Energy Renormalization in a Berry Geometrical Phase: Low-Energy Perturbations of the Strong Interaction and the QCD Mass Gap

1.1. Spontaneous Magnetism and Dual Superconductivity

Evidence for the emergence of spontaneous magnetism and associated superconducting behaviour has been previously reported [1]. The findings are briefly reviewed here with the benefit of additional insights relating to Yang-Mills theory and QCD.

The thermodynamic response functions in Table A4 are calculated by the REFPROP program. These response functions enable a family of critical exponents to be derived. The resulting critical exponents in Table A4 fully satisfy the scaling laws of Fisher, Rushbrooke, Widom and Josephson, to produce a distinctive universality class in 3-dimensional space [20,21].

The response functions and critical exponents of Table A4 allow the magnetic properties of the system to be determined, as recorded in Table A5. The following Maxwell relations then provide for an exchange mechanism to equilibrate frustrated magnetic energy:

$${\left({\displaystyle \frac{\delta V}{\delta \mathrm{M}}}\right)}_{S,P}={\left({\displaystyle \frac{\delta \mathrm{H}}{\delta P}}\right)}_{S,\mathrm{M}} \mathrm{a}\mathrm{n}\mathrm{d} {\left({\displaystyle \frac{\delta \mathrm{M}}{\delta P}}\right)}_{S,\mathrm{H}}={\left({\displaystyle \frac{\delta V}{\delta \mathrm{H}}}\right)}_{S,P}$$

(1)

and expressing in terms of the interaction energy μN:

$${\left({\displaystyle \frac{\delta N}{\delta \mathrm{M}}}\right)}_{S,\mu }={-\left({\displaystyle \frac{\delta \mathrm{H}}{\delta \mu }}\right)}_{S,\mathrm{M}} \mathrm{a}\mathrm{n}\mathrm{d} {\left({\displaystyle \frac{\delta \mathrm{M}}{\delta \mu }}\right)}_{S,\mathrm{H}}={-\left({\displaystyle \frac{\delta N}{\delta \mathrm{H}}}\right)}_{S,\mu }$$

(2)

The results in Table A6 reveal that PV work is only partially accounted for by relaxation of the constraint on specific volume v and total internal energy, as imposed by inhomogeneities and magnetic frustration. However, calculations based upon extensive parameters and intensive parameters only relate to the material system and do not account for the vacuum occupied by that system [22].

The total energy within the spatial region occupied by the system needs to include the vacuum energy value given by:

$$U_{vac}={\mu }_0^{-1}{\mathrm{H}}^{2}V$$

(3)

Where μ₀ is the permeability of flat, free space with an historical value of 4π x 10^-7 T-m A^-1 [22]. However, as detailed in Table A7, the hyperbolic geometry of the system allows for effective values of μ₀ which in fact reflect the ratio of Euclidean space to hyperbolic space. From the geometry of the system, the ‘space density’ is then found to be dependent upon the correlation length scale ξ of 3.05, applicable to approximately 3.5 steps of ‘rolling’ critical response, such that;

$$e^{3.5\xi }={\displaystyle \raisebox{1ex}{${\mu }_{0f}$}\!\left/ \!\raisebox{-1ex}{${\mu }_{0i}$}\right.}$$

(4)

where μ_0f is the final effective value of μ₀ and μ_0i is the initial effective value of μ₀. Variations in effective μ₀ require that a spontaneous magnetism M_s with associated spontaneous magnetic field H_s emerges to conserve magnetic charge.

Application of the scaling laws to the experimental results [1] reveals that pressure perturbation induces low susceptibility χ leading to spontaneous magnetism M_s with associated spontaneous magnetic field H_s. For positive H_s, ordering is attributed to the emergence of the complex parameter field Ψ(r) and the topology associated with magnetic frustration and charge fractionalization. Condensation of fractionalized magnetic charges into a gauge monopole condensate [11,12] would act to exclude the fractionalized magnetic current to provide exchange pathways for the spontaneous magnetism M_s that maximally excludes the electric field E to establish dual superconducting behaviour.

Figure 6 in Appendix C describes the emergence of M_s and H_s. Taking H_s as the ‘auxiliary’ order parameter, a phase transition is apparent at M_c2 where H_s = 0. For M_s > 1, H_s takes positive values as the susceptibility χ falls below unity. A new value for the critical magnetism M_c’ is deemed to be established to the right of the plotted values. It then follows that for this region κ < 1/√2 which is consistent with Type I superconducting behaviour. Here, large values of B_s act to exclude the electric field E to establish dual superconducting behaviour [23]. The initial emergence of this spontaneous magnetism is attributed to the anisotropy of the clathrate hydrate structures and is consistent with proton delocalization [24]. A similar situation is described by Purcell and Pound [25] where opposition of the external field to the magnetizing flux arises from the relaxation time for mutual interaction among spins being less than that between spins and the embedding lattice.

Relatively recent advancements in the understanding of spin ice materials provide an explanation for the superconducting phase transition described above, since effective monopole defects act as both divergent sources and convergent sinks of the magnetic field H_s [8]. Local proton ordering in water ice [26] finds a physical equivalent in the frustrated pyrochlore Ising magnet, as originally identified by Anderson [27]. The Ising spin, as the proton spin, possesses only two discrete orientations (up or down) since it is constrained to point along an axis. The centres of the pyrochlore tetrahedra of these spin ice materials identify with the locations of oxygen atoms in water ice. The Ising spin axes are exactly aligned to the water ice oxygen bonding such that residual Pauling entropy can also be precisely demonstrated in spin ices [28].

