Breakdown of Lorentz Invariance: The Order of Quantum Gravity Phenomenology. Thought of the Day 132.0

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The purpose of quantum gravity phenomenology is to analyze the physical consequences arising from various models of quantum gravity. One hope for obtaining an experimental grasp on quantum gravity is the generic prediction arising in many (but not all) quantum gravity models that ultraviolet physics at or near the Planck scale, MPlanck = 1.2 × 1019 GeV/c2, (or in some models the string scale), typically induces violations of Lorentz invariance at lower scales. Interestingly most investigations, even if they arise from quite different fundamental physics, seem to converge on the prediction that the breakdown of Lorentz invariance can generically become manifest in the form of modified dispersion relations

ω2 = ω02 + (1 + η2) c2k2 + η4(ħ/MLorentz violation)2 + k4 + ….

where the coefficients ηn are dimensionless (and possibly dependent on the particle species under consideration). The particular inertial frame for these dispersion relations is generally specified to be the frame set by cosmological microwave background, and MLorentz violation is the scale of Lorentz symmetry breaking which furthermore is generally assumed to be of the order of MPlanck.

Although several alternative scenarios have been considered to justify the modified kinematics,the most commonly explored avenue is an effective field theory (EFT) approach. Here, the focus is explicitly on the class of non-renormalizable EFTs with Lorentz violations associated to dispersion relations. Even if this framework as a “test theory” is successful, it is interesting to note that there are still significant open issues concerning its theoretical foundations. Perhaps the most pressing one is the so called naturalness problem which can be expressed in the following way: The lowest-order correction, proportional to η2, is not explicitly Planck suppressed. This implies that such a term would always be dominant with respect to the higher-order ones and grossly incompatible with observations (given that we have very good constraints on the universality of the speed of light for different elementary particles). If one were to take cues from observational leads, it is assumed either that some symmetry (other than Lorentz invariance) enforces the η2 coefficients to be exactly zero, or that the presence of some other characteristic EFT mass scale μ ≪ MPlanck (e.g., some particle physics mass scale) associated with the Lorentz symmetry breaking might enter in the lowest order dimensionless coefficient η2, which will be then generically suppressed by appropriate ratios of this characteristic mass to the Planck mass: η2 ∝ (μ/MPlanck)σ where σ ≥ 1 is some positive power (often taken as one or two). If this is the case then one has two distinct regimes: For low momenta p/(MPlanckc) ≪ (μ/MPlanck)σ the lower-order (quadratic in the momentum) deviations will dominate over the higher-order ones, while at high energies p/(MPlanckc) ≫ (μ/MPlanck)σ the higher order terms will be dominant.

The naturalness problem arises because such a scenario is not well justified within an EFT framework; in other words there is no natural suppression of the low-order modifications. EFT cannot justify why only the dimensionless coefficients of the n ≤ 2 terms should be suppressed by powers of the small ratio μ/MPlanck. Even worse, renormalization group arguments seem to imply that a similar mass ratio, μ/MPlanck would implicitly be present also in all the dimensionless n > 2 coefficients, hence suppressing them even further, to the point of complete undetectability. Furthermore, without some protecting symmetry, it is generic that radiative corrections due to particle interactions in an EFT with only Lorentz violations of order n > 2 for the free particles, will generate n = 2 Lorentz violating terms in the dispersion relation, which will then be dominant. Naturalness in EFT would then imply that the higher order terms are at least as suppressed as this, and hence beyond observational reach.

A second issue is that of universality, which is not so much a problem, as an issue of debate as to the best strategy to adopt. In dealing with situations with multiple particles one has to choose between the case of universal (particle-independent) Lorentz violating coefficients ηn, or instead go for a more general ansatz and allow for particle-dependent coefficients; hence allowing different magnitudes of Lorentz symmetry violation for different particles even when considering the same order terms (same n) in regards to momentum. Any violation of Lorentz invariance should be due to the microscopic structure of the effective space-time. This implies that one has to tune the system in order to cancel exactly all those violations of Lorentz invariance which are solely due to mode-mixing interactions in the hydrodynamic limit.

