The Superfluid Universe

The Superfluid Universe

G.E. Volovik Low Temperature Laboratory, Aalto University, Finland
L.D. Landau Institute for Theoretical Physics, Moscow, Russia
August 4, 2019

We discuss phenomenology of quantum vacuum. Phenomenology of macroscopic systems has three sources: thermodynamics, topology and symmetry. Thermodynamics of the self-sustained vacuum allows us to treat the problems related to the vacuum energy: the cosmological constant problems. The natural value of the energy density of the equilibrium self-sustained vacuum is zero. Cosmology is discussed as the process of relaxation of vacuum towards the equilibrium state. The present value of the cosmological constant is very small compared to the Planck scale, because the present Universe is very old and thus is close to equilibrium.

Momentum space topology determines the universality classes of fermionic vacua. The Standard Model vacuum both in its massless and massive states is topological medium. The vacuum in its massless state shares the properties of superfluid He-A, which is topological superfluid. It belongs to the Fermi-point universality class, which has topologically protected fermionic quasiparticles. At low energy they behave as relativistic massless Weyl fermions. Gauge fields and gravity emerge together with Weyl fermions at low energy. This allows us to treat the hierarchy problem in Standard Model: the masses of elementary particles are very small compared to the Planck scale because the natural value of the quark and lepton masses is zero. The small nonzero masses appear in the infrared region, where the quantum vacuum acquires the properties of another topological superfluid, He-B, and 3+1 topological insulators. The other topological media in dimensions 2+1 and 3+1 are also discussed. In most cases, topology is supported by discrete symmetry of the underlying microscopic system, which indicates the important role of discrete symmetry in Standard Model.


I Introduction. Phenomenology of quantum vacuum

i.1 Vacuum as macroscopic many-body system

The aether of the 21-st century is the quantum vacuum. The quantum aether is a new form of matter. This substance has a very peculiar properties strikingly different from the other forms of matter (solids, liquids, gases, plasmas, Bose condensates, radiation, etc.) and from all the old aethers. The new aether has equation of state ; it is Lorentz invariant; and as follows from the recent cosmological observations its energy density is about g/cm (i.e. the quantum aether by 29 orders magnitude lighter than water) and it is actually anti-gravitating.

Quantum vacuum can be viewed as a macroscopic many-body system. Characteristic energy scale in our vacuum (analog of atomic scale in quantum liquids) is Planck energy GeV K. Our present Universe has extremely low energies and temperatures compared to the Planck scale: even the highest energy in the nowadays accelerators is extremely small compared to Planck energy: TeV K. The temperature of cosmic background radiation is much smaller, K.

Cosmology belongs to ultra-low frequency physics. Expansion of Universe is extremely slow: the Hubble parameter compared to the characteristic Planck frequency is . This also means that at the moment our Universe is extremely close to equilibrium. This is natural for any many-body system: if there is no energy flux from environment the energy will be radiated away and the system will be approaching the equilibrium state with vanishing temperature and motion.

According to Landau, though the macroscopic many-body system can be very complicated, at low energy and temperatures its description is highly simplified. Its behavior can be described in a fully phenomenological way, using the symmetry and thermodynamic consideration. Later it became clear that another factor also governs the low energy properties of a macroscopic system – topology. The quantum vacuum is probably a very complicated system. However, using these three sources – thermodynamics, symmetry and topology – we may try to construct the phenomenological theory of the quantum vacuum near its equilibrium state.

i.2 3 sources of phenomenology: thermodynamics, symmetry and topology

Following Landau, at low energy the macroscopic quantum system – superfluid liquid or our Universe – contains two main components: vacuum (the ground state) and matter (fermionic and bosonic quasiparticles above the ground state). The physical laws which govern the matter component are more or less clear to us, because we are able to make experiments in the low-energy region and construct the theory. The quantum vacuum occupies the Planckian and trans-Planckian energy scales and it is governed by the microscopic (trans-Planckian) physics which is still unknown. However, using our experience with a similar two-component quantum liquid we can expect that the quantum vacuum component should also obey the thermodynamic laws, which emerge in any macroscopically large system, relativistic or non-relativistic. This approach allows us to treat the cosmological constant problems. Cosmological constant was introduced by Einstein Einstein1917 (), and was interpreted as the energy density of the quantum vacuum Bronstein1933 (); Zeldovich1967 (). Astronomical observations (see, e.g., Refs. Riess-etal1998 (); Perlmutter-etal1998 (); Komatsu2008 ()) confirmed the existence of cosmological constant which value corresponds to the energy density of order with the characteristic energy scale . However, naive and intuitive theoretical estimation of the vacuum energy density as the zero-point energy of quantum fields suggests that vacuum energy must have the Planck energy scale: . The huge disagreement between the naive expectations and observations is naturally resolved using the thermodynamics of quantum vacuum discussed in Sections II–V of this review. We shall see that the intuitive estimation for the vacuum energy density as is correct, but the relevant vacuum energy which enters Einstein equations as cosmological constant is somewhat different and its value in the fully equilibrium vacuum is zero.

The second element of the Landau phenomenological approach to macroscopic systems is symmetry. It is in the basis of the modern theory of particle physics – the Standard Model, and its extension to higher energy – the Grand Unification (GUT). The vacuum of Standard Model and GUT obeys the fundamental symmetries which become spontaneously broken at low energy, and are restored when the Planck energy scale is approached from below.

This approach contains another huge disagreement between the naive expectations and observations. It concerns masses of elementary particles. The naive and intuitive estimation tells us that these masses should be on the order of the characteristic energy scale in our vacuum, which is the Planck energy scale, . However, the masses of observed particles are many orders of magnitude smaller being below the electroweak energy scale TeV . This is called the hierarchy problem. There should be a general principle, which could resolve this paradoxes. This is the principle of emergent physics based on the topology in momentum space. This approach supports our intuitive estimation of fermion masses of order , but this estimation is not valid for such vacua where the massless fermions are topologically protected.

Let us consider fermionic quantum liquid He. In laboratory we have four different states of this liquid. These are the normal liquid He, and three superfluid phases: He-A, He-B and He-A. Only one of them, He-B, is fully gapped, and for this liquid the intuitive estimation of the gap (analog of mass) in terms of the characteristic energy scale is correct. However, the other liquids are gapless. This gaplessness is protected by the momentum space topology and thus is fundamental: it does not depend much on microscopic physics being robust to the perturbative modification of the interaction between the atoms of the liquid.

i.3 Vacuum as topological medium

Topology operates in particular with integer numbers – topological charges – which do not change under small deformation of the system. The conservation of these topological charges protects the Fermi surface and another object in momentum space – the Fermi point – from destruction. They survive when the interaction between the fermions is introduced and modified. When the momentum of a particle approaches the Fermi surface or the Fermi point its energy necessarily vanishes. Thus the topology is the main reason why there are gapless quasiparticles in quantum liquids and (nearly) massless elementary particles in our Universe.

Topology provides the complementary anti-GUT approach in which the ‘fundamental’ symmetry and ‘fundamental’ fields of GUT gradually emerge together with ‘fundamental’ physical laws when the Planck energy scale is approached from above Froggatt1991 (); Volovik2003 (). The emergence of the ‘fundamental’ laws of physics is provided by the general property of topology – robustness to details of the microscopic trans-Planckian physics. As a result, the physical laws which emerge at low energy together with the matter itself are generic. They do not depend much on the details of the trans-Planckian subsystem, being determined by the universality class, which the whole system belongs to.

