arXiv:2007.04680v1 [cond-mat.str-el] 9 Jul 2020 · [email protected] [email protected] [email protected] well as the doping-induced transition into a spin-glass

Competing orders in a frustrated Heisenberg model on the Fisher lattice

Atanu Maity,1, 2, ∗ Yasir Iqbal,3, † and Saptarshi Mandal1, 2, ‡

1Institute of Physics, Bhubaneswar 751005, Odisha, India2Homi Bhabha National Institute, Mumbai 400 094, Maharashtra, India

3Department of Physics, Indian Institute Of Technology Madras, Chennai 600036, Tamil Nadu, India(Dated: December 8, 2020)

We investigate the Heisenberg model on a decorated square (Fisher) lattice in the presence of firstneighbor J1, second neighbor J2, and third neighbor J3 exchange couplings, with antiferromagneticJ1. The classical ground state phase diagram obtained within a Luttinger-Tisza framework isspanned by two antiferromagnetically ordered phases, and an infinitely degenerate antiferromagneticchain phase. Employing classical Monte Carlo simulations we show that thermal fluctuations failto lift the degeneracy of the antiferromagnetic chain phase. Interestingly, the spin wave spectrumof the Néel state displays three Dirac nodal loops out of which two are symmetry protected whilefor the antiferromagnetic chain phase we find symmetry protected Dirac lines. Furthermore, weinvestigate the spin S = 1/2 limit employing a bond operator formalism which captures the singlet-triplet dynamics, and find a rich ground state phase diagram host to variety of valence-bond solidorders in addition to antiferromagnetically ordered phases.

I. INTRODUCTION

In Mott-Hubbard insulators, a reasonable descriptionof the localized electron state at low-temperatures is pro-vided by the Heisenberg spin Hamiltonian [1]. In thepresence of frustrated interactions, which could be geo-metric or parametric in origin, the determination of theground state and low-energy physics of the Heisenbergmodel poses itself as a highly nontrivial problem. Theprincipal motivation in the investigation of frustratedspin systems lies in the lure of finding either magneti-cally ordered ground states with intricate spin texturesor highly correlated nonmagnetic phases such as spin liq-uids [2–6]. To this end, transition metal oxides have at-tracted much attention as they are found in nature witha rich diversity of geometrically frustrated lattice struc-tures, displaying a wide spectrum of magnetic behav-iors [7, 8]. In particular, in one such family of manganeseoxide compounds (MnO2) such as K1.5(H3O)xMn8O16,Ba1.2Mn8O16, and α–MnO2 [9–11], the Mn ions resideon the vertices of a geometrically frustrated network,namely, the hollandite lattice [9, 12, 13]. Experimen-tal studies on these systems have unveiled the presenceof a plethora of magnetic phases upon variation of tem-perature, magnetic field, and doping, which include, anantiferromagnetic state [14], a ferromagnet, helimagneticorder [15, 16], and spin glass behavior.

In order to understand the origin of this diversity inmagnetic behaviors it is helpful to disentangle the ef-fects of magnetic frustration from those arising due tothe presence of impurities. Recently, theoretical studiesemploying an Ising model on the hollandite lattice [17, 18]successfully explained the origin of the antiferromag-netic ground state in the disorder free system [19] as

∗ [email protected]† [email protected]‡ [email protected]

well as the doping-induced transition into a spin-glassstate [20–22]. However, the Ising model studied inRef. [17] cannot account for helimagnetic (and in gen-eral noncollinear) orders observed in K1.5(H3O)xMn8O16

and K0.15MnO2 at low temperatures [15, 16]. Experi-mental investigations on Manganese compounds [23–25]have provided evidence that these systems have smallmagnetic anisotropies and are thus well described by aHeisenberg model. The zero temperature (T = 0) clas-sical magnetic phase diagram of the Heisenberg modelon the Hollandite lattice allowing for different signs andstrengths of nearest-neighbor couplings was studied inRef. [26].

The hollandite lattice can be viewed either as cou-pled two-dimensional triangular lattices stacked in thez-direction or as decorated square lattices (called Fisherlattice) stacked in the y-direction [26]. An understandingof the magnetic Hamiltonian on a lattice which is a two-dimensional projection of the original three-dimensionallattice often provides valuable insights into how magneticorder develops in the original three-dimensional model,and helps flesh out the structure of the (often intricate)spin configurations. In this regard, investigation of themagnetic phases in kagome lattice as an insightful routetowards understand the complex magnetism in the py-rochlore lattice is noteworthy [27]. Herein, we adopt theroute of understanding the magnetism of the Hollanditelattice by viewing it as coupled Fisher lattices since thenon-trivial mechanism of magnetic order in α-MnO2 ma-terials seems to arise due to the coupling in y-direction[13, 18, 26, 28]. In this work, we carry out a detailed anal-ysis of the magnetic phases present in the T = 0 classicalphase diagram and investigate fluctuation effects beyondthe classical limit via a spin-wave analysis and a bond-operator formalism for spin S = 1/2.

We consider a minimal model on the Fisher lattice [seeFig. 1] such that J1 couples the vertices of neighboringsquares, J2 defines the nearest-neighbour coupling withinthe squares, and J3 is the second nearest-neighbor (diag-

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FIG. 1. (a) The Fisher lattice showing the three exchangecouplings J1 (blue) connecting sites on octagons, J2 (green)connecting the sites of a square, and J3 (black) connectingthe diagonals of a square, of Eq. (1), with the four sites of theunit cell marked by 1, 2, 3, and 4. (b) The first Brillouin zoneof the Fisher lattice with the high-symmetry points marked.

onal) coupling within each square. The inclusion of a J3

coupling has been motivated from recent studies [13, 28]which suggest that it might be necessary to describe themagnetism in Hollandite systems. At the classical level, aLuttinger-Tisza analysis [29] of the (J1, J2, J3) parameterspace reveals the presence of different kinds of antiferro-magnetically (AF) ordered states, an infinitely degener-ate uncorrelated antiferromagnetic chain phase [30, 31],as well as a unique Néel phase which features magnonicDirac nodal lines depending on the sign and strengthof the couplings. Furthermore, we investigate the roleof quantum and thermal fluctuations on these groundstates and find via (numerically) exact classical MonteCarlo simulations that thermal fluctuations fail to lift thedegeneracy of the uncorrelated antiferromagnetic chainphase. We complement our study by going beyond thespin-wave approximation and compute the relative stabil-ity of the semi-classical ground state within a variationalansatz by comparing the energies of competing states andfind that each of them is stable as they feature a finitetriplon excitation gap over suitable singlet states.

Our paper is structured as follows. In Sec. II, wedefine the model Hamiltonian and the Luttinger-Tiszaframework employed to obtain the classical T = 0 phasediagram. In Sec. III A, we discuss the Luttinger-Tiszaground states and study the effect of thermal fluctuationsemploying classical Monte Carlo simulations. In Sec. IVand Sec. V, the impact of quantum fluctuations to har-monic order on the ground states is presented. In Sec. VI,we analyze our model Hamiltonian for spin S = 1/2within the scope of a bond operator formalism and showthe existence of three different types of quantum param-agnetic ground states, namely, a plaquette VBS, and twoother dimer ordered states. Finally, we summarize anddiscuss our results in Sec. VII.

II. MODEL AND METHODS

We consider a two-dimensional plane of the Hollan-dite lattice [see Fig. 1(a)], called the decorated square(Fisher) lattice, which is characterized by a four-site ge-ometrical unit cell [32]. The interactions between thespins localized on the vertices of this lattice are governedby a Heisenberg Hamiltonian

H = J1

∑〈i,j〉1

Si · Sj +J2

∑〈i,j〉2

Si · Sj +J3

∑〈i,j〉3

Si · Sj , (1)

where the J1, J2, and J3 superexchange couplings areschematically illustrated in Fig. 1(a). It is worth not-ing that in earlier studies [17, 26] investigating the mag-netism of the full three-dimensional Hollandite lattice,the consideration of in-plane interactions was restrictedto inter-square (J1) and nearest-neighbor intra-square(J2) couplings only [33], while the inter-plane couplingwas found to yield helimagnetic order. Recent experi-mental studies [13, 28] on Hollandite compounds havepointed to relatively more intricate ground states com-pared to those found in Refs. [15–17, 26]. In particular, inRef. [28], the in-plane magnetic ground state was foundto possess a magnetic unit cell which is a 4 × 4 expan-sion of the geometrical unit cell. Though the materials inquestion potentially involve more complex charge order-ings which are likely to induce further magnetic couplingbetween the Mn atoms, it is understood that a simplemodel which accounts for only the above two mentionedin-plane interactions (J1 and J2 in Fig. 1(a)) is not suffi-cient to explain the formation of a magnetic order with a4× 4 magnetic unit cell. The above fact motivates us toexplore a larger parameter space of exchange couplings,and to this end, we propose the simplest extension byintroducing an additional second nearest-neighbor (diag-onal) coupling within each square, i.e., J3 in Fig. 1(a).In our study we consider all possible signs and strengthsof the (J2, J3) couplings with an antiferromagnetic J1.

