Field-selective classical spin liquid and magnetization plateaus on kagome lattice
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Journal of the Physical Society of Japan
LETTERS
Field-selective classical spin liquid and magnetization plateaus on kagome lattice
Kunio Tokushuku ∗ , Tomonari Mizoguchi , and Masafumi Udagawa , Department of Physics, University of Tokyo, Tokyo, 113-0033, Japan Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan Max-Planck-Institut f¨ur Physik komplexer Systeme, 01187 Dresden, Germany
We obtain a classical spin liquid (CSL) phase by applying a magnetic field to the J - J - J Ising model on a kagomelattice. As we proved in the previous study [Phys. Rev. Lett. , 077207 (2017)], this model realizes one species ofCSL, the hexamer CSL, at the zero magnetic field, which consists of macroscopically degenerate spin configurationswith mixed total magnetization, M . The magnetic field selects its subset, which can be mapped to a trimer covering ofthe dual lattice and forms a magnetization plateau of M = /
9. In addition to this CSL, we find two other magnetizationplateaus at M = / /
27, which are ascribed to the “multimer” superstructures on a dual lattice.
Introduction.-
Realization of quantum spin liquid (QSL) isa central problem of condensed matter physics.
Geometri-cal frustration is considered as one essential ingredient to re-alize this phase, and intensive e ff orts have been focused on thesearch of QSL in materials composed of triangular or tetrahe-dral basis units. Theoretically, although the existence of QSLphases is established in a number of solvable models, itstill remains a di ffi cult task to identify QSL ground state ina specific model, such as the antiferromagnetic Heisenbergmodel on a kagome lattice. One promising strategy to find QSL may be to focus onits high-temperature precursor, classical spin liquid (CSL). Athigh temperatures, the QSL phase is sometimes preceded bycooperative paramagnetic states, composed of a degenerateassembly of classical moments under strong local constraints.When the temperature decreases, CSL is gradually turned intoQSL, as the quantum coherency develops. The nature of QSLcrucially depends on the basic characters of high-temperatureCSL, such as the type of geometrical unit, the rule of localconstraint, and so on. This viewpoint, in turn, implies the pos-sibility of engineering QSL with desirable properties by con-trolling its precedent CSL.For this purpose, the application of a magnetic field pro-vides a simple but practical method to control the local con-straint of CSL. It is particularly promising if a CSL state con-sists of degenerate configurations with di ff erent total magne-tizations. In this case, a magnetic field selects a part of the de-generate assembly and gives rise to a new CSL with di ff erentlocal constraints. The transformation from spin ice to kagomeice
20, 21) gives an example of this mechanism, and it is exper-imentally well confirmed.
22, 23)
The kagome ice state forms a1 / Ti O , and it exhibits fertilephenomenology in thermodynamic and dynamical properties,which are absent in the original spin ice state. In this paper, we focus on one class of CSL, which we ∗ [email protected] named a hexamer CSL, realized in the J - J - J Ising modelon the kagome lattice. This hexamer CSL is composed of theclusters of same-sign gauge charges, involving the configura-tions with mixed values of magnetization at the zero magneticfield. For this state, we examine the state selection by the mag-netic field, and obtained a magnetization plateau at M = / M = / /
9, attributed to superstructures of “multimers”, whichare schematically shown in Fig. 4.Below, we first introduce the language of dimers andmonomers, and clarify the origin of plateaus at M = / / M = / Model.-
We consider the J - J - J Ising model on thekagome lattice in a magnetic field: H = J X h i , j i n . n . σ zi σ zj + J X h i , j i σ zi σ zj + J X h i , j i σ zi σ zj − h X i σ zi . (1)Here, i and j denote sites on the kagome lattice, σ zi = ± i , and h , i n . n . , h , i , and h , i denote, respectively, the nearest-neighbor, the second-neighbor, and the third-neighbor pairs of sites; see Fig. 1(a)for their definitions.In this paper, we focus on the case of J = J = J = J , with small positive J : 0 < J . .