In this scheme, fermions are primary objects. Approaching the Planck energy scale from above, they are transformed to the Standard Model chiral fermions and give rise to the secondary objects: gauge fields and gravity. Below the Planck scale, the GUT scenario intervenes giving rise to symmetry breaking at low energy. This is accompanied by formation of composite objects, Higgs bosons, and tiny Dirac masses of quark and leptons.

In the GUT scheme, general relativity is assumed to be as fundamental as quantum mechanics, while in the second scheme general relativity is a secondary phenomenon. In the anti-GUT scheme, general relativity is the effective theory describing the dynamics of the effective metric experienced by the effective low-energy fields. It is a side product of quantum field theory or of the quantum mechanics in the vacuum with Fermi point.

Vacua with topologically protected gapless (massless) fermions are representatives of the broader class of topological media. In condensed matter it includes topological insulators (see reviews HasanKane2010 (); Xiao-LiangQi2011 ()), topological semimetals (see Abrikosov1971 (); Abrikosov1998 (); Burkov2011 (); XiangangWan2011 (); Ryu2002 (); Manes2007 (); Vozmediano2010 (); Cortijo2011 ()), topological superconductors and superfluids, states which experience quantum Hall effect, and other topologically nontrivial gapless and gapped phases of matter. Topological media have many peculiar properties: topological stability of gap nodes; topologically protected edge states including Majorana fermions; topological quantum phase transitions occurring at ; topological quantization of physical parameters including Hall and spin-Hall conductivity; chiral anomaly; topological Chern-Simons and Wess-Zumino actions; etc.

It appears that quantum vacuum of Standard Model is topologically nontrivial both in its massless and massive states. In the massless state the quantum vacuum is topologically similar to the superfluid He-A and gapless semimetal. In the massive state the quantum vacuum is topologically similar to the superfluid He-B and 3+1 dimensional topological insulator. This is discussed in Sections VI–VIII.

Ii Quantum vacuum as self-sustained medium

ii.1 Vacuum energy and cosmological constant

There is a huge contribution to the vacuum energy density, which comes from the ultraviolet (Planckian) degrees of freedom and is of order . The observed cosmological is smaller by many orders of magnitude and corresponds to the energy density of the vacuum . In general relativity, the cosmological constant is arbitrary constant, and thus its smallness requires fine-tuning. If gravitation would be a truly fundamental interaction, it would be hard to understand why the energies stored in the quantum vacuum would not gravitate at all Nobbenhuis2006 (). If, however, gravitation would be only a low-energy effective interaction, it could be that the corresponding gravitons as quasiparticles do not feel all microscopic degrees of freedom (gravitons would be analogous to small-amplitude waves at the surface of the ocean) and that the gravitating effect of the vacuum energy density would be effectively tuned away and cosmological constant would be naturally small or zero Volovik2003 (); Dreyer2007 ().

ii.2 Variables for Lorentz invariant vacuum

A particular mechanism of nullification of the relevant vacuum energy works for such vacua which have the property of a self-sustained medium KlinkhamerVolovik2008a (); KlinkhamerVolovik2008b (); KlinkhamerVolovik2009b (); KlinkhamerVolovik2008jetpl (); KlinkhamerVolovik2009a (); KlinkhamerVolovik2010 (). A self-sustained vacuum is a medium with a definite macroscopic volume even in the absence of an environment. A condensed matter example is a droplet of quantum liquid at zero temperature in empty space. The observed near-zero value of the cosmological constant compared to Planck-scale values suggests that the quantum vacuum of our universe belongs to this class of systems. As any medium of this kind, the equilibrium vacuum would be homogeneous and extensive. The homogeneity assumption is indeed supported by the observed flatness and smoothness of our universe de Bernardis2000 (); Hinshaw2007 (); Riess2007 (). The implication is that the energy of the equilibrium quantum vacuum would be proportional to the volume considered.

Usually, a self-sustained medium is characterized by an extensive conserved quantity whose total value determines the actual volume of the system LL1980 (); Perrot1998 (). The quantum liquid at is a self sustained system because of the conservation law for the particle number , and its state is characterized by the particle density which acquires a non-zero value in the equilibrium ground state. As distinct from condensed matter systems, the quantum vacuum of our Universe is a relativistic invariant system. The Lorentz invariance of the vacuum imposes strong constraints on the possible form this variable can take. One must find the relativistic analog of the particle density . An example of a possible vacuum variable is a symmetric tensor , which in a homogeneous vacuum is proportional to the metric tensor


This variable satisfies the Lorentz invariance of the vacuum. Another example is the 4-tensor , which in a homogeneous vacuum is proportional either to the fully antisymmetric Levi–Civita tensor:


or to the product of metric tensors such as:


Scalar field is also the Lorentz invariant variable, but it does not satisfy another necessary condition of the self sustained system: the vacuum variable must obey some kind of the conservation law. Below we consider some examples satisfying the two conditions: Lorentz invariance of the perfect vacuum state and the conservation law.

ii.3 Yang-Mills chiral condensate as example

Let us first consider as an example the chiral condensate of gauge fields. It can be the gluonic condensate in QCD Shifman1992 (); Shifman1991 (), or any other condensate of Yang-Mills fields, if it is Lorentz invariant. We assume that the Savvidy vacuum Savvidy () is absent, i.e. the vacuum expectation value of the color magnetic field is zero (we shall omit color indices):


while the vacuum expectation value of the quadratic form is nonzero:


Here is the anomaly-driven topological condensate (see e.g. HalperinZhitnitsky1998 ()):


In the homogeneous static vacuum state, the -condensate violates the and symmetries of the vacuum, but it conserves the combined symmetry symmetry.

ii.3.1 Cosmological term

Let us choose the vacuum action in the form


with given by (6). The energy-momentum tensor of the vacuum field is obtained by variation of the action over :


Using (5) and (6) one obtains


and thus


In Einstein equations this energy momentum tensor plays the role of the cosmological term:


It is important that the cosmological constant is given not by the vacuum energy as is usually assumed, but by thermodynamic potential , where is thermodynamically conjugate to variable, . Below, when we consider dynamcs we shall see that this fact reflects the conservation of the variable .

The crucial difference between the vacuum energy and thermodynamic potential is revealed when we consider the corresponding quantities in the ground state of quantum liquids, the energy density of the liquid and the density of the grand canonical energy, , which enters macroscopic thermodynamics due to conservation of particle number. The first one, , has the value dictated by atomic physics, which is equivalent to in the quantum vacuum. On the contrary, the second one equals minus pressure, , according to the Gibbs-Duhem thermodynamic relation at . Thus its value is dictated not by the microscopic physics, but by external conditions. In the absence of environment, the external pressure is zero, and the value of in a fully equilibrium ground state of the liquid is zero. This is valid for any self-sustained macroscopic system, including the self-sustained quantum vacuum, which suggests the natural solution of the main cosmological constant problem.

ii.3.2 Conservation law for

Equation for in flat space can be obtained from Maxwell equation, which in turn is obtained by variation of the action over the gauge field :


where is the covariant derivative. Since , equation (36) is reduced to


The solution of this equation is


where is integration constant. In thermodynamics, this will play the role of the chemical potential, which is thermodynamically conjugate to . This demonstrates that obeys the conservation law and thus can be the proper variable for description the self-sustained vacuum.

ii.4 4-form field as example

Another example of the vacuum variable appropriate for the self-sustained vacuum is given by the four-form field strength DuffNieuwenhuizen1980 (); Aurilia-etal1980 (); Hawking1984 (); HenneauxTeitelboim1984 (); Duff1989 (); DuncanJensen1989 (); BoussoPolchinski2000 (); Aurilia-etal2004 (); Wu2008 (), which is expressed in terms of in the following way:


where the Levi–Civita tensor density; and the square bracket around spacetime indices complete anti-symmetrization.