The high level of degeneracy expected for the tetrahedrally coordinated crystal-lattice being investigated could lead to similar fractionalization of the magnetic moments into effective, irrationally-charged excitations, i.e. fractionalization of the microscopic spin degrees of freedom [29]. This behaviour is widely reported in low-energy, spin ice magnets [30,31, and references therein]. Crucially, gauge fields also emerge from the geometrical frustration and defects in these fields lead to further monopole excitations [29].

For a frustrated system with charge fractionalization, ordering is generally characterized by the emergent gauge field; i.e. topological ordering [29,32] that opposes Landau ordering and symmetry breaking. Typically, in an ensemble subjected to constrained disorder, violations of local rules (or topological defects) appear as localized excitations of the low-energy states of the system and these local rule violations control the collective dynamics. In spin ices and other water ice manifolds, the violations manifest as effective magnetic monopoles. Castelnovo et al. [29] describe a mechanism for this fractionalizing process. The monopoles cut off the dipolar correlations (i.e. the interaction potential μ → 0) to allow a correlation length ξ to be defined. As a spin ice approaches a critical point, the spin correlation is not defined by an order parameter. Instead, it is defined by deconfinement of a gauge field which, in this case, results from the constrained quasi-micro-canonical ensemble/ constant energy Hamiltonian function. Such an interpretation is consistent with the convention that superconducting states lack any local order parameter but rather exhibit charge fractionalization whilst remaining sensitive to the global topology of the underlying manifold [33,34].

The discussion below proposes that hyperbolic curvature of the vacuum manifold is derived from the electroweak interchange of gluons and W-bosons that give an effective vector boson mass m_V. Relevant calculations are included in Appendix C, Appendix D and Appendix E. This action induces a gauge monopole condensate from the fractionalized charges to establish a macro-scale dual superconductor. The associated Higgs mass m_H is then responsible for the non-extensive volume changes observed in the system. It was previously proposed in [1] that hydrogen bonding-induced curvature on the crystal-fluid lattice initiated the superconducting phase transition. However, the causality is in fact the opposite where electroweak interactions are shown to result in a magnetic condensate with associated Higgs mass m_H.

1.1. Phase Transitions and Collective Phenomena

Loss of homogeneity is characteristic of phase transitions that give rise to critical phenomena. The discovery of a distinctive universality class of critical exponents that satisfy the scaling laws of Fisher, Rushbrooke, Widom and Josephson [21,22] reveals spontaneous magnetic and superconducting properties for the crystal-fluid material investigated. A phase transition from Type-II superconductivity to dual Type-I superconductivity is identified through the Ginzburg-Landau parameter κ where the spontaneous magnetic field H_s transitions from negative to positive (Appendix C) and a complex order parameter field Ψ(r) emerges. Spontaneous magnetism M_s can either increase or decrease dependent upon the change in direction of hyperbolic curvature K. Positive piston displacement results from a reduction in K to give increasing values of M_s. Negative piston displacement results from an increase in K to give reducing values of M_s.

Derivation of the Gaussian hyperbolic curvature K also gives the Gaussian radius R_g as its inverse, as shown below. This enables the hyperbolic surface area of a hollow, walled sphere having effective radius R to be determined (A = 4πsinh²(R/2)) [1]. The hyperbolic surface area maps almost exactly to the negative inverse of the gradient energy, where topological defects are introduced, in an apparent expression of holographic duality [35]. The external pressure perturbations may be interrupted at any point such that both the volume V and the hyperbolic curvature K become fixed and stable. This suggests that the non-equilibrium gradient energy is conserved within the hydrogen bonding interactions of the water ice cages and inhibitor solvent as the system relaxes into a non-critical, stable state (refer to equations (1) and (2)).

Inequalities in the associated Maxwell relations together with calculations of the hyperbolic geometry reveal non-additivity and non-extensivity in the fundamental thermodynamic relation. Additivity can be restored through hyperbolic curvature (i.e. surface area A) whilst extensivity can be restored through gradient energy/ coupling energy (i.e. volume V) [22]. The coupling energy is related to the critical correlation length exponent ν in 3-dimensions combined with values of a scalar field Φ as derived from the gradient energy term -½(∇Φ)² of the Lagrangian such that:

$${\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}\propto \left(e^{\Phi }\right)}^{3v}\to \mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{f}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d}$$

(5)

The critical correlation length ξ associated with the universality class is linked to swept volume dV through Lorentz boosts, i.e. relativistic velocity and the reference frames associated with acceleration and deceleration [1]. The critical correlation length ξ is revealed to be a Lorentz length expansion; a relativistic phenomenon coinciding with the formation of a magnetic condensate [12]. The relativistic time contraction conjugate to the critical correlation length is also revealed in Figure 4 where the y-axis representing scalar field values has units of s^-1, i.e. the reciprocal of time.

Again, the correlation length ξ is associated with a superconducting phase transition and related to a Lorentz boost in 3-dimensional space (Appendix B). Its value reveals the presence of self-organized criticality responsible for a ‘rolling’ critical response in accordance with:

$$\xi ~{│T-T_c│}^{-v}$$

(6)

where T is the system temperature and T_c is the critical temperature. Both temperatures are dynamic under external pressure perturbation to reveal sustained anisotropy in the water ice cage structures in either direction. The magnetic correlation length corresponds to a gap in gradient energy that can be described by a relativistic length expansion having an associated conjugate time contraction [1].

