Reentrant incommensurate order and anomalous magnetic torque in the Kitaev magnet β-Li2IrO3

Mengqun Li,1 Ioannis Rousochatzakis,2 and Natalia B. Perkins1

1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA2Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom

(Dated: March 31, 2020)

We present a theoretical study of the response of β-Li2IrO3 under external magnetic fields in the ab, bc and accrystallographic planes. The results are based on the minimal nearest-neighbor J-K-Γ model and reveal a richintertwining of field-induced phases and magnetic phase transitions with distinctive signatures that can be probeddirectly via torque magnetometry. Most saliently, we observe: i) an unusual reentrance of the incommensuratecounter-rotating order for fields in the ab-plane, and ii) a set of abrupt torque discontinuities which are particularlylarge for fields rotating in the bc plane, and whose characteristic shape resembles closely the ones observed inthe 3D Kitaev magnet γ-Li2IrO3. An experimental confirmation of these predictions will pave the way for anaccurate determination of all relevant microscopic parameters of this 3D Kitaev magnet.

I. Introduction

In recent years, a significant experimental and theoreticaleffort has been devoted to realization of the Kitaev quantumspin liquid phase [1] in 4d and 5d magnetic insulators [2–11]. In these materials, the interplay between the strong spin-orbit coupling and electronic correlations gives rise to highlyanisotropic and spatially dependent interactions between ef-fective moments Jeff = 1/2. In most of the relevant, two-and three-dimensional tri-coordinated lattice structures withedge-sharing octahedra, the dominant exchange interaction isstabilized through the mechanism elucidated by Jackeli andKhaliullin [2] and has a form of the bond-dependent Ising-like Kitaev interaction, which makes these materials potentialcandidates for exploration of the Kitaev quantum spin liquid.However, most of these materials exhibit complex long-rangemagnetic orders at sufficiently low temperatures, indicating thatother subdominant interactions between magnetic moments arepresent and may also have non-trivial bond-dependent charac-ter. Both experiment [12–26] and theory [27–33] have shownthat these orders are fragile and can be efficiently suppressedby external magnetic field. It was also found that the competi-tion between the external field and anisotropic bond-dependentexchange interactions gives rise to highly-anisotropic magneti-zation processes and stabilizes a variety of complex orders atintermediate fields [27–33].

Among various experimental techniques, the torque magne-tometry is especially promising tool for studying field-inducedphases [12–14, 21, 22, 25, 26, 34]. Recently, torque magne-tometry has been used in studying the field response of severalKitaev materials, including α-RuCl3 [14, 21], Na2IrO3 [22],β-Li2IrO3 [26] and γ-Li2IrO3 [12, 13]. Rather generally, themeasured torque in these materials depends strongly on themagnitude of the field, exhibits anomalous behavior at criticalfields associated with field-induced transitions, and shows asawtooth-like angular dependence at high fields.

While the angular and field dependences of the torque inα-RuCl3 has been studied in Ref. [32], the origin of anomaloustorque response in the 3D iridates remains unclear. Therefore,the purpose of this paper is to understand the recent torque mea-surements in β-Li2IrO3 [26] and γ-Li2IrO3 [12, 13]. These twomaterials appear to have closely related local energetics [35],and we will therefore focus entirely on the hyperhoneycomb

β-Li2IrO3 [23–26, 36, 37], whose microscopic minimal model,the nearest-neighbor (NN) J-K-Γ model, has been better un-derstood [30, 31, 38–40].

To understand the behavior of the torque one first needs toelucidate the ground state phase diagram for different fieldorientations and strengths. To this end, we generalize the semi-analytical framework developed previously in [30, 31, 40] tostudy the field-induced phase diagram for fields in the ab, bcand ac crystallographic planes. This framework builds onthe idea that the incommensurate (IC) counter-rotating orderobserved experimentally at zero field can be treated as a long-distance twisting of a nearby commensurate order with sixspin sublattices [40]. This approach has already been appliedsuccessfully for fields along the orthorhombic directions a,b and c [30, 31]. It has been shown in particular that thethree phase diagrams share a number of qualitative features,including: i) a strong intertwining of the modulated, counter-rotating order with a set of uniform orders, some of whichgive rise to a finite torque signal, ii) the disappearance of themodulated order at a critical field H∗, whose value is stronglyanisotropic with H∗b<H∗c�H∗a , and iii) the presence of a robustzigzag phase above H∗. It was further shown that, for H‖c, thedisappearance of the modulated order proceeds via a first-ordertransition which does not restore all broken symmetries, whichin turn gives rise to a second transition at H∗∗c that survives atfinite temperatures [31].

