arXiv:1705.09679v2 [cond-mat.stat-mech] 14 Sep 2017 ‡ † ∗ nitrcinwt poiesg ewe h next- the between sign At opposite . fosters with nearest-neighbors interaction while spins, an neighboring of alignment ferromagnetic favor which interactions nearest-neighbor between competition the in are model the features of ground-state case. These the one-dimensional in appear to etc. the known of transitions, commensurate- one to frustration, incommensurate is lead orders, interactions field magnetic competing transverse modulated which a in models in simplest model ANNNI) (1D be- phase boundary quantum a A ground-states. creating distinct thereby tween ground-state. occur, given may a transition When of fields. change nature dramatically applied the fluctuations the quantum of enough, strength large the system changing the by of constituents or com- the of among strength fluctua- relative interactions These the peting varying by [1]. induced ground- systems are the quantum tions of of characterization properties the state in role important an oe.Telte ossso sn pn ihnearest- Ising with transverse spins Ising the of of consists extension latter The an actually model. is field re- transitions. are phase that triggering systems for magnetic to sponsible that thermal rise as in role gives temperature analogous field of an play magnetic that transverse fluctuations a quantum of presence the unu dlt praht h rudsaepoete ft of properties state ground the to approach fidelity Quantum lcrncades [email protected]ff.br address: Electronic [email protected]ff.br address: Electronic lcrncades bonfi[email protected] address: Electronic rsrto nte1 NN oe rssfo the from arises model ANNNI 1D the in Frustration Ising neighbor next-nearest axial one-dimensional The play fluctuations quantum temperatures, low very At noedmnin h NN oe natransverse a in model ANNNI the dimension, one In natases eduigteqatmfieiyapoc.W n We field transverse approach. the fidelity of quantum function the a using as susceptibility field transverse a in interaction otesailodrn ftesis h rudsaephase ground-state The spins. the of h ordering spatial the to h ouae hssse ob fscn-re,weesthe whereas second-order, of be to seem h phases modulated the ASnmes 75.10.Pq,75.10.Jm numbers: PACS ( 2 2 L , , nti okw nlz h rudsaepoete fthe of properties ground-state the analyze we work this In 2 2 ∞ → .INTRODUCTION I. i i hssi osbyo first-order. of possibly is phases hss n epeita nnt ubro ouae phases modulated of number infinite an predict we and phases, .Prmgeimol cusfrlre antcfils The fields. magnetic larger for occurs only Paramagnetism ). eateto hsc,Uiest fPrln,Prln,O Portland, Portland, of University Physics, of Department J 2 o ytm fu o2 pn.W loeaieteground-sta the examine also We spins. 24 to up of systems for , eatmnod ´sc,Uiesdd eea Fluminense Federal F´ısica, Universidade de Departamento v ioˆnasn ie´i 41-4,R,Brazil RJ, 24210-340, Niter´oi, Litorˆanea s/n, Av. .Boechat B. .F eAcnaaBonfim Alcantara de F. O. rnvrefield transverse a T 0, = † n .Florencio J. and B x enpoe httegon-tt rpriso the of properties also ground-state presence has It the the that [5]. in fields proven spins sim- longitudinal been to Ising and transverse used interacting both was of of lattice chain rubid- optical a of tilted ulate gas a Bose in degenerate confined a ium Fermi Recently, original spinless noninteracting the [4]. of of operators set subsequently set new was a the onto mapping dimension operators by one Pfeuty in by model found the to lution a elzto fta oe nra antcsseswas systems magnetic LiHoF real experimen- in in observed An model transi- that [2]. order-disorder of realization ferroelectrics the tal KDP explain in model observed Ising