Euclidean Q-Balls of Fluctuating SDW/CDW in the ‘Nested’ Hubbard Model of High-T_c Superconductors as the Origin of Pseudogap and Superconducting Behaviors

Round 1

Reviewer 1 Report

As I understand the work, I may not have understood it, the author claims that below Tcr there is a bulk superconductivity, and above this temperature the superconductivity is localized in a Q-balls.

The explanation is simple. Magnetic fluctuations exist in balls only.

It is difficult to accept this, because in the pseudo-gap phase the conductivity is very strange, while we know nothing about conductivity of the sea of balls.

  - The author has determined the volume of the ball considering only the fluctuations of spin. We expect that using Eliashberg's  theory more correct calculation is to treat the spin fluctuations  and superconductivity on equal footing   

Author Response

Response to Reviewer 1:

 

I am grateful to Reviewer 1 for his report and useful comments.

 

1. The answer to the completely reasonable request of the Reviewer about using “Eliashberg's  theory …to treat the spin fluctuations  and superconductivity on equal footing “ is actually contained in the second part of the manuscript, i.e. in Section 4, which is entitled “Eliashberg Equations and Bound States along the Axis of Matsubara Time”. On the basis of self-consistent solution Eqs. (35), (36) of the Eliashberg Equation (27), (34) in Section 4, the pairing- induced part of the spin fluctuations energy U_f  (M) is calculated   in Eqs. (38), (39) and Figure 2, and substituted back into the effective Euclidean action Eq. (1) of the Section 1 for  the spin fluctuations. As a result, in Section 4 a self-consistency equation (40) is obtained and solved for the minimum of the complete effective Euclidean action Eq. (1) of the spin-fluctuations and superconducting pairing fluctuations inside the Q-ball (included via term U_f (M), thus, “on equal footing ”). The solutions of self-consistency Eq. (40) are demonstrated in Fig. 3 in Section 4. The explanation of the procedure presented in Section 4 is given in preceding Section 3 entitled “Free Energy of the Cooper-Pairing Fluctuations inside the Q-Balls”. In this Section 3 the Gor’kov anomalous Green function F of the paired fermions inside the Q-ball is defined in the equations (20), (21). Besides, in Section 3 the ansatz is introduced in Eqs. (22) ,(23) for the anomalous self-energy part \Sigma_2 of the fermionic Gor’kov F, and for the pseudo-gap g_0  inside the Q-balls. The ansatz for the bosonic spin-fluctuation ‘pairing glue’ propagator D(\Omega) as a function of the spin-fluctuation amplitude M is introduced in Section 4 in Eq. (28) after the Eliashberg equation (27). Therefore, procedure described in the manuscript and obtained there results do take spin-fluctuations and superconducting pairing on equal footing.

 

Hence, an argument of the Reviewer 1 “The author has determined the volume of the ball considering only the fluctuations of spin” is declined. Nevertheless, this argument is symptomatic of the difficulty in perception of the narrative of  accomplished derivations presented in the manuscript. Therefore, the argument is very useful, and encourages the author to make an outline of the whole idea of the undertaken derivation already in the Section “Introduction” in the revised version of the manuscript in the form:

 

“We propose here a theory of  Euclidean Q-ball phase of high-T$_c$ superconductors in the 'nested' Hubbard model, that may explain both the high-T$_c$ superconductivity, as well as  the 'pseudo gap' phase, that precedes it.  Namely, it is demonstrated analytically that Euclidean action of the strongly correlated electron system may possess stable saddle-point configurations in the form of finite size bubbles (Q-balls) with superconducting density fluctuations coupled to oscillating in Matsubara time fluctuations of spin or charge. This result is obtained via self-consistent solution of the Eliashberg equations in combination with condition of vanishing first variational derivative of the effective Euclidean action with respect to an amplitude of the spin-/charge fluctuations, see sections IV, V.”

 

2. The request of the Reviewer 1 : “…in the pseudo-gap phase the conductivity is very strange, while we know nothing about conductivity of the sea of balls”, is reflected in the text before Fig. 4 in Section 4, where it is added that:

 

 “The normal conductivity itself in the not superconducting  (pseudogap) state above T* has to be considered using a percolation approach for the electron current  path, that contains 'short-circuits' formed by finite size clusters of  Q-balls possessing Josephson links between them, as well as  resistive parts in the regions outside the Q-balls. This picture will be considered elsewhere and compared with the known properties of the 'strange metal' phase.”

