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Page 1: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Company

LOGO

Jahanfar Abouie

Advanced School on Recent Progress in Cond-Mat

IPM-27-28 June 2012

Quantum Magnetism and Quantum Entanglement

& IPM

Page 2: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Lecture 1

Quantum Magnetism

Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys. Vol 1, 159 Dec. (2005)

Theory of magnetism

Motivations

Some models and Properties

Page 3: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Lecture 2

Background: Supramolecular structure-http://nanotechweb.org/cws/article/lab/43208

Page 4: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of Magnetism

� Magnetic Orders � Antiferromagnet� Ferromagnet� Ferrimagnet� Helimagnet� Spin Liquid� Luttinger Liquid� Spin super-solid� Spin-Flop� …

� Interactions � Dipole-Dipole magnetic interactions� Electrostatic interaction & Pauli exclusion principle

� Ions in Crystals � Crystal fields (Interaction with non-magnetic ions)� Ligands

� Single ions � Diamagnetic ions� Paramagnetic ions

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM4/

Page 5: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of Magnetism-Dia & Para

� Atoms, molecules, ions with an odd number of electrons , likeH, NO, C�C�H�, Na

, etc.� A few molecules with an even number of electrons, like O� and some

organic compounds,

� Atoms or ions with an unfilled electronic shell:

� Transition elements (3d shell incomplete),� The rare earths (Series of Lanthanides-4f shell incomplete),� The series of the actinides (5f sell incomplete).

� Monoatomic rare gas, He, Ne, A, etc.� Most Polyatomic gases, H�, N� , etc.� Ionic Crystals, NaCl� Covalent bonds, C, Se, Ge.� Most Organic Compounds,� Superconductors under some conditions are perfect diamagnets.

Norberto Majilis,The quantum theory of Magnetism, World Scientific, Second Edt. (2007)

� Paramagnets (Permanent dipole moment)

� Diamagnets (No net magnetic moment)

PT

Closed shell

Incomplete shell

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM5/

Page 6: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Poem-Paramagnetism

تا بگويم من برات از مغناطيس گوش دل بشنو نكات مغناطيس تا بگويم من برات از مغناطيس گوش دل بشنو نكات مغناطيس

مغناطيس باشد ممانش دايمي نامند پارامغناطيس وجوديكه همه آن مغناطيس باشد ممانش دايمي نامند پارامغناطيس وجوديكه همه آن

پرنيمه هاي fآكتانيدها با النتانيدها نيمه پر 3dگويم وترنزيشن متال از

آيد پديد چون ميسر نيست ما را النتانيد بررسي هاي آيد پديد 3dچون ميسر نيست ما را النتانيد بررسي هاي

شش تاي بعد آشناتراست استيوا كر چهار فلز اول است

شش تاي بعد آشناتراست however استيوا كر چهار فلز اول است )كروم–واناديوم –تيتانيوم –اسكانديم (

بعد آن منگنز و آهن كبالت ليك نيكل روي و مس در خواب خواببعد آن منگنز و آهن كبالت ليك نيكل روي و مس در خواب خواب

گفت ابويي اينچنين در مكتب اهل يقين تا كه بستايند خداوند كريم را با يقينگفت ابويي اينچنين در مكتب اهل يقين تا كه بستايند خداوند كريم را با يقين

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM6/

Page 7: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of Magnetism-Interactions

Ligands : Non-magnetic ions surrounding one paramagnetic ion.

Crystal Field : Electrostatic interaction between the electrons of paramagnetic ionand electron charge distribution of the ligands.

a) Paramagnetic ion-Nonmagnetic ion interactions

Amount of splitting : depends on the symmetry of the local environment

Effects : Splitting the energy levels of the single magnetic atom.