to tions transverse used The initially field magnetic direction. was a transverse of the presence in the in interactions neighbor ingop[0 1,prubto hoy[2,admatrix and [22], theory renormal- perturbation and 21], [20, renormaliza- bosonization group matrix density tion [18], [19], gap methods behav- groups energy ization scaling the [17], lattice of approach small interface ior of 16], diagonalization quantum [15, exact using systems analysis [14], diagram. are Carlo phase there Monte its studies, establish those to num- Among employed meth- the numerical been to fea- and have intriguing due analytical ods Several and part unusual displays. in with it 13], tures phases [12, quantum of interest ber great work. of this compare ject in not models two shall resem- these we of any Therefore diagrams bear phase other. the models each those ground- to of the blances the that diagrams guarantee of no phase NNN-interactions is state There and weak- [9–11]. NN- the model in the 1D and of field limit transverse in strong coupling exist very only of may equivalence limit such the T, finite at field) verse iesoa sn oe ihatases ed sequiv- ( is the field, to transverse a alent with model Ising dimensional edat field [6–8]. temperatures finite at field magnetic n h tegho h next-nearest-neighbor the of strength the and h rnvre1 NN oe a entesub- the been has model ANNNI 1D transverse The ntecs fte1 NN oe natransverse a in model ANNNI 1D the of case the In s T iga hw ermgei,floating, ferromagnetic, shows diagram 1 = n h DANImdl(ihu trans- (without model ANNNI 2D the and 0 = ∗ mrclydtrie h fidelity the determined umerically / iebtenteflaigadthe and floating the between line ‡ n-iesoa NN model ANNNI one-dimensional 2 d )dmninlIigmdlwtota without model Ising 1)-dimensional + ntetemdnmclimit thermodynamic the in eo 70,USA 97203, regon 4 rniinlnsseparating lines transition na xenlfil 3.A xc so- exact An [3]. field external an in e1 NN oe in model ANNNI 1D he evco ihrespect with vector te d - 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 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 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 . 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. Such scenario is similar to the next-nearest-neighbor interactions introduces frustration one found in the the work of Fisher and Selke [24] on the to the magnetic order. A much richer variety of phases low-temperature phase diagram of an Ising model with becomes possible when one varies the strength of the in- competing interactions. In that study the phase diagram teractions among the spins or their couplings to the mag- shows an infinite number of commensurate phases. netic field. While the identification of the usual thermal phase Given that insofar there is not a definite answer to transitions relies mostly on the behavior of an order pa- the problem of the ground-state properties of the trans- rameter or on an appropriate correlation function, quan- verse ANNNI model, where different approaches yield tum phase transitions can also be characterized solely by distinct phase diagrams, we use quantum fidelity method the properties of the ground-state eigenvectors of the sys- together with direct inspection of the ground-state eigen- tem on each side of the boundary between two competing vector to shed some light into the problem. We believe quantum mechanical states. We use fidelity susceptibility our approach is suitable because both the fidelity sus- to determine the phase boundary lines, as well as a direct ceptibility and ground-state eigenvector provide detailed inspection of the eigenvectors to understand the nature direct information about boundary and nature of the 3