2. In this case, the Hamil-tonian can be written in a form of interacting charges.
The charge variable, Q p , can be defined at each triangle p
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LETTERS such that Q p = η p S p , (2)where S p = P i ∈ p σ zi is the total spin on a triangle p , and η p =+ −
1) for p ∈ △ ( ▽ ), representing the orientation of triangles.The charges live on a dual lattice of the kagome lattice,namely, a honeycomb lattice [Fig. 1(a)]. They take the values, Q p = + , + , − , − Q p , we can rewrite theHamiltonian as H = − J ! X p Q p − J X h p , q i Q p Q q − h X p η p Q p + C , (3)where C = ( J − N p is a constant term, with N p beingthe number of triangles. The first, the second, and the thirdterms of Eq. (3) are, respectively, the self-energy, the nearest-neighbor interaction on the dual lattice, and the staggered po-tential of charges. The competition among these three termsleads to exotic magnetic plateaus, as we will show below.The model with h = It was found that, for J > at-tract to each other, an exotic CSL phase appears, which wenamed a “hexamer CSL”. This CSL has two features: (i) itconsists of Q p = ±
1, and (ii) every same-sign-charge clus-ter contains one closed “loop”; here, a same-sign-charge clus-ter means a set of maximally connected triangles which havesame-sign charges. Importantly, (i) originates from the min-imization of the self-energy term, and (ii) from that of theinteraction term under the geometrical constraint. This pic-ture is well illustrated by the analytical argument based onthe Gauss’ law for the charges, by which the existence ofthe hexamer CSL is proved rigorously.
Magnetization Curve.-
We now start with the overall de-scription of the magnetization process. We show a magneti-zation curve for 0 < J . . Fig. 1. (Color online) (a) Kagome J - J - J Ising model. The black circlesdenote the sites where Ising spins are located. The sites on a dual honeycomblattice with the sign factor η p = + −
1) are represented by the purple (or-ange) dots. (b) The definition of charge variables. The colors correspond tothe values of charges Q p = + + − − Fig. 2. (Color online) Magnetization curve for the Hamiltonian, Eq. (3),for 0 < J . .
2. We assume M jumps directly from 5 / /
27; see themain text. zation plateaus at M = / , / , /
9, and 17 / J =
0. For J =
0, theHamiltonian of Eq. (3), can be transformed as H = X p S p − h ! − h + ! N p . (4)From Eq. (4), one can find that (cid:12)(cid:12)(cid:12) S p − h (cid:12)(cid:12)(cid:12) should be minimizedat each triangle in the ground state. This is achieved by setting S p = + ≤ h ≤
4, and S p = + h ≥
4. The formercorresponds to the “two-up-one-down” state with M = / M = / Q p =+ p ∈ △ and Q p = − p ∈ ▽ . In such a configura- Fig. 3. (Color online) (a) Configuration of the 1 / LETTERSFig. 4. (Color online) Typical charge configurations of (a) the 17 /
27 plateau with a kagome network, (b) the 17 /
27 plateau with a domain-wall structure, (c)the 5 / / tion, every upward triangle shares its minority spin (i.e., a spindown in this case) with one of three neighboring downwardtriangles. We regard a pair of such upward and downward tri-angles sharing the minority spin as a “dimer”, and then thespin configuration can be mapped to a hard-core dimer cov-ering on the dual honeycomb lattice [Fig. 3(a)]. The dimercovering on the honeycomb lattice leads to macroscopic con-figurational degeneracy, so the 1 / in the literature.Meanwhile, the polarized state of M = Q p = + p ∈ △ and Q p = − p ∈ ▽ . We consider these triangleswith | Q p | = Dimer-Monomer Covering for / and / Plateaus.-
In terms of the dimer-monomer representation we introducedabove, we derive the existence of two plateaus, M = / /
