A theoretical model for tellurite-sulfates Na 2Cu5TeO 3SO 43OH 4and K2Cu5TeO 3SO 43OH 4 I. L. Bartolom e1L. Errico1 2V. Fernandez1M. Matera1A.V. Gil Rebaza1and C.A. Lamas1y

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A theoretical model for tellurite-sulfates Na2Cu5(TeO3)(SO4)3(OH)4and

K2Cu5(TeO3)(SO4)3(OH)4

I. L. Bartolom´e,1L. Errico,1, 2 V. Fernandez,1M. Matera,1, ∗A.V. Gil Rebaza,1and C.A. Lamas1, †

1IFLP - CONICET. Departamento de F´ısica, Facultad de Ciencias Exactas.

Universidad Nacional de La Plata,C.C. 67, 1900 La Plata, Argentina.

2Universidad Nacional del Noroeste de la Provincia de Buenos Aires (UNNOBA),

Monteagudo 2772, CP 2700 Pergamino, Buenos Aires, Argentina

A theoretical model for two new tellurite-sulfates, namely Na2Cu5(TeO3)(SO4)3(OH)4and

K2Cu5(TeO3)(SO4)3(OH)4is determined to be compatible with ab-initio calculations. The results

obtained in this work show that some previous speculations in the literature about the couplings

are correct, obtaining a model with a mixture of ferromagnetic and antiferromagnetic couplings.

We use a combination of numerical techniques to study the magnetic properties of the model. Our

numerical calculations based on the density-matrix renormalization group method reveal that the

system presents Ising-like magnetization plateaux at rational values of the saturation magnetization.

PACS numbers: 05.30.Rt,03.65.Aa,03.67.Ac

I. INTRODUCTION

Recently, Yingying Tang et al.1synthesized

by hydrothermal reaction, two new tellurite-

sulfates (TS) with a distorted Kagom´e strip

structure: Na2Cu5(TeO3)(SO4)3(OH)4and

K2Cu5(TeO3)(SO4)3(OH)4(Na-TS and K-TS in the

following). In both compounds, the magnetic behavior

is associated with the single unpaired electron associated

with each Cu+2 ions, localized over a 1D kagom´e strip

sub-lattice. This particular geometry corresponds to

the one dimensional version of the paradigmatic two

dimensional Kagom´e lattice for which some experimen-

tal realizations for S= 1/2 as the Herbertsmithite

ZnCu3(OH)6Cl22, the α-vesignieite BaCu3V2O8(OH)23,

and [NH4]2[C7H14N][V7O6F18]54were estudied.

The crystal structure of the compounds is schematized

in Fig. 1 and the simpliﬁed magnetic geometry we con-

sider is shown in Fig. 2. We show that several magnetic

properties like magnetic plateaux are determined by the

geometry of the plaquette.

The synthesis of these compounds has aroused great in-

terest in the study of the magnetic phase diagram of mod-

els with this Kagom´e strip geometry5–14. In this sense,

the presence of magnetization plateaux15, a Haldane-like

phase9and localized magnon crystal phases have been

detected6,16. The studies carried out so far describe gen-

eral phase diagrams in a parameter space that, a priori, is

not related to the couplings that describe these materials.

Improving the theoretical description then requires esti-

mating the coupling constants of the eﬀective magnetic

model. As proposed by Noodelman17, a way to determine

these coupling constants is by comparing the spectrum of

the reduced model to those obtained by setting the corre-

sponding magnetic conﬁgurations in Density Functional

Theory (DFT) based calculations. The original method

was successfully applied in the literature to compute the

magnetic coupling constants of many compounds. How-

ever, as the number of coupling constants and atoms in

FIG. 1: Crystal structure corresponding to

Na2Cu5(TeO3)(SO4)3(OH)4(Na-TS). The compound

K2Cu5(TeO3)(SO4)3(OH)4is isostructural with Na-TS.

the supercell grows, the direct application of the method

becomes challenging: since the number of possible mag-

netic conﬁgurations grows exponentially with the num-

ber of magnetic atoms, and the evaluation of the energy

of each conﬁguration is computationally expensive, to ex-

haust the full set of magnetic conﬁgurations becomes im-

practical even for a small number of magnetic atoms. On

the other hand, choosing a small set of magnetic conﬁgu-

rations could introduce a large bias in the determination

of the coupling constants. To overcome these issues, a

novel strategy based on Noodelman’s breaking symmetry

method was proposed18. In this work, that methodology

is used to determine the couplings in the magnetic model

describing the tellurite-sulfates. A discussion about this

coupling determination is presented and we show that

the S= 1/2 Heisenberg model with these couplings de-

scribes the magnetic properties of the system and allows

to determine qualitatively the behavior of the magnetic

transitions.

Inspired by the experimental determination of the

atomic distance we propose a model with ﬁve diﬀerent

arXiv:2210.15416v1 [cond-mat.str-el] 27 Oct 2022

Na-TS K-TS

a(˚

A) 7.294(3) 7.467(6)

b(˚

A) 12.005(4) 12.177(9)

c(˚

A) 9.214(3) 9.397(4)

α90.0 90.0

β111.160(6) 111.352(8)

γ90.0 90.0

TABLE I: Crystal Structural parameters for Na-TS and K-TS

compounds.

magnetic couplings and determine the set of couplings

values compatible with the energies calculated by density

functional methods. The resulting model is numerically

studied, determining the zero temperature magnetization

curve by density-matrix renormalization group (DMRG)

calculations. We also determine some thermodynamical

quantities for small systems by exact diagonalization.