The gradient energy term -½(∇Φ)² maps to either positive or negative PV work (i.e. it is described by the least action principle of the Lagrangian) whilst the hyperbolic surface area is a function of the Gaussian radius of hyperbolic curvature R_g. Both properties are calculated from experimental results (Appendix A). In order for the surface of the system to be fully expressed in the gradient energy, it is necessary to introduce topological defects at the superconducting phase transition where the spontaneous magnetic field H_s moves from negative to positive through zero.

1.1. auge Symmetry

The exponential function for the coupling energy exp(3vΦ) (5), as derived from experiment, represents a capturing of the magnetic condensate wavefunction/ gradient energy through the interaction energy μN of the crystal-fluid material such that it becomes real and observable (1,2). Naively, a gauge monopole condensate appears to acquire a temporary critical mass m_H through coupling of the Higgs-like scalar field Φ to the magnetization vector field M_s, as represented by the correlation length exponent ν. The exponential function exp(3vΦ) also expresses the scale-invariant and gauge-invariant properties of the critical system. Conservation of angular momentum requires the existence of a sink/ source for the associated changes in inertia (variable effective radius) together with a corresponding symmetry relation.

Scale-invariance is attributed to the hyperbolic curvature of Lorentz boosts that describe a conformal symmetry at the surface of the embedding vacuum manifold [35,36]. Conformal symmetry is able to represent the tetrahedral, hydrogen bonded, 3-dimensional spatial geometry of the crystal-fluid under non-extensive volume changes whilst its variable hyperbolic surface area maps from the gradient energy term of the Lagrangian (see below). Since a universality class of topologically invariant critical exponents has been determined for the continuous (second order) phase transition, the system can be modelled through conformal field theory in 4-spacetime dimensions, i.e. it is describable by a renormalizable quantum field theory in which the non-perturbative conformal bootstrap is irrelevant [37].

Yang-Mills theory represents such a strongly coupled quantum field theory [16], i.e. a gauge theory in which the low-energy dynamics are far removed from any classical description [38]. It is in turn represented through the mathematical structure of Lie symmetry groups that provide for intricate topologies. The compact, simple Lie group SU(3) describes the strong interaction in QCD, i.e. the binding of quarks and gluons through confinement mechanisms. The mechanical action of the piston expander can be described by the emergence of a scalar field Φ (1.2 ≤ Φ ≤ 3.6 ks^-1) and the critical length exponent ν (0.593) in 3-dimensions, which maps to the relativistic rapidity angle ±φ (±1.779), as equation (8). In QCD the resulting gauge fields are collectively known as gluon fields. The field strength, or curvature F_μν, takes the general form:

$$F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }-i\left[A_{\mu },A_{\nu }\right]$$

(7)

where A_ν provides for Lorentz invariance and A_μ is the gauge connection.

The gauge connection field depends upon a complex scaling symmetry that is exact but not directly observable [39]. In the quantum state Ψ(r)→ e^iθΨ(r), which could be interpreted as a potential sink/ source for the ‘hidden’ inertia of the false vacuum system (although later this is revealed not to be the case). It also represents the complex order parameter field of the Ginzburg-Landau superconducting phase transition included in equation (10).

Experimental results lead to a relativistic manifestation of length expansion and time contraction arising from false vacuum behaviour in a thermodynamically constrained condensed matter system. The local stability conditions maintained through dynamically responsive inhomogeneities in this soft matter are identified with the property of asymptotic freedom, or antiscreening, which accounts for the mechanism of colour confinement in particle physics, i.e. scale-invariance is suggested across the micro- and macro-scales of the system. In micro-scale SU(3) QCD it is the emergence of clouds of virtual gluons that establish the antiscreening phenomenon [3]. In both mechanisms, increasing kinetic/ internal energy ∆u is mirrored by an increasing negative excess energy potential ∆u_e such that total internal energy remains constant.

Whilst the crystal-fluid material displays high stability in total energy and density, the embedding manifold always remains on the threshold of instability. Small positive or negative pressure perturbations produce divergent critical behaviour manifesting as large variations in swept volume dV. However, this is not the specific volume of the material system (density remaining almost constant) but rather the non-extensive volume change associated with the condensation of magnetic entities.

The ‘rolling’ critical response enabled through anisotropy in water ice cage structures appears to deliver net energy gains, either positive or negative, in a display of self-organized criticality [40]. The angular momentum of the material is transferred to or from the embedding vacuum manifold through self-organizing behaviour and high energy degeneracy of the water ice cage structures. However, this brief statement does not provide a full description and a more detailed hypothesis follows.

