This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg)Nanyang Technological University, Singapore.

Magnetization plateaus and supersolid phases inan extended Shastry–Sutherland model

Wierschem, Keola; Zhang, Zhifeng; Wibawa, Andrew; Sengupta, Pinaki

2018

Wierschem, K., Zhang, Z., Wibawa, A., & Sengupta, P. (2018). Magnetization plateaus andsupersolid phases in an extended Shastry–Sutherland model. European Physical Journal B,91(9), 201‑. doi:10.1140/epjb/e2018‑90356‑5

https://hdl.handle.net/10356/137393

https://doi.org/10.1140/epjb/e2018‑90356‑5

© 2018 EDP Sciences, Societ`a Italiana di Fisica, Springer‑Verlag. All rights reserved. Thispaper was published by Springer in European Physical Journal B and is made available withpermission of EDP Sciences, Societ`a Italiana di Fisica, Springer‑Verlag.

Downloaded on 27 Jul 2021 16:33:47 SGT

Magnetization plateaus and supersolid phases in an extended Shastry-Sutherlandmodel

Keola Wierschem1, Zhifeng Zhang1, Andrew Wibawa1, and Pinaki Sengupta11School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371

Obtaining strong magnetoelectric couplings in bulk materials and heterostructures isan ongoing challenge. We demonstrate that manganite heterostructures of the form(Insulator)/(LaMnO3)n/Interface/(CaMnO3)n/(Insulator) show strong multiferroicity in magneticmanganites where ferroelectric polarization is realized by charges leaking from LaMnO3 to CaMnO3

due to repulsion. Here, an effective nearest-neighbor electron-electron (electron-hole) repulsion (at-traction) is generated by cooperative electron-phonon interaction. Double exchange, when a particlevirtually hops to its unoccupied neighboring site and back, produces magnetic polarons that polar-ize antiferromagnetic regions. Thus a striking giant magnetoelectric effect ensues when an externalelectrical field enhances the electron leakage across the interface.

I. INTRODUCTION

The search for complex quantum many body stateswith coexisting long range orders that break differentsymmetries in strongly correlated systems have been atthe center of Condensed Matter research for the past sev-eral decades. Frustrated quantum magnets have longplayed a major role in the discovery of novel quan-tum states as well as for realising magnetic analoguesof many body bosonic states in a more controllable man-ner. The interplay between competing interactions, geo-metric frustration, and external magnetic fields result inthe emergence of a rich variety of magnetic states. Mag-netic systems have the advantage that the magnetic field,which plays role of the chemical potential, can be variedcontinuously over a large range of values.

The supersolid (SS) is a unique state of matter thatpossesses simultaneous diagonal and off-diagonal logrange order. Ever since Penrose and Onsager’s specu-lation on the possibility of such co-existence,1 the su-persolid has been the subject of intense theoretical andexperimental investigation.2–4 However, a conclusive ex-perimental realization of this novel state of matter re-mains elusive to this date. While the supersolid phase ofhelium remains controversial, Quantum magnets providean alternate route via the spin analog of the supersolidphase. Identification of realistic systems with supersolidorder is a valuable step towards the ultimate realizationof this elusive phase.

In this work, we demonstrate the emergence of thespin supersolid phase and magnetization plateaus in ananisotropic S = 1/2 XXZ model with longer rangeinteractions on the geometrically frustrated Shastry-Sutherland lattice (SSL).5 The SSL provides a versa-tile framework to study the emergence of frustrationdriven complex magnetic phases.6–12 The existence ofseveral quasi-2D quantum magnets – including the ex-tensively studied SrCu2(BO3)2 as well as a completelyfamily of rare earth tetraborides, RB4 (R=Tm, Er, Dy,Ho, Eb)13–15 and 221 compounds such as Ce2Pt2Pb,Y b2Pt2Pb and Ce2Mg2Pb – where the magnetic ionsare arranged in a SSL configuration ensures that a wide

range of parameter regimes (such as varying degreesof frustration, strength and nature of interactions) canbe explored experimentally, complementing any theoret-ical studies. Magnetization plateaus are ubiquitous inmany frustrated quantum magnets. Several studies havedemonstrated that a spin SS ground state is stabilizedin the vicinity of magnetization plateaus, including ourown work on the SSL. In this work, we explore a widerange of parameter regimes with varying degrees of frus-tration as well as the nature and strength of longer rangeinteractions.