Originally the quadratic action has been used for this field  DuffNieuwenhuizen1980 (); Aurilia-etal1980 (), which corresponds to the special case of (7) with . For the general one obtains the Maxwell equation


Using (15a) the Maxwell equation is reduced to


The first integral of (17) with integration constant gives again Eq.(14), which reflects the conservation law for .

Variation of the action over gives again the cosmological constant (11) with . This demonstrates the universality of the macroscopic description of the self-sustained vacuum: description of the quantum vacuum in terms of does not depend on the microscopic details of the vacuum and on the nature of the vacuum variable.

ii.5 Aether field as example

Another example of the vacuum variable may be through a four-vector field . This vector field could be the four-dimensional analog of the concept of shift in the deformation theory of crystals. (Deformation theory can be described in terms of a metric field, with the role of torsion and curvature fields played by dislocations and disclinations, respectively; see, e.g., Ref. Dzyaloshinskii1980 () for a review.) A realization of could be also a 4–velocity field entering the description of the structure of spacetime. It is the 4-velocity of “aether” Jacobson2007 (); Gasperini1987 (); Jacobson2001 (); WillNordvedt ().

The nonzero value of the 4-vector in the vacuum violates the Lorentz invariance of the vacuum. To restore this invariance one may assume that is not an observable variable, instead the observables are its covariant derivatives . This means that the action does not depend on explicitly but only depends on :


with an energy density containing even powers of :


According to the imposed conditions, the tensors and depend only on or and the same holds for the other –like tensors in the ellipsis of (19). In particular, the tensor of the quadratic term in (19) has the following form in the notation of Ref. Jacobson2007 ():


for real constants . Distinct from the original aether theory in Ref. Jacobson2007 (), the tensor (20) does not contain a term , as such a term would depend explicitly on and contradict the Lorentz invariance of the quantum vacuum.

The equation of motion for in flat space,


has the Lorentz invariant solution expected for a vacuum-variable –type field:


With this solution, the energy density in the action (18) is simply in terms of contracted coefficients , , and from (19). However, just as for previous examples, the energy-momentum tensor of the vacuum field obtained by variation over and evaluated for solution (22) is expressed again in terms of the thermodynamic potential:


which corresponds to cosmological constant in Einstein’s gravitational field equations.

Iii Thermodynamics of quantum vacuum

iii.1 Liquid-like quantum vacuum

The zeroth order term in (19) corresponds to a “bare” cosmological constant which can be considered as cosmological constant in the “empty” vacuum – vacuum with :


The nonzero value in the self-sustained vacuum does not violate Lorentz symmetry but leads to compensation of the bare cosmological constant in the equilibrium vacuum. This illustrates the important difference between the two states of vacua. The quantum vacuum with can exist only with external pressure . By analogy with condensed-matter physics, this kind of quantum vacuum may be called “gas-like” (Fig. 1). The quantum vacuum with nonzero is self-sustained: it can be stable at , provided that a stable nonzero solution of equation exists. This kind of quantum vacuum may then be called “liquid-like”.

The universal behavior of the self-sustained vacuum in equilibrium suggests that it obeys the same thermodynamic laws as any other self-sustained macroscopic system described by the conserved quantity , such as quantum liquid. In other words, vacuum can be considered as a special quantum liquid which is Lorentz invariant in its ground state. This liquid is characterized by the Lorentz invariant “charge” density – an analog of particle density in non-relativistic quantum liquids.

Figure 1: Vacuum as a medium obeying macroscopic thermodynamic laws. Relativistic vacuum possesses energy density, pressure and compressibility but has no momentum. In equilibrium, the vacuum pressure equals the external pressure acting from the environment. The “gas-like” vacuum may exist only under external pressure. The “liquid-like” vacuum is self-sustained: it can be stable in the absence of external pressure. The thermodynamic energy density of the vacuum which enters the vacuum equation of state does not coincide with the microscopic vacuum energy . While the natural value of is determined by the Planck scale, , the natural value of the macroscopic quantity is zero for the self-sustained vacuum which may exist in the absence of environment, i.e. at . This may explain why the present cosmological constant is small.

Let us consider a large portion of such vacuum liquid under external pressure KlinkhamerVolovik2008a (). The volume of quantum vacuum is variable, but its total “charge” must be conserved, . The energy of this portion of quantum vacuum at fixed total“charge” is then given by the thermodynamic potential


where is the energy density in terms of charge density . As the volume of the system is a free parameter, the equilibrium state of the system is obtained by variation over the volume :


This gives an integrated form of the Gibbs–Duhem equation for the vacuum pressure:


whose solution determines the equilibrium value and the corresponding volume .

iii.2 Macroscopic energy of quantum vacuum

Since the vacuum energy density is the vacuum pressure with minus sign, equation (27) suggests that the relevant vacuum energy, which is revealed in thermodynamics and dynamics of the low-energy Universe, is:


This is confirmed by Eqs. (11) and (23) for energy-momentum tensor of the self-sustained vacuum, which demonstrates that it is rather than , which enters the equation of state for the vacuum and thus corresponds to the cosmological constant:


Figure 2: Contribution of different energy scales into the macroscopic energy of the self-sustained system at . Zero point energy of the effective bosonic and fermionic quantum fields gives rise to the diverging contribution to the energy of the system. In quantum vacuum it is of order of . In equilibrium this contribution is compensated without fine-tuning by microscopic degrees of freedom of the system (by trans-Planckian degrees of quantum vacuum correspondingly).

While the energy of microscopic quantity is determined by the Planck scale, , the relevant vacuum energy which sources the effective gravity is determined by a macroscopic quantity – the external pressure. In the absence of an environment, i.e. at zero external pressure, , one obtains that the pressure of pure and equilibrium vacuum is exactly zero:


Equation determines the equilibrium value of the equilibrium self-sustained vacuum. Thus from the thermodynamic arguments it follows that for any effective theory of gravity the natural value of is zero in equilibrium vacuum.

This result does not depend on the microscopic structure of the vacuum from which gravity emerges, and is actually the final result of the renormalization dictated by macroscopic physics. In the self-sustained quantum liquid the large contribution of zero-point energy of phonon field is naturally compensated by microscopic (atomic) degrees of freedom of quantum liquid. In the same manner, the huge contribution of zero-point energy of macroscopic fields to the vacuum energy is naturally compensated by microscopic degrees of the self sustained quantum vacuum: the vacuum variable is adjusted automatically to nullify the macroscopic vacuum energy, . The actual spectrum of the vacuum energy density (meaning the different contributions to from different energy scales) is not important for the cancellation mechanism, because it is dictated by thermodynamics. The particular example of the spectrum of the vacuum energy density is shown in Fig. 2, where the positive energy of the quantum vacuum, which comes from the zero-point energy of bosonic fields, is compensated by negative contribution from trans-Planckian degrees of freedom VolovikSpectrum () .

