Quantum Magnetism and Quantum Entanglement

Transcription

1 Quantum Magnetism and Quantum Entanglement Jahanfar Abouie Advanced CompanySchool on Recent Progress in Cond-Mat LOGO IPM June 2012 & IPM

2 Lecture 1 Quantum Magnetism Theory of magnetism Motivations Some models and Properties Background: Chiral magnet - PRL 108, (2012); Nature Phys. Vol 1, 159 Dec. (2005)

3 Background: Supramolecularstructure- Lecture 2

4 4/ Theory of Magnetism Single ions Diamagnetic ions Paramagnetic ions Ions in Crystals Crystal fields (Interaction with non-magnetic ions) Ligands Interactions Dipole-Dipole magnetic interactions Electrostatic interaction & Pauli exclusion principle Magnetic Orders Antiferromagnet Ferromagnet Ferrimagnet Helimagnet Spin Liquid Luttinger Liquid Spin super-solid Spin-Flop

5 5/ Theory of Magnetism-Dia & Para Diamagnets (No net magnetic moment) 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. Paramagnets (Permanent dipole moment) Atoms, molecules, ions with an odd number of electrons, like H, 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). Norberto Majilis, The quantum theory of Magnetism, World Scientific, Second Edt. (2007) Closed shell Incomplete shell PT

6 6/ Poem-Paramagnetism f 3d 3d however ( )

7 7/ Theory of Magnetism-Interactions a) Paramagnetic ion-nonmagnetic ion interactions Ligands : Non-magnetic ions surrounding one paramagnetic ion. Crystal Field : Electrostatic interaction between the electrons of paramagnetic ion and electron charge distribution of the ligands. Effects : Splitting the energy levels of the single magnetic atom. Amount of splitting : depends on the symmetry of the local environment

8 8/ Theory of Magnetism-Interaction (Ligands) Common case: Octahedral

9 9/ Theory of Magnetism-Interaction (Ligands) A metal single atom metal M atom in an a M spherical Octahedral field field d d

10 10/ Theory of Magnetism-Interaction (Ligands) Common case: tetrahedral d,d,d d orbitals free ion Average energy of the d orbitals in spherical crystal fields d,d Splitting of the d orbitals in tetrahedral crystal fields Comparison : = 4 9

11 11/ Theory of magnetism-interactions b) Paramagnetic ion - Paramagnetic ion interactions b-1) Dipole-Dipole magnetic 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 3 rd Edt. Springer (2006)

12 12/ Theory of magnetism-interactions b-2) Electrostatic interaction & Pauli principle Direct exchange interaction Origin: Coulomb interaction Expansion in terms of Wannier functions and spinors electron creation operator Heisenberg Hamiltonian

13 13/ Theory of magnetism-interactions exchanged Parallel spins, Ferromagnetic Origin : Orthogonal orbitals Antiparallel spins, Antiferromagnetic Origin: Non-orthogonal orbitals Bethe-Slater curve (Schematic) : radius of an atom : radius of its 3 shell of electrons B. D. Cullity, C. D. Graham, Introduction to magnetic materials 2 nd Edt. IEEE Press (2009)

14 14/ Theory of magnetism-interactions Kinetic exchange interaction Neglect Coulomb interaction between different orbitals (direct exchange), assume one orbital per ion: one-band Hubbard model : amplitude for the single electron hopping process local Coloumb interaction Hubbard model 2nd order perturbation theory for small hopping, :. = 4 allowed forbidden

15 15/ Theory of magnetism-interactions Indirect exchange in ionic solids: superexchange Some oxides (MnO), fluorides (MnF, FeF ), cuprates, have magnetic ground state. (Antiferromagnet) Superexchange: exchange interaction between magnetic ions mediated by non-magnetic ions Why Super? The exchange interaction is normally very short-ranged so that this longer ranged interaction must be in some sense super. Why antiferromagnetic? There is a kinetic energy advantage for antiferromagnetism.

16 16/ Theory of magnetism-interactions

17 17/ Theory of magnetism-interactions Indirect exchange in metals: RKKY interaction or itinerant exchange Exchange interaction between magnetic ions mediated by conduction electrons. ~ (2 ) Oscillatory behavior: depending on the separation, ferromagnetic or antiferromagnetic Double exchange Some oxides (Magnetite and Manganite ( )) have ferromagnetic order Occurs where magnetic ion can show mixed valency Fe Fe Mn Mn

18 18/ 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):

19 19/ Theory of Magnetism - orders Electrons and atoms in quantum world can form many different states of matter. Rotational symmetry Magnets Translation symmetry Crystalline Solids Super conductors The greatest Examples triumph of Cond matt Phys is the classification of these quantum Hall states by the principle of Spontaneous Insulators Symmetry Topological Breaking Insulators Gauge symmetry 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.