9, from the instability analyses of the M = / N + , the number of monomers, N d , the number ofdimers, n ( + , + , the number of monomer-monomer contacts, n ( + , d ) , the number of monomer-dimer contacts, and n ( d , d ) , thenumber of dimer-dimer contacts. By the lattice geometry, thefollowing conditions are imposed between these variables:2 N d + N + = N p , (5)2 n ( + , + + n ( + , d ) = N + , (6)2 n ( d , d ) + n ( + , d ) = N d , (7) n ( + , + + n ( + , d ) + n ( d , d ) = N p − N d . (8)With these variables, we can write the total energy of thesystem as E H = − J ! (9 N + + N d ) + J (cid:2) n ( + , + + n ( + , d ) + n ( d , d ) (cid:3) + JN d − h N + + N d ) . (9) The magnetization is similarly obtained as M = N + + N d N p . (10)Now, let us examine the instability of the M = h . The M = N + = N p , n ( + , + = N p , and N d = n ( + , d ) = n ( d , d ) =
0, and the corre-sponding energy is E M = = (cid:16) + J − h (cid:17) N p . This instabilityof M = ff er-ence, ∆ E : = E H − E M = = h − − J ] N d + Jn ( d , d ) , whichis obtained from the geometrical identities, Eqs. (5)-(8). Thisexpression of ∆ E tells us two things. Firstly, the dimer-dimercontact costs energy, due to the final term, 4 Jn ( d , d ) , thus the in-stability occurs in the sector of n ( d , d ) =
0. And secondly, theinstability of M = h = + J , below whichthe maximal packing of dimers is realized without the dimer-dimer contacts: n ( d , d ) =
0, where N + = N p and N d = N p ,which leads to M = / M takes a rational value with a large de-nominator, which implies a formation of large superstructure.In Figs. 4(a) and (b), we depict two specific dimer configu-rations forming this magnetization plateau. One depicted inFig. 4(a) consists of a kagome network (the bold red lines),whose hexagonal plaquettes contain eight monomers and fivedimers. Since there are three patterns of placing two dimersinside each hexagon [Fig. 4(a)], this configuration has trivialmacroscopic degeneracy of 3 Np . The other type of configura-tion is depicted in Fig. 4(b), where the columnar dimers areseparated by “domain-wall-like” dimers (the bold blue lines).This configuration has semi-macroscopic degeneracy due tothe choice of the positions of domain walls.Similarly, we address an instability of the 1 / / N d = N p , n ( + d , + d ) = N p , and N + = n ( + , + = n ( + , + d ) =
0, resultingin the total energy of the system, E M = / = (cid:16) + J − h (cid:17) N p .Then, the energy di ff erence ∆ E : = E H − E M = / = (4 − J − h ) N + + Jn ( + , + , results in the instability at h = − J ,above which the state is described by the maximal pack-ing of monomers without the monomer-monomer contacts.We display the corresponding configurations in Fig. 4(c). Akagome network appears again (the bold red lines), where
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LETTERS the configurational degeneracy of dimers in each hexago-nal plaquette leads to trivial macroscopic degeneracy, 2 Np .The number of monomers, N + = N p , and that of dimers, N d = N p , amount to M = /
9. Besides the configuration ofFig. 4(c), there also exist the “domain-wall-type” configura-tions [Fig. 4(d)].The boundary between the 17 /
27 and the 5 / h = + J by comparing their en-ergies. However, the subtlety of competing energies betweendimer-dimer contacts and monomer-monomer contacts maylead to the appearance of additional plateaus, which wouldconsist of complicated spin structures including both speciesof contacts – We do not discuss the possibility of intermediateplateaus in this contribution.1 / Plateau as Trimer Covering State.-
So far, we havediscussed two plateaus at M = /
27 and 5 / M = / ff erent variablesfrom those used before: N + , the number of triangles havingthe total spin S p = + N − , the number of triangles having S p = −
1, and n q , q ′ , the number of contacts between S p = q and S p = q ′ . Similarly to Eqs. (5)-(8), these variables are un-der the geometrical constraint: N + + N − = N p , (11) n ( + , − + n ( − , − = N − , (12) n ( + , + + n ( + , − + n ( − , − = N p . (13)Note that we can safely ignore the presence of triangles with S p = ± E L = − J ! ( N + + N − ) + J [ n ( + , + − n ( + , − + n ( − , − ] − h N + − N − ) , (14)and the magnetization as M L = N + − N − N p . (15)The starting point of the analysis is the 1 / N + = N p , n ( + , + = N p , N − = n ( + , − = n ( − , − =
