Superfluid He-3. Thought of the Day 130.0

At higher temperatures ³He is a gas, while below temperature of 3K – due to van der Walls forces – ³He is a normal liquid with all symmetries which a condensed matter system can have: translation, gauge symmetry U(1) and two SO(3) symmetries for the spin (SOS(3)) and orbital (SOL(3)) rotations. At temperatures below 100 mK, ³He behaves as a strongly interacting Fermi liquid. Its physical properties are well described by Landau’s theory. Quasi-particles of the ³He (i.e. ³He atoms “dressed” into mutual interactions) have spin equal to 1/2 and similar to the electrons, they can create Cooper pairs as well. However, different from electrons in a metal, ³He is a liquid without a lattice and the electron-phonon interaction, responsible for superconductivity, can not be applied here. As the ³He quasiparticles have spin, the magnetic interaction between spins rises up when the temperature falls down until, at a certain temperature, Cooper pairs are created – the coupled pairs of ³He quasiparticles – and the normal ³He liquid becomes a superfluid. The Cooper pairs produce a superfluid component and the rest, unpaired ³He quasiparticles, generate a normal component (N -phase).

A physical picture of the superfluid ³He is more complicated than for superconducting electrons. First, the ³He quasiparticles are bare atoms and creating the Cooper pair they have to rotate around its common center of mass, generating an orbital angular momentum of the pair (L = 1). Secondly, the spin of the Cooper pair is equal to one (S = 1), thus superfluid ³He has magnetic properties. Thirdly, the orbital and spin angular momenta of the pair are coupled via a dipole-dipole interaction.

It is evident that the phase transition of ³He into the superfluid state is accompanied by spontaneously broken symmetry: orbital, spin and gauge: SOL(3)× SOS(3) × U(1), except the translational symmetry, as the superfluid ³He is still a liquid. Finally, an energy gap ∆ appears in the energy spectrum separating the Cooper pairs (ground state) from unpaired quasiparticles – Fermi excitations.

In superfluid ³He the density of Fermi excitations decreases upon further cooling. For temperatures below around 0.25T_c (where T_c is the superfluid transition temperature), the density of the Fermi excitations is so low that the excitations can be regarded as a non-interacting gas because almost all of them are paired and occupy the ground state. Therefore, at these very low temperatures, the superfluid phases of helium-3 represent well defined models of the quantum vacua, which allows us to study any influences of various external forces on the ground state and excitations from this state as well.

The ground state of superfluid ³He is formed by the Cooper pairs having both spin (S = 1) and orbital momentum (L = 1). As a consequence of this spin-triplet, orbital p-wave pairing, the order parameter (or wave function) is far more complicated than that of conventional superconductors and superfluid ⁴He. The order parameter of the superfluid ³He joins two spaces: the orbital (or k space) and spin and can be expressed as:

Ψ(k) = Ψ_↑↑(kˆ)|↑↑⟩ + Ψ_↓↓(kˆ)|↓↓⟩ + √2Ψ_↑↓(kˆ)(|↑↓⟩ + |↓↑⟩) —– (1)

where kˆ is a unit vector in k space defining a position on the Fermi surface, Ψ_↑↑(kˆ), Ψ_↓↓(kˆ) a Ψ_↑↓(kˆ) are amplitudes of the spin sub-states operators determined by its projection |↑↑⟩, |↓↓⟩ a (|↑↓⟩ + |↓↑⟩) on a quantization axis z.

The order parameter is more often written in a vector representation as a vector d(k) in spin space. For any orientation of the k on the Fermi surface, d(k) is in the direction for which the Cooper pairs have zero spin projection. Moreover, the amplitude of the superfluid condensate at the same point is defined by |d(k)|² = 1/2tr(ΨΨ^H). The vector form of the order parameter d(k) for its components can be written as:

dν(k) = ∑_μ A_νμk_μ —– (2)

where ν (1,2,3) are orthogonal directions in spin space and μ (x,y,z) are those for orbital space. The matrix components A_νμ are complex and theoretically each of them represents possible superfluid phase of ³He. Experimentally, however, only three are stable.