Work derived from the piston expander can be expressed in terms of an electromagnetic pseudo-scalar gauge-invariantly coupled to the scalar field exponent Φ and the critical correlation length exponent in 3-dimensions 3ν (which is equal in magnitude to the rapidity angle ±φ, as described in Appendix B). The relationship is similar in form to the cosmological inflation model proposed by Ratra [41]:

$$\pm \int P\mathrm{d}\mathit{V}\to {\displaystyle \frac{1}{2}}{F}^{\mu \nu }{F}_{\mu \nu }{\left(e^{\Phi }\right)}^{\left|\pm \mathsf{\phi }\right|}$$

(8)

where the covariant vector F_μν and contravariant gradient potential F^μν combine to produce Lorentz invariance for the pseudo-scalar field when rotated on a hyperbolic manifold, i.e. the electromagnetic field pseudo-scalar enables a non-additive energy contribution to enter the non-equilibrium system in the form of hyperbolic curvature. Expressing the exponential function in terms of the rapidity angle ±φ reveals the action of positive and negative spinors that determine the critical correlation length ξ, as (25,26,27), leading to either an increasing or decreasing negative gradient energy.

In the quasi-micro-canonical ensemble [1], the electromagnetic field pseudo-scalar is involved in the coupling mechanism but contributes no work in itself. It expresses the Berry curvature that is only indirectly derived from the vacuum manifold whilst hosting the magnetic exchange pathways that facilitate energy transfers either to or from the vacuum manifold. The inner-product of the E and B fields remains the same viewed in all relativistic frames [42] with the pseudo-scalar field remaining Lorentz invariant such that:

$${\displaystyle \frac{1}{2}}{F}^{\mu \nu }{F}_{\mu \nu }={\mathrm{B}}^2-{\displaystyle \frac{{\mathrm{E}}^2}{c^2}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$

(9)

where c is the speed of light.

Ginzburg-Landau theory states that the free energy of a superconductor near a phase transition can be expressed in terms of a complex order parameter field [43]:

$$\Psi \left(\mathit{r}\right)=\left|\Psi \left(\mathit{r}\right)\right|e^{i\Phi \left(\mathit{r}\right)}$$

(10)

Then the complex gauge connection field maps to a macroscopic wavefunction of the Berry phase:

$$({e^{i\Phi })}^{3v}\to \left|\Psi \left(\mathit{r}\right)\right|{{(e}^{\Phi \left(r\right)})}^{3v}$$

(11)

where the quantity |Ψ(r)|² reflects the density of superconducting charge carriers; electrons for Type-II and the electroweak monopole counterpart arising from confined fractional magnetic charges for dual Type-I [11]. Appendix C provides a summary of the Ginzburg-Landau theory of superconductors.

In the dual superconductor model of confinement [9,10], the Yang–Mills vacuum is also based on the condensate of a magnetically charged Higgs field. In the current findings, the critical correlation length ξ similarly represents the coherence length ξ’ of a massive macro-scale gauge monopole condensate [12] that manifests as a divergence in the hyperbolic volume V of the system at the superconducting phase transition [43]. In this case the coherence length ξ’ becomes exceptionally large whilst still defining the distance over which the dual superconductor can be represented by a macroscopic wavefunction.

Since the coherence length ξ’ and maximum value for the Ginzburg-Landau parameter κ for the Type-I dual superconductor are known [1], the London penetration depth λ can be derived (see Appendix C). ξ’ and λ are equal to the inverse Higgs mass m_H and inverse vector boson mass m_V, respectively [11]. In normal metallic superconductors λ is the distance within which an externally applied magnetic field disappears inside the superconductor. However, for the dual superconductor λ represents a distance beyond the developing flux tubes (confined fractionalized magnetic charges) within which the magnetic current and electric field are expelled as a result of the dual Meissner effect.

So, the Gaussian hyperbolic curvature K of the embedding vacuum manifold (and subsequently the crystal-fluid geometry) derives from m_V. Then, the resulting magnetic monopole condensate is responsible for m_H, which manifests in the system volume V. Appendix C includes supporting quantitative analysis.

The complex form of the gauge field resembles a macroscopic wavefunction in which the energy spectrum becomes entirely real and observable through dissipative structuring and interaction energy of the crystal-fluid material. Ψ₀(r) initially appears to corresponds to the emergence of the scalar field Φ coupled with a magnetization vector field M_s at the Type-II to dual Type-I superconducting phase transition. Dissipation of either the scalar field Φ or the critical correlation length ξ would represent a collapse in the wavefunction.

Application of de Moivre’s formula and isomorphic mapping of the complex gauge field to hyperbolic rotational matrix form gives:

$${\left(e^{i\Phi }\right)}^{3v}=\cosh 3v\Phi +i\sinh 3v\Phi \to \left[\begin{array}{cc}\cosh 3v\Phi & i\sinh 3v\Phi \\ -i\sinh 3v\Phi & \cosh 3v\Phi \end{array}\right]≕\mathrm{V}$$

(12)

and similarly expressing electromagnetic duality as rotations in the 2-dimensional hyperbolic plane:

$${\displaystyle \frac{1}{2}}{F}^{\mu \nu }{F}_{\mu \nu }\to \left[\begin{array}{cc}\cosh 3v\Phi & -i\sinh 3v\Phi \\ i\sinh 3v\Phi & \cosh 3v\Phi \end{array}\right]≕{\mathrm{V}}^{\mathrm{H}}$$

(13)

The conjugate transpose of (12) is (13) and the determinant of VV^H = 1 suggesting that PV work of the piston expander is contingent upon the decomposition of a Hermitian unitary matrix A into two 2 x 2 non-Hermitian unitary matrices (i.e. two complex matrices V and V^H containing both real and imaginary components such that V^H ≠ V) [44]. The gauge field and the electromagnetic pseudo-scalar are thereby coupled through a reciprocal interaction in the hyperbolic plane.

