Consider paramagnetic ions in a solid with a magnetic moment μ. At a sufficiently high temperature where thermal energy k_BT is larger than the interaction energy between moments ε_m, the ions can be viewed as independent and in paramagnetic phase. Their total angular momentum J contributes a magnetic disorder entropy of S = R · ln(2J + 1). Spontaneous magnetic ordering occurs as the temperature is reduced to k_BT = ε_m. The magnetic moments are oriented in a preferred direction, reducing the magnetic disorder entropy. This can also occur at high temperatures under a magnetic field.

Thus, the refrigeration method exploits the magnetic disorder entropy. Figure 1 shows the process of adiabatic demagnetization refrigeration: the magnetic field is increased from 0 to B_i at the initial temperature T_i to drive the magnetic moments into an ordered state (from A to B). The magnetic disorder entropy S is reduced during this process, and the generated heat is drained to a thermal bath. Then, the demagnetization stage is thermally isolated from the thermal bath, and the magnetic field is ramped down to B_f. By making this process adiabatic, the entropy of the system remains constant, but the entropy of the lattice is transferred to the nuclear spin system, effectively reducing the lattice temperature. Considering an ensemble of N₀ magnetic moments within a magnetic field B, the Zeeman energy of one moment is given by ε_m = gμ_αmB, where the index α = b for the Bohr magneton of electrons, and α = n for the magneton of nuclui; m is the angular momentum quantum number for the z-direction; and g is the Lande factor. Because the partition function for N₀ magnetic moments is a function of B/T, the magnetic disorder entropy is also a function of B/T. Thus, B/T is conserved during the adiabatic process, and the temperature should decrease and settle at . Once the demagnetization finishes, the stage slowly warms because of the heat leakage into the demagnetization stage (C to A).

Fig. 1.

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Fig. 1. Temperature dependence of the molar magnetic disorder entropy of a single crystal of CMN under different magnetic fields. Its angular moment is J = 1/2, with the entropy S = Rln 2. The entropy decreases at a lower temperature as higher ordering emerges. The ordering appears at a higher temperature with a stronger field. The process A → B → C indicates the process of adiabatic refrigeration. From A to B, the magnetic field is increased at the initial temperature Ti. From B to C, the magnetic field is decreased. The temperature decreases to the final temperature Tf. From C to A, the temperature is increased gradually owing to the heat leakage into the demagnetization stage. (Regenerated from Ref. [58].)

The aforementioned relationship B_i/T_i = B_f/T_f is invalid in the limit of B_f → 0 because it neglects the interactions between the magnetic moments, which should give a remnant internal field , where μ_i and r_i are the magnetic moment and the distance of other moments, respectively. In this limit, the magnetic field that one moment feels is , and the final temperature is , yielding a minimum final temperature of .

Demagnetization refrigeration can be realized with different cooling agents. The interaction energy between moments is given by , so that the onset temperature of the spontaneous magnetic ordering is T_c ∝ μ²/r³. Paramagnetic salt can be used to reduce the temperature from the kelvin range to the millikelvin range by exploiting the magnetic moment of electrons. Because the nuclear magnetic moment is on the order of 10⁻²⁷ J/T, which is 34 orders of magnitude lower than that of electrons, the T_c for the nuclear moment is far lower, on the order of 0.1 μ K. Therefore, the ANDR operates at a far lower temperature than paramagnetic salt. It usually starts at T_i ≈ 10 mK and reaches < 1 mK. Because the ANDR requires a low starting temperature, the dilution refrigerator is an ideal facility to precool it. A typical starting field is B_i = 9 T, demagnetized to an effective field b of a few millitesla. A superconducting heat switch connects the mixing chamber and the demagnetization stage (see Fig. 3). The switch is turned on during magnetization to serve as a strong thermal link for dumping heat from the cooling agent into the mixing chamber and turned off during demagnetization to isolate the cooling agent.

Fig. 2.

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	Fig. 2. (color online) Various designs of the copper demagnetization stage.

Fig. 3.

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	Fig. 3. (color online) Schematics of the demagnetization stage attached to the dilution refrigerator: (a) single Cu stage, (b) hybrid double stage with PrNi5 and Cu.

Several criteria should be considered for the choice of a proper cooling agent, as follows. (i) It should be a good conductor, so that free electrons can facilitate thermal equilibrium between the nuclear system and the lattice. (ii) It must have a relatively large nuclear magnetic moment and thus a large magnetic disorder entropy in the paramagnetic phase. (iii) It should not have a strong internal field b at low temperatures. (iv) It should have large mechanical strength and machinability. Considering these criteria with some compromise, copper and PrNi₅ are the most widely used cooling agents for the ANDR.

