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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