Although this interpretation appears at odds with the expression for PV work stated in (8), in fact any 2 x 2 complex symmetric matrix A can be eigendecomposed into a diagonal matrix D sandwiched between two complex unitary matrices, i.e. VDV^H in this case. Minkowski spacetime vectors can be represented by 2 x 2 orthogonally diagonalizable matrices and incorporated into the extended physical VDV^H decomposition to reveal the coupling energy source.

For:

$$\mathrm{D}≔\left[\begin{array}{cc}{\left(e^{\Phi /2}\right)}^{\pm \mathsf{\phi }}& 0\\ 0& {\left(e^{\Phi /2}\right)}^{\pm \mathsf{\phi }}\end{array}\right]$$

(14)

VDV^H can be expressed as the Pauli spin operators acting on a bivector [45]:

$$\left[\begin{array}{cc}{\left(e^{\Phi /2}\right)}^{\mathsf{\phi }}& 0\\ 0& {\left(e^{\Phi /2}\right)}^{\mathsf{\phi }}\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]={\left(e^{\Phi }\right)}^{\mathsf{\phi }}$$

and

$$\left[\begin{array}{cc}{\left(e^{\Phi /2}\right)}^{-\mathsf{\phi }}& 0\\ 0& {\left(e^{\Phi /2}\right)}^{-\mathsf{\phi }}\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]={\left(e^{\Phi }\right)}^{-\mathsf{\phi }}$$

(15)

where ±φ represents the Lorentz rapidity angle (with values of ±1.779) and Φ/2 is the spinor half-angle formulation. Both are incorporated into the Weyl spinors of (25) and (26). Additionally, VV^H corresponds to the Pauli identity matrix σ₀, also included in the Weyl spinors description. The sign of the rapidity angle φ is associated with either a positive or negative spinor, i.e. an increasing or decreasing negative gradient energy.

The diagonal Hermitian matrices exhibit basic 3-dimensional rotation as well as 4-dimensional Lorentz transformation properties consistent with the relativistic length expansion and time contraction associated with the non-extensive element of PV work, as also revealed through the experimental results [1]. Thus, the 2 x 2 unitary matrix A as a member of the compact U(2) symmetry group is decomposed into factors identifiable as both Hermitian and non-Hermitian.

When represented in terms of gauge symmetry groups [44], the U(1) group of electromagnetism (via its mapping to SO(2) in the 2-dimensional real plane) and the SU(2) group of the complex order parameter Ψ(r), are in fact subgroups of the U(2) group such that:

$$U\left(1\right)\otimes SU\left(2\right)/{\mathbb{Z}}_2\to U\left(2\right)$$

(16)

which describes a mapping to a Yang-Mills electroweak symmetry group [46] where ℤ₂ represents the topology associated with a gauge monopole condensate [47]. Formation of the U(2) group is accompanied by critical behaviour and emergence of the gauge field as predicted by the Yang-Mills theory.

The dual superconductor model has several interpretations that require condensation of monopoles, just as normal superconductivity results from the condensation of electric charges (or Cooper pairs – see Appendix C) [11,12]. Theoretical frameworks for the condensation of monopoles have been structured in terms of Abelian projection for the SU(2) gauge symmetry group or non-Abelian gauge-invariance for the SU(3) gauge symmetry group. Recent efforts [12] have sought to extract the Abelian component responsible for gauge-independent quark confinement from the non-Abelian gauge-invariance required for asymptotic freedom, without losing the essential feature of monopole condensation. From the experimental findings [1], an alternative solution emerges where electroweak lepton interactions with fractionalized magnetic charges lead to complex monopole condensation within an overarching U(2) Abelian group, as detailed below.

In the vacuum of a QCD dual superconductor, the dual Meissner effect compresses the chromoelectric flux between a quark and antiquark into a thin flux tube to form the hadronic string [11,48]. As the distance between quark and antiquark increases, the flux tube becomes longer whilst maintaining a minimal thickness. This geometry ensures that the energy increases linearly with length to create a linear confining potential between the quark and antiquark that bears a similarity to the linear oscillating Hamiltonian of the macro-scale system. The flux tube determines the extent of QCD vacuum suppression, i.e. positions where the chromoelectric field is maximally expelled to leave a residual dual superconductivity [49].

Yang-Mills theory requires the existence of both chromomagnetic monopole condensation (given by a coherence length) and the dual Meissner effect (given by a penetration depth) [11,12]. The force carrying gauge bosons of QCD are massless gluons which perform a similar role to photons in electromagnetism. Since the gluon field represents a local expulsion of the QCD vacuum, the (indirect) absorption of gluon emissions into the QCD vacuum would tend to reduce local ‘space density’ and effective magnetic permeability μ₀. The net effect is to increase the hyperbolic curvature K of the embedding manifold, i.e. a quantum mechanical process manifests as an effective vector boson mass m_v or ‘strong gravity’ [50].

1.1. Berry Phase and Parity-Time (PT) Symmetry

The foundations of Berry phase physics lie in the adiabatic theorem of quantum mechanics [51] which provides a formal description for a system coupled to a slowly changing environment. If the system Hamiltonian H(t) varies adiabatically and │Ψ(t)⟩ is an associated eigenstate then, following cyclic evolution of the environmental parameters where H(T) = H(0), the state returns to itself but gains an additional phase factor [51,52]:

$$│\Psi \left(T\right)⟩=e^{i\alpha }│\Psi \left(0\right)⟩$$

(17)

where α represents the angular momentum of the wavefunction. For the macro-scale dual superconductor, it derives from the exclusion of momentum resulting from gauge monopole condensation that is effectively stored in the electromagnetic field pseudo-scalar [38].

