University of Portland Pilot Scholars Physics Faculty Publications and Presentations Physics 2017 Quantum fidelity approach to the ground-state properties of the one-dimensional axial next- nearest-neighbor Ising model in a transverse field O. F. de Alcantara Bonfim University of Portland, [email protected] B. Boechat J. Florencio Follow this and additional works at: https://pilotscholars.up.edu/phy_facpubs Part of the Physics Commons Citation: Pilot Scholars Version (Modified MLA Style) Bonfim, .O F. de Alcantara; Boechat, B.; and Florencio, J., "Quantum fidelity approach to the ground-state properties of the one- dimensional axial next-nearest-neighbor Ising model in a transverse field" (2017). Physics Faculty Publications and Presentations. 61. https://pilotscholars.up.edu/phy_facpubs/61 This Journal Article is brought to you for free and open access by the Physics at Pilot Scholars. It has been accepted for inclusion in Physics Faculty Publications and Presentations by an authorized administrator of Pilot Scholars. For more information, please contact [email protected]. PHYSICAL REVIEW E 96, 042140 (2017) Quantum fidelity approach to the ground-state properties of the one-dimensional axial next-nearest-neighbor Ising model in a transverse field O. F. de Alcantara Bonfim* Department of Physics, University of Portland, Portland, Oregon 97203, USA B. Boechat† and J. Florencio‡ Departamento de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, Niterói, 24210-340, RJ, Brazil (Received 9 February 2017; revised manuscript received 7 September 2017; published 17 October 2017) In this work we analyze the ground-state properties of the s = 1/2 one-dimensional axial next-nearest-neighbor Ising model in a transverse field using the quantum fidelity approach. We numerically determined the fidelity susceptibility as a function of the transverse field Bx and the strength of the next-nearest-neighbor interaction J2, for systems of up to 24 spins. We also examine the ground-state vector with respect to the spatial ordering of the spins. The ground-state phase diagram shows ferromagnetic, floating, and 2,2 phases, and we predict an infinite number of modulated phases in the thermodynamic limit (L →∞). Paramagnetism only occurs for larger magnetic fields. The transition lines separating the modulated phases seem to be of second order, whereas the line between the floating and the 2,2 phases is possibly of first order. DOI: 10.1103/PhysRevE.96.042140 I. INTRODUCTION by Pfeuty by mapping the set of the original spin operators onto a new set of noninteracting spinless Fermi operators [4]. At very low temperatures, quantum fluctuations play an Recently, a degenerate Bose gas of rubidium confined in a tilted important role in the characterization of the ground-state prop- optical lattice was used to simulate a chain of interacting Ising erties of quantum systems [1]. These fluctuations are induced spins in the presence of both transverse and longitudinal fields by varying the relative strength of competing interactions [5]. It has also been proven that the ground-state properties among the constituents of the system or by changing the of the d-dimensional Ising model with a transverse field are strength of the applied fields. When large enough, quantum equivalent to those of the (d + 1)-dimensional Ising model fluctuations dramatically change the nature of a given ground without a magnetic field at finite temperatures [6–8]. state. A quantum phase transition may occur, thereby creating In the case of the 1D ANNNI model in a transverse field at a boundary between distinct ground states. T = 0 and the 2D ANNNI model (without transverse field) at The one-dimensional axial next-nearest-neighbor Ising (1D finite T, such equivalence may only exist in the limit of very ANNNI) model in a transverse field is one of the simplest strong transverse field and in the weak-coupling limit of the models in which competing interactions lead to modulated NN and NNN interactions of the 1D model [9–11]. There is magnetic orders, frustration, commensurate-incommensurate no guarantee that the ground-state phase diagrams of those transitions, etc. These features are known to appear in the models bear any resemblances to each other. Therefore we ground state of the model in the one-dimensional case. shall not compare the phase diagrams of these two models in Frustration in the 1D ANNNI model arises from the compe- this work. tition between nearest-neighbor interactions which favor ferro- The transverse 1D ANNNI model has been the subject magnetic alignment of neighboring spins, while an interaction of great interest [12,13], in part due to the number of with opposite sign between the next-nearest-neighbors fosters quantum phases with unusual and intriguing features that antiferromagnetism. At T = 0, the presence of a transverse it displays. Several analytical and numerical methods have magnetic field gives rise to quantum fluctuations that play been employed to establish its phase diagram. Among