Transcript Document

Exotic Phases in Quantum Magnets
MPA Fisher
KITPC, 7/18/07
Interest: Novel Electronic phases of Mott insulators
Outline:
• 2d Spin liquids: 2 Classes
• Topological Spin liquids
• Critical Spin liquids
• Doped Mott insulators: Conducting Non-Fermi liquids
Quantum theory of solids: Standard Paradigm
Landau Fermi Liquid Theory
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py
Free Fermions
px
particle/hole excitations
Filled Fermi sea
Interacting Fermions
Retain a Fermi surface
Particle/hole excitations are
long lived near FS
Luttingers Thm: Volume of Fermi sea
same as for free fermions
Vanishing decay rate
2
Add periodic potential from ions in crystal
Quic kTime™ and a
TIFF ( Unc ompres s ed) dec ompr es sor
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• Plane waves become Bloch states
• Energy Bands and forbidden energies (gaps)
• Band insulators: Filled bands
• Metals: Partially filled highest energy band
Even number of electrons/cell - (usually) a band
insulator
Odd number per cell - always a metal
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Band Theory
• s or p shell orbitals : Broad bands
Simple (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
Breakdown
• 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”
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:
on-site repulsion
electron creation/annihilation
operators on sites of lattice
inter-site hopping
t
U
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:
Antiferromagnetic Exchange
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
(eg undoped cuprates La2CuO4 )
• Spin
Peierls
(translation symmetry breaking)
Valence Bond (singlet)
=
2 electrons/cell
2 electrons/cell
How to suppress order (i.e., symmetry-breaking)?
• Low spin (i.e., s = ½)
•
Low dimensionality
– e.g., 1D Heisenberg chain
(simplest example of critical
phase)
– Much harder in 2D!
• Geometric Frustration
– Triangular lattice
– Kagome lattice
“almost” AFM order:
S(r)·S(0) ~ (-1) r / r2
?
• 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 Mott
insulator are generally referred to as “spin liquids”
Spin-liquids: 2 Classes
• Topological Spin liquids
RVB state (Anderson)
– Topological degeneracy
Ground state degeneracy on torus
– Short-range correlations
– Gapped local excitations
– Particles with fractional quantum numbers
odd even odd
• 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”
2d square lattice s=1/2 AFM
=
Singlet or a Valence Bond - Gains exchange energy J
Valence Bond Solid
Plaquette Resonance
Resonating Valence Bond “Spin liquid”
Plaquette Resonance
Resonating Valence Bond “Spin liquid”
Plaquette Resonance
Resonating Valence Bond “Spin liquid”
Gapped Spin Excitations
“Break” a Valence Bond - costs
energy of order J
Create s=1 excitation
Try to separate two s=1/2 “spinons”
Valence Bond Solid
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”.
J2
J1=J2=J3 Kagome s=1/2 in easy-axis limit:
Topological spin liquid ground state (Z2)
J1
J3
For Jz >> Jxy have 3-up and 3-down
spins on each hexagon. Perturb in Jxy
projecting into subspace to get ring model
J2
J1=J2=J3 Kagome s=1/2 in easy-axis limit:
Topological spin liquid ground state (Z2)
J1
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
L. Balents, M.P.A.F., S.M. Girvin,
Phys. Rev. B 65, 224412 (2002)
• 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 Balents
Phys. Rev. Lett. 94, 146805 (2005)
Other models with topologically
ordered spin liquid phases (a partial list)
• Quantum dimer models
• Rotor boson models
Moessner, Sondhi
Misguich et al
Motrunich, Senthil
• Honeycomb “Kitaev” model
Kitaev
• 3d Pyrochlore antiferromagnet
Hermele, Balents, M.P.A.F
Freedman, Nayak, Shtengel
■ 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
Key experimental signature:
Non-vanishing magnetic susceptibility in the zero temperature limit
with no magnetic (or other) symmetry breaking
Typically have some magnetic ordering, say Neel, at low temperatures:
T
Frustration parameter:
Triangular lattice critical spin liquids?
• 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
AFM
0
Spin liquid?
0.62K
T
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
Lattice of corner
sharing triangles
• 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
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 hopping
on triangular lattice
pi flux thru each
triangle
boson interactions
Focus on vortices
+
“Vortex”
Vortex number N=1
Due to frustration,
the dual vortices
are at “half-filling”
-
“Anti-vortex”
Vortex number N=0
Boson-Vortex Duality
• Exact mapping from boson to vortex variables.
Dual “magnetic”
field
Dual “electric”
field
Vortex number
Vortex carries
dual gauge charge
• All non-locality is accounted for by dual U(1) gauge force
Duality for triangular AFM
J’
J
Frustrated spins
vortex creation/annihilation ops:
+
“Vortex”
H   J ijeij2  U  (  a)i2
 ij 
i
  tijbi b j e
-
i ( aij  aij0 )
Half-filled bosonic vortices w/
“electromagnetic” interactions
 h.c.
 ij 
vortex hopping
“Anti-vortex”
Vortices see pi flux
thru each hexagon
Chern-Simons Flux Attachment: Fermionic vortices
Difficult to work with half-filled bosonic vortices  fermionize!
•
Chern-Simons
flux attachment
bosonic
vortex
•
fermionic vortex
+ 2 flux
“Flux-smearing” mean-field: Half-filled
fermions on honeycomb with pi-flux
H MF   tij fi f j  h.c.
~
 ij 
•
Band structure: 4 Dirac
points
E
k
Low energy Vortex field theory: QED3 with flavor SU(4)
N = 4 flavors
Linearize around
Dirac points
With log vortex interactions can eliminate Chern-Simons term
Four-fermion interactions:
irrelevant for N>Nc
If Nc>4 then
have a stable:
“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)
Fermionized Vortices for easy-plane Kagome AFM
J’
J
“Decorated” Triangular Lattice XY AFM
J2<0
• 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
J3<0
Flux-smeared mean field: Fermionic
vortices hopping on “decorated”
honeycomb
J1>0
Vortex
duality
Vortex Band Structure: N=8 Dirac Nodes !!
QED3 with SU(8) Flavor Symmetry
Provided Nc <8 will have a stable:
“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”
Doped Mott insulators
High Tc Cuprates
Doped Mott insulator becomes a
d-wave superconductor
Strange metal: Itinerant Non-Fermi liquid with “Fermi surface”
Pseudo-gap: Itinerant Non-Fermi liquid with nodal fermions
Slave Particle approach to
itinerant non-Fermi liquids
Decompose the electron:
spinless charge e boson
and 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)
O. Motrunich/ MPAF cond-mat/0703261
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
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)
4
1
3
2
4
1
Electron singlet pair
“rotation” term
t >> K
t~K
Fermi liquid
D-metal (?)
3
2
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