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18

The Heisenberg spin-1/2 XXZ chain in the presence of electric and magnetic fields

Pradeep Thakur and P. DurganandiniDepartment of Physics, Savitribai Phule Pune University, Pune-411007, INDIA

(Dated: February 1, 2018)

We study the interplay of electric and magnetic order in the one dimensional Heisenberg spin-1/2 XXZ chain with large Ising anisotropy in the presence of the Dzyaloshinskii-Moriya (D-M)interaction and with longitudinal and transverse magnetic fields, interpreting the D-M interactionas a coupling between the local electric polarization and an external electric field. We obtain theground state phase diagram using the density matrix renormalization group method and computevarious ground state quantities like the magnetization, staggered magnetization, electric polarizationand spin correlation functions, etc. In the presence of both longitudinal and transverse magneticfields, there are three different phases corresponding to a gapped Neel phase with antiferromagnetic(AF) order, gapped saturated phase and a critical incommensurate gapless phase. The externalelectric field modifies the phase boundaries but does not lead to any new phases. Both externalmagnetic fields and electric fields can be used to tune between the phases. We also show that thetransverse magnetic field induces a vector chiral order in the Neel phase(even in the absence of anelectric field) which can be interpreted as an electric polarization in a direction parallel to the AForder.

PACS numbers: 75.10.Jm,75.10.Pq,75.85.+t

I. INTRODUCTION

The one dimensional Heisenberg spin S = 1/2 XXZmodel has long served as a paradigm for the study ofquantum magnetism1 in low dimensions. The study ofeffects induced by external magnetic fields has been ofparticular interest since the magnetic behaviour in anexternal magnetic field can be qualitatively differentdepending on the magnitude and direction of the exter-nal field13–16,19,21–23,25,27,28,34. This has led to interestin the study of field-induced quantum phase transi-tions1,13,14,19,21,24–28,34,35. Such models are also believedto describe many quasi-one dimensional compounds likeTlCoCl3

31,CsCoCl333,Cs2CoCl4

20,30,SrCo2V2O831,32.

In recent years, there has also been a lot of interestin the study of the interplay of electric and magneticorder in quantum spin systems, motivated by the directcorrelation between ferroelectricity and non-collinearmagnetic order discovered in perovskite multiferroicslike RMnO3 with R = Tb, Dy, Gd and Eu1−xYx

and in several edge-sharing copper oxide compoundslike LiCu2O2, LiCuVO4, etc2. The direct correlationbetween magnetic order and ferrolectricity implies theexistence of an intrinsic magnetoelectric effect in thesesystems. This has led to a resurgence of focus in thestudy of the magnetoelectric effect due to the interest inbeing able to manipulate magnetism with electric fieldsand vice versa. A microscopic model which connectsthe (ferro-)electric polarization with the non-collinearmagnetic ordering of neighbouring spins is a latticegeometry independent model based on the spin currentmechanism5:

~Pi ∼ γ~eij × (~Si × ~Sj) (1)

where ~Pi is the local polarization at the ith site, ~eijis the unit vector pointing from site i to site i+1, Sa

i

(a = x, y, z) is the ath component of the spin opera-tor at the ith site and γ is a material-dependent con-stant. The electric polarization operator is also inti-mately related to the Dzyaloshinskii-Moriya (D-M) in-

teraction4, ~Dij · ~Si × ~Sj , which arises due to spin-orbitcoupling. The spin-orbit interaction leads to canting ofthe spins and hence frustration in the spin system whichmakes the study of spin orbit effects in spin chain systemsof special interest. Besides, the D-M interaction is knownto modify the dynamic properties and quantum entan-glement of spin chains6,7. It also plays an important rolein explaining the electron spin resonance experiments onone-dimensional antiferromagnets9.Recent studies of the magnetoelectric effect in the

anisotropic XXZ spin chain with large anisotropy in thepresence of magnetic field and the D-M interaction withthe latter being interpreted as an external electric field3,8 showed that the electric field can be used to tune be-tween phases. Different regimes of electric polarizationwith the different regimes being controlled by the mag-netic field were obtained. For a pure transverse magneticfield, there are two different regimes of polarization cor-responding to the Neel phase and the fully magneticallysaturated phase8. For the case of a pure longitudinalmagnetic field, there are three different regimes for thepolarization corresponding to the Neel phase, the Lut-tinger liquid phase and the fully magnetically saturatedphase3,8. An interesting question is that of the effect ofthe electric field when the magnetic field has both lon-gitudinal and transverse components and the spin rota-tional symmetry is completely broken. For example, re-cent bosonization studies on the nearly isotropic spin-1/2Heisenberg chain with longitudinal and transverse mag-netic fields have shown that the competition between D-M interaction and longitudinal and transverse magneticfields can lead to interesting field-induced antiferromag-netic order10,11. The question is also of relevance in the

Typeset by REVTEX

2

context of neutron scattering experiments when the mag-netic field is applied at an angle to the anisotropy axis.In this work, we address this question by studying the in-terplay of electric and magnetic order in the anisotropicspin-1/2 XXZ chain with large anisotropy with longitu-dinal and transverse magnetic fields and with the D-M interaction interpreting it as an electric field. Wehave numerically determined the ground state phase di-agram using the density matrix renormalization group(DMRG) method. Various ground state quantities likethe energy gap, magnetization, staggered magnetization,electric polarization and spin correlation functions havealso been computed. Our main result is that for largeIsing anisotropy, the electric field does not lead to anynew phase but modifies the phase boundaries. With in-creasing electric field, the AF Neel phase region reduceswhile the incommensurate phase region grows. We alsoobtain the electric polarization induced by the externalelectric field and show that both transverse and longi-tudinal magnetic fields can be used to tune the electricpolarization. Interestingly, we show that the transversemagnetic field induces an electric polarization parallel tothe AF order as well as nematic order even in the absenceof the electric field.