6 6 10 10 L = 8 L = 12

Bx= 0.2 Bx= 0.2

4 4 10 10 χ χ

2 2 10 10

0 0 100.3 0.4 0.5 0.6 0.7 100.3 0.4 0.5 0.6 0.7 J2 J2 FIG. 1: Fidelity susceptibility as a function of the next- FIG. 3: Fidelity susceptibility as a function of next-nearest- nearest-neighbor coupling J2 for the transverse ANNNI model neighbor coupling J2, with Bx = 0.2, for the case L = 12 with Bx = 0.2, for the case of a chain with L = 8 spins. Here, spins. The locations of the peaks give the transition points. and also in the next figures, J1 = 1 is set as the energy unit. The locations of the peaks give the transition points.

1.2 L = 12 1.2 L = 8

0.8 0.8 FP P <2,2> Bx 2 1 FP <2,2> Bx 1 0.4 0.4

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 J 2 J 2 FIG. 4: Phase diagram in the (J2, Bx)-plane. The system 1 2 FIG. 2: Phase diagram in the (J2, Bx)-plane for a system of displays four phase regions, ferromagnetic F , P and P , and size L = 8 . The system displays three phase regions, ferro- h2, 2i. Again, the dashed boundary is the Peschel-Emery line. magnetic F , P1, and h2, 2i. No additional phases are present here. The dashed boundary is the exact Peschel-Emery line. fidelity is defined as the absolute value of the overlap between neighboring ground-sates of the system [28, 29], ground-state phases. We investigate how the phase di- agram evolves as we consider larger and larger lattices. F (λ, δ)= | hψ(λ − δ) | ψ(λ + δ)i |. (2) Our results are consistent with some known results, such Here |ψi is the quantum non-degenerate ground-state as the classical multicritical point, the Pfeuty quantum eigenvector that is evaluated at some value of λ, shifted transition point, and the exact Peschel-Emery line which by an arbitrary small quantity δ around it. In addition runs between those two points in the phase diagram [27]. to the dependence on λ and δ, the quantum fidelity is From our results for finite sized systems we can infer also a function of the size of the system. The basic idea which phases will be present in the thermodynamic limit. behind the fidelity approach is that the overlap of the ground-state for values of the parameter λ between the two sides of a quantum transition, exhibits a considerable III. THE FIDELITY METHOD drop due to the distinct nature of the ground states on each side of the phase boundary. Quantum fidelity has Suppose the Hamiltonian of the system depends on a been used in quantum information theory [30] as well as parameter λ, which drives the system through a quantum in condensed matter physics, in particular in the study phase transition at a critical value λ = λc. Quantum of topological phases [31, 32]. 4

100 600 L = 12 L = 12 J = 0.30 J = 0.70 80 2 2

400 60 χ χ

40 200

20

0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Bx Bx FIG. 5: Fidelity susceptibility as a function of the transverse FIG. 7: Fidelity susceptibility as a function of the transverse field Bx for the transverse ANNNI model with J2 = 0.30, for field Bx, with J2 = 0.70, for the case L = 12. The location of the case of a chain with L = 12 spins. The location of the the peak gives the transition point. peak gives the transition point.

6 10 L = 12

J2= 0.46 4 10 the conjugate-gradient methods. The latter is known to be a fast and reliable computational algorithm. It has been used in statistical physics, especially in the context χ 2 10 of Hamiltonian models and of transfer-matrix techniques [33, 34]. Both methods give the same ground-state eigen- values and eigenstates within a given precision. Depend- 0 10 ing on the size of the system, the ground-state energy is calculated with precision between 10−10 and 10−12. We have used δ = 0.001 in all calculations involving the fi- -2 delity susceptibility. For the location of each point at the 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 B critical boundary, we calculated the maximum value of x the fidelity susceptibility as defined by Eq. 3. FIG. 6: Fidelity susceptibility versus the transverse field Bx, In order to identify the nature of the quantum phase, for J2 = 0.46, in the case L = 12. The locations of the peaks give the transition points. we examined how the ground-state eigenvectors are writ- ten in terms of a complete set of appropriate basis vec- tors. To find the eigenstates and corresponding eigen- values of the system we needed to choose a complete set of orthogonal basis vectors and write the Hamiltonian in matrix form using this basis set. The eigenvalues and For a fixed value L and in the limit of very small δ, the eigenstates are found by exact numerical diagonalization. quantum fidelity may be written as a Taylor expansion, A convenient basis consists of the tensor product of L F (λ, δ)=1 − χ(λ)δ2 + O(δ4), (3) eigenstates of the z-component of the local spin-operator acting on each site. Denoting the eigenstates by |s >i, z where the ground-state eigenvector is normalized to where s = 1, is the eigenstate label of the operator σi unity. The quantity χ(λ) is called the fidelity susceptibil- for an up-spin and s = 0 for the a down-spin at site i. A ity and will reach a maximum at the boundary between generic basis eigenstate for the full system with L spins L adjacent quantum phases. We used the fidelity suscepti- can be written as |n >= Qi |s >i, where n labels the bility to find the phase boundary lines the (J2,Bx)-plane basis state and has the values n = 0, 1, ..., N − 1, and and compare them with the results obtained by other where N = 2L represents the dimension of the Hilbert methods. space. The basis index n, if written in binary notation, To determine the ground-state energy and eigenvec- can also be used to specify the configuration of the spins tor as a function of λ, we employed both Lanczos and forming that basis. That is, when n is written in binary 5