Author Response File: Author Response.docx

Reviewer 2 Report

The author of the manuscript proposes a mechanism of pseudogap phase and high temperature superconductivity in High-Tc cuprates by exchange with coherent semiclassical fluctuations inside finite volume Euclidian Q-balls. Before I can decide whether to recommend the paper for publication I would like the author to answers to the questions and comments provided below:

 

1. Can the Cooper pair size which are created according to the proposed pairing mechanism be estimated and compared with the experimentally known values for the cuprates.

2. In Eq. (1) the charge-/spin-fermion coupling constant g is used. However, than in the text above Eq. (19) the author states that he introduces a dimentionless amplitude \alpha as the coupling strength in the spin-fermion interaction. Why such change of notation? Is \alpha something else than g? Can the value of \alpha and/or g used in the calculations be compared with those estimated in experiments?

3. What form of the Q_DW vector has been taken in the calculations and how it was determined? The Fermi surface topology may be important in this context? Is the effect of the Fermi surface topology taken into account in the analysis how the shape of the Fermi surface for the cuprates has been taken into account?

4. The author states that he uses the d-wave symmetric behavior of the superconducting order parameter. Can the explicit form of the momentum dependence of the superconducting gap be provided which is used in the analysis.

5. Is the pseudogap obtained by the author opened only in the proximity of the antinodal directions as it is seen in experiments for the cuprates? Namely can the Fermi arcs creation be reconstructed within the presented theory?

6. An important feature of the high-Tc cuprates is the appearance of the dome-like behavior of Tc as a function of doping and the increase of T* with decreasing doping. Can this be understood by using the pairing mechanism proposed in the manuscript. Can the SC gap and pseudogap as a function of doping be determined here?

7. Another important feature of the cuprates is the appearance of strong Coulomb repulsion. The author writes that he uses the Hubbard model. Is the so-called Hubbard U term taken into acount explicitly? If yes what is the value of the onsite Coulomb repulsion taken in the calculations?

 

Author Response

Response to Reviewer 2:

 

I am grateful to Reviewer 2 for his report and very useful questions and comments, that encouraged the author to add related extra information in the revised manuscript.  Below are the answers in the order of the questions in the review Report.

 

1. Can the Cooper pair size which are created according to the proposed pairing mechanism be estimated and compared with the experimentally known values for the cuprates.

 

This important question is considered in the text added subsection 6.1. Only rough estimate of the Cooper pair size is possible using simplified picture for the gap momentum dependence considered in the present paper. 

 

 

2. In Eq. (1) the charge-/spin-fermion coupling constant g is used. However, than in the text above Eq. (19) the author states that he introduces a dimentionless amplitude \alpha as the coupling strength in the spin-fermion interaction. Why such change of notation? Is \alpha something else than g? Can the value of \alpha and/or g used in the calculations be compared with those estimated in experiments?

 

 

The reason is as follows. There is a standard method of calculation of the diagrammatic expansion of the free energy of a many-body system described in the book Abrikosov, A.A.; Gor'kov, L.P.; Dzyaloshinski, I.E. Methods of Quantum Field Theory in Statistical Physics. Dover Publications: New York, NY, USA, 1963. The essence of the method is to calculate the first derivative of the free energy over artificially introduced dimensionless parameter \alpha, that turns initial coupling constant g into a ‘variable’ g\alpha ranging from 0 to g. After that, the summation of the loop diagrams in the first derivative of the free energy over \alpha avoids an extra prefactor 1/n in the diagrammatic expansion over the number n of the occurrences of the fermion coupling g in the loop diagrams.  Due to the letter fact, the summation to all orders in g could be casted into an averaging with a single exponential function, as is indicated by brackets < > in Eq. (19). After that, a formal integration over the variable \alpha over an interval [0,1] restores the final expression for the free energy itself. This method was used in the manuscript to find expression for the free energy of the Cooper pairing fluctuations presented in Eqs. (38), (39).

 

 

3. What form of the Q_DW vector has been taken in the calculations and how it was determined? The Fermi surface topology may be important in this context? Is the effect of the Fermi surface topology taken into account in the analysis how the shape of the Fermi surface for the cuprates has been taken into account?

 

A particular form of spin fluctuation described by Eq. (16) allows for collective spin degrees of freedom with 'nesting' SDW wave-vector Q_{DW} connecting e.g. 'anti-nodal' points of the bare Fermi-surface in the Brillouin zone. Then, as expressed in Eq. (23) and Fig.1, the fermionic spectral gap g_0 arises inside Euclidean Q-balls in the fermionic energy dispersion E_p in the nested ‘antinodal’ regions of the bare Fermi surface in the vicinity  of the fermi-level. The width of the nested interval along the energy axis is ~2\varepsilon_0, see Eq. (25). The 2D momentum dependence of the superconducting gap was not considered in this simplified picture.