Our analysis of the ground states of the classical ver-sion of Eq. (1) employs the Luttinger-Tisza method. Thecorresponding classical model is obtained by normalizingthe spin operators with respect to their angular momen-tum S and taking the limit S → ∞ [34, 35]. Conse-quently, the spin operators in Eq. (1) are replaced byordinary vectors of unit length at each lattice site i. Fora generic spin interaction, we have the following classicalHamiltonian that needs to be minimized

H =∑i,j,α,β

Jαβ(Rij)Si,α · Sj,β , (2)

where i, j denotes the corresponding Bravais lattice sitesseparated by lattice translation vectors Rij and α, β in-dices denote the sublattice sites. The underlying Bravaislattice of the Fisher lattice is the square lattice. TheLuttinger-Tisza method [29] seeks to find a ground stateof Eq. (2) by enforcing the spin-length constraint at a

3

global level, i.e.,∑i |S2

i | = NS2, where N is the totalnumber of lattice sites, a condition termed as the weakconstraint. This constraint amounts to permitting site-dependent average local moments which take us beyondthe classical limit by approximately incorporating someaspects of quantum fluctuations [36].

A solution of this relaxed problem is achieved by de-composing the spin configuration into its Fourier modesSα(k) on the four sublattices of the Fisher lattice

Si,α =1√N/4

∑k

Sα(k)eık·ri,α . (3)

Inserting this expression into Eq. (2) results in

H =∑k

∑α,β

Jαβ(k)Sα(k) · Sβ(−k), (4)

with the interaction matrix given by

Jαβ(k) =∑i,j

Jαβ(Rij)eık·Rij . (5)

The modes which respect the weak constraint are givenby the wave vector k, for which the lowest eigenvalue ofEq. (5) has its minimum. The eigenvector correspondingto this eigenvalue gives the relative weight of the modeson the sublattices [37], which means that these modesdo not fulfill the strong constraint (|S2

i | = S2, i.e., fixedspin-length constraint on every site) if the components ofthe eigenvector do not have the same magnitude. On theother hand, if this condition is met, the true ground stateof the classical model is a coplanar spiral determined bythe optimal Luttinger-Tisza wave vector [38].

III. CLASSICAL GROUND STATES

A. Luttinger-Tisza analysis

The interaction matrix Jαβ(k) for our model takes theform

0 J2e

ı(kx−ky)a J3eı2kxa + J1e

ıkxb J2eı(kx+ky)a

J2e−ı(kx−ky)a 0 J2e

ı(kx+ky)a J3eı2kxa + J1e

ıkxb

J3e−ı2kxa + J1e

−ıkxb J2e−ı(kx+ky)a 0 J2e

ı(kx−ky)a

J2e−ı(kx+ky)a J3e

−ı2kxa + J1e−ıkxb J2e

−ı(kx−ky)a 0

In the region of parameter space defined by J3 > |J2|,we find that the minimal eigenvalue wave vector (kx, ky)0

is given by

(kx, ky)0 = (2mπ/3, ky) or

(kx, ky)0 = (kx, 2nπ/3) (6)

where m,n ∈ Z, hence, realizing long-range ordering inone direction with an absence of relative ordering in theother direction. Along the line J2 = J3, we find

(kx, ky)0 = (kx, ky), (7)

leading to a degenerate ground state manifold. In theremaining regions of parameter space we find

(kx, ky)0 = ((2n+ 1)π/3, (2m+ 1)π/3), (8)

which corresponds to long range magnetic order withcommensurate ordering wave vectors. The absence of anincommensurate ordering wave vector implies that thedegree to which mutual interactions between spins is sat-isfied is likely to be determined locally, and hence, as astarting point it is helpful to pursue an energy minimiza-tion of a local cluster of spins. To this end, we employ avariational approach which proceeds by first constructingspin configurations of a local cluster of spins that min-imize its energy and subsequently attempt to construct

a global spin configuration which also satisfies the localminimum energy configuration of the cluster of spins. Weverify the accuracy of our global spin configurations fromclassical Monte Carlo simulations.

To start with, we consider a cluster of four spins thatconstitute a unit cell of the Fisher lattice. The spin con-figuration is parameterized by observing that the latticecan be described as a collection of horizontal and verti-cal connections which are coupled via J2 bonds. Eachhorizontal and vertical string of connections hosts twosublattices each. The relative orientation between thespins within both the sublattices are assigned an angleγ, while the relative orientation between the spins be-longing to the same sublattice but in different chains isassigned an angle α [see Fig. 2(b)]. This choice of ansatzgives an energy density

E/NS2 =1

4(J1(cos(γ − kx) + cos(γ − ky)) + J2(2 cosα

+ cos(α+ γ) + cos(α− γ)) + 2J3 cos γ). (9)

The above expression has four free parameters whichneed to be determined to obtain the ground state spinconfiguration. We note that since the antiferromagneticJ1 bond is not frustrated by any other interaction, onemay put forth an ansatz in which the spins connected by

4



-2 -1 0 1 2-2

-1

0

1

2

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J3/J

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FIG. 2. (a) Classical phase diagram of J1–J2–J3 Heisenbergmodel on the Fisher lattice with the couplings as defined inFig. 1(a) and Eq. (1), (b) paramterization of a generic spinconfiguration with γ = kx + π and kx = ky = k.

the J1 bonds are antiparallel, i.e., kx − γ = ky − γ = π,and with Eq. (9) simplifying to

E/NS2 =1

2(J2 cosα(1− cos kx)− J3 cos kx − J1). (10)

Upon minimizing Eq. (10) with respect to kx and αwe get the following two sets of conditions for a spinconfiguration to qualify as a ground state

sin kx(J2 cosα+ J3) = 0 (11)sinα(1− cos kx) = 0. (12)

The solutions (kx, α) satisfying the conditions [Eqs. (11)and (12)] corresponding to different phases are describedbelow

1. Antiferromagnetic chain phase

This phase is characterized by (k, α) = (0, α) wherek = kx = ky, and is stabilized for J3 > |J2|. It is de-picted as phase I (yellow region) in the phase diagramof Fig. 2(a). It features perfect antiferromagnetic orderalong either the horizontal or vertical chains, however,there is a complete absence of spin correlations betweenany two of these ordered chains [see Fig. 3(a)]. This im-plies that within any given four site unit cell, the spinscoupled by J3 bonds are antiferromagnetically correlatedwhile there is no correlation between the spins connectedby J2. Hence, the angle α can take any value, implyingan infinite degeneracy of the ground state manifold. Theground state energy is then independent of α

E/NS2 = −1

2(J3 + J1). (13)

In the above expression, the independence of the en-ergy on α arises due to a cancellation of the contribu-tions from two bonds connected by J2 within a givensquare plaquette. The existence of such a degeneracywithin each square together with long range antiferro-magnetic order along horizontal or vertical chains poses



1 2

3

(c)

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FIG. 3. Spin configuration in (a) phase I (uncorrelated anti-ferromagnetic chain) with (k, α) = (0, α). The individual hor-izontal or vertical chains have perfect antiferromagnetic order,the relative orientations between them is not fixed. With re-spect to lower horizontal chain the orientations of the twovertical chains is α1 and α2, respectively. Also, note that theorientation of upper horizontal chain with respect to lowerhorizontal chain is α3, i.e., the system has an infinite degen-eracy, (b) phase II (Néel phase) with (k, α) = (π, π), and (c)phase III (sublattice Néel phase) with (k, α) = (π, 0).

itself as an interesting platform to investigate the order-by-disorder physics driven by thermal and quantum fluc-tuations. This will be discussed in Sec. III B, Sec. IV andSec. V.

2. Néel phase

For J3 < |J2|, we enter a region of parameter spacewhere the arbitrariness in the choice of the parameterα in the uncorrelated antiferromagnetic chain phase getslifted. In particular, for J2 > 0 and J3 < J2, we obtain aNéel ordered phase [marked as phase II (blue region) inFig. 2(a)]. This phase is characterized by (k, α) = (π, π)which signifies that within any given unit cell (square)there is perfect antiferromagnetic order, and that thespins in neighboring unit cells are aligned antiferromag-netically with respect to each other [see Fig. 3(b)] . How-ever, unlike the familiar Néel phase on the square orhoneycomb lattice, not all antiferromagnetic bonds aresatisfied when J3 > 0, as the spins connected by the J3

couplings remain frustrated. The ground state energy ofthis phase is given by,

E/NS2 = −1

2(2J2 − J3 + J1). (14)

Upon entering the region J3 < 0, this phase is furtherstabilized since the spins coupled via J3 bonds are ferro-magnetically aligned in this Néel phase.