II. MODEL AND BACKGROUND

We consider a spin-1/2 XXZ model defined by theHamiltonian

H =∑〈ij〉

− |Jij∆|(Sxi S

xj + Syi S

yj

)+ JijS

zi S

zj − hz

∑i

Szi ,

(1)with interacting pairs 〈ij〉 defined by the Shastry-Sutherland lattice with extended long range interactionsas depicted in Fig. 1(a). The Hamiltonian is parameter-ized by the bond strengths Jij ∈ {J1, J2, J3, J4}, the ex-change anisotropy ∆, and the applied magnetic field hz.Note that the transverse coupling is taken to be strictlyferromagnetic and the applied magnetic field is longitu-dinal (i.e. parallel to the axis of exchange anisotropy).These are important features for the quantum MonteCarlo methods we will employ, as discussed in the nextsection.

Several versions of the above model have been studiedpreviously16–23. These began with initial studies of theIsing model on the bare Shastry-Sutherland lattice. Itwas quickly recognized that the ground state magneti-zation process in this case includes a single plateau atm/ms = 1/3 only16–18. Incorporating quantum fluctua-tions via the XXZ model, a narrow plateau at m/ms =1/2 can also be stabilized16,17. However, to fully stabilizethe m/ms = 1/2 plateau, extended interactions beyondthe bare Shastry-Sutherland lattice are required, both forthe Ising model21,22 and for the XXZ model19,20,23,24.

2

J2

J3

J1

J4

(a) (b)

3J

4J

I II

III IV

FIG. 1. (Color online) (a) Unit cell of the Shastry-Sutherlandlattice with extended long range interactions. (b) Schematicdepiction of J3–J4 parameter space with strongly frustratedregime (quadrant I) in red and weakly frustrated regime(quadrants II, III, and IV) in blue.

It is worth mentioning that long range interactionscan also lead to novel plateau structures. For example,Huang et al.25 have considered the Ising model on theShastry-Sutherland lattice with added dipole-dipole in-teractions and find the appearance of a m/ms = 1/2plateau. Another way to incorporate long range inter-actions is through the Kondo coupling between local-ized moments and itinerant electrons. Such an inter-action is known to lead to long range oscillating inter-actions i.e. the RKKY interactions. Farkasovsky etal.26 considered a simplified Kondo lattice model on theShastry-Sutherland lattice with local Ising moments tofind plateaus at m = ms/2 and and m = ms/3 as wellas narrow plateaus at m/ms = 1/5 and m/ms = 1/7.Finally, direct consideration of the RKKY interactionscan lead to stabilization of the m/ms = 1/2 plateau forsuitably chosen values of the Fermi momentum27.

III. METHODS AND OBSERVABLES

A. SSE Quantum Monte Carlo

Our primary method is the stochastic series expan-sion (SSE) formulation of quantum Monte Carlo28. Weemploy directed loop updates29 as well as the plaque-tte based vertex expansion30,31. Ground state proper-ties in the thermodynamic limit are accessed via finitesize scaling of the inverse temperature β and system sizeL. Within SSE correlations such as 〈Sxi Sxj + Syi S

yj 〉 and

〈Szi Szj 〉 are easily obtained28,32, and their Fourier trans-forms define corresponding static structure factors Sxy()and Szz(k) that measure transverse and longitudinal or-der, respectively. In the equivalent system of hard-corebosons, these map onto off-diagonal and diagonal order-ing, respectively, and the simultaneous presence of bothtypes of order defines a supersolid phase.

The spin stiffness ρs is defined as the response to

a twist in the boundary conditions33. In simulationsthat sample multiple winding number sectors, as in thepresent implementation of the directed loop algorithm,ρs is simply related to the winding number of the worldlines. For square simulation cells in two dimensions,ρs =

(w2x + w2

y

)/2β, where wx and wy are the winding

numbers in the x and y directions34.

B. Explicit construction of ground states in theIsing limit

We also employ direct calculation of ground state en-ergies for spin configurations in the Ising limit. While inprinciple these calculations are exact, in practice it can bedifficult to construct all of the appropriate ground statespin configurations — especially in the presence of mul-tiple competing interactions. For this reason, we havecarefully compared our results for the Ising limit phasediagram to results for the XXZ model. These are ex-pected to be qualitatively alike in the vicinity of the Isinglimit. The construction of explicit ground states in theIsing limit provides a helpful insight into the nature of thespin configurations in the complex ground state phasesarising in the Hamiltonian (1) for Ising-like anisotropy.

