exotic phases in quantum magnets mpa fisher outline: 2d spin liquids: 2 classes topological spin...
TRANSCRIPT
Exotic Phases in Quantum Magnets
MPA Fisher
Outline:
• 2d Spin liquids: 2 Classes
• Topological Spin liquids
• Critical Spin liquids
• Doped Mott insulators: Conducting Non-Fermi liquids
KITPC, 7/18/07
Interest: Novel Electronic phases of Mott insulators
2
Quantum theory of solids: Standard Paradigm Landau Fermi Liquid Theory
py
pxFree Fermions
Filled Fermi seaparticle/hole excitations
Interacting Fermions
Retain a Fermi surface Luttingers Thm: Volume of Fermi sea same as for free fermions
Particle/hole excitations are long lived near FS Vanishing decay rate
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Add periodic potential from ions in crystal
• Plane waves become Bloch states
• Energy Bands and forbidden energies (gaps)
• Band insulators: Filled bands
• Metals: Partially filled highest energy band
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Even number of electrons/cell - (usually) a band insulator
Odd number per cell - always a metal
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Band Theory
• s or p shell orbitals : Broad bandsSimple (eg noble) metals: Cu, Ag, Au - 4s1, 5s1, 6s1: 1 electron/unit cell
Semiconductors - Si, Ge - 4sp3, 5sp3: 4 electrons/unit cell
Band Insulators - Diamond: 4 electrons/unit cell
Band Theory Works
• d or f shell electrons: Very narrow “bands”
Transition Metal Oxides (Cuprates, Manganites, Chlorides, Bromides,…): Partially filled 3d and 4d bands
Rare Earth and Heavy Fermion Materials: Partially filled 4f and 5f bands
Electrons can ``self-localize”
Breakdown
Mott Insulators:Insulating materials with an odd number of electrons/unit cell
Correlation effects are critical!
Hubbard model with one electron per site on average:
electron creation/annihilation operators on sites of lattice
inter-site hopping
on-site repulsion
t
U
Antiferromagnetic Exchange
Spin Physics
For U>>t expect each electron gets self-localized on a site
(this is a Mott insulator)
Residual spin physics:
s=1/2 operators on each site
Heisenberg Hamiltonian:
Symmetry Breaking
Mott Insulator Unit cell doubling (“Band Insulator”)
Symmetry breaking instability
• Magnetic Long Ranged Order (spin rotation sym breaking)
Ex: 2d square Lattice AFM
• Spin Peierls (translation symmetry breaking)
2 electrons/cell
2 electrons/cellValence Bond (singlet)
=
(eg undoped cuprates La2CuO4 )
How to suppress order (i.e., symmetry-breaking)?
• Low dimensionality– e.g., 1D Heisenberg chain
(simplest example of critical phase)
– Much harder in 2D!
“almost” AFM order:
S(r)·S(0) ~ (-1) r / r2
• Low spin (i.e., s = ½)
• Geometric Frustration– Triangular lattice– Kagome lattice
?
• Doping (eg. Hi-Tc): Conducting Non-Fermi liquids
Spin Liquid: Holy Grail
Theorem: Mott insulators with one electron/cell have low energy excitations above the ground state with (E_1 - E_0) < ln(L)/L for system of size L by L.
(Matt Hastings, 2005)
Remarkable implication - Exotic Quantum Ground States are guaranteed in a Mott insulator with no broken symmetries
Such quantum disordered ground states of a Mottinsulator are generally referred to as “spin liquids”
Spin-liquids: 2 Classes
• Topological Spin liquids
– Topological degeneracyGround state degeneracy on torus
– Short-range correlations– Gapped local excitations– Particles with fractional quantum numbers
RVB state (Anderson)
odd oddeven
• Critical Spin liquids
- Stable Critical Phase with no broken symmetries
- Gapless excitations with no free particle description- Power-law correlations
- Valence bonds on many length scales
Simplest Topological Spin liquid (Z2)Resonating Valence Bond “Picture”
=
Singlet or a Valence Bond - Gains exchange energy J
2d square lattice s=1/2 AFM
Valence Bond Solid
Valence Bond Solid
Gapped Spin Excitations
“Break” a Valence Bond - costsenergy of order J
Create s=1 excitation
Try to separate two s=1/2 “spinons”
Energy cost is linear in separation
Spinons are “Confined” in VBS
RVB State: Exhibits Fractionalization!
Energy cost stays finite when spinons are separated
Spinons are “deconfined” in the RVB state
Spinon carries the electrons spin, but not its charge !
The electron is “fractionalized”.
