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 NonFermi liquids.
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Exotic Phases in Quantum Magnets
MPA Fisher
KITPC, 7/18/07
Interest: Novel Electronic phases of Mott insulators
Outline:
Quantum theory of solids: Standard Paradigm
Landau Fermi Liquid Theory
py
Free Fermions
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Filled Fermi sea
particle/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
Add periodic potential from ions in crystal
Even number of electrons/cell  (usually) a band insulator
Odd number per cell  always a metal
Band Theory
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
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 ``selflocalize”
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:
onsite repulsion
electron creation/annihilation
operators on sites of lattice
U
intersite hopping
t
Spin Physics
For U>>t expect each electron gets selflocalized 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
Ex: 2d square Lattice AFM
(eg undoped cuprates La2CuO4 )
2 electrons/cell
2 electrons/cell
Valence Bond (singlet)
=
?
“almost” AFM order:
S(r)·S(0) ~ (1) r/ r2
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”
RVB state (Anderson)
Ground state degeneracy on torus
odd
even
odd
 Stable Critical Phase with no broken symmetries
 Gapless excitations with no free particle description
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
J3
For Jz >> Jxy have 3up and 3down
spins on each hexagon. Perturb in Jxy
projecting into subspace to get ring model
J2
J1
J3
For Jz >> Jxy have 3up and 3down
spins on each hexagon. Perturb in Jxy
projecting into subspace to get ring model
L. Balents, M.P.A.F., S.M. Girvin,
Phys. Rev. B 65, 224412 (2002)
and tune to soluble RoksharKivelson point
Z2 vortices (visons)  Z2 Topological order
survives in original easyaxis limit
D. N. Sheng, Leon Balents
Phys. Rev. Lett. 94, 146805 (2005)
(a partial list)
Moessner, Sondhi
Misguich et al
Motrunich, Senthil
Kitaev
Freedman, Nayak, Shtengel
Hermele, Balents, M.P.A.F
■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.
Key experimental signature:
Nonvanishing 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:
AFM
Spin liquid?
T
0
0.62K
Fe3+ s=5/2 , Tcw =800K Single crystals
Q=0 Coplaner order at TN = 45K
Cr3+ s=3/2, Tcw = 500K, Glassy ordering at Tg = 3K
C = T2 for T<5K
Cu2+ s=1/2 Tcw = 115K Glassy at T < 2K
Cu2+ s=1/2 , Tcw = 300K, Tc< 2K
Ferromagnetic tendency for T low, C = T2/3 ??
Lattice of corner
sharing triangles
All show much reduced order  if any  and low energy spin excitations present
Boson/Vortex Duality plus vortex fermionization:
(eg: Easy plane triangular/Kagome AFM’s)
+

boson interactions
pi flux thru each
triangle
boson hopping
on triangular lattice
Focus on vortices
“Vortex”
Vortex number N=1
Due to frustration,
the dual vortices
are at “halffilling”
“Antivortex”
Vortex number N=0
Dual “magnetic”
field
Dual “electric”
field
Vortex number
Vortex carries
dual gauge charge
J’
J
+

Frustrated spins
vortex creation/annihilation ops:
Halffilled bosonic vortices w/ “electromagnetic” interactions
“Vortex”
vortex hopping
“Antivortex”
Vortices see pi flux
thru each hexagon
~
ChernSimons flux attachment
bosonic vortex
fermionic vortex + 2 flux
E
k
N = 4 flavors
Linearize around
Dirac points
With log vortex interactions can eliminate ChernSimons term
Fourfermion interactions: irrelevant for N>Nc
If Nc>4 then
have a stable:
“Algebraic vortex liquid”
J’
J
AFM J1, and weak FM J2=J3
J2<0
Vortex
duality
J1>0
J3<0
Fluxsmeared mean field: Fermionic
vortices hopping on “decorated”
honeycomb
QED3 with SU(8) Flavor Symmetry
Provided Nc <8will have a stable:
High Tc Cuprates
Doped Mott insulator becomes a
dwave superconductor
Strange metal: Itinerant NonFermi liquid with “Fermi surface”
Pseudogap: Itinerant NonFermi liquid with nodal fermions
Decompose the electron:
spinless charge e boson
and s=1/2 neutral fermionic spinon,
coupled via compact U(1) gauge field
HalfFilling: One boson/site  Mott insulator of bosons
Spinons describes magnetism (Neel order, spin liquid,...)
Dope away from halffilling: Bosons become itinerant
Fermi Liquid: Bosons condense with spinons in Fermi sea
NonFermi Liquid: Bosons form an uncondensed fluid  a “Bose metal”,
with spinons in Fermi sea (say)
O. Motrunich/ MPAF condmat/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 2particle correlations
Goal: Construct timereversal invariant analog of Laughlin,
(with relative dxy 2particle correlations)
Hint: nu=1/2 Laughlin is a determinant squared
p+ip 2body
``Swave” Bose liquid:
square the wavefunction of Fermi sea
wf is nonnegative and has ODLRO  a superfluid
``Dwave” Bose liquid:
Product of 2 different fermi sea determinants,
elongated in the x or y directions
Nodal structure of DBL wavefunction:

+
+

Dxy relative 2particle correlations
power law decay with direction dependent
wavevectors and exponents, the wavevectors
enclose a kspace volume determined by
the total Bose density (Luttinger theorem)
power laws, also with direction dependent
wavevectors and exponents in
both DBL and DLBL
Itinerant nonFermi liquid phase of 2d electrons
Wavefunction:
tK Ring Hamiltonian
(no double occupancy constraint)
4
3
4
3
2
2
1
1
Electron singlet pair
“rotation” term
t >> K Fermi liquid
t ~ K Dmetal (?)
can be accessed using slave particles, vortex duality/fermionization, ...