To include the effects of weak quantum fluctuations(|∆| � 1), we have applied second order perturbationtheory to find the leading order quantum corrections tothe Ising ground states. In some cases, this can leadto exact results. For example, the saturation field hsatcan often be found as follows. Starting from the fullypolarized ground state at large magnetic field, calculatethe energy cost to flip a single spin. This energy cost willbe zero at hz = hsat for the case of antiferromagneticlongitudinal spin exchanges. This leads to the expression

hsat = (2J1 + J2/2 + J3 + 2J4) (1 + ∆) , (2)

which is exact in the antiferromagnetic parameterregimes of H. In fact, if one or two parameters areweakly ferromagnetic, this formula often holds true bysimply replacing Jα (1 + ∆) by Jα + |Jα∆| for the FMinteractions. Note that the uniform magnetization persite m =

∑j〈Szj 〉/N saturates at ms = 1/2.

C. Spiral Plaquette Representation

To further gain insight into the structure of the magne-tization plateaus and the mechanism behind their emer-gence, we have developed a spiral plaquette representa-tion to describe these states. Generally, plateaus are as-sumed to have crystal structures with various unit cells.Each unit cell could consist of a certain number of trans-lationally invariant smaller cells of the lattice. One ofsuch translationally invariant cell is the spiral plaquetteas shown in Figure 2. The spiral plaquette (splqt) con-

3

FIG. 2. The spiral plaquette contains one central square andfour spirally surrounding dimers. The position of the plaque-tte is denoted at the center of the square.

sists of four dimers surrounding a central square. Eachdimer could be represented by the following four states

|u〉 = | ↑↑〉, |d〉 = | ↓↓〉, |l〉 = | ↓↑〉 and |r〉 = | ↑↓〉, (3)

where the first spin is located at the corner of the centralsquare and we define their dual states:

|u〉 = |u〉, |d〉 = |d〉, |l〉 = |r〉 and |r〉 = |l〉. (4)

The splqt is represented by the direct product of the fourdimer states:

|spr〉 = |a1, a2, a3, a4〉r (5)

where |ai〉 ∈ {|u〉, |d〉, |l〉, |r〉} and r denotes the positionof the splqt. Because of the over-completeness, the splqtstates obey the following constraints:

|a3〉r = |a1〉r+x and |a4〉r = |a2〉r+y. (6)

We introduce Schwinger bosons to represent the fourdimers in the splqt:

u†|∅〉 = |u〉, d†|∅〉 = |d〉, l†|∅〉 = |l〉 and r†|∅〉 = |r〉, (7)

where |∅〉 is the vacuum state. The spin operators at thenth dimers can be represented as:

Szn,1 =1

2(Ln − Rn + Un − Dn),

Szn,2 =1

2(−Ln + Rn + Un − Dn). (8)

The subscript 1 (2) denotes the site at the corner of the

central square (at the edge). Ln = l†n ln is the numberoperator and similar for the other three. After some alge-bra, the Ising Hamiltonian can be expressed in a diagonalform as:

HIsing =∑i,n

(1

8J2 −

1

2h

)Un,i +

(1

8J2 +

1

2h

)Dn,i −

1

8J2(Ln,i + Rn,i)

+1

2J1∑i,n

(Un,i − Dn,i)(Un+1,i + Ln+1,i − Dn+1,i − Rn+1,i)

+1

4(2J4 + J3)

∑〈i,j〉

(U1,i − D1,i)(U1,j − D1,j) + (U2,i − D2,i)(U2,j − D2,j)

+1

4(2J4 − J3)

∑〈i,j〉

(L1,i − R1,i)(L1,j − R1,j) + (L2,i − R2,i)(L2,j − R2,j)

+1

4J3∑〈i,j〉

(L1,i − R1,i)(U1,j − D1,j)− (U1,i − D1,i)(L1,j − R1,j)

+1

4J3∑〈i,j〉

ηij(U2,i − D2,i)(L2,j − R2,j)− ηij(L2,i − R2,i)(U2,j − D2,j) (9)

where 〈i, j〉 means nearest splqt and ηij = 1 if the bondis along x direction and −1 if along y direction.

The number of splqt in a unit cell N is closely relatedto the magnetization q/p:

q

p=

1

4N

N∑i=1

Szi (10)

4

where |Szi | ≤ 4 is the magnetization of the ith splqt. qand p are coprime integers. This relation restricts thepossible number of splqt in a unit cell as 4N/p has to bean integer. For example, for the 1/3 plateau, (p, q)=(1,3);therefore N should be a multiple of 3. The existence ofpossible magnetization plateaus in different parameterranges could be inferred from the Ising Hamiltonian (9)with the constraints (6).

