Geometric Frustration in Magnetism and Ice
Frustrated Magnetism
Ashvin Vishwanath
UC Berkeley
Acknowledgements
:
Fa Wang (MIT),
Frank Pollman
(Dresden), Arun Paramekanti (Toronto), Roger Melko &
Anton Burkov (Waterloo), Donna Sheng (Northridge),
Leon Balents (KITP).
Preview – Quantum Spin Liquids
Recent development:
An electrical insulator, which conducts heat like a metal!
A triangular
lattice magnet
Magnetic Insulators
•
Mott Insulators – Coulomb repulsion localizes
electrons to atomic sites. Only spin degree of
freedom.
U
e
-
t
Mott Insulator: U>t
Typically, oxides of
transition metals
(Fe, Co, Ni, Mn, Cu
etc.)
U
Magnetic Insulators
•
Only spin degree of freedom. Simplest quantum many
body system!
•
Typically, spins order :
•
Except in frustrated systems
•
Example: La
2
Cu
O
4
(parent compound of cuprates)
S=1/2 square lattice anti-ferromagnet
e
-
t
Opposite Spins gain by virtual hopping:
J≈t
2
/U;
H = J S
1
•S
2
.
J ≈
1,000Kelvin
;
t,U ≈10,000 Kelvin
Geometric Frustration
•
Frustration ~ cannot optimize all energetic requirements
simultaneously.
Triangular Lattice:
Kagome Lattice:
#of ground states= (N
sites)
(Classical Spin)
Accidental degeneracy of classical ground states
Geometric Frustration
•
Today, mainly Ising Spins (S=+1, -1)
Note, Heisenberg spins:
Triangular Lattice:
(now
, unfrustrated
–
unique ground state, upto
symmetries)
Kagome Lattice:
Frustrated ((spins
on a triangle must
sum to zero)
Pyrochlore Lattice:
Frustrated (spins on a
tetrahedron must sum to
zero)
Geometrical Frustration in Ice
Residual Entropy of Ice (1936):
Third law of Thermodynamics: S
0
→0
Geometrical Frustration in Ice
Pauling’s Solution:
Structure
:
Bernal Fowler Rules
– 2
H
near, 2
H
far.
O
xygen
H
ydrogen
Alternate
H
site
Idea:
Hydrogens
remain disordered giving entropy
Spin version
: Spin ice
(Ho
2
Ti
2
O
7
)
Ho
Outline
1. Frustration relief by residual
interactions
here
- complex orders from lattice
couplings in CuFeO
2
2. Selection by quantum/thermal
fluctuations
here –
‘supersolid’ order by disorder.
Geometric Frustration & Relief
Classically degenerate
Complex orders - but defined by Landau order parameter.
Outline
3. Quantum spin liquids.
No Landau order parameter.
Alternate descriptions?
Depends on the spin liquid!
eg. Topological order OR
Gapless excitations unrelated to symmetry
Gauge theories emerge
(Tomorrow)
Geometric Frustration & Relief
Strong tunneling
Frustration and Relief - Lattice coupling
Restoring force
Change in
J
Tchernyshyov et al., Penc et al., Bergmann et al. for pyrochlore
Seen in Triangular lattice material
CuFeO
2
Optimal Configuration?
Z state
Z
igzag stripes
Fa Wang and AV, PRL (08)
.
Magnetization Plateaus and CuFeO
2
h/J
c=0.15, D
z
=0.01J
m
1
2
3
Z state
Good agreement with known phases of
Triangular magnet
CuFeO
2.
Prediction for 1/3 plateau structure.
Spin-Phonon model:
Fa Wang and AV, PRL 2008
Complex Phase Structure of Triangular Spin Phonon Model
•
Violations of
“Gibbs Phase
Rule”
(4 phases
meet at a point –
due to
frustration)
•
E
1
- E
2
= E
2
- E
3
= 0
(requires 2 parameters)
•
E
4
- E
3
= E
4
– E
1
= 0
(accidental degeneracy –
frustration)
•
Also
, on Kagome
Spin-phonon Phase Diagram in a field
(numerical Monte Carlo – simulated annealing)
XXZ Model on triangular lattice
•
Anderson and Fazekas (1973) [Resonating Valence
Bond – spin liquid proposal]. Highly anisotropic
triangular lattice antiferromagnet:
•
Project into Ising ground state manifold. Treat
J
as a
perturbation.
Groundstates:
Hardcore
Dimers on
the Honeycomb
lattice (2 to 1 map)
XXZ Model on triangular lattice
•
Anderson-Fazekas proposal:
Ground state superposition of
different dimer coverings (Resonanting Valence Bonds - RVB)
Quantum Dynamics from
J
term
J
Note, J>0 has a sign problem
•
Consider J<0, no sign problem! Solve for ground state.
