : x2n : (5)with aN > 0. If � is small, we show (in the ladder approximation) absence of bound states forspatial dimension d � 3, as well as for d < 3 and a2 � 0. For d < 3 and a2 < 0, there is a uniquebound state. In dimension one, the mass of this bound state isM� = 2M � 9a22�24m4 (1 +O(�)) (6)3

and, for d = 2, M� = 2M � exp "� 4�m23ja2j�(1 +O(�))# (7)Above, M is the mass of the single quasi-particle.This paper is organized as follows. In Section 2, we discuss on the functional integrationrepresentation and the form of the B-S equation which is suitable to handle �eld theories on alattice. The computation of the B-S kernel and the mass spectrum above the one-particle stateare presented in Section 3. Section 4 is devoted for conclusions.2 The Feynman-Kac FormulaConsider the Hamiltonian (4) on a �nite hypercube � � ZZd with periodic boundary conditions:H�f = �12 X~x2� " @2@'(~x)2 f � @S@'(~x) @f@'(~x)# (8)For d'� = Q~x2� d'(~x), let d��(') = 1Z� e�S�d'� (9)with S� given by (1), restricted to � with periodic boundary conditions, and where Z� is anormalization for d�� so that R d�� = 1. With this, the operator H� is Hermitian on the spaceL2(d��). Next, let U� be the unitary operator from L2(d��) to L2(d'�) given by(U�f)(') = Z�1=2� e� 12S�f(') (10)A straightforward calculation shows thatL� = U�H�U�1� = �12 X~x2� @2@'(~x)2 + 14 X~x2� "12 � @S�@'(~x)�2 � @2S�@'(~x)2 # (11)so that L� is a Schr�odinger type operator. Performing the derivatives, we getL� = �12 X~x2� @2@'(~x)2 + 18 X~x2�'(~x)[(��+m2)2'](~x)+�4 X~x2�[(��+m2)'](~x)P 0('(~x)) +X~x2� "�28 P 0('(~x))2 � �4P 00('(~x))� (2d +m2)4 # (12)4

In the above formula, �� is the lattice Laplacian with periodic boundary conditions on � givenby (��')(~x) = 2d'(~x)� Xj~x�~yj=1'(~y) (13)The functional integral associated with (12) can be obtained by standard methods [4]. Iff1; : : : ; fn are functions of the spin con�guration in �, if (') = 1 is the ground state of H� andfor t1 � t2 � : : : tn 2 lR, then we have the Feynman-Kac formula(; f1e�(t2�t1)H�f2 : : : e�(tn�tn�1)H�fn)L2(d��)= (U�; f1e�(t2�t1)L�f2 : : : e�(tn�tn�1)L�fnU�)L2(d'�)= Z f1('(t1)) : : : fn('(tn))d�� (14)where the path space measure d�� is given byd�� = e�W�d��R e�W�d�� (15)withW� = Z 1�1 dtX~x2� "�4P 0('(~x; t))(��+m2)'(~x; t) + �28 P 0('(~x; t))2 � �4P 00('(~x; t))# (16)and d�� is a Gaussian measure with mean zero and variance given byZ '(~x; t)'(~y; t0)d�� = 12�j�j Z 1�1 dp0X~p2~� eip0(t�t0)ei~p�(~x�~y)p20 + hPdi=1(1� cos pi) + m22 i2 (17)In (17), j�j is the number of points in �, ~� is the Fourier dual lattice, ~p = (p1; : : : ; pd) 2 ~� and~p � (~x� ~y) =Pdi=1 pi(xi � yi).The thermodynamic limit � ! ZZd can now be taken in equation (14). The correspond-ing limiting expressions for (12-16) are easily obtained. The normalized sum 1j�jP~p2~� in thepropagator (17) is replaced in the limit by an integral 1(2�)d RTd ddp over the d-dimensional torusTd = [��; �]d. That the limit �! ZZd exists, at least for small �, follows from a cluster expan-sion argument (see [5]). Dropping hereafter the subscript � for the in�nite-volume quantities,5

