Search for novel quantum phases in copper minerals

Hiroi Lab. M1 Ryutaro Okuma27/10/2014

Contents

• Basic notions of magnetism

• Frustration, spin liquid, exotic phases

• Experimental

• Newly synthesized copper compounds with interesting geometry

• Summary and future plans

What is quantum spin systems?

• In 3d transition metal compounds called Mott insulator, each 3d electron is confined to the neighborhood of an atom

• With degrees of freedom of electron spin survived, we can think there exists spin operator in the position of an atom

• Interaction between spin is described by Heisenberg Hamiltonian

1 hole behaves as S=1/2 spin

Cu2+: (Ar)3d9

Ferromagnets and anitiferromagnets

Ferromagnetic order Neel order

At high temperature, spins point arbitrary direction(paramagnetic). Lowering temperature, spins in magnets tends to favor certain ordered states.

J<0 J>0

In this ‘classical’ picture, spin is supposed to be a kind of unit vector. Is it always the case?

easy axis

If the Hamiltonian is “round” there is no special axis. But in real material magnetically easy axis exists. The picture of spontaneous symmetry breaking doesn’t hold true.

Stern-Gerlach’s experiment

Electron spin. A kind of angular momentum?

An electron can be seen as a magnet where only two states exist: up and down. From what does it originate?

Is electron autorotating?

Classically, circular current creates magnetic moment (how strong magnet is)

At least an electron has angular momentum e.g. Einstein de Haas effect

Description of electron spin in Bloch sphere

After projection onto z-axis, the probability of “up” or “down” is given by

1. Electron has intrinsic “angular momentum” called spin 2. Components of take discrete values: ℏ /2 or -ℏ /2 3. State of spin is described by linear combination of up and down

Spin coherent state

Electron’s spin can be regarded as a vector ℏ /2 in length if it is not entangled(like spin singlet).

!S!

S

Geometrical frustration

Typical lattices with frustration a. triangular b. kagome c. pyrochlore Magnetic orders realized in real material are described

Triangular lattice with AF interaction can’t be arranged in Neel order. Singlet pair+1up/down spin is the ground state

Spin liquid material can be compared to liquid Helium. No inorganic material is evident enough for every one to believe it spin liquid so far. However, interesting magnetism still exists in frustrated system.

Resonating Valence Bond state Arbitrary combination of spins form singlet pairs. Ground state is linear combination of such states

spin singlet state

Novel phases in kagome lattice

By contrast, in our grand canonical calculation the sizedependence becomes negligible (less than 10! 3 in two dimen-sions, see Methods) once we enter a cluster size of the propersystem length. Therefore, one could evaluate the spin gap by theonset value of H/J in the magnetization curve near zero field. InFig. 1, we find D¼ 0.05±0.02 (see the red shaded region), whichis obtained on a hexagonal cluster.

We briefly mention that our results are fully consistent with thedata of the previous conventional DMRG studies: in our grandcanonical DMRG on a cylinder with fixed small circumference(see Supplementary Fig. S1), the spin gap gives more than twice aslarge values as the value mentioned above. This value should becompared with the data in ref. 9 on a long cylinder with the samecircumference. For a proper extrapolation of the cylindricalresults to a bulk two-dimension, one needs to enlarge both thelength and the circumferences simultaneously23. In fact, ourgrand canonical spin gap on a hexagonal cluster is very close tothose of ref. 22 on a square cluster.

Zero and 1/9 plateaus. The zero plateau ranging at 0rH/Jr0.05is the continuation of the zero-field ground state. Correspond-ingly, in our calculation the spin structure in real space turned outto be completely structureless (see Supplementary Fig. S2). Oneway to identify the nature of the spin liquid is to calculate the vonNeumann entropy, S¼ !Tr(r ln r), defined on a subsystem of along open cylinder by the conventional DMRG, where r is thereduced density matrix of the subsystem. The value should follow,SBZ Ly! g, where Ly is the circumference, Z is a constant andg¼ ln(D) is the topological entropy. In ref. 9, the topologicaldimension, D, of the ground state is given as DB2, whichsupports the gapped Z2 spin liquid.

In the 1/9-plateau state, the real space profile of the spinstructure is rather intriguing, several geometries breaking the

translational symmetry are quasi-degenerate (see SupplementaryFig. S2 and Supplementary Note 3), and their stability is sensitiveto the shape and size of the cluster. We consider this to be thegood reason that the symmetry-breaking long order is absent.Therefore, we perform the conventional DMRG and calculate theentanglement entropy of the 1/9-plateau state in the samemanner as refs 2 and 9, as shown in Fig. 2; to have the 1/9magnetization, we need to keep the system size at the multiple ofnine, and thus the choice of the clusters are limited comparedwith the calculation on the M/N¼ 0 ground state. Thetopological dimension obtained in the Ly¼ 0 limit seeminglygives the value D¼ 3. Thus, the spin-gapped state at 1/9magnetization is possibly a Z3 spin liquid, and is the firstexample of a spin-liquid plateau induced by the magnetic field.Even a Z3 spin liquid itself has so far been observed only in aspecified bosonic model24, and the present model gives a morerealistic setup. Further examination is required to identify thedetailed nature of this phase.

