Quantum Phase Transitions Lecture (MVSpec) - Quiz Questions - ST 2020 Thomas Gasenzer First-order phase transitions in water Let us consider an isotherm across the 1st-order phase transition from water ice to vapour. No other substances be mixed into the water. Which of the following statements concerning the thermodynamic potentials are correct? (A) The free energy changes by an amount which is solely determined by the pressure and the change in volume. (B) The internal energy E is constant accross the transition. (C) As the Gibbs free enthalpy is constant, the enthalpy H changes by − TΔS − SΔT. (D) The change in energy is given by the latent heat and the work done. (E) At the critical point, no work is done across the phase transition, but latent heat can be freed. (F) The enthalpy H is convex across the phase transition. (G) The convexity of the enthalpy is due to the specific heat at constant temperature being positive. Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer First-order phase transitions in water Let us consider an isotherm across the 1st-order phase transition from water ice to vapour. No other substances be mixed into the water. Which of the following statements concerning the thermodynamic potentials are correct? (A) The free energy changes by an amount which is solely determined by the pressure and the change in volume. ✓ F = − PΔV (B) The internal energy E is constant across the transition. ✘ ΔE = TΔS − PΔV > 0 as ΔS > 0 and ΔV < 0 when going from ice to liquid water. From ice to vapour this could be different, but there is no reason why in that case work and latent heat will always cancel. (C) As the Gibbs free enthalpy is constant, the enthalpy H changes by − TΔS − SΔT. ✓ - Yes, while T is constant on an isotherm and thus the second term is zero. (D) The change in energy is given by the latent heat and the work done. ✓ Yes, check the Legendre transforms from G to E via F or H and use that G is constant. (E) At the critical point, no work is done across the phase transition, but latent heat can be nd freed. ✘ - At the critical point, the p.t. is of 2 order and thus (∂G/∂T)P is continuous and therefore the entropy change is zero. 2 2 (F) The enthalpy H is convex across the phase transition. ✓ Yes, because (∂ H/∂S )P = (∂T/∂S)P = cP/T > 0. (G) The convexity of the enthalpy is due to the specific heat at constant temperature being positive. ✘ - It is the specific heat at constant pressure which matters. Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Critical point in water and superfluid Helium The upper figures show the isothermal compressibility and the isobaric heat capacity of water near the critical point. The lower figure depicts the isobaric heat capacity near the lambda-transition to superfluid Helium. Which of the following statements are correct? (A) The isobaric heat capacity of water diverges at the critical point. (B) At the critical point, the entropy as a function of temperature, at constant pressure, shows a cusp. (C) The isothermal compressibility remains finite at the critical point. (D) When, instead, considering the solid-to-gas transition of water, the jump in isothermal compressibility is from a smaller to a larger value. (E) The isobaric heat capacity is finite at the lambda point of liquid helium. Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer < Critical point in water and superfluid Helium The upper figures show the isothermal compressibility and the isobaric heat capacity of water near the critical point. The lower figure depicts the isobaric heat capacity near the lambda-transition to superfluid Helium. Which of the following statements are correct? (A) The isobaric heat capacity of water diverges at the critical point. ✓ ‒ Yes, cP ‒ α diverges as cP ~ |t| at the critical point, where t = |T ‒ Tc|/Tc, with the 3D Ising exponent α ≈ 0.11. (B) At the critical point, the entropy as a function of temperature, at constant pressure, shows a cusp. ✘ ‒ That could be the case in principle but is not the case in general. For example, in water, the slope of S diverges and thus there will not be a cusp. In a superfluid, there is, however. (C) The isothermal compressibility remains finite at the critical point. ✘ ‒ No, the ‒ γ isothermal compressibility of water diverges as κT ~ |t| at the critical point, with the 3D Ising exponent γ ≈ 1.24. (D) When, instead, considering the solid-to-gas transition of water, the jump in isothermal compressibility is from a smaller to a larger value. ✓ ‒ Yes (E) The isobaric heat capacity is finite at the lambda point of liquid helium. ✓ ‒ Yes, the lambda point belongs to the XY universality class the critical exponent of which in 3D is negative, α ≈ ‒ 0.01. Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Critical point in water and superfluid Helium Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ising ferromagnetic transition The left figure shows the phases of a ferromagnetic material in the space spanned by temperature T and magnetic field H, while the right figure shows the magnetization M as a function of temperature. Which of the following statements are correct? (A) The ferromagnetic transition is of second order. (B) Below the critical temperature Tc , the isotherms of the ferromagnet equation of state show cusps. (C) When tuning the magnetic field from negative to positive, the magnetization changes continuously only at the critical point. (D) Putting your hard disk into the heated oven in your kitchen is not recommended unless you are pursued by The Force. (E) The magnetic susceptibility diverges at the critical point. (F) The critical exponent beta characterizing the scaling of the magnetization with temperature near criticality is negative. Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ferromagnets Ising ferromagnetic transition The left figure shows the phases of a ferromagnetic material in the space spanned by temperature T and magnetic field H, while the right figure shows the magnetization M as a function of temperature. Which of the following statements are correct? (A) The ferromagnetic transition is of second order. ✓ ‒ Yes, the Ising phase transition occuring at the critical temperature Tc is continuous. (B) Below the critical temperature Tc , the isotherms of the ferromagnet equation of state show cusps. ✓ ‒ Yes, they end at positive magnetization for h → 0+ and at negative M for h → 0−. In between, all magnetizations are in principle possible. (C) When tuning the magnetic field from negative to positive, the magnetization changes continuously only at the critical point. ✘ ‒ No, it also does so for temperatures above the critical temperature. (D) Putting your hard disk into the heated oven in your kitchen is not recommended unless you are pursued by The Force. ✓ ‒ Yes, rather to prevent low melting metals (in particular Sn-Pb at > ~180°C) to stay in place. Demagnetization temperatures, however, are above ca. 300°C which are hard to reach in a household oven. (E) The magnetic susceptibility diverges at the critical point. ✓ ‒ Yes, with critical exponent γ ≈ 1.24. (F) The critical exponent beta characterizing the scaling of the magnetization with temperature near criticality is negative. ✘ ‒ No, it is β ≈ 0.33, otherwise the magnetization would not diverge. Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ising ferromagnetic transition Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Quantum phase transitions: Introduction Consider a nonrelativistic, ideal Bose gas of point particles of mass m in its ground state. Which of the following statements are correct? (A) In the ground state different particles in the gas behave different from each other. (B) The lowest excitation of the gas is a spin flip. (C) When considering N particles in a finite-size container, the gap parameter is non- zero. (D) The dynamical exponent z is different from 1. (E) When taking into account repulsive interactions between the particles, the dynamical exponent z at the quantum critical point of the Bose gas remains invariant as compared to that of the ideal gas. Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Quantum phase transitions: Introduction Consider a nonrelativistic, ideal Bose gas of point particles of mass m in its ground state. Which of the following statements are correct? (A) In the ground state different particles in the gas behave different from each other. ✘ ‒ No, they are bosons in a Bose-Einstein condensate. (B) The lowest excitation of the gas is a spin flip. ✓ ‒ Yes, that could be the case if the bosons had spin and no external magnetic field is applied. Without spin, the lowest excitation would be the motion of a single bosons with momentum k, with |k| > 0. (C) When considering N particles in a finite-size container, the gap parameter is non- zero. ✓ ‒ Yes, the gap in that case is given by the lowest possible kinetic energy, Δ = k2/2m which is non-zero in a box. (D) The dynamical exponent z is different from 1. ✓ ‒ Yes, in this state, it is z = 2 because the gap scales quadratically in k. (E) When taking into account repulsive interactions between the particles, the dynamical exponent z at the quantum critical point of the Bose gas remains invariant as compared to that of the ideal gas. ✘ ‒ No, it will change, in a 3D gas to z = 3/2, as we will lateron see (XY-model quantum critical point).