we have the representation(; '̂(~x1)e�(t2�t1)H '̂(~x2) : : : e�(tn�tn�1)H '̂(~xn)) == Z '(~x1; t1) : : : '(~xn; tn)d� � S(n)� (~x1; t1; : : : ; ~xn; tn) (18)where the inner product (�; �) on the lhs is taken on the physical Hilbert space L2(d�), '̂(x) isthe zero time �eld at ~x 2 ZZd, and t1 � t2 � � � � tn 2 lR. Since the in�nite volume theory istranslational invariant, we can introduce momentum operators ~P , commuting with H, such that(writing '̂(0) = '̂)(; '̂(~x1)e�(t2�t1)H '̂(~x2) : : : e�(tn�tn�1)H '̂(~xn))= (; '̂e�(t2�t1)H+i ~P �(~x2�~x1)'̂ : : : e�(tn�tn�1)H+i ~P �(~xn�~xn�1)'̂) (19)For p = (p0; ~p), p0 2 lR and ~p 2 Td, let~S(2)� (p) = Z 1�1 dt X~x2ZZd S(2)� (~0; 0; ~x; t)e�i(p0t+~p�~x) (20)It follows from (18), (19) and the spectral theorem that~S(2)� (p) = Z 10 ZTd 2EE2 + p20 (2�)d�(~q � ~p)d(; '̂E(E; ~q)'̂) (21)where E(E; ~p) is the spectral projection associated with the operators (H; ~P ). The integral overE runs from 0 to 1 and that over ~q is on Td. We can write (21) in the form~S(2)� (p) = Z 10 2EE2 + p20d��(E; ~p) (22)where the positive measure d��(E; ~p) is supported on the spectrum of H restricted to the oddstates with momentum ~p. When � = 0, we have from (17)~S(2)� (p) = 1p20 +E0(~p)2 ; E0(~p) = dXi=1(1� cos pi) + 12m2 (23)E0(~p) is identi�ed with the energy of an elementary excitation (quasiparticle) with momentum~p and mass E0(~0) =M0 = m2=2 in the free, i.e. the � = 0, case.6

When � is small, ~S(2)� (p) has the representation (see Section 3)~S(2)� (p) = c�(~p)p20 +E�(~p)2 + Z 12M0 2EE2 + p20d��(E; ~p) (24)E�(~p) is the dispersion function in the interacting theory and, as will be shown in the nextsection, it di�ers from E0(~p) by O(�2). Thus, if m is large, E�(~p) is isolated from the rest ofthe spectrum. The mass of the interacting quasiparticle is M� = E�(~0) =M0 +O(�2).To study the spectrum of H on even states, consider the truncated four-point functionD�(x1; x2;x3; x4) = S(4)� (x1; x2; x3; x4)� S(2)� (x1; x2)S(2)� (x3; x4) (25)where xi = (ti; ~xi). From translation invariance, D� depends only on di�erence variables. Let� = x2 � x1, � = x4 � x3 and � = x3 � x2. Writing � = (�0; ~�), etc, it follows from (19) that if�0 = �0 = 0, D�(�; �; �) = (�(�~�); e�j� jHei ~P �~��(~�)) (26)where �(~�) = '̂(~0)'̂(~�)� (; '̂(~0)'̂(~�)) (27)Let f : ZZd ! C be an arbitrary function vanishing outside a �nite set and let ~f(~p) and ~D�(p; q; k)be respectively the Fourier transforms of f(~x) and D�(�; �; �), de�ned as in (20). A simplecalculation shows thatZ Z 1�1 Z ZTd dd+1p dd+1q ~f(~p) ~f(~q) ~D�(p; q; k) == Z 10 ZTd 2Ek20 +E2 (2�)3d+2�(~q � ~k)d(�(f); E(E; ~q)�(f)) (28)where �(f) = X~x2ZZd f(~x)�(�~x) (29)Formula (28) is similar to (21). The singularities in k0, for �xed ~k, of the lhs of (28) givedirect information about the spectrum of H on the even subspace of states with momentum ~k.7

To test for the presence of bound states, it is su�cient to study the spectrum on the subspaceof zero-momentum states.When � = 0 and ~k = 0, a direct calculation shows that( ~f; ~D0(k0; ~k = 0); ~f)L2 = �4 (2�)d+1 ZTd ddp j ~f(~p) + ~f(�~p)j2E0(~p)[E0(~p)2 + 14k20] (30)where the lhs of (30) is a short notation for the lhs of (28). The rhs in (30) is analytic in k0 forjImk0j < 2M0. Then, for the free (� = 0) theory, there is no energy spectrum in the interval(0; 2M0) for even states with zero momentum, as it should be.We will show in the next Section that, if � > 0 and the (normal ordered) interacting polyno-mial has a negative quartic term, then the left hand side of (28) has a singularity on the positiveimaginary axis below 2M� if d � 2. Therefore, in this case, we do get two-particle bound states.3 Analysis of Bound StatesThe analysis of bound states in local relativistic �eld theories using the Bethe-Salpeter equationis well known. In our particular problem, we deal with a slightly non-local �eld theory on alattice space. The non-locality makes the Bethe-Salpeter kernel more complicated and the latticemakes unsuitable the use of canonical relative coordinates, as in e.g. [6]. Nevertheless, usingthe coordinates �, � and � , de�ned before (26) most of the analysis can be done in the standardway. Therefore we will be brief.For ease of computation, the interacting polynomial (5) is written as an Hermite expansion,with the generating function for the monomials : xk : given by: ei�x := ei�xe� 12�2c (31)where � 2 lR and c = 1(2�)d+1 Z 1�1 dp0 ZTd ddp 1p20 +E0(~p)2 = S(2)0 (~0; 0;~0; 0) (32)8