The plan of our paper is as follows: In Sec. II, webriefly discuss the XXZ model in the presence of thelongitudinal and transverse magnetic fields and presentthe results here for the phase diagram obtained usingthe DMRG study and compare with previous studieswhich were mainly based on the mean field approxima-tion. There are some earlier numerical studies on theXXZ model in the presence of a longitudinal field23 and atransverse field34 but as far as we are aware, there do notseem to be earlier detailed numerical studies of the phasediagram for the model in the presence of both transverseand longitudinal magnetic fields. We also show that thetransverse magnetic field leads to nematic order as wellas vector chirality in a direction parallel to the AF orderin the Neel phase. In Sec. III, we present the results forthe phase diagram obtained in the presence of the elec-tric field when both the longitudinal and transverse fieldsare present. We discuss the behaviour of various groundstate quantities and their dependences on the externalfields. We summarize our work in Sec. IV.

II. XXZ CHAIN WITH LONGITUDINAL AND

TRANSVERSE MAGNETIC FIELDS

The anisotropic S = 1/2 XXZ-chain in the presenceof longitudinal and transverse magnetic fields and witha D-M interaction along the z direction is described bythe Hamiltonian H:

H =

N∑

n=1

[J(SxnS

xn+1 + Sy

nSyn+1 +∆Sz

nSzn+1)

+E(SxnS

yn+1 − Sy

nSxn+1)− hzS

zn − hxS

xn] (2)

where Sai , with a = x, y, z, denote the components of

the spin operator at the ith site along the chain, ∆ isthe easy-axis anisotropy (the xy-plane being the easy-plane), hx (hz) the external magnetic field along the x(z)-direction and E denotes the strength of the D-M inter-action which we interpret as an external electric field Ealong the y direction coupling to the polarization opera-tor: P y

i ≡ Syi S

xi+1 − Sx

i Syi+1(We have taken the direction

along the chain to be the x direction, ~eij = ~x). TheXXZ model with periodic boundary conditions(PBC)and in a pure longitudinal magnetic field is known tobe exactly solvable by the Bethe ansatz method 28. Forthe anisotropy parameter ∆ > 1, the longitudinal mag-netic field hz leads to a sequence of three different phaseswith increasing field strength: (i) a gapped antiferro-magnetic phase with Neel order along the z directionfor 0 < hz < hc1; (ii) critical Luttinger liquid behaviourfor hc1 < hz < hc2 and (iii) a fully saturated phase forhz > hc2. The critical fields hc1 and hc2 can be obtainedfrom the Bethe ansatz solution(for periodic boundaryconditions) to be

hc1 = sinh η∞∑

k=−∞

(−1)k

coshkη; hc2 = 1 +∆ (3)

where cosh η = ∆.The XXZ model in a transverse field is not integrableand therefore not amenable to an exact Bethe ansatzsolution. For the XXZ model in a pure transversefield, the phase diagram has been obtained by studyingthe model through diagonalization and DMRG meth-ods13,15,16 as well as approximate analytic methods13.Mean field studies and exact diagonalization studieson small length chains13 showed that two differentphases are possible: the antiferromagnetic phase withNeel order along the z-direction for hx < hc3(∆) andthe magnetically saturated phase (along x-direction)for hx > hc3(∆) where hc3(∆) is the critical field.

(hc3(∆) ≈√

2(1 + ∆), the classical value)13. The effectof both longitudinal and transverse magnetic fieldswere studied using the mean field approximation14 fora chain with periodic boundary conditions (PBC) butas far as we are aware, there are no reports of exactphase diagram studies of the anisotropic model (∆ > 1)in the presence of both longitudinal and transversefields. In the presence of the electric field and in a purelongitudinal field (i.e. hx = 0), the electric field termcan be ’gauged’ away by a rotation transformation ofthe spin operators: S±

m → exp (±imα), tanα = E/J .This leads to a transformed XXZ Hamiltonian withparameters J =

√J2 +∆2 and ∆ = ∆√

1+E2and with

twisted boundary conditions. However such a rotationtransformation is not possible when hx 6= 0 and hencethe problem cannot be solved using Bethe Ansatz.

In this work, we study the model(Eq. 2) using the nu-merical DMRG method. We have obtained the energygap and also studied the behaviour of various physical

3

FIG. 1. (Color online) The schematic of the phase diagramobtained by DMRG for a chain of length L = 64 and withopen boundary conditions. Here ∆ = 4.5 and E = 0. RegionI depicts a gapped antiferromagnetic (AF) phase, region II

a gapless incommensurate (IC) phase and the region III , agapped paramagnetic (PM) phase. These have been describedin the text.

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5

hz

hx

∆ = 4.5, E = 0.0

0

0.5

1

1.5

2

2.5

3

3.5

I

II

III

FIG. 2. (Color online) The hx − hz dependence of the energygap in units of J(= 1) (white depicts zero gap). The gapped(I and III) and the gapless (II) regions are clearly seen. Forcalculating the energy eigenvalues and the gaps, we set themaximum number of states to 70 and the number of sweepsto 10, from which we obtained a truncation error of the orderof 10−15.

quantities like the uniform and staggered magnetization,spin correlations, electric polarization, etc. The numer-ical calculations have been carried out using the ALPSDMRG application12. Further, we restrict to the case∆ > 1 and have used open boundary conditions(OBC).We start by discussing the model (Eq. 2) in the ab-sence of the electric field. From the behaviour of vari-ous physical quantities, we identify three different phases:a gapped Neel antiferromagnetic (AF) phase, a gapped

paramagnetic (PM) phase and a gapless critical incom-mensurate Luttinger (IC) phase depending on the rel-ative strengths of the longitudinal and transverse mag-netic fields as shown in the schematic phase diagram inFig. 1. The Neel and Luttinger phases are separated bythe transition curve L while the Luttinger and param-agnetic phases are separated by the transition curve U .The transition curve G separates the Neel and the para-magnetic phase. In Fig. 1, critical points HL lyingon the transition curve L denote the critical transverseand longitudinal fields (hLx, hLz) while points HU on thetransition curve U represents the critical transverse andlongitudinal fields (hUx, hUz). A critical point hG on thetransition curveG denotes the critical transverse and lon-gitudinal fields (hGx, hGz). At field strengths (hTx, hTz),there is a tricritical point T between the three phases.For transverse field strengths, 0 ≤ hx < hTx, the longi-tudinal field hz gives rise to a sequence of three differentphases: a gapped AF phase with Neel order along bothz and x direction for 0 ≤ hz < hLz, a gapless Luttingerphase for hLz < hz < hUz and a magnetically saturatedphase for hz > hUz. The dependence of the lower criticallongitudinal field, hLz, and the upper critical longitudi-nal field, hUz on the transverse field can be determinedfrom the transition curves L and U respectively. At zerotransverse field, hLz = hl and hUz = hu. The lower crit-ical longitudinal field increases with the transverse field,i.e., hLz(hx 6= 0) > hl while the upper critical longi-tudinal field hUz decreases with the transverse field orhUz(hx 6= 0) < hu. For transverse fields, hx > hTx, twophases are possible: a gapped AF phase with Neel orderfor 0 ≤ hz < hGz, and a magnetically saturated phasefor hz > hGz. The longitudinal field dependence of thecritical field, hGx separating the Neel and magneticallysaturated phases is determined by the transition curveG. Also, hGx(hz = 0) = hc = hc3(∆).The above three phases have been identified from the

