Quantum Phase Transitions in Magnetic Systems Stefan Wessel Institut für Theoretische Physik III Universität Stuttgart SFB/TR 21 Summer school Blaubeuren 2008
Quantum Phase Transitions
in Magnetic Systems
Stefan Wessel
Institut für Theoretische Physik III
Universität Stuttgart
SFB/TR 21 Summer school Blaubeuren 2008
Outline
� Quantum vs. Thermal Phase Transitions� Critical phenomena� Quantum criticality� Example: Transverse-field Ising model
� Quantum Magnetism� Quantum Heisenberg model� Spin dimers and spin liquids � Magnetic-field-induced BEC of triplons� Pressure-induced QPT� Impurity effects
� Exotic Phases and Criticality� Frustration� Exotic quantum phases� Deconfined quantum critical points
Thermal Phase Transitions
First-Order TransitionsCoexistence of phases at transition temperature TC
Second-Order TransitionsEx: ferromagnetism in iron
Order vanishes continuously
Magnetic order
Order Parameter Fluctuations
Thermodynamic quantity to identify phases� Ferromagnet: magnetization m
Phenomenological description � Ginzburg-Landau theory
Correlation length of
local order parameter
fluctuations around
its mean value: ξ
Critical Phenomena
Long ranged spatial correlations at TC
Characteristic timescale of the fluctuations
CC TTTt /)( −=
υξ
−∝ t
zz t υ
ξτ−
∝∝
Correlations on all length- and time-scales
dynamical critical exponent
Scale Invariance
Power-law behavior of observables
Critical exponents – depend on � symmetry of the order parameter
� dimensionality of the system
� interaction range
Renormalization group approach
Universalityclasses
iron
lnχ
Ising Model
Z2 symmetry universality class
1 ,,
±=−= ∑ ij
ji
iJH σσσ Critical fluctuations
2D: 4/1 1 15 4/7 8/1 0 ====== ηνδγβα
BC kJTT / 269.2==
Relevance of Quantum Mechanics
Near Thermal Transitions
Characteristic energy of large-scale order parameter fluctuations
0/1 0→∝∝ →tz
C tυ
τωh
? TkBC <<ωhyes
no
thermal fluctuationsdominate
Critical fluctuations behave classical close to TC
quantum mechanicsimportant
Critical behavior is of purely classical nature
Quantum Phase Transitions
Varying a non-thermal control parameter� magnetic field, pressure, composition
level crossing:
discontinuous transition
avoided level crossing:
continuous transition
r
r
E E
rPhase A Phase B Phase A Phase B
narrows for largesystems
Continuous QPT
Crr
υξ
−−∝ Crr
z
C
−∝ ξωh
quantum critical point
Phase A Phase B
βCrrm −∝
Characteristicenergy scale (e.g. gap)
(ordered) (disordered)
Phase Diagram near a QCP
crossover to the quantum critical regime:z
CCB rrTk
ν
ω −∝≈ h
Quantum Critical Regime
Extends up to high temperatures
meV 100≈J
KTJTkB 1000 ≈⇒≈
Ex.: Magnetic exchange in cuprates:
Quantum-Classical Mapping
d-dimensionalquantum system
Temperature T
(d+1)-dimensionalclassical system
additional dimension of extend
TkB/1=β
( )TkH
QBeZ
/Tr
−= ∑=C
C CWZ )(
εττε iee HiH −== −− ,
imaginary time-step
…
imaginary time TkB/1=β
What makes QPT special?
� Non-standard effective classical models� topological effects e.g. from Berry-phases� effects to disorder
� Dynamical critical properties cannot directly be extract from the quantum-classical mapping
� No quasi-classical particle description of the excitations inside the quantum critical regime
� New states of matter with strong quantum and thermal fluctuations inside the broad quantum critical temperature regime (e.g. non-Fermi liquids)
→ Talk by A. Muramatsu
� Genuine new time-scale: quantum phase-coherence timeCτ
Phase Coherence Time
Time scale over which the system retains
phase coherence
- quantum effects relevant on scales below
Cr
T
r
Cτ
TkBC /h≈τ
Example: Ising Chain in a
Transverse Field
Chain of (many) coupled qbits
∑∑ −−=i
x
i
ji
z
j
z
i ShSSJH,
ii↓↔↑
align neighborsquantum tunneling
∑∑ −−=i
x
i
ji
z
j
z
i ShSSJH,
( )iiiN
Jh ↓+↑=→→⊗⊗→⊗→>>2
1 ,... :
10
Ground states in simple limits:
NNhJ ↓⊗⊗↓⊗↓↑⊗⊗↑⊗↑>> ... or ,... :
1010
=
01
10
2
hx
iS
LiHoF4
h
Ground State Phase Diagram
N
N
↓⊗⊗↓⊗↓
↑⊗⊗↑⊗↑
...
,...
10
10
Jh /long ranged order
Z2 symmetry breaking
∑∑ −−=i
x
i
ji
z
j
z
i ShSSJH,
z
i
z
i
x
i
x
i
SS
SS
−→
→
N→⊗⊗→⊗→ ...
