16 April 2001

Physics Letters A 282 (2001) 169–174www.elsevier.nl/locate/pla

Noise-induced bound states

Emmanuel PereiraDepartamento de Física-ICEx, UFMG, CP 702, 30.161-970 Belo Horizonte MG, Brazil

Received 1 December 2000; accepted 12 March 2001Communicated by A.R. Bishop

Abstract

We investigate the presence of two-particle bound states in the stochastic dynamics generator of weakly coupled Ginzburg–Landau models, and study their dependence on the noise strength. For the space dimensiond 3, by analyzing the Bethe–Salpeter equation in the ladder approximation, we show that a bound state appears but disappears again at a higher value of thenoise intensity. Furthermore, we show that in the case of the polynomial interaction with a negative quartic term the bound stateappears and disappears for the noise intensity much smaller than that for the interaction with the quartic term positive. We alsodescribe the curves giving the bound states masses in terms of the noise strength, which show the effect that, for a suitable noiseintensity, the two-particle bound state mass becomes smaller than the one-particle mass. 2001 Published by Elsevier ScienceB.V.

PACS: 05.40.Ca; 05.45.-aKeywords: Time-dependent Ginzburg–Landau models; Two-particle bound states; Noise strength

The presence of noise in nonlinear dynamics systems produces remarkable and counterintuitive phenomenadirectly related to physics, chemistry, biology and several fields. In some situations, instead of disorder, noiseproduces a transition to an ordered state [1], not observed in the absence of noise, which may still appear anddisappear according to the noise strength [2,3]. Sometimes, the addition of a precise amount of noise may improve(instead of disturbing) the performance of some devices (stochastic resonance) [4].

In this Letter we investigate some effects, also due to changes in the noise intensity, in the dynamics of somescalar field models extensively used in physics. Precisely, we analyze the low-lying spectrum of the stochasticLangevin dynamics generator associated to weakly coupled (time-dependent) Ginzburg–Landau (GL) models,searching for the presence of bound states of two quasi-particles and studying their dependence on the noisestrength. Using a Feynman–Kac formalism, we map the stochastic problem into a quantum field theory and then,by analyzing the Bethe–Salpeter (BS) equation in the ladder approximation (details below), we show that noiseproduces bound states which however disappear at a higher value of the noise intensity. Furthermore, for the spacedimensiond 3, we show that, when the polynomial interaction (in the GL potential) has a negative quartic term,a bound state appears and disappears for the noise intensity much smaller than that necessary to create a similarphenomenon for the interaction with a positive quartic term. We still show that, for a suitable noise intensity, thetwo-particle bound state mass becomes smaller than the one-particle mass.

E-mail address: emmanuel@fisica.ufmg.br (E. Pereira).

0375-9601/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0375-9601(01)00177-3

170 E. Pereira / Physics Letters A 282 (2001) 169–174

We remark that the presence of bound states in the spectrum of these dynamical systems shows up directly in theapproach to equilibrium. Thus, one of the consequences of our results is to show how the approach to equilibriumof even states of physical systems described by these GL models changes with the addition of a precise amount ofnoise. We give examples describing concrete physical systems below.

We now present the model and the approach to be used. To avoid ultraviolet problems (irrelevant for the lowspectrum) we consider systems in a (space) lattice with action given by

(1)S(ϕ)=∑x∈Zd

1

2ϕ(x)((−+m2)ϕ)(x) + λP(

ϕ(x)),

whereϕ(x) ∈ R, is the lattice Laplacian,λ andm are positive parameters (λ 1 m), andP an evenpolynomial bounded from below. The stochastic dynamics ofϕ is given by the Langevin equation

(2)∂

∂tϕ(x, t) = −1

2

δ

δϕ(x, t)S + η(x, t),whereη is a Gaussian white-noise random variable with the expectationsE(η(x, t)) = 0, E(η(x, t)η(y, t ′)) =γ δx,yδ(t − t ′), γ positive. Such models frequently appear in the study of dynamical critical phenomena: e.g., theycan be used to describe the time evolution of an order parameter (e.g., magnetization) for a statistical mechanicalsystem [5,6]. The GL interaction itself is a recurrent theme in physics, which makes of general interest the study offundamental properties of such stochastic dynamics generator [5–7]. The search of bound states for these modelshave been considered in a previous paper [8], but there the dependence on the noise intensity is lost:γ is fixed as 1,which makes impossible any question related to changes in the noise.

Here, as in [8], we will associate the dynamical system to a quantum field theory using an integral representationfor the field correlation functions, namely, a Feynman–Kac formula (which is standard in quantum physics andfield theory [9,10]). Then, analyzing the associated BS equation (also standard in field theory [9,10]) we studythe bound states. To avoid a technically involved presentation we skip mathematical details which may be foundin [8,11].