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Tantric Initiation. Thought of the Day 131.0

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Man, universe, gods and ritual are not considered separate entities but rather different manifestations of the same Śakti. Therefore, during a particular ritual every element of it is symbolic of something else. The flowers are representative of something else, the incense is representative of something else and so on. This viewpoint is based upon the crucial teaching that “worldly and spiritual” are the two faces of a same coin. One often thinks that “spirituality” is associated with something which is “within”, while “worldliness” is associated with something which is “without”. So, if you see a light “within”, that is a “spiritual” experience, while if you see a light “without”, that is a “worldly” experience. Besides, the worldliness is based on “day-to-day experiences”. It is approximately so. Tantricism considers all to be the manifestation of Śakti, the Divine Mother. So, an external light is as spiritual as an internal one and vice versa. In fact, there is neither spirituality nor worldliness because only one Supreme Consciousness is permeating everything and everyone.

Śakti or the Divine Mother is the core of all tantric practices. She is known as Kuṇḍalinī when residing in a living being. She is the bestower of the Supreme Bliss for all those followers that worship Her according to the sacred rituals and meditations contained in the Tantra-s. Her importance has been emphasized in Niruttaratantra:

बहूनां जन्मनामन्ते शक्तिज्ञानं प्रजायते।
शक्तिज्ञानं विना देवि निर्वाणं नैव जायते॥

Bahūnaṁ janmanāmante śaktijñānaṁ prajāyate|
Śaktijñānaṁ vinā devi nirvāṇaṁ naiva jāyate||

After (ante) many (bahūnām) births (janmanām), the knowledge (jñānam) of Śakti (śakti) is born (in oneself) (prajāyate). Oh goddess (devi)!, without (vinā) the knowledge (jñānam) of Śakti (śakti), Nirvāṇa — final Liberation — (nirvāṇam) does not (na eva) spring up (jāyate).

However, Tantricism should not be “strictly” equated to Shaktism, because there are groups of Śākta-s (followers of Śakti) which are not “tantric” at all. In turn, there are tantric groups that worship Śiva, Viṣṇu, etc. as well as Śakti.

Consequently, one may use a set of elements as representative of other realities. For example: a man represents Śiva and a woman represents Śakti. Then, their union is representative of that of Śiva and Śakti. Microcosm and macrocosm are closely allied to each other, because the two are the manifestation of only one Power. The following fragment extracted from the ancient Tantra-s clearly shows the aforesaid correlation between man, universe, gods and ritual. The sādhaka or practitioner is meditating on the Divine Mother (Śakti) in his heart lotus. He forms a mental image of Śakti there, and begins worshipping Her this way:

हृत्पद्मासनं दद्यात् सहस्रारच्युतामृतैः।
पाद्यं चरणयोर्दद्यान्मनसार्घ्यं निवेदयेत्॥

तेनामृतेनाचमनं स्नानीयमपि कल्पयेत्।
आकाशतत्त्वं वसनं गन्धं तु गन्धतत्त्वकम्॥

चित्तं प्रकल्पयेत् पुष्पं धूपं प्राणान् प्रकल्पयेत्।
तेजस्तत्त्वं च दीपार्थे नैवेद्यं च सुधाम्बुधिम्॥

अनाहतध्वनिं घण्टां वायुतत्त्वं च चामरम्।
नृत्यमिन्द्रियकर्माणि चाञ्चल्यं मनसस्तथा॥

पुष्पं नानाविधं दद्यादात्मनो भावसिद्धये।
अमायामनहङ्कारमरागममदं तथा॥

अमोहकमदम्भं च अद्वेषाक्षोभके तथा।
अमात्सर्यमलोभं च दशपुष्पं प्रकीर्तितम्॥

अहिंसा परमं पुष्पं पुष्पमिन्द्रियनिग्रहम्।
दयाक्षमाज्ञानपुष्पं पञ्चपुष्पं ततः परम्॥

इति पञ्चदशैर्पुष्पैर्भावपुष्पैः प्रपूजयेत्॥

Hṛtpadmāsanaṁ dadyāt sahasrāracyutāmṛtaiḥ|
Pādyaṁ caraṇayordadyānmanasārghyaṁ nivedayet||

Tenāmṛtenācamanaṁ snānīyamapi kalpayet|
Ākāśatattvaṁ vasanaṁ gandhaṁ tu gandhatattvakam||