Looking at the phase diagram of ³He we can see the presence of two main superfluid phases: A – phase and B – phase. While B – phase consists of all three spin components, the A – phase does not have the component (|↑↓⟩ + |↓↑⟩). There is also a narrow region of the A1 superfluid phase which exists only at higher pressures and temperatures and in nonzero magnetic field. The A1 – phase has only one spin component |↑↑⟩. The phase transition from N – phase to the A or B – phase is a second order transition, while the phase transition between the superfluid A and B phases is of first order.

The B – phase occupies a low field region and it is stable down to the lowest temperatures. In zero field, the B – phase is a pure manifestation of p-wave superfluidity. Having equal numbers of all possible spin and angular momentum projections, the energy gap separating ground state from excitation is isotropic in k space.

The A – phase is preferable at higher pressures and temperatures in zero field. In limit T → 0K, the A – phase can exist at higher magnetic fields (above 340 mT) at zero pressure and this critical field needed for creation of the A – phase rises up as the pressure increases. In this phase, all Cooper pairs have orbital momenta orientated in a common direction defined by the vector lˆ, that is the direction in which the energy gap is reduced to zero. It results in a remarkable difference between these superfluid phases. The B – phase has an isotropic gap, while the A – phase energy spectrum consists of two Fermi points i.e. points with zero energy gap. The difference in the gap structure leads to the different thermodynamic properties of quasiparticle excitations in the limit T → 0K. The density of excitation in the B – phase falls down exponentially with temperature as exp(−∆/k_BT), where k_B is the Boltzman constant. At the lowest temperatures their density is so low that the excitations can be regarded as a non-interacting gas with a mean free path of the order of kilometers. On the other hand, in A – phase the Fermi points (or nodes) are far more populated with quasiparticle excitations. The nodes orientation in the lˆ direction make the A – phase excitations almost perfectly one-dimensional. The presence of the nodes in the energy spectrum leads to a T³ temperature dependence of the density of excitations and entropy. As a result, as T → 0K, the specific heat of the A – phase is far greater than that of the B – phase. In this limit, the A – phase represents a model system for a vacuum of the Standard model and B – phase is a model system for a Dirac vacuum.

In experiments with superfluid ³He phases, application of different external forces can excite the collective modes of the order parameter representing so called Bose excitations, while the Fermi excitations are responsible for the energy dissipation. Coexistence and mutual interactions of these excitations in the limit T → 0K (in limit of low energies), can be described by quantum field theory, where Bose and Fermi excitations represent Bose and Fermi quantum fields. Thus, ³He has a much broader impact by offering the possibility of experimentally investigating quantum field/cosmological theories via their analogies with the superfluid phases of ³He.

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Conjuncted: Speculatively Accelerated Capital – Trading Outside the Pit.

High Frequency Traders (HFTs hereafter) may anticipate the trades of a mutual fund, for instance, if the mutual fund splits large orders into a series of smaller ones and the initial trades reveal information about the mutual funds’ future trading intentions. HFTs might also forecast order flow if traditional asset managers with similar trading demands do not all trade at the same time, allowing the possibility that the initiation of a trade by one mutual fund could forecast similar future trades by other mutual funds. If an HFT were able to forecast a traditional asset managers’ order flow by either these or some other means, then the HFT could potentially trade ahead of them and profit from the traditional asset manager’s subsequent price impact.

There are two main empirical implications of HFTs engaging in such a trading strategy. The first implication is that HFT trading should lead non-HFT trading – if an HFT buys a stock, non-HFTs should subsequently come into the market and buy those same stocks. Second, since the HFT’s objective would be to profit from non-HFTs’ subsequent price impact, it should be the case that the prices of the stocks they buy rise and those of the stocks they sell fall. These two patterns, together, are consistent with HFTs trading stocks in order to profit from non-HFTs’ future buying and selling pressure. 

While HFTs may in aggregate anticipate non-HFT order flow, it is also possible that among HFTs, some firms’ trades are strongly correlated with future non-HFT order flow, while other firms’ trades have little or no correlation with non-HFT order flow. This may be the case if certain HFTs focus more on strategies that anticipate order flow or if some HFTs are more skilled than other firms. If certain HFTs are better at forecasting order flow or if they focus more on such a strategy, then these HFTs’ trades should be consistently more strongly correlated with future non-HFT trades than are trades from other HFTs. Additionally, if these HFTs are more skilled, then one might expect these HFTs’ trades to be more strongly correlated with future returns. 

