Q-Ball Mechanism of Electron Transport and SPIN/Lattice Excitations Properties of High-T_c Superconductors

1. Introduction

It was demonstrated previously [1,2] that electron/hole Fermi surface possessing ’nested’ parts with high enough density of states , e.g., extended Van Hove singularity [3], makes the system unstable to formation of the Euclidean Q-balls (nontopological solitons) that constitute Bose condensed collective spin/charge density wave fluctuations (SDW/CDW) with zero static mean that arise inside a Q-ball simultaneously with Cooper/local pairs superconducting condensate for which they serve as the ’pairing glue’. Bose condensed collective spin/charge density wave fluctuations (SDW/CDW) possess the wave-vectors Q_DW that connect corresponding ’nested’ regions of the Fermi surface near the van Hove singularities, e.g., in the doped cuprates. Consideration of the self-consistency equations for the quantum order parameters in the form of periodic in the imaginary Matsubara time SDW/CDW semiclassical fields with zero mean actually signifies a new development beyond the Landau static mean-field order parameter approach. Corresponding free energy of the system is introduced then using Euclidean action instead of Landau free energy functional. In a particular case considered below the Bose condensation of the Matsubara time dependent SDW/CDW collective fluctuations forming a Q-ball is accompanied self-consistently by formation of the Cooper/local pair superconducting condensate formed of the $Q_{DW}$ - ’nested’ fermionic states, thus making Q-ball formation energetically favourable. The Q-ball SDW/CDW semiclassical field with $U(1)$ symmetric bare Hamiltonian rotates with Matsubara bosonic frequency $\Omega =2\pi T$, thus, breaking chiral symmetry along the Matsubara time axis in Euclidean space-time and possessing zero mean. This rotation provides conservation of the Noether "charge" Q proportional to the number of elementary SDW/CDW excitations (spin-waves/phonons) condensed inside the Q-ball. The extra conserved quantity Q (besides the energy) makes the Q-ball volume finite. The temperature T* of the Q-balls condensation is proportional to the inverse correlation length (that defines the fluctuation’s ’mass’) of the SDW/CDW field and is much higher than static bulk SDW/CDW transition temperature [2]. The latter falls down below the dome of superconducting phase transition temperatures T_c defined by the Q-ball induced Cooper/local pairing mechanism of superconductivity [2]. The quintessence of why T* and T_c are relatively high with respect to the Bardeen-Cooper-Schrieffer (BCS) predicted superconducting transition temperature $T_c^{BCS}$ lies in that T* and T_c characterise instability with respect to condensation of the semiclassical Q-ball SDW/CDW field coherently mediating Cooper/local pairing of the ’nested’ fermionic states inside the Q-balls. This picture is drastically different from the BCS scenario of condensation of the Cooper pairs due to coupling of fermions via incoherent gas of single spin waves/phonons. Opening of the pairing energy gap in the fermionic spectrum of the Q-balls on the ’nested’ parts of the Fermi surface simultaneously transforms the whole system into ’pseudogap’ (PG) phase. The condition that the Q-balls field solutions minimise the free energy of the whole system provides the self-consistency equations solved in [1,2], that extend the Ginzburg-Landau static order parameter approach. Previously real and imaginary parts of ’rotating’ in real time order parameter field, that constituted two scalar fields with nonderivative interactions in Minkowski space-time, were considered by Sidney Coleman as a Q-ball ansatz for baryons [4] in the supersymmetric standard model, where conserved Noether charge Q counted the number of baryons associated with the $U(1)$ symmetry of the squarks field [4,5,6]. The present paper extends farther theoretical investigation of the Euclidean Q-ball model predictions to the electron transport, phonon/spin dispersions splitting and diamagnetic properties of high-T_c superconductors above T_c. In particular, the T-linear temperature dependence of electrical resistivity in the interval of temperatures $T_c<T<T^{*}$ above superconducting dome maximum T_c is derived analytically. The diamagnetic behaviour observed experimentally in the "strange metal" phase [7,8] is derived analytically as well. Here we also investigate the changes in the spectra of the quantum spin and phonon excitations due to their scattering on the superconducting condensates inside the Q-balls. In particular, magnetic and phonon dispersions, analytically derived below, demonstrate direct correspondence with the experimentally found famous ’hourglass’ spin-waves dispersion in the vicinities of the SDW antiferromagnetic wave-vectors [9] and splitting into hardened and softened modes of the optical phonons found in the inelastic neutron scattering studies [10,11] in the high-T_c cuprates in the vicinities of the CDW ’nesting’ wave vectors of the Fermi surface regions in the PG phase. It is important to stress here, that unlike in the usual static ’nesting’ scenario with resulting long-ranged SDW/CDW ordering competing with superconducting pairing [12,13], the Q-ball semiclassical fields possess zero static mean values and therefore may belong to the ’local hidden orders’. Hence, the Q-ball induced magnetic and phonon spectral anomalies differ from the conventional Peierls-like pictures of static SDW/CDW formation, where electron-phonon/magnon coupling drives to zero corresponding dispersion branches of the excitations at the ordering wave vector of a static (spin)lattice distortion. Inelastic x-ray scattering studies of the phonon softening in high-T_c cuprates were also considered in relation with phonon coupling to CDW fluctuations, see [14,15,16] and references therein. Previously, the Q-balls theory predictions for X-ray scattering [17] were found in favourable accord with experimental results of X-ray diffraction in high-Tc cuprates superconductors in the heterogenous/PG phase [18,19,20].

The plan of the paper is as follows. In the next Sections II and III a quintessence of the Euclidean Q-balls picture is presented. Analytical derivations of the major parameters of the Q-balls, including temperature dependence of their energy, semiclassical Q-ball field amplitude and related phase diagram of the system possessing Q-balls are presented. In particular, it is demonstrated that the first order transition temperature T* is proportional to the inverse correlation length of the short-range spin/charge density wave fluctuations. An idea of a Matsubara time-dependent semiclassical ‘pairing glue’ between fermions in cuprates, but for an itinerant case, was proposed earlier [24]. Described here superconducting pairing mechanism mediated by semiclassical Q-ball field is distinct from the usual phonon- [25,26] or spin-fermion coupling models [27] considered previously for high-T_c cuprates, based upon exchange with infinitesimal spin- and charge-density excitations [15] or polarons [28] in the usual Fröhlich picture. In section IV the self-consistently determined temperature dependence of the Q-ball field amplitude is used for an analytical derivation of the T-linear temperature dependence of the inverse life-time of fermionic excitations due to scattering on Q-balls in the normal phase. This inverse life-time T-linear temperature dependence is in accord with experimentally measured linear temperature dependence of electrical resistivity in the ’strange metal’ phase [7]. In Section V the paraconductivity calculation method by Alex Abrikosov [30] is used for derivation of electrical resistivity dominated by Q-balls slide. In Section VI diamagnetic response of Q-balls ’gas’ is calculated and qualitative accord with experimental data of L. Li et al. [8] is found. In Section VII the ’hourglass’ like dispersion in the vicinity of antiferromagnetic SDW wave vectors of the magnetic excitations, and splitting of the Cu-O in-plane phonon brunches into the softened and hardened ones close to CDW fluctuations wave vectors in the Brillouin zone are derived analytically and plots generated by Wolfram Mathematica program are presented. Conclusions follow in Section VIII.

2. Euclidean Extension of Landau’s Free Energy Functional Approach

This section contains an overview of the previous work on Q-balls in Euclidean space-time that helps a reader to navigate from the origins of the main expressions to the derivation of the temperature dependences of the field amplitude used to achieve the main results of the present paper. More detailed information is incorporated in the works published previously [1,2,17,20].

The usual Hamiltonian describing interaction between electrons and phonons/spin waves is written in the form [31]:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill H=\sum _{\overrightarrow{p},\sigma }{\epsilon }_{\overrightarrow{p}}{c}_{\overrightarrow{p},\sigma }^{†}{c}_{\overrightarrow{p},\sigma }+\sum _{\overrightarrow{q}}{\omega }_{\overrightarrow{q}}{b}_{\overrightarrow{q}}^{†}{b}_{\overrightarrow{q}}+g\sum _{\overrightarrow{p},\overrightarrow{q},\sigma }{\left({\displaystyle \frac{{\omega }_{\overrightarrow{q}}}{2V}}\right)}^{1/2}\left(c_{\overrightarrow{p},\sigma }^{†}{c}_{\overrightarrow{p}+\overrightarrow{q},\sigma }{b}_{\overrightarrow{q}}^{†}+h.c.\right)\end{array}\end{array}$$

(1)

leading to the effective interaction Hamiltonian used by Eliashberg [26] in the Fröhlich picture [31] of superconductivity mechanism:

$$\begin{array}{c}\hfill H_{int}(\tau )={\displaystyle \frac{g^2}{V}}\sum _{\mathbf{p},{\mathbf{p}}^{\prime },\mathbf{q},\sigma ,{\sigma }^{\prime }}{\int }_0^{\beta }d{\tau }^{\prime }{D}_{\mathbf{q}}(\tau -{\tau }^{\prime })\left(c_{\mathbf{p},\sigma }^{+}(\tau )c_{\mathbf{p}+\mathbf{q},\sigma }(\tau )c_{{\mathbf{p}}^{\prime }+\mathbf{q},{\sigma }^{\prime }}^{+}({\tau }^{\prime })c_{{\mathbf{p}}^{\prime },{\sigma }^{\prime }}({\tau }^{\prime })\right)\end{array}$$

(2)

where phonon/spin-wave propagator $D_{\mathbf{q}}(\tau -{\tau }^{\prime })$ equals [31]:

$$\begin{array}{cc}& D_{\mathbf{q}}(\tau )=-{\displaystyle \frac{{\omega }_{\mathbf{q}}}{2}}\left[\left(N(\mathbf{q})+1\right)exp\left\{{\omega }_{\mathbf{q}}|\tau |\right\}+N(\mathbf{q})exp\left\{-{\omega }_{\mathbf{q}}|\tau |\right\}\right];\\ & N(\mathbf{q})={\left[exp\left\{{\omega }_{\mathbf{q}}/T\right\}-1\right]}^{-1}\end{array}$$

(3)

where the Boltzmann constant is taken for unity $k_B=1$. The Q-ball picture of superconductivity proposed recently [1,2,17] considers possibility of Bose-condensation of phonons/spin-waves in some particular points $\left\{\Omega ,\mathbf{q}\right\}$ of their $D+1$ Euclidean frequency-momentum phase-space thus leading to a substitution of the ${\left({\displaystyle \frac{{\omega }_{\overrightarrow{q}}}{2V}}\right)}^{1/2}{b}_{\overrightarrow{q},\Omega }^{†},{\left({\displaystyle \frac{{\omega }_{\overrightarrow{q}}}{2V}}\right)}^{1/2}{b}_{\overrightarrow{q},\Omega }$ creation/annihilation phonon/spin-wave Bose-operators with the Matsubara time - dependent order parameter represented by the Bose-condensate c-number operators ${\zeta }_0^{+},{\zeta }_0$:

$$\begin{array}{c}\hfill {\zeta }_0^{+}={M^{*}}_{\overrightarrow{q},\Omega }exp\left\{-i\mathbf{q}·\mathbf{r}+i\Omega \tau \right\};{\zeta }_0=M_{\overrightarrow{q},\Omega }exp\left\{i\mathbf{q}·\mathbf{r}-i\Omega \tau \right\}\end{array}$$