10
exponential decay of the correlation function
ξ/ji xxz
j
z
i eSS−−
∝
QPT
Quantum Critical Point
Critical decay of the correlation function
4/1
1 :
ji
z
j
z
i
xxSShJ
−≈=
Quantum-classical mapping
→ 2D Ising universality class (z = 1)
Low-Energy Excitations
Quasi-classical particle description
→→→→→→→→→>> :Jh →→→←→→→→
Flipped spinMobile for finite J
Dispersing quasi-particle excitations
Many flipped spin states
( )iii
↓−↑=←2
1
→→→←→→←→
Low-Energy Excitations
A different quasi-particle description
→↑↑↑↑↑↑↑↑↑↑↑↑>> :hJ ↓↓↑↑↑↑↑↓↓↓↓↓
↓↓↑↑↑↑↑↑↓↓↓↓
↓↓↑↑↑↑↑↑↑↓↓↓
↓↓↑↑↑↑↑↑↑↑↓↓
Domain wall Mobile for finite h
Dispersing quasi-particle excitations
Many domain wall states ↑↑↑↑↑↑↑↑↓↓↓↓↑
Phase Diagram
Jh /
↓↓↑↑↑↑↑↓↓↓↓↓ →→→←→→→→
flipped spinquasi-particles
domain wallquasi-particles
de-Broglie wavelength≈
particle spacing
TTkBC /h≈τ
Universal response function inside the quantum critical regime
)/(),0( 4/7 TkTk Bωωχ hΦ== −
scalingT/ω
Outline
� Quantum vs. Thermal Phase Transitions� Critical phenomena� Quantum criticality� Example: Transverse-field Ising model
� Quantum Magnetism� Quantum Heisenberg model� Spin dimers and spin liquids � Magnetic-field-induced BEC of triplons� Pressure-induced QPT� Impurity effects
� Exotic Phases and Criticality � Frustration� Exotic quantum phases� Deconfined quantum critical points
Quantum Antiferromagnet
La2CuO4
Cu2+
spin-1/2
Band theorie: half filling → metall
High-TC Superconductor
La2-xSrxCuO4
Tem
perature
Doping x
Superconductor
Antiferromagnet
300 K
40 K
5% 30%
Strange metal
Sr
0
→ Talk by A. Muramatsu
Hubbard ModelScreened
coulomb
repulsionMobile electrons +
UU
t
t
Bosonic version → Talk by HP. Büchler
UtU >>
Effective spin model ?
Half
filling
Kinetic Exchange
t t
U
Anti-parallel
U
tE
2
−∝∆→ energy gain
blocked
Pauli principle
Parallel
Quantum Heisenberg
Antiferromagnet
04 2
>=U
tJ
Antiferromagnetic exchange
= ( - )/√2
Two spins bound in a spin singlet state
…at low temperaturs?
J
Antiferromagnetic order…
Mermin-Wagner Theorem
No spontaneous breaking of
a continuous symmetry
at finite temperatures
→ order only in the ground state (T= 0)
spin-rotation symmetry
broken
NéelENéelH 0≠
Not An Eigenstate
Staggered Moment
)0()()( SrSrC ⋅=rr
quantum fluctuations→ m ≈ 0.3
)(rCr
0
0.25
2m
r
rr
Correlation function
Excitations
Spin waves (Goldstone modes)
Spectrum accessible via neutron scattering
FM AFM
Co0.92Fe0.08 MnF3
How to destroy the AF order?
• Enhance quantum fluctuations
• Frustration
• Both !
JJm
Jm≥ J
Bond Modulations
Weak ModulationsJm ≈ J
antiferromagnetic order
Strong Modulations→ Singlets
Jm » J
= ( - )/√2
Spin Liquid
)0()()( SrSrCvvvr
⋅=
No long range magnetic order
Finite correlation length
r
ξ
)(rCr
0
0.25
ξ/re−∝
Spin Liquid
Finite energy gap to magnetic excitations
(spin gap)
- breaking up a singlet
triplet
→ triplons
Phase DiagramT
Jm / J1 ≈1.9
quantum critical point
quantum criticial
regime
0
thermallydisordered
quantum disordered
Quantum Critical Scaling
Uniform susceptibility of the uniform Heisenberg model
T-linearscaling inthe quantumcritical regime
Kim, Troyer PRL (98)
Scalingin the low-Tthermallydisorderedregime
Universality Class of the QCP
Quantum-classical mapping:
QPT is in the 3D O(3) universality class
Recent Quantum Monte Carlo results call this into question:
3D O(3) 3D O(3)????
Topological effect from Berry-phases in the quantum action?