The dynamics given by (2) is determined by a Markov semigroup: precisely, for any functionf (ϕ) with timeevolution given byft (ψ)= E(f (ϕ(t))), whereϕ(0)= ψ (initial condition in (2)), we haveft determined by theMarkov semigroupe−tH with the generator given by

(3)Hf = ∑

x∈Zd

−1

2γ

∂2

∂ϕ(x)2 + 1

2

∂S

∂ϕ(x)∂

∂ϕ(x)f,

whereH is (Hermitian) positive onL2(dµ), dµ ≡ e−S(ϕ)/γ dϕ/normalization. Its eigenfunction with zeroeigenvalue is given byf = 1. Using the unitary operatorU from L2(dµ) to L2(dϕ) given by (Uf )(ϕ) =Z−1/2e−S/2γ f (ϕ), we have (the Schrödinger type operator)

(4)L=UHU−1 =∑x∈Λ

−1

2γ

∂2

∂ϕ(x)2 + 1

4

[1

2γ

(∂S

∂ϕ(x))2

− ∂2S

∂ϕ(x)2],

L in L2(dϕ). Hence, standard procedures [9] lead to the Feynman–Kac representation

(Ω,f1e

−(t2−t1)Hf2 . . . e−(tn−tn−1)HfnΩ

)L2(dµ)

= (UΩ,f1e

−(t2−t1)Lf2 . . . e−(tn−tn−1)LfnUΩ

)L2(dϕ)

(5)=∫f1

(ϕ(t1)

). . . fn

(ϕ(tn)

)dρ,

E. Pereira / Physics Letters A 282 (2001) 169–174 171

whereΩ(ϕ) = 1 is the ground state ofH , f1, f2, . . . are functions ofϕ, t1 t2 · · · tn, anddρ = e−W dν/∫e−W dν, with

(6)W =∞∫

−∞dt

∑x∈Zd

λ

4γP ′(ϕ(x, t))[(−+m2)ϕ](x, t) + λ2

8γP ′(ϕ(x, t))2 − λ

4P ′′(ϕ(x, t)),

where′ means the derivative in relation toϕ, anddν is a Gaussian measure with mean zero and covariance givenby

(7)γC(x, t; y, t ′) ≡ γ

(2π)d+1

∞∫−∞

dp0

∫Td

ddpeip0(t−t ′)+i p·(x−y)

p20 + (∑d

i=1(1− cospi)+m2/2)2 ,

Td is the torus(−π,π]d .From the expression above, for largem and smallλwe shall obtain, first in the low spectrum, a one quasi-particle

state with massm2/2+O(λ2). Here, we intend to study the bound states of two quasi-particles in first order inλ,i.e., the spectrum below 2×m2/2.

We consider the truncated four-point functionDλ(x1, x2;x3, x4)≡ 〈ϕ(x1)ϕ(x2)ϕ(x3)ϕ(x4)〉 − 〈ϕ(x1)ϕ(x2)〉 ×〈ϕ(x3)ϕ(x4)〉, wherex = (x0, x), x0 ∈ R, x ∈ Z

d , and〈. . .〉 is the average respect todρ (5). Due to translationinvariance,Dλ depends only on difference variables. Hence, we introduceξ = x2 − x1, η = x4 − x3, τ = x3 − x2.Writing ξ = (ξ0, ξ), etc., and takingξ0 = η0 = 0, from (5) we get

(8)Dλ(ξ, η, τ )=(θ(−ξ), e−|τ0|H+i P ·τ θ

(η)),where P is the momentum operator (which commutes withH ), andθ(η)= ϕ(0)ϕ(η)Ω−(Ω, ϕ(0)ϕ(η)Ω)Ω ; ϕ(x)is the zero time field atx ∈ Z

d . Now, for any arbitrary functionf (with compact support), the spectral theoremgives us

(9)

∫ ∞∫−∞

∫ ∫Td

dd+1pdd+1q f( p)f

(q)Dλ(p, q, k)=∞∫

0

∫Td

2E

k20 +E2

(2π)3d+2δ(q − k)d(θ(f ),E(

E, q)θ(f )),where f is the Fourier transform off and f the complex conjugate;θ(f ) = ∑

x f (x)θ(−x); E(E, q) is thespectral projection associated with the operators(H, P ), andE runs from 0 to∞. For a fixedk, the singularitiesin k0 determine the spectrum ofH on the even subspace of states with momentumk. To analyze only the existenceof bound states we can takek = 0.