(4)

describing a finite density of the corresponding lattice charge/spin density distortion: ${\zeta }_0^{+}{\zeta }_0={|M_{\overrightarrow{q},\Omega }|}^2$. Importantly, the Bose c-number operators in Equation (4) possess chirality along the Matsubara time axis, i.e., the symmetry $\Omega \to -\Omega$ is broken. This is necessary condition for nonzero conserved Noether "charge" Q to exist along the winding trajectory of the Bose-condensate in Euclidean space leading to a finite Q-ball volume. It is shown in [1,2,17] that the free energy of the system may possess a minimum with respect to $|M_{\overrightarrow{q},\Omega }|$ amplitude and Q-ball volume $V_Q$. Such a nontrivial minimum arises for a finite volume Q-balls due to condensation of superconducting Cooper/local fermionic pairs, for which the operators ${\zeta }_0^{+},{\zeta }_0$ play the role of the ’pairing glue’ bosons. Thus, this new mechanism of superconductivity describes the picture of fluctuating semiclassical charge/spin-density waves with a particular Bose frequency $\Omega =2\pi nT$ inside the Q-balls, represented by operators ${\zeta }_0^{+},{\zeta }_0$, that are not at all competing with superconductivity, but on the contrary, are creating it in a self-consistent manner. To see how this happens, one may start from a simple t-U Hubbard Hamiltonian:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill H=-t\sum _{\langle i,j\rangle ,\sigma }{c}_{i,\sigma }^{†}{c}_{j,\sigma }+U\sum _i{\widehat{n}}_{i,\uparrow }{\widehat{n}}_{i,\downarrow }-\mu \sum _{i,\sigma }{\widehat{n}}_{i,\sigma },\end{array}\end{array}$$

(5)

and use formally the Hubbard-Stratonovich decoupling procedure with a scalar complex field $M_{\mathbf{q}}(\tau ,\mathbf{r})={\zeta }_0$ defined in Equation (4): $M_{\mathbf{q}}(\tau ,\mathbf{r})=M_{\mathbf{q},\Omega }exp\left\{i\mathbf{q}·\mathbf{r}-i\Omega \tau \right\}$, that leads either to electrons/holes scattering on SDW field $\sigma M_{\mathbf{q}}(\tau ,\mathbf{r})$:

$$\begin{array}{c}\hfill H_{int}^{SDW}(\tau )=\sum _{\mathbf{q},\mathbf{Q},\sigma }\left(c_{\mathbf{q}+\mathbf{Q},\sigma }^{+}(\tau )M_{\mathbf{Q}}(\tau )\sigma c_{\mathbf{q},\sigma }(\tau )+H.c.\right)\end{array}$$

(6)

or to electrons/holes scattering on CDW field $M(\tau ,\mathbf{r})$ in the crystal lattice:

$$\begin{array}{c}\hfill H_{int}^{CDW}(\tau )=\sum _{\mathbf{q},\mathbf{Q},\sigma }\left(c_{\mathbf{q}+\mathbf{Q},\sigma }^{+}(\tau )M_{\mathbf{Q}}(\tau )c_{\mathbf{q},\sigma }(\tau )+H.c.\right)\end{array}$$

(7)

where the shorthand $M_{\mathbf{Q}}(\tau )\equiv M_{\mathbf{Q},\Omega }exp\left\{-i\Omega \tau \right\}$, and $\sigma$ spin factor is missing in the charge - fermion coupling vertex $c^{+}Mc$ and interaction representation is implied for the fermionic creation/annihilation operators with the hamiltonian (5), but without U-term. The Hubbard-Stratonovich field of a Q-ball nontopological soliton $M(\tau ,\mathbf{r})$ is sought for in the form:

$$\begin{array}{c}\hfill M(\tau ,\mathbf{r})=e^{i\mathbf{Q}·\mathbf{r}-i\Omega \tau }M\Theta \left\{\mathbf{r}\right\};\Theta (\mathbf{r})\equiv \left\{\begin{array}{c}1;\mathbf{r}\in V;\hfill \\ 0;\mathbf{r}\notin V.\hfill \end{array}\right.,\Omega =2\pi nT,n=1,2,...\end{array}$$

(8)

where V is the Q-ball volume that minimises the Euclidean action found below. Thus, besides Matsubara time periodicity $M(\tau +1/T,\mathbf{r})=M(\tau ,\mathbf{r})$, the field is assumed to break chirality along the Matsubara time axis. A simple model Euclidean action $S_M$ for the scalar field $M(\tau ,\mathbf{r})$, could be written as:

$$\begin{array}{ccc}& S_M^0& ={\int }_0^{\beta }{\int }_{V}d\tau d^D\mathbf{r}{\displaystyle \frac{1}{g}}\left\{|{\partial }_{\tau }{M|}^2+s^2{|{\partial }_{\mathbf{r}}M|}^2+{\mu }_0^2{|M|}^2\right\},\hfill \\ & & M\equiv M(\tau ,\mathbf{r})\hfill \end{array}$$

(9)

where s is bare propagation velocity, and the ‘mass’ term ${\mu }_0^2\sim 1/{\xi }^2$ imposes finite correlation length $\xi$ of the fluctuations. Considering now the Fermi surface with the ’nested’ parts with high enough density of states, e.g., Van Hove singularity [3], being connected by a wave vector ${\mathbf{Q}}_{DW}$, one would customary declare as the next step that field $M(\tau ,\mathbf{r})$ is Matsubara time $\tau$ independent, with a ’nesting’ wave-vector $\mathbf{Q}={\mathbf{Q}}_{DW}$ becoming a SDW or CDW wave vector formed under (Peierls-like) phase transition, that forms corresponding energy gap in the fermionic spectrum. Hence, superconductivity would be then considered as a ’competing order’, see e.g., [12,13] for a recent overview. The Q-ball picture approach is drastically different.

2.1. Hubbard-Stratonovich Q-Ball Field as the ’Pairing Glue’

It was shown [1,2] that Euclidean action of the Hubbard-Stratonovich field may develop a semiclassical minimum of the Q-ball universality class (nontopological Euclidean soliton) (8) when simultaneously the fermions that are ’decoupled’ by this field in Equations (6), (7) start locally condense into a Cooper/local pair superconducting condensate. The soliton field $M(\tau ,\mathbf{r})$ is periodic in Matsubara time with zero mean value, and, therefore, is called ’thermodynamic quantum time crystal’ [21,22]. But, most important, besides Matsubara time periodicity $M(\tau +1/T,\mathbf{r})=M(\tau ,\mathbf{r})$, the field is assumed to break chirality along the Matsubara time axis, thus taking the form in Equation (8). The Q-balls with ’counterclockwise’ chiral combination $e^{i{\mathbf{Q}}_{\mathbf{DW}}·\mathbf{r}+\Omega \tau }$ are also allowed as separate objects. After the Hubbard-Stratonovich SDW/CDW field $M(\tau ,\mathbf{r})$ in the above form is inserted into decoupled parts of the t-U Hubbard Hamiltonian Equations (6) and (7) the Cooper paired fermions are integrated out creating an additional energy term $U_f$ in the effective action of the $M(\tau ,\mathbf{r})$ field:

$$\begin{array}{cc}\hfill & V{U}_f(|M(\tau ,\mathbf{r})|)=\Delta {\Omega }_s=-Tln{\displaystyle \frac{Tr\left\{e^{-{\int }_0^{\beta }{H}_{int}(\tau )d\tau }\mathcal{G}(0)\right\}}{Tr\left\{\mathcal{G}(0)\right\}}}\equiv {\Omega }_s-{\Omega }_0;\mathcal{G}(0)\equiv e^{-\beta H_0};\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & H_0=\sum _{\mathbf{q},\sigma }{\epsilon }_q{c}_{\mathbf{q},\sigma }^{+}{c}_{\mathbf{q},\sigma }\hfill \end{array}$$

(11)

Here $H_0$ is Hamiltonian of the noninteracting fermions on a lattice with the bare dispersion ${\epsilon }_q$ and $\Delta {\Omega }_s$ is the electron pairs contribution to the free energy. The latter is calculated via standard procedure [31] and its result is presented in detail in [1,2]:

$$\begin{array}{cc}\hfill & U_f=-{\displaystyle \frac{4\nu {\epsilon }_0\Omega }{3}}I\left({\displaystyle \frac{M}{\Omega }}\right),M\equiv |M(\tau )|\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & I\left({\displaystyle \frac{M}{\Omega }}\right)={\int }_{\gamma }^{M/\Omega }d\alpha {\displaystyle \frac{\alpha \sqrt{2\alpha \left(\alpha -\gamma \right)}}{(1+8\alpha \left(\alpha -\gamma \right))}}tanh{\displaystyle \frac{\sqrt{2\alpha \left(\alpha -\gamma \right)}\Omega }{{\epsilon }_0}}tanh{\displaystyle \frac{\sqrt{2\alpha \left(\alpha -\gamma \right)}\Omega }{2T}},\gamma \approx 1/2\hfill \end{array}$$

(13)

where $2{\epsilon }_0$ signifies the width of the interval along the energy axis, $|{\epsilon }_p-\mu |\le {\epsilon }_0$, with nonzero density of fermionic states $\nu ({\epsilon }_p)\approx \nu$ obeying the ’nesting’ relation ${\epsilon }_{p-Q_{DW}}=-{\epsilon }_p$ around the chemical potential $\mu$. Expression (13) is remarkable, since it bears result of the important two step self-consistent ’bootstrap procedure’ [1,2]. Namely, at the first step the following relation is derived using mentioned above procedure for the free energy calculation via integration over a variable coupling strength $\alpha$ [31]:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\Omega }_s}{\partial \alpha }}=T{\int }_0^{\beta }〈{\displaystyle \frac{\partial H_{int}(\tau )}{\partial \alpha }}〉d\tau =-{\displaystyle \frac{T}{\alpha }}{\int }_0^{\beta }{\int }_0^{\beta }d\tau d{\tau }_1〈H_{int}(\tau )H_{int}({\tau }_1)〉=\hfill \\ \hfill & -{\displaystyle \frac{TV}{\alpha }}{|M|}^{2}T\sum _{\omega ,\mathbf{p},\sigma }\sigma \overline{\sigma }{\overline{F}}_{\sigma ,\overline{\sigma }}(\omega ,\mathbf{p})F_{\overline{\sigma },\sigma }(\omega -\Omega ,\mathbf{p}-{\mathbf{Q}}_{\mathbf{DW}}){\alpha }^2,\hfill \end{array}$$

(14)

where the loop of Gor’kov anomalous functions $F^{†},F$ describing the condensed paired fermions contains the Q-ball semi-classical field propagator $D(\tau -{\tau }^{\prime },\mathbf{r}-{\mathbf{r}}^{\prime })\sim {M({\tau }^{\prime },{\mathbf{r}}^{\prime })}^{*}·M(\tau ,\mathbf{r})$ instead of the usual phonon-/spin-wave propagator in the Fröhlich picture [31].