0. Fromthis, we obtain the energy di ff erence, ∆ E : = E L − E M = / = [6 J − h ] N − + Jn ( − , − , which results in the phase boundaryat h = J . The phase below the M = / S p = − S p = −
1. Wecall these overlapping dimers a trimer, and the overlapping S p = − (a) (b) Fig. 5. (Color online) (a) Correspondence between the charges and thedimers. (b) Schematic picture of the spin configuration of the M = / the contact between S p = − S p = +
1. The lowest energy state can then be described asa “trimer covering” under the condition that hinges can nottouch each other [Fig. 5(b)]. A similar trimer covering phaseis obtained in the previous work on a checkerboard lattice.
Since each trimer consists of two triangles with S p = + S p = −
1, we obtain N + = N p and N − = N p . Therefore, the magnetization of this phase is M = / and estimated its value S / ∼ . Connection with Hexamer CSL.-
The trimer covering phaseis, in fact, a submanifold of the hexamer CSL realized atthe zero magnetic field.
The hexamer CSL is defined asthe state where the whole lattice is covered with the same-sign-charge clusters containing one loop. Figure 5(b) illus-trates the relation between the trimer covering and the hex-amer CSL. Firstly, due to the staggered sign in the definitionof charge, each trimer is composed of three same charges.Secondly, the hinges of two trimers cannot neighbor with eachother. Accordingly, the hinge of one trimer is always insidethe same-sign-charge cluster, i.e., it has the same charge withall its neighbors. This second property results in the pres-ence of one and only one loop in a cluster, as illustrated inFig. 5(b). Algorithmically, starting from one endpoint lobeof a same-sign-charge cluster (the blue circle in Fig. 5(b)),one can trace touching (i.e., not overlapping) dimers to findthe position of the loop. Since this dimer string always endswith an open hinge, if it does not end with a loop, it contra-dicts the second property above. Rigorous but rather involved
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LETTERS (a) (b) (7) (8) (11)
Fig. A · (a) Definition of pair-triangles (the dotted squares) used as a unitin the transfer matrix method. (b) The twelve possible trimer configurationson pair-triangles in the trimer covering phase. proof can be available with the help of the Gauss’ law, whichis given in Sec. II of Appendix. It means the configuration atthe M = / Summary.-
We have investigated the magnetization processof the J - J - J Ising model on the kagome lattice, and foundthree magnetization plateaus at M = / , / / M = / ∼ .
12. This state results from the selection of a maxi-mally polarized subset of hexamer CSL at the zero magneticfield. This selection by the magnetic field gives a general strat-egy to engineer a new classical spin liquid state.The current study can be extended to diverse directions.Quantum fluctuations will give rise to exotic quantum super-position states of degenerate configurations, which can openup a way to novel quantum spin liquids at the magnetizationplateaus. Indeed, various exotic plateaus have been found inquantum kagome magnets,
42, 43) and extensive theoretical andnumerical studies have revealed that the formations ofsuch plateaus are often attributed to the superstructures ofmagnons and / or valence bonds. In particular, the authors ofRef. discussed the possible realization of topological or-dered state at M = / ff erent, our trimer state proposes one mechanismto stabilize the M = / Acknowledgements.-
This work was supported by the JSPSKAKENHI (Grants No. JP15H05852 and No. JP16H04026),MEXT, Japan. K. T. was supported by the Japan Society forthe Promotion of Science through the Program for LeadingGraduate Schools (MERIT).
