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
Plaquette Resonance
Resonating Valence Bond “Spin liquid”
Plaquette Resonance
Resonating Valence Bond “Spin liquid”
Plaquette Resonance
Resonating Valence Bond “Spin liquid”
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