2.2. Eliashberg-like Equation with Q-Ball Field as the ’Pairing Glue’

The second step of the ’bootstrap’ self-consistent procedure would be then to solve the Eliashberg-like equation for the Gor’kov anomalous fermionic Green’s functions $F^{†},F$ with the Q-ball field (8) playing role of the ’pairing glue’ that couples ’nested’ fermionic states on the bare Fermi surface [1,2]:

$$\begin{array}{c}\hfill F_{p,\sigma }(\omega )=-{\Sigma }_{2p,\sigma }(\omega )K_p(\omega )=-K_p(\omega )\left[T\sum _{\pm \Omega }{\mathcal{D}}_{Q_{DW}}(\Omega )F_{p-Q_{DW},\sigma }(\omega -\Omega )\right],\end{array}$$

(15)

where :

$$\begin{array}{c}\hfill K_p(\omega )={\left\{|i\omega -{\epsilon }_p-{\Sigma }_{1p,\sigma }{(\omega )|}^2+{|{\Sigma }_{2p,\sigma }(\omega )|}^2\right\}}^{-1}\approx {\left\{{\omega }^2+{\epsilon }_p^2+{|{\Sigma }_{2p,\sigma }(\omega )|}^2\right\}}^{-1}.\end{array}$$

(16)

Here:

$$\begin{array}{c}\hfill {\mathcal{D}}_{Q_{DW}}(\Omega )\equiv {\displaystyle \frac{M^2}{T}},{\mathcal{D}}_{Q_{DW}}(\tau )\equiv 2M^{2}cos(\Omega \tau ).\end{array}$$

(17)

The self-energy function ${\Sigma }_{2p,\sigma }(\omega )$ is approximated with parabolic function of the bare fermionic dispersion ${\epsilon }_p$ in the vicinity of the Fermi energy:

$$\begin{array}{c}\hfill |{\Sigma }_{2p,\sigma }{(\omega )|}^2=g_0^2-{\epsilon }_p^2.\end{array}$$

(18)

As it was found previously [24], compare [26], the normal self-energy function ${\Sigma }_{1p,\sigma }(\omega )$ leads to renormalisation of the Fermi velocity and the chemical potential shift and could be neglected. Then, using approximation (18) for the self-energy function ${\Sigma }_{2p,\sigma }(\omega )$ it is straightforward to convert Equation (15) directly into differential equation of the Mathieu type [1,2]:

$${\partial }_{\tau }^2{F}_{p,\sigma }\left(\tau \right)+\left(2M^{2}cos\left(\Omega \tau \right)-g_0^2\right)F_{p,\sigma }\left(\tau \right)=0,F_{p,\sigma }\left(\tau +{\displaystyle \frac{1}{T}}\right)=-F_{p,\sigma }\left(\tau \right),\Omega =2\pi nT,$$

(19)

exact solutions of which are well known, and the anti-periodicity condition [31] for the sought for solution for the fermionic Green’s function $F_{p,\sigma }\left(\tau \right)$ is explicitly indicated. The lowest possible eigenvalue $-g_0^2$ is then [32]:

$$g_0^2=2M\left(M-\gamma \Omega \right);\gamma \approx 1/2,$$

(20)

Hence, a non-zero anti-symmetric solution $F(\tau )$ of Equation (19) with real eigenvalue $g_0^2$ exists when $M>\gamma \Omega$. This result is remarkable: it follows from relations (15), (18) and (20), that Q-ball of ’glue boson’ condensate possessing amplitude $M>\gamma \Omega$ also possesses a Cooper pair condensate characterised with non-zero anomalous Green’s function $F_{p,\sigma }$:

$$\begin{array}{c}\hfill F_{p,\sigma }(\omega )=-{\Sigma }_{2p,\sigma }(\omega )K_p(\omega )\approx -{\displaystyle \frac{\sqrt{g_0^2-{\epsilon }_p^2}}{{\omega }^2+g_0^2}},\end{array}$$

(21)

Since non-zero solution for the superconducting gap arises starting from finite amplitude $M>\gamma \Omega$, the Q-ball phase transition is of the first order with respect to the Q-ball field amplitude M. On the other hand, at finite M, that obeys: $M=\gamma \Omega$ the density of superconducting condensate inside the Q-ball equals zero. Hence the Q-ball phase transition is of the second order with respect to the density of superconducting condensate inside the Q-ball. Next, at the last step, an amplitude M is substituted with $\alpha M$ in (20) according to the definition of the formal dimensionless coupling parameter $\alpha$ in Equation (14), thus, leading to the final expression in Equations (12), (13):

$$g_0^2(\alpha )=2M\alpha \left(M\alpha -\gamma \Omega \right),$$

(22)

which is used in the final ’bootstrap’ expression for the effective Q-ball field energy $U_f$ in Equation (13). Hence, now one obtains the effective Euclidean action of the field $M(\tau ,\mathbf{r})$, that includes contribution from condensed superconducting fermionic pairs, $U_f(|M(\tau ,\mathbf{r})|)$. The action has to be searched for the assumed Q-balls quasi-classical minima:

$$\begin{array}{c}\hfill S_M={\int }_0^{\beta }{\int }_{V}d\tau d^D\mathbf{r}{\displaystyle \frac{1}{g}}\left\{|{\partial }_{\tau }{M|}^2+s^2|{\partial }_{\mathbf{r}}{M|}^2+{\mu }_0^2{|M|}^2+g{U}_f{(|M|}^2)\right\},M\equiv M(\tau ,\mathbf{r}),\end{array}$$

(23)

where $g=U^{3}V$, V - is the system volume, ${\mu }_0^2\propto U^2$. Under these values choice the action (23) would indeed serve to decouple the interaction term $U{\sum }_i{\widehat{n}}_{i,\uparrow }{\widehat{n}}_{i,\downarrow }$ in the t-U Hubbard Hamiltonian in Equation (5), that results then in appearance of the term $c_{\mathbf{q}+\mathbf{Q},\sigma }^{+}(\tau )M_{\mathbf{Q}}(\tau )\sigma c_{\mathbf{q},\sigma }(\tau )$ or similar in the $H_{int}$ Hamiltonians in Equations (6),(7). Hence, the amplitude of the field $M(\tau ,\mathbf{r})$ relates with the local condensed spin/charge densities as: $|{\widehat{n}}_{i,\uparrow }\pm {\widehat{n}}_{i,\downarrow }|\sim M(\tau ,\mathbf{r})/U$. This in turn imposes upper limit on the amplitude of the field: $|M(\tau ,\mathbf{r})|<U$ allowing for the value of a local electron spin/charge that might condense in the SDW/CDW Q-ball fluctuation considered here [29]. The model (23) is $U(1)$ invariant under the global phase rotation $\varphi$: $M\to M{e}^{i\varphi }$. Hence, corresponding ‘Noether charge’ is conserved along the Matsubara time axis [1,2]:

$$\begin{array}{c}\hfill Q={\int }_V{j}_{\tau }{d}^D\mathbf{r}={\int }_V{\displaystyle \frac{i}{2}}\left\{M^{*}(\tau ,\mathbf{r}){\partial }_{\tau }M(\tau ,\mathbf{r})-M(\tau ,\mathbf{r}){\partial }_{\tau }{M}^{*}(\tau ,\mathbf{r})\right\}{d}^D\mathbf{r}=\Omega M^{2}V,\end{array}$$

(24)

The ‘Noether charge’ conservation causes occurrence of Matsubara time periodic, finite volume Q-ball semiclassical solutions, that otherwise would be banned in D$>2$ by Derrick theorem [33] in the $\tau$-independent static case. Indeed, substitution of relation for the charge Q (24) into Euclidean action (23) leads to the expression that possesses minimum at finite Q-ball volume $V_Q$:

$$\begin{array}{c}\hfill S_M={\displaystyle \frac{1}{gT}}\left\{{\displaystyle \frac{Q^2}{V{M}^2}}+V[{\mu }_0^2{M}^2+g{U}_f]\right\},\end{array}$$

(25)

Provided the $\propto V$ term above is positive, the action $S_M$ is minimised by Q-ball volume $V_Q$:

$$\begin{array}{c}\hfill V_Q={\displaystyle \frac{Q}{M\sqrt{{\mu }_0^2{M}^2+g{U}_f(M)}}};\end{array}$$

(26)

for which the free energy of the Q-ball field equals:

$$\begin{array}{c}\hfill E_Q=T{S}_M^{min}={\displaystyle \frac{2Q\sqrt{{\mu }_0^2{M}^2+g{U}_f(M)}}{gM}}={\displaystyle \frac{2Q\Omega }{g}},\end{array}$$

(27)

After cancellation of Q-factor in the last relation in Equation (27) above the following self-consistency equation emerges [2]:

$$\begin{array}{c}\hfill 0=({\mu }_0^2-{\Omega }^2)M^2+g{U}_f(M).\end{array}$$

(28)

One then substitutes $U_f(M)$ from Equations (12), (13) into self-consistency equation Equation (28) and finds:

$$\begin{array}{c}\hfill ({\mu }_0^2-{\Omega }^2)M^2-{\displaystyle \frac{4\Omega g\nu {\epsilon }_0}{3}}I\left({\displaystyle \frac{M}{\Omega }}\right)=0.\end{array}$$

(29)

The above self-consistency equation was investigated in [1,2] numerically. Below, using approximate expression for the potential energy $U_f$ the $M(T)$ dependences will be found analytically and used for calculation of the temperature dependences of electrical resistivity and diamagnetic response of the Q-ball gas above superconducting transition temperture T_c. Simultaneously, condition $E_Q=0$ at $Q\ne 0$ marks transition to the bulk superconducting state since the Q-ball volume $V_Q$ then diverges according to Equation (26). One important observation is in order. Expression (29) could be obtained in two different cases. Namely, there are two possibilities for the self-energy function ${\Sigma }_{2p,\sigma }$: the d-wave symmetric behaviour of superconducting order parameter ${\Sigma }_{2p-Q_{DW},\sigma }=-{\Sigma }_{2p,\sigma }$ and the s-wave behaviour of it ${\Sigma }_{2p-Q_{CDW},\sigma }={\Sigma }_{2p,\sigma }$. The first case realises when spin (SDW) fluctuations couple to the fermions via interaction Hamiltonian (6), while the second case realises when charge (CDW) fluctuations instead of spin fluctuations couple to the fermions via interaction Hamiltonian (7). The particular choice of the two symmetries: ${\Sigma }_{2p-Q_{DW},\sigma }=\pm {\Sigma }_{2p,\sigma }$, is governed by the demand that contribution to the free energy in (14) due to pairing would be negative. The $\sigma$ spin factor is missing in the charge - fermion coupling vertex $c^{+}Mc$ in (7). This leads to the absence of the factor $\sigma \overline{\sigma }=-1$ in the Equation (14) in the CDW Q-ball field case. Hence, in order to keep $U_f<0$, as is necessary for the Q-ball formation, one has to compensate for this sign change in the CDW Q-ball field case by the change of the sign of the Green’s functions product ${\overline{F}}_{\sigma ,\overline{\sigma }}(\omega ,\mathbf{p})F_{\overline{\sigma },\sigma }(\omega -\Omega ,\mathbf{p}-{\mathbf{Q}}_{\mathbf{DW}})$ in Equation (14). Then, allowing for the structure of the Gor’kov’s anomalous Green’s function in Equations (15), (16) one concludes, that relation between the values of superconducting order parameter in the points connected by the ’nesting’ wave vector $Q_{CDW}$ should be altered with respect to Q-ball of SDW fluctuation, i.e in case of CDW Q-ball mediated pairing the ’nesting’ wave vector should couple points with the same sign of superconducting order parameter.