J1=J2=J3 Kagome s=1/2 in easy-axis limit: Topological spin liquid ground state (Z2)
J1
J2
J3
For Jz >> Jxy have 3-up and 3-down spins on each hexagon. Perturb in Jxy
projecting into subspace to get ring model
J1=J2=J3 Kagome s=1/2 in easy-axis limit: Topological spin liquid ground state (Z2)
J1
J2
J3
For Jz >> Jxy have 3-up and 3-down spins on each hexagon. Perturb in Jxy
projecting into subspace to get ring model
Properties of Ring Model
• No sign problem!
• Can add a ring flip suppression term and tune to soluble Rokshar-Kivelson point
• Can identify “spinons” (sz =1/2) and Z2 vortices (visons) - Z2 Topological order
• Exact diagonalization shows Z2 Phase survives in original easy-axis limit
D. N. Sheng, Leon BalentsPhys. Rev. Lett. 94, 146805 (2005)
L. Balents, M.P.A.F., S.M. Girvin, Phys. Rev. B 65, 224412 (2002)
Other models with topologically ordered spin liquid phases
• Quantum dimer models
• Rotor boson models
• Honeycomb “Kitaev” model
• 3d Pyrochlore antiferromagnet
Moessner, Sondhi Misguich et al
Motrunich, Senthil
Hermele, Balents, M.P.A.F
Freedman, Nayak, ShtengelKitaev
(a partial list)
■ Models are not crazy but contrived. It remains a huge challenge to find these phases in the lab – and develop theoretical techniques to look for them in realistic models.
Critical Spin liquids
T
Frustration parameter:
Key experimental signature: Non-vanishing magnetic susceptibility in the zero temperature limitwith no magnetic (or other) symmetry breaking
Typically have some magnetic ordering, say Neel, at low temperatures:
• Organic Mott Insulator, -(ET)2Cu2(CN)3: f ~ 104
– A weak Mott insulator - small charge gap– Nearly isotropic, large exchange energy (J ~ 250K)– No LRO detected down to 32mK : Spin-liquid ground state?
• Cs2CuCl4: f ~ 5-10– Anisotropic, low exchange energy (J ~ 1-4K)– AFM order at T=0.6K
T0.62K
AFM Spin liquid?
0
Triangular lattice critical spin liquids?
Kagome lattice critical spin liquids?
• Iron Jarosite, KFe3 (OH)6 (SO4)2 : f ~ 20
Fe3+ s=5/2 , Tcw =800K Single crystals
Q=0 Coplaner order at TN = 45K
• 2d “spinels” Kag/triang planes SrCr8Ga4O19 f ~ 100
Cr3+ s=3/2, Tcw = 500K, Glassy ordering at Tg = 3K
C = T2 for T<5K
• Volborthite Cu3V2O7(OH)2 2H2O f ~ 75
Cu2+ s=1/2 Tcw = 115K Glassy at T < 2K
• Herbertsmithite ZnCu3(OH)6Cl2 f > 600
Cu2+ s=1/2 , Tcw = 300K, Tc< 2K
Ferromagnetic tendency for T low, C = T2/3 ??
All show much reduced order - if any - and low energy spin excitations present
Lattice of corner sharing triangles
Theoretical approaches to critical spin liquids
Slave Particles:
• Express s=1/2 spin operator in terms of Fermionic spinons • Mean field theory: Free spinons hopping on the lattice• Critical spin liquids - Fermi surface or Dirac fermi points for spinons• Gauge field U(1) minimally coupled to spinons • For Dirac spinons: QED3
Boson/Vortex Duality plus vortex fermionization: (eg: Easy plane triangular/Kagome AFM’s)
Triangular/Kagome s=1/2 XY AF equivalent to bosons in “magnetic field”
boson hoppingon triangular lattice
boson interactionspi flux thru each triangle
Focus on vortices
Vortex number N=1
Vortex number N=0
“Vortex”
“Anti-vortex”
+
-
Due to frustration,the dual vortices are at “half-filling”
Boson-Vortex Duality• Exact mapping from boson to vortex variables.
• All non-locality is accounted for by dual U(1) gauge force
Dual “magnetic” field
Dual “electric” field
Vortex number
Vortex carriesdual gauge charge
J
J’
“Vortex”
“Anti-vortex”
+
-
∑∑ ×+=⟨ i
iijij
ij aUeJH 22 )(
..)( 0
chebbt ijij aaiji
ijij +− +
⟨∑
Half-filled bosonic vortices w/ “electromagnetic” interactions
Frustrated spins
vortex hopping
vortex creation/annihilation ops:
Vortices see pi flux thru each hexagon
Duality for triangular AFM
• Difficult to work with half-filled bosonic vortices fermionize!