3. Sublattice Néel phase

In the region J2 < 0, when J3 < |J2|, the free pa-rameter α characterizing phase I is determined to zeroimplying that all spins within a given unit cell are ferro-magnetically aligned. Furthermore, these four-site unitcells form Néel order throughout the lattice, and hence,

5

1.×10-7

5.6×10-6

3.2×10-4

1.8×10-2

1.

FIG. 4. The static (equal-time) spin structure factor obtained from classical Monte Carlo simulations for the three differentphases in the classical phase diagram of Fig. 2(a) evaluated at T/J1S2 = 0.001 for a system of 1600 spins, (a) phase I at(J2/J1, J3/J1) = (1, 4), (b) phase II at (J2/J1, J3/J1) = (4, 1), and (c) phase III (J2/J1, J3/J1) = (−4, 1).

each sublattice is Néel ordered [see Fig. 3(c)], we hence-forth refer to this phase as a sublattice Néel ordered phase[marked as phase III (orange region) in Fig. 2(a)]. Thisstate is thus characterized by (k, α) = (π, 0). The groundstate energy density can be written as,

E = −1

2(2|J2| − J3 + J1). (15)

When J3 < 0, the spin confiuration satisfies all the cou-plings.

B. Classical Monte Carlo analysis

Since the Luttinger-Tisza approach is not a priori ex-pected to give the exact ground state spin configura-tion on a non-Bravais lattice such as ours, and given thefact that our approach is based on a variational ansatz[Eq. (9)], we perform classical Monte Carlo simulations toinvestigate the accuracy of our analysis as well as to studythe role of thermal fluctuations. It is worth mention-ing that a two-dimensional system of Heisenberg spinswith finite-range antiferromagnetic or ferromagnetic in-teractions cannot feature long-range order at any finitetemperature by virtue of the Hohenberg-Mermin-Wagnertheorem [39]. Our discussion thus refers to the behav-ior of order parameters (here spin orientations) at shortlength-scales, i.e., at distances r less than the correlationradius ξ. We consider a system of 1600 (= 20 × 20 × 4)spins, employ parallel tempering, and carry out simula-tions at temperatures T/J1S

2 = 0.001 [40]. We find thatin phases II and III both the angles (γ, α) [see Fig. 2(b)]lock into the values (0, π) and (0, 0), respectively, as ex-pected from the Luttinger-Tisza result. In contrast, inphase I, the angle γ settles into a value of π, while thermalorder-by-disorder mechanism fails to lift the degeneracyin the angles α1, α2 and α3 [see Fig. 3(a)] which thereforecontinue to exhibit a fluctuating behavior with time evo-lution/Monte Carlo steps [41]. Furthermore, as the tem-perature T → 0 the specific heat C tends to a value lessthan one, pointing to the important fact that the role of

anharmonic fluctuation modes cannot be neglected. Ouranalysis thus provides evidence for the stabilization ofan uncorrelated antiferromagnetic chain phase at finitetemperatures. This is also reflected in the finding thatwithin the region of parameter space occupied by phaseI, the ground state energy obtained from classical MonteCarlo simulations is independent of J2 indicating an ab-sence of spin correlations between the different horizontaland vertical chains. Hence, the spin configurations deter-mined from the (numerically) exact classical Monte Carlosimulations are in complete agreement with those deter-mined from the variational ansatz [Eq. (9)], validatingthe classical phase diagram of Fig. 2(a).

Having discussed the classical ground states occupy-ing the (J1, J2, J3) parameter space, it is instructive tocalculate the magnetic structure factor which can be ex-perimentally measured in a neutron scattering experi-ment [42] to reveal the signatures of the magnetic groundstates. As discussed above, we have found three differentmagnetic ground states, namely, an uncorrelated antifer-romagnetic chain phase, Néel phase and sublattice Néelphase, depending on the sign and magnitude of the ex-change parameters in the Hamiltonian. Here, we calcu-late the static (equal-time) spin structure factor

S(k) =1

N

∑i,j

e−ık·Rij 〈Si · Sj〉 (16)

via classical Monte Carlo simulations for three differentsets of parameter values. Here, N is the total numberof sites, and i, j run over all the sites of the lattice, andthe nearest-neighbour distance between two sites is set tounity. This implies that the distance between the neigh-bouring unit cells is three units which makes the periodic-ity of S(k) as 2π/3. In Fig. 4 we show the structure factorof the different phases in Fig. 2(a). We note that in phaseI the horizontal and vertical chains are uncorrelated, andthereby have no global ordering in any direction, on av-erage. The perfect antiferromagnetic order in a givenchain yields a vanishing contribution when summed overall chains. Hence, we expect a featureless structure factor

6

with the presence of subdued peaks at the zone bound-ary (corresponding to an ordering vector k which is zeroupon a statistical averaging) as can be seen in Fig. 4(a).For the magnetic ground states given by phase II andphase III, we note that a global antiferromagnetic (Néel)order should yield peaks at k = (mπ

3 , nπ3 ) where m,n

are odd integers. However, the peaks at the above men-tioned points are modulated due to the form factor ofthe magnetic ordering in a given unit cell of Néel phase(phase II) and subattice Néel phase (phase III). Theseform factors lead to the appearance of blue lines alongthe diagonal for the phase II making a cross like controurand for phase III a square pattern. The origin of suchlow intensity contours can be understood by calculatingthe structure factor for the Néel and the sublattice Néelphases analytically. An exact expression of the staticstructure factor for the different classical spin configu-rations obtained within a Luttinger-Tisza formalism [seeSec. III A] can be determined by evaluating Eq. (16) an-alytically. The static structure factor F = S2|f |2 (whereS denotes the form factor of the magnetization in a givenunit cell and f arises due to global Néel ordering) thusobtained is given by

f =(1− eıNLkx)(1− eıNLky )

(4N)2(1 + eıLkx)(1 + eıLky ), L = 3a (17)

S(k) = 4(cos kx ∓ cos ky)2 (18)

for a lattice of N × N unit cells, and −(+) correspondsto the form factor for the Néel (sublattice Néel) phase.

The form factor S is zero along the line kx = 2m′π±kyfor phase II and kx = (2m′+1)π±ky for phase III wherem′ is any integer including zero. Thus, the expectedpeaks at k = (mπ

3 , nπ3 ) will be modulated due to such

lines of zero intensity and it would remove some peaksin structure factor which are expected due to globalNéel ordering. Figure 4(b) represents the structurefactor for the Néel order (phase II). We see that thepeaks appear at k = (mπ

3 , nπ3 ) with m 6= n due to the

form factor modulation. On the other hand, such adestructive modulation is absent for the sublattice Néelphase and the peaks appear at the expected locationsk = (mπ

3 , nπ3 ) with m and n being odd integers [see

Fig. 4(c)]. In practice, a particular material may nothave the exact symmetry of the lattice we have consid-ered. For example, the unit cell may not be a square astaken here and also the distances between different sitesof different neighbors connected by exchange couplingswould also be different, in which case the experimentallyobtained structure factor would be modified comparedto that shown in Fig. 4. However, the structure factor inthat case can be easily compared by evaluating Eq. (17)with modified lattice parameters, mainly the differentvalues of Rij .

IV. SPIN WAVE ANALYSIS

A. Antiferromagnetic chain phase

The T = 0 classical antiferromagnetic chain phase[phase I in Fig. 2(a)] has an infinitely degenerate groundstate manifold. In particular, on a lattice consisting of Nhorizontal and M vertical lines [see Fig. 2(b)], there areN×M independent α parameters each of which can takeon values ranging from 0 to 2π. It is thus natural to askthe question whether quantum fluctuations can lift thisdegeneracy via an order-by-disorder mechanism [43, 44]and select a unique configuration parameterized by a cer-tain value of α. To investigate the role of quantum fluc-tuations, we carry out a linear spin wave analysis. To thiseffect, we rotate our coordinate system in such a way thatthe z-axis of the local coordinate system coincides withthe axis of the local spin orientation,

Sαi = Rx

(π2

)Rz(φi)S

α′

i , α = x, y, z. (19)

The Holstein-Primakoff transformation [45] can now bewritten as

Sz′

i,m ≈ s− a†i,mai,m

Sx′

i,m ≈√s

2

(a†i,m + ai,m

)Sy

′

i,m ≈ ı√s

2

(a†i,m − ai,m

). (20)