Because of its good machinability, electrical and thermal conductivities, affordable price, and commercial availability, copper is widely used for demagnetization refrigeration. It has a nuclear magnetic moment of I = 3/2. Given its superior conductivity, a large bulk amount of copper should be avoided to prevent eddy currents and heating during demagnetization. One way to do this is by using a bundle of fine copper rods with insulation. However, the defect density is usually high in thin rods, which limits their thermal conductivity. Moreover, the connections to flanges at both ends are difficult to make owing to thermal boundary resistance. Another effective method is to cut slits in a large copper rod and keep the entire stage as one piece (see Fig. 2). This method has the advantages of maintaining the superior mechanical properties and avoiding the trouble of making proper soldering connections. Defects and impurities affect the thermal conductivity of copper (as well as the electrical conductivity, according to the Wiedemann–Franz law). To improve its thermal conductivity, copper should be carefully annealed before installation. With proper annealing, the residual resistivity ratio can be as high as 1000. Additionally, H₂ has been found in copper, and its ortho–para conversion is a significant source of heat leakage at ultralow temperatures even when its concentration is low. Even one week after the initial cool down, 1 ppm of H₂ in 1 kg of copper can give rise to 5 nW of heat leakage.^[9] Annealing is also effective for decreasing the H₂ concentration. The annealing should be conducted in vacuum. It is advisable to introduce the proper amount of O impurities to seize magnetic impurities, weakening the internal field b. The amount of O should be fine-tuned to prevent extensive oxidation.

The main reduction of the magnetic disorder entropy of copper occurs below 10 mK, requiring an initial temperature of T_i <10 mK. On the other hand, PrNi₅, with a nuclear magnetic moment of I = 5/2 and thus a larger entropy reduction, undergoes its main entropy reduction between 100 mK and 10 mK, as shown in Fig. 4. Thus, PrNi₅ is usually used as an intermediate stage between the mixing chamber and the copper stage in two-stage ANDRs (as shown in Fig. 3(b)) if the base temperature of the dilution fridge is not sufficiently low. In addition to its large magnetic moment, it has a high specific heat, yielding a long holding time at low temperatures. However, it has poor machinability and thermal conductivity. To resolve these disadvantages, PrNi₅ is usually used as a bundle of thin rods, and high-purity copper wires should be soldered along the entire length to improve the thermal conductance.

Fig. 4.

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	Fig. 4. Nuclear spin entropy reduction of 1.4 mol of Nb (solid line), 1.0 mol of In (dotted line), 2.0 mol of Cu (dotted-dashed line), and 0.3 mol of PrNi5 (dashed line). Panels (a) and (b) plot the same set of data within a different temperature range. The aforementioned amount of each material yields the same volume of refrigerant. (Regenerated from Ref. [9].)

The ANDR operates in the temperature range of 10 mK to sub-mK. An appropriate thermometry within this range should have not only sufficient sensitivity but also superior thermal contact to the experimental stage. Common types of thermometry include ³He melting curve thermometry (³He-MCT), cerium magnesium nitrate thermometry (CMN), and Pt nuclear magnetic resonance (Pt-NMR) thermometry. ³He-MCT and Pt-NMR thermometry, which are briefly reviewed in this section, will be performed on the sub-mK station at the SECUF.

³He-MCT Both ⁴He and ³He remain liquid till absolute zero unless pressurized into a solid phase. The extensive liquid phase down to absolute zero makes both isotopes ideal refrigerants. Their melting curves, which also extend to absolute zero, can be used for proper thermometry. For ⁴He, the melting curve flattens out below ∼ 1 K (with a very shallow minimum around 0.7 K), limiting its use in thermometry to higher temperatures. In contrast, the melting curve of ³He does not have a flat region (see Fig. 5); it decreases with decreasing temperature, reaches a minimum of P_min = 2.93113 MPa at T_min = 315.24 mK, and then starts to increase again. In addition, liquid ³He transitions from a normal liquid to the superfluid A-phase (P_A = 3.43407 MPa, T_A = 2.444 mK) and then to the superfluid B-phase (P_AB = 3.43609 MPa, T_AB = 1.896 mK). Furthermore, solid ³He experiences a Néel transition (P_N = 3.43934 MPa, T_N = 0.902 K). These fixed points can be used as reference for thermometry calibration. The complete melting curve of ³He is given by the new provisional low-temperature scale (PLTS-2000) down to 0.9 mK.^[10]

Fig. 5.