20 20/ Magnetic orders and order parameters Phases with one order parameter Ferromagnetic magnetizations = 0 The GS is fully aligned GS Antiferromagnetic staggered magnetizations = ( 1) 0 The GS is not fully aligned Tentative GS does not lead back to not even eigenstate! This is a quantum effect

21 21/ Magnetic orders and order parameters Helimagnetic (Chiral order) ~ Competition between exchange coupling (Alignment) & DM interaction (Screw like arrangement) Chiral antiferromagnet Chiral ferromagnet Nature Physics, Vol 1, 159 (2005) PRL 108, (2012) and refs. therein

22 22/ Magnetic orders and order parameters String Order (Hidden Order) Phases with two independent order parameters Ferrimagnetic magnetizations & staggered magnetizations (same direction 0, 0) Ferrite MO.Fe O, M is divalent cationco,fe,ni,cu,mn Garnet R Fe O, R is trivalent rare earth atom. Spin-Flop magnetization & staggered magnetizations (2 diff. directions 0, 0)

23 23/ Magnetic orders and order parameters Spin Supersolid spin structure factor = ( ), & spin stiffness 0 0 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. 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.

24 24/ Magnetic orders and order parameters Phases with unknown order parameter but correlation functions and energy gap Luttinger liquid 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. Good candidate for studying the Luttinger Liquid phase * 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

25 25/ Magnetic orders and order parameters is a unique system for controlling and probing the physics of LL

26 26/ Motivations-High Tc Superconductors Bednorz and Muller Z. Phys. B 1986 Mott insulator Charge stripes and AF domains Experiment: Tranquada et al Nature 1995 Theory: Emery & Kivelson 1995

27 27/ Motivations-Spin Orbit Separations

28 28/ Motivations-Spin Orbit Separations Electron, as an elementary particle + = charge spin When binding to the atomic nucleus + + = 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

29 29/ Motivations-Spin Orbit Separations

30 30/ Motivations-Spin Orbit Separations Spin-charge separation process in an antiferromagnetic spin chain Generated in processes of angle-resolved photoemission spectroscopy 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 ; H. Fujisawa et al PRB ; B. J. Kim et al Nature Phys

31 31/ Motivations-Spin Orbit Separations Spin-orbital separation process in an antiferromagnetic spin chain emerging after exciting an orbital Generated in processes of resonant inelastic X-ray scattering (RIXS) Excited state orbital Ground state orbital A second order scattering technique and can excite transition between the copper 3d of different symmetry (orbital excitations) Theory : K. Wohlfeld et al PRL (2011) (IFW Dresden, and MPI Stuttgart) Experiment : J. Schlappa et al, Nature, published 18 April 2012.

32 32/ Motivations-Spin Orbit Separations Arbitrary units Lattice constant 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.

33 33/ Motivations-Spin Orbit Separations

34 34/ Motivations-Spin Orbit Separations

35 35/ Magnetism and Topological Insulators

36 36/ 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.

37 37/ Physical properties of electrons in solids Hˆ = Kˆ + Uˆ ˆK ˆ +U Kˆ 1 U ˆ >> Itinerant electrons The typical time spent near a specific atom in the crystal lattice is very short Wave-like picture Large bandwidth ˆK + Uˆ Kˆ 1 U ˆ << Localized electrons The typical time spent near a specific atom in the crystal lattice is large Particle-like picture Narrow bandwidth

38 38/ Model Hamiltonian - quantum magnetism Kˆ Uˆ + Hesitant electrons Hubbard Model The simplest model Hamiltonian Kinetic term KINETIC TERM INTERACTING POTENTIAL 2 2 LL r * r r h r r trr ~ dr χl( R) χl ( R ) RR 2m r r r r r r r r U ~ dr dr χ ( R) U ( ) χ ( R) * 2 2 L s L