In the above m = 1, 2, 3, 4 denote the sublattice indices.Within a quadratic approximation to the boson opera-tors we find that a uniform choice of α (when all N ×Mvalues of α are the same) is energetically favorable com-pared to disordered configurations of α (when all N ×Mvalues of α are different), indicating an order-by-disordermechanism at work. This lifting of the degeneracy is onlypartial as α (which is now the same for all N ×M sites)can still take on any value between 0 and 2π yieldingthe same ground state energy, and the hence there stillremain an infinite number of degenerate ground states.However, the magnon spectrum for each value of α neednot be the same and thus we investigate the α depen-dence of the spin-wave spectrum. In Appendix A weprovide the expressions of the resulting Hamiltonian af-ter implementing the Holstein-Primakoff transformationcorresponding to Eqs. (19) and (20). As expected, theHamiltonian is invariant under PT -symmetry which isdefined as PT = σx⊗ σ0K (where K is the complex con-jugation operator). In phase I, the magnetic and crys-tallographic unit cells are identical, as a result of whichwe get a 8 × 8 Hamiltonian matrix [see Eq. (A1)] in k-space. As a consequence of PT symmetry we obtain fourdoubly-degenerate magnon branches shown in Fig. 5 with

7

0.0

0.5

1.0

1.5

ϵ μ

0.0

0.5

1.0

1.5ϵ μ

0.0

0.5

1.0

1.5

ϵ μ

Γ X M Γ M Y Γ X M Γ M Y Γ X M Γ M Y

α = 0 , π α = π / 4 α = π / 2(c)(b)(a)

FIG. 5. Free magnon spectrum corresponding to Eqs. (21) and (22) in phase I for the parameter values (J2/J1, J3/J1) = (1, 2)plotted along the high-symmetry path Γ(0, 0), X(π, 0), M(π, π), and Y (0, π) for three choices of degeneracy parameters, (a)α = 0 or π, (b) α = π/4 and (c) α = π/2.

the following dispersion relations

ε(k)1,2=1

2

√pk ±

1

2

√Q(0)fk + gk(0) + hk (21)

ε(k)3,4=1

2

√pk ±

1

2

√Q(α)fk + gk(α) + hk (22)

where Q(α) = 2(J2)2 cos2 α−(J3)2

4 and

pk =J3

4(2− cos kx − cos ky) (23)

fk = (cos(kx + ky) + cos(kx − ky)) (24)gk(α) = (J2)2 cos2 α(1− cos kx − cos ky) (25)

hk = (J3)2(2 + cos(2kx) + cos(2ky))/8 (26)

From Eq. (21) we see that there are two modes whichare independent of α, and Eq. (22) shows that the othertwo modes are α dependent. For any given value ofα there are two Goldstone modes originating from thespontaneously broken U(1) symmetry, and we observethe presence of zero-energy modes along the segmentsΓX and ΓY . The presence of zero-energy modes can beunderstood from Eqs. (21) and (22) which upon substi-tution of kx(ky) = 0 and ky(kx) = k yields ε(k) = 0 andε(k) =

√J3 sin k/2. Hence, there are two doubly degener-

ate modes along the kx = 0 and ky = 0 axes out of whichone doubly degenerate mode has zero excitation energyand another linearly dispersing in k for small values.

It is worth noting that along the kx = 0 and ky = 0axes we have linear band crossings along the line seg-ments ΓX and ΓY in the Brillouin zone, thus formingDirac nodal lines. We now discuss how the spin wavespectrum depends on α. (i) For α = 0 or π: in this casethe system has two sublattice Néel order. The Hamilto-nian is block diagonal [see Appendix A], and each blockis PT invariant which gives rise to two 4-fold degeneratebands [see Fig. 5(a)]. (ii) For α = π/4: The block diag-onal structure disappears and as a result the two 4-folddegenerate bands split into four 2-fold degenerate bands[see Fig. 5(b)]. (iii) For α = π/2: In this case we have aband touching of the α dependent bands at the M pointwhich is a consequence of the underlying mirror reflec-tion symmetry about the kx, ky and kx = ky axes. Wenote that there is a Dirac nodal line along the segmentΓM [see Fig. 5(c)]. Low energy expansion of dispersion

along ΓM gives

ε(κ)1,2= 2(J3 ± J2 cosα)| sinκ/2| (27)ε(κ)3,4= 2(J3 ± J2)| sinκ/2| (28)

In the above expressions, we have substituted kx = ky =κ. It is evident that the number of independent modesalong the ΓM depends on α. For α = 0, π there aretwo doubly-degenerate optical modes as represented byviolet and blue lines in Fig. 5(a). When α = π/4, thetwo-fold degeneracy of both the optical modes gets lifted[see Fig. 5(b)], however, when α = π/2 only the degen-eracy of the lower optical mode gets lifted. The aboveobservation may have experimental relevance in decidingthe exchange parameter set for the model Hamiltonianor the selected angle α.The appearance of doubly degenerate zero energy modesalong the ΓY and ΓX is reminsicent of the fact that ir-respective of the one-dimensional order along a certaindirection, the perpendicular direction can adjust itselffree of energy cost. We have also found that the inclu-sion of anisotropy or magnetic field perpendicular to thespin alignment plane leads to a lifting of the degeneracyof the optical modes along YM or ΓM , however, the zeroenergy mode along the kx(ky) = 0 survives on ΓY andΓX segments. The inclusion of higher order terms maylift this degeneracy [46] but we expect the zero energymodes at high symmetry points to survive the inclusionof magnon-magnon interaction terms [47, 48].

B. Phase II and Phase III

This phase has perfect antiferromagnetic structurewhich is the same as for the unfrustrated case (J3 = 0 andJ1, J2 > 0) studied in Ref. [49] where the presence of twoDirac nodal loops was found. The magnetic unit cell (8-sites) is twice the size of the crystallographic unit cell (4sites). In the bipartite representation, the Hamiltoniancan be written as in Eq. (A5), and is seen to be blockdiagonal with each block being PT invariant. This re-sults in four four-fold degenerate bands shown in Fig. 6.Similar to what was found for the unfrustrated model(J3 = 0), we find two nodal loops [marked by blue linein Fig. 7(a)], namely, in Figs. 6(a) and 6(f) we see that

8

FIG. 6. Free magnon spectrum for phase II (Néel phase): (a), (b) (J2, J3) = (2, 1) and (f),(g) (J2, J3) = (3, 2), (c), (d), (e) and(h), (i), (j) represent the band touchings between different bands, which form the Dirac nodal loops for the two above-mentionedchoices of parameters, respectively. Notice that the nodal loop due to touching of the bands denoted by green and blue colorsappears for some choices of parameters and is gapped out for other choices. Hence, this nodal loop is not protected by anysymmetry whereas the nodal loops due to touching of the upper and lower two bands survive for all choices of parameters,thereby rendering these loops symmetry protected.

the black circled points form one nodal loop while thecyan circled points form the second nodal loop. In theabsence of a J3 coupling it was shown in Ref. [49] that theDirac nodal loops are topologically protected, and thatthere is a triple band touching at the Γ and M points,while in the presence of an additional J3 coupling wefind a quadruple band touching at the X and Y pointsas shown in Fig. 6(a). Furthermore, we find that a J3

coupling leads to the appearance of an additional Diracnodal loop along the Brillouin zone boundary and alsoalong the kx(ky) = 0 axes [see Fig. 6(d) and the orangesegment in Fig. 7(a)]. However, this additional loop ap-pears only for particular choices of parameters shown inFig. 7(b), and is not protected by any symmetry except atthe time-reversal invariant momentum points, i.e., the ΓandM points [see Fig. 6(f) and Fig. 6(i)], where there re-mains a two-fold degeneracy since the Hamiltonian givenby Eq. (A5) is invariant under T = ıσyK operator.

0 1 2 3 4 50

1

2

3

|J2|

J3

FIG. 7. (a) Projection of Dirac nodal loop on the first Bril-louin zone. The blue line denotes the symmetry protectednodal loops where as the orange one appers for some specificchoices of parameters. (b) denotes the choices of parameterswhere the additional nodal loop appears which is not pro-tected by any symmetry.

In phase III, whose magnetic structure has an eight siteunit cell, we similarly find that the Hamiltonian Eq. (A8)is block diagonal with each block being PT invariant,leading to four four-fold degenerate bands as in phase II.However, the symmetry protected Dirac nodal loops ofphase II are not found in phase III, on the other hand,the additional nodal loop along the zone boundary andkx(ky) = 0 axes found for phase II also appears in phaseIII for the same choice of parameters shown in Fig. 8.We note that the surface states for a ribbon geometry inphase II are gapless but the edge states for phase III aregapped.

FIG. 8. Free magnon spectrum for phase III (sublattice Néelphase) for (a) (J2, J3) = (−2, 1) and (b) (J2, J3) = (−0.8, 0.6).Unlike Néel phase there is no robust Dirac nodal loop in thisphase. However similar to Néel phase in some choices of pa-rameters a nodal loop (not protected by any symmetry) ap-pears along the zone boundary and kx = 0 and ky = 0 axesas can be seen from figure (a).

9

=

(b) (c)(a)

FIG. 9. A schematic illustration of the pattern of singlet dimer formations for the parameter regimes of Eq. (1), (a) |J1| (|J2|, |J3|) (VBS1), (b) |J3| (|J1|, |J2|) (VBS2), (c) |J2| (|J1|, |J3|) (Plaquette RVB) wherein the quantum ground state isgiven by a superposition of two states with dimer formation on the opposite sides of the square as shown in the right panel.