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Fig. 5. (color online) Molar volume (top) and melting curve (bottom) of liquid and solid 3He. The solid circles represent data from Ref. [59], and the empty squares represent data from Ref. [60]. The orange curve is generated according to PLTS-2000.[10] The area between the two horizontal dashed lines indicates the density range for MCT. To make the density fall within this range, the temperature at which the sample hits the melting curve should be between the two temperatures indicated by the vertical dashed lines.

³He-MCT works when a solid–liquid coexisting state of ³He is present. The density of the coexisting sample should be chosen appropriately to maximize its working range. To prepare a coexisting sample, liquid ³He is pressurized to a relatively high pressure at a high temperature. The temperature is then decreased and hits the melting curve when a solid starts to form. Freezing usually occurs within the capillary, stopping liquid ³He from entering the cell; thus, the density within the cell is kept constant. The solid fraction increases as the temperature cools towards T_min. Below T_min, the solid starts to melt owing to the negative slope of the melting curve (thus negative latent heat). Complete freezing before reaching T_min and complete melting before reaching the lowest temperature should be avoided to keep the sample on the melting curve. Hence, the density should fall into a proper range: it should be between that of liquid at the lowest temperature (molar volume V_m = 254 cm³/mol) and that of solid at the melting-curve minimum (V_m = 2495 cm³/mol). As shown in Fig. 5, this restriction is translated into a starting zone where liquid can hit the melting curve, i.e., between (765 mK, 3.436 MPa) and (915 mK, 3.781 MPa). The practical upper end should be lower because a large solid fraction, although not 100%, might completely cover the pressure gauge and apply anisotropic stress on it, preventing it from measuring the real hydrostatic pressure.

Figure 6 schematically shows a device for ³He-MCT. ³He is injected into the MCT cell through a thin capillary, and a Straty–Adams capacitive pressure gauge^[11] is used to detect the pressure of the coexisting sample. Here, we wish to emphasize several points. First, coaxial cables must be used for the pressure gauge to avoid extra capacitance between leads. The two bare wires next to the plates should be shielded from each other. Second, because of the notoriously large specific heat of liquid/solid ³He and the significantly large Kapitza resistance at low temperatures, Ag sinters must be installed within the cell to improve the thermal links between the ³He and the MCT cell. Ag particles with average diameter of 70 nm should be packed on the opposite side of the pressure gauge to guarantee proper pressure measurement. Fine copper gauze could be buried into the particles before packing to prevent the sinter from cracking after multiple thermal cycles. Third, the mounting base should be made of copper with gold coating in order to minimize the thermal boundary resistance between the MCT cell and the experimental stage. Fourth, the capillaries to the MCT cell should be carefully heat-sunk at every stage, and their inner diameter should be minimized to reduce the thermal leakage along the liquid He in the capillary. Lastly, the Straty–Adams pressure gauge should be calibrated using a pressure gauge at room temperature by pressurizing the MCT cell with liquid ³He at a relatively high temperature. However, there should be a pressure gradient along the thin capillary, which is presumably constant. A correction should be made to shift the measured melting curve according to its fixed minimum point.

Fig. 6.

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	Fig. 6. Design of a cell for 3He-MCT.

Pt-NMR thermometry The ³He MCT temperature range extends to the Néel temperature of solid ³He: T_N = 0904 mK. This barely covers the sub-mK range in which the ANDR mainly operates. To date, the only type of thermometry that covers the sub-mK temperature range is NMR thermometry.^[12] Pt, which obeys Curie’s law down to this range with a large spin–lattice interaction time constant (τ₁) and a small spin–spin interaction time constant (τ₂), is an ideal material for NMR thermometry.