FIG. 1. Orthorhombic unit cell of the hyperhoneycomb lattice. Thefive NN bonds of the J-K-Γ model are marked in red for t ∈ {x, x′},green for t ∈ {y, y′}, and blue for t ∈ {z}.

arX

iv:2

003.

1310

2v1

[co

nd-m

at.s

tr-e

l] 2

9 M

ar 2

020

2

FIG. 2. Global phase diagram as a function of magnetic field strength H (vertical axes) and angular orientation (horizontal axes) as we rotate inthe bc plane (a), ab plane (b), and ac plane (c). The color coding tracks the evolution of the Bragg peak intensity II of the modulated IC order,with the thick blue solid line showing its boundary, i.e., the critical field H∗. The red vertical line at H ‖c depicts the fully polarized state atH>H∗∗c . The insets show the field dependence of II at some representative angles.

The extension of this approach for fields applied in the ab,bc and ac planes shows that the interpolation of the phasediagram from one crystallographic axis to another comes witha number of unexpected results, including: i) a reentrant ICphase that survives in a finite region of field strengths andorientations for fields in the ab plane (and to a lesser extent inthe ac plane), ii) a region featuring a two-step disappearanceof the IC order, proceeding via two consecutive metamagneticjumps, and iii) large discontinuous reversals of the magnetictorque, with characteristic sawtooth-like angular profiles thatresemble closely the ones reported for γ-Li2IrO3. These resultswill be elucidated in Secs. III and IV, but we first recap themain aspects of the microscopic model in Sec. II.

II. Microscopic model

The Hamiltonian of the J-K-Γ model reads [30, 31, 38–40]

H =∑

t∑〈i j〉∈tH

ti j +HZ ,

H ti j = JSi · S j + KS αt

i S αtj + σtΓ(S βt

i S γtj + S γt

i S βtj ) ,

HZ = −µBH ·∑

i gi · Si .

(1)

Here t ∈ {x, x′, y, y′, z} labels the different types of NN bonds(see Fig. 1), H t

i j includes the couplings between NN sites iand j of a type-t bond, αt is the Cartesian component x, y,or z associated with t, and (βt, γt) label the remaining twoCartesian components coupled by the Γ interaction. The pref-actor σt alternates between +1 and −1 for t ∈ {x, y′, z} andt ∈ {y, x′}, respectively. The termHZ stands for the Zeeman in-teraction, where the g-tensors carry a staggered, site-dependent

off-diagonal element gab [25] (in the orthorhombic frame),

gi = gdiag + pigoff−diag ≡

gaa pigab 0pigab gbb 0

0 0 gcc

(2)

where pi =1 for spins on the xy chains and −1 for spins on thex′y′ chains (see Fig. 1). More specifically, for fields in the bc,ab and ac planes, the Zeeman terms read (with h=µBH)

HZbc =−h

∑i

[cos θbc(gbbS b

i +gab piS ai )+sin θbcgccS c

i

], (3)

HZab =−h

∑i

[cos θab(gbbS b

i +gab piS ai )+sin θab(gaaS a

i +gab piS bi )], (4)

HZac =−h

∑i

[cos θac(gaaS a

i +gab piS bi )+sin θacgccS c

i

], (5)

where θbc and θab denote the angle between H and b, whereasθac is the angle between H and a, see insets of Fig. 2. Thecoupling parameters are taken as in [31], namely J =0.4 meV,K =−18 meV, Γ=−10 meV, gaa =gbb =gcc =2 and gab =0.1.