The adiabatic theorem is based upon a single, non-degenerate eigenstate to which the system ‘clings’ as the environment is slowly changed [53]. However, for the pressure-perturbed system being examined, asymptotic freedom constrains innumerable, degenerate and excited eigenstates to a singular value of total energy in the Hamiltonian function. External pressure perturbations applied to the crystal-fluid material see changes in kinetic/ internal energy ∆u mirrored by changes in negative energy potential ∆u_e such that total internal energy remains constant. Integration of the scalar potential ∇Φ over a Hamiltonian cycle reveals the gradient energy term -½(∇Φ)² that becomes observable in the PV work extracted from the piston expander (see equation (29)).

When perturbations cease, the gradient energy appears to be conserved in the sterically-induced interaction energy of the crystal-fluid, subject to some limited dielectric relaxation. Since hyperbolic geometry is retained in the crystal-fluid, there is negligible net loss of mass as the Higgs mass m_H and vector boson mass m_V dissipate. The phase factor Ψ(T) exists only for the duration of pressure perturbations with the Hamiltonian remaining constant under these perturbations (as Figure 3). However, the resting values of total energy in the Lagrangian function L are ‘propped’ and metastabilized through dissipative structuring of water ice cages, as quantified by Stage 2-3 and Stage 4-1 (Figure 3). Following metastable decay, which can last several weeks, the mass associated with sterically-induced interaction energy (recorded at a maximum of 0.3 kg, approx.) eventually dissipates. For the metastable system, L remains approximately constant:

$$\mathrm{L}\stackrel{metastable}{\to }-{{\displaystyle \frac{1}{2}}\left(\nabla \Phi \right)}^2+\mu N$$

(18)

During acceleration (Stage 1-2) the effective material radius decreases, and during deceleration (Stage 3-4) the effective material radius increases. However, the associated ‘hidden’ inertia is deemed not to be responsible for the Berry curvature term within the geometrical phase (11) since the externally-induced momentum appears to manifest entirely in non-additive hyperbolic curvature K of the embedding vacuum manifold. Instead, Berry curvature is linked to the condensation of magnetic entities whereby the excluded charge momentum is incorporated into the energy potential of the electromagnetic field pseudo-scalar (9). The Berry curvature is subsequently captured to be made real and observable in the variable hyperbolic volume V of the system. Again, this hyperbolic volume is metastabilized by the dissipative structuring of water ice cages within the crystal-fluid material such that the complex Berry phase is transformed into real work done.

For a classical thermodynamic system, changes in inertia ½mr² represent changes in kinetic/ internal energy. However, since both internal energy and specific volume are highly constrained parameters within a false vacuum system, the energy of acceleration/ deceleration is prevented from manifesting in the internal energy of the crystal-fluid material. Thus, Pv work is limited to interactions with the walls of the vessel whilst the variable effective radius R results from changes in inertia. For the synchronized U(2) symmetry group identified below, conservation of angular momentum may be understood in terms of accelerating/ decelerating quarks that result in the emission/ absorption of gluons, i.e. changes in negative energy potential.

Gluons emitted by quarks are indirectly absorbed by the QCD vacuum manifold whilst the gluons absorbed by quarks indirectly emerge from the QCD manifold, thereby tending to effect local ‘space density’ and effective permeability μ₀. A gluon-induced process for gg → W*W* → leptons (where W*W* represents an intermediate W-boson/ vector boson pair) is described in [54]. At some stage in the process, the inverse fine structure constant 1/α (≈ 137) appears to play a key role (see Appendix D for analysis).

Lepton interactions may act to transform fractionalized microscopic magnetic spin degrees of freedom into SU(2) electroweak monopole entities [31]. A sufficiently large number of SU(2) electroweak monopoles could then generate a Planck mass such that a U(2) chromomagnetic condensate having Higgs mass m_H emerges together with a chromoelectric confining potential. The associated gauge monopole topology facilitates the spontaneous magnetism M_s that maximally excludes the electric field E and establishes the macro-scale dual superconductor (see Appendix D and Appendix E for analysis). This phase-change is represented by the Ginzburg-Landau parameter κ with values ≤ 1/√2 that describe the ratio of the Higgs mass m_H to the vector boson mass m_V.

Pv is insignificant in comparison to PV such that it represents the negative energy potential of the crystal-fluid material. Therefore, for a constant Hamiltonian of constant mass m, ½r² ∝ 1/Pv, as described in Appendix B of [1]. The average 1-dimensional radius r_x of the critical system is then found:

$$r_x\propto \sqrt{{\displaystyle \frac{2}{Pv}}}$$

(19)

The Gaussian curvature K for the 2-dimensional, hyperbolic surface of the critical system for the principal curvature relationship of r_x = - r_y , can then be determined:

$$K={\displaystyle \frac{1}{r_x{r}_y}}$$

(20)

or

$$K\propto -\left({\displaystyle \frac{Pv}{2}}\right)$$

(21)

Then, the average Gaussian radius of hyperbolic curvature (1/K or R_g) is given by:

$$R_g=\left({\displaystyle \frac{2}{Pv}}\right)$$

(22)

