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

(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

(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

(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).

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Quantum phase transitions: Introduction

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Entropy

In the video, the partition sum and the Boltzmann factor were used to recover the Legendre transformation relating internal energy and free energy via the entropy. This relation was given in two different forms, one with Hamiltonian H, (1) exp(−βTS) = exp[β(F − H)], and one involving the mean energy, (2) TS = ⟨H⟩ − F. Which of the following statements are correct?

(A) The form (1) of the relation is the right one. (B) The form (2) of the relation is the right one. (C) Both, (1) and (2) describe the relation, but (2) is only valid in quantum theory. (D) Both, (1) and (2) describe the relation, while (1) is the one used for microcanonical ensembles. (E) Neither (1) nor (2) are strictly speaking correct.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Entropy

(A) The form (1) of the relation is the right one. ✘ ‒ No, see below (B) The form (2) of the relation is the right one. ✘ ‒ No, see below (C) Both, (1) and (2) describe the relation, but (2) is only valid in quantum theory. ✘ ‒ No, both are correct but the 2nd one can also be taken as the relation in the classical canonical ensemble, if you interpret ρ as the probability of a given microstate and the trace as the sum over all microstates. (D) Both, (1) and (2) describe the relation, while (1) is the one used for microcanonical ensembles. ✓ ‒ Yes, in classical statistical mechanics. In QM it would be an operator relation, and one there is no entropy operator used in QM. (E) Neither (1) nor (2) are strictly speaking correct. ✘ ‒ No, see above.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Entropy

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Von Neumann Entropy

The von Neumann entropy SvN= − kBTr( ρ ln ρ ) was defined in the last lecture. It is defined in both, quantum mechanics and classical statistical physics. Please tick all the correct statements made about it in the following.

(A) If ρ denotes a pure state, the entropy S is unity. (B) For a maximally mixed state, S is at its maximum, the log of the dimension of the Hilbert space. (C) S depends on the particular choice of basis.

(D) For a canonical ensemble ρc, S is additive in the entropies of the microstates ρmc

being averaged over in ρc.

(E) ρ is strongly subadditive, i.e., for a state ρAB on the combined subsystems A and

B, the entropy satisfies S(ρAB) ≤ S(ρA) + S(ρB), where ρA,B = TrB,A ρAB are the

reduced density matrices of ρAB.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Von Neumann Entropy

(A) If ρ denotes a pure state, the entropy S is unity. ✘ ‒ No, it is zero, as is seen by n 2 expanding ln ρ = ln(1 − 1 + ρ) = − ∑n (1 − ρ) /n and using that ρ = ρ and Tr ρ = 1. (B) For a maximally mixed state, S is at its maximum, the log of the dimension of the Hilbert space. ✓ ‒ Yes, in this case, ρ is diagonal and there is at most one entry per

dimension, each that of a pure state but with trace Tr ρi = pi < 1 such that Tr(ρi ln ρi ) < 0. In

the maximally mixed state, all pi = 1/N, such that S = kB ln N. (C) S depends on the particular choice of basis. ✘ ‒ No, since the trace is invariant under unitary transformations of ρ, see in particular the expansion in (A).

(D) For a canonical ensemble ρc, S is additive in the entropies of the microstates ρmc

being averaged over in ρc. ✓ ‒ Yes, ρc is a diagonal operator, with each of the diagonal entries containing the Boltzmann factor of one energy microstate, such that S becomes a weighted sum of the entropies of the microstates. Additivity applies to all product states.

(E) ρ is strongly subadditive, i.e., for a state ρAB on the combined subsystems A and

B, the entropy satisfies S(ρAB) ≤ S(ρA) + S(ρB), where ρA,B = TrB,A ρAB are the Von Neumann Entropy

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ising model in one and two spatial dimensions

Low-dimensional systems are often special in that exact solutions can be found even for an infinite number of degrees of freedom. But before bothering about finding such a solution it is often easier to explore simpler scaling arguments. Which of the following statements about the 1D and 2D Ising models are correct?