Using the quantum-liquid counterpart of the self-sustained quantum vacuum as example, one may predict the behavior of the vacuum after cosmological phase transition, when is kicked from its zero value. The vacuum will readjust itself to a new equilibrium state with new so that will again approach its equilibrium zero value KlinkhamerVolovik2008a (). The process of relaxation of the system to the equilibrium state depends on details of dynamics of the vacuum variable and its interaction with matter fields, and later on we shall consider some examples of dynamical relaxation of .

iii.3 Compressibility of the vacuum

Using the standard definition of the inverse of the isothermal compressibility, (Fig. 1), one obtains the compressibility of the vacuum by varying Eq.(27) at fixed KlinkhamerVolovik2008a ():


A positive value of the vacuum compressibility is a necessary condition for the stability of the vacuum. It is, in fact, the stability of the vacuum, which is at the origin of the nullification of the cosmological constant in the absence of an external environment.

From the low-energy point of view, the compressibility of the vacuum is as fundamental physical constant as the Newton constant . It enters equations describing the response of the quantum vacuum to different perturbations. While the natural value of the macroscopic quantity (and ) is zero, the natural values of the parameters and are determined by the Planck physics and are expected to be of order and correspondingly.

iii.4 Thermal fluctuations of and the volume of Universe

The compressibility of the vacuum , though not measurable at the moment, can be used for estimation of the lower limit for the volume of the Universe. This estimation follows from the upper limit for thermal fluctuations of cosmological constant Volovik2004 (). The mean square of thermal fluctuations of equals the mean square of thermal fluctuations of the vacuum pressure, which in turn is determined by thermodynamic equation LL1980 ():


Typical fluctuations of the cosmological constant should not exceed the observed value: . Let us assume, for example, that the temperature of the Universe is determined by the temperature of the cosmic microwave background radiation. Then, using our estimate for vacuum compressibility , one obtains that the volume of our Universe highly exceeds the Hubble volume – the volume of visible Universe inside the present cosmological horizon:


This demonstrates that the real volume of the Universe is certainly not limited by the present cosmological horizon.

Iv Dynamics of quantum vacuum

iv.1 Action

In section II a special quantity, the vacuum “charge” , was introduced to describe the statics and thermodynamics of the self-sustained quantum vacuum. Now we can extend this approach to the dynamics of the vacuum charge. We expect to find some universal features of the vacuum dynamics, using several realizations of this vacuum variable. We start with the 4-form field strength DuffNieuwenhuizen1980 (); Aurilia-etal1980 (); Hawking1984 (); HenneauxTeitelboim1984 (); Duff1989 (); DuncanJensen1989 (); BoussoPolchinski2000 (); Aurilia-etal2004 (); Wu2008 () expressed in terms of . The low-energy effective action takes the following general form:


where denotes the Ricci curvature scalar; and is matter action. Throughout, we use the same conventions as in Ref. Weinberg1988 (), in particular, those for the Riemann curvature tensor and the metric signature .

The vacuum energy density in (34a) depends on the vacuum variable which in turn is expressed via the 3-form field and metric field in (34b). The field combines all the matter fields of the Standard Model. All possible constant terms in matter action (which includes the zero-point energies from the Standard Model fields) are absorbed in the vacuum energy .

Since describes the state of the vacuum, the parameters of the effective action – the Newton constant and parameters which enter the matter action – must depend on . This dependence results in particular in the interaction between the matter fields and the vacuum. There are different sources of this interaction. For example, in the gauge field sector of Standard Model, the running coupling contains the ultraviolet cut-off and thus depends on :


where is the field strength of the particular gauge field (we omitted the color indices). In the fermionic sector, should enter parameters of the Yukawa interaction and fermion masses.

iv.2 Vacuum dynamics

The variation of the action (34a) over the three-form gauge field gives the generalized Maxwell equations for -field,


Using (34c) for , we find that the solutions of Maxwell equations (36) are still determined by the integration constant


iv.3 Generalized Einstein equations

The variation over the metric gives the generalized Einstein equations,


where is the invariant d’Alembertian; and is the energy-momentum tensor of the matter fields, obtained by variation over at constant , i.e. without variation over , which enters .

Eliminating and from (38) by use of (37), the generalized Einstein equations become




For the special case when the dependence of the Newton constant and matter action on is ignored, (39) reduces to the standard Einstein equation of general relativity with the constant cosmological constant .

iv.4 Minkowski-type solution and Weinberg problem

Among different solutions of equations (36) and (38) there is the solution corresponding to perfect equilibrium Minkowski vacuum without matter. It is characterized by the constant in space and time values and obeying the following two conditions:


The two conditions (41a)–(41b) can be combined into a single equilibrium condition for :


with the derived quantity


In order for the Minkowski vacuum to be stable, there is the further condition: where corresponds to the isothermal vacuum compressibility (31KlinkhamerVolovik2008a (). In this equilibrium vacuum the gravitational constant can be identified with Newton’s constant .

Let us compare the conditions for the equilibrium self-sustained vacuum, (42) and (43), with the two conditions suggested by Weinberg, who used the fundamental scalar field for the description of the vacuum. In this description there are two constant-field equilibrium conditions for Minkowski vacuum, and , see Eqs. (6.2) and (6.3) in  Weinberg1988 (). These two conditions turn out to be inconsistent, unless the potential term in is fine-tuned (see also Sec. 2 of Ref. Weinberg1996 ()). In other words, the Minkowski vacuum solution may exist only for the fine-tuned action. This is the Weinberg formulation of the cosmological constant problem.

The self-sustained vacuum naturally bypasses this problem KlinkhamerVolovik2010 (). Equation  corresponds to the equation (42). However, the equation is relaxed in the -theory of self-sustained vacuum. Instead of the condition , the conditions are , which allow for having with an arbitrary constant . This is the crucial difference between a fundamental scalar field and the variable describing the self-sustained vacuum. As a result, the equilibrium conditions for and can be consistent without fine-tuning of the original action. For Minkowski vacuum to exist only one condition (42) must be satisfied. In other words, the Minkowski vacuum solution exists for arbitrary action provided that solution of equation (42) exists.

Figure 3: A set of Minkowski equilibrium vacua emerging for a particular choice of the vacuum energy density function in (44). In each vacuum the huge bare cosmological constant is compensated by the -field. The curves of the top panel show the left-hand side of (45a) for those values of that obey the stability condition (45b). The curves of the bottom panel show the corresponding positive segments of the inverse of the dimensionless vacuum compressibility . Minkowski-type vacua are obtained at the intersection points of the curve of the top panel with a horizontal line at the value [for example, the dashed line at gives the value corresponding to the heavy dot in the top panel]. Each such vacuum is characterized, in part, by the corresponding value of the inverse vacuum compressibility from the bottom panel shown by the heavy dot for the case chosen in the top panel]. Minkowski vacua with positive compressibility are stable and become attractors in a dynamical context (cf. next Section).

iv.5 Multiple Mnkowski vacua

It is instructive to illustrate this using a concrete example. The particular choice for the vacuum energy density function is considered in KlinkhamerVolovik2010 ():


It contains the higher-order terms in addition to the standard quadratic term . With (44), the expressions for the equilibrium condition (42) and the stability condition (31) become


where dimensionless quantities and are introduced. A straightforward graphical analysis (Fig. 3) shows that, for any , there are infinitely many equilibrium states of quantum vacuum, i.e. infinitely many values which obey both (45a) and (45b). Each of these vacua has its own values of the Newton constant and Standard Model parameters. But all these vacua have zero cosmological constant: the Planck-scale bare cosmological constant is compensated by the field in any equilibrium vacuum. The top panel of Fig. 3 shows that the values on the one segment singled-out by the heavy dot already allow for a complete cancellation of any value between and .