We only consider antiferromagnetic (AFM) J1 and J2.In the strong ferromagnetic (FM) J3 and J4 range, thethird quadrant, the third line of the Ising Hamiltonian(9) dominates. Identical adjacent splqt formed by u ord type dimers are preferred, which suggests singlet splqtunit cell. And in the strong FM J4 and AFM J3 range,the fourth quadrant, the fourth line of the Ising Hamil-tonian dominates and it also implies identical adjacentsplqt formed by l or r type dimers. On the other hand,in the strong AFM J4 and FM J3 range, the second quad-rant, the fourth line of the Ising Hamiltonian dominatesand prefers unit cell with two splqt of configuration A B

B Aformed by l and r types dimers. In the first quadrant,the extremely frustrated regime, complicated large unitcells are possible.

IV. ZERO-FIELD PHASE DIAGRAM IN ISINGLIMIT

There are four possible zero-field magnetically orderedground states of H in the Ising limit as shown in Fig. 3.In addition to these states there is a non-magnetic dimerproduct state. This exponentially degenerate state iscomposed of the direct products of dimers on the J2bonds. The limits of stability for these five Ising groundstates are obtained as follows.

For J3 = J4 = 0 we have an Ising antiferromagneticground state for J1 > J2/2, otherwise the ground stateis the exponentially degenerate Ising dimer state. Fornon-zero J3 and / or J4, this degeneracy is genericallylifted (except along portions of the line J3 = 2J4). Newground states are stabilized as follows. For large AFMJ3 and FM (or weak AFM) J4 the columnar state with(π, 0) ordering is stabilized. Conversely, the “J3 Chess-board” state, with all four spins up / down on alternateJ3 plaquettes, is stabilized for large AFM J4 and FM(or weak AFM) J3. When both J3 and J4 are stronglyAFM, the “J2 Chessboard” state of alternately polarizedJ2 plaquettes commences. Finally, the Neel state domi-nates the region where both J3 and J4 are FM.

The constraint for the Ising dimer state in the absenceof longer range interactions, J1 ≤ J2/2, is modified inthe presence of non-zero J3 and J4. The requirementsinstead become (i) J3 = 2J4 (ii) J1 ≤ J2/2 + J3 and (iii)J3 ≤ J2/2. When weak quantum fluctuations are takeninto account 0 < ∆ � 1 the stability of the quantumIsing dimer expands to a finite region of the J3–J4 planedue to the formation of spin-zero triplets on each J2 bond.

In Fig. 4 we show the ground state Ising phase diagram

(a) (b)

(c) (d)

FIG. 3. Magnetically ordered zero-field Ising ground states:(a) Neel antiferromagnet, (b) Columnar antiferromagnet, (c)J2 Chessboard antiferromagnet, and (d) J3 Chessboard anti-ferromagnet. Not shown is the exponentially degenerate non-magnetic dimer product state in which each J2 bond consistsof one up and one down spin.

in the J3–J4 plane for J1 = J2 and 5J1 = 2J2. The quali-tative features of these two phase diagrams are preservedfor any J1 ≥ J2/2 and J1 < J2/2, respectively. Themain effect of increasing/decreasing J1 relative to J2 isto enhance/diminish the extent of the Neel state, respec-tively. Note, however, that for J1 > J2 a phase boundarydevelops between the Neel state and the J2 Chessboardstate, similar to the Columnar to J3 Chessboard phaseboundary for J1 < J2.

V. WEAKLY FRUSTRATED REGIME

When J3 or J4 (or both) are ferromagnetic, the systemis in the weakly frustrated regime. In the limit that theferromagnetic terms dominate the system approaches asimple ferromagnet. We first consider this weakly frus-trated regime, corresponding to quadrants II, III, and IVof Fig. 1(b). We will demonstrate how J3 < 0 and J4 < 0stabilize 1/2 plateaus in this regime.

By the construction of exact Ising ground states andthe use of the splqt representation the spin configurationsfor the ground state phase are identified for the plateausat m/ms = 1/3, 1/2 and 5/9. By comparison of theirenergies at a fixed magnetization, including phase sepa-rated mixtures of states, we construct the phase diagramat m/ms = 1/3 and m/ms = 1/2 in the J3–J4 planefor J1 = J2 (see Fig. 5). At m/ms = 1/3 it is clearthat either FM J3 or J4 will act to destabilize the thirdplateau states. In contrast, the half plateau states atm/ms = 1/2 are only destabilized by J3 and J4 both FMor AFM. Thus there are significant regions of the J3–J4plane that display a 0– 1

2–1 magnetization sequence, allof which contain at least one FM interaction (i.e. they

5

-2 -1 0 1 2J3

-1

-0.5

0

0.5

1J 4

J3 Chessboard (d)

