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