The Ising model in a transverse field on comb-like ramified linear structures

2014 Combining a real space renormalization group method and exact results, the Ising model with a transverse field has been studied at T = 0 for comb-like ramified linear structures. Tome 40 No 12 15 JUIN 1979 LE JOURNAL DE PHYSIQUE LETTRES Classification Physics Abstracts 03.65 04.90 75. 10D The Ising model with a transverse field described by the Hamiltonian where Sf, Sf, Si are Pauli spin 1/2 matrices, provides a good representation for physical systems having constituents with two low-lying energy levels which can be sponsored by states 5’f = ± 1. These include for instance phase transitions in magnetic materials with a pair of singlet crystal field levels [1] and in hydrogen-bonded ferroelectrics [2]. This model has been extensively studied when the sites i, j are nearest neighbours on a regular D dimensional lattice [3], [4]. In one dimension (D = 1) the model can be solved exactly [5]. In higher dimensions only approximate calculations exist at the moment either from series expansions [6] or from real space renormalization group methods [7], [8]. At finite temperature T 0 0 this quantum model behaves similary to the classical Ising model and spontaneous order appears below a critical temperature 7p(r) which is non zero for D > 2. At low temperature (T -~ 0) quantum effects show up. At T = 0 a new behaviour is expected. At T = 0 when the transverse field r varies the ground state displays a second order phase transition. For r/J less than a critical value (r/J)c the ground state 0 ~ is asymptotically (in the limit of an infinite system) doubly degenerate with a spontaneous (*) On leave from the Institute of Theoretical Physics, Freie Univ.. Berlin, W. Germany. Work supported by a grant from Deutscher Akademischer Austauschdienst (DAAD). ° (**) Laboratoire associe au C.N.R.S. order parameter At" = ( 0 I Sf 0 ) ; for r/J larger than (r/7)c the ground state becomes non-degenerate and the order disappears. This transition exists for D ~ 1. For D = 1 the critical field is equal to 1 [5] (duality exists) and the exact solution maps onto the Onsager exact solution [9] for the D = 2 classical Ising model (at finite temperature). This analogy extends to D dimensions as shown by Suzuki [10] : the D-dimensional Ising model in a transverse field at T = 0 is equivalent to a D + 1 dimensional classical Ising model (r = 0) at T =t= 0 (the transverse field plays the role of a temperature). The dilute Ising model in a transverse field has also been considered [11], [12]. It is related to phase transitions in a number of dilute systems [13]. The model is defined as follows : the sites i, j are defined on a regular periodic lattice but a proportion p of the sites are missing (the non-dilute model corresponds to p = 0). At T = 0 for D > 2 when the concentration p of non-magnetic sites increases the critical transverse field (r/7)~ (p) goes down and when p is larger than the percolation threshold Pc (Pc depends on the lattice considered ; for instance for a cubic lattice~ ~ 0.693), there is no more infinite cluster and no ordered ground state exists leading to (r/Y)c (p) = 0 for p > pc. It has been argued [11] that at p = Pc the critical field (~7~)c (Pc) is not zero and is larger than 1, the value for the linear chain. This is due to the fact that the Ising model in a transverse field has always a transition if the lattice structure (regular or not) is infinite and if, due to inequalities for spin quantum systems [14], (r/Y)c increases when new bonds are added. This situation is different from the usual dilute classical Ising model [15] where Tc(p) goes continuousArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019790040012023700 L-238 JOURNAL DE PHYSIQUE LETTRES ly to zero (for classical Ising models there is no ordered phase on a linear chain) as shown in recent experiments [16]. For the dilute transverse field Ising model, the critical field (~/y)c (p = Pc) reflects the detailed structure of the infinite percolating cluster (concentration of free ends, loops, dead ends... etc.). This critical field can be in principle evaluated (up to now series expansions have failed to give a definite answer [12] but real space renormalization group methods are presently used in this perspective). To relate (T/J)~ (p = pc) to the structure of the cluster it is necessary to understand how the structure affects the critical field. This is partially achieved in this letter where we show how one can study a quantum model on a ramified lattice. We shall consider here a regular ramified lattice but as shown later the method can be extended to more general non regular structures. The method consists in using first a block real-space renormalization-group iteration scheme introduced for quantum systems at T = 0 [17], [18] and the Hamiltonian (1) on a ramified linear structure is reduced to a similar Hamiltonian with new parameters on a linear chain which then is solved exactly using results of ref. [5]. This will be illustrated with simple comb-like structures. Let us first study the comb structure shown in figure 1. The Hamiltonian (1) defined on such an infinite translationally invariant lattice is divided into interand intra-block parts (the blocks are made of six sites labelled from 1 to 6). The transverse field Ising model is solved exactly for the block part and the two lowest levels + > and I > with energies E+ and Eare retained. A new set of spin 1/2 operators sa’ is associated with each block (the eigenstates of Sz’ are I + > and I )) and the intra-block parts are rewritten in the new spin representations. Taking the matrix elements of the old spins 5~ ( j being the label of the block and p the label of the spin in the block) between the block states I + > and > we get the relation (see reference [18]) Fig. 1. An example of block partitioning in the case C = 1. Table I. Summary of the results. For each different ramified structure studied (corresponding to different concentrations C of free ends) and each block used in the partitioning method, the result for (T/J)~ is given and the corresponding label in figure 1 is precised. System, C block (r/,I)~ Symbol .~......~. 1 /7 = 0.142 8 ’"!’" 1.073 0