To find the classical ground states we follow the strategydeveloped in [30, 31] and use a general six-sublattice ansatzwhich is also cross-checked by independent unconstrainedminimizations of Eq. (1) on finite-size clusters. As it turns out,the classical ground state for H in the ab plane can be obtainedby using the ansatz reported in [30, 31] for H‖a. This is dueto the fact that the ansatz for H‖b is a special case of the moregeneral ansatz for H‖a. The same situation takes place for Hin the bc plane, where the classical ground state is obtained byusing more general ansatz for H‖c. By contrast the classicalground states in the ac plane cannot be obtained from either theansatz for H ‖a or the ansatz for H ‖c, but by a more generalunconstrained six-sublattice ansatz.

3

0 20 40 60 80 100 120H(T)

0

10

20

30

40

50T(K

)3ab = 69/

0.1

0.2

0.3

0.4

0.5

0.6

0.7OQ=2a=3(N = 144)

IC+uniform

uniform(ZZ+FM)

IC+uniform

FIG. 3. The field-temperature phase diagram for H in the ab planewith θab =69◦ obtained from classical MC simulations. The criticaltemperatures at which the modulated order disappears are shown byred dots and the black line is a guide to the eye.

III. Phase diagram

Figure 2 summarizes the emerging picture for the phasediagram as we rotate from one crystallographic axis to another.The blue solid line shows the boundary of the IC order, i.e., thecritical field H∗, as obtained by tracking the associated struc-ture factor II (see definition in App. A and [31]), whose fieldevolution is shown in the insets of Fig. 2 for a number of repre-sentative field orientations. Altogether, the interpolation of thephase diagram between different crystallographic axes comeswith a number of unexpected results. Most notably, we findthat while the critical field H∗ grows very fast as we approachthe a axis, reflecting the inequalities H∗b <H∗c �H∗a found in[31], the way H∗ grows in the ab-plane gives rise to a reentrantIC order in a finite range of field orientations (θab ∈ [62◦, 69◦]and θab ∈ [111◦, 118◦]), see inset of Fig. 2 (b). In this range,the magnetic field drives the system through four differentphases: the low-field six-sublattice phase (including the ICplus the uniform orders), followed by a two-sublattice order(including only the uniform orders), followed by the reentrantsix-sublattice order, and then back to the two-sublattice order(whose robust zigzag component diminishes only at H→∞).

Our classical Monte Carlo (MC) simulations shown in Fig. 3for θab = 69◦ demonstrate that this quite unusual reentrantbehavior remains robust in a finite temperature range (up to18 K for the parameters chosen in the present study) and shouldtherefore be observable experimentally, given in addition thatthe reentrance can occur at a field as low as 22 T (for θab =69◦).The critical temperatures shown by red circles in Fig. 3 arecomputed using the crossing of the Binder cumulant [31, 41]

BOQ=2a/3 = 1 − 〈O4Q=2a/3〉

/(3〈O2

Q=2a/3〉2), (6)

where OQ=2a/3 is the modulated order (see App. A and [31]).A similar reentrant behavior is also observed for fields in the

ac plane, for θac ∈ [34.8◦, 35.1◦] (see e.g. the dashed verticalline in Fig. 4), although this region is likely too narrow to

survive at high enough temperatures to be observed.Figure 4 shows in addition another interesting feature, which

will show up in our discussion of the torque signal below.Namely that, in the region extending for θac ∈ [17.0◦, 34.8◦], thedisappearance of the IC order proceeds with two consecutivemetamagnetic jumps. Such jumps can be seen in the inset ofFig. 2 (c). The characteristic fields at which these jumps takeplace are denoted by H∗ac,1 and H∗ac,2 in Fig. 4. The two-stepdisappearance of the IC order is accompanied by a similartwo-step behaviour of the uniform (zigzag and ferromagnetic)orders, as seen in the evolution of the associated structurefactor IV (see Fig. 9 (c)). This arises from the fact that the totalintensity from all intertwined orders obeys the intensity sumrule 2II +IV =S 2 [30, 31].