 

 

The model is simplified in as much as it makes it possible to solve it analytically to demonstrate just a matter of principle. The antiferromagnetic fluctuations are considered for definiteness in the part of Euclidean action S_f  (spin-fermion interaction) Eq. (15), using ,for the spin-fermion coupling the standard Hamiltonian, see e.g.  Abanov, A.; Chubukov, A.V.; Schmalian, J., Quantum-critical theory of the spin-fermion model and its application to cuprates: Normal state analysis,  Adv. Phys. {\bf 52}, 119 - 218 (2003). The charge fluctuations could be considered as well, see Eq. (26) and text after it. The nesting wave-vector Q_{DW }connects the opposite anti-nodal points.

 

 

 

 

4. The author states that he uses the d-wave symmetric behavior of the superconducting order parameter. Can the explicit form of the momentum dependence of the superconducting gap be provided which is used in the analysis.

 

An explicit form of the momentum dependence of the superconducting gap is provided in Eq. (20) in the form of relation between values of anomalous self-energy function \Sigma_2 at the opposite antinodal points connected with the wave vector Q_{DW}. This relation depends on which type of fluctuations, i.e. of the spin or of the charge is considered. It is of ‘d-wave type,’ see Eq. (20), in the case of spin-fermion interaction  Eq. (15), and of  ‘s-wave type’ in the case of charge-fermion interaction, see Eq.  (26) and text after. The explicit 2D momentum dependence of the superconducting gap was not considered in this simplified picture.

 

5. Is the pseudogap obtained by the author opened only in the proximity of the antinodal directions as it is seen in experiments for the cuprates? Namely can the Fermi arcs creation be reconstructed within the presented theory?

 

The system of equations considered in the manuscript: the Eliashberg equation (27)-(29) and equation for the minimum of Q-ball volume in the Euclidean action (12), (40) stems from the ‘nesting’ assumption performed via fluctuation field M with wave-vector Q_{DW} that connects just vicinities of the antinodal points in momentum space, in the energy window of width ~2\varepsilon_0 around the Fermi-level.   Hence, the superconducting (pseudo)gap in the fermionic spectrum of the fermions populating the Q-balls arises under such scheme only in the antinodal directions in the vicinities of the antinodal points. A problem of creation of the Fermi arcs is, in principle, treatable in the presented Q-ball gas picture, but demands elaboration of the self consistency equation for momenta $p$ also outside the antinodal points. This remark is added in the text after Eq. (23) in Section 3.

 

6. An important feature of the high-Tc cuprates is the appearance of the dome-like behavior of Tc as a function of doping and the increase of T* with decreasing doping. Can this be understood by using the pairing mechanism proposed in the manuscript. Can the SC gap and pseudogap as a function of doping be determined here?

 

 

An extra text is added at the and of Section 4. :

 

The phase diagram in Fig. \ref{T} obtained in the 2D plane {temperature, coupling constant} is actually in qualitative correspondence with the right half of the diagram for the stripe-phase bubbles formation experimentally found in the 2D plane {temperature, micro-strain $\epsilon$ in the CuO2 plane of all high-T$_c$ cuprates }, see A. Bianconi, N. L. Saini, S. Agrestini, D. Di Castro "The strain quantum critical point for superstripes in the phase diagram of all cuprate perovskites", International Journal of Modern Physics B, {\bf {14}}, Nos. 29-31,  3342-3355 (2000). The other half is assumed to be possible to find by considering the percolative behaviour for the Cooper-pairs of the Q-balls gas. On the other hand, to calculate phase diagram in the plane $\{$temperature, doping concentration$\}$  one has to solve an extra microscopic problem to find a relation between the effective coupling constant $g$ and the doping concentration of holes. This is an interesting problem to be solved in the future.

 

 

7. Another important feature of the cuprates is the appearance of strong Coulomb repulsion. The author writes that he uses the Hubbard model. Is the so-called Hubbard U term taken into acount explicitly? If yes what is the value of the onsite Coulomb repulsion taken in the calculations?

 

 

A corresponding explanation is added in the text after Eq. (15):

 

Actually, the model (1), (15) could be obtained e.g. from a bare on-site repulsive-U Hubbard Hamiltonian after decoupling of the inter- fermion interaction via auxiliary Hubbard-Stratonovich  field M(r,\tau) , see e.g.  H. J. Schulz, Effective Action for Strongly Correlated Fermions from Functional Integrals, Phys. Rev. Lett. {\bf 65}, 2462-2465 (1990); Mukhin, S.I., Negative Energy Antiferromagnetic Instantons Forming Cooper-Pairing Glue and Hidden Order in High-Tc Cuprates,  Condens. Matter {\bf 3(4)}, 39 (2018). Then, before renormalizations, the constant g is actually: g~U.