Width: L Entropy (per spin)2 0.134826093 0.131596534 0.125901435 0.125597766 0.12642098
Table A · Residual entropy per spin for the 1 / L . Appendix A: Estimation of residual entropy by transfermatrix method
We estimate the residual entropy of the trimer coveringstate by using the transfer matrix method on a stripe geom-etry.
We take a pair-triangle as a unit, and consider thestripe consists of L × N units [Fig. A · · We obtain the result shown in Table A ·
1, indicating S trimer − covering ∼ . Appendix B: Alternative derivation of 1 / We address another derivation of the M = / From ∆ E in the main text, the groundstate below the M = / S p = − n ( − , − =
0. From Eqs. (12)-(13) in the main text, we obtain n ( + , + + n ( + , − = N p , (B · N − = n ( + , − . (B · ·
2) means that seeking maximal packing of S p = − n ( + , − under Eq. (B · D + and D − shown in Fig. B ·
1, which aremaximal sets of connected triangles satisfyingcluster D + : S p = + p ∈ △− p ∈ ▽ , (B · · D − : S p = − p ∈ △ + p ∈ ▽ . (B · D + ( − ) consists of same-signcharges. In addition, all inner spins of a cluster D (either ∈ D + or D − ) contribute to n ( + , − and that of boundary spins n ( D ) b contribute to n ( + , + : n ( D ) i = n ( D )( + , − , (B ·
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LETTERSFig. B · Schematic picture of the definition of the cluster D + / − . The boldline represents a maximal set of plus-sign charges, i.e., D ∈ D + . The spinsrepresented by the purple circles (magenda square) are classified as inner(boundary) spins. The dotted circle represents a loop structure. In this cluster,the number of inner spins n ( D ) i is 12 and the number of triangles N ( D ) is 12.Therefore the number of loop N loop( D ) is 1 from Eq. (B · n ( D ) b = n ( D )( + , + . (B · n ( D ) i is deter-mined by topology of each cluster: n ( D ) i = N ( D ) + N loop( D ) − , (B · N ( D ) is the number of triangles in a cluster D and N loop( D ) is the number of loop structures of that. Algebraically,Eq. (B ·
8) gives the definition of N loop( D ) (see the caption ofFig. B · ·
6) and (B · n ( + , − = X D n ( D )( + , − = N p + X D ( N loop( D ) − . (B · · n ( + , − is nothing butmaximizing N loop( D ) .In the following, we show that the maximum number ofloop-structures is one in this phase. In fact, this can be doneby using the lattice analogue of the Gauss’ law: X p ∈D Q p = X i ∈ ∂ D η p D ( i ) σ zi , (B · ∂ D is a boundary of the cluster D . Here the boundarysite belongs to the two triangles, one inside, and one outside D , and p D stands for the former. From the Gauss’ law, thefollowing triangle inequality holds: | X p ∈ D Q p | ≤ X i ∈ ∂ D | η pD ( i ) σ zi | = n ( D ) b . (B · ·
8) and the geometrical identity 3 N ( D ) = n ( D ) i + n ( D ) b , we obtain n ( D ) b = N ( D ) + − N loop( D ) . (B · ·
11) can be expressed bythe number of triangles. Namely, since every cluster consists only of triangles with Q p = + −
1, we obtain | X p ∈ D Q p | = N ( D ) . (B · · · N loop( D ) ≤ . (B · · ·
9) and (B · n ( + , − ≤ N p . (B · ·
15) holds, i.e., n ( + , − = N p , (B · n ( + , + = N p , (B · N + = N p , (B · N − = N p . (B · ·
18) and (B ·
19) lead to M = / ·
18) and (B · S p = + S p = −
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