Néel (a) Columnar (b)

J2 Chessboard (c)

-2 -1 0 1 2J3

-1

-0.5

0

0.5

1

J 4

J3 Chessboard (d)

Néel (a)Columnar (b)

J2 Chessboard (c)

FIG. 4. Ground state Ising phase diagram in the J3–J4 planefor J1 = J2 (upper panel) and 5J1 = 2J2 (lower panel). Inboth cases we set ∆ = 0 and J2 = 1. In the lower panel, wesketch the extent of the quantum dimer state (dashed blueboundary) for ∆ = 0.1 up to first order in ∆.

lie within the weakly frustrated regime considered in thepresent section). The plateauless 0–1 magnetization se-quence occurs for J3 and J4 both strongly FM. At smallvalues of J3 there is a 0– 1

3– 12–1 magnetization sequence,

as long as J4 > −1/4. In the strongly frustrated regimeconsidered in the next section, we shall see two addi-tional Ising limit magnetization sequences that includea plateau at m/ms = 5/9 which appears whenever allfour interactions are AFM. They are the 0– 1

3– 12– 5

9–1 and

0– 13– 5

9–1 magnetization sequences.The Ising structures of the third plateau are shown in

Fig. 6 and those of the one-half and five-ninths plateausare shown in Fig. 7.

Finally, we show the possible magnetization sequencesfor J1 = J2 in the J3–J4 plane in Fig. 8.

A. Supersolids near 1/2 plateau

In Figs. 9 and 10 we show the appearance of super-solid phases just below the 1/2 plateau in the Columnar(AFM J3) and J3 Chessboard (AFM J4) regimes, re-

-2 -1 0 1 2J

3

-1

-0.5

0

0.5

1

J 4

TP (d)PS (1)

PS (2) TP (a)

PS (3) PS (4)PS (5)

TP (c)

TP (b)

-2 -1 0 1 2J

3

-1

-0.5

0

0.5

1

J 4HP (b)

HP (c)

PS (4)

HP (a)

PS (5)

PS (2)

PS (3)

PS (1)

FIG. 5. Ground state Ising phase diagram in the J3–J4plane for J1 = J2 = 1 at m/ms = 1/3 (upper panel) andm/ms = 1/2 (lower panel). In both cases ∆ = 0 and thephase separated regions indicate plateau instability.

(a) (b)

(c) (d)

FIG. 6. The four allowed Third Plateaus (TP) atm/ms = 1/3when J1 = J2.

6

(a) (b)

(c) (d)

FIG. 7. (a), (b) and (c) The three allowed Half Plateaus(HP) at m/ms = 1/2 when J1 = J2. (d) Five-ninths Plateauat m/ms = 5/9.

-2 -1 0 1 2J

3

-1

-0.5

0

0.5

1

J 4

0 - 1

0 - 1/2 - 1

0 - 1/3 - 1/2 - 1

0 - 1/2 - 1

0 - 1/3 - 1/2 - 5/9 - 1

0 - 1/3 - 5/9 - 1

0 -

1/3

- 1/

2 -

5/9

- 1

ErB4

TmB4

SS1

SS2

FIG. 8. Ising ground state magnetization sequences in theJ3–J4 plane for J1 = J2. The parameter sets that capture themagnetic properties of ErB4 and TmB4 are marked, as well asthe location of two emergent supersolid phases (for non-zero∆) considered later in the text.

spectively. We refer to these as SS1 and SS2, and theirlocations on the J3–J4 plane are shown in Fig. 8. Thezero field ground states exhibit long range magnetic or-dering – columnar AFM order for J3 > 0, J4 < 0 andchessboard AFM for J3 < 0, J4 > 0 – with a finite gap tolowest magnetic excitations. This is indicated by finitem2c (m2

b) for the former (latter) situation and vanishingspin stiffness for both. The ground state remains un-changed at weak applied fieds, but at a critical field ofhz ∼ 1.2, the magnetic ordering is quenched via a discon-tinuous transition and the system enters a gapless spinsuperfluid phase. The AFM order parameters vanish in-dicating the suppression of AFM ordering, while the spin

0

0.05

0.1

0.15

0.2

0.25

mc2

L=12L=16

0

0.1

0.2

0.3

0.4

0.5

mz

0.5 1 1.5 2 2.5h

z

0

0.01

0.02

0.03

0.04

0.05

ms2

0.5 1 1.5 2 2.5h

z

0

0.2

0.4

0.6

ρ s

β=64, ∆=0.4

J3=1, J

4=-0.5

FIG. 9. (Color online) Evidence for the supersolid phase SS1with columnar magnetic order below the half plateau in theColumnar regime of the model when J1 = J2.