•
Map J>0 to J<0 problem, in Hilbert space of Ising ground states.
Solve model for J<0;
more naturally viewed as a boson model with hopping
t=-2J
Spin-Boson Mapping
Spin-boson mapping:
Repulsion
Quantum Fluctuation:
Boson hopping
Bosons on the Triangular lattice:
t=0 highly frustrated
Physical Realization:
Ultra-cold Dipolar
Atoms in an optical lattice?
Supersolid order on the triangular lattice
Case 2. If
J
z
>>t
Case 1. If
t >>J
z
uniform superfluid
Melko, Paramekanti, Burkov, A.V., Sheng and Balents, PRL 06; Haiderian and Damle; Wessel and Troyer
Expect a solid. Charge order
(m)
No sign problem – large system sizes can be studied with Quantum Monte Carlo
Supersolid order on the triangular lattice
Case 2. If
J
z
>>t
Case 1. If
t >>J
z
uniform superfluid
J
z
/t
9
superfluid
lattice
super
solid
(Quantum Monte Carlo)
AND
superfluid (
ρ
s
)
high &
low
density
Charge order
(m)
t=1/2
Triangular Lattice bosons with
Frustrated
Hopping
•
However,
in the limit
J
z
>>t
,
a unitary transformation exists – that reverses
the sign. Only works in the space of Ising ground states (dimer states) –
projector P.
+
XXZ antiferromagnet on the triangular lattice
Sign Problem
– cannot use Quantum Monte Carlo.
Fa Wang, Frank Pollman, AV, PRL 09.
&
D. Sheng et al. PRB 09.
Triangular Lattice XXZ Model at J
z
>>J
•
Unitary transformation:
–
Diagonal in S
z
basis.
–
Thermodynamics of +t and –t models identical
Only in the limit
J
z
>>t
(ground state sector of Ising antiferromagnet)
Triangular Lattice bosons with
Frustrated
Hopping
Immediate Implications for +t
–
Ground state also has same solid order (U diagonal in density/S
z
basis)
as -t
–
Same superfluid density at +t as –t (free energy with vector potential:
F[A] identical)
–
Also a
SuperSolid
•
Nature of Supersolid order: obtained using a variational
wavefunction approach
(A. Sen et al. PRL (08), for the unfrustrated
case)
–
Excellent energetics/correlations for unfrustrated case.
•
Correlation functions evaluated using Grassman techniques
invented to solve 2D dimer stat-mech.
Phase Diagram of the XXZ Antiferromagnet
•
Ordering pattern obtained near t/J
z
=0. If we include
t/J
z
=-1/2 (Heisenberg point) with 120
o
order. Can be
connected smoothly.
•
Anisotropic XXZ S=1/2 Heisenberg magnet is ordered
– deformed 120
o
order. Not a spin liquid.
XXZ antiferromagnet
Connection to Lattice Gauge Theories
•
Hardcore Dimer model (on bond
ij
)
•
On a
bipartite
lattice:
–
Define a vector field
e
Realized quantum electrodynamics on D=2 Lattice. Also called U(1) gauge theory.
Gauss Law
Confinement and Spin Liquid Phase
•
Lattice gauge theories – two possible phases:
–
Confined phase – electric fields frozen
. (magnetic order)
–
Coulomb phase (gapless photon excitation as in Maxwell’s
electrodynamics)
(Spin liquid)
•
Remarkable general result
(A. Polyakov)
–
In D=2, lattice electrodynamics (U(1) gauge theory), has
only one phase – confined phase.
–
Spin liquid very unlikely in Anderson-Fazekas model.
Beating confinement
•
To obtain deconfinement
–
Consider other gauge groups like Z
2
(eg. non-bipartite
dimer models)
–
Go to D=3 [spin ice related models]
–
Add other excitations. [deconfined critical points, critical
spin liquids]
References:
J. Kogut, “Introduction to Lattice Gauge Theories and Spin
Systems” RMP, Vol 51, 659 (1979).
S. Sachdev, “Quantum phases and phase transitions of Mott
insulators “, page 15-29
[mapping spin models to gauge theories]
We will discuss each of these tomorrow
Explore the concept of geometric frustration in magnetism and ice, where energetic requirements cannot be optimized simultaneously. Learn about frustrated magnetic insulators, quantum spin liquids, and the unique properties of geometrically frustrated systems such as triangular and Kagome lattices. Delve into the interplay of spin degrees of freedom and geometric arrangements that give rise to complex behaviors in these materials.