When a static magnetic field is applied along the z-axis , the nuclear magnetization of Pt is aligned along the field , where M_n = χ_nH_z = B_zλ_n/μ₀T, and λ_n is the Curie constant. As a short pulse of sinusoidal field is applied in the xy-plane, for instance, with ω matching the Zeeman splitting (ħω = μμ₀B_z/I, where μ and I are the relative permeability and the nuclear magnetic moment of Pt, respectively, and μ₀ is the permeability of free space), M_n is tipped from by an azimuthal angle of θ = πB₁/B_z and precesses around at the resonance frequency ω. The precession of the in-plane component causes a potential difference in a pick-up coil U(t) = αωM_n sin θ sin (ωt)e^−t/τ₂, where α is a coil-related constant, and M_n∝ 1/T. After being amplified and integrated over the decay time, U(t) is transformed into a signal S∝ 1/T, making it a proper candidate for thermometry. To determine the other parameters, one should calibrate the thermometry at a higher known temperature with the assumption that the spin–spin interaction time constant τ₂ is temperature-independent. A block diagram of Pt-NMR thermometry is shown in Fig. 8, and additional details regarding a typical pulsed NMR measurement are presented in Ref. [13]. The same coil can be used for tipping and pick-up, as the tipping and measurement are not simultaneous.

To conduct pulsed Pt-NMR thermometry in the sub-mK range and achieve reliable results, several requirements must be satisfied. The NMR signal scales with sin θ, i.e., the tipping angle of the nuclear magnetization away from the z-direction. A larger tipping angle should yield better resolution. However, the tipping field induces an eddy current within Pt, potentially introducing self-heating. Additionally, the tipping M_n effectively increases the temperature of the nuclear system from T_n to T_ncos⁻¹θ, with an increment of ΔT = T_n(cos⁻¹θ−1) ≈ T_nθ²/2 for small θ. Therefore, a tradeoff between the signal amplitude and the heating effect should be achieved. The radio-frequency field of the tipping pulse also introduces eddy-current heating and increases the electron temperature,^[14,15] which should be carefully considered and minimized. Moreover, small samples should always be used, not only to keep the eddy-current heating small but also to make the electromagnetic field fully penetrate the sample. The penetration depth is given by , where ρ is the resistivity of Pt. The typical size of Pt is on the order of 10 μm; thus, bundles of thin wires or powders are often used. On the other hand, the thermal link between the Pt sample and the experimental stage should be strong, and a highly pure Ag post is often used (see Fig. 7). Additionally, proper magnetic shielding is crucial for the thermometry. A double-layer Nb shield is used for the Pt-NMR thermometer of the ANDR at the SECUF.

Fig. 7.

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	Fig. 7. Design of the pulsed Pt-NMR thermometer.

Fig. 8.

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	Fig. 8. (a) Schematic of pulsed Pt-NMR thermometry. (b) Block diagram of Pt-NMR thermometry and a drawing of the coil assembly. The tipping and pick-up coils are separate in the drawing but can be realized with one coil in principle. The figure is adapted from Ref. [9].

³He shares some features with its isotope ⁴He, for example, large zero-point motion, remaining in the liquid state down to absolute zero, and superfluidity at low temperatures. However, being a fermion with a nuclear spin of 1/2, it features significant uniqueness: it obeys Fermi liquid theory in its normal liquid phase, and its bulk superfluid phase is divided into two phases: 1) an A-phase at the high-temperature and high-pressure corner and 2) a B-phase at a low pressure or temperature. A phase diagram of the superfluid ³He is shown in Fig. 9. With zero magnetic field, liquid ³He enters the A-phase first at T_c for pressures higher than 2.15 MPa. T_c increases with the pressure until it terminates at the melting curve at (T = 2.79 mK, P = 3.44 MPa). With further cooling, it enters into the B-phase at T_AB. For a pressure lower than 2.15 MPa, it directly enters the B-phase, at which T_c extends to 1.04 mK under zero pressure. The P–T_c curve is smooth at the polycritical point (PCP: T = 2.56 mK, P = 2.15 MPa), where normal liquid, A-phase, and B-phase coexist. The phase diagram is very sensitive to the magnetic field. Once a small field is applied, the PCP disappears, and the A-phase extends to zero pressure through a narrow stripe, meaning that normal liquid ³He has to enter the A-phase before reaching the B-phase for all pressures. Meanwhile, the A-phase splits into the A₁-phase and A₂-phase. The A₁-phase is sandwiched between the normal phase and the A₂-phase, stemming from the melting pressure at B = 0, and its width increases with the magnetic field. The A₂-phase is the same as the A-phase at zero field while A₁-phase is half of it, as discussed below.

Fig. 9.

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	Fig. 9. (color online) (a) Phase diagram of 3He under zero magnetic field. (b) Phase diagram of liquid 3He with variable temperature and magnetic field (figure credit: http://ltl.tkk.fi/images/archive/ab.jpg).