0.80 0.20 J = 0.565 L = 12 J2 = 0.345 L = 12 2 B = 0.200 Bx= 0.200 x 0.60 0.15 ) )

n 0.10 n ( ( 0

0 0.40 a a

0.05 0.20

0.00 0.00 0 1000 2000 3000 4000 0 1000 2000 3000 4000 n n FIG. 8: Ground-state amplitude versus the basis state index n for (J2, Bx)=(0.345, 0.200), within the phase F for L = 12. 0.20 J = 0.565 L = 12 The two largest amplitudes correspond to the ferromagnetic 2 B = 2.000 phase. The smaller amplitudes are induced by the transverse x magnetic field. 0.15 )

n 0.10 ( 0 0.20 a J = 0.438 L = 12 2 0.05 Bx= 0.200 0.15 0.00

) 0 1000 2000 3000 4000

n 0.10

( n 0 a 0.20 0.05 J2 = 0.565 L = 12 Bx= 20.00 0.15 0.00

0 1000 2000 3000 4000 )

n n 0.10 ( 0

FIG. 9: Ground-state amplitude for each basis state index n, a with (J2, Bx)=(0.438, 0.200), located inside the phase region P2, for L = 12. The two largest amplitudes correspond to a 0.05 ferromagnetic phase. The next largest amplitudes are from states with a single-kink separating ferromagnetic domains. 0.00

0 1000 2000 3000 4000 n FIG. 10: Amplitude of the ground-state against the basis state index n for the case L = 12, J2 = 0.565, and differ- ent values of Bx. (Top) Bx = 0.200, which lies in the phase region P1 in Fig. 4. The six highest amplitudes correspond to a h3, 3i-phase, while the next highest amplitudes belong to states without sequential order for the spins. (Middle) Bx = 2.000, here there are no noticeable prominent ampli- notation, the position and value of a bit will indicate tudes, since the system is already in an induced paramagnetic whether the spin at that position (site) is up (1) or down state, where the spins are mostly aligned to the transverse (0). For instance, for a chain of 12 spins the state |1755> field. (Bottom) Case Bx = 20.00, now nearly all the spins are in binary notation is written as |011011011011>, which aligned with the transverse field. represents a periodic configuration with one down-spin (0) followed by two up-spins (11). In this notation, an 6

0.60 1.2 L = 16 J2 = 0.675 L = 12 B = 0.200 0.50 x

0.40 0.8

) FP P P <2,2> n B 3 2 1 ( x 0 0.30 a 0.4 0.20

0.10

0.0 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 1000 2000 3000 4000 n J 2 FIG. 11: Amplitude of the ground-state for each of the basis FIG. 13: Phase diagram in the (J2, Bx)-plane for the case state index n when (J2, Bx)=(0.675, 0.200), within the phase L = 16. The figure shows the phase regions: F, P1 , P2, P3, h2, 2i for L = 12. The four largest amplitudes correspond to and h2, 2i. The dashed boundary is the exact Peschel-Emery the h2, 2i-phase. The transverse magnetic field is responsible line. for the appearance of the smaller amplitudes.

6 10 6 10 L = 20

L = 16 Bx= 0.2 B = 0.2 x 4 10 4 10 χ χ 2 10 2 10

0 10 0 0.3 0.4 0.5 0.6 0.7 10 0.3 0.4 0.5 0.6 0.7 J J 2 2 FIG. 14: Fidelity susceptibility as a function of next-nearest- FIG. 12: Fidelity susceptibility as a function of the next- neighbor coupling J2 for Bx = 0.2 in the case L = 20. The nearest-neighbor coupling for the case L = 16. The four peaks five peaks are centered at the transition points. shown are centered at the transition points. Here Bx = 0.2.

IV. RESULTS arbitrary eigenstate of the Hamiltonian may be cast as

N−1 In the following we present our results for system sizes L = 8, 12, 16, 20, and 24. We chose those sizes in order to |φα >= X aα(n)|n >, (4) n=0 avoid the effects of frustration and preserve the symmetry of the h2, 2i antiphase, which has periodicity of 4 lattice where α = 0, ..., N − 1, labels the quantum states, with spacings. Still we are able to draw reliable conclusions α = 0 assigned to the ground-state. Since the matrix as well as predictions about the quantum model in the Hamiltonian is real and symmetric, the coefficients aα(n) thermodynamic limit. are real. As a result, the quantum state |φα > can be vi- Let us consider first the case L = 8. Figure 1 shows the sualized in a single graph by plotting aα(n) as a function fidelity susceptibility plotted against the next-nearest- of the quantum state index n. The graph will completely neighbor interaction J2 for a fixed transverse field Bx = identify the spatial distribution of spins in the quantum 0.2. The two peaks in the graph give the locations of state [35–37]. the critical points where quantum phase transitions oc- 7