bosonic vortex
fermionic vortex + 2 flux
Chern-Simons flux attachment
• “Flux-smearing” mean-field: Half-filled fermions on honeycomb with pi-flux
..chfftH jiij
ijMF +−= ∑⟨
~
E
k
• Band structure: 4 Dirac points
Chern-Simons Flux Attachment: Fermionic vortices
With log vortex interactions can eliminate Chern-Simons term
Four-fermion interactions: irrelevant for N>Nc
“Algebraic vortex liquid”– “Critical Phase” with no free particle description
– No broken symmetries - but an emergent SU(4)
– Power-law correlations
– Stable gapless spin-liquid (no fine tuning)
N = 4 flavors
Low energy Vortex field theory: QED3 with flavor SU(4)
Linearize aroundDirac points
If Nc>4 then have a stable:
“Decorated” Triangular Lattice XY AFM
• s=1/2 on Kagome, s=1 on “red” sites• reduces to a Kagome s=1/2 with AFM J1, and weak FM J2=J3
J’
J
J1>0
J2<0
J3<0
Flux-smeared mean field: Fermionicvortices hopping on “decorated”honeycomb
Vortex duality
Fermionized Vortices for easy-plane Kagome AFM
QED3 with SU(8) Flavor Symmetry
“Algebraic vortex liquid” in s=1/2 Kagome XY Model–Stable “Critical Phase”
–No broken symmetries
– Many gapless singlets (from Dirac nodes)
– Spin correlations decay with large power law - “spin pseudogap”
Vortex Band Structure: N=8 Dirac Nodes !!
Provided Nc <8 will have a stable:
Doped Mott insulators
High Tc Cuprates
Doped Mott insulator becomes ad-wave superconductor
Strange metal: Itinerant Non-Fermi liquid with “Fermi surface”
Pseudo-gap: Itinerant Non-Fermi liquid with nodal fermions
Slave Particle approach toitinerant non-Fermi liquids
Decompose the electron:spinless charge e bosonand s=1/2 neutral fermionic spinon,coupled via compact U(1) gauge field
Half-Filling: One boson/site - Mott insulator of bosons Spinons describes magnetism (Neel order, spin liquid,...)
Dope away from half-filling: Bosons become itinerant
Fermi Liquid: Bosons condense with spinons in Fermi sea
Non-Fermi Liquid: Bosons form an uncondensed fluid - a “Bose metal”, with spinons in Fermi sea (say)
Uncondensed quantum fluid of bosons: D-wave Bose Liquid (DBL)
Wavefunctions:
N bosons moving in 2d:
Define a ``relative single particle function”
Laughlin nu=1/2 Bosons:
Point nodes in ``relative particle function”Relative d+id 2-particle correlations
Goal: Construct time-reversal invariant analog of Laughlin,(with relative dxy 2-particle correlations)
Hint: nu=1/2 Laughlin is a determinant squared
p+ip 2-body
O. Motrunich/ MPAF cond-mat/0703261
Wavefunction for D-wave Bose Liquid (DBL)
``S-wave” Bose liquid: square the wavefunction of Fermi sea wf is non-negative and has ODLRO - a superfluid
``D-wave” Bose liquid: Product of 2 different fermi sea determinants,elongated in the x or y directions
Nodal structure of DBL wavefunction:
+
+
-
-
Dxy relative 2-particle correlations
Analysis of DBL phase
• Equal time correlators obtained numerically from variational wavefunctions
• Slave fermion decomposition and mean field theory
• Gauge field fluctuations for slave fermions - stability of DBL, enhanced correlators
• “Local” variant of phase - D-wave Local Bose liquid (DLBL)
• Lattice Ring Hamiltonian and variational energetics
Properties of DBL/DLBL• Stable gapless quantum fluids of uncondensed itinerant bosons
• Boson Greens function in DBL has oscillatory power law decay with direction dependent wavevectors and exponents, the wavevectors enclose a k-space volume determined by the total Bose density (Luttinger theorem)
• Boson Greens function in DLBL is spatially short-ranged
• Power law local Boson tunneling DOS in both DBL and DLBL
• DBL and DLBL are both ``metals” with resistance R(T) ~ T4/3
• Density-density correlator exhibits oscillatory power laws, also with direction dependent wavevectors and exponents in both DBL and DLBL
D-Wave Metal
Itinerant non-Fermi liquid phase of 2d electrons
Wavefunction:
t-K Ring Hamiltonian (no double occupancy constraint)
1 2
34
1 2
34
Electron singlet pair“rotation” term
t >> K Fermi liquidt ~ K D-metal (?)
Summary & Outlook
• Quantum spin liquids come in 2 varieties: Topological and critical, and
can be accessed using slave particles, vortex duality/fermionization, ...
• Several experimental s=1/2 triangular and Kagome AFM’s are candidates for critical spin liquids (not topological spin liquids)
• D-wave Bose liquid: a 2d uncondensed quantum fluid of itinerant bosons with many gapless strongly interacting excitations, metallic type transport,...
• Much future work:– Characterize/explore critical spin liquids– Unambiguously establish an experimental spin liquid– Explore the D-wave metal, a non-Fermi liquid of itinerant electrons