The superfluidity of ³He was accidentally observed by Osheroff, Richardson, and Lee during the study of the magnetic ordering in solid ³He along the melting curve in a Pomeranchuk cell.^[16] Making use of the inverted entropy difference between liquid and solid ³He, a solid–liquid coexisting sample was cooled to 1 mK via mechanical compression. Figure 10 shows a compression–release cycle. The temperature was reduced to its minimum (point C) during the compression and increased during the release. There was a kink at point A (P = 339053 MPa) during the cooling and another kink at A′ during the warming at the same pressure/temperature. Similarly, two jumps occurred at points B and B′ (P = 339279 MPa). These kinks were first interpreted as the entrance of the solid into a new magnetic-ordering state. However, the following NMR experiment unambiguously proved that it was the liquid, rather than the solid, that transitioned to new phases.^[17] The continuity at A and the abrupt jump at B suggest the nature of the second-order transition from the normal phase to the A-phase and the first-order transition from the A-phase to the B-phase. It was confirmed by a specific-heat experiment^[18] that there exists a jump in the specific heat at T_c and a mild kink at T_AB (see Fig. 11). With a magnetic field, the specific heat exhibits two jumps near T_c, signifying transitions into the A₁ and A₂ phases, respectively, both being second-order transitions. Although superfluid ³He was first observed in a dilution fridge under Pomeranchuk cooling, the ANDR is a far more reliable and convenient cooling method for studies in this field. The ANDR at the SECUF, extending below 1 mK, could be an ideal platform for studies on superfluid ³He.

Fig. 10.

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	Fig. 10. Pressure and temperature variations during the compression–release cycle within the Pomeranchuk cell containing liquid–solid coexisting 3He. (Regenerated from Ref. [16].)

Fig. 11.

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	Fig. 11. Specific heat of liquid 3He under (a) zero magnetic field and (b) a field of 8.8 kOe. The arrow in (a) indicates the transition to the B-phase. The arrows in (b) indicate the transition from the normal phase to the A1 phase and from the A1 phase to the A2 phase. (Regenerated from Ref. [18].)

The theory of superfluid ³He was first developed by Leggett,^[19] arguing that it is realized by pairing two ³He atoms into a Cooper pair. However, in contrast to electron pairing in Bardeen–Cooper–Schrieffer (BCS) theory, ³He atoms are regarded as rigid spheres and thus cannot occupy the same position as the s-wave bound state. Instead, they may form bound states with higher angular momentum, starting from p-wave with angular momentum l = 1 and l_z = 0, ± 1. Being fermions, spins must form a triplet instead of a singlet to preserve antisymmetry, giving s = 1, s_z = 0, ± 1. The order parameter is then represented by a complex 3 × 3 tensor A_μi with the index μ in the spin space and i in the orbital space. A_μi is composed of three parts: the superfluid gap with a phase according to U(1) symmetry Δe^iΦ, as in the s-wave superfluid state in ⁴He; the anisotropy in the orbital space described by , where and are mutually orthogonal unit vectors, and represents the direction of the orbital angular momentum; and the anisotropy in the spin space, which is described by the vector d as The second equation makes it obvious that d · s = 0, i.e., d lies in the plane perpendicular to the spin orientation.

Different superfluid phases have different forms of order parameters. The A-phase consists of only |↑↑⟩ and |↓↓⟩. Its order parameter is given as^[20] which conserves spin-rotation symmetry U(1)_S and a reduced orbital-rotation symmetry U(1)_L−N/2 as the rotation of can be compensated by gauge transformation Φ → Φ + δΦ. Its Hamiltonian is given by with the quasiparticle spectrum where τ and σ are the Pauli matrices in the particle–hole space and spin space, respectively; ; m is the mass of a ³He atom; μ is the chemical potential; c_⊥ = Δ_A/p_F; and p_F is the Fermi momentum. The energy diagram for the A-phase is shown in Fig. 12(a). There are two nodes where the energy gap closes on the Fermi surface: . The A-phase fits the state proposed by Anderson, Brinkman, and Morel, which was later called the ABM state.^[21,22] The A₁-phase exists when a magnetic field is applied. It is spin-polarized, composed of only |↑↑⟩ and |↓↓⟩; thus, it can be viewed as half of the A-phase.

Fig. 12.

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Fig. 12. (color online) (a) Spectrum of A-phase of superfluid 3He with the energy gap closed at two nodes p± and maximized at the equator. There are preferential orientations for the spontaneous orbital angular momentum l and the spin anisotropy . (b) Spectrum of the B-phase with an isotropic energy gap. The spin anisotropy orientation is locked to that of the angular momentum, but there is no preferential orientation for the composite.