Cittaṁ prakalpayet puṣpaṁ dhūpaṁ prāṇān prakalpayet|
Tejastattvaṁ ca dīpārthe naivedyaṁ ca sudhāmbudhim||

Anāhatadhvaniṁ ghaṇṭāṁ vāyutattvaṁ ca cāmaram|
Nṛtyamindriyakarmāṇi cāñcalyaṁ manasastathā||

Puṣpaṁ nānāvidhaṁ dadyādātmano bhāvasiddhaye|
Amāyāmanahaṅkāramarāgamamadaṁ tathā||

Amohakamadambham ca adveṣākṣobhake tathā|
Amātsaryamalobhaṁ ca daśapuṣpaṁ prakīrtitam||

Ahiṁsā paramaṁ puṣpamindriyanigraham|
Dayākṣamājñānapuṣpaṁ pañcapuṣpaṁ tataḥ param||

Iti pañcadaśairpuṣpairbhāvapuṣpaiḥ prapūjayet||

He gives (dadyāt… dadyāt) (his) heart (hṛt) lotus (padma) as the seat (āsanam), and the water for washing (pādyam) the feet (caraṇayoḥ) in the form of the nectars (amṛtaiḥ) flowing (cyuta) from Sahasrāra — the supreme Cakra placed at the crown of the head– (sahasrāra). He presents (nivedayet) the offering — lit. water offered to a guest — (arghyam) in the form of (his) mind (manasā).

He also (api) prepares (kalpayet) the water to be sipped from the palm of the hand — a purificatory ceremony that is performed before any ritual or meal — (ācamanam) (as well as) the water to be used in ablutions (snānīyam) by means of that very (tena) nectar (amṛtena). (He gives) the principle (tattvam) of Ākāśa — ether or space– (ākāśa) as the dress (vasanam), and the power of smelling (gandhatattvakam) as the odor (gandham).

He prepares (prakalpayet) (his) mind (manas) as the flower (vai) (and) arranges (prakalpayet) (his) vital energies (prāṇān) as incense (dhūpam). (He) also (ca) (arranges) the principle (tattvam) of Tejas — fire — (tejas) for it to act as (arthe) the lamp (dīpa), and (ca) the ocean (ambudhim) of nectar (sudhā) as the offering of food (naivedyam).

(He prepares) the Anāhata (anāhata) sound — which keeps sounding constantly in the heart lotus — (dhvanim) as the bell (ghaṇṭām), and (ca) the principle (tattvam) of Vāyu –air– (vāyu) as the fly-whisk made of tail of Yak (cāmaram). (He offers) the actions (karmāṇi) of the senses (indriya) as well as (tathā) the unsteadiness (cāñcalyam) of mind (manasaḥ) as dance (nṛtyam).

For realizing (siddhaye) the state (bhāva) of the Self (ātmanaḥ), he gives (dadyāt) flower(s) (puṣpam) of various sorts (nānāvidham): absence of delusion (amāyām), nonegotism (anahaṅkāram), dispassion and detachment (arāgam) as well as (tathā) absence of arrogance (amadam);…… absence of both bewilderment (amohakam) and (ca) deceit (adambham), as well as (tathā) nonmalevolence (adveṣa) and freedom from agitation (akṣobhake); absence of envy (amātsaryam) and (ca) liberty from covetousness (alobham)” — (these virtues) are named (prakīrtitam) the ten (daśa) flower(s) (puṣpam) –.

The supreme (paramam) flower(s) (puṣpam) (known as) Áhiṁsā — nonviolence and harmlessness — (ahiṁsā) and subjugation (nigraham) of the senses (indriya) (along with) the flower(s) (puṣpam) (known as) compassion (dayā), patience (kṣamā) and knowledge (jñāna), (are) therefore (tatas) the highest (param) five (pañca) flowers (puspam). Thus (iti), through (these) fifteen (pañcadaśaiḥ) flowers (puṣpaiḥ), (which are actually fifteen) flowers (puṣpaiḥ) formed from feelings (bhāva), he performs the worship (prapūjayet).