Experimental Realization
TlCuCl3 – a 3D array of coupled spin dimers
Spin gap ≈ 0.8 meV ≈ 8 K
Triplon Excitations
Neutron scattering
N. Cavadini et al., PRB 2001
Magnetic Field Driven QPT
hoppinginteractions
Isolated dimer
h
Energy ↓↓
( )↓↑+↑↓2
1
( )↓↑−↑↓2
1
↑↑
Coupled dimers
Magnetic Field Driven QPT
hoppinginteractions
Isolated dimer
wave vector
ω
Coupled dimers
Magnetic Phase Diagram
BEC of Triplons
Magnetization curves
T=0
BEC of Triplons
Magnetic Order and
ExcitationsNeutron scattering
Goldstone
mode
ordered
moment
C. Ruegg et al., Nature (2003)
Pressure Driven QPT
� Pressure modifies the exchange constantsInter-dimer exchange enhanced
� Can drive a pressure induced QPT to an ordered state
TlCuCl3
C. Ruegg et al., PRL (2004)
Excitations
� Gapped longitudinal mode emerges in the ordered phase� Amplitude modulations
Gap
C. Ruegg et al., PRL (2008)
Impurities in Spin Liquids
„dangling spins“
weak
couplingCu2+ → Mg2+
Order by Disorderinduced AF order
impurity
concentrationpercolation
treshold
0
only finite clusters
Oosawa et al.,PRB (2002)
Outline
� Quantum vs. Thermal Phase Transitions� Critical phenomena� Quantum criticality� Example: Transverse-field Ising model
� Quantum Magnetism� Quantum Heisenberg model� Spin dimers and spin liquids � Magnetic-field-induced BEC of triplons� Pressure-induced QPT� Impurity effects
� Exotic Phases and Criticality� Frustration� Exotic quantum phases� Deconfined quantum critical points
J
JJ
Frustrated Quantum Spins
� Triangular lattice: Long range order
survives quantum fluctuations (in theory…)
: no long range order down to 5mK
Kagome Lattice
ZnCu3(OH)6Cl2
No long-range orderNo apparent spin gap
What is the nature of themagnetic ground state?
Dimensional Reduction
at a QCP - driven by frustration
Han purple pigment
S. Sebastianet al.,Nature 2006
SrCu2(BO3)2
Another Prominent Example
spin-1/22+
2-
2+
3+
K. Kodama, et al., Science (2002)
Orthogonal Spin Dimers
Strong quantum fluctuations + frustration
JD / J ≈ 1.57
J
JD
Exact Ground State
Singlet product state: ( )∏ ↓↑−↑↓=Ψ dimers
0 2
1
Magnetic Excitations
spin gaplokalized
tripletsK 35meV 5.3 ≈≈
magnetic
susceptiblity
TkBe/∆−∝KkB 35/ ≈∆
Fractional Magnetization
Plateaus
Mott insulator of magnetic excitations
Magnetic super structureat m/ms=1/8
Quantum Phase Diagram
Maximum frustration
Néel singlets
JD / J?1 1.5
New Quantum Phase
Maximum frustration
Valence Bond CrystalNéel Singlets
JD / J
Plaquette - ordering
SrCu2(BO3)2
1.471.4
Valence Bond Crystals
� Realized in a number of compounds
Spontaneous breaking of thelattice symmetry
2D Spin-Peierls transition
Continuous Quantum Phase
Transition?
Valence Bond CrystalNéel
?
Spin rotation symmetry
spontaneously brokenLattice symmetry
spontaneously broken
JD / J
order
Landau-Ginzburg Theory: generically first-order transition
Effective Quantum Field
Theory
Berry-phase
)(ˆ τrn
rA
3D Nonlinear sigma-model + Berry-phases
Fluktuations around the Néel-state
Controlls the strength of the fluctuations
Spin-quantization
Configuration of the Vector
Field
Skyrmion - Configuration
Skyrmion number (topological invariant)
= 1
Hedgehogs
Change the configuration’s skyrmion number
Give rise to the Valence Bond Solid order
time
Continuous QPT
Anti-
ferromagnet
Valence
Bond
Crystal
S=1
S=1/2S=1/2
Relevance of hedgehog excitations
Deconfinement
of spin exciations
over large scales
Quantum critical point
Deconfined QCP
Exotic critical exponents
S=1
S=1/2S=1/2
Deconfinement at the
critical point
New paradigma for
quantum phase transitions?
→ currently being investigated
(thus far not conclusive ...)
Doping With Mobile Charges
FL
strange metal
pseudogap
Doping
Temperature
d-SCAF
Phase diagram of the High-TC materials
→ Talk by A. Muramatsu
Further Readings
� S. Sachdev: Quantum Phase Transitions
Cambridge Univ. Press (2002)
� M. Voijta: Quantum Phase Transitions
Prep. Prog. Phys. 66 (2003) 2069-2110
� S. Sachdev: Quantum magnetism and criticality
Nature Physics, March 2008
� T. Giamarchi, C. Rüegg, and O. Tchernyshvov:
Bose-Einstein condensation in magnetic insulators
Nature Physics, March 2008
FF
Seltsames Metall
Pseudogap
Dotierung
Temperatur
SupraleiterAF
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