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Quantum Fidelity Approach to the Ground State Properties of the 1D Quantum fidelity approach to the ground state properties of the 1D ANNNI 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 F´ısica, Universidade Federal Fluminense Av. Litorˆanea s/n, Niter´oi, 24210-340, RJ, Brazil In this work we analyze the ground-state properties of the s = 1/2 one-dimensional ANNNI 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, h2, 2i 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 h2, 2i phases is possibly of first-order. PACS numbers: 75.10.Pq,75.10.Jm I. INTRODUCTION neighbor interactions in the presence of a magnetic field in the transverse direction. The transverse Ising model At very low temperatures, quantum fluctuations play was initially used to explain the order-disorder transi- an important role in the characterization of the ground- tions observed in KDP ferroelectrics [2]. An experimen- state properties of quantum systems [1]. These fluctua- tal realization of that model in real magnetic systems was tions are induced by varying the relative strength of com- observed in LiHoF4 in an external field [3]. An exact so- peting interactions among the constituents of the system lution to the model in one dimension was subsequently or by changing the strength of the applied fields. When found by Pfeuty by mapping the set of the original spin large enough, quantum fluctuations dramatically change operators onto a new set of noninteracting spinless Fermi the nature of a given ground-state. A quantum phase operators [4]. Recently, a degenerate Bose gas of rubid- transition may occur, thereby creating a boundary be- ium confined in a tilted optical lattice was used to sim- tween distinct ground-states. ulate a chain of interacting Ising spins in the presence The one-dimensional axial next-nearest neighbor Ising of both transverse and longitudinal fields [5]. It has also (1D ANNNI) model in a transverse field is one of the been proven that the ground-state properties of the d- simplest models in which competing interactions lead to dimensional Ising model with a transverse field, is equiv- modulated magnetic orders, frustration, commensurate- alent to the (d + 1)-dimensional Ising model without a incommensurate transitions, etc. These features are magnetic field at finite temperatures [6–8]. known to appear in the ground-state of the model in the In the case of the 1D ANNNI model in a transverse one-dimensional case. field at T = 0 and the 2D ANNNI model (without trans- Frustration in the 1D ANNNI model arises from the verse field) at finite T, such equivalence may only exist in competition between nearest-neighbor interactions which the limit of very strong transverse field and in the weak- favor ferromagnetic alignment of neighboring spins, while coupling limit of the NN- and NNN-interactions of the an interaction with opposite sign between the next- 1D model [9–11]. There is no guarantee that the ground- nearest-neighbors fosters antiferromagnetism. At T = 0, state phase diagrams of those models bear any resem- the presence of a transverse magnetic field gives rise to blances to each other. Therefore we shall not compare arXiv:1705.09679v2 [cond-mat.stat-mech] 14 Sep 2017 quantum fluctuations that play an analogous role as that the phase diagrams of these two models in this work. of temperature in thermal magnetic systems that are re- The transverse 1D ANNNI model has been the sub- sponsible for triggering phase transitions. ject of great interest [12, 13], in part due to the num- In one dimension, the ANNNI model in a transverse ber of quantum phases with unusual and intriguing fea- field is actually an extension of the transverse Ising tures it displays. Several analytical and numerical meth- model. The latter consists of Ising spins with nearest- ods have been employed to establish its phase diagram. Among those studies, there are analysis using quantum Monte Carlo [14], exact diagonalization of small lattice systems [15, 16], interface approach [17], scaling behav- ∗Electronic address: bonfi[email protected] ior of the energy gap [18], bosonization and renormal- †Electronic address: [email protected]ff.br ization groups methods [19], density matrix renormaliza- ‡Electronic address: [email protected]ff.br tion group [20, 21], perturbation theory [22], and matrix 2 product states [23]. of the phases. In our work, paragnetism only occurs at The phase diagrams