Gaussian curvature K has units of m² s^-2 that map directly to the effective vector boson mass m_V as the inverse of the penetration depth λ (as described in Appendix C). Through this mechanism, the negative energy potential of gluons is conserved in the associated hyperbolic curvature quantifiable by the non-equilibrium values of pressure P and specific volume v. Thereby, a quantum mechanical action is tuned thermodynamically under false vacuum conditions. Expressed in terms of the holographic principle, or the anti-de Sitter/ Conformal Field Theory (AdS/ CFT) correspondence, this hyperbolic geometry results from a renormalization group flow/ scaling flow. In other words, a scaling flow from the boundary surface (described by a 2-dimensional CFT) to the interior is encoded in the geometrical properties of the hyperbolic manifold (described by a 3-dimensional AdS) in accordance with the Einstein field equations [35] (see Appendix B).

Decomposition of the complex gauge connection field (e^iΦ)^3ν in equations (12) and (13) suggests that complex Berry curvature is necessary for emergence of the coupling energy (5). It also describes the phase of electromagnetic duality, which in the extreme leads to dual superconducting behaviour, i.e. condensation of magnetic entities resulting in the exclusion of fractionalized magnetic current and the electric field. The cyclic evolution of the gauge field results from the effective adiabatic property of the constrained false vacuum system (as revealed in the constant Hamiltonian function of Figure 3) to establish a novel form of the Berry phase [51], one responsible for topological ordering in the macro-scale dual Type-I superconductor [8]. As with the conventional ground-state Berry phase, this ‘excited-states’ variant exposes the gauge structure in quantum mechanics [13,55].

In addition to describing the emergence of a scalar field Φ, the gradient energy term -½(∇Φ)² of the Lagrangian also maps to the complex order parameter field Ψ(r) in accordance with Ginzburg-Landau theory. The PV work generated in the piston expander suggests that the associated macroscopic wavefunction is made real and observable, a phenomenon recently uncovered by Gu et al. [13]. More precisely, the VDV^H decomposition reveals that the wavefunction becomes entirely real as the gauge field is exposed through the diagonal matrix D in the VDV^H decomposition (15).

Since the system can be described through a combination of Hermitian and non-Hermitian matrices, it resembles a PT symmetric system [14]. Such systems are characterized as not being isolated from the environment (i.e. non-adiabatic) but subject to highly constrained interactions. This description is consistent with the false vacuum behaviour of the crystal-fluid material where both specific volume and internal energy are highly constrained. Energy and entropy gains and losses to the environment (including the embedding vacuum manifold in this case) are exactly balanced, i.e. a renormalized, scale-invariant interaction between condensed matter and macroscopic wavefunction becomes evident in the gradient energy and interaction energy terms of the Lagrangian function (18).

PT symmetry requires both space reflection and time reversal symmetries. The upside-down potential of the quartic term as identified by Bender [14] is consistent with the marginal interaction and negative gradient energy term derived from experiment [1]. However, the results presented here reveal the symmetry of Lorentz boosts, i.e. symmetries in the expansion and contraction of both space and time, which may represent a more generalized form of PT symmetry.

1.1. Gapped and Gapless Topologies

The results in Figure 4 show emergence of the scalar field Φ as a gap (∆ 2.4 ks^-1) between Type-II superconductivity on the left and dual Type-I superconductivity on the right. This represents a transition between the gapped state of the magnetically ordered Type-II superconductor and gapless state of the topologically ordered dual Type-I superconductor. At this point, the gauge monopole condensate forms to facilitate spontaneous magnetism M_s. The electric field E is excluded to be confined on the surface of the system (including the additional surface created through gauge topological defects that form the complex magnetic condensate). That is, a gapless surface is established so that the Berry phase manifests as a non-trivial topological insulator [51].

The gapless surface may be protected from external perturbations tending to re-open the gap through Abelian topology, as represented by the ℤ₂ Chern number in the symmetry group decompositions of (16) and (24). In a review of topological superconductors [65], Sato and Ando explore the connection between ℤ₂ and time reversal symmetry that is consistent with the symmetry of Lorentz boosts described above. The ℤ₂ Chern number can be interpreted as a U(2) group that fibres over a circle as a 3-sphere bundle, i.e. a Hopf fibration results [66].

Typically, a topological insulator is characterized by a non-robust, non-degenerate ground-state in which energy bands coincide and exceptional, or ‘diabolical’, points occur. However, the Berry phase variant identified above displays the following features: asymptotic behaviour (robustness against perturbations); a critical correlation length ξ (long-range entanglement); conformal geometry (describable through quantum field theory); and degeneracy in non-trivial topology on a hyperbolic manifold [66]. Thus, the system also appears to be topologically ordered and so describable by an effective, low-energy topological quantum field theory (TQFT) in which many-body states would typically have topological ground-state degeneracy [67]. In TQFT the critical correlation length ξ is topologically invariant and therefore insensitive to the geometry of the embedding manifold, i.e. the critical exponents within the universality class are background independent under Lorentz boosts.

Within the research field of topological phases of matter, as investigated to date, all the topologically ordered states realized experimentally or investigated theoretically are established through strong electron-electron interactions. The coinciding valence bands of gapless ‘diabolical’ points allow for degenerate electron movements between the bands. In a crystal structure, the electronic band structures are described by Bloch’s theorem as expressed by:

$$\Psi \left(\mathit{r}\right)=e^{ik.\mathit{r}}u\left(\mathit{r}\right)$$

(30)

where Ψ is the wavefunction, r is position, u is a periodic function, and k is the crystal momentum vector.