3. The Phase Diagram of the Q-Balls Gas

Summarising, Equation (23) was used to describe effective theory of the Fourier components of the leading Q-ball (i.e., short-range) SDW/CDW fluctuations. Explicit expression for $U_f(|M(\tau ,\mathbf{r})|)$ was derived and investigated in detail previously [1,2] by integrating out Cooper/local-pairs fluctuations in the ‘nested’ Hubbard model with charge-/spin-fermion interactions. As a result, Q-ball self-consistency Equation (28) was solved and investigated, and it was established that Euclidean Q-balls describe stable semiclassical short-range charge/spin-ordering fluctuations of finite energy that appear at finite temperatures below some temperature T*, found to be $T^{*}={\mu }_0/2\pi$ [1,2]. Next, it was also found that transition into pseudogap phase at the temperature T* is of the 1st order with respect to the amplitude M of the Q-ball SDW/CDW fluctuation and of 2nd order with respect to the superconducting gap $g_0$. In particular, the following temperature dependences of these characteristics of the Q-balls were derived from Equations (13), (20) and (28) in the vicinity of the transition temperature T* into Q-ball phase [2] for the CDW/SDW amplitude:

$$\begin{array}{c}\hfill M={\displaystyle \frac{\Omega }{2}}\left(1+{\left({\displaystyle \frac{\left(T^{*}-T\right)15\sqrt{2}{\mu }_0}{g\nu }}\right)}^{{\displaystyle \frac{2}{5}}}\right),T^{*}={\displaystyle \frac{{\mu }_0}{2\pi }},\end{array}$$

(30)

and for the pseudogap $g_0$:

$$g_0^2={\left(T^{*}-T\right)}^{{\displaystyle \frac{2}{5}}}{\Omega }^2{\left({\displaystyle \frac{15{\mu }_0}{4g\nu }}\right)}^{{\displaystyle \frac{2}{5}}},$$

(31)

which follows after substitution of Equation (30) into Equation (20). The above results differ from those presented in [2] by the change of the value of $\gamma =1$, used previously, for the more precise value $\gamma =1/2$ in (20).

An expression for the fermionic pairing contribution to the potential energy, $U_f{(|M|}^2)$, found in Equations (12) and (13) could be approximated away from the $T^{*}$ temperature in the form:

$$g{U}_f(M)=-{\displaystyle \frac{4g\nu {\epsilon }_0\Omega }{3}}I\left({\displaystyle \frac{M}{\Omega }}\right)\approx -{\displaystyle \frac{g\nu {\epsilon }_0}{3\sqrt{2}}}\left(M-{\displaystyle \frac{\Omega }{2}}\right),$$

(32)

where now $\gamma \approx 1/2$ is used for the superconducting gap function $g_0$ :

$$g_0^2\approx 2M\left(M-{\displaystyle \frac{\Omega }{2}}\right).$$

(33)

Hence, substituting the above expression into the pairing-induced effective potential energy of SDW/CDW field in Equation (12), one now finds:

$$\begin{array}{cc}\hfill & U_{eff}(M)={\mu }_0^2{M}^2+g{U}_f\approx {\mu }_0^2{M}^2-{\displaystyle \frac{g\nu {\epsilon }_0}{3\sqrt{2}}}\left(M-{\displaystyle \frac{\Omega }{2}}\right)\equiv \hfill \\ \hfill & \equiv {\mu }_0^2\left\{{\left(M-{\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}{\mu }_0^2}}\right)}^2+{\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}{\mu }_0^2}}\left(\Omega -{\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}{\mu }_0^2}}\right)\right\}.\hfill \end{array}$$

(34)

Then, one finds the value of the highest superconducting transition temperature $T_c={\Omega }_c/2\pi$ from the Q-ball volume divergence condition in Equation (26):

$$\begin{array}{cc}\hfill & U_{eff}(M)={\mu }_0^2{M}^2+g{U}_f=0={\mu }_0^2\left\{{\left(M-{\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}{\mu }_0^2}}\right)}^2+{\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}{\mu }_0^2}}\left({\Omega }_c-{\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}{\mu }_0^2}}\right)\right\}.\hfill \end{array}$$

(35)

From where it is straightforward to find:

$$\begin{array}{c}\hfill T_c={\displaystyle \frac{{\Omega }_c}{2\pi }}={\displaystyle \frac{g\nu {\epsilon }_0}{12\pi \sqrt{2}{\mu }_0^2}};M_c\equiv M({\Omega }_c)={\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}{\mu }_0^2}}\equiv {\Omega }_c\end{array}$$

(36)

Now, substituting results from Equation (36) into Equation (33) one finds directly the following relation between the superconducting gap $g_c$ and the temperature $T_c$:

$$g_c=\sqrt{2M_c\left(M_c-{\displaystyle \frac{{\Omega }_c}{2}}\right)}={\Omega }_c;{\displaystyle \frac{2g_c}{T_c}}\equiv 2\pi {\displaystyle \frac{2g_c}{{\Omega }_c}}=4\pi \approx 12,57.$$

(37)

The above results in Equations (36), (37) are remarkable from two points of view. First, the ratio $\approx 12,57$ of reduced superconducting gap in Equation (37) drastically differs from the BCS ratio $=3.5$ [25] and compares batter with $\approx 7.4$ found for BSCCO high-$T_c$ compounds [34]. Second, expression for $T_c$ in Equation (36) permits, in principle, to infer relation with the isotope effect, since besides the product $g\nu {\epsilon }_0$ the expression contains also the SDW/CDW characteristic parameter $\sim {\mu }_0^{-2}$. Simultaneously, the pseudogap/strange metal transition temperature $T^{*}={\mu }_0/2\pi$ possesses the same parameter ${\mu }_0$ in the numerator. Hence, the isotope effects for $T^{*}$ and $T_c$ would contain exponents of the opposite signs related with Q-ball parameter ${\mu }_0$. The next application of the approximate expression for the fermionic pairing contribution to the potential energy, $U_f{(|M|}^2)$, found in Equation (32) is even more impressive.

Namely, substituting Equation (32) into the self-consistency equation (28) one finds:

$$\begin{array}{cc}\hfill & 0=({\mu }_0^2-{\Omega }^2)M^2+g{U}_f(M)=({\mu }_0^2-{\Omega }^2)\left\{{\left(M-{\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}({\mu }_0^2-{\Omega }^2)}}\right)}^2-\right.\hfill \\ \hfill & \left.-{\left({\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}({\mu }_0^2-{\Omega }^2)}}\right)}^2\left(1-\Omega {\displaystyle \frac{6\sqrt{2}({\mu }_0^2-{\Omega }^2)}{g\nu {\epsilon }_0}}\right)\right\},\hfill \end{array}$$

(38)

that trivially leads to the two-branches solution $M_{\pm }(\Omega )$ obtained previously (see Figure3 in [1,2])from numerics:

$$\begin{array}{c}\hfill M_{\pm }={\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}({\mu }_0^2-{\Omega }^2)}}\left(1\pm \sqrt{1-\Omega {\displaystyle \frac{6\sqrt{2}({\mu }_0^2-{\Omega }^2)}{g\nu {\epsilon }_0}}}\right).\end{array}$$

(39)

First of all, introducing notation:

$$\begin{array}{c}\hfill \kappa ={\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}}}\equiv {\displaystyle \frac{4g\nu {\epsilon }_0}{3·8\sqrt{2}}}\equiv c{\displaystyle \frac{4g\nu {\epsilon }_0}{3}}\equiv \kappa ;c={\displaystyle \frac{1}{8\sqrt{2}}}\approx 0.09\end{array}$$

(40)

one finds the following equation that defines the boundary of the regions in the $\{\Omega ,\kappa \}$ plane where both brunches $M_{\pm }$ in Equation (39) are real:

$$\begin{array}{c}\hfill 1-\Omega {\displaystyle \frac{({\mu }_0^2-{\Omega }^2)}{\kappa }}=0.\end{array}$$

(41)

Thus, one finds that Equation (41) coincides with condition previously obtained numerically [1,2] for the existence of solution of the self-consistency equation (29), see Figure 1. In particular, a straightforward algebra gives from Equation (41) the result for the strength ${\kappa }^{*}=2{\mu }_0^3/(3\sqrt{3})$ corresponding to the touching point of the temperature boundaries of the strange metal and superconducting dome $T^{*}({\kappa }^{*})=T_c({\kappa }^{*})={\mu }_0/(2\pi \sqrt{3})$. Here one notes that at ${\kappa }^{*}=2{\mu }_0^3/(3\sqrt{3})$ the bulk superconductivity transition temperature ${T^b}_c$ found from Equation (35) is below the Q-balls temperature $T_c({\kappa }^{*})$: ${T^b}_c({\kappa }^{*})=2{\mu }_0/(6\pi \sqrt{3})<T_c({\kappa }^{*})={\mu }_0/(2\pi \sqrt{3})$, which is obvious from the Figure (35), where bulk superconductivity temperatures are plotted with dotted line. It is remarkable, that using neutron scattering results for ${\mu }_0$ related with effective superexchange coupling $J\approx 140meV$ between neighbouring magnetic moments in high-T_c cuprates [35] one finds ’why $T_c$ is high’:

$$\begin{array}{c}\hfill T_c({\kappa }^{*})={\mu }_0/(2\pi \sqrt{3})\sim {100}^{\circ }K-{200}^{\circ }K\end{array}$$

(42)

Last, but not the least, it follows from Equation (39) that in the interval ${\mu }_0/(2\pi \sqrt{3})\le T\le {\mu }_0/(2\pi )$, i.e., in the temperatures interval $\{T_c({\kappa }^{*}),T^{*}\}$ with $T^{*}={\mu }_0/(2\pi )$, the square root in Equation (39) could be expanded in the powers of the second term, which is smaller then unity, leading to the following expression for $M_{-}$ brunch:

$$\begin{array}{c}\hfill M_{-}\approx {\displaystyle \frac{g\nu {\epsilon }_0}{6\sqrt{2}({\mu }_0^2-{\Omega }^2)}}{\displaystyle \frac{\Omega 3\sqrt{2}({\mu }_0^2-{\Omega }^2)}{g\nu {\epsilon }_0}}\equiv {\displaystyle \frac{\Omega }{2}}=\pi T.\end{array}$$

(43)

This remarkable result, meaning the linear temperature dependence of the Q-ball field amplitude M, leads to the linear temperature dependence of the electric resistivity due to scattering of electrons on the $M_{-}$ field Q-balls, as is shown in the next Section IV. The numerical results of finding $M(T)$ dependences after substitution into Equation (35) and Equation (38) of the complete expression for the effective potential energy $U_f$ from Equations (12), (13) are plotted in Figure 2.

4. Electron Scattering and Resistivity of Q-Ball Gas

The Q-ball mechanism of the high-T_c superconductivity and pseudo-gap phase in cuprates introduced previously [1,2,17] is in essence a mechanism of Cooper-pairing that occurs due to pairing of fermions via exchange with bosonic fluctuations of spin- or charge density waves (SDW/CDW) condensed locally into Q-balls, the nontopological solitons of thermodynamic quantum time crystals. The conserved Noether charge Q counts the total number of condensed bosonic fluctuations inside the Q-ball, and the basic internal rotation frequency of the Q-ball is bosonic Matsubara frequency $\Omega =2\pi T$ of the fundamental Fourier component of the SDW/CDW semiclassical fluctuation. The heterogeneous phase of Q-balls appears below T* temperature and exists down to the temperature ${\mathrm{T}}_1^{*}$, that bounds from below the ’strange metal’ phase. In the optimally doped case ${\mathrm{T}}_1^{*}$ coincides with the top of the superconducting dome T_c of the high-T_c cuprates phase diagram [1,2]. Below we demonstrate that influence of Q-balls on the electrical transport in the "strange metal" phase causes "Planckian" [36] linear temperature dependence of the normal metal resistivity [7]. In short, since a Q-ball occupies finite space, there are outside electrons, that are not Cooper paired, and are scattered by the Q-ball SDW/CDW fluctuation. Demonstration of the fact that the T-linear temperature dependence of electrical resistivity of the "strange metal" phase occurs due to electrons scattering on the Q-balls is the focus of the present work. Besides, dragged by electric field (unpinned) CDW Q-balls become sliding charge ’droplets’, and hence, also contribute to the resistive normal current. This effect is considered below as well. Stability of Q-balls was proven for finite temperatures in [1,2] and long before that for the ground state of quantum matter [4,5].