V Cosmology as approach to equilibrium

v.1 Energy exchange between vacuum and gravity+matter

In the curved Universe and/or in the presence of matter, becomes space-time dependent due to interaction with gravity and matter (see (37)). As a result the vacuum energy can be transferred to the energy of gravitational field and/or to the energy of matter fields. This also means that the energy of matter is not conserved. The energy-momentum tensor of matter , which enters the generalized Einstein equations (39), is determined by variation over at constant . That is why it is not conserved:


The matter energy can be transferred to the vacuum energy due to interaction with -field. Using (37) and the equation (40) for cosmological constant one obtains that the vacuum energy is transferred both to gravity and matter with the rate:


The energy exchange between the vacuum and gravity+matter allows for the relaxation of the vacuum energy and cosmological “constant”.

v.2 Dynamic relaxation of vacuum energy

Let us assume that we can make a sharp kick of the system from its equilibrium state. For quantum liquids (or any other quantum condensed matter) we know the result of the kick: the liquid or superconductor starts to move back to the equilibrium state, and with or without oscillations it finally approaches the equilibrium VolkovKogan1974 (); Barankov2004 (); Yuzbashyan2005 (); Yuzbashyan2008 (); Gurarie2009 (). The same should happen with the quantum vacuum. Let us consider this behavior using the realization of the vacuum field in terms of the 4-form field, when serves as the overall integration constant. We start with the fully equilibrium vacuum state, which is characterized by the values and in (41). The kick moves the variable away from its equilibrium value, while still remains the same being the overall integration constant, . In the non-equilibrium state which arises immediately after the kick, the vacuum energy is non-zero and big. If the kick is very sharp, with the time scale of order , the energy density of the vacuum can reach the Planck-scale value, .

For simplicity we ignore the interaction between the vacuum and matter. Then from the solution of dynamic equations (37) and (39) with one finds that after the kick does return to its equilibrium value in the Minkowski vacuum. At late time the relaxation has the following asymptotic behavior: KlinkhamerVolovik2008b ()


where oscillation frequency is of the order of the Planck-energy scale . The gravitational constant approaches its Newton value also with the power-law modulation:


The vacuum energy relaxes to zero in the following way:

For the Planck scale kick, the vacuum energy density after the kick, i.e. at , has a Planck-scale value, . According to (50a), at present time it must reach the value
where is the Hubble parameter. This value approximately corresponds to the measured value of the cosmological constant.
This, however, can be considered as an illustration of the dynamical reduction of the large value of the cosmological constant, rather than the real scenario of the evolution of the Universe. We did not take into account quantum dissipative effects and the energy exchange between vacuum and matter. Indeed, matter field radiation (matter quanta emission) by the oscillations of the vacuum can be expected to lead to faster relaxation of the initial vacuum energy Starobinsky1980 (),

with a decay rate .

Nevertheless, the cancellation mechanism and example of relaxation provide the following lesson. The Minkowski-type solution appears without fine-tuning of the parameters of the action, precisely because the vacuum is characterized by a constant derivative of the vacuum field rather than by a constant vacuum field itself. As a result, the parameter emerges in (41a) as an integration constant, i.e., as a parameter of the solution rather than a parameter of the action. Since after the kick the integration constant remains intact, the Universe will return to its equilibrium Minkowski state with , even if in the non-equilibrium state after the kick the vacuum energy could reach . The idea that the constant derivative of a field may be important for the cosmological constant problem has been suggested earlier by Dolgov Dolgov1985 (); Dolgov1997 () and Polyakov Polyakov1991 (); PolyakovPrivateComm (), where the latter explored the analogy with the Larkin–Pikin effect LarkinPikin1969 () in solid-state physics.

v.3 Minkowski vacuum as attractor

The example of relaxation of the vacuum energy in Sec. V.2 has the principle drawback. Instead of the fine-tuning of the action, which is bypassed in the self-sustained vacuum, we have the fine-tuning of the integration constant. We assumed that originally, before the kick, the Universe was in its Minkowski ground state, and thus the specific value of the integration constant has been chosen, that fixes the value of the original Minkowski equilibrium vacuum. In the 4-form realization of the vacuum field, any other choice of the integration constant () leads asymptotically to a de-Sitter-type solution KlinkhamerVolovik2008b ().

To avoid this fine-tuning and obtain the natural relaxation of to , which as we know occurs in quantum liquids, we must relax the condition on . It should not serve as an overall integration constant, while remaining the conjugate variable in thermodynamics. Then using the condensed matter experience one may expect that the Minkowski equilibrium vacuum becomes an attractor and the de-Sitter solution with will inevitably relax to Minkowski vacuum with . This expectation is confirmed in the aether type realization of the vacuum variable in terms of a vector field as discussed in Sec. II.5.

The constant vacuum field there appears as the derivative of a vector field in the specific solution corresponding to the equilibrium vacuum, . In this realization, the effective chemical potential plays a role only for the equilibrium states (i.e., for their thermodynamical properties), but does not appear as an integration constant for the dynamics. Hence, the fine-tuning problem of the integration constant is overcome, simply because there is no integration constant.

The instability of the de-Sitter solution towards the Minkowski one has been already demonstrated by Dolgov Dolgov1997 (), who considered the simplest quadratic choices of the Lagrange density of . But his result also holds for the generalized Lagrangian with a generic function in Sec. II.5 KlinkhamerVolovik2010 ().

Figure 4: Aaether-field evolution and Minkowski attractor in a spatially flat Friedmann–Robertson–Walker universe in Dolgov model Dolgov1997 () (see KlinkhamerVolovik2010 () for details). The bare cosmological constant is . Four numerical solutions correspond to different boundary conditions, but all approach the Minkowski-spacetime solution (51). The Minkowski vacuum is an attractor because the vacuum compressibility (31) is positive, .

The Dolgov scenario does not require the variable gravitational coupling parameter, so that we use . In this scenario, for a spatially flat Robertson–Walker metric with cosmic time and scale factor , the initial de-Sitter-type expansion evolves towards the Minkowski attractor by the following asymptotic solution for the aether-type field :


where the Hubble parameter . At large cosmic times , the curvature terms decay as and the Einstein equations lead to the nullification of the energy-momentum tensor of the field: . Since (51) with satisfies the –theory Ansatz , the energy-momentum tensor is completely expressed by the single constant : . As a result, the equation leads to the equilibrium condition (42) for the Minkowski vacuum and to the equilibrium value in (51).

Figure 4 shows explicitly the attractor behavior for the simplest case of Dolgov action, with the numerical value of in (51) appearing dynamically. This simple version of Dolgov scenario does not appear to give a realistic description of the present Universe RubakovTinyakov1999 () and requires an appropriate modification EmelyanovKlinkhamer2011 (). It nevertheless demonstrates that the compensation of a large initial vacuum energy density can occur dynamically and that Minkowski spacetime can emerge spontaneously, without setting a chemical potential. In other words, an “existence proof” has been given for the conjecture that the appropriate Minkowski value can result from an attractor-type solution of the field equations. The only condition for the Minkowski vacuum to be an attractor is a positive vacuum compressibility (31).

