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Phase Transitions in Condensed MatterSpontaneous Symmetry Breaking and Universality
Hans-Henning Klauss
Institut für FestkörperphysikTU Dresden
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References[1] Stephen Blundell, Magnetism in Condensed Matter, Oxford University Press
[2] Igot Herbut, A Modern Approach to Critical Phenomena, Cambridge University Press
[3] Eugene Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford Science Pub.
[4] Roser Valenti, Lecture Notes on Thermodynamics, U Frankfurt
[5] Matthias Vojta, Lecture Notes on Thermal and Quantum Phase Transitions, Les Houches 2015
[6] Thomas Palstra, Lecture Notes on Multiferroics: Materials and Mechanisms, Zuoz 2013
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Outline• Phase transitions in fluids
- Phase diagram, order parameter and symmetry breaking
- Microscopic van-der-Waals theory universality
• Magnetic phase transitions in condensed matter
- Ferromagnetic phase transition
- Interacting magnetic dipole moments “spins”
- Weiss model for ferromagnetism, phase diagram
- Landau theory
• Consequences of symmetry breaking
- Critical phenomena and universality
- Excitations, Nambu-Goldstone-, Higgs-modes
• More complex ordering phenomena
- Multiferroics, competing order
- [Quantum phase transitions]
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• What is a thermodynamic phase?
- Equilibrium state of matter of a many body system
- Well defined symmetry
- Thermodynamic potential changesanalytically for small parameter changes(temperature, pressure, magnetic field)
• What is a phase transition?
- Point in parameter space where the equilibrium properties of a system change qualitatively.
- The system is unstable w.r.t. small changes of external parameters
Introduction
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Introduction
• What is a thermodynamic phase?
- Equilibrium state of matter (many body system)
- Well defined symmetry
- Thermodynamic potentials changeanalytically for small parameter changes(temperature, pressure, magnetic field)
• What is a phase transition?
- Point in parameter space where the equilibrium properties of a system change qualitatively.
- The system is unstable w.r.t. small changes of external parameters
Phase diagram of water
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Introduction
• Many, many phase diagrams in nature….
Mathur et al., Nature 1998
Fernandes et al., Nature Phys. 2014
structural phases electronic phases
combined electronic and structuralelectronic
Luetkenset al., Nature Mat. 2009
wikipedia
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Introduction
liquid -- solid
liquid -- gas
• What is an order parameter?
- observable f which distiguishes between phases
< f > = 0 in the disordered phase (high temperature phase)
≠ 0 in the ordered phase
fourier component of charge density rG
paramagnet -- ferromagnetBose-Einstein condensation
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Phase Transitions in Fluids
• What is an order parameter?
- observable f which distiguishes between phases
< f > = 0 in the disordered phase
≠ 0 in the ordered phase
liquid gas
water
fourier component of charge density rG
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Phase Transitions in Fluids
• First order transitionorder parameter changes discontinuously
(Ehrenfest definition: first derivative of Gibbs free enthalpy G is discontinuous)
• Continuous transitionorder parameter varies continuously
• Critical pointtransition point of a continuous transition
for water @ 647 K and 22.064 MPa
liquid to gas phase transitioncan be of first order or continuous!
liquid -- gas
water
Phase diagram of water
••
[1]
[google]
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Microscopic Model
• Van- der-Waals-model:
Attractive particle – particle interactionfully rotational invariant V = V(r)
+Finite particle volume
van-der-Waals equation
• Maxwell construction isotherms andphase coexistencebelow TC
• Universal mixed liquid vapor regionin p-V diagram for many materials
via normalization p/pc and V/Vc!
[2]
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• Spontaneous symmetry breaking always leads to a phase transition
no critical end point since symmetry cannot change continuously!Example: solid – liquid phase transition (path A)
• Phase transitions can occur without spontaneous symmetry breaking
Example liquid – gas phase transition (path B)
• Continuous crossoverfrom liquid to gas without phase transitionvia supercritical fluid (path C)
Spontaneous Symmetry Breaking and Phase Transitions
supercritical fluid
[1]
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Outline• Phase transitions in fluids
- Phase diagram, order parameter and symmetry breaking
- Microscopic van-der-Waals theory universality
• Magnetic phase transitions in condensed matter
- Paramagnet
- Interacting magnetic dipole moments, “spins”
- Weiss model for ferromagnetism universality
- Landau theory
• Consequences of symmetry breaking
- Critical phenomena and universality
- Excitations, Nambu-Goldstone-, Higgs-modes
• More complex ordering phenomena
- Multiferroics, competing order
- [Quantum phase transitions]
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Paramagnet
• Periodic lattice of localized non-interacting magnetic moments
� = gL µB �
• in external field �� =gL µB ∑ ���� Brillouin function
• Magnetic susceptibility :
Curie-law for small B ( << 1)
[1]
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• Interacting magnetic moments:
Exchange term + Zeeman term
• Origin of exchange: spin-dependent Coulomb interaction
e.g. superexchange
Ferromagnet
[1]
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Weiss (1907):
Define an effective magnetic field at site i caused by neighbors j
„molecular field“
single particle problem
Weiss-Model
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Ansatz: Bmf ~ Magnetization M
with
Two linear independent equations
BJ = Brillouin function
Magnetization of a paramagnet
in total magnetic field B +lM
Weiss-Model
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Graphical Solution M=0 always possible
M ≠ 0 only if T < TC
TC ~ exchange energy J,number of neighborssize of moments
Typical:
Continuous phase transition
Huge!