We analyze the magnetic plateaux at zero temperature

in the context of the Oshikawa-Yamanaka-Aﬄeck (OYA)

theorem19 which provides the necessary condition for the

existence of these magnetization plateaux as

NS(1 −m) = integer

where Nis the number of spins in the ground state (G. S.)

cell presenting spatial periodicity and m=M/Msat is the

normalized magnetization per site. If the translational

symmetry in the G. S. is preserved, then N= 5 and

the magnetization curve may have plateaux at m= 1/5

and m= 3/5. In the following, we show that the G. S.

periodicity is enlarged to N= 10, but still only the semi-

classical plateux at m= 1/5 and m= 3/5 are present.

The paper is organized as follows; in Sec. II basic prop-

erties of the lattice and magnetic degrees of freedom are

discussed. In Sec. III we discuss details of the couplings

estimation by following the methodology of ref18. Esti-

mated values of the coupling constants are also presented.

Then, in Sec. IV, we study the magnetic properties aris-

ing from the ﬁtted model, both for large systems in the

zero-temperature limit, by DMRG calculations, and at

ﬁnite temperature, through full diagonalization of the

quantum model for small systems. Finally in Sec. V,

we present the conclusions and perspectives.

II. THE MAGNETIC MODEL

Na-TS and K-TS are isostructural compounds that

crystallize in a monoclinic structure with space group

P21/m, see Figure 1. The structural information for both

compounds is reported in Table I. Atomic positions of

each atom in the structure for both Na-TS and K-TS

can be found in Ref.1. In its three inequivalent crystallo-

graphic sites, the Cu+2 ions form distorted CuO6octahe-

dral with bond lengths ranging from 1.85 to 2.31 ˚

A(Na-

TS) and 1.88 to 2.50 ˚

A(K-TS), exhibiting a Kagom´e-

strip arrangement which can be regarded as a dimen-

sional reduction of Kagom´e-lattice.

FIG. 2: a) Distance between Kagom´e-strip lattice for Na-TS.

b) labels of the Cu atoms in the 2D Kagom´e-strip.

Na-TS K-TS Coupling constant

dCu2−Cu6(˚

A) 3.01 3.07 Jd

dCu5−Cu9(˚

A) 2.84 2.89 Ju

dCu1−Cu2(˚

A) 3.08 3.11 J0

dCu1−Cu3(˚

A) 3.07 3.11 J1

dCu2−Cu3(˚

A) 2.94 2.96 J2

TABLE II: Distances between Cu atoms corresponding to

each coupling constant.

Our aim in this work is to study the magnetic behavior

of Na-TS and K-TS. Since the Te, S, O, H, and Na/K

ions do not present spin polarization, the spin-lattice is

determined by the Cu+2 ions that form the Kagom´e-strip

lattice, as can be seen in Figure 2. Since each magnetic

ion has a single localized unpaired electron, its magnetic

degree of freedom can be described as a spin S= 1/2.

In order to build a simple eﬀective model for the mag-

netic degrees of freedom, we propose a symmetric Heisen-

berg model

H=−X

(i,j)

Ji,j~

Si·~

Sj.(1)

with ~

Si=1

2(σx,i, σy,i, σz,i) the spin vector, and Ji,j the

coupling constants. To determine them, we impose the

constraint that the diﬀerence between the DFT energy

and the energy of the Heisenberg model for a given set

of couplings must be lower than the DFT energy error

(1mRy). This is our compatibility criterium.

The Kagom´e-strip lattice formed by the Cu ions in

the Na-TS and K-TS compounds is quite distorted,

showing ﬁve diﬀerent Cu-Cu bond lengths (see Table II).

The nearest Cu −Cu distance between Kagom´e-strips

is in the order of 4.34 ˚

A(Na-TS) and 4.44 ˚

A(K-TS),

respectively, while the shortest distance between the

layers are 6.4 ˚

A(Na-TS) and 6.7 ˚

A(K-TS), respectively

see Figure 2a.

In the present work, we considered interactions up to

3.1 ˚

A, Figure 2b, i.e. we discard the interactions be-

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Atheoreticalmodelfortellurite-sulfatesNa2Cu5(TeO3)(SO4)3(OH)4andK2Cu5(TeO3)(SO4)3(OH)4I.L.Bartolome,1L.Errico,1,2V.Fernandez,1M.Matera,1,A.V.GilRebaza,1andC.A.Lamas1,y1IFLP-CONICET.DepartamentodeFsica,FacultaddeCienciasExactas.UniversidadNacionaldeLaPlata,C.C.67,1900LaPlata,Argentina.2Universida...

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A theoretical model for tellurite-sulfates Na 2Cu5TeO 3SO 43OH 4and K2Cu5TeO 3SO 43OH 4 I. L. Bartolom e1L. Errico1 2V. Fernandez1M. Matera1A.V. Gil Rebaza1and C.A. Lamas1y.pdf

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A theoretical model for tellurite-sulfates Na 2Cu5TeO 3SO 43OH 4and K2Cu5TeO 3SO 43OH 4 I. L. Bartolom e1L. Errico1 2V. Fernandez1M. Matera1A.V. Gil Rebaza1and C.A. Lamas1y

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