39 39/ Heisenberg spin models U t >> 1 Hubbard model t- J model half filling Heisenberg model r r r ur r r ˆ x x x y y y z z z H = Ji, jsi S j + Ji, jsi S j + Ji, jsi S j + h ( Si + S j ) + D.( Si S j ) i, j Coupling constants, Ferromagnetism J < 0 Anti-ferromagnetism J > 0 External magnetic field Spin operators: Homogeneous Si = S j Inhomogeneous (Ferrimagnetisms) S i S j Spin-orbit coupling DM interaction

40 40/ Heisenberg spin models Questions? Specific heat Magnetic susceptibilities Ground states? Response Functions? Quantum phases? Changing Coupling constant,, Magnetic field h Spin value DM interaction Quantum fluctuations Critical fields Order Parameters Energy gap = lim ( )

41 Any Questions?

42 41/ Heisenberg spin models-energy gap Numerical method: DMRG: Steven R. White, PRL (1992) Exact diagonalization Lanczos method Field theory: Nonlinear Sigma model 1983 Haldane: AF spin-s Heisenberg chain O(3) NLSM F. D. M. Haldane, Phys. Lett. A, 93, 464 (1983); F. D. M. Haldane, Phys. Rev. Lett. 59, 1153 (1983). Demonstrate: Integer spin NLSM gapped Conjecture: Half-integer NLSM+ topological Berry phase gapless 1990 (LSM) Spin systems Quantum Hall Effect Topological insulators Tunneling effects

43 42/ Heisenberg spin models-nlsm 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. One spin systems Partition function and spin coherent state Classical Hamiltonian Berry phase Geometrical Berry phase

44 43/ Heisenberg spin models-nlsm AF Heisenberg spin chain Separation between slow and fast spin wave fluctuations. Unimodular Neel field Transverse canting field, describes the ferromagnetic fluctuations around the local Neel field Classical Hamiltonian Berry phase Topological Berry phase

45 44/ Heisenberg spin models-nlsm Integrating out Coupling constant Topological winding number or Pontryagin index Θ=π Θ=0 Θ 0,π

46 45/ Heisenberg spin models-nlsm Effects of alternation Coupling constant Spin wave velocity Topological term S. Mahdavifar, and J. Abouie, J. Phys. Condensed Matter (2011)

47 46/ Exact ground state and critical fields (s=1/2) Ising and XY model in a transverse field, = ( ) = ( h ) Fermionization: Jordan-Wigner transformation, Model mapped to a non-interacting fermion model S. Suchdev, Quantum Phase Transition Cambridge University press (1998) Anisotropic Heisenberg spin-1/2 chain, Bethe Ansatz solutions Coupled Nonlinear Int. ˆ x x x y y y z z z = i i+ 1 + i i+ 1 + i i+ 1 i H J S S J S S J S S N. M. Bogoliubov, A. G. Izergin, and V. E. Korepin, Nucl. Phys. B 275, 687 (1986). C. N. Yang and C. P. Yang, Phys. Rev. B 150, 321 (1966); 327 (1966). XXZ in longitudinal field (S=1/2) ˆ x x x y y y z z z = i i+ 1 + i i+ 1 + i i+ 1 i H J S S J S S J S S S. Kimura, et al Phys. Rev. Lett. 100, (2008).

48 47/ Exact ground state and critical fields Except of a few particular model Hamiltonians the exact GS of many models are not known spin chains 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, J α F J α A F J α F J α A F Anisotropic tetramerized chains in a field, ( CH NH Cu Cl 3 ) 2 3 J α F J α F J α A F J α A F

49 48/ Exact ground state and critical field Ladder geometry Anisotropic spin-1/2 ladders in transverse field, Ferrimagnetic ladders, J α J α J α J α J α J α J α J α J α J α J α Anisotropic 2D and 3D lattices Square, Honeycomb and Triangular lattices.

50 49/ Factorizing field Many thanks to Josef Kurmann, Harry Thomas and Gerhard Muller J. Kurmann, H. Thomas, and G. Muller, Physica 112A, 235 (1982). Magnetic field Suppresses quantum fluctuations Induces an order in the system 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 h f h c The GS at this point is a factorized classical state GS = Si Single particle state i

51 50/ Factorizing field: example ˆ x x y y z z x i i i i i i i i xxz chain in transverse field. H = S S S S S S h S h f = 2(1 + ) Increasing magnetic field Anisotropic case = 0.25 h = 1.58 f h c = 1.6 Isotropic case = 1 h = h = 2 f c