Another implication of the anticipatory trading hypothesis is that the correlation between HFT trades and future non-HFT trades should be stronger at times when non-HFTs are impatient. The reason is anticipating buying and selling pressure requires forecasting future trades based on patterns in past trades and orders. To make anticipating their order flow difficult, non-HFTs typically use execution algorithms to disguise their trading intentions. But there is a trade-off between disguising order flow and trading a large position quickly. When non-HFTs are impatient and focused on trading a position quickly, they may not hide their order flow as well, making it easier for HFTs to anticipate their trades. At such times, the correlation between HFT trades and future non-HFT trades should be stronger. 

If there are common trading patterns among firms, then the HFT and non-HFT trading measures will be contemporaneously correlated. This might be the case if, for instance, firms in the HFT and non-HFT samples use the same trading signals. If there is also serial correlation in these trading patterns, then HFT trading will predict non-HFT trading simply because it is a noisy proxy for lagged non-HFT trading. If this explanation is driving a lead-lag relationship between HFT and non-HFT trading and the form is such that HFT trading is non-HFT trading plus noise, then the lead-lag relationship between the two variables will go away after controlling for lagged non-HFT trading.

Cosmology: Friedmann-Lemaître Universes

Cosmology starts by assuming that the large-scale evolution of spacetime can be determined by applying Einstein’s field equations of Gravitation everywhere: global evolution will follow from local physics. The standard models of cosmology are based on the assumption that once one has averaged over a large enough physical scale, isotropy is observed by all fundamental observers (the preferred family of observers associated with the average motion of matter in the universe). When this isotropy is exact, the universe is spatially homogeneous as well as isotropic. The matter motion is then along irrotational and shearfree geodesic curves with tangent vector ua, implying the existence of a canonical time-variable t obeying u_a = −t_,a. The Robertson-Walker (‘RW’) geometries used to describe the large-scale structure of the universe embody these symmetries exactly. Consequently they are conformally flat, that is, the Weyl tensor is zero:

C_ijkl := R_ijkl + 1/2(R_ikg_jl + R_jlg_ik − R_il g_jk − R_jkg_il) − 1/6R(g_ikg_jl − g_ilg_jk) = 0 —– (1)

this tensor represents the free gravitational field, enabling non-local effects such as tidal forces and gravitational waves which do not occur in the exact RW geometries.

Comoving coordinates can be chosen so that the metric takes the form:

ds² = −dt² + S²(t)dσ², u^a = δ^a₀ (a=0,1,2,3) —– (2)

where S(t) is the time-dependent scale factor, and the worldlines with tangent vector u^a = dx^a/dt represent the histories of fundamental observers. The space sections {t = const} are surfaces of homogeneity and have maximal symmetry: they are 3-spaces of constant curvature K = k/S²(t) where k is the sign of K. The normalized metric dσ² characterizes a 3-space of normalized constant curvature k; coordinates (r, θ, φ) can be chosen such that

dσ² = dr² + f²(r) dθ² + sin²θdφ² —– (3)

where f (r) = {sin r, r, sinh r} if k = {+1, 0, −1} respectively. The rate of expansion at any time t is characterized by the Hubble parameter H(t) = S ̇/S.

To determine the metric’s evolution in time, one applies the Einstein Field Equations, showing the effect of matter on space-time curvature, to the metric (2,3). Because of local isotropy, the matter tensor T_ab necessarily takes a perfect fluid form relative to the preferred worldlines with tangent vector u^a:

T_ab = (μ + p/c²)u_au_b + (p/c²)g_ab —– (4)

, c is the speed of light. The energy density μ(t) and pressure term p(t)/c² are the timelike and spacelike eigenvalues of T_ab. The integrability conditions for the Einstein Field Equations are the energy-density conservation equation

T^ab_;b = 0 ⇔ μ ̇ + (μ + p/c²)3S ̇/S = 0 —– (5)

This becomes determinate when a suitable equation of state function w := pc²/μ relates the pressure p to the energy density μ and temperature T : p = w(μ,T)μ/c² (w may or may not be constant). Baryons have {p_bar = 0 ⇔ w = 0} and radiation has {p_radc² = μ_rad/3 ⇔ w = 1/3,μ_rad = aT⁴_rad}, which by (5) imply