numerical results for the energy gap and the various mag-netic orders. We show the hx−hz dependence of the en-ergy gap in Fig. 2. Here, the first excitation determinesthe gap for non-degenerate ground state and the secondexcitation determines the gap for the degenerate groundstate. The hx − hz dependence plots for the staggeredmagnetization Mz

s and Mxs along the z and x direction

are shown in Figs. 3(a) and 3(b), respectively while thecorresponding plots for the uniform magnetization Mz

and Mx are shown in Fig. 3(c) and Fig. 3(d) respectively.The uniform magnetization, Ma and the staggered mag-netization, Ma

s have been defined as:

Ma =1

N

N∑

i=1

〈Sai 〉; a = x, y, z (4)

Mas =

1

N

N∑

i=1

(−1)i+1〈Sai 〉; a = x, y, z (5)

4

hz

hx

∆ = 4.5, E = 0.0

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.1

0.2

0.3

0.4Ms

z

(a)

hz

hx

∆ = 4.5, E = 0.0

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.04

0.08

0.12

0.16

0.2

Msx

(b)

hz

hx

∆ = 4.5, E = 0.0

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 0

0.1

0.2

0.3

0.4

0.5

Mz

(c)

hz

hx

∆ = 4.5, E = 0.0

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 0

0.1

0.2

0.3

0.4

0.5

Mx

(d)

FIG. 3. (Color online) The hx−hz dependence of (a) staggered magnetization Mz

s (b) Mx

s (Mx

s = 0 along hz = 0), the uniformmagnetization (c) Mz and (d) Mx, obtained by DMRG for a chain of length L = 64 and with open boundary conditions. Hereagain, ∆ = 4.5 and E = 0. For calculating the observables like the magnetization, polarization, etc, we used up to a maximumof 70 states, the number of sweeps was set to 10, and we kept a truncation tolerance of 10−6.

We distinguish between the three different regionsshown in Fig. 1 as follows:

(i) Region I corresponds to a gapped phase with anti-ferromagnetic(AF) Neel order (non-zero Mz

s ) along the zdirection as can be seen from the plot for the gap shownin Fig. 2 and the plot for the staggered magnetizationMz

s along the z direction shown in Fig. 3(a). We fur-ther distinguish between three subregions in this phaseas described below:

Subregion Ia corresponds to the case of a pure lon-gitudinal magnetic field. From the plots for the en-ergy gap and staggered magnetization Mz

s (Fig. 2 andFig. 3(a)), it can be seen that here, the lower criticalfield hLz = hl ∼ 1

2hc1 where hc1 is the lower critical fieldas given by the Bethe ansatz relation for PBC (Eq. 3).This is because we are considering the chain with OBC.This subregion corresponds to an AF phase with orderonly along the z direction as can be seen from Figs. 3(a)and (b). Also, it can be seen from Figs. 3(c) and (d),thatboth Mz and Mx are zero in this subregion.

Subregion Ib corresponds to the case of a pure trans-verse magnetic field, representing an AF phase with orderonly along the z direction as can be seen from Figs. 3(a)and (b). Also, from Figs. 3(c) and (d), it can be seenthat Mx is finite in this subregion which tends towards

saturation at hx = hc. However, Mz is zero in this sub-region.

The subregion Ic corresponds to the case with bothlongitudinal and transverse fields, representing an AFphase with staggered order along both z and x directionas observed from the plots for the staggered magnetiza-tion(Fig. 3(a) and (b)). The staggered order along the xdirection arises due to the oscillations in the local mag-netization 〈Sx

i 〉 about a nonzero mean value. It can alsobe seen from Fig. 3(a) and (b) that the staggered ordersin both the z and x directions are nearly equal in theregion near the tricritical point T . Further in this case,where both longitudinal and transverse fields are present,there is a finite uniform magnetization along both z andx directions as can be seen from Figs. 3(c) and 3(d).

(ii) The region II corresponds to a gapless incommensu-rate (IC) Luttinger liquid phase. The energy gap van-ishes in this region as can be seen from Fig. 2. FromFig. 3(a) and (b), it can be seen that the staggered mag-netization along z and x direction are both zero in thisregion. We do not show this but the local magnetiza-tion 〈Sz

i 〉 and 〈Sxi 〉 show (in)commensurate oscillatory

behaviour.We distinguish two cases here:

Subregion IIa corresponds to the case of a pure lon-gitudinal field. We can see from Fig. 3(c) and (d) that

5

Mz s

hz

∆ = 4.5, E = 0.0hx = 0.0 = 0.8 = 1.6 = 2.4 = 3.2 = 4.0

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6

(a)

Mz s

hx

∆ = 4.5, E = 0.0hz = 0.0 = 0.8 = 3.0 = 4.7 = 6.0

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5 3 3.5 4

(b)

Mx s

hz

∆ = 4.5, E = 0.0hx = 0.0 = 0.8 = 1.6 = 2.4 = 3.2 = 4.0

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5 6

(c)

Mx s

hx

∆ = 4.5, E = 0.0hz = 0.0 = 0.8 = 3.0 = 4.7 = 6.0

0

0.05

0.1

0.15

0.2

0 0.5 1 1.5 2 2.5 3 3.5 4

(d)

FIG. 4. (Color online) The dependence of the staggered magnetization on hz and hx in the absence of the electric field. (a)M

z

s as a function of hz, for different values of hx. (b) Mz

s as a function of hx, for different hz. (c) Mx

s as a function of hz, fordifferent hx. (d) M

x

s as a function of hx, for different hz. All other parameters are as in Fig. 3.

there is a finite uniform magnetization Mz for hz > hl

which saturates at the upper critical field hUz = hu =1 + ∆. The uniform magnetization Mx is zero in thissubregion.