First, let us write (for ease of computation) the polynomial interactionP in terms of Wick ordered monomials(in respect to the covarianceγC): P(ϕ)= ∑N

n=4(an/n!) :ϕn:, n even.We use the Bethe–Salpeter equationDλ = D0λ + DλKλD0λ, whereD0λ(x1, x2;x3, x4) ≡ 〈ϕ(x1)ϕ(x3)〉 ×

〈ϕ(x2)ϕ(x4)〉 + 〈ϕ(x1)ϕ(x4)〉〈ϕ(x2)ϕ(x3)〉, to study the truncated four-point function. The BS kernelKλ(x1, x2;x3, x4) is given by the sum of all (channel) two-particle irreducible connected Feynman diagrams with four(amputated) external lines. From BS equation we obtain

(10)Dλ(k)= D0λ(k)[1− (2π)−2(d+1)Kλ(k)D0λ(k)

]−1,

where the notation means(Dλ(k)f )(p)≡∫ ∞−∞ dq0

∫Tdddq Dλ(p, q, k)f (q), etc., wherep,q, k are the conjugate

variables (respect to Fourier transform) ofξ, η, τ . Recall from (9) that what we want to study is(f, Dλ(k0)f ). Wemake the calculations up to first order inλ, which we call the ladder approximation (some known rigorous results[12] show that such an approximation essentially contains the information about the low spectrum; more comments

172 E. Pereira / Physics Letters A 282 (2001) 169–174

below). We get

(11)(f , D0λ

(k0, k = 0

), f

)L2 = γ 2π

4(2π)d+1

∫Td

ddp|f ( p)+ f (− p)|2

E0( p)[E0( p)2 + (1/4)k20],

whereE0( p) = ∑dj=1(1 − cospj )+m2/2. The expression above is analytic ink0 on the whole complex plane,

except on the segments[m2,m2 + 4d] and [−m2 − 4d,−m2] in the imaginary axis. Thus, the singularities on| Imk0|<m2/2 must come from[1− (2π)−2(d+1)Kλ(k0)Dλ(k0)]−1. We have

(12)(Kλ(k0)f

)(p)= −3a4λ

2γ

[E0

( p) ∞∫−∞

∫Td

dd+1q f (q)+∞∫

−∞

∫Td

dd+1q E0(q)f (q)

],

i.e., Kλ(k0, 0) is a rank-two operator, generated by the functions 1 andE0( p) (whereas, we must emphasize it, ina local genuine field theory the rank is one). From these previous expressions, the singularities come only from thezeroes of[1− (2π)−2(d+1)Kλ(k0)Dλ(k0)]. Thus, calculating the eigenvalues of(2π)−2(d+1)Kλ(k0)Dλ(k0) on thespace generated by 1 andE0( p) we obtain

eigenvalues= −3

4a4λγ (2π)−dµ±, µ± = α ± √

βδ,

α(k0)≡∫Td

ddq1

E0(q)2 + k20/4, β(k0)≡

∫Td

ddq1

E0(q)[E0(q)2 + k20/4] ,

(13)δ(k0)≡∫Td

ddqE0(q)

E0(q)2 + k20/4.

Hence,(·, Dλ(k0)·) acting on the eigenvectors√δ± √

βE0(·), which correspond to theµ± described above, givesus

(14)(√δ± √

βE0(·), Dλ(k0)[√δ± √

βE0(·)]) = ±(2π)d+2γ 2√βδµ±

1+ (3/4)(2π)−da4λγµ±.

A simple analysis shows that√βδ is analytic on| Imk0| < m2. Thus, the singularities come from the zeroes of

1 + (3/4)(2π)−da4λγµ±. Writing k0 = iχ = i(m2 − ε), we haveα(iχ), β(iχ), δ(iχ) positive andα √βδ,

hence:µ+ is positive andµ− is negative. In short, the equation for the masses of the bound states in terms of thenoise intensityγ is given by

(15)µ±(ε)= −4(2π)d/3a4λγ,

whereε runs in[0,m2] (recall thatε comes fromχ =m2 − ε). And, sinceµ+ is positive andµ− negative, boundstates associated toµ+ may appear just fora4< 0 (for stability, the theory must contain, of course, another termof higher power inϕ with positive coefficient); and those associated toµ−, for a4 > 0. Let us now describe theresults.d = 1: As ε→ 0, α, β andδ diverge:µ+ diverges as 1/

√ε, and so, for smallγ (λ is taken small and fixed)

there is always a bound state associated to it; butµ− is finite, thus, for smallγ there is no bound state associatedto it. As ε→m2 (recall,m 1),µ+ → c/m4, i.e.,a4λγ ≈m4 (c is constant, multiple ofπ ), and−µ− runs fromc/m6, asε→ 0, toc/m8 asε→m2. We keep the same notationc for different constants. The curves for the boundstate massM∗ (M∗ = 2Mλ− ε;Mλ =m2/2 is the one-particle mass) versusγ are depicted in Figs. 1 and 2 for thestates associated toµ+ and−µ−, respectively. The dependence on the noise intensity is transparent: turning up thenoise, we first get a bound state associated toµ+ with mass close to 2×m2/2. Increasing the noise intensityγ ,