However, the original formulation of the Berry phase was not specifically related to Bloch electrons. Instead, it was based on the general idea that quantum adiabatic transport of particles in slowly varying fields (e.g. electric, magnetic, or strain) could in principle modify the wavefunction by terms other than just the dynamical phase. So, equation (30) is seen to map to the experimentally derived equation (11) where variable electromagnetic duality of the pseudo-scalar field gives the periodic function u(r) and e^ik.r represents the vector field of the gauge connection [38].

Since the degeneracy associated with the crystal-fluid material occurs in metastable excited-states (with non-zero temperatures), the gapless degeneracy of the crystal-fluid material cannot be attributed to Bloch electrons. So, whilst the gapless surface of the dual superconductor is protected through ℤ₂ topology, an additional mechanism is necessary to supress excited-state fluctuations such that Bloch-wave behaviour can emerge.

A potential solution is presented in Figure 5 where the synchronized continuous symmetry group U(2) leads to descriptions of asymptotic freedom in both quantum and condensed matter systems. As conservation of angular momentum extends into the microscopic quantum realm, so confinement mechanisms extend out into the macroscopic condensed matter of the dual superconductor under a renormalized Noether symmetry. The non-Abelian SU(3) subgroup of QCD remains dominant so that excited-state fluctuations due to changes in momentum are suppressed through confinement mechanisms and the constant Hamiltonian function is preserved. Topological defects may be created by chromoelectric flux tubes, or penetrating vortices resulting from chromomagnetic gauge monopole condensation, that enable the formation of quark-antiquark pairs together with an inherent confinement mechanism.

In a non-Abelian gauge theory, the expectation value of the confinement phase can be detected by the area law of a Wilson Loop [68]. The Wilson loop average W(C) is related the energy of the interaction of static (i.e. infinitely heavy) quarks. For large loops, the potential energy is a linear function of the distance between quarks since the gluon field contracts to a flux tube that establishes a string tension K.

$$W\left(C\right)\stackrel{\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}C}{\to }{e}^{-\mathrm{K}{A}_{min}\left(C\right)}$$

(31)

If A_min(C) is taken to be the hyperbolic area formed by the rapidity angle exponents |±φ| of the Weyl spinors (25) and (26), then - K maps to the reciprocal of the scalar field Φ, i.e. a temporal component with units ks^-1 is also incorporated, as shown in Figure 4. The results in Table A3 and Table A8 reveal that the expectation value is expressed in the gauge connection field with associated gauge monopole defects manifesting in hyperbolic surface area:

$$W\left(C\right)\to 4\mathsf{\pi }{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^2\left({\displaystyle \frac{R}{2}}\right)\stackrel{\begin{array}{c}topological\\ defects\end{array}}{\to }{{(e}^{\Phi })}^{\left|\pm \mathsf{\phi }\right|}$$

(32)

The rapidity angle exponents ±φ of the Weyl spinors lead to the same invariant expectation value in both local Lorentz spacetime and the local non-Abelian Wilson loop associated with the critical system (|±φ| ≡ 3v). Background independence across the entire U(2) symmetry group is thereby demonstrated. Such an interpretation is consistent with the requirements of a TQFT [69].

The Wilson loop operator is comprised of either time-like loops and space-like loops. There is no intrinsic difference between these objects which are related through Lorentz invariance [4] and combined in equations (31) and (32). So, in Yang-Mills theory, the Wilson loop can describe the confined quark potential as an autonomous phenomenon free from matter. However, the gauge monopole condensate as identified represents a deconfined Higgs phase which, for the exceptional correlation length ξ established, is expected to satisfy the area law of the ‘t Hooft loop in ℤ₂ gauge theory. These two loop operators describe opposite behaviours. Under such circumstances, a mixed phase that is both confined and deconfined is described and the non-Abelian SU(3) can exist as a subgroup within the Abelian U(2) group. Where Wilson loop and ‘t Hooft loop operators exhibit these anti-commutation relations, it is possible to construct a TQFT [70].

Acceleration and deceleration of the crystal-fluid material thereby generate confined interactions responsible for the splitting and recombining of gluons. Gluons are either indirectly absorbed by or indirectly emerge from the QCD vacuum manifold [54]. The Higgs-like scalar field Φ also emerges in the transition from gapped Type-II superconductivity to gapless dual Type-I superconductivity to establish a reciprocal gap in the gradient energy, as revealed in Figure 4. For the gapless state, the spin-1 vector gluons are the force-carrying SU(3) gauge bosons that indirectly generate the Bloch-wave description in the U(2) gauge monopole condensate [71]. These indirect gluon interactions with the embedding QCD vacuum manifold enable fictitious forces to emerge in non-inertial reference frames which lead to PV work in the piston expander.

Whilst the massless spin-1 gluons either emitted or absorbed by accelerating or decelerating quarks give rise to gauge particles with effective spin-½ behaviour, gauge invariance is only established where the associated gauge field can emerge under U(2) ‘symmetry synchronization’ at critical correlation length ξ in the low-energy system. That is, a gauge monopole topology [54] is necessary so that the scalar field potential can induce magnetic flux such that ∇. B_s = 0, where B_s is the spontaneous magnetic flux density.