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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

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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

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

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

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”

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


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 )

2 electrons/cell

  • Spin Peierls(translation symmetry breaking)

2 electrons/cell

Valence Bond (singlet)



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

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

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

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”.

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

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. 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

  • 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

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?

  • 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?

  • 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

  • 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 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

  • Exact mapping from boson to vortex variables.

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

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

  • 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


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


  • “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 !!

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”

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 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)

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)

``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


t-K Ring Hamiltonian

(no double occupancy constraint)









Electron singlet pair

“rotation” term

t >> K Fermi liquid

t ~ 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

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