V. THERMAL AND QUANTUMORDER-BY-DISORDER EFFECTS

For a two-dimensional system of Heisenberg spins, theHohenberg-Mermin-Wagner theorem dictates that evenat infinitesimally small temperature, the deviation of thespin orientation, i.e., the spin fluctuation, is infinitelylarge. Thus, the assumption of small fluctuations aboutthe classical ground state, i.e., the harmonic approxima-tion is, in principle, not applicable. Nonetheless, theentropic selection of the dominant magnetic fluctuationtendencies at low temperatures carried out within a har-monic analysis may provide information on the nature ofthe short range correlations. It is worth mentioning thatthe harmonic order results can be significantly altered ifthe anharmonic modes play a decisive role.

We now proceed to carry out such an analysis for phaseI, which at zero temperature is infinitely degenerate be-ing characterized any value of α ∈ [0, 2π]. However, atfinite temperature, the fluctuation of the spins explic-itly contribute to the α dependent free energy. Thiscould possibly lead to a lifting of the degeneracy inthe parameter α via a thermal order-by-disorder mech-anism [50–52]. To this effect, we introduce a spin devi-ation θR,µ → θ0

R,µ + δθR,µ, where θR,µ is the groundstate spin configuration of the µth sublattice of a unitcell with radius vector R and δθR,µ is the deviationfrom ground state spin configuration. Substituting thisin Eq. (1) and expanding around the ground state upto quadratic order in δθR,µ, in Fourier space we obtainH = EGS + Hfluctuation with

Hfluctuation =∑q

ψ†qAq(α)ψq (29)

where ψq = [δθq,1 δθq,2 δθq,3 δθq,4]T , and EGS is theground state energy.

In this (harmonic) approximation, the fluctuations canbe integrated out in the partition sum, and give rise to alinear-T dependence in the free energy F(α, T ). Follow-ing Ref. [52], F(α, T ) can be written as

F(α, T ) = EGS −NT lnT + T∑q∈BZ

ln(detAq(α)), (30)

where the last term is the α dependent part of the low-temperature entropy density. The state which mini-mizes this term corresponds to the minimum of the freeenergy—this is the entropic order-by-disorder selectionmechanism discussed in Refs. [43, 44, 51, 52]. In the re-gion of the phase diagram occupied by phase I, we findthat thermal fluctuations select a value of α equal to 0or π. This is in contrast to classical Monte Carlo resultwhich finds that the order-by-disorder mechanism failsto lift the degeneracy in the angle α. Our results thuspoint to the non-negligible impact of anharmonic orderfluctuations which seem to alter the harmonic order pic-ture sharply in favor of an uncorrelated antiferromagneticchain phase.

Furthermore, the energy of the spin-wave modes is~Sεq,µ(α) in the semi-classical description for spins oflength S 1. Quantum fluctuations then choose thestate with the lowest zero-point energy

EZP(α) =∑q∈BZ

~S2εq,µ(α). (31)

The EZP(α) behaves qualitatively like the the last termof Eq. (30) and selects the same ordering vectors. Inthe region of the phase diagram occupied by phase I,we find that at zero-temperature quantum fluctuationsselect a value of α equal to 0 or π. It will be interesting toinvestigate the impact of anharmonic order terms, whichwe leave for a future study.

VI. BOND OPERATOR ANALYSIS: VALENCEBOND SOLID PHASES

In the extreme quantum limit of S = 1/2, there arisesthe possibility of zero-point quantum fluctuations de-stroying long-range antiferromagnetic orders when theamplitude of the fluctuations becomes of the order of thespin length. Furthermore, the presence of frustrated in-teractions enhances quantum fluctuations thereby aidingthe stabilization of quantum paramagnetic phases suchas quantum spin liquids and valence bond solids (VBS).Here, we investigate the effect of quantum fluctuations

10

beyond the spin-wave approximation by employing res-onating valence bond variational wave functions. Wefirst investigate our J1-J2-J3 model in parameter regimeswhere one of the couplings is overwhelmingly strongercompared to the remaining two. In this limiting regime,we note that at zeroth order the strongest bonds willform a singlet or triplet dimer and the locally excitedstates correspond to singlet to triplet excitations or viceversa for antiferromagnetic or ferromagnetic bonds, re-spectively [see Fig. 9]. The effect of non-zero values ofthe remaining two (subdominant) couplings is to dynam-ically create local excitations on the strong singlet andtriplet dimer bonds. Hence, an effective Hamiltonian ofinteracting singlet or triplet bonds between neighboringdimers can be constructed within this approach. Thisprocedure is known to be effective in explaining the lowenergy physics of interacting spin systems [53–61]. Wenow discuss the different VBS phases which are found tobe realized as variational quantum ground states of theJ1-J2-J3 model.

A. VBS1 and VBS2 phases

In the limit when |J1| (|J2|, |J3|) we have a VBSconfiguration consisting of dimers on J1 type bonds [seeFig. 9(a)]. The local Hilbert space is four dimensionalspanning the singlet ground state and the three tripletsas the excited states. Following Ref. [55], we define ψ†iand χ†i as the creation operators of the singlet and tripletstates, respectively, on the ith bond within a given unitcell with the accompanying constraint on the dimension-ality of the Hilbert space

ψ†i ψi + χ†i,αχi,α = 2S, (32)

where i = 1 or 2 corresponds to the two J1 type bondsin a unit cell, and α = 1, 2, 3 denotes the three typesof triplets in a given dimer. The interaction term be-tween the two neighboring unit cells is obtained by writ-ing down the spin components in terms of the above men-tioned valence bond operators. This is achieved by cal-culating 〈m|Sνµ|n〉 where µ = 1, 2 denotes the two spinsin a given dimer, ν = x, y, z labels the three spin com-ponents, and |m〉 (|n〉) represents the singlet or tripletstates. In general, a spin operator at a given site hasthe form Si ≈ 1

2 (ψ†i χi + h.c.) where i labels a givendimer [see Appendix B for details]. As a result of thistransformation we land up with a Hamiltonian which isquartic in the field operators χ and ψ. We carry out amean-field decoupling such that the resulting quadraticHamiltonian [Eq. (B2)] separates into singlet and tripletsectors with no mixing terms. The ground state energy isobtained by extremizing the Hamiltonian with respect tothe mean-field parameter Ni (where

√Ni = 〈ψ†i 〉 = 〈ψi〉)

and the Lagrange multiplier λ needed to implement theconstraint of Eq. (32), which yields two sets of self consis-tent equations which are solved. The resulting expression

Phase-III (LRO)

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Phas

e-II

(LR

O)

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Plaquette VBS

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VBS1

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VBS2

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Phase-I (LRO)

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FIG. 10. Quantum phase diagram of the S = 1/2 J1-J2-J3Heisenberg model on the Fisher lattice (Fig. 1(a)) obtainedby a bond operator analysis. We see the appearance of threedifferent VBS phases in addition to the three long-range or-dered (LRO) phases also present in the classical phase dia-gram (Fig. 2(a)).

for the ground state energy per unit cell for VBS state inthe parameter regime J1 (J2, J3) is,

Ea = (Na,1 +Na,2)Ea − λ[(Na,1 +Na,2)− 1]

+ 32N

∑k

∑2i=1(Θk,i −Ak,i), (33)

where a labels the VBS1 state, Θk,i is the eigenenergyobtained in the triplon sector of the interacting Hamil-tonian Eq. (B2) signifying the excitation of the tripletstate [59], and Ak,i is the diagonal element of Eq. (B2).A similar procedure can be followed for the VBS2 state inthe parameter regime J3 (J1, J2). For the VBS2 statethe unit cell has been conveniently chosen as an elemen-tary square. In a similar manner as above, we obtain theground state energy per square,

Eb = (Nb,1 +Nb,2)Eb − λ[(Nb,1 +Nb,2)− 1]

+ 32N

∑k

∑2i=1(Πk,i − Bk,i). (34)

where b labels the VBS2 state, and the remaining termsall have identical meaning to that in Eq. (33) [see Ap-pendix B for further details].