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM7/

Page 8: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of Magnetism-Interaction (Ligands)

Common case: Octahedral

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM8/

Page 9: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

A single metal atom M A metal atom M in a spherical field

Theory of Magnetism-Interaction (Ligands)

A metal atom M in an Octahedral field

d�����

d��

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM9/

Page 10: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Common case: tetrahedral

d��, d��, d��

d����� , d��

��

d orbitals free ion

Average energy of the d orbitals

in spherical crystal fields

Splitting of the d orbitals in tetrahedral crystal

fields

∆�=49∆#Comparison :

Theory of Magnetism-Interaction (Ligands)Jahanfar AbouieAdv Cond Mat Smr School2012-IPM10/

Page 11: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of magnetism-Interactions

b-1) Dipole-Dipole magnetic interactions

b) Paramagnetic ion - Paramagnetic ion interactions

Order of magnitude of this effect:

Many materials order at ~ 1000 Κ Dipole interaction must be too weak to account for the ordering of most magnetic materials

Important in the properties of those materials which order at mΚ

Robert M. White, “Quantum Theory of Magnetism” 3rd Edt. Springer (2006)

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM11/

Page 12: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of magnetism-Interactions

b-2) Electrostatic interaction & Pauli principle

� Direct exchange interaction

Origin: Coulomb interaction

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM12/

Expansion in terms of Wannier functions & and spinors '

electron creation operator

Heisenberg Hamiltonian

Page 13: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of magnetism-Interactions

exchanged

B. D. Cullity, C. D. Graham, “Introduction to magnetic materials” 2nd Edt. IEEE Press (2009)

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM13/

Bethe-Slater curve (Schematic)Bethe-Slater curve (Schematic)

() : radius of an atom

(�* : radius of its 3,shell of electrons

� -./ 0 1 → Parallel spins, Ferromagnetic� Origin : Orthogonal orbitals

� -./ 3 1 → Antiparallel spins, Antiferromagnetic� Origin: Non-orthogonal orbitals

Page 14: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of magnetism-Interactions

� Kinetic exchange interaction

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM14/

Neglect Coulomb interaction between different orbitals (direct exchange),assume one orbital per ion: one-band Hubbard model

Hubbardmodel

local Coloumb interaction45 : amplitude for the single electron hopping process

2nd order perturbation theory for small hopping, 5 ≪ 7:

forbiddenallowed

89:;. =45�

7

Page 15: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of magnetism-Interactions

� Indirect exchange in ionic solids: superexchange

� Some oxides (MnO), fluorides (MnF� , FeF�), cuprates, … have magnetic ground state.

(Antiferromagnet)

@AB

� Superexchange: exchange interaction between magnetic ions mediated by non-magnetic ions

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM15/

� Why Super?� Why antiferromagnetic?

The exchange interaction is normally very short-ranged so that this longer ranged interaction must be in some sense “super”.

There is a kinetic energy advantage for antiferromagnetism.

Page 16: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of magnetism-InteractionsJahanfar AbouieAdv Cond Mat Smr School2012-IPM16/

Page 17: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of magnetism-Interactions

� Indirect exchange in metals: RKKY interaction or itinerant exchange

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM17/

� Exchange interaction between magnetic ions mediated by conduction electrons.

8CDDE ( ~GHI(2KL()

(�� Oscillatory behavior: depending on the separation, ferromagnetic or antiferromagnetic

� Double exchange

� Some oxides (Magnetite M�NOP and Manganite QRS�TUVTWXON(1 ≤ / ≤ S)) have ferromagnetic order

Mn� ↔ Mn[ Fe� ↔ Fe�

� Occurs where magnetic ion can show mixed valency

Page 18: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Theory of magnetism-Interactions

Higher orders in perturbation theory (and dipolar interaction) result in magnetic anisotropies:

• on-site anisotropy: (uniaxial),(cubic)

• exchange anisotropy: (uniaxial)

• dipolar:

• Dzyaloshinskii-Moriya:

as well as further higher-order terms

• biquadratic exchange:

• ring exchange (square):

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM18/

Page 19: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Magnets

Superconductors

Topological Insulators

Hall Insulators

Crystalline Solids

Theory of Magnetism - ordersJahanfar AbouieAdv Cond Mat Smr School2012-IPM19/

Electrons and atoms in quantum world can form many different states of matter.