Let ~��(p) = ~S(2)� (p)�1 and let ~k�(p) = ~�0(p)� ~��(p), so that Dyson's equation~S(2)� = ~S(2)0 + ~S(2)0 ~k� ~S(2)� (33)is satis�ed.Diagrammatically, ~k� is the sum of all one-particle-irreducible Feynman graphs with twoexternal legs. This implies ~k� and hence ~�� to be analytic on jImp0j < 2M0 for real ~p. Also,since ~k� is O(�) near the zeroes (�iE0(~p); ~p) of ~�0(p), it follows that ~��(p) has zeroes nearby,which we call (�iE�(~p); ~p) with E�(~p) � E0(~p) = O(�). That these zeroes can only be locatedon the imaginary axis follows from general principles, see eq. (21). The representation (24) withc�(~p) = 1 +O(�) follows immediately from the above facts. Actually, we have E�(~p)�E0(~p) =O(�2) since @@� ~��(p)j�=0 = 0 by explicit computation.We next study the truncated four point function (25) using the Bethe-Salpeter equationD� = D0� +D0�K�D� (34)where D�, etc, are operators de�ned by the kernels D�(x1; x2;x3; x4), etc, andD0�(x1; x2;x3; x4) = S(2)� (x1; x3)S(2)� (x2; x4) + S(2)� (x1; x4)S(2)� (x2; x3) (35)The Bethe-Sapeter kernel K�(x1; x2;x3; x4) is the sum of all connected Feynman diagramswith four (amputated) external lines which are (channel) two-particle irreducible. Introducingthe relative coordinates �, � and � as in Section 2, it follows that the Fourier transform of thekernels of D�, D0� and K� satisfy an equation similar to (34):~D�(k) = ~D0�(k) + (2�)�2(d+1) ~D0�(k) ~K�(k) ~D�(k) (36)where e.g. ~D�(k) is de�ned by the kernel ~D�(p; q; k), i.e.,( ~D�(k)f)(p) = Z 1�1 dq0 ZTd ddq ~D�(p; q; k)f(q) (37)The ladder approximation which we adopt here consists of replacing ~K� by its �rst orderterm ~L� in the perturbation expansion. Explicit calculation shows that~L�(p; q; k) = �34a2�[E0(~p) +E0(~q) +E0(~p� ~k) +E0(~q � ~k)] (38)9

At zero total momentum, ~L�(p; q; (k0;~0)) = �32a2�[E0(~p) +E0(~q)] (39)We see that ~L�(k0;~0) is a rank two operator, in contrast with what happens in a genuine local�eld theory, where the rank is just one.Equation (36) with ~K� replaced by ~L� can be solved for ~D� to yield~D�(k0) = [1� (2�)�2(d+1) ~D0�(k0)~L�(k0)]�1 ~D0�(k0)= ~D0�(k0)[1 � (2�)�2(d+1) ~L�(k0) ~D0�(k0)]�1 (40)where ~D�(k0) = ~D�((k0;~0)), etc.>From (35), one can show that the action of ~D0�(k0) on functions f(p) depending only on ~pis ( ~D0�(k0)f)(p) = (2�)d+1 ~S(2)� (p) ~S(2)� (k � p)[f(~p) + f(�~p)] (41)Therefore, if f depends only on ~p, we have(~L�(k0) ~D�(k0)f)(p) = �3a2�(2�)d+1[�0(f) + �1(f)E0(~p)] (42)where �n(f) = 12 ZTd ddq G(~q; k0)E0(~q)n[f(~q) + f(�~q)] ; n = 0; 1 (43)and G(~q; k0) = Z 1�1 dq0 ~S(2)� (q) ~S(2)� (k0 � q0; ~q) (44)It follows from (24) and from a simple analytic continuation argument that G(~q; k0) is analyticon jImk0j < 2E�(~0). This result depends on the fact that E�(~0) � E�(~p) for any ~p 2 Td, whichholds because S(2)� (x; y) > 0.Recall, from (28), that the basic object we want to analyze is (f; ~D�(k0)f), which has theform (f; ~D�(k0)f) = 2(2�)d+1 ZTd ddp f(~p)G(~p; k0)g(~p; k0) (45)10