Long-range ordered plateaus. In contrast to the first two pla-teaus, the rest of the plateaus have symmetry-breaking long-rangeorders. Figure 3a–c shows the real space profiles of the magne-tization density for 1/3, 5/9 and 7/9 plateaus. All of them arebased on a same unit of a hexagram, which holds nine lattice sites.This magnetic (extended) unit cell is three times as large as theoriginal unit cell, namely, Qmag¼Q# 3¼ 9, with the spin densityshown in Fig. 3d. Such symmetry-breaking requires stronginteraction between bosons, and the emergence of three suchplateaus in a single system is already a quite unexpected matter tohappen.

DiscussionIn spin-1/2 quantum magnets, a conventional (non-topological)non-magnetic state basically comprises a singlet, a unit ofspin 0, often represented by the quantum fluctuation between twospins, (|mkS–|kmS)/O2. A breaking of singlet yields a bosonicelementary particle carrying spin 1, which is called a magnon.The magnetic field controls the density of these bosons, serving asa chemical potential. As in the Mott insulator, there are particularvalues of the boson densities commensurate with the latticeperiodicity25, at which the gapped states are strongly pinned.

M/M

sat

1

0.5

00 1 2 3

H/J

Q = 3

Kagome

7/9

5/9

1/3

1/9

Figure 1 | Magnetization curve of the spin-1/2 kagome Heisenbergantiferromagnet in a uniform magnetic field. The saturation value of themagnetization density per site is Msat/N¼ 1/2. The inset shows thegeometry of the kagome lattice. The shaded hexagon is the original latticeunit cell including three sites (Q¼ 3). Data points are obtained by the grandcanonical analysis on a hexagonal cluster with N¼ 114 and 132, whichdirectly gives the curve of the thermodynamic limit without any size scaling.The range of each plateau is highlighted.

a b

LyLx

4

3

2

1

00 0 2 4 6 8

–In3

–2

–1

0.05 0.1 0.15

Ly= 4

1/Lx Ly

Ly= 8

Ly= 6

PeriodicOpenS

(Lx,

Ly)

S(∞

, Ly)

0

1

2

3

4

Figure 2 | Entanglement entropy of a 1/9 magnetization plateau. Theresults here are calculated on a long cylinder by the conventional DMRG.(a) S(Lx, Ly) as a function of 1/Lx is given for Ly¼4, 6, 8, where Lx and Ly

denote the number of sites along the leg and the circumference of thecylinder, respectively. (b) The value extrapolated for the infinite length,Lx¼N, is given as a function of circumference Ly. The best fit to S¼ ZLy! g gives g¼ 1.18±0.3. We estimate the error in scaling S(Lx, Ly) toLx-N, which gives the uncertainty of linear scaling S(N, Ly) against Ly,displayed by grey shading in b.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3287 ARTICLE

NATURE COMMUNICATIONS | 4:2287 | DOI: 10.1038/ncomms3287 | www.nature.com/naturecommunications 3

& 2013 Macmillan Publishers Limited. All rights reserved.

Calculated M-H plot for S=1/2 kagome lattice

hexagram magnetization patterns appear at plateaus

Magnetic Field

M

H

Typical M-H curve for Antiferromagnets

All spins point in the direction of magnetic field

Features 1. 4 plateaus & 1 jump 2. spin liquid state at M/Ms=0, 1/9 3. hexagram patterns at M/Ms=1/3, 5/9, 7/9

We want to discover unusual

magnetism in frustrated system

not limited to spin liquid

easy axis

Nishimoto, Satoshi et al., Nature communications 4 (2013).

T=0K

Vesignieite: S=1/2 Structurally perfect kagomeBaCu3(VO4)2(OH)2 Space Group:R-3m Antiferromagnetic Insulator w/J~60K

Cu2+ forms perfect kagome network by edge sharing of octahedron

Orange tetrahedrons consists of four O2- around V5+ Usually in vanadate minerals As5+ and P5+ partially substitute for V5+. So I tried to synthesize an Arsenic substitute. Natural vesignieite→

Photo Copyright © Vincent Bourgoin

Experimental

Synthesis: Hydrothermal method

1. Ingredient is sealed with solvent in a teflon(or Au) tube 2. Heated up to 150~250℃(Au can bear over 600℃) 3. At high temperature, water becomes highly reactive

Features of hydrothermal method

Pros • Can make material contains H • Likely to get new substance • And single crystals with way less time

Cons • Must tune more parameters (pH, density, Temp, time…) • Often neglects molar ratio of chemical formula • Tends to produce many byproducts

Compared with solid state reaction, hydrothermal synthesis

Distorted kagome lattice

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Spin flop transition around M/Ms=1/2

Features • Very low saturation field~7T

• FM interaction nearly cancels AFM. • Hysteresis around M/Ms=1/2

• usually AFmagnets have no hysteresis

• spin flop transition is 1st order, but hysteresis is usually small(ΔB<<1T)Spin flop transition

Phase transition at just M/Ms = 1/2 may be related to interesting

magnetismB

When the magnetic order is perpendicular to B, spins can’t change their direction easily. If they do, that will be a sudden transition.

T=2K

Quasi 1 dimensional chain

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Future Plans

• Uncovering the magnetic state of As-vesignieite at M/Ms=1/2

• Synthesis of a large crystal of perfect kagome mineral Vesignieite

• Making more and more new material with interesting properties!