1.2 1.2 L = 20 L = 24

0.8 0.8

FP P P P <2,2> FP P P P P <2,2> Bx 4 3 2 1 Bx 5 4 3 2 1

0.4 0.4

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 J 2 J 2 FIG. 15: Phase diagram in the (J2, Bx)-plane when the sys- FIG. 17: Phase diagram in the (J2, Bx)-plane when the sys- tem size is L = 20. In addition to the phases F and h2, 2i, tem size is L = 24. In addition to the phases F and h2, 2i, at the left and right of the diagram, respectively, there are at the left and right of the diagram, there are five phases in four phases in between them, namely P1, P2, P3 and P4. The between them, namely P1, P2, P3, P4, and P5. The dashed dashed boundary is the exact Peschel-Emery line. boundary is the exact Peschel-Emery line, which lies very close to transition line between F and P5.

6 10

L = 24 6 10 Bx= 0.2

4 10

4 χ 10

2 10 (max)

χ 2 10

0 100.3 0.4 0.5 0.6 0.7 J 2 0 FIG. 16: Fidelity susceptibility as a function of next-nearest- 10 1 10 100 neighbor coupling J2 for the case L = 20. The six peaks are L centered at the transition points. Here Bx = 0.2. FIG. 18: Fidelity susceptibility at criticality as a function of the lattice size L for two different transition lines. Squares are for the transition line bordering the ferromagnetic phase (Peschel-Emery line) while circles are for the antiphase. cur. By calculating the susceptibility for several values of Bx and J2, we obtain the phase diagram shown in Fig. 2. There, we readily identify three distinct phases for low magnetic fields. The region farthest to the left gram shown, which covers relatively low magnetic fields, (F) is ferromagnetic, while the middle (P1) has a mod- where lies the interesting physics. That is also true for ulated phase, and the region farthest to the right has all the following phase diagrams below, which are valid the antiphase (h2, 2i). The transition line bordering the at the low field region, where we are concerned with the ferromagnetic phase is close to the exact Peschel-Emery onset and further evolution of modulated phases as the line [27]. As we shall see, for larger system sizes we ob- system sizes increases. tain results which closer to that line. Notice that all Consider now L = 12. Figure 3 shows the fidelity sus- the phase boundary lines meet at (J2,Bx) = (0.5, 0.0), ceptibility versus J2, for Bx = 0.2. The three peaks on the known multicritical point. Finally, for large enough the graph give the locations where the phase transitions magnetic fields, the modulated phase becomes paramag- occur. Proceeding in a similar way for various values netic. Such a feature does not appear in the phase dia- of Bx we determine the phase diagram, which is shown 8 in Fig. 4. Alternately, by keeping J2 fixed and sweeping periodic. As the field becomes sufficiently large, the am- with Bx we obtain the same phase diagram. As an exam- plitudes for the ordered phase disappear, while all the ple of this we present Figs. 5, 6, and 7, which shows the other amplitudes becomes larger, as can be seen in the susceptibilities along Bx. The peaks are at the same lo- middle figure of Fig. 10. There, most of the spins are cations as those obtained earlier with J2 sweeps. As can equally likely to align themselves with the transverse be seen, there appears an additional phase boundary line, field. Finally, for very large fields (e. g., Bx = 20.00), as compared to the case L = 8. There is a modulated nearly all the spins align themselves with the field, re- phase in the region P2, and a floating phase P1. These sulting in a more evenly distributed amplitudes of the phases are separated by the boundary line that meets basis vectors. Clearly the system is in an induced para- at the multicritical point. For very large fields Bx we magnetic phase. As we shall see later, when we consider expect the system to be paramagnetic. The ferromag- larger lattices, nonperiodic configurations will dominate netic and antiphase regions remain basically the same, the low-Bx phase. That amounts to the so-called floating apart from a slight shift in their borders, due to finite phase. In that phase there is not any periodic spin order size effects. The boundary line between the ferromag- commensurate with the underlying lattice. netic and its neighboring modulated phase is now closer Finally, the ground-state of the rightmost phase in to the Peschel-Emery line than that of the case L = 8. Fig. 4 is dominated by four amplitudes corresponding The spin configurations in each of the phases can be in- to the h2, 2i-phase. The