where g(�; k0) = [1� (2�)�2(d+1) ~L�(k0) ~D0�(k0)]�1f (46)The only singularities of (45) on jImk0j < 2E�(~0) must come from those of g(�; k0), which in turncome from the zeroes of 1���(k0), where ��(k0) are the eigenvalues of (2�)�2(d+1) ~L�(k0) ~D0�(k0)on the space generated by the functions 1 and E0(~p).We �nd ��(k0) = �3a2�(2�)�(d+1)(�(k0)� [�(k0) (k0)]1=2) (47)where�(k0) = ZTd E0(~q)G(~q; k0)ddq; �(k0) = ZTd G(~q; k0)ddq; (k0) = ZTd E0(~q)2G(~q; k0)ddq (48)Now, from (24), G(~q; k0) can be written asG(~q; k0) = �2 c�(~q)2E�(~q)[E�(~q)2 + 14 (k0)2] +G1(~q; k0) (49)where G1(~q; k0) is analytic on jImk0j < E�(~0) + 2M0.>From general principles, the singularities of (45) can only be located on the imaginaryk0 axis. Writing k0 = i� with � � 0, it is possible to show (using an explicit formula) thatG(~q; i�) > 0 for 0 � � < 2E�(~0). It follows then that �(i�), �(i�) and (i�) are positive and,by Schwarz's inequality, � � [� ]1=2 on 0 � � < 2E�(~0).For space dimension d � 3, then �(i�), �(i�) and (i�) increase to a �nite limit as � !2E�(~0) because the singularity generated by G(~q; i�) is quadratic and therefore integrable. Thus,if � is small enough, 1 � ��(i�) cannot be zero on 0 < � < 2E�(~0) so that, in the ladderapproximation, there are no bound states.If d < 3, �, � and diverge as �! 2E�(~0), but �� [� ]1=2 remains �nite. This yields thenonvanishing of 1� ��(i�). Finally, 1� �+(i�) is nonzero if a2 > 0, and has a unique zero onthe interval 0 < � < 2E�(~0), if a2 < 0. This implies the existence of one bound state for thelast case. 11

Let M� = E�(~0) be the mass for a single quasiparticle in the interacting theory. The massM� of the bound state is the solution of (assuming a2 < 0)F (�; iM�) = �(2�)d+13a2� (50)where F (�; k0) = �(�; k0) + [�(�; k0) (�; k0)]1=2, and we have made explicit the � dependenceof �, � and . Let E = 2M� �M�. Performing an asymptotic analysis of the coe�cients �, �,and we �nd E(�) = 8>><>>: 94 �2m4 a22(1 +O(�)) ; if d = 1exp h� 4�m23ja2j�(1 +O(�))i ; if d = 2 (51)4 ConclusionsIn this paper, we have analyzed the existence of bound states for the generator of stochastic dy-namics in purely relaxational lattice Landau-Ginzburg models. This problem is directly relatedto decay rates of some observables.We have shown the existence of a bound state for small coupling if the polynomial interactionhas a negative quartic term and the space dimension is one or two. This result was obtainedby analyzing the Bethe-Salpeter kernel of the non-local lattice quantum �eld theory associatedwith the stochastic model. Computations have been done in the ladder approximation, whichproved to be quite reliable in cases where a rigorous analysis could be performed [6].Acknowledgements: This work was partially supported by CNPq and PRONEX (Brazil).References[1] P. C. Hohenberg, B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).[2] H. Spohn, Large Scale Dynamics of Interacting Particles, Springer Verlag, Berlin (1991).[3] Yu. G. Kondratiev and R. A. Minlos, J. Stat. Phys. 87, 613 (1997).12

[4] J. Glimm, A. Ja�e, Quantum Physics: A Functional Integral Point of View, SpringerVerlag, New York (1987).[5] J. Dimock, J. Stat. Phys. 58, 1181 (1990).[6] J. Dimock, J. P. Eckmann, Commun. Math. Phys. 51, 41 (1976).

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