stiffness acquires a finite value. The uniform magnetiza-tion jumps to a finite value (mz ≈ 0.2) and increasesmonotonically with increasing field. At a field strengthof hz ∼ 2, there is a second transition whereby long rangeAFM magnetic sets in. Unlike the low field AFM orer,the ground state in this field range develops a staggered(π, π) AFM order. This is true for both the parameterregimes under consideration. However, the spin stiffnessdoes not vanish for 2 . hz . 2.1, indicating a simul-taneous spin superfluid ordering. Thus, for this rangeof applied field, the ground state has co-existing longrange diagonal (staggered AFM) and off-diagonal (spinsuperfluid) ordering – in other words, the ground state isa spin supersolid. In both cases, the corresponding FMparameter acts to stabilize the solid ordering, while finiteexchange ∆ enhances superfluid ordering. One intrigu-ing difference between SS1 and SS2 is that while theyboth emerge from superfluid phases as a half plateau isapproached from below, in SS1 this occurs via a discon-tinuous phase transition while in SS2 this occurs as acontinuous phase transition. It is tempting to speculatethat this difference is related to the fact that SS1 is adi-abatically connected to the supersolid phase of the ErB4

parameter set, which has been shown to occur within anunfrustrated sublattice in the Ising limit23.

VI. STRONGLY FRUSTRATED REGIME

In this section we focus on the complex magnetic phasediagram that emerges in the strongly frustrated regime,where Ji > 0 ∀ i. Although we have tried to capture allof the relevant plateaus using ground state energy com-parison of diverse Ising structures, this method may misssome complicated plateau structures with large unit cells.Additionally, new plateaus may emerge when quantum

7

0

0.05

0.1

0.15

0.2

0.25

mb2

L=12L=16

0

0.1

0.2

0.3

0.4

0.5

mz

0.5 1 1.5 2 2.5h

z

0

0.01

0.02

0.03

0.04

0.05

ms2

0.5 1 1.5 2 2.5h

z

0

0.2

0.4

0.6

ρ s

β=16, ∆=0.4

J3=-1, J

4=0.5

FIG. 10. (Color online) Evidence for the supersolid phase SS2with chessboard magnetic order below the half plateau in theJ3 Chessboard regime of the model when J1 = J2.

fluctuations ∆ > 0 are added to this highly frustratedregime via the quantum order by disorder mechanism.The situation at finite temperature becomes even morecomplicated. The competition between AFM nearest andnext-nearest neighbor interactions (in particular, J1 andJ4) is similar to that found in the axial next nearestneighbor Ising (ANNNI) model35. As such, a contin-uum of modulated phases can be expected to exists atintermediate temperatures.

For the above reasons, we have mapped out the groundstate magnetic phase diagram in this regime via extensivequantum Monte Carlo simulations, in the two regimesof the bare Shastry-Sutherland interactions, namely theNeel regime J1 > 2J2 and the dimer regime J1 < 2J2.The plateau sequence is found to be much more com-plex when J1 < 2J2. The magnetic phase diagram inthe hz − J3 parameter space for ∆ = 0.1 and J4 = 0.2is shown in Fig. 11 for a representative values of J1and J2. However, some of these putative plateaus maybe metastable states and more in-depth investigation isneeded to confirm their existence in the thermodynamiclimit. In this study, we focus on a few stable and promi-nent plateaus.5/9 plateau: One interesting feature to be found in

Fig. 11 is the broad stability of the 5/9 plateau for J3 > 0,replacing the 1/2 plateau found when J3 < 0.

This new plateau is ubiquitous throughout the stronglyfrustrated region. The appearance of this plateau can beexplained by the following argument. The largest mag-netic plateau should simultaneously maximize the uni-form magnetization (in order to gain energy from thecoupling to magnetic field) and minimize the number offrustrated bonds. Starting from the fully polarized phase,all bonds are frustrated, yet the uniform magnetization isglobally maximized. From here, if we start to flip spinsin an effort to minimize frustrated bonds, the best we

FIG. 11. (Color online) Magnetic phase diagram in the hz–J3plane for 5J1 = 2J2. We set ∆ = 0.1 and J4 = 0.2 at β = 32using a 18 × 18 simulation cell.

can do turns out to be forming the 5/9 plateau. In thiscase, each flipped spin now lies on unfrustrated bonds,and any additional spin flip will actually create morefrustrated bonds by forcing at least two flipped spins tocorrelate ferromagnetically (remember that we are in theregime where all interactions are antiferromagnetic, andtherefore ferromagnetic correlations increase the energy).Hence we arrive at the conclusion that starting at thefully polarized state and lowering the magnetic field, wemust always cross a finite regime of 5/9 plateau.