(A) To create the lowest-energy excitation above the zero-temperature ground state in the 1D Ising model (with h = 0) an infinite number of spins must flip. (B) To create the lowest-energy excitation state above the ground state in the 2D Ising model (with h = 0) an infinite number of spins must flip. (C) From the validity/incorrectness of (A) and (B), it directly follows that there is a phase transition at non-zero T in the 2D system while there is not in 1D. (D) The argument leading to a non-zero critical temperature in 2D is also applicable in 3D.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ising model in one and two spatial dimensions

(A) To create the lowest-energy excitation above the zero-temperature ground state in the 1D Ising model (with h = 0) an infinite number of spins must flip. ✓ ‒ Yes, the ground state is one with all spins aligned, no matter if h vanishes or not. If h = 0, the lowest- energy excitation is that with two neighbouring spins being oppositely aligned while all other neighbours remain coaligned. For this all spins to one side of that pair must be flipped. (B) To create the lowest-energy excitation state above the ground state in the 2D Ising model (with h = 0) an infinite number of spins must flip. ✘ ‒ No, in contrast to (A), in this case, the lowest-energy excitation is that with four frustrated links around one flipped spin. (C) From the validity/incorrectness of (A) and (B), it directly follows that there is a phase transition at non-zero T in the 2D system while there is not in 1D. ✘ ‒ No, first, the possibility of a phase transition at T > 0 is not due to the energy of the lowest excitation (or the second lowest – which, in 1D also only requires one spin to be flipped), but due to the entropy scaling as the energy of the excitation. (D) The argument leading to a non-zero critical temperature in 2D is also applicable in 3D. ✓ ‒ Yes, the argument also works in 3D where the energy and the entropy scale as the third root of the volume of flipped spins.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ising model in one and two spatial dimensions

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Mean-field description of the critical Ising model

Recall that the Ising critical point describes a second-order phase transition and consider its mean-field description. Tick the correct statements.

(A) At mean-field order, the Ising transition as a function of T at zero h is of first order. (B) In the mean-field approximation, the magnetic susceptibility at constant temperature diverges as a function of temperature. (C) In the mean-field approximation, the magnetic susceptibility is continuous across the phase transition. (D) At mean-field order, the susceptibility at constant temperature diverges as a function of magnetic field h. The divergence is the same for h > 0 and h < 0.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Mean-field description of the critical Ising model

Recall that the Ising critical point describes a second-order phase transition and consider its mean-field description. Tick the correct statements.

(A) At mean-field order, the Ising transition as a function of T at zero h is of first order. ✘ ‒ No, also within the mean-field approximation, the transition at the critical point is of

second order, as can also be seen by computing f(T,0;m0(T)) across T = Tc which gives that

the free energy density is continuous in T and has a continuous derivative ∂T f. (B) In the mean-field approximation, the magnetic susceptibility at constant temperature diverges as a function of temperature. ✓ ‒ Yes, it diverges as χ(t) ~ |t|‒γ

with γMF = 1. (C) In the mean-field approximation, the magnetic susceptibility is continuous across the phase transition. ✘ ‒ No, it diverges, and it does so differently for t < 0 and t > 0, as ~ (2|t|)‒γ and ~ |t|‒γ, respectively. (D) At mean-field order, the susceptibility at constant temperature diverges as a function of magnetic field h. The divergence is the same for h > 0 and h < 0. ✓ ‒ 1/δ Yes, one finds m ~ |h| , with δMF = 3.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Mean-field description of the critical Ising model

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Berezinskii-Kosterlitz-Thouless transition

(A) The phase transition involves spontaneous symmetry breaking. (B) The phase transition is of first order. (C) None of the above applies but it is a continous phase transition. (D) In the disordered phase, φ is restricted to a bound interval.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Berezinskii-Kosterlitz-Thouless transition