The sādhaka or practitioner uses every object in the ritual as representative of a virtue, state and so on. Therefore, one “must” be initiated in order to understand the Truth according to the Tantra-s, since only then the well-known vedic spirit of renunciation could be replaced for “a reintegration of the worldly life to the purposes of Enlightenment”. The “desire” and all values associated with it are then employed to achieve final Liberation. The tantric practitioner is both a master in spiritual matters and a master in worldly matters, because, in fact, there is no difference between “spiritual” and “worldly”. They are the two aspects in which the Divine Mother (Śakti) is manifested. So, a freed person is one who has transcended all pains and Saṁsāra (transmigration of the souls, that is, to be born and then to die, and to die and then to be born), and one who has acquired astonishing skills to lead a mundane life which is full of fulfillments.

मद्यपानेन मनुजो यदि सिद्धिं लभेत वै।
मद्यपानरताः सर्वे सिद्धिं गच्छन्तु पामराः॥११७॥

मांसभक्षणमात्रेण यदि पुण्यगतिर्भवेत्।
लोके मांसाशिनः सर्वे पुण्यभाजो भवन्त्विह॥११८॥

स्त्रीसम्भोगेन देवेशि यदि मोक्षं व्रजन्ति वै।
सर्वेऽपि जन्तवो लोके मुक्ताः स्युः स्त्रीनिषेवणात्॥११९॥

Madyapānena manujo yadi siddhiṁ labheta vai|
Madyapānaratāḥ sarve siddhiṁ gacchantu pāmarāḥ||117||

Māṁsabhakṣaṇamātreṇa yadi puṇyagatirbhavet|
Loke māṁsāśinaḥ sarve puṇyabhājo bhavantviha||118||

Strīsambhogena deveśi yadi mokṣaṁ vrajanti vai|
Sarve’pi jantavo loke muktāḥ syuḥ strīniṣevaṇāt||119||

If (yadi) a man (manujaḥ) really (vai) could attain (labheta) to Perfection (siddhim) by drinking (pānena) wine (madya), (then) may all (sarve) (those) vile (pāmarāḥ) people who are addicted to drinking (pānaratāḥ) wine (madya) achieve (gacchantu) Perfection (siddhim)!||117||

If (yadi) the achievement (gatiḥ) of Virtue (puṇya) would result (bhavet) from merely (mātreṇa) eating (bhakṣaṇa) meat (māṁsa), (then) may all (sarve) carnivorous beings (māṁsāśinaḥ) in this world (loke… iha) be (bhavantu) virtuous (puṇyabhājaḥ)!||118||

Oh goddess (deveśi)!, if (yadi) (the beings) indeed (vai) attain (vrajanti) to Liberation (mokṣam) through the enjoyment (sambhogena) of women (strī), (then) all (sarve) creatures (jantavaḥ) in this world (loke) would become (syuḥ) liberated (muktāḥ) by frequenting (niṣevaṇāt) women (strī)||119||

How Black Holes Emitting Hawking Radiation At Best Give Non-Trivial Information About Planckian Physics: Towards Entanglement Entropy.

The analogy between quantised sound waves in fluids and quantum fields in curved space-times facilitates an interdisciplinary knowhow transfer in both directions. On the one hand, one may use the microscopic structure of the fluid as a toy model for unknown high-energy (Planckian) effects in quantum gravity, for example, and investigate the influence of the corresponding cut-off. Examining the derivation of the Hawking effect for various dispersion relations, one reproduces Hawking radiation for a rather large class of scenarios, but there are also counter-examples, which do not appear to be unphysical or artificial, displaying strong deviations from Hawkings result. Therefore, whether real black holes emit Hawking radiation remains an open question and could give non-trivial information about Planckian physics.

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On the other hand, the emergence of an effective geometry/metric allows us to apply the vast amount of universal tools and concepts developed for general relativity (such as horizons), which provide a unified description and better understanding of (classical and quantum) non-equilibrium phenomena (e.g., freezing and amplification of quantum fluctuations) in condensed matter systems. As an example for such a universal mechanism, the Kibble-Zurek effect describes the generation of topological effects due to the amplification of classical/thermal fluctuations in non-equilibrium thermal phase transitions. The loss of causal connection underlying the Kibble-Zurek mechanism can be understood in terms of an effective horizon – which clearly indicates the departure from equilibrium. The associated breakdown of adiabaticity leads to an amplification of thermal fluctuations (as in the Kibble-Zurek mechanism) as well as quantum fluctuations (at zero temperature). The zero-temperature version of this amplification mechanism is completely analogous to the early universe and becomes particularly important for the new and rapidly developing field of quantum phase transitions.