In constructing our model Hamiltonian H in Eq. 1,we have neglected the second cross diagonal term on J2plaquettes. Labeling this term as J ′

2, it is clear that the5/9 plateau is only possible when J ′

2 ≤ 0. However, forAFM values J ′

2 > 0 we can still apply the same argumentas above, only now it will lead to a 3/5 plateau. Such adistinction is important when applying H as an effectivemodel of rare earth tetraboride magnetism.

5/9 supersolid: Looking at the 5/9 plateau in Fig. 11,there appears to be a related supersolid phase for mag-netizations just below this plateau. This putative super-solid phase exists in the intervening field range betweenthe 1/3 and 5/9 plateaus. The 1/3 and 5/9 plateaus donot share the same ordering wave vectors. It is likely thatthe mechanism is similar to that discussed in the previoussection. The magnetic order at the 1/3 plateau melts ata critical field, to be replaced by a spin superfluid phase.At a higher field, the magnetic ordering correspondingto the 5/9 plateau sets in, without quenching the su-perfluid order completely. The two orders co-exist for afinite range of fields beyond which the offdiagonal orderis suppressed and the 5/9 plateau is stabilised. We arecurrently investigating this putative supersolid phase.

2/9 plateau: This plateau was previously observed in

SrCu2(B)3)2 and found theoretically in the canonical

8

Shastry-Sutherland model12. Later it was found in theextended Shastry-Sutherland model by Suzuki et al.20

in the field range 0.6 < hz < 0.74 for parameters ∆ =

0.2, 2J1 = J2, J3 = 0 and J1 =√

23J4. They found a

magnetic structure similar to that of the 5/9 plateau, butwith additional blocks of spins that form plaquette singletstates. In the parameter ranges that we have consideredin this study, a weak 2/9 plateau emerges for J1 < 2J2,whereas it is absent in the dimer regime (J1 < 2J2).1/9 plateau: This plateau appears for a narrow win-dow of parameter values, as seen in the upper panel ofFig. 11. Due to the finite size of our simulation cell, how-ever, this phase may or may not be metastable. This rel-atively small plateau is of importance in the understand-ing of similar small plateaus in the rare earth tetraborideTmB4

15.

VII. CONCLUSION

We have investigated the emergence of field inducedmagnetization plateaus and spin supersolid phase in anextended Shastry Sutherland model with longer rangeinteractions and Ising-like exchange anisotropy. Themodel is motivated by its relevance to understandingthe magnetic behavior in rare earth tetraborides. Ourresults demonstrate that the sequence of field inducedplateaus depends strongly on the nature and strengthof the longer range interactions, and that multiple spinsupersolid phases are stabilized in different parameter

regimes. By explicitly constructing the spin configura-tions at the plateaus in the Ising limit, we are able togain insight into the nature of the different supersolidphases ad the underlying mechanism behind their emer-gence. One interesting aspect of highly frustrated XXZmodels in the Ising limit that we did not touch on inthis work is that in some cases spin liquid phases canbecome stabilized. This is the case, for example, in theJ1 − J2 − J3 kagome lattice XXZ model in the Isinglimit36. Similar spin liquid phases have also been pro-posed for Ising limit XXZ models on the J1− J2 squarelattice and the J1 − J2 − J3 honeycomb lattice37,38.

VIII. ACKNOWLEDGMENTS

It is a pleasure to thank Ernest Siang for helpful discus-sions and help in preparing the figures. Financial supportfor the work was provided by the Ministry of Eduction,Singapore, through grant MOE2014-T2-2-112. This re-search used resources of the National Energy ResearchScientific Computing Center, which is supported by theOffice of Science of the U.S. Department of Energy underContract No. DE-AC02-05CH11231.

IX. AUTHORS CONTRIBUTIONS

All the authors were involved in the preparation of themanuscript. All the authors have read and approved thefinal manuscript.

1 O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).2 M. Boninsegni and N. V. Prokof’ev, Rev. Mod. Phys. 84,

759 (2012).3 D. Y. Kim and M. H. W. Chan, Phys. Rev. B 90, 064503

(2014).4 A. Eyal, X. Mi, A. V. Talanov, and J. D. Reppy, Proc.