Figure 5: Dashed curve: relaxation according to the relation in Eq.(50a). Full curve: Sketch of the relaxation of the vacuum energy density during the evolution of the Universe according to Ref. KlinkhamerVolovik2011a (). The origin of the current plateau in the vacuum energy is discussed in Sec. V.4.

v.4 Remnant cosmological constant

Figure 5 demonstrates the possible more realistic scenario with a step-wise relaxation of the vacuum energy density KlinkhamerVolovik2011a (). The vacuum energy density moves from plateau to plateau responding to the possible phase transitions or crossovers in the Standard Model vacuum and follows, on average, the steadily decreasing matter energy density. The origin of the current plateau with a small positive value of the vacuum energy density is still not clear. It may result from the phenomena, which occur in the infrared. It may come for example from anomalies in the neutrino sector of the quantum vacuum, such as non-equilibrium contribution of the light massive neutrinos to the quantum vacuum KlinkhamerVolovik2011a (); reentrant violation of Lorentz invariance Volovik2001 () and Fermi point splitting in the neutrino sector KlinkhamerVolovik2005b (); KlinkhamerVolovik2011b () (see Sec. VII.7). The other possible sources include the QCD anomaly Schutzhold2002 (); KlinkhamerVolovik2009a (); UrbanZhitnitsky2009 (); Ohta2011 (); Holdom2011 (); torsion Poplawski2011 (); relaxation effects during the electroweak crossover KlinkhamerVolovik2009b (); etc. Most of these scenarios are determined by the momentum space topology of the quantum vacuum.

v.5 Summary and outlook

To study the problems related to quantum vacuum one must search for the proper extension of the current theory of elementary particle physics – the Standard Model. However, many properties of the quantum vacuum can be understood by extending of our experience with self-sustained macroscopic systems to the quantum vacuum. A simple picture of quantum vacuum is based on three assumptions: (i) The quantum vacuum is a self-sustained medium – the system which is stable at zero external pressure, like quantum liquids. (ii) The quantum vacuum is characterized by a conserved charge , which is analog of the particle density in quantum liquids and which is non-zero in the ground state of the system, . (iii) The quantum vacuum with is a Lorentz-invariant state. This is the only property which distinguishes the quantum vacuum from the condensed-matter quantum liquids.

These assumptions naturally solve the main cosmological constant problem without fine-tuning. In any self-sustained system, relativistic or non-relativistic, in thermodynamic equilibrium at the zero-point energy of quantum fields is fully compensated by the microscopic degrees of freedom, so that the relevant energy density is zero in the ground state. This consequence of thermodynamics is automatically fulfilled in any system, which may exist without external environment. This leads to the trivial result for gravity: the cosmological constant in any equilibrium vacuum state is zero. The zero-point energy of the Standard Model fields is automatically compensated by the –field that describes the degrees of freedom of the deep quantum vacuum.

These assumptions allow us to suggest that cosmology is the process of equilibration. From the condensed matter experience we know that the ground state of the system serves as an attractor: starting far away from equilibrium, the quantum liquid finally reaches its ground state. The same should occur for the particular case of our Universe: starting far away from equilibrium in a very early phase of universe, the vacuum is moving towards the Minkowski attractor. We are now close to this attractor, simply because our Universe is old. This is a possible reason of the small remnant cosmological constant measured in present time.

The –theory transforms the standard cosmological constant problem into the search for the proper decay mechanism of the vacuum energy density and for the proper mechanism of formation of small remnant cosmological constant. For that we need the theory of dynamics of quantum vacuum. The latter is a new topic in physics waiting for input from theory and observational cosmology. Using several possible realozation of the vacuum variable we are able to model some features of the vacuum dynamics in a hope that this will allow us to find the generic features and construct the phenomenology of equilibration.

Vi Vacuum as topological medium

vi.1 Topological media

There are two schemes for the classification of states in condensed matter physics and relativistic quantum fields: classification by symmetry and classification by topology.

For the first classification method, a given state of the system is characterized by a symmetry group which is a subgroup of the symmetry group of the relevant physical laws. The thermodynamic phase transition between equilibrium states is usually marked by a change of the symmetry group . This classification reflects the phenomenon of spontaneously broken symmetry. In relativistic quantum fields the chain of successive phase transitions, in which the large symmetry group existing at high energy is reduced at low energy, is in the basis of the Grand Unification models (GUT) UnificationModel (); Unification (). In condensed matter the spontaneous symmetry breaking is a typical phenomenon, and the thermodynamic states are also classified in terms of the subgroup of the relevant group (see e.g, the classification of superfluid and superconducting states in Refs. VolovikGorkov1985 (); VollhardtWoelfle ()). The groups and are also responsible for classification of topological defects, which are determined by the nontrivial elements of the homotopy groups TopologyReview1 ().

The second classification method deals with the ground states of the system at zero temperature (). In particle physics it is the classification of quantum vacua. Topological media are systems whose properties are protected by topology and thus are robust to deformations of the action. The universality classes of topological media are determined by momentum-space topology. The latter is also responsible for the type of the effective theory which emerges at low energy. In this sense, topological classification reflects the tendency opposite to GUT , which is called the anti Grand Unification (anti-GUT). In the GUT scheme, the fundamental symmetry of the vacuum state is primary and the phenomenon of spontaneous symmetry breaking gives rise to topological defects. In the anti-GUT scheme the topology is primary, while effective symmetry gradually emerges at low energy Froggatt1991 (); Volovik2003 ().

Different aspects of physics of topological matter have been discussed, including topological stability of gap nodes; classification of fully gapped vacua; edge states; Majorana fermions; influence of disorder and interaction; topological quantum phase transitions; intrinsic quantum Hall and spin-Hall effects; quantization of physical parameters; experimental realization; connections with relativistic quantum fields; chiral anomaly; topological Chern-Simons and Wess-Zumino actions; etc.

vi.2 Gapless topological media

There are two big groups of topological media: with fully gapped fermionic excitations and with gapless fermions.

In 3+1 spacetime, there are four basic universality classes of gapless fermionic vacua protected by topology in momentum space Volovik2003 (); Horava2005 ():

(i) Vacua with fermionic excitations characterized by Fermi points (Dirac points, Weyl points, Majorana points, etc.) – points in 3D momentum space at which the energy of fermionic quasiparticle vanishes. Examples are provided by the spin triplet -wave superfluid He-A, Weyl semimetals, and also by the quantum vacuum of Standard Model above the electroweak transition, where all elementary particles are Weyl fermions with Fermi points in the spectrum. This universality class manifests the phenomenon of emergent relativistic quantum fields at low energy: close to the Fermi points the fermionic quasiparticles behave as massless Weyl fermions, while the collective modes of the vacuum interact with these fermions as gauge and gravitational fields.

(ii) Vacua with fermionic excitations characterized by lines in 3D momentum space or points in 2D momentum space. We shall characterize zeroes by their co-dimension – the dimension of -space minus the dimension of the manifold of zeros. Lines in 3D momentum space and points in 2D momentum space have co-dimension 2: since ; compare this with zeroes of class (i) which have co-dimension . Zeroes of co-dimension 2 are topologically stable only if some special symmetry is obeyed. Examples are provided by the vacuum of the high cuprate superconductors where the Cooper pairing into a -wave state occurs Campuzano2008 () and graphene Volovik2007 (); Manes2007 (); Vozmediano2010 (); Cortijo2011 (). Nodes in spectrum are stabilized there by the combined effect of momentum-space topology and discrete symmetry.