To conclude this section, we comment on the red dashedlines shown on top of the boundaries of the IC phase in Fig. 2.These lines correspond to the following approximations for thecritical fields

bc-plane: H∗bc ' H∗b/ cos θbc , (7)ab-plane: H∗ab ' H∗b/ cos θab , (8)ac-plane: H∗ac ' H∗c/ sin θac . (9)

These expressions arise from simple geometrical considera-tions, i.e., by associating the onset of the transition with thefield at which its projection along the axes b or c reaches thecritical fields H∗b or H∗c , respectively. The expressions worksurprisingly well in a a wide range of angles, especially forfields in the bc and ab planes. Importantly, the above linearrelationship between H∗bc and 1/ cos θbc has been reported byModic et al for γ-Li2IrO3 [12].

IV. Magnetic torque

Having established the general form of the phase diagramwe now turn to the magnetic torque, whose behavior as afunction of field strength and orientation is summarized inFig. 5. The torque τ= (τa, τb, τc) is given as τ=m×H, where

34 34.5 35 35.5 363ac(

/)

38

40

42

44

46

48

H(T

)

0

0.2

0.4

0.6

0.8

12II=S

2

uniform

IC+uniform

IC+uniform

H$ac

H$ac;2

H$ac;1

FIG. 4. Evolution of the Bragg peak intensities, II , when the fielddirection in the ac-plane is within the range of angles, θac ∈ [34◦, 36◦].

4

0 30 60 90H(T)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6= a

=H

(7B=Ir

)(a) 0/(180/)

15/(165/)30/(150/)45/(135/)

60/(120/)75/(105/)90+/(90!/)

0 30 60 90 120 150 1803bc(

/)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

= a=H

(7B=Ir

)

(d) 1T2T3T4T

15T25T35T45T

3bc

b

c

H

0 30 60 90 120 150H(T)

-1.5

-1

-0.5

0

0.5

1

1.5

= c=H

(7B=Ir

)

(b) 0/(180/)25/(155/)50/(130/)

60/(120/)70/(110/)80/(100/)

90+/(90!/)

0 30 60 90 120 150 1803ab(

/)

-1

-0.5

0

0.5

1

= c=H

(7B=Ir

)

(e)

1T2T3T4T25T75T

0 30 60 90 120 150H(T)

-1

-0.5

0

0.5

1

= b=H

(7B=Ir

)

(c) 0/(90/; 180/)5/(175/)15/(165/)

40/(140/)60/(120/)75/(105/)

0 30 60 90 120 150 1803ac(

/)

-1

-0.5

0

0.5

1

= b=H

(7B=Ir

)

(f) 15T20T40T60T80T100T

3ac

a

c

H3ab

b

a

H

FIG. 5. Field (a-c) and angular (d-f) dependence of the ratio of the torque to the magnetic field: (a,d) τa/H for the field applied in the bc plane;(b,e) τc/H for the field applied in the ab plane; (c,f) τb/H for the field applied in the ac plane. The insets in (d, e) zoom-in the jumps of τa/H atH‖c and τc/H at H‖a, respectively. These jumps originate from the explicit breaking of some of the discrete symmetries by the field componentalong b [31].

m= (ma,mb,mc) is the magnetization per site. The data shownin Fig. 5 reveal a wealth of complex features that are distinc-tive for each crystallographic plane. Some of these featuresreflect naturally the phase transitions discussed above, butthere are again some additional striking features of experi-mental relevance. Among these, the most notable is the largediscontinuous reversal of the torque at H ‖ c as the field ro-tates in the bc plane, and its characteristic ‘sawtooth’ shape(see Fig. 5 (d)) that resembles closely the one reported in γ-Li2IrO3 [12, 13]. This discontinuity was already predicted in[31] and is attributed to the fact that the uniform (zigzag andferromagnetic) orders that drive the torque are chosen spon-taneously inside the region 0 < H ≤ H∗∗c for H ‖ c. A torquereversal of similar origin occurs at H‖a when H rotates in theab plane (and for 0<H≤H∗a), but the height of this reversal islikely too small to be observed.

Before we discuss these features in detail let us first brieflyrecall the main aspects of the torques for H ‖ a, H ‖ b andH‖c [31], which are included in the insets of Fig. 5.