Natl. Acad. Sci., USA 113, E3203 (2016).5 B. S. Shastry and B. Sutherland, Physica B+C 108, 1069

(1981).6 H. Kageyama, K. Yoshimura, R. Stern, N. Mushnikov,

K. Onizuka, M. Kato, K. Kosuge, C. Slichter, T. Goto,and Y. Ueda, Physical review letters 82, 3168 (1999).

7 S. Miyahara and K. Ueda, Physical review letters 82, 3701(1999).

8 K. Onizuka, H. Kageyama, Y. Narumi, K. Kindo, Y. Ueda,and T. Goto, Journal of the Physical Society of Japan 69,1016 (2000).

9 H. Kageyama, M. Nishi, N. Aso, K. Onizuka, T. Yosihama,K. Nukui, K. Kodama, K. Kakurai, and Y. Ueda, Physicalreview letters 84, 5876 (2000).

10 K. Kodama, M. Takigawa, M. Horvatic, C. Berthier,H. Kageyama, Y. Ueda, S. Miyahara, F. Becca, andF. Mila, Science 298, 395 (2002).

11 S. Miyahara and K. Ueda, J. Phys.: Condens. Matter 15,R327 (2003).

12 S. E. Sebastian, N. Harrison, P. Sengupta, C. D. Batista,S. Francoual, E. Palm, T. Murphy, N. Marcano, H. A.Dabkowska, and B. D. Gaulin, Proceedings of the NationalAcademy of Sciences 105, 20157 (2008).

13 S. Yoshii, T. Yamamoto, M. Hagiwara, A. Shigekawa,S. Michimura, F. Iga, T. Takabatake, and K. Kindo, J.Phys.: Conf. Ser. 51, 59 (2006).

14 S. Michimura, A. Shigekawa, F. Iga, M. Sera, T. Taka-batake, K. Ohoyama, and Y. Okabe, Physica B 378-380,596 (2006).

15 K. Siemensmeyer, E. Wulf, H.-J. Mikeska, K. Flachbart,S. Gabani, S. Mat’as, P. Priputen, A. Efdokimova, andN. Shitsevalova, Phys. Rev. Lett. 101, 177201 (2008).

16 Z. Y. Meng and S. Wessel, Phys. Rev. B 78, 224416 (2008).17 F. Liu and S. Sachdev, “Magnetization of the shastry-

sutherland antiferromagnet near the ising limit,” (2009),arXiv:0904.3018.

18 M.-C. Chang and M.-F. Yang, Phys. Rev. B 79, 104411(2009).

19 T. Suzuki, Y. Tomita, and N. Kawashima, Phys. Rev. B80, 180405 (2009).

20 T. Suzuki, Y. Tomita, N. Kawashima, and P. Sengupta,Phys. Rev. B 82, 214404 (2010).

21 Y. I. Dublenych, Phys. Rev. Lett. 109, 167202 (2012).22 Y. I. Dublenych, Phys. Rev. E 88, 022111 (2013).

9

23 K. Wierschem and P. Sengupta, Phys. Rev. Lett. 110,207207 (2013).

24 K. Wierschem and P. Sengupta, J. Korean Phys. Soc. 63,486 (2013).

25 W. C. Huang, L. Huo, G. Tian, H. R. Qian, X. S. Gao,M. H. Qin, and J.-M. Liu, J. Phys.: Condens. Matter 24,386003 (2012).

26 P. Farkasovsky, H. Cencarikova, and S. Mat’as, Phys. Rev.B 82, 054409 (2010).

27 T. Suzuki, Private communication.28 A. W. Sandvik and J. Kurkijarvi, Phys. Rev. B 43, 5950

(1991).29 O. F. Syljuasen and A. W. Sandvik, Phys. Rev. E 66,

046701 (2002).

30 K. Louis and C. Gros, Phys. Rev. B 70, 100410 (2004).31 R. G. Melko, J. Phys.: Condens. Matter 19, 145203 (2007).32 A. Dorneich and M. Troyer, Phys. Rev. E 64, 066701

(2001).33 A. W. Sandvik, Phys. Rev. B 56, 11678 (1997).34 E. L. Pollock and D. M. Ceperley, Phys. Rev. B 36, 8343

(1987).35 W. Selke, Physics Reports 170, 213 (1988).36 L. Balents, M. P. A. Fisher, and S. M. Girvin, Phys. Rev.

B 65, 224412 (2002).37 A. Kalz, A. Honecker, S. Fuchs, and T. Pruschke, Phys.

Rev. B 83, 174519 (2011).38 A. Kalz, M. Arlego, D. Cabra, A. Honecker, and

G. Rossini, Phys. Rev. B 85, 104505 (2012).