Furthermore, these analogue models might provide the exciting opportunity of measuring the analogues of these exotic effects – such as Hawking radiation or the generation of the seeds for structure formation during inflation – in actual laboratory experiments, i.e., experimental quantum simulations of black hole physics or the early universe. Even though the detection of these exotic quantum effects is partially very hard and requires ultra-low temperatures etc., there is no (known) principal objection against it. The analogue models range from black and/or white hole event horizons in flowing fluids and other laboratory systems over apparent horizons in expanding Bose–Einstein condensates, for example, to particle horizons in quantum phase transitions etc.

However, one should stress that the analogy reproduces the kinematics (quantum fields in curved space-times with horizons etc.) but not the dynamics, i.e., the effective geometry/metric is not described by the Einstein equations in general. An important and strongly related problem is the correct description of the back-reaction of the quantum fluctuations (e.g., phonons) onto the background (e.g., fluid flow). In gravity, the impact of the (classical or quantum) matter is usually incorporated by the (expectation value of) energy-momentum tensor. Since this quantity can be introduced at a purely kinematic level, one may use the same construction for phonons in flowing fluids, for example, the pseudo energy-momentum tensor. The relevant component of this tensor describing the energy density (which is conserved for stationary flows) may become negative as soon as the flow velocity exceeds the sound speed. These negative contributions explain the energy balance of the Hawking radiation in black hole analogues as well as super-radiant scattering. However, the (expectation value of the) pseudo energy-momentum tensor does not determine the quantum back-reaction correctly.

One should not neglect to mention the occurrence of a horizon in the laboratory – the Unruh effect. A uniformly accelerated observer cannot see half of the (1+1- dimensional) space-time, the two Rindler wedges are completely causally disconnected by the horizon(s). In each wedge, one may introduce a set of observables corresponding to the measurements made by the observers confined to this wedge – thereby obtaining two equivalent copies of observables in one wedge. In terms of these two copies, the Minkowski vacuum is an entangled state which yields the usual phenomena (thermo-field formalism) including the Unruh effect – i.e., the uniformly accelerated observer experiences the Minkowski vacuum as a thermal bath: For rather general quantum fields (Bisognano-Wichmann theorem), the quantum state ρ obtained by restricting the Minkowski vacuum to one of the Rindler wedges behaves as a mixed state ρ = exp{−2πHˆτ/κ}/Z, where Hˆτ corresponds to the Hamiltonian generating the proper (co-moving wristwatch) time τ measured by the accelerated observer and κ is the analogue to the surface gravity and determines the acceleration.

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Space-time diagram with a trajectory of a uniformly accelerated observer and the resulting particle horizons. The observer is confined to the right Rindler wedge (region x > |ct| between the two horizons) and cannot influence or be influenced by all events in the left Rindler wedge (x < |ct|), which is completely causally disconnected.

The thermal character of this restricted state ρ arises from the quantum correlations of the Minkowski vacuum in the two Rindler wedges, i.e., the Minkowski vacuum is a multi-mode squeezed state with respect the two equivalent copies of observables in each wedge. This is a quite general phenomenon associated with doubling the degrees of freedom and describes the underlying idea of the thermo-field formalism, for example. The entropy of the thermal radiation in the Unruh and the Hawking effect can be understood as an entanglement entropy: For the Unruh effect, it is caused by averaging over the quantum correlations between the two Rindler wedges. In the black hole case, each particle of the outgoing Hawking radiation has its infalling partner particle (with a negative energy with respect to spatial infinity) and the entanglement between the two generates the entropy flux of the Hawking radiation. Instead of accelerating a detector and measuring its excitations, one could replace the accelerated observer by an accelerated scatterer. This device would scatter (virtual) particles from the thermal bath and thereby create real particles – which can be interpreted as a signature of Unruh effect.