these an analogous role to that of temperature in thermal magnetic studies, there are analysis using quantum Monte Carlo [14], systems that are responsible for triggering phase transitions. exact diagonalization of small lattice systems [15,16], the In one dimension, the ANNNI model in a transverse field is interface approach [17], scaling behavior of the energy gap actually an extension of the transverse Ising model. The latter [18], bosonization and renormalization group methods [19], consists of Ising spins with nearest-neighbor interactions in the density matrix renormalization group [20,21], perturbation presence of a magnetic field in the transverse direction. The theory [22], and matrix product states [23]. transverse Ising model was initially used to explain the order- The phase diagrams from those works do not necessarily disorder transitions observed in KDP ferroelectrics [2]. An agree with each other. In the following we discuss the common experimental realization of that model in real magnetic systems features as well as some of the differences between them. In was observed in LiHoF in an external field [3]. An exact 4 most of the studies, there is ferromagnetism for J < 0.5 and solution to the model in one dimension was subsequently found 2 the 2,2 antiphase for J2 > 0.5. The transition lines usually end at the multicritical point (J2,Bx ) = (0.5,0.0). The phase diagram of Dutta and Sen shows antiferromagnetism instead of * bonfi[email protected] the 2,2 antiphase for J2 > 0.5[19]. This is a rather surprising †[email protected] result, not to show the antiphase, since even in the classical ‡ [email protected] case, Bx = 0, that antiphase is energetically favorable. Some 2470-0045/2017/96(4)/042140(9) 042140-1 ©2017 American Physical Society BONFIM, BOECHAT, AND FLORENCIO PHYSICAL REVIEW E 96, 042140 (2017) authors obtain diagrams with 5 phases, namely, ferromag- interaction J2 > 0. Bx is the strength of a transverse applied netic, paramagnetic, modulated paramagnetic, floating, and magnetic field along the x direction. We set J1 = 1 as the unit antiphase. Such are the diagrams of Arizmendi et al. [14], of energy. Sen et al. [15], and Beccaria et al. [20,21]. On the other At T = 0 and in the absence of an external magnetic field hand, Rieger and Uimin [16], Chandra and Dasgupta [22], (Bx = 0), the model is trivially solvable and presents several and Nagy [23] present diagrams with 4 phases, ferromagnetic, ordered phases. For J2 < 0.5, the ground-state ordering is paramagnetic, floating, and antiphase. In Refs. [16,23]the ferromagnetic, and for J2 > 0.5, the ordering changes to a boundary lines meet at the multicritical point, whereas in periodic configuration with two up-spins followed by two Ref. [22] the paramagnetic phase is restricted to sufficiently down-spins which is termed the 2,2 phase, or antiphase. high Bx ; thus its boundary lines do not reach the multicritical In this work we have used the notation p,q to represent a point. In the studies by Sen [17] and Guimarães et al. periodic phase, with p up-spins followed by q down-spins. At [18], one finds diagrams with 3 phases only, ferromagnetic, J2 = 0.5, the model has a multiphase point where the ground paramagnetic, and antiphase, where their transition lines end state is infinitely degenerate and a large number of p,q at the multicritical point. The phase diagram of Dutta and Sen phases are present, as well as other spin configurations. The [19] displays ferromagnetism, a spin-flop phase, a floating number of phases increases exponentially with the size of the phase, and an antiferromagnetic phase. In that work, the system [25,26]. On the other hand, for a nonzero external mag- floating phase lies between the antiferromagnetic and the netic field and J2 = 0, the model reduces to the Ising model spin-flop phases. Such spin-flop and antiferromagnetic phases in a transverse field, which was solved exactly by Pfeuty [4]. do not appear in any of the other phase diagrams in the The transverse magnetic field induces quantum fluctuations literature. In addition, their transition lines do not end at the that eventually drive the system through a quantum phase multicritical point. As one can see, there is no consensus on transition. Its ground state undergoes a second-order quantum the ground-state phase diagram of the model. The number, phase transition at Bx = 1, separating ferromagnetic from nature, or location of the phases usually vary from one work paramagnetic phases. In the 1D transverse ANNNI model, to another. In any case, all the studies in the literature report next-nearest-neighbor interactions introduce frustration to the on a finite number of phases. As we shall see below, our magnetic order. A much richer variety of phases becomes phase diagram agrees with some of the works in the literature possible when one varies the strength of the interactions among with regard to the existence of ferromagnetic, floating, and the spins or their couplings to the magnetic field.