To proceed one uses the Q-ball - fermion interaction Hamiltonian in the form [2]:

$$\begin{array}{c}\hfill {\widehat{H}}_{int}=8\pi \kappa \sum _{\overrightarrow{p},\overrightarrow{q},i}{e}^{-i\overrightarrow{q}{\overrightarrow{R}}_i}\left({\displaystyle \frac{M{c}_{\mathbf{p},\sigma }^{+}{c}_{\mathbf{p}-\mathbf{q},\sigma }{e}^{-i\Omega (\tau +{\tau }_0)}}{{({\kappa }^2+{(\overrightarrow{q}-\overrightarrow{Q})}^2)}^2}}+{\displaystyle \frac{M^{*}{c}_{\mathbf{p}-\mathbf{q},\sigma }^{+}{c}_{\mathbf{p},\sigma }{e}^{i\Omega (\tau +{\tau }_0)}}{{({\kappa }^2+{(\overrightarrow{q}+\overrightarrow{Q})}^2)}^2}}\right)\end{array}$$

(44)

where $\overrightarrow{Q}$ is either antiferromagnetic Brillouin zone SDW nesting wave-vector, or CDW wave-vector connecting the hot spots of the Fermi surface, and $\kappa =1/R\propto V^{-1/3}$, and R, V, M are Q-ball radius, volume and amplitude defined in Equations (8), (24) and found self-consistently. Summation over random coordinates ${\overrightarrow{R}}_i$ of the Q-ball centres is assumed in Equation (44). The Dyson equation for the Green’s function of electrons scattered on the Q-balls potential is presented in Figure 3. It follows from the well-known impurity scattering procedure [31], that averaging over the coordinates ${\overrightarrow{R}}_i$ of the Q-ball centres leads to the sum over double-scattered fermions on each Q-ball separately.

In Figure 3 the heavy and thin lines are fermionic temperature Green’s functions $G(r-r^{\prime },\tau -{\tau }^{\prime })$ and $G_0(r-r^{\prime },\tau -{\tau }^{\prime })$ respectively, that depend on the differences of the $D+1$ coordinates after averaging over positions of the Q-balls in space and Matsubara time origin ${\tau }_0$. Dots are vertices of fermion- Q-ball field M interaction introduced in Equation (44). The M-field bosonic Green’s function $D_M$, that follows from Equation (44) after averaging over positions of the centres of the Q-balls ${\overrightarrow{R}}_i$ and Matsubara time zero-origin ${\tau }_0$ is:

$$\begin{array}{c}\hfill D_M(\overrightarrow{q},\omega )={\left(8\pi M\kappa \right)}^2\left\{{\displaystyle \frac{{\delta }_{\omega ,\Omega }}{{({\kappa }^2+{(\overrightarrow{q}-\overrightarrow{Q})}^2)}^4}}+{\displaystyle \frac{{\delta }_{\omega ,-\Omega }}{{({\kappa }^2+{(\overrightarrow{q}+\overrightarrow{Q})}^2)}^4}}\right\}\end{array}$$

(45)

It is remarkable that due to semiclassical nature of the Q-ball fluctuation its Green’s function $D_M(\overrightarrow{q},\omega )$ possesses only single frequency $\Omega$ which is self-consistently determined from Equations (28) and (12), (13). Then, taking the Green’s function of the scattered fermions in the form:

$$\begin{array}{c}\hfill G(\overrightarrow{p},\omega )={\displaystyle \frac{1}{i\omega -\xi (\overrightarrow{p})-\overline{G}(\overrightarrow{p},\omega )}};\xi (\overrightarrow{p})=\epsilon (\overrightarrow{p})-\mu ,\end{array}$$

(46)

where $\mu$ is the chemical potential, and using the Dyson’s equation in Figure 3, one finds the following equation for the self-energy function $\overline{G}$:

$$\begin{array}{c}\hfill \overline{G}(\overrightarrow{p},\omega )=\sum _Q{\overline{n}}_Q{M}^2{\displaystyle \frac{{(8\pi \kappa )}^2}{{(2\pi )}^3}}\int d^3\overrightarrow{q}{\displaystyle \frac{G(\overrightarrow{p}-\overrightarrow{q},\omega -\Omega )+G(\overrightarrow{p}+\overrightarrow{q},\omega +\Omega )}{{({\kappa }^2+{(\overrightarrow{q}-\overrightarrow{Q})}^2)}^4}}.\end{array}$$

(47)

where ${\overline{n}}_Q$ is density of Q-balls with "charge" Q defined as:

$$\begin{array}{ccc}& & {\overline{n}}_Q={\displaystyle \frac{1}{V}}{G}_{Q}Cexp-{\displaystyle \frac{E_Q}{k_{B}T}}={\displaystyle \frac{C}{V_Q}}exp-{\displaystyle \frac{2Q\Omega }{g{k}_{B}T}}={\displaystyle \frac{4\pi }{g{V}_Q}}exp-{\displaystyle \frac{4\pi Q}{g}},\hfill \\ & & G_Q={\displaystyle \frac{V}{V_Q}},C={\displaystyle \frac{4\pi }{g}},\hfill \end{array}$$

(48)

Below it is assumed for simplicity that major scattering involves the fermions that occupy hole pockets in the Brillouin zone of doped cuprates with $\overrightarrow{Q}={\overrightarrow{Q}}_{SDW}$ being approximately magnetic Brillouin zone wave vector [1,2], or $\overrightarrow{Q}={\overrightarrow{Q}}_{CDW}$ and connects hot spots on the Fermi surface. Therefore, it is assumed that both $\overrightarrow{Q}$-vectors connect quasiparticle states of the opposite energies with respect to Fermi level, i.e., quasi-holes with quasi-electrons and vice versa : $\xi (\overrightarrow{p}\pm \overrightarrow{Q})=-\xi (\overrightarrow{p})$. Hence, using the latter equalities it is straightforward to change integration vector in the integral equation (47): $\overrightarrow{q}-\overrightarrow{Q}\to \overrightarrow{q}$, that leads to:

$$\begin{array}{c}\hfill \overline{G}(\overrightarrow{p},\omega )=\sum _Q{\overline{n}}_Q{M}^2{\displaystyle \frac{{(8\pi \kappa )}^2}{{(2\pi )}^3}}\int d^3\overrightarrow{q}{\displaystyle \frac{G(\overrightarrow{p}-\overrightarrow{q},\omega -\Omega )+G(\overrightarrow{p}+\overrightarrow{q},\omega +\Omega )}{{({\kappa }^2+q^2)}^4}}.\end{array}$$

(49)

Then, assuming: $\xi =\epsilon (\overrightarrow{p})-\mu =p^2/2m-\mu$, and changing the integration variables (compare [31]):

$$\begin{array}{c}\hfill \int d^3\overrightarrow{q}={\displaystyle \frac{2\pi m}{p}}{\int }_0^{\infty }qdq{\int }_{\xi -}^{{\xi }_{+}}d\xi ;{\xi }_{\pm }=\xi (p\pm q),\end{array}$$

(50)

one rewrites Equation (49) in the form:

$$\begin{array}{ccc}\hfill \overline{G}(\overrightarrow{p},\omega )=& & \sum _Q{\overline{n}}_Q{M}^2{\displaystyle \frac{{(8\pi \kappa )}^2}{{(2\pi )}^2}}{\displaystyle \frac{m}{p}}{\int }_0^{\infty }qdq{\int }_{\xi -}^{{\xi }_{+}}d\xi {\displaystyle \frac{1}{{({\kappa }^2+q^2)}^4}}\left\{{\displaystyle \frac{1}{i(\omega -\Omega )+\xi -{\overline{G}}_{-}}}+\right.\hfill \\ & & \left.{\displaystyle \frac{1}{i(\omega +\Omega )+\xi -{\overline{G}}_{+}}}\right\};{\overline{G}}_{\mp }=\overline{G}(\xi ,\omega \mp \Omega )\hfill \end{array}$$

(51)

Now, allowing for the relation justified aposteriori: ${\overline{G}}_{\mp }=\overline{G}$, Equation (51) reads:

$$\begin{array}{ccc}& & \overline{G}=\sum _Q{\overline{n}}_Q{M}^2{\displaystyle \frac{{(8\pi \kappa )}^2}{{(2\pi )}^2}}{\displaystyle \frac{m}{p}}{\int }_0^{\infty }{\displaystyle \frac{qdq}{{({\kappa }^2+q^2)}^4}}{\int }_{\xi -}^{{\xi }_{+}}d\xi {\displaystyle \frac{2(i\omega +\xi -\overline{G})}{{(i\omega +\xi -\overline{G})}^2+{\Omega }^2}}\hfill \end{array}$$

(52)

Next, analytic continuation of Equation (52) to the real axis of frequencies, $i\omega \to \omega$, gives:

$$\begin{array}{ccc}& & \overline{G}(\overrightarrow{p},\omega )=\sum _Q{\overline{n}}_Q{M}^2{\displaystyle \frac{{(8\pi \kappa )}^2}{{(2\pi )}^2}}{\displaystyle \frac{m}{p}}{\int }_0^{\infty }{\displaystyle \frac{qdq}{{({\kappa }^2+q^2)}^4}}ln{\displaystyle \frac{{(\omega +{\xi }_{+}-\overline{G})}^2+{\Omega }^2}{{(\omega +{\xi }_{-}-\overline{G})}^2+{\Omega }^2}}\hfill \end{array}$$

(53)

The Q-ball form factor $\propto {({\kappa }^2+q^2)}^{-4}$ reduces integration over q to the interval $0\le q\le \kappa$ and, therefore, allowing for the mesoscopic Q-ball sizes [17]: $R_Q^{-1}\sim \kappa \ll p$, it is fare to approximate the above relation expanding ${\xi }_{\pm }$ to the first order in $q\sim \kappa$:

$$\begin{array}{c}\hfill {\xi }_{\pm }=\xi (p\pm q)\approx \xi (p)\pm vq;v\equiv {\displaystyle \frac{\partial \xi (p)}{\partial p}}\end{array}$$

(54)

Hence, one finds from Equation (53) the following ’on shell’, $\omega +\xi (p)=0$, equation for $\overline{G}(\overrightarrow{p},\omega )$:

$$\begin{array}{c}\hfill \overline{G}(\overrightarrow{p},\omega )=\sum _Q{\overline{n}}_Q{M}^2{\displaystyle \frac{{(8\pi \kappa )}^2}{{(2\pi )}^{2}v}}{\int }_0^{\infty }{\displaystyle \frac{qdq}{{({\kappa }^2+q^2)}^4}}ln{\displaystyle \frac{{\Omega }^2-2vq\overline{G}}{{\Omega }^2+2vq\overline{G}}}\end{array}$$