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Presentation Transcript
Frustrated Magnetism Ashvin Vishwanath UC Berkeley Acknowledgements: Fa Wang (MIT), Frank Pollman (Dresden), Arun Paramekanti (Toronto), Roger Melko & Anton Burkov (Waterloo), Donna Sheng (Northridge), Leon Balents (KITP).
Preview Quantum Spin Liquids A triangular lattice magnet Recent development: An electrical insulator, which conducts heat like a metal!
Magnetic Insulators Mott Insulators Coulomb repulsion localizes electrons to atomic sites. Only spin degree of freedom. t e- Mott Insulator: U>t U U Typically, oxides of transition metals (Fe, Co, Ni, Mn, Cu etc.)
Magnetic Insulators Only spin degree of freedom. Simplest quantum many body system! t e- Opposite Spins gain by virtual hopping: J t2/U; H = J S1 S2 Typically, spins order : Except in frustrated systems T J Example: La2CuO4 (parent compound of cuprates) S=1/2 square lattice anti-ferromagnet .J 1,000Kelvin; t,U 10,000 Kelvin
Geometric Frustration Frustration ~ cannot optimize all energetic requirements simultaneously. neighbors = E J S S i j , i j = , 0 1 J S (Classical Spin) i N . 0 502 e Kagome Lattice: Triangular Lattice: #of ground states= (N sites) N . 0 323 e Accidental degeneracy of classical ground states
Geometric Frustration Today, mainly Ising Spins (S=+1, -1) neighbors = H J S iS Note, Heisenberg spins: j , j i Triangular Lattice: Kagome Lattice: Pyrochlore Lattice: Frustrated ((spins on a triangle must sum to zero) Frustrated (spins on a tetrahedron must sum to zero) (now, unfrustrated unique ground state, upto symmetries) J = + + 2 ( ) H S S S 1 2 3 2 triangles
Geometrical Frustration in Ice T C 0 T = ( ) ( ) S T S T dT 0 T cal/molo = K . 0 82 . 0 05 S Residual Entropy of Ice (1936): 0 Third law of Thermodynamics: S0 0
Geometrical Frustration in Ice Pauling s Solution: Bernal Fowler Rules 2 H near, 2 H far. chemistryicevoet2Structure: Oxygen = log S B k Hydrogen Alternate H site 0 cal/molo = K . 0 806 S Pauling Idea: Hydrogens remain disordered giving entropy Spin version: Spin ice (Ho2Ti2O7) Ho
Geometric Frustration & Relief Outline 1. Frustration relief by residual interactions here - complex orders from lattice couplings in CuFeO2 Classically degenerate 2. Selection by quantum/thermal fluctuations here supersolid order by disorder. Complex orders - but defined by Landau order parameter.
Geometric Frustration & Relief Outline 3. Quantum spin liquids. No Landau order parameter. Strong tunneling Alternate descriptions? Depends on the spin liquid! eg. Topological order OR Gapless excitations unrelated to symmetry Gauge theories emerge (Tomorrow)
Frustration and Relief - Lattice coupling j nbrs = + ( [ ]) E J J r r S S r i j i j , i i 2 + K r i Tchernyshyov et al., Penc et al., Bergmann et al. for pyrochlore Restoring force Change in J Seen in Triangular lattice material CuFeO2 Optimal Configuration? Z state Zigzag stripes Fa Wang and AV, PRL (08).
Magnetization Plateaus and CuFeO2 Spin-Phonon model: 2 i = S S F H J c i j ij i 2 2 JS 2 = = e F S S c i ij i j K m j 1 3 Good agreement with known phases of Triangular magnetCuFeO2. 1 5 Prediction for 1/3 plateau structure. h/J 2 3 1 c=0.15, Dz=0.01J Z state Fa Wang and AV, PRL 2008
Complex Phase Structure of Triangular Spin Phonon Model Violations of Gibbs Phase Rule (4 phases meet at a point due to frustration) E1 - E2= E2 - E3= 0 (requires 2 parameters) E4 - E3= E4 E1= 0 (accidental degeneracy frustration) Also, on Kagome Spin-phonon Phase Diagram in a field (numerical Monte Carlo simulated annealing)
XXZ Model on triangular lattice Anderson and Fazekas (1973) [Resonating Valence Bond spin liquid proposal]. Highly anisotropic triangular lattice antiferromagnet: S S J H + = J ( ) + z z x x y y J S S S S z i j i j i j ij ij J z Project into Ising ground state manifold. Treat J as a perturbation. Groundstates: Hardcore Dimers on the Honeycomb lattice (2 to 1 map)