The Berezinskii-Kosterlitz-Thouless (BKT) transition in 2D superfluids is described by a classical potential sinusoidal in the order parameter φ which is a real unbounded scalar and represents an angle, V(φ) = g cos φ , with some coupling constant g. A phase transition occurs as a function of temperature across which g changes smoothly from a positive value to zero. φ is, in fact, a disorder parameter which is nonzero in the phase without order. Which of the following statements are correct? (A) The phase transition involves spontaneous symmetry breaking. ✘ ‒ No, the model is in general symmetric under a shift by 2πn, with integer n. Note that the omitted terms are derivatives and thus invariant under a constant shift, too. But also the ground state, in either phase, bears the same symmetry. When g = 0, the continuous U(1) symmetry is restored. (B) The phase transition is of first order. ✘ ‒ No, it is said to be of „infinite order“ because there is an essential singularity, e.g., of the correlation length at vanishing reduced 1/2 temperature, t = 0, where it scales as ξ(T) ~ l exp{(Eξ/[T ‒ Tc]) }. (C) None of the above applies but it is a continous phase transition. ✓ ‒ Yes, also an infinite order phase transition is continuous as the disorder parameter does not have any meaningful mean value below the transition. (D) In the disordered phase, φ is restricted to a bound interval. ✘ ‒ No, the transition is described by the non-compact Sine-Gordon model.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Berezinskii-Kosterlitz-Thouless transition

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ginsburg-Landau-Wilson functional

(A) The leading-order quadratic derivative term takes into account a potential energy due to the spatial variations of the field. (B) There are no other quadratic derivative terms in the action. (C) There are no odd powers of derivatives appearing.

4 (D) Quartic derivative terms ~ k would occur in the "mass" term r0 ~ t only. (E) *The GLW functional is valid also in one spatial dimension.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ginsburg-Landau-Wilson functional

The Ginsburg-Landau-Wilson (GLW) functional S[φ] of the Ising model without coupling to an external magnetic field represents an effective action for the auxiliary field φ. Expanded to fourth order in the fields it takes the form of the standard λφ4 model for the scalar field φ, for the first in a static approximation without time deriva-tives. It is furthermore an expansion of the exact functional in spatial derivatives of the field. Which of the following statements are correct close to the critical point, i.e., in leading order in the reduced temperature t ? (A) The leading-order quadratic derivative term takes into account a potential energy due to the spatial variations of the field. ✘ ‒ No, the leading term in k2 and t represents a kinetic energy due to the spatial field variations. However, there is also a next-to-leading- 2 order term ~ O(t k ) contributing to the „mass“ r0, which one should, however, also regard as a kinetic term. Such terms will represent a "wave function renormalisation". (B) There are no other quadratic derivative terms in the action. ✘ ‒ No, see above. There is also one contributing to the coupling u. (C) There are no odd powers of derivatives appearing. ✓ ‒ Yes, odd powers would not form scalars for the given model (parity symmetry, isotropy). 4 ✘ (D) Quartic derivative terms ~ k would occur in the "mass" term r0 ~ t only. ‒ No, there is also one at O(t 0), both, in the φ2 and the φ4 terms contributing to c and u, respectively. (E) *The GLW functional is valid also in one spatial dimension. ✘ ‒ No, in 1D, one has a vanishing critical temperature and for T → 0, the static approximation fails.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ginsburg-Landau-Wilson functional

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ising nearest-neighbour coupling

The Ising model with nearest-neighbour coupling is defined by the Hamiltonian

H = – J Σ⟨ i, j ⟩ Si Sj

We compute the Ginsburg-Landau-Wilson functional of this model, which includes a derivative expansion of the field to quadratic order. Consider instead an Ising model with nearest- and next-to-nearest-neighbour interactions. Which answer is correct?

