Company LOGO Jahanfar Abouie Advanced School on Recent Progress in Cond-Mat IPM-27-28 June 2012 Quantum Magnetism and Quantum Entanglement & IPM
Company
LOGO
Jahanfar Abouie
Advanced School on Recent Progress in Cond-Mat
IPM-27-28 June 2012
Quantum Magnetism and Quantum Entanglement
& IPM
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
Lecture 2
Background: Supramolecular structure-http://nanotechweb.org/cws/article/lab/43208
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/
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/
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/
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/
Theory of Magnetism-Interaction (Ligands)
Common case: Octahedral
Jahanfar AbouieAdv Cond Mat Smr School2012-IPM8/
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/
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/
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/
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
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
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
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.
Theory of magnetism-InteractionsJahanfar AbouieAdv Cond Mat Smr School2012-IPM16/
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
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/
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.
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
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/
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.
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/
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
Magnetic orders and order parameters
is a unique system for controlling and probing the physics of LL
Jahanfar AbouieAdv Cond Mat Smr School2012-IPM25/
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/
Motivations-Spin Orbit SeparationsJahanfar AbouieAdv Cond Mat Smr School2012-IPM27/
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/
Motivations-Spin Orbit SeparationsJahanfar AbouieAdv Cond Mat Smr School2012-IPM29/
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/
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/
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/
Motivations-Spin Orbit SeparationsJahanfar AbouieAdv Cond Mat Smr School2012-IPM33/
Motivations-Spin Orbit SeparationsJahanfar AbouieAdv Cond Mat Smr School2012-IPM34/
Magnetism and Topological InsulatorsJahanfar AbouieAdv Cond Mat Smr School2012-IPM35/
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/
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
1ˆ
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/
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/
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
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
Any Questions?
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)
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
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
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/
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)
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/
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/
||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/
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/
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/
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/
Factorized GS properties
Entanglement,Quantum Discord
Spin models
Molecular Spintronics
Jahanfar AbouieAdv Cond Mat Smr School2012-IPM52/
Condensed matter physics
Magnetism
Quantum Information
theory
EntanglementJahanfar AbouieAdv Cond Mat Smr School2012-IPM53/
A kind of non-local quantum correlation
EntanglementJahanfar AbouieAdv Cond Mat Smr School2012-IPM54/
Entanglement-Pure and mixed stateJahanfar AbouieAdv Cond Mat Smr School2012-IPM55/
Classical vs quantum correlationsJahanfar AbouieAdv Cond Mat Smr School2012-IPM56/
Measures of entanglementJahanfar AbouieAdv Cond Mat Smr School2012-IPM57/
W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)
Jahanfar AbouieAdv Cond Mat Smr School2012-IPM58/
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
Entanglement and Berry phase
Entanglement and DOS
Entanglement of RVB states, liquid state, …..
Jahanfar AbouieAdv Cond Mat Smr School2012-IPM60/
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/
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/
Factorized state propertiesJahanfar AbouieAdv Cond Mat Smr School2012-IPM63/
2D Ising model
1D XY model
Key point
Transfer matrix
Equivalence of 1D Q and 2D CJahanfar AbouieAdv Cond Mat Smr School2012-IPM64/
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
Quantum discord and factorization
Quantum discord and mutual correlations
Jahanfar AbouieAdv Cond Mat Smr School2012-IPM66/
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/
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/
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/
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/
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/
Conditions of factorized state
2) Imposing the condition to have a factorized state
Jahanfar AbouieAdv Cond Mat Smr School2012-IPM72/
Factorizing point for a many body system
HamiltonianHamiltonian
constraint
Jahanfar AbouieAdv Cond Mat Smr School2012-IPM73/
Factorized ground stateJahanfar AbouieAdv Cond Mat Smr School2012-IPM74/
Examples
Triangular lattice
Honeycomb lattice
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Examples
Ladder geometry
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Examples
Bond alternating AF-F chain
Other models
� Spin-Peirels model
� Nersesyan-Luthur model
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Experimental results for
M. Kenzelmann, et. al, Phys. Rev. B, 65, 144432 (2002)
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Order parameters
Entanglement or concurrence Magnetization and Staggered magnetization
Jahanfar AbouieAdv Cond Mat Smr School2012-IPM79/
Spin wave theory around the factorizing pointJahanfar AbouieAdv Cond Mat Smr School2012-IPM80/
Specific heat
The number of bosons are controlled by this constraint
Existence of two energy scales at hf<h<hc
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Thermal entanglementJahanfar AbouieAdv Cond Mat Smr School2012-IPM82/
Experiment and Theory
Lanczos method
J. Abouie, A. Langari and M. Siahatgar, J. Phys.: Condensed Matter, 22, (2010)
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Thanks for your attentions