The B-phase is composed of all three spin configurations. Its order parameter is given by^[20] where Δ_B is the energy gap, and R_μi represents the combined action of the spin and orbital rotation. The spin and orbital momenta are locked in the B-phase. The locking reduces two independence symmetries, SO(3)_L in the orbital space and SO(3)_S in the spin space, into a combined symmetry SO(3)_L+S. Its Hamiltonian is given by with c = Δ_B/p_F, and its spectrum gives an isotropic energy gap, as shown in Fig. 12(b). The B-phase fits the state proposed by Balian and Werthamer, which is called the BW state.^[23]

Searching for non-Abelian anyons is an interesting topic in contemporary condensed-matter research. Such quasiparticles are predicted to exist in topological superconductors, which differ from the trivial s-wave pairing superconductors. Proposals have been made to investigate the 5/2 fractional quantum Hall state^[24] and Sr₂RuO₄,^[25–27] which are suspected as p-wave (-like) superconductors. However, superfluid ³He, as an unambiguous p-wave pairing superfluid, has been overlooked. B-phase ³He is a superfluid counterpart of the three-dimensional topological insulator.^[28] With a fully gapped bulk form and a gapless surface, theoretical prediction shows that it can host the Majorana state at the surface.^[29] The existence of the surface state has been experimentally detected via acousticimpedance^[30,31] and specific-heat measurements,^[32] and this state is interpreted as an Andreev bound state. Further experiments, such as electron spin relaxation,^[29] have been proposed to search for evidence of the Majorana state.

On the other hand, the A-phase is a chiral p_x+ip_y superfluid. Dirac and Weyl quasiparticles can be realized in the low-energy approximation near the two gapless nodes. Although the A-phase exists at a high pressure and high temperature in bulk, magnetic field and spatial confinement can be used to extend it to zero temperature at a low pressure. Searching for its chirality and time-reversal symmetry breaking will pave the way for further studies on its topological properties. Clues of the chirality have been found in a pioneering experiment studying electron bubbles in liquid ³He.^[33] A two-dimensional (2D) confinement can fix its orbital angular momentum normal to the substrate surface and manifest a one-dimensional edge current. An experimental search for such a chiral edge current has been proposed.^[34]

There are some preliminary plans for superfluid ³He research to be conducted using the ANDR at the SECUF. As mentioned previously, the Majorana state is predicted to naturally exist at the edge/surface of superfluid ³He. The transport of Majorana along edges can carry heat, creating a temperature gradient. Owing to the spin–orbit interactions, the transport can be manipulated by a magnetic field. By using the experimental magnet of the ANDR at the SECUF, we expect to detect thermal transport and its variation under a magnetic field in order to find signatures of Majorana modes. The plan is still in a preliminary stage. Additional details will be confirmed during and after the construction of the ANDR.

The Hall effect was discovered in 1879. It demonstrates that a voltage perpendicular to the current is established between two edges of a conductor when a magnetic field is present. The voltage is proportional to the field. Nearly a century later, in 1980, its quantized version was observed by von Klitzing et al. in a 2D electron gas (2DEG) system.^[3] The perpendicular voltage, referred to as the Hall voltage, exhibits a stepwise increase with the magnetic field instead of a linear dependence. The corresponding Hall (transverse) resistance ρ_xy exhibits plateaus strictly at h/ne ² for an integer n (see Fig. 13), which is described as the IQHE. When the applied magnetic field is sufficiently strong, electrons follow quantized circular orbitals with discrete energy levels E_n = ħω_c(n + 1/2). The energy levels are called Landau levels, and the energy gap ħω_c is the cyclotron energy quantum. Landau levels remain discrete in the bulk but approach infinity near the edge. Correspondingly, the electron trajectories are complete circles in the bulk but incomplete at edges, giving rise to edge currents, as shown in Fig. 14. This can be explained by the fact that the Fermi level lies in the gap between two neighboring Landau levels in the bulk but cuts through the edge levels that bend upwards. The gap in which the Fermi level lies is determined by the filling factor. The number of the magnetic flux can be defined as n_Φ = B · A/Φ₀, where B is the magnetic-field magnitude, A is the area of the 2DEG, and Φ₀ = hc/e is the flux quantum. The filling factor is given as the ratio of the electron number n_e to n_Φ: ν = n_e/n_Φ = ρΦ₀/B. Obviously, an increase in B reduces the filling factor.