from those works do not neces- high fields Bx, hence it does not appear in our phase dia- sarily agree with each other. In the following we dis- gram, which covers the low field region only. In addition, cuss the common features as well as some of the dif- our numerical analysis points to the the existence of a ferences between them. In most of the studies, there region of finite width for the floating phase. is ferromagnetism for J2 < 0.5 and h2, 2i antiphase for J2 > 0.5. The transition lines usually end at the mul- II. THE MODEL ticritical point (J2,Bx) = (0.5, 0.0). The phase diagram of Dutta and Sen shows antiferromagnetism instead of the h2, 2i antiphase for J2 > 0.5 [19]. That is a rather The one-dimensional ANNNI model in the presence of surprising result not to show the antiphase, since even a transverse magnetic field is defined as in the classical case, Bx = 0, that antiphase is ener- z z z z x getically favorable. Some authors obtain diagrams with H = −J1 X σi σi+1 + J2 X σi σi+2 − Bx X σi . (1) 5 phases, namely, ferromagnetic, paramagnetic, modu- i i i lated paramagnetic, floating, and antiphase. Such are The system consists of L spins, with s = 1/2, where the diagrams of Arizmendi et al. [14], Sen et al. [15], and α σi (α = x,y,z) is the α-component of a Pauli oper- Beccaria et al. [20, 21]. On the other hand, Rieger and ator located at site i in a chain where periodic bound- Uimin [16], Chandra and Dasgupta [22], and Nagy [23] ary conditions are imposed. We considered ferromagnetic present diagrams with 4 phases, ferromagnetic, param- nearest-neighbor Ising coupling J1 > 0 and antiferromag- agnetic, floating, and antiphase. In Refs.[16] and [23] the netic next-nearest-neighbor interaction J2 > 0. Bx is the boundary lines meet at the multicritical point, whereas strength of a transverse applied magnetic field along the in Ref. [22] the paramagnetic phase is restricted to su- x-direction. We set J1 = 1 as the unit of energy. ficiently high Bx, thus its boundary lines do not reach At T = 0 and in the absence of an external magnetic the multicritical point. In the studies by Sen [17] and field (Bx = 0), the model is trivially solvable and presents Guimar˜aes et al. [18], one finds diagrams with 3 phases several ordered phases. For J2 < 0.5, the ground-state only, ferromagnetic, paramagnetic, and antiphase, where ordering is ferromagnetic, and for J2 > 0.5, the order- their transition lines end at the multicritical point. The ing changes to a periodic configuration with two up- phase diagram of Dutta and Sen [19] displays ferromag- spins followed by two down-spins which is termed the netism, a spin-flop phase, a floating phase, and an anti- h2, 2i-phase, or antiphase. In this work we have used ferromagnetic phase. In that work, the floating phase lies the notation hp, qi to represent a periodic phase, with between the antiferromagnetic and the spin-flop phases. p up-spins followed by q down-spins. At J2 = 0.5, the Such spin-flop and antiferromagnetic phases do not ap- model has a multiphase point where the ground-state is pear in any of the other phase diagrams in the literature. infinitely degenerate and a large number of hp, qi-phases In addition, their transition lines do not end at the mul- are present, as well as other spin configurations. The ticritical point. As one can see, there is not a consensus number of phases increases exponentially with the size of on the ground-state phase diagram of the model. The the system [25, 26]. On the other hand, for a non-zero number, nature, or location of the phases usually vary external magnetic field and J2 = 0, the model reduces to from one work to another. In any case, all the studies in the Ising model in a transverse field, which was solved the literature report on a finite number of phases. As we exactly by Pfeuty [4]. The transverse magnetic field in- shall see below, our phase diagram agrees with some of duces quantum fluctuations that eventually drive the sys- the works in the literature with regard to the existence tem through a quantum phase transition. Its ground- of ferromagnetic, floating, and the antiphase. However, state undergoes a second-order quantum phase transi- our numerical results suggest that there are an infinite tion at Bx = 1, separating ferromagnetic from para- number of modulated phases between the ferromagnetic magnetic phases. In the 1D transverse ANNNI model, and the floating phase.
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