dependence of the amplitudes ferred from a plot of the amplitudes a0(n) of the ground- with the state index for (J2,Bx)=(0.675, 0.200) in that state eigenvector versus the basis index n for a point deep phase, is depicted in Fig. 11. Again, small amplitudes within a given phase. For instance, consider the point in are due to the transverse magnetic field and, as in the the phase diagram (J2,Bx) = (0.345, 0.200), which is in other cases, and they get larger as Bx increases. the F-phase. Figure 8 shows a0(n) vs n for that point. Both the F-phase and the h2, 2i-phase are present in The two largest contributions to the ground-state cor- all the cases we considered (Bx ≤ 1.2), for all lattice respond to the ferromagnetic spin configurations, n = sizes L. They are expected to be present in the ther- 0 and n = 4095, which have binary representations modynamic limit. This is in agreement with the results |000000000000i and |111111111111i, respectively. The found by other methods [14, 18, 20–23]. However, as we other basis states with smaller amplitudes are induced consider larger lattices, other modulated phases appear by the transverse magnetic field. Those amplitudes in- in between the ferromagnetic and the floating phase. It crease with Bx. Consider now (J2,Bx) = (0.438, 0.200), should be noted that all the transition lines start out at which lies in the region P2 of Fig. 4. The amplitudes of the multiphase point and then spread outwards as Bx the ground-state basis vectors are depicted in Fig. 9. The increases. For sufficiently large Bx the phase is expected largest contributions come from ferromagnetic orderings, to be paramagnetic. while the second largest amplitudes are from the basis Let us consider now the model with size L = 16. Fig- state |000000111111i and its cyclic permutations of the ure 12 shows the fidelity susceptibility as a function of spins. The third largest amplitudes are very close to J2, for Bx =0.2. The susceptibility exhibits four peaks, the second. They come from the states |000000011111i, thus indicating five distinct phases. Again, by numeri- |111111100000i, and all the others were obtained by their cally varying Bx and J2, we obtained the phase diagram cyclic relatives. The boundary line separating the F- for the system, depicted in Fig. 13. At the two far sides phase from the neighboring modulated phase starts out of the diagram we obtained the F- and h2, 2i-phases, as at the multiphase point (J2,Bx) = (0.5, 0.0) and ends in the previous case. The positions of the boundaries of close to the Pfeuty transition point (J2,Bx)=(0.0, 1.0). the F- and h2, 2i-phases with their neighboring phases We find that as the transverse field becomes sufficiently are weakly dependent on the system size, especially the large the system enters a paramagnetic phase, where the boundary of the F-phase. The slope of the boundary spins tend to point in the same direction as the field. line of the h2, 2i-phase for L = 16 diminishes a little as That is a general feature of the model. No matter which compared with the previous case L = 12. We find an ad- phase the system is in when Bx is small, eventually it ditional modulated phase, which is dominated by states will become paramagnetic as the field increases. We do with the ordered pattern h4, 4i. There appears to be not find any evidence of a sharp transition to paramag- other contributions to the ground-state of much smaller netism. It seems that paramagnetism is achieved through weights which are not ordered, but which will increase a crossover mechanism, so that no transition line is ob- with the applied field Bx. Again, all the transition lines served. Fig. 10 shows the ground-state eigenvector am- start at the multicritical point. plitudes for 3 cases: Bx = 0.200, 2.000, and 20.00. The For L = 20 and Bx = 0.2 the fidelity susceptibility figures were obtained for L = 12 and J2 =0.565, but sim- shows 5 peaks, as seen in Fig. 14. The plot indicates ilar behavior is expected for any other set of parameters the existence of five phase transitions for this lattice size. L and J2. The top figure (Bx =0.2) shows 6 largest am- The phase diagram J2–Bx is shown in Fig. 15. We ob- plitudes that correspond to that basis vectors containing serve that another modulated phase has appeared. Now, periodic sequences of 3 up- followed by 3 down-spins. The in addition to the ferromagnetic, floating, and h2, 2i- next largest amplitudes stem from spin arrangements not antiphase, the system now has three modulated phases. 9