(55)

Now, assuming $\overline{G}$ to be responsible for electrons damping rate and thus purely imaginary, one finally finds after integration in (55) an equation for $\overline{G}$:

$$\begin{array}{c}\hfill \overline{G}=-\sum _Q{\overline{n}}_Q{M}^2{\displaystyle \frac{4\pi }{2{\Omega }^2{\kappa }^3}}\left[\overline{G}+{\displaystyle \frac{4v^2{\kappa }^2{\overline{G}}^3}{3{\Omega }^4}}\right]\equiv -\left[I_1\overline{G}+I_2{\overline{G}}^3\right]\end{array}$$

(56)

Using now definition for ${\overline{n}}_Q$ from Equation (48) and relation $\kappa =1/R_Q$, where $R_Q$ is Q-ball radius, substituting summation over Q by integration, expressing $V_Q$ via M and Q using Equation (24), and allowing for the scaling of the Q-ball amplitude with temperature in Equations (30), (43): $M=s\Omega$, $s>1$, one finds:

$$\begin{array}{c}\hfill I_1=\sum _Q{\overline{n}}_Q{M}^2{\displaystyle \frac{4\pi }{2{\Omega }^2{\kappa }^3}}={\int }_0^{\infty }{\displaystyle \frac{6M^{2}P(Q)dQ}{{\Omega }^{2}4}}={\displaystyle \frac{3M^2}{2{\Omega }^2}}={\displaystyle \frac{3s^2}{2}};{\displaystyle \frac{1}{{\kappa }^3}}={\displaystyle \frac{3V_Q}{4\pi }}\end{array}$$

(57)

The coefficient $I_2$ in front of ${\overline{G}}^3$ in Equation(56) is more elaborate:

$$\begin{array}{c}\hfill I_2=\sum _Q{\overline{n}}_Q{M}^2{\displaystyle \frac{4\pi }{2{\Omega }^2{\kappa }^3}}{\displaystyle \frac{4v^2{\kappa }^2}{3{\Omega }^4}}=4{\int }_0^{\infty }{\displaystyle \frac{M^{2}P(Q)v^{2}dQ}{2{\Omega }^6{V}_Q^{\frac{2}{3}}}}{\left({\displaystyle \frac{4\pi }{3}}\right)}^{{\displaystyle \frac{2}{3}}};{\kappa }^2={\left({\displaystyle \frac{4\pi }{3V_Q}}\right)}^{{\displaystyle \frac{2}{3}}}\end{array}$$

(58)

Hence,

$$\begin{array}{c}\hfill I_2=4{\left({\displaystyle \frac{4\pi }{3}}\right)}^{{\displaystyle \frac{2}{3}}}{\displaystyle \frac{v^2{s}^{\frac{10}{3}}}{2{\Omega }^2}}{\int }_0^{\infty }{\displaystyle \frac{P(Q)dQ}{Q^{\frac{2}{3}}}}={\displaystyle \frac{\tilde{C}}{{\Omega }^2}};\tilde{C}\equiv 4{\displaystyle \frac{{(4\pi )}^{\frac{4}{3}}{v}^2{s}^{\frac{10}{3}}}{2{(3g)}^{\frac{2}{3}}}}{\int }_0^{\infty }{\displaystyle \frac{e^{-x}dx}{x^{\frac{2}{3}}}}\end{array}$$

(59)

Solving Equation (56) with the aid of relations (57) and (59) one finds the following relation for the fermionic quasiparticle lifetime due to Q-ball scattering (± sign below is chosen depending on retarded- or advanced Green’s function is considered), ${\tau }_Q$:

$$\begin{array}{c}\hfill \overline{G}=\pm {\displaystyle \frac{i}{{\tau }_Q}};{\displaystyle \frac{1}{{\tau }_Q}}=\sqrt{{\displaystyle \frac{1+I_1}{I_2}}}={\displaystyle \frac{\Omega }{\sqrt{\tilde{C}}}}\sqrt{1+3s^2/2}\propto T.\end{array}$$

(60)

The above result is remarkable, since it demonstrates that linear temperature dependence of the fermionic inverse lifetime arises due to Q-ball scattering in the whole temperature interval $T_1^{*}<T<T_0^{*}$, thus providing origin of the "strange metal" behaviour. The bosonic frequency $\Omega =2\pi T$ of the quantum thermodynamic Q-ball time crystal plays the role of a scattering rate $1/\tau \propto \Omega$ for the fermions in the Q-ball semiclassical field, manifesting the prominent ’Planckian’ scattering rate behaviour [36]. It follows also from Equation (45), that $D(\pm \overrightarrow{q})$ plays the role of $\pm \Omega$ Fourier components of the Q-ball field propagator modulo Q-ball density ${\overline{n}}_Q$. Simultaneously, the CDW/SDW wave vector $\overrightarrow{Q}$ entering propagator $D(\overrightarrow{q})$, causes anisotropy of the scattering rate, thus explaining ’quantum nematic’ behaviour known for high-$T_c$ cuprates [37]:

$$\begin{array}{c}\hfill {\sigma }_{i,j}\propto {\displaystyle \frac{Q_i{Q}_j{\tau }_Q}{{\overrightarrow{Q}}^2}},\end{array}$$

(61)

where ${\sigma }_{i,j}$ is electron conductivity tensor.

5. Electron Resistivity Due to Q-Balls Slide

The picture of ’free’ fermions scattered by a gas of randomly distributed in space Q-balls considered in the previous Section might be not complete in the case of Q-balls, that are not pinned to the lattice. Namely, one may consider contribution to the electrical resistivity coming from the slide of the Q-ball CDW as a whole in a weak electric field. To calculate this contribution one may use a method described in [30] by adding potential energy term of a Q-ball charge density in a homogeneous constant electric field: $\varphi =-e\overrightarrow{r}\overrightarrow{E}$, thus adding an extra term to the Euclidean action in Equation (9) and, correspondingly, modifying the saddle-point equation, that becomes then:

$$\begin{array}{ccc}& & {\displaystyle \frac{\delta S_M}{\delta M^{*}(\tau ,\mathbf{r})}}=-{\partial }_{\tau }^{2}M(\tau ,\mathbf{r})-s^2\sum _{\alpha =\mathbf{r}}{\partial }_{\alpha }^{2}M(\tau ,\mathbf{r})+{\mu }_0^{2}M(\tau ,\mathbf{r})+gM(\tau ,\mathbf{r}){\displaystyle \frac{\partial U_f}{{\partial |M(\tau ,\mathbf{r})|}^2}}\hfill \\ & & -2i\Omega ({\partial }_{\tau }+{\displaystyle \frac{ie\varphi }{\hslash }})M(\tau ,\mathbf{r})=0\hfill \end{array}$$

(62)

Solving this equation expressed via Fourier transformed function $M(\Omega ,\overrightarrow{p})$ to the first order in potential $\varphi$ Fourier component, one finds:

$$\begin{array}{c}\hfill M=M_0+M_1;M_1(\pm \Omega ,\overrightarrow{p})={\displaystyle \frac{2\Omega e\varphi M_0(\pm \Omega ,\overrightarrow{p})}{\hslash ({\mu }_0^2-{\Omega }^2)}}={\displaystyle \frac{2\Omega e\overrightarrow{E}}{\hslash ({\mu }_0^2-{\Omega }^2)}}{\displaystyle \frac{i\partial M_0(\pm \Omega ,\overrightarrow{p})}{\partial \overrightarrow{p}}}\end{array}$$

(63)

where $M_0$ reads:

$$\begin{array}{c}\hfill M_0(\pm \Omega ,\overrightarrow{p})={\displaystyle \frac{8\pi \kappa M}{{({\kappa }^2+{(\overrightarrow{p}\mp \overrightarrow{Q})}^2)}^2}}\end{array}$$

(64)

Then, to the first order in electric field $\overrightarrow{E}$ the Q-ball sliding CDW current density reads:

$$\begin{array}{ccc}& & \overrightarrow{j}=-{\displaystyle \frac{ie\hslash }{4m}}\sum _q(M^{*}\overrightarrow{\nabla }M-M\overrightarrow{\nabla }{M}^{*})={\displaystyle \frac{e^2}{2m}}{\displaystyle \frac{\Omega }{({\mu }_0^2-{\Omega }^2)}}\sum _p\overrightarrow{p}\overrightarrow{E}·{\displaystyle \frac{\partial }{\partial \overrightarrow{p}}}\left[M_0{(\Omega ,\overrightarrow{p})}^2+\right.\hfill \\ & & \left.M_0{(-\Omega ,\overrightarrow{p})}^2\right]\equiv \overrightarrow{E}{\sigma }_{CDW}\hfill \end{array}$$

(65)

and hence:

$$\begin{array}{c}\hfill {\sigma }_{CDW}\propto {\displaystyle \frac{e^2\Omega M^2}{m({\mu }_0^2-{\Omega }^2){\kappa }^3}}\end{array}$$

(66)

First, expression in Equation(66) is remarkably different from expression for the electrical conductivity due to scattering of the ’free electrons’ on the Q-balls. Namely, the pronounced nematicity of the conductivity tensor in Equation (61) is manifestly absent in Equation (66). This points to a hydrodynamic character of the Q-ball CDW slide in external electric field. Next, it is instructive to apply above result to the vicinity of T* temperature, since the power indices for the temperature dependencies of the Q-ball parameters where found earlier [1,2]. Taking into account that superconducting gap approaches zero at T* according to Equation (31), and following Ginzburg-Landau theory of the superconducting order parameter, see e.g., Chapt.17 [30], one finds that minimal radius $R_{min}$ of the Q-ball with superconducting condensate inside diverges as [2]:

$$\begin{array}{c}\hfill R_{min}\propto {\displaystyle \frac{1}{g_0}}\sim {\displaystyle \frac{1}{{(T^{*}-T)}^{\frac{1}{5}}}}.\end{array}$$

(67)

Then, using definition of T* in Equation (30), one rewrites expression Equation (66) in the asymptotic form:

$$\begin{array}{c}\hfill {\sigma }_{CDW}\propto {\displaystyle \frac{e^2\Omega M^2}{m({\mu }_0^2-{\Omega }^2){\kappa }^3}}\sim {\displaystyle \frac{{R_{min}}^3}{T^{*}-T}}\propto {\displaystyle \frac{1}{{(T^{*}-T)}^{8/5}}}\equiv {\displaystyle \frac{1}{{(T^{*}-T)}^{1.6}}}\end{array}$$

(68)

This critical behaviour significantly differs from Ginzburg-Landau theory prediction for the 3D case in the vicinity of superconducting transition temperature T_c [30]:

$$\begin{array}{c}\hfill {\sigma }_{GL}\propto {\displaystyle \frac{1}{{(T-T_c)}^{\gamma }}};\gamma =1/2\end{array}$$

(69)

and is most close to the 1D case, $\gamma =3/2$, [30].