ExamplesTranslation symmetry

Gauge symmetry

Rotational symmetry

The greatest triumph of Cond matt Phys is the classification of these quantum states by the principle of Spontaneous Symmetry Breaking

� The pattern of symmetry breaking leads to a unique order parameter.� Order parameter has a nonvanishing expectation value only in the ordered phase.� A general effective field theory (Landau Ginzburge) can be formulated based on

the order parameter.

Page 20: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Magnetic orders and order parametersJahanfar AbouieAdv Cond Mat Smr School2012-IPM20/

Ferromagnetic → magnetizations @\ = ]^∑ :\ a 0

Phases with one order parameter

• The GS is fully aligned

• GS →

Antiferromagnetic → staggered magnetizations `@\ = ]^∑(41): :\ a 0

• The GS is not fully aligned

• Tentative GS →

does not lead back to→ not even eigenstate!

This is a quantum effect

Page 21: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Magnetic orders and order parameters

Helimagnetic (Chiral order) →

Chiral antiferromagnetChiral ferromagnet

Competition between exchange coupling (Alignment) & DM interaction (Screw like arrangement)

Nature Physics, Vol 1, 159 (2005)PRL 108, 107202 (2012) and refs. therein

d~ : e f \

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM21/

Page 22: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Magnetic orders and order parameters

Ferrimagnetic → magnetizations & staggered magnetizations (same direction @\ a 0, `@\ a 0)

Spin-Flop → magnetization & staggered magnetizations (2 diff. directions @\ a 0, `@g a 0)

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM22/

Phases with two independent order parameters

String Order (Hidden Order)

Ferrite MO. Fe�O� , M is divalent cation Co� , Fe� , Ni� , Cu� ,Mn� Garnet R�FeO]� , R is trivalent rare earth atom.

Page 23: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Magnetic orders and order parameters

Spin Supersolid → spin structure factor `�� = ]^∑ l�:m(f�n) f � n �f,n a 0

& spin stiffness op qr�st�u

ru�a 0

The simultaneous breaking of two independent symmetries is counterintuitive and unusual, because normally a spontaneously broken order locks the system into a single phase.

Only when the remaining fluctuations are large enough, two independent order parameters may exist in one phase, e.g. due to frustration.

A bosonic supersolid phase is characterized by the coexistence of two seemingly contradictory order parameters, a solid crystalline order and a superfluid density. This reflects the spontaneous breaking of two independent symmetries, translational and a U(1) rotational symmetry, diagonal and off diagonal order.

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM23/

Page 24: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Magnetic orders and order parameters

Luttinger liquid

Phases with unknown order parameter but correlation functions and energy gap

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM24/

1) The paradigm for the description of interacting one-dimensional(1D) quantum systems.

2) The correlation functions decay as power laws.

3) The ground state has a quasi-long range order .

* One dimensional quantum spin systems such as antiferromagnetic spin-1/2 chains

* Quantum spin ladder

* Bond alternating spin AF-F spin chains

* Organic conductors* Quantum wires* Carbon nanotubes

Good candidate for studying the Luttinger Liquid phase

Page 25: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Magnetic orders and order parameters

is a unique system for controlling and probing the physics of LL

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM25/

Page 26: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Motivations-High Tc Superconductors

Bednorz and Muller Z. Phys. B 1986

Charge stripes and AF domainsExperiment: Tranquada et al Nature 1995Theory: Emery & Kivelson 1995

Mott insulator

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM26/

Page 27: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Motivations-Spin Orbit SeparationsJahanfar AbouieAdv Cond Mat Smr School2012-IPM27/

Page 28: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

When binding to the atomic nucleus

charge spin

+ =

orbit

+ =

Even if electrons in solids form bands and delocalize from the nuclei, in Mott insulators they retain their three fundamental

quantum numbers: spin, charge and orbital

+

Motivations-Spin Orbit Separations

Electron, as an elementary particle

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM28/

Page 29: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Motivations-Spin Orbit SeparationsJahanfar AbouieAdv Cond Mat Smr School2012-IPM29/

Page 30: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Motivations-Spin Orbit Separations

Generated in processes of angle-resolved photoemission

spectroscopy

Spin-charge separation process in an antiferromagnetic spin chain

Predicted : decades ago – Ref: T. Giamarchi, “Quantum Physics in 1D” (2004), and references therein.