Fig. 13.

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	Fig. 13. Longitudinal and transverse resistances of the IQHE. (Regenerated from Ref. [61].)

Fig. 14.

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	Fig. 14. (color online) (a) Discrete orbitals with an edge current. (b) Landau levels with distortion at the edges. When the Fermi level lies between two Landau levels, the bulk is insulating, but the edge is conducting.

The FQHE was discovered by Tsui, Stormer, and Gossard in 1982.^[4] They observed that the Hall resistance is quantized at h/νe ², where ν is a fractional number. The first observation was ν = 1/3, and most of the states observed thus far have odd denominators, with a few exceptions. In contrast to the IQHE, where electrons are treated independently, electron–electron interactions are the main reason for the FQHE. A high mobility and low temperature are the key prerequisites for realizing the FQHE experimentally.

Laughlin proposed a series of wave-functions that successfully describe states with ν = 1/k (k being odd integer).^[35] Later, Jain proposed the composite-fermion approach and generalized the filling factor to a broader range.^[36] A composite fermion consists of an electron attached to an even number (α = 2p) of flux quanta. The effective field felt by the composite fermions is B^* = B − 2pρΦ₀, where ρ is the particle density. The effective filling factor for composite fermions is ν* = ρΦ₀/|B^*| = 1/|ν⁻¹ − 2p|, which gives ν = ν^*/(2pν^*±1), where ± indicates B^* being parallel/antiparallel to B. For an integer ν^*, these FQHE states can be viewed as the IQHE of composite fermions: the primary sequence of the FQHE. The FQHE of composite fermions, the secondary sequence, is also observed with ν^* as a fractional number. The hierarchical structure is shown in Fig. 15. In addition, flux quanta can be attached to holes, giving a series of filling factors ν = 1−ν^*/(2pν^*±1). Together with the electron-flux composites, this covers most of the states for ν < 1. The fractional filling can also occur at higher Landau levels, leading to , where n is the number of fully filled Landau levels, and is the effective filling factor on the partially filled level.

Fig. 15.

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	Fig. 15. (color online) Hierarchy of the FQHE. The structures of FQHE states near ν = 1/2 and 1/4 can match those near ν = 1 and 1/2, respectively, for the same set of data. (Regenerated from Ref. [62].)

The states in the aforementioned hierarchy all have odd denominators. However, ν with even denominators has also been observed. The most famous case is the ν = 5/2 state,^[37] where the lowest two Landau levels are occupied, and the third is half-filled. For the hierarchical filling factors, lim_{p → 1,ν ^* → +∞}(ν) = 1/2. The half-filled state can be considered as a Fermi sea of composite fermions, which should not exhibit the FQHE.^[38] Obviously, the FQHE at ν = 5/2 is not simply a combination of two inert occupied and one half-filled Landau levels. One theory proposes that it is a “superconducting” state of composite fermions.^[39] The formation of composite fermions overcomes the Coulomb repulsion and makes the interactions between composite fermions attractive, thus allowing pairing.^[40] Several possible states have been proposed for ν = 5/2. Moore and Read proposed a p-wave superconducting state of polarized composite fermions.^[39,41] The Moore–Read state (also known as the Pfaffian state because of its mathematical description) follows non-Abelian statistics, potentially being useful for topological quantum computations. The anti-Pfaffian state, a particle–hole conjugate of the Pfaffian state, was proposed by Morf,^[42] which also obeys non-Abelian statistics. A series of states obeying Abelian statistics have also been proposed, including one with d-wave pairing of composite fermions proposed by Haldane and Rezayi,^[43] and the “331” state proposed by Halperin et al. ^[38,44] Wen proposed two possible states: the non-Abelian U(1)×SU₂(2) state^[45,46] and the Abelian K = 8 state.^[47]