A. Magnetic torque for fields along the crystallographic axes

The torque for H‖b is zero for all the fields. For H‖a andH ‖ c, the torques are, correspondingly, along c and a. Thetorque for H ‖ a remains non-zero up to the critical field H∗a,whereas the torque for H‖c remains non-zero up to the secondcritical field H∗∗c , which marks the transition into the fullypolarized state [31]. In both cases, the sign of the torque ischosen spontaneously and its magnitude is proportional to themb-component of the magnetization, τc/H =mb and τa/H =mb.

Note, however, that mb is significantly different in the twocases, leading to a much larger |τa| for fields along c comparedto |τc| for fields along a [31]. Note also that for H‖c, the zigzagand ferromagnetic orders are comparable to each other, butthe dominant contribution to τa comes from the ferromagneticorder, since gbb � gab. Moreover, the uniform orders have anon-monotonic behavior as a function of H and exhibit a finitejump at the critical field H∗c , leading to a sharp kink in τa, seedark blue line in Fig. 5 (a) associated with θbc =90◦.

B. Magnetic torque for fields away from the crystallographicaxes

Fields within the bc plane. This is the setup explored experi-mentally in γ-Li2IrO3 [12, 13]. Here the torque is along the aaxis, with

τa/H = mb sin θbc − mc cos θbc . (10)

The sign of τa for H ‖ c (θbc = 90◦) is chosen spontaneouslybelow H∗∗c , is zero for H‖b (θbc =0◦) [31] and is chosen by thefield for any intermediate angles.

The corresponding torque data are shown in Figs. 5 (a, d).At low θbc and H we find τa ∝ H2 sin θbc, which is the ex-pected behavior in the linear magnetization regime, where mb =

χbH cos θbc, mc = χcH sin θbc and τa = 12 (χb−χc)H2 sin 2θbc,

where χb and χc are the magnetic susceptibilities. This behav-ior ends with a kink at the critical field H∗bc associated withthe disappearance of the IC order, which happens via a secondorder phase transition.

5

With increasing θbc, the low-field behavior of τa/H turnsfrom linear to nonlinear (see Fig. 5 (a)), and for θbc close to 90◦

the kink at H∗ turns to a discontinuity, consistent with previousresults for a metamagnetic transition at H ‖ c [31]. The sizeof the discontinuity depends non-monotonously on θbc, withthe largest jump taking place at θbc =90◦. At this angle, thereis a second phase transition at H∗∗c , above which the classicalmagnetization saturates and the torque vanishes identically(dark blue line in Fig. 5 (a)). Recall that the presence ofthe second transition at high field is due to the fact that thezigzag order for H‖c can take two possible orientations, andthe system chooses one of them spontaneously. For all otherfields in the bc-plane, the orientation of the zigzag order isdetermined by the field and the second phase transition isabsent. For all θbc , 90◦, the torque is slowly approachingzero with increasing field, and becomes identically zero onlyat infinite field.

Turning to the angular dependence of the torque (see Fig. 5(d)), the low-field sinusoidal behavior turns into a characteristicsawtooth shape at high fields, featuring a large torque reversalat θbc =90◦. Such a reversal is also present at lower fields, butbecomes much more pronounced at higher fields. The reversalis due to the explicit breaking of the discrete symmetries ΘC2band C2c by the component of the field along b [31]. Notethat the sawtooth torque profile will give rise to a pronouncedanomaly in the magnetotropic coefficient k = (1/H)∂τa/∂θbc(see Fig. 6). This thermodynamic quantity characterizes thecurvature of the free energy with respect to the field orientation,and can be measured directly via torsion magnetometry [21].