E. Pereira / Physics Letters A 282 (2001) 169–174 173

Fig. 1. CurveM∗ (bound state mass) versusγ (noise intensity). The one-particle mass is taken as 50.cλ = −3a4λ/4(2π)d , whereλ is the

coupling constant anda4 the coefficient of the quartic term of the polynomial interaction.µ+ is an eigenfunction associated to the BS equation.Note thatM∗ becomes smaller than the one-particle mass for largeγ .

Fig. 2. CurveM∗ (bound state mass) versusγ (noise intensity). The one-particle mass is taken as 50.cλ = −3a4λ/4(2π)d , whereλ is the

coupling constant anda4 the coefficient of the quartic term of the polynomial interaction.µ− is an eigenfunction associated to the BS equation.There is a bound state only for largeγ . Note thatM∗ becomes smaller than the one-particle mass for largeγ .

the rhs of (15) becomes smaller and so the bound state mass, i.e., it becomes more tightly bound asa4λγ goes toO(m4). Then, the bound state disappears. However, fora4λγ running fromO(m6) toO(m8) it appears again, nowassociated toµ−, i.e., for a system witha4> 0.d = 2: The picture is quite similar.α, β andδ diverge asε→ 0: µ+ diverges, however as ln(1/ε), andµ− is

finite. Asε→m2, µ+ → c/m4; andµ− also runs fromc/m6 (ε = 0) to c/m8 (ε =m2). Again, the curves for thebound state massesM∗ are quite similar to Figs. 1 and 2.d = 3: α, β andδ become finite (hence, there is no bound state for smallγ ) andµ+(0) = c/m2. Discarding

this point, the curves forM∗ are still similar to Figs. 1 and 2 (as said, now without the divergence ofµ+ at ε = 0,which means no bound state for smallγ ).

A very interesting phenomenon described by the curves forM∗ is that, for a suitable noise intensity, the two-particle bound state mass becomes smaller than the one-particle massMλ (Mλ =m2/2+O(λ2)).

We now illustrate some of our predicted results with experimentally observable effects in concrete physicalsystems. For the case of a magnetic system governed by this time-dependent GL model (the fieldϕ describingthe magnetization), our results show that the time relaxation of the magnetization fluctuation given by〈ϕ(t)ϕ(0)〉goes as exp[−Mt], whereM is the one-particle mass, while the relaxation in the fluctuation of susceptibility

174 E. Pereira / Physics Letters A 282 (2001) 169–174

〈ϕϕ(t);ϕϕ(0)〉 goes as exp[−M∗t], whereM∗ is the two-particle bound state mass, and it changes with the noisestrength, going from 2M − ε to values smaller thanM as the noise increases. As a second example, we considerthe diffusion of small interacting balls, e.g., proteins, in a fluid. If we experimentally analyze this system usinglight scattering (nowϕ is related to the electric fieldE scattered by the diffusing particles, and we assume theinteractions are such that the dynamics is governed by the time-dependent GL model), according to our results,we will see the temporal autocorrelation function of the fluctuations in this scattered electric fieldE decaying asexp[−Mt], and the temporal autocorrelation function of the intensity of scattered light decaying as exp[−M∗t].Moreover, as stated above, withM∗ being quite sensitive to the noise strength, becoming smaller thanM for largenoise.

To emphasize, once more, the physical relevance of our results, we remark that the spectral properties describedhere still hold for related dynamical problems in different scenarios: e.g., imaginaryt in Eq. (2) gives systemsgoverned by a nonlinear Schrödinger equation (e.g., a nonlinear optical system).

Finally, as a comment on the reliability of the ladder approximation, we remark that the complete nonperturbativeanalysis for smallγ (γ < m2) follows, due to simple rescaling arguments, the caseγ = 1, which is rigorouslyproved in [11] adapting techniques of constructive field theory [13] and using the convergence of the clusterexpansion for stochastic lattice field models established in [14]. For this caseγ = 1, the complete analysis givesonly negligible corrections to the ladder approximation results. Note that the range ofγ < m2 is enough to coverthe main features of the bound-state mass behavior for the case of a negative quartic term in the polynomialinteraction (namely, this range covers the decay of the bound-state mass with the noise strength). For largeγ , thenonperturbative treatment is currently being investigated.

Acknowledgements

This work was partially supported by CNPq and PRONEX (Brazil). The author thanks R. Schor and M. O’Carrollfor useful discussions.

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