(iii) Vacua with fermionic excitations characterized by Fermi surfaces. The representatives of this universality class are normal metals and normal liquid He. This universality class also manifests the phenomenon of emergent physics, though non-relativistic: at low temperature all the metals behave in a similar way, and this behavior is determined by the Landau theory of Fermi liquid – the effective theory based on the existence of Fermi surface. Fermi surface has co-dimension 1: in 3D system it is the surface (co-dimension ), in 2D system it is the line (co-dimension ), and in 1D system it is the point (co-dimension ; in one dimensional system the Landau Fermi-liquid theory does not work, but the Fermi surface survives).

(iv) The Fermi band class, where the energy vanishes in the finite region of the 3D momentum space, and thus zeroes have co-dimension 0. The possible states of this kind has been discussed in Khodel1990 (); NewClass (); Shaginyan2010 (). In particle physics, the Fermi band or the Fermi ball appears in a 2+1 dimensional nonrelativistic quantum field theory which is dual to a gravitational theory in the anti-de Sitter background with a charged black hole Sung-SikLee2009 (). Topologically stable flat band exists on the surface of the materials with lines of zeroes in bulk SchnyderRyu2010 (); HeikkilaKopninVolovik2011 (); SchnyderBrydonTimm2011 () and in the spectrum of fermion zero modes localized in the core of some vortices KopninSalomaa1991 (); Volovik1994 (); Volovik2011a ().

vi.3 Fully gapped topological media

The gapless and gapped vacuum states are interrelated. For example, the quantum phase transition between the fully gapped states with different topology occurs via the intermediate gapless state. The related phenomenon is that the interface between the fully gapped states with different values of topological invariant contains gapless fermions.

The most popular examples of the fully gapped topological matter are topological insulators Kane2005 (); HasanKane2010 (); Xiao-LiangQi2011 (). The first discussion of the possibility of 3+1 topological insulators can be found in Refs. Volkov1981 (); VolkovPankratov1985 (). The main feature of such materials is that they are insulators in bulk, where electron spectrum has a gap, but there are 2+1 gapless edge states of electrons on the surface or at the interface between topologically different bulk states as discussed in Ref. VolkovPankratov1985 (). The spin triplet -wave superfluid He-B is another example the fully gapped 3+1 matter with nontrivial topology in momentum space. It has 2+1 gapless quasiparticles living at interfaces between vacua with different values of the topological invariant describing the bulk states of He-B SalomaaVolovik1988 (); Volovik2009 (). The only difference from the topological insulators is that the gapless fermions living on the surface of the topological superfluid and superconductor or at the interface are Majorana fermions. The quantum vacuum of Standard Model below the electroweak transition, i.e. in its massive phase, is the relativistic counterpart of the topological insulators and gapped topological superfluids Volovik2010a ().

Examples of the 2+1 topological fully gapped systems are provided by the films of superfluid He-A with broken time reversal symmetry VolovikYakovenko1989 (); Volovik1992b () and by the planar phase which is time reversal invariant VolovikYakovenko1989 (); Volovik1992b (). The topological invariants for 2+1 vacua give rise to quantization of the Hall and spin-Hall conducticity in these films in the absence of external magnetic field (the so-called intrinsic qauntum and spin-quantum Hall effects) VolovikYakovenko1989 (); SQHE (), see Sec. VIII.1.3.

vi.4 Green’s function as an object

For study the topological properties of condensed matter systems, the ideal noninteracting systems are frequently used. Sometimes this is justified, if one can find the effective single-particle Hamiltonian, which emerges at low energy and which reflects the topological properties of the real interacting many-body system. However, in general the primary object for the topological classification of the real systems is the one-electron propagator – Green’s function . In principle one can construct the effective Hamiltonian by proper simplification of the Green’s function at zero frequency, . Though in the interacting case the propagator determines correctly only the zero energy states, see e.g. Haldane2004 (), in some cases it can be used for the construction of the topological invariants alongside with the full Green’s function . On the other hand there are situations when the Green’s function does not have poles (see Volovik2007 (); FaridTsvelik2009 (); Giamarchi2004 ()). In these cases no well defined energy spectrum exists, and the effective low energy Hamiltonian cannot be introduced. In particle physics, interaction may also lead to the anomalous infrared behavior of propagators. For example, the pole in the Green’s function is absent for the so-called unparticles Georgi2007 (); LuoZhu2008 (); the phenomenon of quark confinement in QCD can lead to the anomalous infrared behavior of the quark and gluon propagators Gribov1978 (); Chernodub2008 (); Burgio2009 (); marginal Green’s function of fermions may occur at the black hole horizon Faulkner2010 (); etc. Thus in the interacting systems, all the information on the topology is encoded in the topology of the Green’s function matrix, and also in its symmetry. The latter is important, because symmetry supports additional topological invariants, which are absent in the absence of symmetry, see below.

Green’s function topology has been used in particular for classification of topologically protected nodes in the quasiparticle energy spectrum of systems of different dimensions including the vacuum of Standard Model in its gapless state Froggatt1991 (); Volovik2003 (); Horava2005 (); Volovik2007 (); for the classification of the topological ground states in the fully gapped systems, which experience intrinsic quantum Hall and spin-Hall effects VolovikYakovenko1989 (); Yakovenko1989 (); SenguptaYakovenko2000 (); ReadGreen2000 (); Volovik2003 (); Volovik2007 (); in relativistic quantum field theory of massive Dirac fermions So1985 (); IshikawaMatsuyama1986 (); IshikawaMatsuyama1987 (); Matsuyama1987 (); Jansen1996 () and massive Dirac fermions Volovik2010 (); etc. (see also recent papers EssinGurarie2011 (); ZubkovVolovik2012 ()).

For the topological classification of the gapless vacua, the Green’s function is considered at imaginary frequency . This allows us to consider only the relevant singularities in the Green’s function and to avoid the singularities on the mass shell, which exist in any vacuum, gapless or fully gapped.

vi.5 Fermi surface as topological object

Figure 6: Fermi surface in 2+1 systems represents the nodes of co-dimension 1. In this case, the Green’s function has singularities on line , in the three-dimensional space . Stability of Fermi surface is protected by the invariant (52) which is represented by integral over an arbitrary contour around the Green’s function singularity. This is applicable to nodes of co-dimension 1 in any +1 dimension. For the nodes form conventional Fermi surface in metals and in normal He.

Let us start with gapless vacua. The Green’s 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). The general analysis Horava2005 () demonstrates that topologically stable nodes of co-dimension 1 (Fermi surface in 3+1 metal, Fermi line in 2+1 system or Fermi point in 1+1 system) are described by the group of integers. The winding number , which is responsible for the topological stability of these node, is expressed analytically in terms of the Green’s function Volovik2003 ():


Here the integral is taken over an arbitrary contour around the Green’s function singularity in the momentum-frequency space. See Fig. 6 for D=2. Example of the Green’s function in any dimension is scalar function . For , the singularity with winding number is on the line , , which represents the one-dimensional Fermi surface.

Due to nontrivial topological invariant, Fermi surface survives the perturbative interaction and exists even in marginal and Luttinger liquids without poles in the Green’s function, where quasiparticles are not well defined.