Field within the ab plane. Because of a very stronganisotropy of the critical fields, H∗a/H

∗b'50, a rather different

picture is observed for H in the ab-plane. In this setup thetorque is along the c axis, with

τc/H = ma cos θab − mb sin θab . (11)

The corresponding data are shown in Fig. 5 (b, e). Here thelow-field behavior τc∝H2 sin θab persists up to a critical fieldH∗ab'H∗b. At this field the torque shows a kink which persistsfor a large range of angles, θab 6 50◦ and θab > 130◦. This

0 30 60 90 120 150 1803bc(

/)

-3

-2

-1

0

1

(1=H

)@= a

=@3bc

(7B=Ir

=rad

)

1T2T3T4T

15T25T35T45T

3bc

b

c

H

FIG. 6. Magnetotropic coefficient k = (1/H)∂τa/∂θbc as a functionof θbc for several field strengths. The magnetotropic coefficient atθbc =90◦ is not well-defined due to the discontinuity of τa.

0 20 40 60 80H(T)

-0.6

-0.3

0

0.3

0.6

= a=H

(7B=I

r)

(a)(a) 0/

5/

15/

40/

60/

75/

0 20 40 60 80H(T)

-0.5

0

0.5

= c=H

(7B=I

r)

(b) 0/

5/

15/

40/

60/

75/

FIG. 7. Evolution of (a) τa/H and (b) τc/H as a function of H forfields rotating in the ac plane, for several field orientations θac. Toeach angle there correspond two curves with opposite torque signs,reflecting the fact that this sign is chosen spontaneously when H ⊥ b.

behavior is similar to the case of the field sweeping in thebc plane except that at H>H∗ab the torque shows almost field-independent behavior up to high fields, H∼H∗a , and then slowlydecreases. This behavior reflects the robustness of the uniformorders contributing to ma and mb.

As the field gets closer to a, the contribution from the a-component of the field increases. For the ranges θab ∈ [50◦, 62◦]and θab ∈ [118◦, 130◦], τc/H then exhibits two sharp kinksappearing due to the competition between the two contributionsin the Zeeman term in Eq. (4). In particular, Fig. 5 (b) showsthat τc/H for θab = 60◦(120◦) has two kinks, one at low fieldclose to H∗b and the other at intermediate field close to H∗a.

Especially interesting is the behavior of the torque forθab ∈ [62◦, 69◦] and θab ∈ [111◦, 118◦]. Here, the competitionbetween the two contributions in Eq. (4) leads to the unusualreappearance of the modulated order at intermediate fields, asdiscussed above. As the field further approaches the a-axis(70◦6θab 6110◦), these two kinks disappear but another inter-esting feature is observed. At the angles and magnitudes of thefield at which the first and the second term in Eq. (11) becomeequal, the magnetic torque simply vanishes. In Fig. 5 (e) thissituation is shown for θab = 70◦(110◦) and θab = 80◦(100◦).Finally, when θab =90◦ (H‖a), the torque signal is barely visi-ble since the contribution of the first term in Eq. (11) becomesvanishingly small for all values of H. Also, for H‖a the sign ofthe ferromagnetic uniform order (‖ b) is spontaneously chosen,which results in a small jump in the torque as shown in theinset of Fig. 5(e).

Field within the ac plane. Unlike the two previous cases,when we sweep the field in the ac plane the torque is no longerperpendicular to the plane of the field. In particular, all threecomponents τa, τb and τc remain finite for all angles θac <{0◦, 90◦, 180◦}. This aspect leads to a complex non-monotonicevolution of the torque. Here, the torque components arerelated to the magnetization components via

τa/H =mb sin θac,

τb/H =mc cos θac−ma sin θac, (12)τc/H =−mb cos θac.

6

38 40 42 44 46 48H(T)

0

0.2

0.4

0.6j=

aj=

H(7

B=Ir

)(a) 3ac = 34:5/

3ac = 35:0/

38 40 42 44 46 48H(T)

0

0.2

0.4

0.6

j=bj=

H(7

B=Ir

)

(b) 3ac = 34:5/

3ac = 35:0/

38 40 42 44 46 48H(T)

0

0.2

0.4

0.6

j=cj=

H(7

B=Ir

)

(c) 3ac = 34:5/

3ac = 35:0/

FIG. 8. Non-monotonous evolution of |τa|/H (a), |τb|/H (b) and|τc|/H (c) as a function of H at θac = 34.5◦ and θac = 35◦, where thedestruction of the IC order happens via two-step process. At θac =35◦

the reentrant behavior is observed in addition to the two-step process.