B. Plaquette VBS

In the parameter regime (J2, J3) J1 where we expecta plaquette VBS phase [see Fig. 9(c)], we have two choicesof forming two dimers inside a square. More precisely, if

11

we denote the four vertices of a square as V1, V2, V3, andV4, then we have two possibilities for dimer formation,namely V1–V2 and V3–V4 or V1–V4 and V2–V3. Thecorresponding Hamiltonian is

Hp = J2(S1 + S3) · (S2 + S4) +J3(S1 · S3 + S2 · S4) (35)

The diagonalization of the above Hamiltonian gives thefollowing two lowest energy plaquette singlet states,

|Ψp,(±)〉 =1√2

(|ψ1,2〉|ψ3,4〉 ± |ψ1,4〉|ψ2,3〉) (36)

where |ψi,j〉 denotes a singlet state formed between thesites i and j, |Ψp,+〉 is the ground state wavefunction withenergy Es+ = −2J2 + J3

2 and |Ψp,−〉 is the first (singlet)excited state with energy Es,− = − 3

2J3. Above thesestates lie the nine triplet states of the plaquette VBSwith energies,

Etµ,µ = (−J2 +J3

2)δµ,3 −

J3

2(δµ,1 + δµ,2) (37)

where µ, ν = 1, 2, 3. The five quintet states have a degen-erate energy Ed = J2 + J3

2 . To capture the low-energydynamics we have restricted our analysis to within thesinglet-triplet manifold. The low-energy dynamics nowincludes, in addition to the triplet excited states con-sidered for the VBS1 and VBS2 states, the singlet ex-cited states. Within this approximation the effective low-energy Hamiltonian for a single plaquette can be writtenas

Hp =∑i=±

Es,iψ†p,iψp,i +

3∑µ,ν=1

Et(µ,ν)χ†(µ,ν)χ(µ,ν). (38)

The Eqs. (35), (36), and (38) together with the constraintof Eq. (32) provide a complete description of the low-energy spectrum at zeroth order, i.e., in the absence of aJ1 interaction, leading to isolated plaquettes. The pres-ence of a finite J1 introduces interactions between neigh-boring plaquettes which induce transitions between dif-ferent states of Eq. (38). The inter-plaquette interactionsare obtained by writing the spin components in terms ofthe above-mentioned plaquette operators as was done forthe VBS1 and VBS2 states [see Appendix C for details].The effective low-energy Hamiltonian for the interactingplaquette-VBS state thus obtained is given in Eq. (C3)[see Appendix C for details]. The ground state energyper plaquette is

Ec = Nc,+(Ec − λ) + λ+3

2N

∑k

3∑i=1

(Ωk,i − Ck,i) (39)

where√Nc,+ = 〈ψ†p,+〉 = 〈ψp,+〉, Ωk,i is the eigenenergy

obtained in the triplon sector of the interacting plaquetteHamiltonian [Eq. (C3)].

Employing the expressions [Eq. (33), Eq. (34) andEq. (39)] of the ground state energies of the three phases

0

0.5

1.0

1.5

2.0

2.5

J2/J1

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J3/J

1

<latexit sha1_base64="LwT4s/5nHowyP5Qs9oHvM+h5yzA=">AAAB8nicbVDLSgMxFM3UV62vqks3wSK4qjNaUHdFN+Kqgn3AdCiZNG1DM8mQ3BHKMJ/hxoUibv0ad/6NaTsLbT2QcDjnXu69J4wFN+C6305hZXVtfaO4Wdra3tndK+8ftIxKNGVNqoTSnZAYJrhkTeAgWCfWjEShYO1wfDv1209MG67kI0xiFkRkKPmAUwJW8u976UV2Zn8v65UrbtWdAS8TLycVlKPRK391+4omEZNABTHG99wYgpRo4FSwrNRNDIsJHZMh8y2VJGImSGcrZ/jEKn08UNo+CXim/u5ISWTMJAptZURgZBa9qfif5ycwuApSLuMEmKTzQYNEYFB4ej/uc80oiIklhGpud8V0RDShYFMq2RC8xZOXSeu86tWq1w+1Sv0mj6OIjtAxOkUeukR1dIcaqIkoUugZvaI3B5wX5935mJcWnLznEP2B8/kDh3WQyQ==</latexit>

FIG. 11. Density plot of triplet excitation gap obtained byplaquette and bond operator analysis.

-1.2

-1.0

-0.8

-0.6

-0.4

<latexit sha1_base64="X0iOH1rNdlQgElOo3fKH03TUw/Y=">AAAB9HicbVDLSsNAFL2pr1pfVZduBlvBVU0CosuiG3FVwT6gDWUynbRDJ5M4MymUkO9w40IRt36MO//GaZuFth64l8M59zJ3jh9zprRtf1uFtfWNza3idmlnd2//oHx41FJRIgltkohHsuNjRTkTtKmZ5rQTS4pDn9O2P76d+e0JlYpF4lFPY+qFeChYwAjWRvKq9/3UzS5Md7Jqv1yxa/YcaJU4OalAjka//NUbRCQJqdCEY6W6jh1rL8VSM8JpVuolisaYjPGQdg0VOKTKS+dHZ+jMKAMURNKU0Giu/t5IcajUNPTNZIj1SC17M/E/r5vo4NpLmYgTTQVZPBQkHOkIzRJAAyYp0XxqCCaSmVsRGWGJiTY5lUwIzvKXV0nLrTmXNfvBrdRv8jiKcAKncA4OXEEd7qABTSDwBM/wCm/WxHqx3q2PxWjByneO4Q+szx8/YZEa</latexit>

J2/J1

<latexit sha1_base64="xLN/xISYGIAmTqP88UTjZcJ9CyI=">AAAB9HicbVDLTgIxFL2DL8QX6tJNI5i4whmI0SXRjXGFiTwSmJBO6UBDpzO2HRIyme9w40Jj3Pox7vwbC8xCwZPcm5Nz7k1vjxdxprRtf1u5tfWNza38dmFnd2//oHh41FJhLAltkpCHsuNhRTkTtKmZ5rQTSYoDj9O2N76d+e0JlYqF4lFPI+oGeCiYzwjWRnLL9/2kll6Y7qTlfrFkV+w50CpxMlKCDI1+8as3CEkcUKEJx0p1HTvSboKlZoTTtNCLFY0wGeMh7RoqcECVm8yPTtGZUQbID6UpodFc/b2R4ECpaeCZyQDrkVr2ZuJ/XjfW/rWbMBHFmgqyeMiPOdIhmiWABkxSovnUEEwkM7ciMsISE21yKpgQnOUvr5JWteJcVuyHaql+k8WRhxM4hXNw4ArqcAcNaAKBJ3iGV3izJtaL9W59LEZzVrZzDH9gff4AQO2RGw==</latexit> J3/J

1

FIG. 12. Density plot of ground state energy obtained byplaquette and bond operator analysis.

we map out the resulting phase diagram. The mostsalient feature of our phase diagram is the appearance ofthree quantum paramagnetic phases, namely, a plaquetteVBS, and two other types of dimer ordered states dubbedVBS1 and VBS2 as shown in Fig. 10. The classical re-gion of existence of the uncorrelated AF chain phase isnow found to be divided into two region under the influ-ence quantum fluctuations. For J1 (J2, J3), the VBS1

state is stabilized while for J3 (J1, J2), the VBS2 stateis stabilized. At the phase boundary of these two VBSstates, a long range ordered (LRO) state is found to bestabilized in a sliver of parameters space as inferred bythe vanishing singlet-triplet excitation gap. We label thislong range ordered state as phase I (LRO) and it is char-acterized by the ordering vector Q = (0, ky), (kx, 0). Theordering wave vectors of the three LRO phases shown inFig. 10 are determined by the momenta associated withthe singlet-triplet gaps of the VBS phases that vanish atthe Néel-VBS quantum phase transition

Another interesting aspect of the quantum phase dia-

12

gram is that the Néel phase (labeled as phase II in Fig.2(a)) gives was to a plaquette-RVB state irrespective ofthe sign of the J3 coupling. On the other hand, the sub-lattice Néel phase (labeled as phase III in Fig. 2(a)) islargely immune to quantum fluctuations, and is referredto as phase III (LRO) in Fig. 10. Apart from thesephases, we find another long range ordered phase in asmall region for negative J3 sandwiched between VBS1

and plaquette VBS phases and label it as phase II (LRO)in Fig. 10. For the quantum paramagnetic phases such asVBS1, VBS2 and plaquette RVB, the singlet-triplet ex-citation gap is used as a measure of determining theirstability as ground states with respect to inclusion ofhigher order corrections. For the three long range or-dered phases shown in Fig. 10, the respective wave vec-tors which characterize these phases correspond to thevanishing of the singlet-triplet excitation gap of the VBSphases.

The singlet-triplet excitation gap for the various VBSphases is shown in Fig. 11. In Fig. 12, we prsesent adensity plot of the ground state energy density for allthe phases in the J2 − J3 plane. It indicates that theground state energy of the LRO phases is higher com-pared to the VBS phases which highlights the role ofquantum fluctuations in stabilizing quantum paramagnetphases. In Fig. 13, we present the excitation spectrumfor different VBS phases at a few representative param-eter values. In Fig. 13(a) and Fig. 13(b) we notice thatdepending on the value of J3, the minimum of the dis-persion occurs at either at the M or the Γ point. Thelocal minima at the M point changes to a local max-ima upon increasing the value of J3. This points to thefact that when J3 < J2 it is easier to create excitationsthrough J2 bonds which require an antiferromagnetic or-dering between the dimers formed on J1 bonds. Thisvirtual antiferromagnetic ordering shows up as minimaat the M points [see Fig. 13(a)]. However, when J3 > J2

such virtual antiferromagnetic ordering is not favorableas indicated by the local maxima at the M point [seeFig. 13(b)]. In Fig. 13(c) we present the excitation spec-trum for VBS2 which shows a maxima at the M point.The excitation spectrum for plaquette RVB is shown inFig. 13(d) is a gapped quadratic dispersion with minimaat the M points.