Weiss-Model
[1]
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Graphical Solution M=0 always possible
M ≠ 0 only if T < TC
TC ~ exchange energy J,number of neighborssize of moments
• Universal T dependenceof order parameter
• depends only on total angular momentummultiplicity J
Continuous phase transition
Weiss-Model
[1]
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Spontaneous Symmetry Breaking
• Hamiltonian has full rotational symmetry in space(scalar product is invariant)
• Ferromagnetic state has a reduced symmetry(invariant only under rotation around M)
� =∑μ��
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crf Weiss theory:
Landau Theory of Ferromagnetism
[1]
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In magnetic field: Magnetization parallel to field is always > 0 No phase transition !
Explicit symmetry breaking due to Zeeman term
��
F
BM < 0 metastable solution exists for small B only First order transition below TC as a function of external field
Weiss and Landau Theory of Ferromagnetism
[1][1]
[Web]
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Solution in magnetic field: Magnetization parallel to field always > 0 No phase transition !
��
M
BM < 0 metastable solution exists for small B only First order transition below TC as a function of external field
Weiss and Landau Theory of Ferromagnetism
[1]
[Web]
Explicit symmetry breaking due to Zeeman term
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for fluid
Comparison of fluid and magnet phase diagrams
for magnet
Sometimes density r = N m / V used
Gibbs free energy
G = U –TS + p V
Gibbs free energy
G = U –TS – M B
[3,5]
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Outline• Phase transitions in fluids
- Phase diagram, order parameter and symmetry breaking
- Microscopic van-der-Waals theory universality
• Magnetic phase transitions in condensed matter
- Paramagnet
- Interacting magnetic dipole moments, “spins”
- Weiss model for ferromagnetism universality
- Landau theory
• Consequences of symmetry breaking
- Critical phenomena and universality
- Excitations, Nambu-Goldstone-, Higgs-modes
• More complex ordering phenomena
- Multiferroics, competing order
- [Quantum phase transitions]
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Result of renormalisation group theory (Wilson) and of numerical calculations:
For continuous phase transitions the behavior close tothe critical point (Tc ) (i.e. the critical exponents a,b,g,d )depends only on a few parameters:
• Dimensionality of the order parameter Ordnungsparameters d
• Dimensionality of the interaction D
• Is the interaction long-range (power law decay r-n , i.e. no length scale) or short range (exponential decay exp (-r/r0) , i.e. length scale r0) ?