Subregion IIb corresponds to the case where both lon-gitudinal and transverse fields are present(hLz < hz <hUz , 0 < hx < hTx). In this subregion, both Mz and Mx

are finite as can be seen from Fig. 3(c) and (d).

(iii) Region III is a gapped phase described by a magnet-ically saturated state with uniform magnetization alongthe direction of the applied field. The energy gap canbe seen from Fig. 2. Here we distinguish between threesubregions:

Subregion IIIa corresponds to the case of a pure lon-gitudinal field (hx = 0, hz > hu). In this subregion, theuniform magnetization Mz is fully saturated as can beseen from Fig. 3(c).

Subregion IIIb corresponds to the case of a pure trans-verse field. In this subregion, the uniform magnetizationis along the x direction as can be seen from Fig. 3(d).However,Mx does not saturate completely because of thecompetition between the Neel ordering due to the Isinganisotropy along the z-direction on one hand and thequantum fluctuations caused by the effect of the strongtransverse field in the easy-plane, on the other.

Subregion IIIc corresponds to the case where bothlongitudinal and transverse magnetic fields are present.

As can be seen from Figs. 3((d) and (e)), there is uniformmagnetization along both the z- and the x-directions,i.e, along the direction of the applied field, which tendstowards saturation at very large fields.

The behaviour of the uniform and staggered mag-netization in the different phases and the nature ofthe transitions between the phases can be understoodbetter by studying their dependence on the longitudinal(transverse) field for fixed values of transverse (longi-tudinal) field. In Figs. 4(a), we plot Mz

s as a functionof hz for fixed hx. Mz

s is non-zero only in the Neelphase (region I). For a pure longitudinal field, we seethat Mz

s is a constant which drops sharply to zero atthe lower longitudinal critical field hl. In the presenceof the transverse field, the magnitude of Mz

s decreases,also the drop to zero at the lower longitudinal criticalfield hLz gets smoothened. We show the hx dependenceof Mz

s in Fig. 4(b) from which we find a non-zero valueonly for longitudinal fields hz < hTz. From Fig. 4(a)and (b), we also observe that for small transverseand longitudinal fields Mz

s − const ∼ −ah2x − bh2

z. InFigs. 4(c) and (d), we plot Mx

s as a function of hx(hz)for fixed hz(hx). It can be seen from the figure thatMx

s is non-zero for longitudinal fields hz < hTz andshows a linear dependence on the longitudinal field, i.e,Mx

s ∼ hz. For very small transverse and longitudinalfields, it can be also seen from Figs. 4(c) and (d) that

6

Mz

hz

∆ = 4.5, E = 0.0hx = 0.0 = 0.8 = 1.6 = 2.4 = 3.2 = 4.0

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6

(a)

Mz

hx

∆ = 4.5, E = 0.0hz = 0.0 = 0.8 = 3.0 = 4.7 = 6.0

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5 3 3.5 4

(b)

Mx

hz

∆ = 4.5, E = 0.0hx = 0.0 = 0.8 = 1.6 = 2.4 = 3.2 = 4.0

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6

(c)

Mx

hx

∆ = 4.5, E = 0.0hz = 0.0 = 0.8 = 3.0 = 4.7 = 6.0

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5 3 3.5 4

(d)

FIG. 5. (Color online) The dependence of the uniform magnetization on hz and hx in the absence of the electric field. (a) Mz

as a function of hz, for different values of hx. (b) Mz as a function of hx, for different hz. (c) M

x as a function of hz, fordifferent hx. (d) M

x as a function of hx, for different hz. All other parameters are as in Fig. 3.

Mxs ∼ hzhx. Further, we see from Fig. 4 (a) and (c) that

hLz increases with the transverse field for hx < hTx. Fortransverse fields hx > hTx, the longitudinal critical fieldhGz can be seen to decrease with increasing transversefield. Figs. 4(b) and (d) show that the lower criticaltransverse field hLx increases with the longitudinal fieldwhile the upper critical transverse field hGx decreaseswith the longitudinal field.

We next study the behaviour of the uniform magneti-zation. In Fig. 5(a), we plot the uniform magnetizationMz as a function of the longitudinal field for different(fixed) transverse field values. The small step seen inFig. 5(a) for Mz at hl = hc1/2 for the case of a purelongitudinal field (subregion Ia) is due to the finite sizeeffect arising in chains with OBC. Otherwise, the fielddependence of Mz is similar to that for a chain withPBC and in a pure longitudinal field28: the magnetiza-tion vanishes in the Neel phase, increases monotonicallyin the Luttinger phase reaching saturation at the uppercritical field hc2 = 1 + ∆. It shows square root singu-lar behaviour36 in the vicinity of the transitions from theNeel phase to the Luttinger phase and from the Lut-tinger phase to the magnetically saturated phase. In fi-nite transverse fields, 0 < hx < hTx (hTx ∼ 2.0 for ourcase), the behaviour is similar, however Mz is now non-zero in the Neel phase. Also, the singular behaviour near

the transition from the Neel phase to the Luttinger phasegets smoothened by the transverse field. For transversefields, hx > hTx, Mz increases with hz monotonically,however it no longer reaches saturation value as can beseen from Fig. 5(a). Further, we observe that in general,for small longitudinal fields, Mz ∼ hz. We show the cor-responding dependence of Mz on the transverse field inFig. 5(b). For small transverse fields and for longitudi-nal fields hz < hLz, it can be seen that Mz ∼ h2

x. ForhLz < hz < hUz, M

z increases slowly with the transversefield, showing a maximum near the transition from theLuttinger IC phase to the magnetically saturated phase,and then decreases with further increase in hx as can beseen from Fig. 5(b). For hz > hUz, M

z decreases withincreasing hx. From Fig. 5(a) and (b), we conclude thatfor small transverse and longitudinal fields, Mz ∼ hzh

2x

which is in agreement with mean field theory results14.In Fig. 5(c), we plot the uniform magnetization Mx asa function of the longitudinal field for different (fixed)transverse field values, while in Fig. 5(d), we plot thetransverse field dependence of Mx. From the two plots,we can see that Mx ∼ hx for small transverse fields andthat it tends towards saturation in subregions IIIb andIIIc.