(A) With nearest- and next-to-nearest-neighbour interactions one always obtains a term quadratic in momentum. (B) With nearest- and next-to-nearest-neighbour interactions one only gets a term quartic in the momentum. (C) With nearest- and next-to-nearest-neighbour interactions one can get, quadratic or quartic terms as leading order momentum terms depending on how the interactions are chosen.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ising nearest-neighbour coupling

The Ising model with nearest-neighbour coupling is defined by the Hamiltonian

H = – J Σ⟨ i, j ⟩ Si Sj

(A) With nearest- and next-to-nearest-neighbour interactions one always obtains a term quadratic in momentum. ✘ ‒ No, see (C). (B) With nearest- and next-to-nearest-neighbour interactions one only gets a term quartic in the momentum. ✘ ‒ No, see (C). (C) With nearest- and next-to-nearest-neighbour interactions one can get, quadratic or quartic terms as leading order momentum terms depending on how the

interactions are chosen. ✓ ‒ Yes, this is possible. Consider the Hamiltonian H = – Jn Σi Si 2 Si + 1 – Jnn Σi Si Si + 2. This gives, in momentum space, Jk = 2(Jn+Jnn) – (Jn+4Jnn)(ka) + 4 O((ka) ). Hence, for Jn+4Jnn = 0, the quadratic term vanishes, otherwise there is one.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Ising nearest-neighbour coupling

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Critical scaling in Gaussian approximation

Taking into account Gaussian fluctuations around the mean-field magnetisation of the Ising model yields the critical exponent α = 2 – d/2 of the specific heat.

Which of the following statements are correct?

(A) Within the leading-order mean-field approximation, one finds α = 0. (B) The Gaussian fluctuations contribute a shift to the mean-field free energy density 2 f ≈ – 3r0 /(2u0), which scales as ~ |t| G0(r = a;t) near vanishing reduced temperature |t| ≈ 0.

the Gaussian fluctuations, ~ r0 G0(a), vanishes if d < 4.

(D) Close to criticality, the shift from Gaussian fluctuations, ~ r0 G0(a), can cause the specific heat at vanishing magnetic field to diverge. (E) The exponent α vanishes always in d > 4 dimensions.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Critical scaling in Gaussian approximation

Taking into account Gaussian fluctuations around the mean-field magnetisation of the Ising model yields the critical exponent α = 2 – d/2 of the specific heat. Which of the following statements are correct?

(A) Within the leading-order mean-field approximation, one finds α = 0. ✓ ‒ Yes, close 2 to the critical point, the mean-field free energy density at h = 0 is f = – 3r0 /(2u0) for t < 0, and 2 f = 0 for t > 0, such that the specific heat ch = (∂t f)h jumps from one constant to another one. (B) The Gaussian fluctuations contribute a shift to the mean-field free energy density 2 f ≈ – 3r0 /(2u0), which scales as ~ |t| G0(r = a;t) near vanishing reduced 2 temperature |t| ≈ 0. ✓ ‒ Yes, in the broken phase one obtains φ0 = − 6(r0/u0 + G0(a)/2) for 1/a the mean field including the Gaussian 1-loop shift ∫k G0(k) = G0(r = a). Inserting this into the 2 free energy density gives the above 1-loop shift, as well as a subdominant term ~ u0 G0(a) .

(C) Close to criticality, the second derivative with respect to r0 of the shift caused by ✘ the Gaussian fluctuations, ~ r0 G0(a), vanishes if d < 4. ‒ No, it diverges, since it scales as |t|−α, with α = 2 – d/2.

(D) Close to criticality, the shift from Gaussian fluctuations, ~ r0 G0(a), can cause the specific heat at vanishing magnetic field to diverge. ✓ ‒ Yes, see (C). (E) The exponent α vanishes always in d > 4 dimensions. ✓ ‒ Yes, because the upper

critical dimension is dup = 4, above which mean-field theory is exact.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Critical scaling in Gaussian approximation

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Validity of the Gaussian approximation

The validity of the Gaussian approximation is controlled by the Ginsburg criterion, which compares the effect of fluctuations with the corresponding mean-field approximated result.

