PHYSICAL REVIEW B VOLUME 45, NUMBER 1 1 JANUARY 1992-I Low-frequency vibrational spectrum and low-temperature specific heat of Bethe lattices Davide Cassi Dipartimento de Fisica, Universita di Parma, viale delle Scienze, 43100 Parma, Italy (Received 18 March 1991) By mapping the harmonic-oscillations equations onto the random-walk equations, we calculate analyt- ically the low-frequency vibrational spectrum of a Bethe lattice and show that it has a gap between zero frequency and a critical frequency. Using these results, we then obtain a low-temperature asymptotic ex- pression for the vibrational specific heat and find that it goes exponentially to zero as T~O. I. INTRODUCTION — mco x;=k g A;i(x. — x;)=k g (A;J — z5;~)x~ J j (3) The dynamical properties of physical systems are deep- ly affected by their internal geometry. Fractal structures are perhaps the most common example of how usual physical laws have to be modified in order to hold in the case of unusual geometrical structures. ' Recently, highly branched structures, for which Bethe lattices are the simplest models, have been the subject of growing interest, ' since they show dynamical proper- ties that are different from those of both regular and frac- tal lattices. In this Brief Report we shall calculate the low- frequency vibrational spectrum of Bethe lattices for har- monic oscillations and then, using a Debye-like model, the low-temperature specific-heat contribution due to "phonons. " The results, that markedly differ from the analogous ones for Euclidean and fractal geometries, lead to a deeper comprehension of the physics on these in- teresting structures. II. HARMONIC OSCILLATIONS ON A BETHE LATTICE Let us consider a Bethe lattice (BL) with coordination number z & 2, i.e. , an infinite treelike network each site of which has exactly z nearest neighbors (NN). On each site there is a mass m connected to its NN by z springs with elastic constant k. If we call x; the displacement of the ith mass from its equilibrium position, the equations of motion for the system read X- m =k g A, "(x — x, ) . (2) System (2) can be reduced to an algebraic system by Fourier transforming with respect to the time: d x; m =k g A; (x — x, ) for i EBL, J where A, =1 if sites i and j are NN and A, =0 other- wise ( A, ~ is the adjacency matrix of the BL). Since there are no couplings between different corn- ponents of the vectors x, , we can limit ourselves to study the scalar version of (1), describing each component in- dependently: so that the frequencies co of the normal modes are related to the eigenvalues a of A; by 2 k 2 co = — (z — a)— = coo(z — a) . m Now we are interested in the density of modes per de- gree of freedom p(co) defined in such a way that p(co)dco is the fraction of modes with frequencies between co and co+ d co [note that according to this definition f 0"p(co)dco= 1]. p(co) is related to the autocorrelation function or prob- ability of returning to the origin Po(t) of random walks on the lattice and we shall use this relation to calculate it. III. RANDOM WALKS AND VIBRATIONAL SPECTRUM Let us consider the random-walk problem on the BL described by the master equations dP, (t) =iJ g A; [P (t) — P;(t)], dt where P;(t) is the probability for the walker of being at site i at time t and w is the transition rate. The solution of (5) can be formally written as P; ( t ) = g I exp [ tio ( A zI ) ] ];i'(0), — J (6) where A is the matrix A; and I is the unit matrix. The probability of returning to the origin relative to the site i is the solution P, ( t ) of (5) corresponding to the initial conditions P, (0)=1 and P i&, i(0)=0. For a BL such a function is independent of the given site i (Ref. 7) and we shall call it simply Po(t). Now it follows from (3), (4), and (6) that Po(t) is related to p(co) by Po(t)= I p(co) exp( twco Ico02)dco—(7) so that if we know Po(t) we can in principle calculate p(co). However we are interested in the behavior of p(co) for sma11 co, which can be obtained from the asymptotic expansion of Po(t) for large t This asymptoti. c behavior is given by ' 45 454 1992 The American Physical Society