We also study the behaviour of the spin-correlation de-cays in order to confirm the above identification of thedifferent phases. We define the two-spin correlation func-

7

0.001

0.01

0.1

1

0 10 20 30 40 50 60r

hx=0.5, hz=0.0, E=0.0

|<Six Si+r

x>||<Si

y Si+ry>|

|<Siz Si+r

z>|

(a)

0.001

0.01

0.1

1

0 10 20 30 40 50 60r

hx=2.0, hz=1.8, E=0.0

|<Six Si+r

x>||<Si

y Si+ry>|

|<Siz Si+r

z>|

(b)

0.001

0.01

0.1

1

0 10 20 30 40 50 60r

hx=0.5, hz=4.3, E=0.0

|<Six Si+r

x>||<Si

y Si+ry>|

|<Siz Si+r

z>|

(c)

0.001

0.01

0.1

1

0 10 20 30 40 50 60r

hx=4.0, hz=0.0, E=0.0

|<Six Si+r

x>||<Si

y Si+ry>|

|<Siz Si+r

z>|

(d)

FIG. 6. (Color online) Semilog plots for the spin correlation decays as a function of the distance for L = 64, ∆ = 4.5 andE = 0.0, for representative values of hx − hz in the different phases. We only plot here the absolute values. Panel (a) showsthe long range (AF) order of the longitudinal spin correlations in subregion Ib (hx = 0.5, hz = 0.0) of the Neel phase. A smallinduced paramagnetic order in the tranverse x component spin correlations can also be observed. Panel (b) describes the effectof both transverse and longitudinal field on the spin correlations in subregion Ic (hx = 2.0,hz = 1.8) of the Neel phase. Thereare additional fluctuations in the paramagnetic x spin correlations due to the longitudinal field. Panel (c) shows the absenceof long range order in the subregion IIb (hx = 0.5, hz = 4.3) of the critical incommensurate phase while panel (d) shows theinduced long range ferromagnetic order for the spin correlations in the subregion IIIb (hx = 4.0, hz = 0.0) of the magneticallysaturated phase.

tions Ca(n) as:

Ca(n) ≡ 〈Cai,i+n〉 = 〈Sa

i Sai+n〉; a = x, y, z (6)

where n is the distance between the two spins. In gen-eral, in the gapped regions I and III, the spin corre-lations show long range AF order and long range FMorder respectively while in the gapless phase II, thespin correlations show an algebraic decay. For a purelongitudinal field, the transverse spin correlations areisotropic in nature because of the spin rotation symme-try about the z axis. In the subregion Ia(Neel phase),the longitudinal spin correlations (along the z direction)show AF staggered long range order while the trans-verse spin correlations(x and the y components) decayexponentially to zero. In subregion IIa (IC phase), thespin correlations show a power-law decay. In subre-gion IIIa(magnetically saturated phase), the longitudi-nal spin correlations show long range FM order while thetransverse spin correlations decay exponentially. Theseresults are described in Fig. 6, where we show the semilogplots of the absolute values of the two spin correlation de-cays |Ca(n)|, a = x, y, z as a function of the distance n

for representative field values in the various phases fo-cussing on the effect of transverse field. Panel (a) ofthe figure shows the spin correlation decays for a puretransverse field (subregion Ib). It can be seen that thelongitudinal spin correlations show long-range (AF) or-der. The spin correlations along the x direction decay toa finite uniform value while the spin correlations alongthe y-direction decay exponentially to zero (the spin ro-tation symmetry in the x− y plane is broken due to thetransverse field). In the presence of both longitudinaland transverse fields (subregion Ic), as can be seen fromFig. 6(b), both longitudinal and transverse spin correla-tions along the x direction show long range order withsmall oscillations about the mean values. The transversespin correlations along the y direction decay exponen-tially. Panel (c) of Fig. 6 depicts the two spin correlationdecay behaviour for representative values of longitudi-nal and transverse fields in the gapless incommensuratephase (subregion IIb). We see here that the transverseand longitudinal spin correlations decay in accordancewith a power-law, albeit with fluctuations about the

8

mean decay. Also, the x− y isotropy is broken. In panel(d) of the figure we show the behaviour of the spin cor-relations for a pure transverse field in the fully saturatedphase (subregion IIIb). There is long range ferromag-netic order of the spin correlations along the direction ofthe applied field, hx. The y- and the z-correlations decayexponentially. We do not show this but, in the presence ofboth transverse and longitudinal fields(subregion IIIc),the spin correlations show long range FM order along thedirection of the applied field.While the above phase diagram is in general consistent

with earlier mean field approximation studies 14, thereare some differences. The phase boundary of the criticalphase which we have determined shows that in a smallapplied transverse field hx, the sequence of phase tran-sitions with increasing longitudinal magnetic field hz issimilar to that in the absence of the transverse field. Thisis in contrast to what was suggested in Ref.14 that thereis an intermediate AF phase between the critical phaseand the saturated phase due to the induced anisotropyoperator

∑

i[Sxi S

xi+1−Sy

i Syi+1] becoming relevant beyond

a certain field strength hz > hz1. We do observe such aninduced anisotropy as discussed in the next paragraph,however, it does not lead to any intermediate AF phase.We argue that this is because, at these field strengths,the Zeeman coupling hz

∑

i Szi becomes much larger and

hence there is a transition directly from the IC phase tothe saturated phase. For higher transverse field strengths(hx > hTx), with increasing hz, there is only one phasetransition from the AF Neel phase to the fully saturatedphase.The transverse magnetic field also leads to other kinds

of order. One such effect is the nematic order 〈S+i S+

i+1〉which arises due to the x− y anisotropy induced by thetransverse field. The real and the imaginary parts of theuniform nematic order are defined as29

ReNu =1

N

∑

i

〈Sxi S

xi+1 − Sy

i Syi+1〉

ImNu =1

N

∑

i

〈Sxi S

yi+1 + Sy

i Sxi+1〉 (7)