Tick the correct statements.

(A) The Ginsburg criterion defines the lower critical dimension. (B) The Ginsburg criterion represents a first estimate based on evaluating the long- wave-length fluctuations as compared to the total quadratic magnetisation of the system. (C) The upper critical dimension defines the limit above which mean-field theory breaks down. (D) The difference between the upper and the lower critical dimension is related to the universal answer 42. (E) The Ginsburg criterion is violated where higher-order perturbative corrections lead to a non-vanishing anomalous dimension η

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Validity of the Gaussian approximation

The validity of the Gaussian approximation is controlled by the Ginsburg criterion, which compares the effect of fluctuations with the corresponding mean-field approximated result. Tick the correct statements.

(A) The Ginsburg criterion defines the lower critical dimension. ✘ ‒ No, it provides the upper critical dimension which depends on the scaling of the interaction term as compared the quadratic terms in the action. (B) The Ginsburg criterion represents a first estimate based on evaluating the long- wave-length fluctuations as compared to the total quadratic magnetisation of the system. ✓ ‒ Yes, this is how we introduced in in the lecture. (C) The upper critical dimension defines the limit above which mean-field theory breaks down. ✘ ‒ No, above the upper critical dimension mean-field scaling is exact. (D) The difference between the upper and the lower critical dimension is related to the universal answer 42. ✘ ‒ No, there is, to my knowledge, no fundamental relation between the lower and upper crit. dimensions. The former requires a deeper analysis and indicates, at least for continuous symmetries subject to the Mermin-Wagner theorem, the dimension below which long-range order is impossible. (E) The Ginsburg criterion is violated where higher-order perturbative corrections Validity of the Gaussian approximation

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Critical exponents

The universal behaviour of the Ising spin system close to criticality is characterised by universal critical exponents of which we have encountered, so far, α, β, γ, δ, ν, η.

Which of the following statements concerning their relevance are correct?

(A) The critical exponents imply that all physical observables can be expressed as power laws close to criticality. (B) The critical exponents are independent of each other. (C) The values of the critical exponents are independent of the spatial dimension of the system. (D) The scaling behaviour of each observable involves precisely one critical exponent. (E) Correlation functions involving transcendent functions can not show universal scaling behaviour. (F) Ticking statement (F) is the only correct answer in this quiz.

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Critical exponents

The universal behaviour of the Ising spin system close to criticality is characterised by universal critical exponents of which we have encountered, so far, α, β, γ, δ, ν, η. Which of the following statements concerning their relevance are correct?

(A) The critical exponents imply that all physical observables can be expressed as power laws close to criticality. ✘ ‒ No, correlation functions can involve transcendent functional behaviour (e.g. exponential). (B) The critical exponents are independent of each other. ✘ ‒ No, they are not, which can already be seen from the dependence on d to Gaussian order. We will later find scaling relations linking the different exponents with each other. There is a maximum number of independent exponents depending on the number of relevant operators in the system. (C) The values of the critical exponents are independent of the spatial dimension of

the system. ✘ ‒ No, e.g. α as well the scaling exponent of G0 depend on d. (D) The scaling behaviour of each observable involves precisely one critical exponent. ✘ ‒ No, the scaling of correlation functions (scaling forms) involves exponents for the arguments as well as for the function itself. (E) Correlation functions involving transcendent functions can not show universal scaling behaviour. ✘ ‒ No, see (D). Suitable rescaling of their arguments leaves them invariant. (F) Ticking statement (F) is the only correct answer in this quiz. ✓ ‒ Yes!

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer Critical exponents

Lecture · Quantum Phase Transitions · Heidelberg University · ST 2020 Thomas Gasenzer