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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.

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Exotic phases in quantum magnets l.jpg

Exotic Phases in Quantum Magnets

MPA Fisher

KITPC, 7/18/07

Interest: Novel Electronic phases of Mott insulators


  • 2d Spin liquids: 2 Classes

  • Topological Spin liquids

  • Critical Spin liquids

  • Doped Mott insulators: Conducting Non-Fermi liquids

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Quantum theory of solids: Standard Paradigm

Landau Fermi Liquid Theory


Free Fermions


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

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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

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 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


  • 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”

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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


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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

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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 )

2 electrons/cell

  • Spin Peierls(translation symmetry breaking)

2 electrons/cell

Valence Bond (singlet)


How to suppress order i e symmetry breaking l.jpg


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!

“almost” AFM order:

S(r)·S(0) ~ (-1) r/ r2

  • Geometric Frustration

    • Triangular lattice

    • Kagome lattice

  • Doping (eg. Hi-Tc): Conducting Non-Fermi liquids

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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”

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Spin-liquids: 2 Classes

RVB state (Anderson)

  • Topological Spin liquids

    • Topological degeneracy

      Ground state degeneracy on torus

    • Short-range correlations

    • Gapped local excitations

    • Particles with fractional quantum numbers




  • 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

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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

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Plaquette Resonance

Resonating Valence Bond “Spin liquid”

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Plaquette Resonance

Resonating Valence Bond “Spin liquid”

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Plaquette Resonance

Resonating Valence Bond “Spin liquid”

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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

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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”.

J 1 j 2 j 3 kagome s 1 2 in easy axis limit topological spin liquid ground state z 2 l.jpg
J1=J2=J3 Kagome s=1/2 in easy-axis limit: Topological spin liquid ground state (Z2)




For Jz >> Jxy have 3-up and 3-down

spins on each hexagon. Perturb in Jxy

projecting into subspace to get ring model

J 1 j 2 j 3 kagome s 1 2 in easy axis limit topological spin liquid ground state z 218 l.jpg
J1=J2=J3 Kagome s=1/2 in easy-axis limit: Topological spin liquid ground state (Z2)




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.jpg
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 l.jpg
Other models with topologically ordered spin liquid phases

(a partial list)

  • Quantum dimer models

  • Rotor boson models

  • Honeycomb “Kitaev” model

  • 3d Pyrochlore antiferromagnet

Moessner, Sondhi

Misguich et al

Motrunich, Senthil


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.

Critical spin liquids l.jpg
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:


Frustration parameter:

Triangular lattice critical spin liquids l.jpg
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


Spin liquid?




Kagome lattice critical spin liquids l.jpg
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 SrCr8Ga4O19f ~ 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)6Cl2f > 600

    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

Theoretical approaches to critical spin liquids l.jpg
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 l.jpg



Triangular/Kagome s=1/2 XY AF equivalent to bosons in “magnetic field”

boson interactions

pi flux thru each


boson hopping

on triangular lattice

Focus on vortices


Vortex number N=1

Due to frustration,

the dual vortices

are at “half-filling”


Vortex number N=0

Boson vortex duality l.jpg

Boson-Vortex Duality

Dual “magnetic”


Dual “electric”


Vortex number

Vortex carries

dual gauge charge

  • All non-locality is accounted for by dual U(1) gauge force

Duality for triangular afm l.jpg





Duality for triangular AFM

Frustrated spins

vortex creation/annihilation ops:

Half-filled bosonic vortices w/ “electromagnetic” interactions


vortex hopping


Vortices see pi flux

thru each hexagon

Chern simons flux attachment fermionic vortices l.jpg


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


  • Band structure: 4 Dirac points


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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)

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Fermionized Vortices for easy-plane Kagome AFM


  • “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






Flux-smeared mean field: Fermionic

vortices hopping on “decorated”


Vortex band structure n 8 dirac nodes l.jpg
Vortex Band Structure:N=8 Dirac Nodes !!

QED3 with SU(8) Flavor Symmetry

Provided Nc <8will 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”

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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 l.jpg
Slave Particle approach toitinerant 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 l.jpg
Uncondensed quantum fluid of bosons:D-wave Bose Liquid (DBL)

O. Motrunich/ MPAF cond-mat/0703261


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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
D-Wave Metal

Itinerant non-Fermi liquid phase of 2d electrons


t-K Ring Hamiltonian

(no double occupancy constraint)









Electron singlet pair

“rotation” term

t >> K Fermi liquid

t ~ K D-metal (?)

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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