Confirmed : in the mid 1990s – Ref: C. Kim et al PRL 77 4054; H. Fujisawa et al PRB 59 7358; B. J. Kim et al Nature Phys. 2 397.

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM30/

Page 31: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Motivations-Spin Orbit Separations

Generated in processes of resonant inelastic X-ray scattering (RIXS)

Ground state orbitalGround state orbital

Excited state orbitalExcited state orbital

Spin-orbital separation process in an antiferromagnetic

spin chain emerging after exciting an orbital

Theory : K. Wohlfeld et al PRL (2011) (IFW Dresden, and MPI Stuttgart)

Experiment : J. Schlappa et al, Nature, published 18 April 2012.

A second order scattering technique

and can excite transition between the copper 3d of

different symmetry (orbital excitations)

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM31/

Page 32: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Motivations-Spin Orbit Separations

RIXS intensity map of the dispersing spin and orbital excitations in Sr2CuO3 as functions of photon momentum transfer along the chains and photon energy transfer.

Arbitrary units

Lattice constant

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM32/

Page 33: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Motivations-Spin Orbit SeparationsJahanfar AbouieAdv Cond Mat Smr School2012-IPM33/

Page 34: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Motivations-Spin Orbit SeparationsJahanfar AbouieAdv Cond Mat Smr School2012-IPM34/

Page 35: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Magnetism and Topological InsulatorsJahanfar AbouieAdv Cond Mat Smr School2012-IPM35/

Page 36: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Magnetism and Topological Insulators

They demonstrate that the edge states of the S=1 spin chain is nicely captured if one starts with the edge state of the dimerized 1D topological band insulator.

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM36/

Page 37: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Physical properties of electrons in solids

ˆ ˆ ˆH K U= +

K U+Itinerant electrons

The typical time spent near a specific atom in the crystal lattice is very short

Wave-like pictureˆ

K

U>>

Large bandwidth

K +U Localized electrons

The typical time spent near a specific atom in the crystal lattice is large

Particle-like picture

ˆ1

ˆK

U<<

Narrow

bandwidth

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM37/

Page 38: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Model Hamiltonian - quantum magnetism

ˆ ˆK U+ Hesitant electrons

* 2 2~ | ( ) | ( ) | ( ) |L s LU dr dr r R U r r r Rχ χ′ ′ ′− − −∫r rr r r r r r

Hubbard ModelThe simplest model Hamiltonian

Kinetic term

INTERACTINGPOTENTIAL

KINETICTERM

2 2*~ ( ) ( )

2LL

L LRRt dr r R r R

mχ χ′

′′

∇ ′− −∫r r

r rr r rh

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM38/

Page 39: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Heisenberg spin models

Hubbard model t- J model1U

t>>

, , ,,

ˆ ( ) .( )x x x y y y z z zi j i j i j i j i j i j i j i j

i j

H J S S J S S J S S h S S D S S= + + + ⋅ + + ×∑urr r r r r

External magnetic field

half filling

Coupling constants, Ferromagnetism

Anti-ferromagnetism

0J <

0J >Spin operators:Homogeneous

Inhomogeneous (Ferrimagnetisms)i jS S=

i jS S≠

Heisenberg model

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM39/

Spin-orbit couplingDM interaction

Page 40: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Heisenberg spin models

Quantumphases ?

Groundstates?

ResponseFunctions?

Quantum fluctuations

Questions?

Changing� Coupling constant 8v,w,�� Magnetic field ℎ� Spin value `� DM interaction y

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM40/

• Critical fields• Order Parameters• Energy gap = lim

^→z({]−{z)

• Specific heat• Magnetic susceptibilities

Page 41: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Any Questions?