Our numerical data for τb are shown in Figs. 5 (c, f), whilethose for τa and τc are shown in Figs. 7 (a, b). Note thatwhile the sign of τb is fixed by the field, the signs of τa andτc are chosen spontaneously for all values of θac. These twocomponents are proportional to the magnetization along theb-axis, mb. Since the sign of mb is spontaneously chosen, thesigns of τa and τc are spontaneously chosen as well. The twochoices of signs are shown by the two sets of branches in Fig. 7associated with each θac.

As we can see from Figs. 5 (c) and 7, at low fields all threecomponents of the torque are small. At intermediate fields,the overall magnitude of the torque is growing and the threecomponents exhibit a strong angular dependence. At smallangles θac, τb/H first grows very slowly with the field and thenexhibits a kink at H∗ac where H∗∗c < H∗ac . H∗a (see, e.g., blueline for θac =5◦(175◦) in Fig. 5 (c)). At intermediate and largeangles, at which the field direction is relatively close to the c-axis, τb/H exhibits a jump at H∗ac&H∗c , indicating a first orderphase transition, and then exhibits a kink at H∗∗ac . H∗a (see,e.g., rose line for θac = 40◦(140◦) in Fig. 5 (c)). The angularbehavior of τb/H shown in Fig. 5 (f) for various field strengthsalso reflects this behavior.

Two critical fields at some field directions are also clearlyseen in τa/H and τc/H (see Fig. 7). At angles close to the c-axis, both τa/H and τc/H show a jump at H∗ac&H∗c and then goto zero above the critical field H∗∗ac 'H∗∗c . When the orientationof the field is closer to the a-axis, both τa/H and τc/H remainnegligibly small, and the total torque is oriented almost alongthe b-axis.

Finally, let us discuss what happens with the torque in the re-gion depicted in Fig. 4, which includes a narrow range (aroundθac =35◦) featuring a reentrant IC phase and a more extendedrange (θac ∈ [17.0◦, 34.8◦]) featuring a two-step disappearanceof the IC order. Figure 8 shows what happens at two represen-tative angles, θac = 35◦ and θac = 34.5◦, one from each range.As expected, the torque components show three abrupt dis-continuities at θac =35◦ and two discontinuities at θac =34.5◦,reflecting the respective evolution of the IC order.

V. Discussion

The above numerical results demonstrate a rather complexinterplay between the modulated and uniform (zigzag andferromagnetic) orders of β-Li2IrO3 as we rotate the field indifferent crystallographic planes. The resulting phase diagramshows a number of unexpected results. Most saliently, wefind a rare example of a reentrant IC phase for fields rotatingin the ab plane. The stability region of this reentrant phaseextends in a finite range of field orientations (θab ∈ [62◦, 69◦]and θab ∈ [111◦, 118◦]), field strengths and temperatures. Thisunusual behavior is yet another manifestation of the stronglyanisotropic character of the underlying microscopic interac-tions in β-Li2IrO3, and we have provided numerical estimatesfor its experimental verification. A similar reentrant behavioris also observed for fields sweeping in the ac-plane but onlyfor a very narrow range of angles.

Our numerical results for the behavior of the magnetic torqueoffers further insights and reveals a number of distinctive sig-natures that are unique for each crystallographic plane. Quitegenerally, the sequence of field-induced phase transitions giverise to torque anomalies (jumps and kinks) which can be di-rectly seen with torque magnetometry. Among the distinctiveaspects of the torque is the observation of a large discontin-uous reversal as the field rotates in the bc plane and passesthrough the c axis. The field orientation profile of this sharpdiscontinuity resembles closely the sawtooth-like behaviour re-ported by Modic et al in γ-Li2IrO3 [12, 13]. In that compoundthe discontinuity extends well beyond H∗c '16 T and persistsup to 55 T, which likely corresponds to the specific value ofH∗∗c in γ-Li2IrO3 [13]. While Modic et al [42] attributed thistorque anomaly of γ-Li2IrO3 to the presence of a chiral spin-order, the more conventional symmetry breaking mechanismpresented here for β-Li2IrO3 appears more likely, given theclose similarity of the two compounds.