The floating phase P1 for this lattice size is dominated lar, the transition between the modulated phase P2 and by the orderings h3, 2i and h2, 3i. Again, the modulated the floating phase (P1) seems to be of second-order, con- phases eventually become paramagnetic for large enough trary to the claims that it is of BKT type. Finally, the transverse fields. transition line separating the floating and h2, 2i antiphase For larger system sizes, we observe a pattern that al- is of first-order, since the behavior of the susceptibility lows us to make inferences about the phases of the system deviates from power-law, as can be seen in Fig. 18. in the thermodynamic limit. Due to computer limita- The scaling behavior of the fidelity susceptibility in the tions, the largest system studied is L = 24. Figure 16 vicinity of a quantum critical point has been found to be shows the fidelity susceptibility as a function of J2, for [39, 40]: Bx = 0.2. There are 6 peaks, indicating an equal num- 2/ν ber of phase transitions. The phase diagram is shown χ(λc) ∼ L . (5) in Fig. 17. We now identify 4 modulated phases in the figure, P2,P3,P4, and P5, in addition to the floating where ν is the critical exponent describing the divergence P1, ferromagnetic F, and the h2, 2i phases. The param- of the correlation function. For the case of the transi- agnetic phase only occurs for high Bx, where the phases tion line closest to the ferromagnetic phase (see Fig. 18), lose their characteristics as the spins tend to align with the behavior of the fidelity susceptibiliy at criticality is the transverse field. The modulated phases are charac- quadratic implying that ν = 1. Hence, in this region the terized by several periodicities, among them h4, 4i for P3, model is in the same universality class as the transverse and h3, 3i for P2. The floating phase P1 is now dominated Ising model. by configurations which do not exhibit any periodicity within the system size. No particular ordering seems to take place as L increases, hence no commensurate order emerges in the floating phase. V. SUMMARY AND CONCLUSIONS As the system size increases, more modulated phases appear. For sufficiently large transverse magnetic fields We have studied the ground-state properties of the one expects the system to become paramagnetic. The one-dimensional ANNNI model in a transverse magnetic origin of the modulated phases follows from the degen- field. The phase diagrams in the (J2, Bx) plane were eracy of the ground-state at J2 = 0.5 and Bx = 0.0. obtained using the quantum fidelity method for several There, the ground-state is highly degenerate, with the lattice sizes. A new picture emerged that is distinct from number of configurations exponentially increasing with previously reported results. In addition to the known the size of the system, as mentioned before. The trans- phases, namely, ferromagnetic, floating, and the h2, 2i verse magnetic field lifts the degeneracies, thus separat- phase, it seems that there will be an infinite number of ing the phases. At finite sizes, some of the phases become modulated phases of spin sequences commensurate with visible. As one consider larger systems, more of those the underlying lattice in the thermodynamic limit. We phases appear. The ferromagnetic as well as the h2, 2i do not find paramagnetism for small values of the ap- phases should be obviously present for any system size plied field. Paramagnetism is expected to occur at suffi- in the cases J2 < 0.5 and J2 > 0.5, respectively, since ciently high fields, not shown in our phase diagrams. The they are energetically favorable in those situations. Our transitions between the modulated phases seem to be of numerical analysis was done with a maximum of 24 spins second-order. On the other hand, the transition between due to computer limitations. Yet, we can expect that the floating and h2, 2i phase appears to be of first-order. as the number of spins increases there will appear more and more distinct modulated phases. We predict that at the thermodynamic limit there will be a (denumerable) ACKNOWLEDGEMENTS infinite number of modulated phases. At criticality the fidelity susceptibilty shows power-law behavior with the lattice size, indicating that the transi- We thank C. Warner for critical reading of the tion is of second-order; otherwise it is of first-order [38]. manuscript. We also thank FAPERJ, CNPq and For instance, for the transition line closest to the fero- PROPPI (UFF) for financial support. O.F.A.B. ac- magnetic phase we observe a power-law behavior, which knowledges support from the Murdoch College of Sci- is shown in Fig. 18. The line is the numerical fit ence Research Program and a grant from the Research χ = 59.2L2. It seems that all the transition lines be- Corporation through the Cottrell College Science Award tween modulated phases are of second-order. In particu- No. CC5737.

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