The 5/2 state was first observed by Willett et al. in Hall measurements.^[24] Subsequently, numerous experiments focused on its physical mechanism, owing to its exception to the hierarchical sequence and potential application in quantum computation. The effective fractional charge of e/4 has been observed in shot-noise measurements^[48] and by coupling 2DEG with a single-electron transistor.^[49] In addition, an interference experiment performed by Willett revealed Aharonov–Bohm oscillation with alternating periods of e/2 and e/4 ^[24] The alternation is considered as a demonstration of the non-Abelian property of the e/4 quasiparticles. Thus far, whether the ν = 5/2 state is Abelian or non-Abelian remains an open question. Various experiments have been conducted to probe the statistics, but no concensus has been reached. Although all the proposed states have an effective fractional charge of e/4, the predicted quasiparticle interaction factor g differs in each model. The tunneling experiment is an effective way to probe the interaction factor, and a previous investigation showed that the measured g favors the Abelian 331 state.^[50] However, a recent experiment indicated that both the Abelian and non-Abelian states can be realized in the same device and switched by adjusting the confinement.^[51] In addition to the magnetic field and temperature, the hydrostatic pressure affects the 5/2 state. It was observed that a quantum Hall nematic phase emerges at a certain applied pressure, causing the vanishing of the topological order and emergence of a broken symmetry.^[52]

In addition to the 5/2 state, the 12/5 state was proposed to be non-Abelian by Read and Rezayi,^[53,54] although it falls into the hierarchy in the form of 2+2/5. They proposed “parafermion” states stemming from the 1/3 Laughlin state and the 1/2 Pfaffian state. The parafermion series features a filling factor of . The filling factor 12/5 can be viewed as 2+(1−3/5), meaning that it is the particle–hole conjugate state of k = 3 with the lowest two Landau levels filled. Theories predicted that it hosts Fibonacci anyons, which permits universal topological quantum computations. The 12/5 state has been observed in a few experiments thus far.^[55–57] However, its particle–hole conjugate, the 13/5 state, has not yet been observed.

The ultralow-temperature condition is essential for studies on the FQHE because the FQHE states should be isolated from the excited states by energy gaps larger than thermal energy Δ ≫ k_BT, but Δ is already small for some states. The Δ values of FQHE states near ν = 1/2 and ν = 1/4 exhibit a linear dependence^[5] (see Fig. 16), which is consistent with the composite-fermion model: ħω_c = ħeB ^*/m ^*, where ω_c and m ^* are the cyclotron frequency and the effective mass of the composite fermions, respectively; and B ^* is the effective field that composite fermions feel. The linear dependencies shown in Fig. 16 have negative intercepts at ν = 1/2 and 1/4, owing to the broadening o the energy levels, which effectively reduces the energy gaps.

Fig. 16.

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Fig. 16. (color online) (a) Gap energies with various filling factors for two samples, plotted with closed and open circles. The horizontal scale of the magnetic field has been shifted to match the filling factors between the two samples (regenerated from Ref. [5]). (b) Dependence of the energy gaps on the effective magnetic field Beff for filling factors of 2+1/3 < ν < 2+2/3 (regenerated from Ref. [56]).

We are particularly interested in the two FQHE states that are the potential candidates for quantum computation: ν = 5/2 and 12/5. The ν = 5/2 state is difficult to observe compared with other states such as ν = 1/3. Its energy gap is strongly sample-dependent and sensitive to the electron density and mobility, as well as impurities. The energy gap can be as high as ∼ 0.6 K for samples with high mobility but can also be undetectable for non-ideal samples at the other extreme. A usual requirement to observe such states is a mobility higher than 10⁷ cm²/V·s. This requires extremely strict or challenging methods and protocols for sample growth. Only a few labs worldwide can grow samples with such high quality. In addition to strict and challenging sample-preparation techniques, light-emitting diode illumination at low temperatures has been demonstrated as an effective way to increase the mobility. Compared with the 5/2 state, the 12/5 state is more elusive and fragile. Its energy gap is also dependent on the mobility and electron density, with the optimized value measured as approximately 80 mK.^[56]

Because of the small energy gaps, experimental research on the FQHE, especially for the 5/2 and 12/5 states, requires ultralow-temperature conditions. Unfortunately, the coupling between electrons and lattices is weak at ultralow temperatures. For a dilution refrigerator without careful filtering, the electron temperature can be significantly higher than the base lattice temperature of ∼ 10 mK. Various methods have been proposed and implemented to reduce the temperature limit, such as ³He immersion and careful filtering along signal lines. Adiabatic nuclear demagnetization can cool the lattice to the sub-mK range. The low lattice temperature certainly helps in cooling electrons. However, careful signal filtering is necessary and of great importance. An electron temperature of ∼ 4 mK was realized for a 2DEG sample in the 7/2 state (Δ was measured to be 22 mK) with the lattice at 0.5 mK using the ANDR at the Institute of Physics, Chinese Academy of Sciences. The filtering will be improved for the ANDR at the SECUF.