Vii Vacuum in a semi-metal state

For our Universe, which obeys the Lorentz invariance, only those vacua are important that are either Lorentz invariant, or acquire the Lorentz invariance as an effective symmetry emerging at low energy. This excludes the vacua with Fermi surface and Fermi lines and leaves the class of vacua with Fermi point of chiral type, in which fermionic excitations behave as left-handed or right-handed Weyl fermions Froggatt1991 (); Volovik2003 (), and the class of vacua with the nodal point obeying topology, where fermionic excitations behave as massless Majorana neutrinos Horava2005 (); Volovik2007 ().

vii.1 Fermi points in 3+1 vacua

For relativistic quantum vacuum of our 3+1 Universe the Green’s function singularity of co-dimension 3 is relevant. They are described by the following topological invariant expressed via integer valued integral over the surface around the singular point in the 4-momentum space Volovik2003 ():


If the invaraint is nonzero, the Green’s function has point singularity inside the surface – the Fermi point. If the topological charge is or , the Fermi point represents the so-called conical Dirac point, but actually describes the chiral Weyl fermions. This is the consequence of the so-called Atiyah-Bott-Shapiro construction Horava2005 (), which leads to the following general form of expansion of the inverse fermionic propagator near the Fermi point with or :


Here are Pauli matrices (or Dirac matrices in the more general case); the expansion parameters are the vector indicating the position of the Fermi point in momentum space where the Green’s function has a singularity, and the matrix ; ellipsis denote higher order terms in expansion.

vii.2 Emergent fermionic matter

The equation (54) can be continuously deformed to the simple one, which describes the relativistic Weyl fermions


where the position of the Fermi point is shifted to and ellipsis denote higher order terms in and ; the matrix is deformed to unit matrix. This means that close to the Fermi point with , the low energy fermions behave as right handed relativistic particles, while the Fermi point with gives rise to the left handed particles.

The equation (55) suggests the effective Weyl Hamiltonian


However, the infrared divergences may violate the simple pole structure of the propagator in Eq.(55). In this case in the vicinity of Fermi point one has


with . This modification does not change the topology of the propagator: the topological charge of singularity is for arbitrary parameter Volovik2007 (). For fermionic unparticles one has , where is the scale dimension of the quantum field Georgi2007 (); LuoZhu2008 ().

For , the spectrum of (quasi)particles in the vicinity of singularity depends on symmetry. One may obtain either two Weyl fermions or exotic massless fermions with nonlinear dispersion at low energy: semi-Dirac fermions with linear dispersion in one direction and quadratic dispersion in the two others Volovik2001 (); Volovik2003 (); Volovik2007 ().


Similar consideration for the 2+1 systems may lead to semi-Dirac fermions and to fermions with quadratic dispersion at low energy Volovik2007 (); Dietl-Piechon-Montambaux2008 (); Banerjee2009 ()


For the higher values of topological charge, the spectrum becomes even more interesting (see e.g. Refs. HeikkilaVolovik2010 (); HeikkilaVolovik2011 () for 2+1 systems). But if the relativistic invariance is obeyed, or under the special discrete symmetry, the nonzero invariant corresponds to species of Weyl fermions near the Fermi point.

The main property of the vacua with Dirac points is that according to (55), close to the Fermi points the massless relativistic fermions emerge. This is consistent with the fermionic content of our Universe, where all the elementary particles – left-handed and right-handed quarks and leptons – are Weyl fermions. Such a coincidence demonstrates that the vacuum of Standard Model in its massless phase is the topological medium of the Fermi point universality class. This solves the hierarchy problem, since the value of the masses of elementary particles in the vacua of this universality class is strictly zero.

Let us suppose for a moment, that there is no topological invariant which protects massless fermions. Then the Universe is fully gapped and the natural masses of fermions must be on the order of Planck energy scale: GeV. In such a natural Universe, where all masses are of order , all fermionic degrees of freedom are completely frozen out because of the Bolzmann factor , which is about at the temperature corresponding to the highest energy reached in accelerators. There is no fermionic matter in such a Universe at low energy. That we survive in our Universe is not the result of the anthropic principle (the latter chooses the Universes which are fine-tuned for life but have an extremely low probability). Our Universe is also natural and its vacuum is generic, but it belongs to a different universality class of vacua – the vacua with Fermi points. In such vacua the masslessness of fermions is protected by topology (combined with symmetry, see below).

vii.3 Emergent gauge fields

The vacua with Fermi-point suggest a particular mechanism for emergent symmetry. The Lorentz symmetry is simply the result of the linear expansion: this symmetry becomes better and better when the Fermi point is approached and the non-relativistic higher order terms in Eq.(55) may be neglected. This expansion demonstrates the emergence of the relativistic spin, which is described by the Pauli matrices. It also demonstrates how gauge fields and gravity emerge together with chiral fermions. The expansion parameters and may depend on the space and time coordinates and they actually represent collective dynamic bosonic fields in the vacuum with Fermi point. The vector field in the expansion plays the role of the effective gauge field acting on fermions.

For the Fermi points with topological charge the situation depends on the symmetry of the system. In the case, when the spectrum corresponds to several species of relativistic Weyl fermions, the shift becomes the matrix field; it gives rise to effective non-Abelian (Yang-Mills) gauge fields emerging in the vicinity of Fermi point, i.e. at low energy Volovik2003 (). For example, the Fermi point with may give rise to the effective gauge field in addition to the effective gauge field


where are Pauli matrices corresponding to the emergent isotopic spin. This is what happens in superfluid He-A. In the case, when the symmetry leads to exotic fermions with the non-linear spectrum , the quantum electrodynamics with anisotropic scaling emerges KatsnelsonVolovik2012 (); Zubkov2012b (), which is similar to the quantum gravity with anisotropic scaling suggested by Hořava HoravaPRL2009 (); HoravaPRD2009 (); Horava2010 ().

vii.4 Emergent gravity

The matrix field in (60) acts on the (quasi)particles as the field of vierbein, and thus describes the emergent dynamical gravity field. As a result, close to the Fermi point, matter fields (all ingredients of Standard Model: chiral fermions and Abelian and non-Abelian gauge fields) emerge together with geometry, relativistic spin, Dirac matrices, and physical laws: Lorentz and gauge invariance, equivalence principle, etc. In such vacua, gravity emerges together with matter. If this Fermi point mechanism of emergence of physical laws works for our Universe, then the so-called “quantum gravity” does not exist. The gravitational degrees of freedom can be separated from all other degrees of freedom of quantum vacuum only at low energy.

In this scenario, classical gravity is a natural macroscopic phenomenon emerging in the low-energy corner of the microscopic quantum vacuum, i.e. it is a typical and actually inevitable consequence of the coarse graining procedure. It is possible to quantize gravitational waves to obtain their quanta – gravitons, since in the low energy corner the results of microscopic and effective theories coincide. It is also possible to obtain some (but not all) quantum corrections to Einstein equation and to extend classical gravity to the semiclassical level. But one cannot obtain “quantum gravity” by quantization of Einstein equations, since all other degrees of freedom of quantum vacuum will be missed in this procedure.

vii.5 Topological invariant for specific Fermi surface

Figure 7: Fermi surface is formed from the Fermi point at finite chemical potential of chiral fermions, when the Fermi point is moved away from the zero energy level.

If the symmetry which fixes the position of conical (Dirac) point at zero energy level is violated, the conical point moves from the zero energy position upward or downward from the chemical potential and the Fermi surface is formed. This is shown in Fig. 7. This Fermi surface has specific property: in addition to the local charge in (52), which characterizes singularities at the Fermi surface, it is described by the global charge in (53). The integral in (53) is now over the surface which embrace whole Fermi sphere. The Fermi surface with the global topological charge appears in superfluid He-A in the presence of mass flow Volovik2003 (); it is also discussed for the 2+1 systems in relation to the gapless states on the surface of 3+1 insulators