Page 42: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Heisenberg spin models-energy gapJahanfar AbouieAdv Cond Mat Smr School2012-IPM41/

Field theory: Nonlinear Sigma model

Numerical method:

F. D. M. Haldane, Phys. Lett. A, 93, 464 (1983);F. D. M. Haldane, Phys. Rev. Lett. 59, 1153 (1983).

1983 Haldane: AF spin-s Heisenberg chain → O(3) NLSM

→ →→ →

Demonstrate: Integer spin → NLSM → gapped Conjecture: Half-integer → NLSM+ topological Berry phase → gapless

Spin systemsQuantum Hall EffectTopological insulatorsTunneling effects

Steven R. White, PRL 69 2863 (1992)� DMRG:� Exact diagonalization Lanczos method

1990 (LSM)1990 (LSM)

Page 43: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Heisenberg spin models-NLSMJahanfar AbouieAdv Cond Mat Smr School2012-IPM42/

Mapping:• Generalizing the Hubbard-Stratonovich formula in the partition function,• Applying gradient expansions in the Hamiltonian formalism, • Using spin coherent states in the path integral formalism.Using spin coherent states in the path integral formalism.

Partition function and spin coherent state

Geometrical Berry phase

One spin systems

Berry phase

Classical Hamiltonian

Page 44: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Heisenberg spin models-NLSMJahanfar AbouieAdv Cond Mat Smr School2012-IPM43/

AF Heisenberg spin chain

Unimodular Neel field Transverse canting field, describes the ferromagnetic fluctuations around the local Neel field

Separation between slow and fast spin wave fluctuations.

Topological Berry phase

Classical Hamiltonian

Berry phase

Page 45: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Heisenberg spin models-NLSM

Integrating out |

Topological winding number or Pontryagin index

Coupling constant

Θ = 0 → ~A5l�(���l → ����l,Θ = π → ~A5l�(���l → ����lII

Θ ∈ 0, π → qHA�A5l�(���l → ����l,

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM44/

Page 46: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Heisenberg spin models-NLSM

Effects of alternation

Spin wave velocity

Coupling constant

Topological term

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM45/

S. Mahdavifar, and J. Abouie, J. Phys. Condensed Matter 23 246002 (2011)

Page 47: Quantum Magnetism and Quantum EntanglementQuantum Magnetism and Quantum Entanglement & IPM Lecture 1 Quantum Magnetism Background: Chiral magnet - PRL 108, 107202 (2012); Nature Phys

Exact ground state and critical fields

(s=1/2) Ising and XY model in a transverse field,

� =�( : � : ]� � � : v)

� =�(8v : v : ]v � 8w :w : ]w � ℎ� :�)Fermionization: Jordan-Wigner transformation,Model mapped to a non-interacting fermion model

S. Suchdev, “Quantum Phase Transition” Cambridge University press (1998)

N. M. Bogoliubov, A. G. Izergin, and V. E. Korepin, Nucl. Phys. B275, 687 (1986).C. N. Yang and C. P. Yang, Phys. Rev. B 150, 321 (1966); 327 (1966).

1 1 1ˆ x x x y y y z z z

i i i i i ii

H J S S J S S J S S+ + += + +∑

Anisotropic Heisenberg spin-1/2 chain, Bethe Ansatz solutions → Coupled Nonlinear Int.

1 1 1ˆ x x x y y y z z z

i i i i i ii

H J S S J S S J S S+ + += + +∑

XXZ in longitudinal field (S=1/2)

S. Kimura, et al Phys. Rev. Lett. 100, 057202 (2008).

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM46/

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Exact ground state and critical fields

Antiferromagnetic spin-1/2 Heisenebrg XYZ in a field,

Anisotropic ferrimagnetic (S,s) models in a field.

