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Phase Transitions in Condensed Matter Hans-Henning Klauss Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality Hans-Henning Klauss Institut für Festkörperphysik TU Dresden 1 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 2 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] 3 Introduction • What is a thermodynamic phase? - Equilibrium state of matter of a many body system - Well defined symmetry - Thermodynamic potential changes analytically 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 4 Introduction • What is a thermodynamic phase? - Equilibrium state of matter (many body system) Phase diagram of water - Well defined symmetry - Thermodynamic potentials change analytically 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 5 Introduction • Many, many phase diagrams in nature…. structural phases electronic phases Mathur et al., Nature 1998 wikipedia electronic combined electronic and structural Luetkens et al., Nature Mat. 2009 Fernandes et al., Nature Phys. 2014 6 Introduction • 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 liquid -- solid fourier component of charge density rG Bose-Einstein condensation paramagnet -- ferromagnet liquid -- gas 7 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 fourier component of charge density rG water liquid gas 8 Phase Transitions in Fluids • First order transition order parameter changes discontinuously Phase diagram of water (Ehrenfest definition: first derivative • of Gibbs free enthalpy G is discontinuous) • • Continuous transition order parameter varies continuously • Critical point transition point of a continuous transition for water @ 647 K and 22.064 MPa liquid to gas phase transition can be of first order or continuous! water [1] liquid -- gas [google] 9 Microscopic Model • Van- der-Waals-model: Attractive particle – particle interaction fully rotational invariant V = V(r) + Finite particle volume van-der-Waals equation • Universal mixed liquid vapor region in p-V diagram for many materials • Maxwell construction isotherms and via normalization p/p and V/V ! phase coexistence c c below TC [2] 10 Spontaneous Symmetry Breaking and Phase Transitions • 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) supercritical fluid • Continuous crossover from liquid to gas without phase transition via supercritical fluid (path C) 11 [1] 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] 12 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] 13 Ferromagnet • Interacting magnetic moments: Exchange term + Zeeman term • Origin of exchange: spin-dependent Coulomb interaction e.g. superexchange [1] 14 Weiss-Model Weiss (1907): Define an effective magnetic field at site i caused by neighbors j „molecular field“ single particle problem 15 Weiss-Model Ansatz: Bmf ~ Magnetization M with Two linear independent equations BJ = Brillouin function Magnetization of a paramagnet in total magnetic field B +lM 16 Weiss-Model Graphical Solution M=0 always possible M ≠ 0 only if T < TC TC ~ exchange energy J, number of neighbors Continuous phase transition size of moments Typical: [1] Huge! 17 Weiss-Model Graphical Solution M=0 always possible M ≠ 0 only if T < TC TC ~ exchange energy J, number of neighbors Continuous phase transition size of moments • Universal T dependence of order parameter • depends only on total angular momentum multiplicity J [1] 18 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 = 19 Landau Theory of Ferromagnetism crf Weiss theory: [1] 20 Folie 20 Weiss and Landau Theory of Ferromagnetism In magnetic field: Magnetization parallel to field is always > 0 No phase transition ! [1] [1] Explicit symmetry breaking due to Zeeman term F M < 0 metastable solution exists for small B only [Web] 21 First order transition below TC as a function of external field B Weiss and Landau Theory of Ferromagnetism Solution in magnetic field: Magnetization parallel to field always > 0 No phase transition ! [1] Explicit symmetry breaking due to Zeeman term M M < 0 metastable solution exists for small B only [Web] 22 First order transition below TC as a function of external field B Comparison of fluid and magnet phase diagrams for fluid Sometimes density r = N m / V used Gibbs free energy G = U –TS + p V for magnet Gibbs free energy G = U –TS – M B [3,5] 23 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] 24 Critical phenomena and universality Result of renormalisation group theory (Wilson) and of numerical calculations: For continuous phase transitions the behavior close to the 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) ? Response functions magnetic susceptibility Order parameter 25 Critical phenomena and universality dimensionality of the interaction dimensionality of the order parameter [2] [2] [2] 26 Excitations in the symmetry broken state of a continuous symmetry Excitations = time dependent fluctuations of the order parameter . massive excitation massless Nambu-Goldstone-Bosons Variation of the absolute Continuous rotation of the order parameter value of the order parameter connecting different equivalent „amplitude mode“ ground states with the same absolute value „Higgs mode“ „phase mode“ 27 Nambu-Goldstone excitation: k=0 magnon in the Heisenberg model Short range interactions: 1-D chain w (q=0) = 0 28 Magnons in the Heisenberg Model EuS: Ferromagnet (TC = 16,5 K) with localized magnetic moments face centered cubic Eu2+ (4f7): J=S=7/2 ions Isotropic Heisenberg interaction of nearest neighbor spins: H J nm SnSm n,m 29 Magnons and spin correlations Magnons are low temperature excitations excitations in the critical region are spin-spin correlations divergent in space and time, with correlation length and correlation time 30 Magnons and spin correlations Magnons are low temperature excitations excitations in the critical region are spin-spin correlations divergent in space and time, with correlation length and correlation time Numerical simulations of 2-D Ising system 31 Ginzburg-Landau 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
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