Superfluid He-3. Thought of the Day 130.0

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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 0.25Tc (where Tc 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(ΨΨH). The vector form of the order parameter d(k) for its components can be written as:

dν(k) = ∑μ Aνμkμ —– (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 Aνμ are complex and theoretically each of them represents possible superfluid phase of 3He. Experimentally, however, only three are stable.

phasediagramLooking 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(−∆/kBT), where kB 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.

Fermi Surface Singularities

In ideal Fermi gases, the Fermi surface at p = pF = √2μm is the boundary in p-space between the occupied states (np = 1) at p2/2m < μ and empty states (np = 0) at p2/2m > μ. At this boundary (the surface in 3D momentum space) the energy is zero. What happens when the interaction between particles is introduced? Due to interaction the distribution function np of particles in the ground state is no longer exactly 1 or 0. However, it appears that the Fermi surface survives as the singularity in np. Such stability of the Fermi surface comes from a topological property of the one-particle Green’s function at imaginary frequency:

G-1 = iω – p2/2m + μ —– (1)

Let us for simplicity skip one spatial dimension pz so that the Fermi surface becomes the line in 2D momentum space (px,py); this does not change the co-dimension of zeroes which remains 1 = 3−2 = 2−1. The Green’s function has singularities lying on a closed line ω = 0, p2x + p2y = p2F in the 3D momentum-frequency space (ω,px,py). This is the line of the quantized vortex in the momentum space, since the phase Φ of the Green’s function G = |G|e changes by 2πN1 around the path embracing any element of this vortex line. In the considered case the phase winding number is N1 = 1. If we add the third momentum dimension pz the vortex line becomes the surface in the 4D momentum-frequency space (ω,px,py,pz) – the Fermi surface – but again the phase changes by 2π along any closed loop empracing the element of the 2D surface in the 4D momentum-frequency space.

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Fermi surface is a topological object in momentum space – a vortex loop. When the chemical potential μ decreases the loop shrinks and disappears at μ < 0. The point μ = T = 0 marks the Lifshitz transition between the gapless ground state at μ > 0 to the fully gapped vacuum at μ < 0.

The winding number cannot change by continuous deformation of the Green’s function: the momentum-space vortex is robust toward any perturbation. Thus the singularity of the Green’s function on the Fermi surface is preserved, even when interaction between fermions is introduced. The invariant is the same for any space dimension, since the co-dimension remains 1.

The Green function is generally a matrix with spin indices. In addition, it may have the band indices (in the case of electrons in the periodic potential of crystals). In such a case the phase of the Green’s function becomes meaning-less; however, the topological property of the Green’s function remains robust. The general analysis demonstrates that topologically stable Fermi surfaces are described by the group Z of integers. The winding number N1 is expressed analytically in terms of the Green’s function:

N= tr ∮C dl/2πi G(μ,p) ∂lG-1(μ,p) —– (2)

Here the integral is taken over an arbitrary contour C around the momentum- space vortex, and tr is the trace over the spin, band and/or other indices.

The Fermi surface cannot be destroyed by small perturbations, since it is protected by topology and thus is robust to perturbations. But the Fermi surface can be removed by large perturbations in the processes which reproduces the processes occurring for the real-space counterpart of the Fermi surface – the loop of quantized vortex in superfluids and superconductors. The vortex ring can continuously shrink to a point and then disappear, or continuously expand and leave the momentum space. The first scenario occurs when one continuously changes the chemical potential from the positive to the negative value: at μ < 0 there is no vortex loop in momentum space and the ground state (vacuum) is fully gapped. The point μ = 0 marks the quantum phase transition – the Lifshitz transition – at which the topology of the energy spectrum changes. At this transition the symmetry of the ground state does not changes. The second scenario of the quantum phase transition to the fully gapped states occurs when the inverse mass 1/m in (1) crosses zero.

Similar Lifshitz transitions from the fully gapped state to the state with the Fermi surface may occur in superfluids and superconductors. This happens, for example, when the superfluid velocity crosses the Landau critical velocity. The symmetry of the order parameter does not change across such a quantum phase transition. In the non-superconduting states, the transition from the gapless to gapped state is the metal-insulator transition.

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The Lifshitz transitions involving the vortex lines in p-space may occur be- tween the gapless states. They are accompanied by the change of the topology of the Fermi surface itself. The simplest example of such a phase transition discussed in terms of the vortex lines is provided by the reconnection of the vortex lines.

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