In the absence of the electric field, the transverse fieldinduces only the real part of the uniform nematic orderRe[N ]u. From Fig. 7(a), where we show the dependenceof the uniform average of the real part Re[N ]u on thetransverse field for fixed hz, we can see that the nematicorder increases monotonically with increasing transversefield tending to saturation in the magnetically saturated(along x direction) phase. We plot the longitudinal de-pendence of Re[N ]u in Fig. 7(b) for fixed transversefields. We observe that for small transverse fields, thenematic order increases slowly with hz, reaches a max-imum near the transition to the magnetically saturated(along z direction) phase, and then decreases in the fullysaturated phase. For larger transverse fields, the nematicorder is large for small hz and then eventually decreaseswith increasing hz. The combined effects of both the

Re[

N] u

hx

∆ = 4.5, E= 0.0hz=0.0 2.0 4.0 5.0 6.0

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6

(a)

Re[

N] u

hz

∆ = 4.5, E= 0.0hx=0.0 0.5 1.0 2.0 3.0 4.0

0

0.04

0.08

0.12

0.16

0 1 2 3 4 5 6 7 8

(b)

hz

hx

∆ = 4.5, E = 0.0

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6-0.05

0

0.05

0.1

0.15

0.2

0.25

Re[N]uRe[N]u

(c)

FIG. 7. (Color online) The behaviour of the uniform aver-age of the real part of the nematic correlations Re[N ]u as afunction of the longitudinal and transverse fields and in theabsence of the electric field. The different panels describe: (a)transverse field dependence of Re[N ]u for fixed longitudinalfield values; (b) longitudinal field dependence of Re[N ]u forfixed transverse field values; (c) the combined hx − hz depen-dence of Re[N ]u . Re[N ]u is identically zero for hx = 0.

transverse and longitudinal fields on Re[N ]u are shownin Fig. 7(c). It can be seen from the figure that thenematic order is small in the Neel and Luttinger (IC)phases, increases with the transverse field and attainsits maximum value in the magnetically saturated phase(subregions IIIb and IIIc).The transverse field also induces a small real part of

the staggered nematic order(with magnitude two orderssmaller than the uniform part) where the real and imag-inary parts of the staggered nematic order are defined

9

as

ReNs =1

N

∑

i

(−1)i〈Sxi S

xi+1 − Sy

i Syi+1〉

ImNs =1

N

∑

i

(−1)i〈Sxi S

yi+1 + Sy

i Sxi+1〉 (8)

The presence of both the uniform and staggered nematicorders suggests that the transverse field induces a bond-dimerization29. Further, the breaking of the spin U(1)rotational symmetry by the transverse field also leads to a

small nematic order of the type: N±u =

1

N

∑

i〈(Szi S

±i+1+

S±i Sz

i+1)〉.

More interestingly, the transverse field also induces avector chirality in the x − y plane where we define theuniform and staggered in-plane vector chirality as

K±u =

1

N

∑

i

〈(Szi S

±i+1 − S±

i Szi+1)〉 (9)

K±s =

1

N

∑

i

(−1)i〈(Szi S

±i+1 − S±

i Szi+1)〉 (10)

The real and imaginary parts of K+u can be written as:

Kyu ≡ ReK+

u =1

N

∑

i

〈(Szi S

xi+1 − Sx

i Szi+1)〉 (11)

Kxu ≡ −ImK+

u = − 1

N

∑

i

〈(Szi S

yi+1 − Sy

i Szi+1)〉 (12)

and similarly for the staggered chirality. The vector chi-ral order arises due to the competition between the AFordering due to the Ising anisotropy and the magneticordering due to the transverse field hx. From Fig. 8(a),where we show the hx − hz dependence of Ky

s , it canbe seen that there is a large staggered vector chiral or-der in the Neel phase. For fixed hz, and in the Neelphase, the vector chirality Ky

s shows a linear dependenceon the transverse magnetic field, hx, as can be seen fromFig. 8(b). Further, we can see from Fig. 8(b), that Ky

s

goes to zero sharply at the transition from the Neel phaseto the magnetically saturated phase. The hz-dependenceof Ky

s for fixed hx is shown in Fig. 8(c). In the Neelphase, Ky

s shows almost plateau-like behaviour and thengoes to zero in the critical phase. We also find a smalluniform Ky

u (two orders of magnitude smaller than thestaggered chiral order)in the Neel phase. The AF or-der along the z direction is responsible for the fact thatthe magnitude of the staggered chiral order Ky

s is muchlarger than the uniform chiral order Ky

u. Within the spincurrent mechanism, Ky can be considered as an electricpolarization along the z direction.

hz

hx

∆ = 4.5, E = 0.0

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.05

0.1

0.15

0.2

Kys

(a)

Ky s

hx

∆ = 4.5, E= 0.0hz=0.1 2.0 3.0 4.0 5.0

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 3 3.5 4

(b)K

y s

hz

∆ = 4.5, E= 0.0hx=0.0 0.5 1.0 2.0 3.0 4.0

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5

(c)

FIG. 8. (Color online) The behaviour of the real part of thestaggered vector chirality K

y

s as a function of the longitudinaland transverse magnetic fields and in absence of the electricfield. The different panels describe (a) hx −hz dependence ofK

y

s ; (b) transverse field dependence of Ky

s for fixed longitu-dinal field values; (c) longitudinal field dependence of Ky

s forfixed transverse field values. Ky

s is zero for hx = 0.

III. XXZ CHAIN WITH MAGNETIC AND

ELECTRIC FIELDS

In this section, we discuss the combined effect ofthe external electric field along the y direction and thelongitudinal and transverse magnetic fields. We presentour numerical DMRG results for the phase diagram andthe physical observables when both the longitudinal andtransverse magnetic fields are present and there is anelectric field as well. The main result we find is that

10

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

hz

hx

∆ = 4.5, E = 0.1

0

0.5

1

1.5

2

2.5

3

3.5

I

II

III

(a)

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

hz

hx

∆ = 4.5, E = 1.0

0

0.5

1

1.5

2

2.5

I

II

III

(b)

hz

hx

∆ = 4.5, E = 0.1

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.1

0.2

0.3

0.4Ms

z

(c)

hz

hx

∆ = 4.5, E = 1.0

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.1

0.2

0.3

0.4

Msz

(d)

hz

hx

∆ = 4.5, E = 0.1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 0

0.006

0.012

0.018Py

M

(e)

hz

hx

∆ = 4.5, E = 1.0

0

1

2

3

4

5

6

7

0 1 2 3 4 5 0

0.03

0.06

0.09

0.12

0.15Py

M

(f)