XXZ Model on triangular lattice J Quantum Dynamics from J term Anderson-Fazekas proposal: Ground state superposition of different dimer coverings (Resonanting Valence Bonds - RVB) Note, J>0 has a sign problem Consider J<0, no sign problem! Solve for ground state. Map J>0 to J<0 problem, in Hilbert space of Ising ground states. Solve model for J<0; more naturally viewed as a boson model with hopping t=-2J
Spin-Boson Mapping 1 1 nbr j , nbr j , Spin-boson mapping: z i z j = H J J ( )( ) S S n n Repulsion z z i j 2 2 n = n = 1 0 z = 2 / 1 + i i S * j * i x i x j y y j Quantum Fluctuation: Boson hopping + + b b b b S S S S z = 2 / 1 S i j i Physical Realization: Ultra-cold Dipolar Atoms in an optical lattice? Bosons on the Triangular lattice: = ij * j * i + t b b b b H i j ij + ) 2 / 1 ) 2 / 1 J (n (n z i j t=0 highly frustrated
Supersolid order on the triangular lattice ij ij * j * i = + + ) 2 / 1 ) 2 / 1 t - b b b b J (n (n H i j z i j Case 1. If t >>Jz uniform superfluid Case 2. If Jz >>t Expect a solid. Charge order (m) No sign problem large system sizes can be studied with Quantum Monte Carlo Melko, Paramekanti, Burkov, A.V., Sheng and Balents, PRL 06; Haiderian and Damle; Wessel and Troyer
Supersolid order on the triangular lattice ij ij * j * i = + + ) 2 / 1 ) 2 / 1 t b b b b J (n (n H i j z i j Case 1. If t >>Jz uniform superfluid s Case 2. If Jz >>t Charge order (m) AND superfluid ( s) high & low density 2 m superfluid lattice supersolid Jz/t (Quantum Monte Carlo) t=1/2 9
Triangular Lattice bosons with Frustrated Hopping + = + + ) 2 / 1 ) 2 / 1 * j * i t - b b b b J (n (n H i j z i j ij ij XXZ antiferromagnet on the triangular lattice Sign Problem cannot use Quantum Monte Carlo. However, in the limitJz >>t , a unitary transformation exists that reverses the sign. Only works in the space of Ising ground states (dimer states) projector P. ij * j * i = + + t b b b b ( ) H P P i j ij + * j * i = + U U t b b b b ( ) H P P i j Fa Wang, Frank Pollman, AV, PRL 09. & D. Sheng et al. PRB 09.
Triangular Lattice XXZ Model at Jz>>J Unitary transformation: N = U dimer dimer i Diagonal in Sz basis. Thermodynamics of +t and t models identical brown Only in the limitJz >>t (ground state sector of Ising antiferromagnet)
Triangular Lattice bosons with Frustrated Hopping Immediate Implications for +t Ground state also has same solid order (U diagonal in density/Sz basis) as -t Same superfluid density at +t as t (free energy with vector potential: F[A] identical) Also a SuperSolid Nature of Supersolid order: obtained using a variational wavefunction approach (A. Sen et al. PRL (08), for the unfrustrated case) Excellent energetics/correlations for unfrustrated case. V[dimer] - = e dimer | | = 2 | ' | U dimer Correlation functions evaluated using Grassman techniques invented to solve 2D dimer stat-mech.
Phase Diagram of the XXZ Antiferromagnet XXZ antiferromagnet J z Ordering pattern obtained near t/Jz=0. If we include t/Jz=-1/2 (Heisenberg point) with 120o order. Can be connected smoothly. Anisotropic XXZ S=1/2 Heisenberg magnet is ordered deformed 120o order. Not a spin liquid.
Connection to Lattice Gauge Theories Hardcore Dimer model (on bond ij) ; 1 , 0 = j = 1 (Constrain t) n n ij ij On a bipartite lattice: Define a vector field e j ) B = = e ( / ); e 1 (Constrain t) n i A B ij ij ij = div e 1 ( / i A Gauss Law i Realized quantum electrodynamics on D=2 Lattice. Also called U(1) gauge theory.
Confinement and Spin Liquid Phase Lattice gauge theories two possible phases: Confined phase electric fields frozen. (magnetic order) Coulomb phase (gapless photon excitation as in Maxwell s electrodynamics) (Spin liquid) Remarkable general result (A. Polyakov) In D=2, lattice electrodynamics (U(1) gauge theory), has only one phase confined phase. Spin liquid very unlikely in Anderson-Fazekas model.
Beating confinement To obtain deconfinement Consider other gauge groups like Z2 (eg. non-bipartite dimer models) Go to D=3 [spin ice related models] Add other excitations. [deconfined critical points, critical spin liquids] We will discuss each of these tomorrow References: J. Kogut, Introduction to Lattice Gauge Theories and Spin Systems RMP, Vol 51, 659 (1979). S. Sachdev, Quantum phases and phase transitions of Mott insulators , page 15-29 [mapping spin models to gauge theories]