Critical phenomena and universality
Order parameter
Response functionsmagnetic susceptibility
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Critical phenomena and universality
dimensionality of the interactiondimensionality of the order parameter
[2][2] [2]
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Excitations in the symmetry broken state of a continuous symmetry
massive excitation massless Nambu-Goldstone-Bosons
Variation of the absolute value of the order parameter„amplitude mode“„Higgs mode“
Continuous rotation of the order parameterconnecting different equivalentground states with the same absolute value„phase mode“
Excitations = time dependent fluctuations of the order parameter
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1-D chain
Short range interactions:
w (q=0) = 0
Nambu-Goldstone excitation: k=0 magnon in the Heisenberg model
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EuS: Ferromagnet (TC = 16,5 K) with localized magnetic moments
face centered cubicEu2+ (4f7): J=S=7/2 ions
Isotropic Heisenberg interactionof nearest neighbor spins:
mnmn
nm SSJH ,
Magnons in the Heisenberg Model
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Magnons arelow temperature excitations
Magnons and spin correlations
excitations in the critical regionare spin-spin correlationsdivergent in space and time, with correlation length andcorrelation time
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Magnons arelow temperature excitations
excitations in the critical regionare spin-spin correlationsdivergent in space and time, with correlation length andcorrelation time
Magnons and spin correlations
Numerical simulations of 2-D Ising system
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Energy increase through spatial fluctuations of the order parameter
For a Ferromagnet one direction of M is spontaneously choosen Spatial fluctuations are e.g. rotations of the local order parameter
Energy increase D E ~ (��)�
This holds in general for order parameters and is described by theGinzburg-Landau-Theory (crf. Superconductivity, Brout-Engert-Higgs):For charged particles (here Cooper pairs with charge 2e) this leads to the canonical momentum term (principle of minimal coupling)
Ginzburg-Landau
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Overview spontaneous symmetry breaking
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Outline• Phase transitions in fluids
- Phase diagram, order parameter and symmetry breaking
- Microscopic van-der-Waals theory universality
• Magnetic phase transitions in condensed matter
- Paramagnet
- Interacting magnetic dipole moments, “spins”
- Weiss model for ferromagnetism universality
- Landau theory
• Consequences of symmetry breaking
- Critical phenomena and universality
- Excitations, Nambu-Goldstone-, Higgs-modes
• More complex ordering phenomena
- Multiferroics, competing order
- [Quantum phase transitions]
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Multiferroics
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Multiferroics
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Conventional Superconductivity
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Type I superconductor
• Superconductivity is a thermodynamic phase
B = 0 inside for B < Bc (Meißner phase)
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Thermodynamics: specific heat
• exponential low T behavior in conventional sc (BCS)power law low T behavior in unconventional sc
• matching areas entropy conserved at TC
consistent with second order phase transition
Electronic specific heat around superconducting transition
Vanadium
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Thermodynamics: entropy
• Superconducting state is the more ordered state• Description using the concept of an order parameter useful
Ginzburg-Landau theory
Entropy S versus temperature
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Superconducting Transition Temperatures
1986 Cuprate high-TC
systems
?Liquid nitrogenboiling temp
2007 Fe-basedSuperconductorsHosono et al.
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Coexistence of Superconductivity and Magnetic Order
BaFe2-xCoxAs2Ba1-xKxFe2As2
Coexistence
D.K. Pratt et al., PRL ´09
Nandi et al., PRL 2010
Critical evidence for coexistence:• Bulk magnetic order • Bulk superconductivity• Coupling of order parameters
Structural order parameter
Magnetic order parameter
E. Wiesenmayer et al., Phys. Rev. Lett. 107, 237001 (2011)T. Goltz et al, Phys. Rev. B 89, 144511 (2014)Ph. Materne et al., Phys. Rev. B 92, 134511 (2015)
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Electronic Instabilities
SDW magnetism Superconductivity
Γ Γ
M
MQ Q
-Q
Resonant single electron scattering on the Fermi surface with nesting vector Q
Resonant electron pair scattering on the Fermi surface with nesting vector Q
Competition for free electrons on the Fermi surface
Susceptibilites depend differently on details of the Fermi surfaces (size, shape,…)
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Landau-Theory for coupled order parameters
Magnetism and superconductivity compete for the same electrons at the Fermi surface d positive
Linear suppression of the magnetic order parameteras a function of the ratio of the critical temperatures TC /TN
Conditions for non-zero order parameters
Ph. Materne et al., Phys. Rev. B (2015)
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Universal suppression of magnetic order parameter
Ph. Materne et al., Phys. Rev. B (2015)
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Quantum Phase transitions• What happens, when for a continuous phase transition TC is suppressed to zero temperature
via some external parameter p ?
Critical temperature becomes quantum critical point (QCP)
• What destroys the ordered state at T0 as a function of p?
Enhanced quantum critical fluctuations, e.g. antiferromagnetic spin fluctuations
• Often new order emerges driven by these quantum fluctuations, e.g. superconductivity
Mathur et al., Nature 1998
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Summary• Phase transitions in fluids
- Phase diagram, order parameter and symmetry breaking
- Microscopic van-der-Waals theory universality
• Magnetic phase transitions in condensed matter
- Paramagnet
- Interacting magnetic dipole moments, “spins”
- Weiss model for ferromagnetism universality
- Landau theory
• Consequences of symmetry breaking
- Critical phenomena and universality
- Excitations, Nambu-Goldstone-, Higgs-modes
• More complex ordering phenomena
- Multiferroics, competing order
- [Quantum phase transitions]
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Energy increase through spatial fluctuations of the order parameter
For a Ferromagnet one direction of M is spontaneously choosen Spatial fluctuations are e.g. rotations of the local order parameter
Energy increase D E ~ (��)�
This holds in general for order parameters and is described by theGinzburg-Landau-Theory (crf. Superconductivity, Brout-Engert-Higgs):For charged particles (here Cooper pairs with charge 2e) this leads to the canonical momentum term (principle of minimal coupling)
Ginzburg-Landau