FIG. 9. (Color online) The hx − hz dependence of (a),(b) the energy gap; (c),(d) staggered magnetization Mz

s ; and (e),(f)polarization P

y for E = 0.1 and E = 1.0 respectively. All other parameters are as in Fig.3. The white colour depicts zero forall observables.

the electric field does not give rise to any new order, ithowever tends to destroy the AF order and increase thefluctuations of the transverse spin components, leadingto a modification of the phase boundaries. This can beseen from Fig. 9(a,b,c,d) where we show the hx − hz

dependence of the energy gap and the staggered mag-netization for two representative values of the electricfield E = 0.1 and E = 1.0. It can be seen from panels(a) and (b) of the figure that with increasing electricfield strengths, there is an increase in the extent of theIC gapless Luttinger liquid phase at the cost of the AFNeel and the magnetically saturated phases. This canalso be corroborated from Figs. 9(c) and (d), where wesee that the staggered magnetization Mz

s is non-zero in

a smaller parameter region as compared to that in theabsence of the electric field. Also, the magnitude of theorder parameter decreases with increasing strength ofthe electric field. The behaviour of all other quantitieslike Mz, Mx, Mx

s is consistent with the above phasediagrams. The spin correlation decay also behave as ex-pected in the different phases. The only difference is thatthe electric field induces small fluctuations in the spincorrelations along the y direction in the critical IC phase.

Besides the uniform and staggered magnetization, inthe presence of an electric field, the other observableof interest is the uniform electric polarization P y =1

N

∑

i〈Pyi 〉. While the polarization P y is induced by

11

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7

Py

hz

∆ = 4.5, E = 1.0

hx = 0.0 0.8 1.6 2.0 3.8

(a)

0

0.1

0.2

0.3

0 1 2 3 4 5

Py

hx

∆ = 4.5, E = 1.0

hz = 0.0 1.8 2.8 4.3 5.0 5.4 6.0

(b)

FIG. 10. (Color online) The polarization Py as (a) a function of the longitudinal field for different transverse field values, and

as (b) a function of the transverse field for different longitudinal field values. The electric field value is set at E = 1.0. Allother parameters are as in Fig.3.

Mz

hz

∆ = 4.5, hx = 1.0

E=0.0 0.5 1.0 2.0 3.0

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 7 8

(a)

Msz

hz

∆ = 4.5, hx = 1.0

E=0.0 0.01 0.1 0.5 1.0 2.0 3.0

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 7 8

(b)

Mx

hz

∆ = 4.5, hx = 1.0

E=0.0 0.5 1.0 2.0 3.0

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6 7 8

(c)

Py

hz

∆ = 4.5, hx = 1.0

E=0.0 0.01 0.1 0.5 1.0 2.0 3.0

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8

(d)

FIG. 11. (Color online) The combined effect of the transverse field hx and the electric field E on the hz-dependence of theobservables (a) M

z, (b) Mz

s , (c) Mx and (d) P

y. Here the transverse field is kept fixed at hx = 1.0 and E takes zero anddifferent non-zero values, all other parameters being as in Fig. 3. The effect of hx (smoothening of the transitions) is suppressedby the electric field, clearly seen in panels (a) and (b).

the external electric field (P y = 0 when E = 0), itsbehaviour is influenced both by the electric and mag-netic fields. We show the hx − hz dependence of P y

in Fig. 9(e) and (f) for the two representative values ofE = 0.1 and E = 1.0, respectively. Since the electric po-larization is due to the canting of the x− y componentsof the spins, P y is significant in the AF Neel and theIC critical phase i.e., in the regions where the nearest

neighbour spin correlations are antiferromagnetic. P y

takes its maximum value in the IC phase (region M ofFig. 9(e, f)). It is very small in the magnetically sat-urated phase. It can also be seen from the figure thatthe magnitude of the electric polarization increases withthe electric field. We can understand the behaviour ofthe electric polarization in the different phases betterby studying the hz(hx) dependence of the electric polar-

12

ization for fixed transverse(longitudinal) field values asshown in Fig. 10(a(b)) for E = 1.0. It can be seen fromFig. 10(a), that in a pure longitudinal field(hx = 0), P y

shows a plateau in the Neel phase,non-monotonic depen-dence on hz in the critical IC phase and then eventuallyvanishes in the fully saturated phase. This is consis-tent with earlier results3,8. The non-monotonic hz de-pendence of the polarization in the IC phase can be re-lated to the fact that in the critical spin liquid phase for∆ > 1, there is a crossover from an Ising-like phase withdominant incommensurate longitudinal spin correlationsto an XY-like phase with dominant staggered transversespin correlations17,18. The crossover takes place at a fieldhl < h∗ < hu. We suggest that the non-monotonic be-haviour of the polarization in the incommensurate LLphase is also due to this crossover. The polarizationincreases with decreasing longitudinal spin correlationsand then decreases with increasing transverse spin cor-relations (The transverse spin correlations also decreasein magnitude but the longitudinal spin correlations goto zero faster and before the transverse correlations do).The maximum value of the polarization is near the fieldvalue where the crossover takes place. From Fig. 10(b),we see that in the presence of a pure transverse magneticfield, P y is maximum at zero field and then decreasesmonotonically with increasing hx until it vanishes in themagnetically saturated phase at hx = hGx. Since a finitetransverse field favours alignment of spins along the fielddirection, this leads to a decrease in the AF spin corre-lations and hence a decrease in the canting of the spinsor the electric polarization. Such a monotonic decreasewith increasing hx is also observed in the presence of fi-nite longitudinal fields in the Neel phase(0 < hz < hLz).This can also be seen from Fig. 10(a) where finite trans-verse leads to a decrease in the magnitude of the polariza-tion. However, the maximum value that P y can attain,changes non-monotonically for longitudinal fields in therange hLz < hz < hUz, again for reasons described in theprevious paragraph.