The experimental observation of the above distinctive fea-tures can be used to map out a quantitative phase diagramand extract accurate values of the microscopic exchange cou-plings of β-Li2IrO3. This compound is rather unique in thisrespect, because its microscopic model has been better under-stood [30, 31, 38–40] compared to its 2D and 3D analogues,α- and γ-Li2IrO3.

Acknowledgments: We thank J. Analytis, J. Betouras, K. Modic,A. Ruiz and A. A. Tsirlin for helpful discussions. This workwas supported by the U.S. Department of Energy, Office of Sci-ence, Basic Energy Sciences under Award No. DE-SC0018056.We also acknowledge the support of the Minnesota Supercom-puting Institute (MSI) at the University of Minnesota.

Appendix A: Static structure factors and Bragg peakintensities

In this appendix, we briefly review our definitions of thesymmetry-resolved static structure factors. For the variouscomponents of the static structure factor, we follow the conven-tion of Ref. [40] and denote the modulated components withQ = 2a/3 by the letter M and the uniform components with

7

FIG. 9. Evolution of the individual Bragg peak intensities from the uniform orders with field in (a) bc-plane, (b) ab-plane, and (c) ac-plane.

Q=0 by M′. Specifically, the modulated Q=2a/3 componentsread

i M(A)i M(C)M(F)i M(G)

≡ 14

S1(Q) − S2(Q) − S3(Q) + S4(Q)S1(Q) + S2(Q) − S3(Q) − S4(Q)S1(Q) + S2(Q) + S3(Q) + S4(Q)S1(Q) − S2(Q) + S3(Q) − S4(Q)

Q=2a/3

, (A1)

and the uniform Q= 0 components are defined asM′(A)M′(C)M′(F)M′(G)

≡ 14

S1(0) − S2(0) − S3(0) + S4(0)S1(0) + S2(0) − S3(0) − S4(0)S1(0) + S2(0) + S3(0) + S4(0)S1(0) − S2(0) + S3(0) − S4(0)

, (A2)

where Sν=1−4 represent the four spin sites in each primitive unitcell. The relative amplitudes of Sν are characterized by thefour-component symmetry basis vectors

A=

1−1−11

, C =

11−1−1

, F =

1111

, G =

1−11−1

, (A3)

where A, C, F, and G denote the Neel, stripy, ferromagnetic,and zigzag order, respectively [36].

Given the static structure factor components, we can com-pute the corresponding contribution to the Bragg peak inten-sities Itot = 2II + IV . We first examine the modulated ordercontribution which is given by

II =

4∑i=1

II, Γi ' II,Γ4 = |Ma(A)|2 + |Mb(C)|2 + |Mc(F)|2, (A4)

where Γi=1−4 are the irreducible representations associated withthe basis vectors in Eq. (A3). It turns out that Γ4 = (A,C, F)always provides the dominant contribution [31]. The evolutionof II for field within each crystallographic plane is given inFig. 2 of the main text.

The uniform orders can have six different components,which correspond to the zigzag (ZZ) order or ferromagnetic(FM) order along a, b, or c direction, i.e., M′a(G), M′a(F),M′b(G), M′b(F), M′c(G), and M′c(F). As discussed in Ref. [31],when the field is applied along one of the orthorhombic axes,the uniform components are:

H ‖ a : M′a(G),M′a(F),M′b(G),M′b(F),H ‖ b : M′a(G),M′b(F),H ‖ c : M′a(G),M′b(F),M′c(F). (A5)

Note that for H ‖ c, M′a(G) and M′b(F) only remain finite forH < H∗∗c . Correspondingly, when the field is rotated on thecrystallographic planes, there is always a competition betweenthose components in Eq. (A5). Figure 9 shows the individ-ual non-vanishing contributions IV,i = |M′i |

2 of each uniformcomponent to the total Bragg peak intensity IV =

∑6i=1 IV,i.

This complicated evolution clearly signifies the anisotropiccharacter of the Hamiltonian in Eq. (1).

8

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