Anisotropic dimerized AF-F chains in a field,

Anisotropic tetramerized chains in a field,

spin chains

A FJα

FJα

A FJα

FJα

A FJα

A FJα

FJα

FJα

323 )( ClCuNHCH

Except of a few particular model Hamiltonians the exact GS of many models are not known

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM47/

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||J α

||J α

J α⊥

||J α

||J α

J α⊥ J α

||J α||J α

||J α||J α

Ladder geometry

Anisotropic spin-1/2 ladders in transverse field,

Ferrimagnetic ladders ,

Anisotropic 2D and 3D lattices

Square, Honeycomb and Triangular lattices.

Exact ground state and critical fieldJahanfar AbouieAdv Cond Mat Smr School2012-IPM48/

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Many thanks to Josef Kurmann, Harry Thomas and Gerhard Muller

J. Kurmann, H. Thomas, and G. Muller, Physica112A, 235 (1982).

Is there a field where the quantum fluctuations be uncorrelated and the exact ground state be well known?

Isotropic cases : At critical field

Anisotropic cases : At factorizing field

f ch h≤

The GS at this point is a factorized classical state

ii

GS S= ⊗ Single particle state

Factorizing field

Magnetic field

Suppresses quantum fluctuations

Induces an order in the system

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM49/

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Factorizing field: example

xxz chain in transverse field. 1 1 1ˆ x x y y z z x

i i i i i i ii

H S S S S S S h S+ + += + + ∆ +∑

2(1 )fh = + ∆Increasing magnetic field

0.25 1.58

1.6f

c

h

h

∆ = ⇒ =

=

Anisotropic case

1 2f ch h∆ = ⇒ = =Isotropic case

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM50/

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Factorizing point – Homogenous spin-1/2 model

,,

l ll l

H H ′′

=∑

,l l l l l lH ψ ψ ε ψ ψ′ ′ ′=

2 2cos sin2 2

i i

l e eϕ ϕθ θψ

−= ↑ + ↓

Bloch sphere

ϕ

Conditions for factorization

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM51/

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Factorized GS properties

Entanglement,Quantum Discord

Spin models

Molecular Spintronics

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM52/

Condensed matter physics

Magnetism

Quantum Information

theory

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EntanglementJahanfar AbouieAdv Cond Mat Smr School2012-IPM53/

A kind of non-local quantum correlation

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EntanglementJahanfar AbouieAdv Cond Mat Smr School2012-IPM54/

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Entanglement-Pure and mixed stateJahanfar AbouieAdv Cond Mat Smr School2012-IPM55/

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Classical vs quantum correlationsJahanfar AbouieAdv Cond Mat Smr School2012-IPM56/

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Measures of entanglementJahanfar AbouieAdv Cond Mat Smr School2012-IPM57/

W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)

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Jahanfar AbouieAdv Cond Mat Smr School2012-IPM58/

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Entanglement and correlations

Concurrence Negativity

Density Matrix

1

2S = spin model

11,

2S = spin model

In addition of one and two-point correlationtriad and quad correlations

N. Askari and J. Abouie, submitted

S=1

L. Amico, et al, Phys. Rev. A 69, 022304 (2004)

Magnetization and Two-point correlation functions

S=1/2

G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002)

1S ≥spin model

Negative eigen-values of

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM59/

Mixed state

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Entanglement and Berry phase

Entanglement and DOS

Entanglement of RVB states, liquid state, …..

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM60/

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Concurrence at the factorizing point

QMC simulation

T. Roscilde, et al, Phys. Rev. Lett, 93, 167203 (2004)J. Abouie, A. Langari and M. Siahatgar, J. Phys.: Condebsed Matter, 22 (2010)

Lanczos method

At the factorizing point

Entanglement is zero,Ground state has a product form

At the factorizing point

Entanglement is zero,Ground state has a product form

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM61/

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Entanglement and factorizing line

Anti-parallel entanglement

|↑↓� + |↓↑�

Anti-parallel entanglement

|↑↓� + |↓↑�

Parallel entanglement

|↑↑� + |↓↓�

Parallel entanglement

|↑↑� + |↓↓�

Factorized line

Critical line

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM62/

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Factorized state propertiesJahanfar AbouieAdv Cond Mat Smr School2012-IPM63/

2D Ising model

1D XY model

Key point

Transfer matrix

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Equivalence of 1D Q and 2D CJahanfar AbouieAdv Cond Mat Smr School2012-IPM64/

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Equivalence of 1D and 2D-boundary

Factorized line

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM65/

Bond alternation spin-1/2 chainIn collaboration with R. Sepehrinia

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Quantum discord and factorization

Quantum discord and mutual correlations

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM66/

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Factorizing point in spin models

Determining the factorizing conditions

Why is it important finding the factorized ground state and factorizing field?