μ_bar ∝ S⁻³, μ_rad ∝ S⁻⁴, T_rad ∝ S⁻¹ —– (6)

The scale factor S(t) obeys the Raychaudhuri equation

3S ̈/S = -1/2 κ(μ + 3p/c²) + Λ —– (7)

, where κ is the gravitational constant and Λ is the cosmological constant. A cosmological constant can also be regarded as a fluid with pressure p related to the energy density μ by {p = −μc² ⇔ w = −1}. This shows that the active gravitational mass density of the matter and fields present is μ_grav := μ + 3p/c². For ordinary matter this will be positive:

μ + 3p/c² > 0 ⇔ w > −1/3 —– (8)

(the ‘Strong Energy Condition’), so ordinary matter will tend to cause the universe to decelerate (S ̈ < 0). It is also apparent that a positive cosmological constant on its own will cause an accelerating expansion (S ̈ > 0). When matter and a cosmological constant are both present, either result may occur depending on which effect is dominant. The first integral of equations (5, 7) when S ̇≠ 0 is the Friedmann equation

S ̇²/S² = κμ/3 + Λ/3 – k/S² —– (9)

This is just the Gauss equation relating the 3-space curvature to the 4-space curvature, showing how matter directly causes a curvature of 3-spaces. Because of the spacetime symmetries, the ten Einstein Filed Equations are equivalent to the two equations (7, 9). Models of this kind, that is with a Robertson-Walker (‘RW’) geometry with metric (2, 3) and dynamics governed by equations (5, 7, 9), are called Friedmann-Lemaître universes (‘FL’). The Friedmann equation (9) controls the expansion of the universe, and the conservation equation (5) controls the density of matter as the universe expands; when S ̇≠ 0 , equation (7) will necessarily hold if (5, 9) are both satisfied. Given a determinate matter description (specifying the equation of state w = w(μ, T) explicitly or implicitly) for each matter component, the existence and uniqueness of solutions follows both for a single matter component and for a combination of different kinds of matter, for example μ = μ_bar + μ_rad + μ_cdm + μ_ν where we include cold dark matter (cdm) and neutrinos (ν). Initial data for such solutions at an arbitrary time t₀ (eg. today) consists of,

• The Hubble constant H₀ := (S ̇/S)₀ = 100h km/sec/Mpc;

• A dimensionless density parameter Ω_i0 := κμ_i0/3H₀² for each type of matter present (labelled by i);

• If Λ ≠ 0, either Ω_Λ0 := Λ/3H²₀, or the dimensionless deceleration parameter q := −(S ̈/S) H⁻²₀.

Given the equations of state for the matter, this data then determines a unique solution {S(t), μ(t)}, i.e. a unique corresponding universe history. The total matter density is the sum of the terms Ω_i0 for each type of matter present, for example

Ω_m0 = Ω_bar0 + Ω_rad0 + Ω_cdm0 + Ω_ν0, —– (10)

and the total density parameter Ω₀ is the sum of that for matter and for the cosmological constant:

Ω₀ = Ω_m0 + Ω_Λ0 —– (11)

Evaluating the Raychaudhuri equation (7) at the present time gives an important relation between these parameters: when the pressure term p/c² can be ignored relative to the matter term μ (as is plausible at the present time, and assuming we represent ‘dark energy’ as a cosmological constant.),

q₀ = 1/2 Ω_m0 − Ω_Λ0 —– (12)

This shows that a cosmological constant Λ can cause an acceleration (negative q₀); if it vanishes, the expression simplifies: Λ = 0 ⇒ q = 1 Ω_m0, showing how matter causes a deceleration of the universe. Evaluating the Friedmann equation (9) at the time t₀, the spatial curvature is

K₀:= k/S₀² = H₀² (Ω₀ − 1) —– (13)

The value Ω₀ = 1 corresponds to spatially flat universes (K₀ = 0), separating models with positive spatial curvature (Ω₀ > 1 ⇔ K₀ > 0) from those with negative spatial curvature (Ω₀ < 1 ⇔ K₀ < 0).

The FL models are the standard models of modern cosmology, surprisingly effective in view of their extreme geometrical simplicity. One of their great strengths is their explanatory role in terms of making explicit the way the local gravitational effect of matter and radiation determines the evolution of the universe as a whole, this in turn forming the dynamic background for local physics (including the evolution of the matter and radiation).