6. Diamagnetic Response of Q-Ball Gas

It is straightforward to apply presented above picture of Q-ball gas in high-T_c superconductors for description of experimentally discovered diamagnetic behaviour above T_c in cuprates [8,38]. Again, as in Equation (48) using the concept of the phase space of the Q-balls formed by the values of the ’Noether charge’ Q and discrete values of the Matsubara frequencies ${\Omega }_n\equiv 2\pi nT$, $n=1,2,...$, and counting the number of the different ’positions’ of a Q-ball in the real space as $V/V_{Q,n}$, where V is the volume of the system and the Q-ball volume is determined using the ’charge’ Q conservation law Equation (24):

$$\begin{array}{c}\hfill V_{Q,n}\equiv {\displaystyle \frac{4\pi R^3}{3}}={\displaystyle \frac{Q}{{\Omega }_n{M}^2}},\end{array}$$

(70)

one finds the following expression for the partition function of the Q-balls gas in the temperature range where it exists, $T_1^{*}(\kappa )<T<{\mu }_0/(2\pi )$, see Figure 1:

$$\begin{array}{c}\hfill Z_Q=\sum _{Q,n}{\displaystyle \frac{1}{N!}}{\left[{\int }_{Q_m}^{Q_H}dQ{\displaystyle \frac{V}{V_{Q,n}}}exp\left\{-\left[{\displaystyle \frac{2Q{\Omega }_n}{gT}}-{\displaystyle \frac{M_{Q}H}{T}}\right]\right\}\right]}^N,\end{array}$$

(71)

The Q-ball energy in the first term of the Boltzmann’s expression in the brackets in Equation (71), $E_Q/T$, is taken from the self-consistency Equation (27). The lower and upper bounds in the integral over $dQ$ are as follows. The smallest value of $Q=Q_m$ is obtained from Equation (70) for the Q-ball of the size $R_m$ bound from below by the Landau correlation length $\xi$, see Equation (67):

$$\begin{array}{cc}& Q_m=\Omega M^2{\displaystyle \frac{4\pi R_m^3}{3}},R_m=\xi \equiv \pi \sqrt{{\displaystyle \frac{{\hslash }^2}{4mb{g}_0^2}}}.\end{array}$$

(72)

with $g_0$ defined by Equation (31). The upper bound $Q_H$ in the integral in Equation (71) is obtained as follows:

$$\begin{array}{cc}& Q_H=\Omega M^2{\displaystyle \frac{4\pi R_H^3}{3}},R_H={\displaystyle \frac{{\delta }_L{H}_c\sqrt{20}}{H}},{\delta }_L={\displaystyle \frac{\sqrt{m{c}^2}}{\sqrt{4\pi n_s{e}^2}}},\end{array}$$

(73)

where $R_H\ll {\delta }_L$ is the maximum radius of a small superconducting sphere [39], at which it remains superconducting in magnetic field H, and ${\delta }_L$ is London penetration depth, $H_c$ is critical magnetic field of the bulk superconductor material, $n_s\approx 2\pi \nu T_c/3$ is superconducting electrons density, as derived in [1] in accord with Uemura plot behaviour [40], with $2\pi T_c\equiv {\Omega }_c$ given in Equation (37), $\nu$ is the bare fermionic density at the Fermi level, m is electron mass, and c is light velocity. The next term, $-M_{Q}H/T$, in the Boltzmann’s expression in the brackets in Equation (71) is the energy of diamagnetic moment $M_Q$ in magnetic field H:

$$\begin{array}{cc}& M_Q=-{\displaystyle \frac{R^{5}H}{30{\delta }_L^2}}H=-{\left({\displaystyle \frac{3Q}{4\pi M^2\Omega }}\right)}^{{\displaystyle \frac{5}{3}}}{\displaystyle \frac{H^2}{30{\delta }_L^2}},\end{array}$$

(74)

where $M_Q$ is projection of diamagnetic moment of a Q-ball on the magnetic field direction $\overrightarrow{H}$. The Q-ball is regarded as a small superconducting sphere of radius $R\ll {\delta }_L$ possessing diamagnetic moment in magnetic field H [39]. In the last equality in Equation (74) R is substituted via the expression $R=R(Q)$ obtained from the Q-ball ’charge’ Q conservation relation Equations (24), (70). Composing altogether the above relations one finds the following expression for the free energy of the Q-ball gas:

$$\begin{array}{cc}& {\displaystyle F=-Tln{Z}_Q,Z_Q=\sum _{n,N}{\displaystyle \frac{G_n^N}{N!}}\equiv exp{G}_n,}\end{array}$$

(75)

$$\begin{array}{cc}& {\displaystyle G_n={\int }_{Q_m}^{Q_H}dQ{\displaystyle \frac{V{\Omega }_n{M}^2}{Q}}exp\left\{-\left[{\displaystyle \frac{2Q{\Omega }_n}{gT}}+{\left({\displaystyle \frac{3Q}{4\pi M^2{\Omega }_n}}\right)}^{\frac{5}{3}}{\displaystyle \frac{H^2}{30{\delta }_L^{2}T}}\right]\right\},}\end{array}$$

(76)

$$\begin{array}{cc}& Q_H={\displaystyle \frac{{\delta }_L^3{H}_c^3}{H^3}}{\displaystyle \frac{4\pi {\Omega }_n{M}^2{20}^{\frac{3}{2}}}{3}}\end{array}$$

(77)

In the highest temperature interval $T_1^{*}(\kappa )<T<{\mu }_0/(2\pi n)$ one takes integer $n=1$, see Equation (41) and Figure 1, and then for the free energy of the Q-balls gas and density of its diamagnetic moment $<M_Q>/V$ one finds:

$$\begin{array}{cc}& F=-T{G}_{n=1}\equiv -TG,<M_Q/V>=T{\displaystyle \frac{\partial G}{V\partial H}}\equiv -M_1-M_2,\end{array}$$

(78)

$$\begin{array}{cc}& {\displaystyle M_1={\displaystyle \frac{2H{3}^{5/3}}{30{\delta }_L^2{(4\pi )}^{5/3}{(M^2\Omega )}^{2/3}}}{\int }_{Q_m}^{Q_H}dQ{Q}^{2/3}exp\left\{-\left[{\displaystyle \frac{2Q\Omega }{gT}}+{\left({\displaystyle \frac{3Q}{4\pi M^2\Omega }}\right)}^{\frac{5}{3}}{\displaystyle \frac{H^2}{30{\delta }_L^{2}T}}\right]\right\},}\end{array}$$

(79)

$$\begin{array}{cc}& M_2={\displaystyle \frac{3\Omega M^2}{H}}exp\left\{-\left[{\displaystyle \frac{2Q_H\Omega }{gT}}+{\left({\displaystyle \frac{3Q_H}{4\pi M^2\Omega }}\right)}^{{\displaystyle \frac{5}{3}}}{\displaystyle \frac{H^2}{30{\delta }_L^{2}T}}\right]\right\},\end{array}$$

(80)

where one has to substitute solution $M=M(\Omega )$ of the self-consistency Equation (27) using e.g., solutions from Equation (30), or in the form of the two-branches solution Equation (39). This leads to the following dependence found numerically from Equations (79), (80) above, see Figure 4.

7. "Hourglass" Magnetic Spectrum and Splitting of the In-Plane Phonon Brunches into the Softened and Hardened Ones Due to Q-Balls Scattering

The scattering of quantum spin excitations and phonons on the condensates of Cooper/local pairs inside the Q-balls induces self-energy, see Figure 5, that consists of closed fermionic loop of Gor’kov’s anomalous Green’s functions F, $F^{+}$, the latter are found in Equation (21) above. The difference of the two cases of the Q-balls formed by SDW and CDW fluctuations causes different patterns of the magnons and phonons dispersions in the vicinities of the $Q_{SDW}$ and $Q_{CDW}$ wave vectors in the brillouin zone. We start from the $Q_{SDW}$ case.

7.1. Hourglass Magnetic Spectrum Due to SDW Q-Balls Scattering

The Dyson equation for the Green’s function $D(\omega ,\overrightarrow{q})$ of magnons/phonons, that will be defined below by a particular choice of the vicinity of the wave-vector $\overrightarrow{q}$ and corresponding bare dispersion ${\omega }_0(\overrightarrow{q})$, takes the form [31]:

$$\begin{array}{cc}& D(\omega ,\overrightarrow{q})=D_0(\omega ,\overrightarrow{q})\left(1-D(\omega ,\overrightarrow{q}){\Pi }_F(\omega ,\overrightarrow{q})\right),\end{array}$$

(81)

$$\begin{array}{cc}& D_0(\omega ,\overrightarrow{q})=-{\displaystyle \frac{{{\omega }_0}^2(\overrightarrow{q})}{{\omega }^2+{{\omega }_0}^2(\overrightarrow{q})}},\omega =2\pi n;\end{array}$$

(82)

$$\begin{array}{cc}& {\Pi }_F(\omega ,\overrightarrow{q})=g_{bf}^{2}T\sum _{\overrightarrow{k},{\omega }_1}{\overline{F}}_{\sigma ,\overline{\sigma }}({\omega }_1,\overrightarrow{k})F_{\overline{\sigma },\sigma }({\omega }_1+\omega ,\overrightarrow{k}+\overrightarrow{q}),\end{array}$$

(83)

where Matsubara bosonic frequency $\omega$ is defined by any integer n and coupling constant $g_{bf}$ describes coupling of spin/phonon excitations to the fermions. Here one important observation is in order.

In the case of the coupling with magnons and the Green’s function with momenta close to the anti-ferromagnetic wave-vector ${\overrightarrow{Q}}_{SDW}$ corresponding to SDW fluctuations, one considers $D_{\sigma ,\overline{\sigma }}(\omega ,\overrightarrow{q})$ component of the magnons Green’s function in the Dyson’s equation (81) above and correspondently the polarisation loop in Equation (83) contains then pre-factor $\sigma \overline{\sigma }=-1$, which is compensated by discussed previously, see end of Section 3 [2], d-wave symmetry of the superconducting order parameter known for cuprates [3]: $F_{\sigma ,\overline{\sigma }}(\omega ,\mathbf{k})=-F_{\sigma ,\overline{\sigma }}(\omega ,\overrightarrow{k}-{\overrightarrow{Q}}_{SDW})$. On the other hand, when coupling with phonons is considered close to the ’nesting’ wave-vector ${\overrightarrow{Q}}_{CDW}$ in the Brillouin zone corresponding to the CDW fluctuations, then the pre-factor $\sigma \overline{\sigma }$ is missing in the polarisation loop in Equation (83), and corresponding superconducting order parameter is of s-wave symmetry [20]: $F_{\sigma ,\overline{\sigma }}(\omega ,\overrightarrow{k})=F_{\sigma ,\overline{\sigma }}(\omega ,\overrightarrow{k}-{\overrightarrow{Q}}_{CDW})$, see Figure 6. Hence, the expressions for the polarisation loop and Green’s function of magnons with the wave vectors close to the antiferromagnetic wave-vector $Q_{AF}=\left\{\pi ,\pi \right\}$ in the inverse crystal lattice units take the following form obtained with the use of Equation (21):

$$\begin{array}{c}\hfill D(\omega ,\overrightarrow{q})=-{\displaystyle \frac{{\omega }_0^2(\overrightarrow{q})}{{\omega }^2+{\omega }_0^2(\overrightarrow{q})\left(1-{\Pi }_F(\omega ,\overrightarrow{q})\right)}},\omega =2\pi n;\end{array}$$