The above results obtained within bond operator for-malism suggests that the large degeneracy in the disor-dered antiferromagnetic chain phase in the classical limitmay not lead to a ground state degeneracy in the exactquantum limit as indicated by the stability of the VBS2

phase due to large values of the singlet-triplet excita-tion gap as shown in Fig. 11. On the other hand, theNéel phase which is known to be susceptible to quantumfluctuations is gives way to a plaquette VBS state. Itis to be noted that the singlet-triplet excitation gap forVBS1 state is smaller compared to that of the VBS2 state.In fact, Fig. 11 suggests that the singlet-triplet excita-tion gap gradually decreases with decreasing J3. Whilethe uncorrelated antiferromagnetic chain phase and Néel

0.6

0.8

1.

1.2

1.4

1.6

ϵ μ

0.8

1.

1.2

1.4

1.6

ϵ μ

2.

2.5

3.

3.5

4.

ϵ μ

0.8

1.2

1.6

2.

ϵ μ

FIG. 13. Dispersion spectra for different VBS phases for rep-resentative values of (J2/J1, J3/J1) (a) VBS1(±0.5, 0.3), (b)VBS1(±0.5, 0.7), (c) VBS2(±1.0, 1.5) and (d) Plaquette VBS(1.5, 1.1).

phase yield to quantum paramagnetic states under quan-tum fluctuations, remarkably the sublattice Néel phaseis quite stable as evident from Fig. 10. This may be at-tributed to the fact that the number of nearest neighbourbonds with ferromagnetic alignment is three times thanthe number of bonds with antiferromagnetic alignment.The other possible explanation is that each square pla-quette can be thought of as a large spin with magnitudeof 4S which protects it from quantum fluctuations [62].Finally, we note that similar observations of a plaque-tte VBS, and competing magnetic phases on a variant ofthe model, namely, the square kagome lattice Heisenbergmodel have previously been made [63–68].

VII. DISCUSSION

We have investigated the ground state phase diagramof the Heisenberg model on the Fisher lattice in the pres-ence of first neighbor J1, second neighbor J2, and thirdneighbor J3 Heisenberg couplings, as a route towrads pro-viding a magnetic model of a two dimensional layer ofthe Hollandite lattice. At the classical level, a Luttinger-Tisza analysis shows that the phase diagram is host tothree different phases, namely, (i) an uncorrelated anti-ferromagnetic chain phase wherein each horizontal andvertical chain has perfect one-dimensional antiferromag-netic order but the relative orientations between any twochains is not fixed at T = 0 [30, 31, 69]. This uncor-related antiferromagnetic chain phase exists for J2 > J3

and antiferromagnetic J2, J3 only. Furthermore, thereexist two different Néel phasse depending on the signof J2. For antiferromagnetic J2, we find a Néel phase(phase II) with the four spins within a unit cell beingantiferromagnetically ordered. On the other hand, forferromagnetic J2, we find that the four spins within aunit cell are aligned ferromagnetically with such a clus-ter of ferromagnetic spins forming Néel order, namely the

13

sublattice Néel order (phase III).We have investigated the role of thermal and quan-

tum order-by-disorder effects. Interestingly, our classicalMonte Carlo analysis finds that the uncorrelated antifer-romagnetic chain phase survives at finite temperature,i.e., the order-by-disorder mechanism fails to lift the de-generacy. On the other hand, a harmonic order analy-sis of quantum fluctuations reveals an order-by-disordertransition by selecting a common angle α between all theone dimensional chains. We find that although the zerothorder energy in the spin wave approximation is identicalfor each α, the details of the spin wave spectrum dependon the value of α, e.g., the spectrum could be linear orquadratic for different values of α, and the number ofzero energy modes depends on α. Interestingly, quantumfluctuations (within a harmonic order treatment) lift thedegeneracy and select α = 0, π. For specific choices ofα, the spin-wave spectrum shows Dirac nodal lines alongΓX and ΓY segments. The spin-wave spectrum for phaseII reveals the presence of three Dirac nodal loops out ofwhich two are symmetry protected and do not depend onthe value of J3.

Finally, we have employed a bond operator formalismto analyze the model Hamiltonian beyond the spin-wavewave approximation. The analysis for spin S = 1/2shows that most of the classical phases except the sub-lattice Néel phase give way to different types of valencebond states. The region of parameter space classically oc-cupied by the uncorrelated antiferromagnetic chain phaseis stabilized into VBS1 dimer order with an appreciablesinglet-triplet excitations gap. This VBS1 phase appearsmostly for large positive values of J3. In a triangularshaped region around the centre in J2−J3 plane, a VBS2

dimer state is stabilized. The Néel phase largely givesway to a plaquette VBS state, while the sublattice Néelphase is found to be stable under quantum fluctuationswithin the bond operator formalism.

We expect that our study would set the stage for fur-ther investigations into the magnetic phases on the Hol-landite lattice [13, 17, 26, 28]. The experimental real-ization of two dimensional layers of α-MnO2 (if possi-ble) might serve as a platform to confirm the existenceof some of the phases that have been found here. Apossible extension of the present study is to include acoupling between such two dimensional layers yieldinga three dimensional model of magnetism in Hollanditelattice. As a future endeavor, it would be interestingto study the spin S = 1/2 quantum phase diagram em-ploying state-of-the-art numerical quantum many-bodyframeworks such as pseudofermion functional renormal-ization group [70, 71] and variational quantum MonteCarlo methods [72] which have already been applied onthe square [73] and other and other two- and three- di-mensional lattices [74, 75]. In particular, for S = 1/2there exists the likely possibility of quantum spin liq-uid(s) occupying a finite region of parameter space [76].It will be worthwhile to carry out a projective symmetrygroup classification [77] of U(1) and Z2 quantum spin liq-

uids on the Fisher lattice. The resulting Ansätze and thecompetition between them could then be studied eitherby combining the projective symmetry group classifica-tion framework with a functional renormalization groupapproach [78] or employing variational Monte Carlo onthe corresponding Gutzwiller projected wave functionssupplemented by a few Lanczos steps [79–84]. In partic-ular, it would be important to investigate their stabil-ity and energetic competitiveness with the valence-bondsolid orders, similar to what has been done on the kagomelattice [85, 86]. Finally, we would like to mention thatan interesting avenue for further exploration would beto possibly destabilize ferromagnetic order on the Fisherlattice which could potentially give rise to a plethoraof nematic orders as has been found in the square lat-tice [87, 88]

ACKNOWLEDGEMENTS

Y.I. thanks B. Dabholkar and K. Penc for help-ful discussions. Y.I. acknowledges financial supportby the Science and Engineering Research Board, De-partment of Science and Technology, Ministry of Sci-ence and Technology, India through the Startup Re-search Grant No. SRG/2019/000056 and MATRICSGrant No. MTR/2019/001042. This research was sup-ported in part by, the National Science Foundation underGrant No. NSF PHY-1748958, the Abdus Salam Inter-national Centre for Theoretical Physics (ICTP) throughthe Simons Associateship scheme funded by the SimonsFoundation, the International Centre for Theoretical Sci-ences (ICTS), Bengaluru, India during a visit for par-ticipating in the program “Novel phases of quantummatter” (Code: ICTS/topmatter2019/12) and “The 2ndAsia Pacific Workshop on Quantum Magnetism” (Code:ICTS/apfm2018/11). The Monte Carlo simulations wereperformed at SAMKHYA: High Performance ComputingFacility provided by Institute of Physics, Bhubaneswar.

Appendix A: Hamiltonian for spin-wave spectrum

In the appendix, we explicitly write the Hamiltonianwhich is used to find the spectrum of spin wave excita-tions.

1. Phase I

For phase I, which we refer to as the uncorrelated an-tiferromagnetic chain phase, the Hamiltonian matrix is,

Hk =

[A(k) B(k)B(k) A(k)

]. (A1)

The basis vector is chosen to be ˆχk = [χk, χ†−k]T , with

χk = (ak,1, ak,2, ak,3, ak,4). In the above, Ak and Bk

14

are 4× 4 matrices

Ak =

a b 0 −cb a −c 00 −c a b−c 0 b a

(A2)

Bk =

0 c bx(k) −bc 0 −b by(k)

b∗x(k) −b 0 c−b b∗y(k) c 0

. (A3)

Among the various parameters that appear in the abovetwo equations a, b, c are constants with a = 1+J3

2 , b =J24 (1− cosα), c = J2

4 (1 + cosα). b(x/y)(k) is given below,

bx(k) = −1

2(J3 + eıkx), by(k) = −1

2(J3 + e−ıky ). (A4)

2. Phase II

For the Néel phase, the Hamiltonian becomes an8 × 8 hermitian matrix due to its four sublatticestructure and the presence of global antiferromag-netic order. The basis vector which is used to de-fine the Hamiltonian is ˆχk = [χk, χ

†−k]T , with χk =

(ak,1, ak,2, ak,3, ak,4, b†−k,1, b

†−k,2, b

†−k,3, b

†−k,4), where

ak and bk respectively denotes the up spin and down spinin momentum space. The Hamiltonian is obtained as,

Hk = I2×2 ⊗[A(k) B(k)B(k) A(k)

], (A5)

where Ak and Bk are given below.