Incoming slides

1. It manifests zero entanglement which is necessary to be identified for reliable manipulating of quantum computing.

2. A factorizing field can be also a quantum critical point in certain condition.

3. The information about the factorizing field is attractive for the study of quantum phase transition.

4. Study of the physical properties around the factorizing field.

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM67/

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Factorizing point for ferrimagnets

realize both AF and F interactions.

HamiltonianHamiltonian

1) Consider a two-spin (1,1/2) model1) Consider a two-spin (1,1/2) model

2) The factorized state should be satisfied by2) The factorized state should be satisfied by

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM68/

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Factorizing point for (1, 1/2) ferrimagnets

σθ

ϕ

ρβ

α

2 21 1

cos sin2 2 2 2

i ie e

ϕ ϕθ θσ−

= + + −

1 2 1(1 cos ) 1 sin 0 (1 cos ) 1

2 2 2i ie eα αρ β α β−= + + + + − −

3)3)

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM69/

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Factorizing point for (1,1/2) ferrimagnets

4) Finding the conditions to have 4) Finding the conditions to have

2 2

2 2

2 2

2 2

2

2

2

2

( ) ( )2cos

( ) ( )2

( ) ( )2cos ,

( ) ( )2

0 ,

2

0 .

,

2

2

2

x xy y z z y

f f

y yx x z z x

f f

x yy y z z x

f f

y xx x z z y

f f

J Jh J J J h J J

J Jh J J J h J J

J Jh J J J h J J

J Jh J J J h J J

θ

β

α ϕ

′ ′+ − + + +=−

′ ′+ − + + +

′ ′+ − + + +=−

′ ′+ − + + +

= =

5) The ordering of the spins in factorized state 5) The ordering of the spins in factorized state

σ θ

ρβ

x yJ J>y xJ J>

x zy z

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM70/

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Factorizing point for ferrimagnets

1) Make a rotation

Generalization Two spins model of arbitrary spin values Generalization Two spins model of arbitrary spin values

( , )σ ρJahanfar AbouieAdv Cond Mat Smr School2012-IPM71/

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Conditions of factorized state

2) Imposing the condition to have a factorized state

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM72/

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Factorizing point for a many body system

HamiltonianHamiltonian

constraint

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM73/

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Factorized ground stateJahanfar AbouieAdv Cond Mat Smr School2012-IPM74/

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Examples

Triangular lattice

Honeycomb lattice

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM75/

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Examples

Ladder geometry

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM76/

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Examples

Bond alternating AF-F chain

Other models

� Spin-Peirels model

� Nersesyan-Luthur model

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM77/

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Experimental results for

M. Kenzelmann, et. al, Phys. Rev. B, 65, 144432 (2002)

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM78/

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Order parameters

Entanglement or concurrence Magnetization and Staggered magnetization

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM79/

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Spin wave theory around the factorizing pointJahanfar AbouieAdv Cond Mat Smr School2012-IPM80/

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Specific heat

The number of bosons are controlled by this constraint

Existence of two energy scales at hf<h<hc

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM81/

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Thermal entanglementJahanfar AbouieAdv Cond Mat Smr School2012-IPM82/

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Experiment and Theory

Lanczos method

J. Abouie, A. Langari and M. Siahatgar, J. Phys.: Condensed Matter, 22, (2010)

Jahanfar AbouieAdv Cond Mat Smr School2012-IPM83/

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Thanks for your attentions