52 51/ Factorizing point Homogenous spin-1/2 model = H H l, l l, l Bloch sphere H ψ ψ = ε ψ ψ l, l l l l l ϕ ϕ θ i θ i 2 2 ψ l = cos e + sin e 2 2 ϕ Conditions for factorization

53 52/ Factorized GS properties Quantum Information theory Entanglement, Quantum Discord Magnetism Spin models Condensed matter physics Molecular Spintronics

54 53/ Entanglement A kind of non-local quantum correlation

55 54/ Entanglement

56 55/ Entanglement-Pure and mixed state

57 56/ Classical vs quantum correlations

58 57/ Measures of entanglement W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)

59 58/

60 59/ Entanglement and correlations Concurrence Mixed state Density Matrix Negativity 1 S = spin model 2 1 S = 1, spin model 2 spin S 1 model S=1/2 Magnetization and Two-point correlation functions L. Amico, et al, Phys. Rev. A 69, (2004) S=1 Negative eigen-values of In addition of one and two-point correlation triad and quad correlations N. Askari and J. Abouie, submitted G. Vidal and R. F. Werner, Phys. Rev. A 65, (2002)

61 60/ Entanglement and Berry phase Entanglement and DOS Entanglement of RVB states, liquid state,..

62 61/ Concurrence at the factorizing point At the factorizing point QMC simulation Entanglement is zero, Lanczos method Ground state has a product form T. Roscilde, et al, Phys. Rev. Lett, 93, (2004) J. Abouie, A. Langari and M. Siahatgar, J. Phys.: Condebsed Matter, 22 (2010)

63 62/ Entanglement and factorizing line Anti-parallel entanglement + Parallel entanglement + Factorized line Critical line

64 63/ Factorized state properties 2D Ising model 1D XY model Key point Transfer matrix

65 64/ Equivalence of 1D Q and 2D C

66 65/ Equivalence of 1D and 2D-boundary Factorized line Bond alternation spin-1/2 chain In collaboration with R. Sepehrinia

67 66/ Quantum discord and factorization Quantum discord and mutual correlations

68 67/ Factorizing point in spin models Determining the factorizing conditions Incoming slides Why is it important finding the factorized ground state and factorizing field? 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.

69 68/ Factorizing point for ferrimagnets Hamiltonian realize both AF and F interactions. 1) Consider a two-spin (1,1/2) model 2) The factorized state should be satisfied by

70 69/ Factorizing point for (1, 1/2) ferrimagnets 3) β ρ θ σ α ϕ σ ϕ ϕ θ i 1 θ i = cos e + + sin e iα 2 1 iα ρ = (1 + cos β ) e sinα 0 + (1 cos β ) e

71 70/ Factorizing point for (1,1/2) ferrimagnets 4) Finding the conditions to have 5) The ordering of the spins in factorized state x x y J y z z y J h f J + ( J + J ) + h f J ( J + ) cos θ = 2 2, y y x J x z z x J h f J + ( J + J ) + h f J ( J + ) 2 2 x y y J y z z x J h f J + ( J + J ) + h f J ( J + ) cos β = 2 2, y x x J x z z y J h f J + ( J + J ) + h f J ( J + ) 2 2 α = 0, ϕ = 0. ρ β J J y x > J > J x y σ θ x z y z

72 71/ Factorizing point for ( σ, ρ) ferrimagnets Generalization Two spins model of arbitrary spin values 1) Make a rotation

73 72/ Conditions of factorized state 2) Imposing the condition to have a factorized state

74 73/ Factorizing point for a many body system Hamiltonian constraint

75 74/ Factorized ground state

76 75/ Examples Triangular lattice Honeycomb lattice

77 76/ Examples Ladder geometry

78 77/ Examples Bond alternating AF-F chain Other models Spin-Peirels model Nersesyan-Luthur model

79 78/ Experimental results for M. Kenzelmann, et. al, Phys. Rev. B, 65, (2002)

80 79/ Order parameters Magnetization and Staggered magnetization Entanglement or concurrence

81 80/ Spin wave theory around the factorizing point

82 81/ Specific heat The number of bosons are controlled by this constraint Existence of two energy scales at hf<h<hc

83 82/ Thermal entanglement

84 83/ Experiment and Theory Lanczos method J. Abouie, A. Langari and M. Siahatgar, J. Phys.: Condensed Matter, 22, (2010)

85 Thanks for your attentions