Ak =

d 0 −J3 00 d 0 −J3

−J3 0 d 00 −J3 0 d

, (A6)

Bk =

0 J2 eıkx J2

J2 0 J2 eıkx

e−ıkx J2 0 J2

J2 e−ıkx J2 0

. (A7)

In the above d = J1 + 2J2 − J3 and denotes the groundstate energy per plaquette in phase II.

3. Phase III

For the sublattice Néel phase, the basis vector usedto define the Hamiltonian matrix for each momentum isidentical to phase II. The Hamiltonian contains a few ad-ditional parameters. The Hamiltonian has the followingexpression,

Hk = I2×2 ⊗[A(k) B(k)B(k) A(k)

](A8)

where Ak and Bk are given below.

Ak =

d J2 −J3 J2

J2 d J2 −J3

−J3 J2 d J2

J2 −J3 J2 d

, (A9)

Bk =

0 0 eıkx 00 0 0 eıkx

e−ıkx 0 0 00 e−ıkx 0 0

, (A10)

where d = J1− 2J2−J3 denotes the ground state energyper plaquette in phase III.

Appendix B: Valence Bond Operator Analysis

Here, we provide the detailed procedure followed in thebond operator formalism. First, we give the definition ofspins in terms of the field operators ψ and χ associatedwith the singlet and triplet excitations [55, 59], respec-tively.

S1,α ≈1

2(χ†αψ + h.c.), S2,α ≈ −

1

2(χ†αψ + h.c.)(B1)

In the above, the subscript 1, 2 refers to two spins withina dimer and α = x, y, z represents the three componentsof the spin. In defining the above transformation we haverestricted ourselves up to quadratic order in the fields.The above definitions can be used to write down the ef-fective Hamiltonian in terms of the field operators. Theeffective Hamiltonian contains a Lagrangian multiplier λin order to ensure that the magnitude of total spin of agiven dimer is 2S. It is straightforward to observe thatthe use of Eq. (B1) yields quartic terms in field operators.Mean-field type decomposition has been used to reducethese quartic terms into appropriate quadratic terms insinglet and triplet sectors, while neglecting the mixing be-tween them. Furthermore, as we are interested in findingthe excitations due to triplets over singlet condensation,we introduce

√Na,i =

⟨ψ†a,i

⟩=⟨ψa,i

⟩as the singlet oc-

cupation number which is used to define the zeroth ordercondensate energy Ea. Here ‘a’ denotes the VBS1 config-uration and ‘i = 1, 2’ refers to two dimers within an unitcell. Similar definition holds for VBS2 which is labelledby the subscript ‘b’. Hence, we can write the effectiveHamiltonian for the VBS1 as

Ha = Ea +1

2

∑k

φ†k,αHk,aφk,α. (B2)

In the above expression, the singlet condensate energyEa is

Eg,a= (Na,1 +Na,2)Esa − λ(Na,1 +Na,2 − 1)

− 3

2N

∑k

∑i=1,2

Ak,a,i (B3)

15

where, Ak,a,i represents the ith diagonal element of Hk,a

and Esa = −3J1/4 is energy of the singlet states per pla-quette. The second term in Eq. (B2) refers to tripletexcitations. The basis vector used to obtain Eq. (B2) isφk,α = [ξk, ξ

†−k]T , with ξk = (χa,1,α,k, χa,2,α,k), where α

denotes different states within triplet sector. It is clearthat Hk,a is a 4× 4 matrix which can be written as,

Hk,a =

[Vk,a +Da Vk,aVk,a Vk,a +Da

], (B4)

where V and D are 2× 2 matrices

Vk,a =

[−J32 Na1 cos 2kx J2 sin kx sin ky

J2Na12 sin kx sin ky −J32 Na2 cos 2ky

], (B5)

Da =

[Eta,α − λ 0

0 Eta,α − λ

]. (B6)

In the above we have used Na,12 =√Na1Na2. Et refers

to energy of triplet states with Eta,α = J1/4 i.e., all thetriple states are degenerate in energy. To obtain the cor-responding representations for VBS2 we use the singletand triplet state energies as Esb = −3J3/4, E

tb,α = J3/4.

All other expressions of Vk,a as given in Eq. (B4) will bereplaced by Vk,b which is given below,

Vk,b =

[−J12 Nb1 cos kx 0

0 −J12 Nb2 cos ky

]. (B7)

Appendix C: Plaquette Operator Analysis

For the plaquette VBS state represented in Fig. 9(c),the spin operators are written as [54],

Sδ,α≈ cδ,µ(χ†µ,αψ+ + h.c.)

+ dδ,ν(χ†µ,αψ− + h.c.), (C1)

where α = x, y, z, µ = 1, 2, 3 denotes the nine tripletsand δ = 1, 2, 3, 4 denotes site indices inside a plaque-tte. A summation over the repeated indices is implied inEq. (C1). The matrix cδ,µ and dδ,ν are given below.

cδ,µ =1√6

1√2

0 1

0 1√2−1

− 1√2

0 1

0 − 1√2−1

, dµ,ν =1

2

0 1 01 0 00 −1 0−1 0 0

.(C2)To derive the effective Hamiltonian as obtained for thevalence bond singlet states in Appendix B in Eq. (B2),we follow a procedure similar to that explained before

Eq. (B2). After doing elementary algebra we obtain theeffective Hamiltonian in this case

Hc= Ec +1

2

∑k

φ†k,αHkφk,α +∑k

(Es− − λ)ψ†k−ψk−.(C3)

The first term Ec in the above equation corresponds toground state condensate energy per plaquette. For thesecond term, we have used the basis vector as, φk,α =

[ξk, ξ†−k]T , with ξk = (χ1,α,k, χ2,α,k, χ3,α,k). Below we

provide explicit expressions of the various terms presentin Eq. (C3). First we provide Ec,

Ec= Nc+Es+ − λ(Nc+ − 1)− 3

2N

∑k

∑i=1,2,3

Ck,i (C4)

where we have used√Nc+ =

⟨ψ†+

⟩=⟨ψ+

⟩and λ is

the Lagrange multiplier to satisfy the constraint of totalangular momentum of the dimer to be 2S. Es± is theplaquette singlet state energy for the state |Ψ±〉 withEs+ = −2J2 + J3

2 , Es− = − 3J3

2 . Ck,i is the ‘i’th diagonalelement of Hk. The 6× 6 Hamiltonian matrix Hk in thesecond term of Eq. (C3) is

Hk =

[Wk +Dc Wk

Wk Wk +Dc

](C5)

where

Wk =Nc+

3

−12 cos kx 0 −ı√

2sin kx

0 −1√2

cos ky−ı√

2sin ky

ı√2

sin kxı√2

sin ky cos kx + cos ky

(C6)

Dc =

Et1,α − λ 0 00 Et2,α − λ 00 0 Et3,α − λ

(C7)

where i, α = 1, 2, 3 denotes the nine triplet states. Afterdiagonalization, the fluctuation due to triplon excitationcontributes to the ground state energy and the final ex-pression for the ground state energy can be written asgiven in Eq. (39). We note that the above analysis hasbeen carried out for J2 > J3 where |Ψ+〉 is the groundstate and |Ψ−〉 is the first excited states. When J3 > J2

with J1 = 0, |Ψ−〉 becomes the ground state and |Ψ+〉becomes the first excited state. Thus, there are parame-ter regimes where analogous analysis needs to be followedconsidering |Ψ−〉 as the ground state. However after do-ing that we find that the final energy obtained in theformer case, i.e., when |Ψ+〉, is the ground state is lowercompared to the case when |Ψ−〉 is the ground state. Toobtain the energy expression when |Ψ−〉 is the groundstate one needs to replace the ‘−’ subscript of the thirdterm in Eq. (C3) by ‘+’, ‘+’ subscript in Eq. (C4) by‘−’. The expression of Wk as given in Eq. (C6) have thefollowing form,

Wk =−Nc−

4

cos ky 0 00 cos kx 00 0 0

(C8)

16

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arXiv:2007.04680v1 [cond-mat.str-el] 9 Jul 2020 · [email protected] [email protected] [email protected] well as the doping-induced transition into a spin-glass