(84)

$$\begin{array}{cc}& {\Pi }_F(\omega ,\overrightarrow{q})=g_{bf}^2{\displaystyle \frac{tanh\left({\displaystyle \frac{g_0}{2T}}\right)}{g_0(4g_0^2+{\omega }^2)}}\int {\displaystyle \frac{d^3\overrightarrow{k}}{{(2\pi \hslash )}^3}}\sqrt{{g_0}^2-{\epsilon }^2(\overrightarrow{k})}\sqrt{{g_0}^2-{\epsilon }^2(\overrightarrow{k}+\overrightarrow{q})}\equiv \\ & \equiv g_{bf}^2{\displaystyle \frac{tanh\left({\displaystyle \frac{g_0}{2T}}\right)}{g_0(4g_0^2+{\omega }^2)}}·I_{\overrightarrow{q}}\end{array}$$

(85)

Here we first take the bare fermionic dispersion $\epsilon (\overrightarrow{k})$ along the ’nesting’ vector ${\overrightarrow{Q}}_{SDW}$ close to the bare Fermi surface at the extended Van Hove singularity for a one-dimensional dispersion function: $\epsilon (\overrightarrow{k})=v_f(k-p_F)$ and ’nesting’ condition along the ${\overrightarrow{Q}}_{SDW}$ would be then: $\epsilon (\overrightarrow{k}-{\overrightarrow{Q}}_{SDW})=-v_f(k-2p_F+p_F)\equiv -v_f(k-p_F)$ with the corresponding $2p_F$ : $Q_{SDW}=2p_F$, see Figure 6. Hence, calculation of the polarisation loop ${\Pi }_F(\omega ,\overrightarrow{q})$ for $\overrightarrow{q}$ along the nesting vector direction ${\overrightarrow{Q}}_{SDW}$ in the Van Hove singularity region of energies $\epsilon (\overrightarrow{k})$, i.e., for $q=Q_{SDW}+\tilde{q}$ with $\tilde{q}\ll Q_{SDW}$ could be done along the energy axis with corresponding density of states near the Van Hove singularity $\nu$. Then, the integral $I_{\overrightarrow{q}}$ in the Equation (85) takes the form:

$$\begin{array}{c}\hfill I_{\overrightarrow{q}}=\nu {\int }_{-g_0+v_f\tilde{q}}^{g_0}d\epsilon \sqrt{{g_0}^2-{\epsilon }^2}\sqrt{{g_0}^2-{(\epsilon -v_f\tilde{q})}^2}\approx \nu \left[{\displaystyle \frac{1}{3}}{(v_f\tilde{q})}^3-g_0{(v_f\tilde{q})}^2+{\displaystyle \frac{4}{3}}{g}_0^3\right]\end{array}$$

(86)

Next, making analytic continuation to the axis of real frequencies: $i\omega \to \omega +i\delta ,\delta \to +0$ in Equations (84), (85) one finds the following expression for the retarded magnons Green’s function $D^R(\omega ,\overrightarrow{q})$, where a new notation $\omega +i\delta \equiv \tilde{\omega }$ is used for brevity:

$$\begin{array}{cc}& D^R(\omega ,\overrightarrow{q})={\displaystyle \frac{{\omega }_0^2(\overrightarrow{q})(4g_0^2-{\tilde{\omega }}^2)}{-{\tilde{\omega }}^4+{\tilde{\omega }}^2\left(4g_0^2+{\omega }_0^2(\overrightarrow{q})\right)+{\omega }_0^2(\overrightarrow{q})\left(E(\overrightarrow{q})-4g_0^2\right)}},\tilde{\omega }\equiv \omega +i\delta ;\\ & E(\overrightarrow{q})\equiv {\displaystyle \frac{g_{bf}^{2}tanh\left({\displaystyle \frac{g_0}{2T}}\right)I_{\overrightarrow{q}}}{g_0}}.\end{array}$$

(87)

The above result is remarkable: the poles of the $D^R(\omega ,\overrightarrow{q})$ expression in Equation (87) provide two dispersion branches of magnons near the ${\overrightarrow{Q}}_{SDW}$ in the Brillouin zone:

$$\begin{array}{c}\hfill {\omega }_{\pm }(\overrightarrow{q})=\sqrt{{\displaystyle \frac{{\omega }_0^2(\overrightarrow{q})+4g_0^2}{2}}\pm \sqrt{{\left({\displaystyle \frac{{\omega }_0^2(\overrightarrow{q})-4g_0^2}{2}}\right)}^2+{\omega }_0^2(\overrightarrow{q})E(\overrightarrow{q})}}\end{array}$$

(88)

When wave-vector of the magnons $\overrightarrow{q}$ is taken along the antiferromagnetic SDW nesting vector direction ${\overrightarrow{Q}}_{SDW}$ in the vicinity of the bottom of the antiferromagnetic excitations band, i.e., for $q=Q_{SDW}+\tilde{q}$ with $\tilde{q}\ll Q_{SDW}$, one may approximate the bare magnon dispersion in the form:

$$\begin{array}{c}\hfill {\omega }_0^2(\overrightarrow{q})=s^2{\tilde{q}}^2+{\mu }_0^2\end{array}$$

(89)

introduced already in Equation (9), where s is spin-wave velocity and ${\mu }_0$ is the ’magnon mass’ characterising the short range character of antiferromagnetic fluctuations in the doped cuprates. Substituting the above expression from Equation (89) into expression for the magnon dispersion branches in Equation (88) one finds dispersion plotted in Figure 7, that follows famous experimental data for high-T_c cuprates reported long ago [9].

7.2. Splitting of the In-Plane Phonon Dispersion Brunches Due to CDW Q-Balls Scattering

Now, we turn to the splitting of the in-plane phonon dispersion brunches into the softened and hardened ones due to scattering of phonons on the CDW Q-balls close to the $Q_{CDW}$ wave vectors designated in Figure 6. Obtained theoretical curves of the phonon brunch splitting that follow from Equation (88) are presented in Figure 8. Substituting the bare phonons dispersion:

$$\begin{array}{c}\hfill {\omega }_0(\overrightarrow{q})=sq\end{array}$$

(90)

close to the ’nesting’ wave-vectors $Q_{CDW}$ into the general expression in Equation (88) one obtains splitting of the in-plane phonon dispersion brunches in the vicinity of these wave-vectors that was studied experimentally [10], compare Figure 8a and Figure4 in [10]. The pole strength of the phonons Green’s function in Equation (87) corresponding to the "hardened" dispersion brunch ${\omega }_{+}(q)\approx 2g_0$ at $q\approx Q_{CDW}$ in Figure 8a is small and corresponds to the excitation of the phonon by the virtual collapse of two fermions into the Cooper /local pairs condensate inside the Q-ball with transferring the energy twice the superconducting gap $2g_0$ to the lattice distortion, see ${\nu }_{+}$ line in the insert Figure 8b. Simultaneously, the "softened" dispersion brunch:

$$\begin{array}{c}\hfill {\omega }_{-}(q)\approx {\omega }_0(q)\sqrt{1-{\displaystyle \frac{2E(q)}{4g_0^2-{\omega }_0^2(q)}}}\end{array}$$

(91)

possesses maximum softening at $q\approx Q_{CDW}$, see Figure 8a, due to the $\sim E(q)$ term defined in Equations (86), (87). The higher pole strength of this ${\omega }_{-}(q)$ brunch is plotted in the insert Figure 8b, see ${\nu }_{-}$ line. It is remarkable, that rather different dispersion curves in the figures Figure 7 and Figure 8 are obtained from one and the same Dyson equation (88), but with the different bare dispersions ${\omega }_0(\overrightarrow{q})$ in Equations (89) and (90), and with the different Gor’kov’s functions F, $F^{†}$ that describe d-wave and s-wave Cooper-pairs condensates inside the SDW and CDW Q-balls respectfully.

7.3. Dynamical Structure Factor $S(\overrightarrow{Q},\omega )$ of the Q-Balls ’Gas’

Since inelastic pulsed neutron scattering technique provides experimental data for the dynamical structure factor $S(\overrightarrow{Q},\omega )$, where $\overrightarrow{Q}$ is the momentum transfer and $\omega$ is the energy transfer [10], it is possible to compare experiment with the present theoretical results. Namely, $S(\overrightarrow{Q},\omega )$ is imaginary part of the density-density correlation function, that could be expressed via phonon Green’s function found above in Equation (87) to the lowest order of the perturbation theory with respect to electron-phonon coupling in the following form:

$$\begin{array}{cc}& S(\overrightarrow{Q},\Omega )=-{\displaystyle \frac{2}{\pi }}\Im {\Pi }^R(\overrightarrow{Q},\Omega ),\end{array}$$

(92)

$$\begin{array}{cc}& \Pi (\overrightarrow{Q},\Omega )={\langle n(\overrightarrow{r},\tau )n({\overrightarrow{r}}^{\prime },{\tau }^{\prime })\rangle }_{\overrightarrow{Q},\Omega }=T\int {\displaystyle \frac{d\overrightarrow{q}}{{(2\pi \hslash )}^3}}\sum _{\omega }D(\omega +\Omega ,\overrightarrow{q}+\overrightarrow{Q})D(\omega ,\overrightarrow{q}).\end{array}$$

(93)

Substituting phonon Green’s functions from Equation (87) and turning to retarded expression in Equation (93) after analytical continuation to real frequency axis, one finds the dynamical structure factor $S(\overrightarrow{Q_{CDW}},\omega )\equiv I(\omega )$ at different temperatures indicated with two arrow bars $T_{c1,2}$ in the Q-balls phase diagram in Figure 1. Numerical results obtained from Equation (92) with superconducting gap $g_0$ from Equation (31) are presented in Figure 9 and Figure 10.

8. Discussion

To summarise, presented above theory of the quantum orders in the form of periodic in the imaginary Matsubara time SDW/CDW fields signifies a new development beyond the Landau static mean-field order parameter approach. Favourable comparison of the theory with experiment [7,8,9,10,11] indicates that described above picture of free fermions outside the gas of Q-balls with Cooper pairs condensates inside of them below temperature T* opens an avenue for direct investigation of the thermodynamic quantum time crystals [21,22] of SDW/CDW dynamic quantum condensates forming space heterogeneous fluctuations [20]. This realm of new phenomena possesses direct relation to observed physical properties of high-Tc superconductors. In a particular picture related with high-T_c Q-balls scenario, the vanishing density of superconducting condensate at T* leads to inflation of Q-balls sizes, that self-consistently suppresses X-ray Bragg’s peak intensity close to Q-ball phase transition temperature [18]. Linear temperature dependence of electrical resistivity in the Q-balls phase due to scattering of electrons on the SDW/CDW condensates forming the Q-balls is also demonstrated. The T-linear dependence of electrical resistivity arises due to inverse temperature dependence of the Q-ball radius and linear dependences of SDW/CDW Q-ball amplitudes as functions of temperature in the "strange metal" phase. Simultaneously, the Cooper-pairs condensates inside the Q-balls give rise to diamagnetic response in the "strange metal" phase in accord with experiments [8,38]. It is also demonstrated above that scattering of antiferromagnetic spin-waves (magnons) on the superconducting condensates of the Q-balls creates famous "hourglass" dispersion of magnetic excitations close to the antiferromagnetic wave-vectors in the Brillouin zone in accord with experiments [9]. Splitting of the in-plane phonon brunches into the softened and hardened ones close to Q_CDW wave vectors in the Brillouin zone, as found in the inelastic pulsed neutron scattering experiments [10], is explained above by the scattering of phonons on the superconducting Q-ball condensates and is manifested by the calculated above phonons two-brunch Green’s function $D(\omega ,\overrightarrow{q})$ and dynamical structure factor $S(\overrightarrow{q},\omega )$, where $\overrightarrow{q}$ and $\omega$ is momentum and energy transfer respectively.