Solving the Schrödinger equation for the Sherrington–Kirkpatrick model in a transverse field

By numerically solving the Schrödinger equation for small sizes we investigate the quantum critical point of the infinite-range Ising spin glass in a transverse field at zero temperature. Despite its simplicity the method yields accurate information on the value of the critical field and critical exponents. We obtain 0c = 1.47± 0.01 and check that exponents are in agreement with analytical approaches. There has recently been renewed interest in the study of quantum phase transitions in disordered systems [1]. In particular, Ising spin glass models in a transverse field are simple systems in which to study the effect of competition between randomness and quantum fluctuations. The case of infinite-range models is especially interesting because they show non-trivial quantum phase transitions yet are to some extent amenable to analytical computations. The canonical example in this family of models is the quantum Sherrington– Kirkpatrick (SK) model in a transverse field. At zero transverse field this reduces to the usual SK model which has a finite-temperature transition to low-temperature phase where replica symmetry is broken [2]. As the transverse field is turned on, spin-glass ordering occurs at lower temperatures and above a certain critical field the spin-glass order is completely suppressed at the expense of ordering in the transverse direction. Our understanding of this model was significantly extended by the non-perturbative analysis of Miller and Huse [3] and also by the different approach of Ye et al [4]. The critical behaviour is now well established, and values for exponents, predictions for logarithmic corrections and estimates of the value of the critical field are known. The model is therefore well adapted as a testing ground for numerical methods to investigate quantum phase transitions. From this point of view, the phase diagram of the quantum SK model in a transverse field and its zerotemperature critical behaviour have been studied using numerical techniques such as spin summation [5], perturbation expansions [6] and quantum Monte Carlo methods [7]. It is the purpose of this letter to introduce a new numerical approach based on the intuitive method of directly solving the Schrödinger equation for finite systems. Despite its simplicity, this method is able to give quantitative information on the value of the critical field and critical exponents even for the very small size systems we consider. § E-mail addresses: [email protected], [email protected] 0305-4470/97/040041+07$19.50 c © 1997 IOP Publishing Ltd L41 L42 Letter to the Editor The SK model in a transverse field is defined by the Hamiltonian,