We can also study the combined effect of the magneticand electric fields of the various physical observables likethe uniform and staggered magnetization and the po-larization. In the absence of the transverse field, thelongitudinal magnetic field dependence of the staggeredmagnetization Mz

s and Mz are similar to that shownin the absence of the electric field(Fig. 4 and Fig. 5)),only the critical field values hl and hu change due tothe electric field. The effect of a transverse field on thehz dependence of the uniform and staggered magneti-zation, Mz,Mx,Mz

s , and polarization P y are shown inFig. 11(a, b, c, d) for different E values. It can be seenfrom Figs. 11(a) and (b), that for small electric fields, theuniform (Mz) and staggered magnetization(Mz

s ) showssimilar qualitative behaviour in the three different phasesas in the absence of the transverse field. The main effectof the transverse field is that we now no longer observethe cusp square root singular behaviour near the criticalfields hLz and hUz . Interestingly, it can also be observed

from the figures that the singular behaviour is restoredfor electric field strength values comparable to that of thetransverse field. Finite electric fields also tend to reducethe magnitude ofMx as can be seen from Fig. 11(c). Thepolarization P y (Fig. 11(d)) also shows similar qualita-tive behaviour as in the absence of the transverse field.Again, for electric fields small compared to the trans-verse field, there is a smoothening of the cusp singularitynear the critical fields. Further, since the transverse fieldtends to align the spins along the x direction while theelectric field tends to rotate the spins in the x− y plane,the magnitude of the electric polarization decreases forelectric fields small compared to the transverse fields. Forelectric field strength values large compared to that of thetransverse field, one finds that the magnitude of the po-larization does not change very much compared to thatin the absence of the transverse field and also the squareroot singular behaviour near the critical fields is restored.Or in other words, the effect of the transverse field is re-duced by large electric fields.

Ky s

hx

∆ = 4.5, hz= 2.0

E=0.0 0.1 0.5 1.0

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6

(a)

Ky s

hz

∆ = 4.5, hx= 2.0

E=0.0 0.1 0.5 1.0

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5

(b)

FIG. 12. (Color online) (a) The transverse field dependence ofK

y

s for fixed longitudinal field value hz = 2.0 and for differentE values; (b) the longitudinal field dependence of Ky

s for fixedtransverse field value hx = 2.0 and for different E values.

We next discuss the effect of the electric field on thenematic order and the vector chirality. Qualitatively,the behaviour of the real part of the nematic order andthe vector chirality as a function of the longitudinal andtransverse field does not change very much in the pres-ence of the electric field. The main difference is that the

13

hx − hz parameter regimes where the nematic and chiralorder become significant change due to the modificationof the phase boundaries by the electric field. We demon-strate this in Fig. 12. From Fig. 12(a), where we plot thestaggered vector chirality as a function of hx for differ-ent E values at a fixed value of hz, we can see that withincreasing electric field strengths, while larger thresholdtransverse fields are required to induce the vector chiralorder, the functional dependence on the transverse fieldin the IC phase does not change very much in the pres-ence of the electric field. Electric field strengths largecompared to that of the transverse field tend to reducethe chiral order. This can be seen from Fig 12(b), wherehave shown the vector chirality as a function of hz fordifferent E values at a fixed value of hx. It can be seenfrom the figure that the magnitude of the chiral orderdecreases with increasing electric field strengths. Also,larger electric field values tend to restore the sharpnessof the transition. We do not show this, but in addition,the electric field also induces the imaginary part of boththe nematic order and the vector chirality; however, incomparison to their real counterparts, these are two or-ders of magnitude smaller. ImNu(s) is non-zero only in

the Neel phase while Kxu(s) = −ImK+

u(s) is non-zero only

in the IC phase.

IV. CONCLUSIONS

We have studied the joint influence of longitudinal andtransverse magnetic and electric fields on the groundstate properties of the anisotropic Heisenberg spin-1/2XXZ chain with large Ising anisotropy using DMRGmethod. We have obtained the ground state phase dia-gram in the presence of both the longitudinal and trans-verse magnetic fields. There are three different phasescorresponding to a gapped Neel phase with AF order,gapped saturated phase and a critical incommensurategapless phase when both longitudinal and transversefields are present. In the presence of the transverse field,the nature of the behaviour of the uniform and staggeredmagnetizations near the critical fields changes from acusp square-root singular behaviour for pure longitudi-nal field to a smooth behaviour. The external electricfield does not lead to any new phase, however, the phaseboundaries get modified. With increasing electric field,

the AF Neel phase region reduces while the IC criticalregion grows. Electric field strengths comparable andlarger than the transverse field tend to reduce the effectof the transverse field and restore the sharpness of thetransitions near the phase boundaries. The electric fieldinduces an electric polarization which takes its maximumvalue in the IC phase. We argue that the maximumvalue of the electric polarization occurs at field valueswhere there is a crossover from an Ising like phase withdominant longitudinal spin correlations to an XY likephase with dominant transverse spin correlations. Inter-estingly, we show that even in the absence of an elec-tric field, the transverse magnetic field induces a uniformand staggered nematic order in the fully saturated phaseand a staggered vector chiral order in the Neel phase.Within the spin current mechanism, the induced vectorchiral order can be identified with a staggered electricpolarization in a direction parallel to the AF order. Thepresence of both uniform and staggered nematic ordersuggests that the transverse magnetic field may be usedto induce a bond-order dimerization.Since our results indicate that the main effect of the

electric field is to change the critical magnetic fieldstrengths and the nature of the transitions between thedifferent phases, it implies that both external magneticfields and electric fields can be used to tune between dif-ferent kinds of order and phases in the spin system. Thiscan be tested experimentally by investigating the mag-netization and electric polarization in quasi-one dimen-sional Ising-like magnetic systems 20,30–33,37 in the pres-ence of applied magnetic and electric fields. Also, neu-tron scattering31,32,39 and electron spin resonance (ESR)experiments38,40 can be used to study the interplay ofexternal electric and magnetic fields. Previous experi-mental studies showed that magnetic fields can inducean incommensurate order due to a crossover in quasi-1dantiferromagnets due to the crossover from an Ising likephase to a XY like phase 18. Our results suggest thatan external electric field can be also used to tune such acrossover from an Ising like phase to a XY like phase andhence induce incommensurate order.

ACKNOWLEDGMENTS

PD thanks BCUD, SP Pune University and SERB,DST India for financial support through research grant.

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