 

 

 

Round 2

Reviewer 1 Report

The motivation letter of the authors is detailed and comments on all my questions. The corrections made answer all my remarks.

The paper is suitable for publication. 

   

Author Response

The author is grateful to the Reviewer 1 for positive response on the revised manuscript.

Reviewer 2 Report

I am satisfied by most of the answers to my report provided by the author. However, I still have some comments regarding two of the points raised in my first report. Below I provide those two points and the author’s answer in blue. My new comments are provided in the bylines in red.

1. The author states that he uses the d-wave symmetric behavior of the superconducting order parameter. Can the explicit form of the momentum dependence of the superconducting gap be provided which is used in the analysis.

Author’s answer:

An explicit form of the momentum dependence of the superconducting gap is provided in Eq. (20) in the form of relation between values of anomalous self-energy function \Sigma_2 at the opposite antinodal points connected with the wave vector Q_{DW}. This relation depends on which type of fluctuations, i.e. of the spin or of the charge is considered. It is of ‘d-wave type,’ see Eq. (20), in the case of spin-fermion interaction Eq. (15), and of ‘s-wave type’ in the case of charge-fermion interaction, see Eq. (26) and text after. The explicit 2D momentum dependence of the superconducting gap was not considered in this simplified picture.

 

Referee comment:

I think that it should be clearly stated in the main text how the fact that the mechanism is based on spin- or charge-fermion interaction influences the gap symmetry within the proposed approach. Here, I am refering to the sentence “It is of ‘d-wave type,’ see Eq. (20), in the case of spin-fermion interaction Eq. (15), and of ‘s-wave type’ in the case of charge-fermion interaction”. Such information is important since it is well established by the ARPES data that the d-wave pairing scenario is realized in the copper based superconductors.

Also, in my opinion the comment that the “The explicit 2D momentum dependence of the superconducting gap was not considered in this simplified picture.” should also be in some form included somewhere in the paper for the sake of clarity

 

2. Another important feature of the cuprates is the appearance of strong Coulomb repulsion. The author writes that he uses the Hubbard model. Is the so-called Hubbard U term taken into acount explicitly? If yes what is the value of the onsite Coulomb repulsion taken in the calculations?

 

Author’s answer:

A corresponding explanation is added in the text after Eq. (15):

Actually, the model (1), (15) could be obtained e.g. from a bare on-site repulsive-U Hubbard Hamiltonian after decoupling of the inter- fermion interaction via auxiliary Hubbard-Stratonovich field M(r,\tau) , see e.g. H. J. Schulz, Effective Action for Strongly Correlated Fermions from Functional Integrals, Phys. Rev. Lett. {\bf 65}, 2462-2465 (1990); Mukhin, S.I., Negative Energy Antiferromagnetic Instantons Forming Cooper-Pairing Glue and Hidden Order in High-Tc Cuprates, Condens. Matter {\bf 3(4)}, 39 (2018). Then, before renormalizations, the constant g is actually: g~U.

Referee comment:

Since g~U than I believe that it would be convenient to comment in the text on the comparison between the value of g (or the range of values) taken in the calculations and the U value which is appropriate for the cuprates - the value of U for the single band Hubbard model as applied to the description of cuprates is believed to be 4-6 eV.

 

One additional remark. Namely I believe that it would be beneficial for the reader to provide a broader perspective in the introduction when it comes to the theoretical concepts related with the theoretical description of the cuprates which have been analyzed over the years. For example the mechanism based on the antiferromagnetic spin fluctuations has been analyzed in the review article D. J. Scalapino. Rev. Mod. Phys. 84 (2012), 1383. Also, general information about the theoretical description of the cuprates is provided in the review article M. Ogata, H. Fukuyama, Rep. Prog. Phys. 71 (2008), 036501 and relatively recent article studying the pairing mechanism within the paradigm of strong electronic correlations which also could be useful is J. Spałek et al., Phys Rev. B 95, 024506 (2017).

 

Author Response

Referee comment:

1)  I think that it should be clearly stated in the main text how the fact that the mechanism is based on spin- or charge-fermion interaction influences the gap symmetry within the proposed approach. Here, I am refering to the sentence “It is of ‘d-wave type,’ see Eq. (20), in the case of spin-fermion interaction Eq. (15), and of ‘s-wave type’ in the case of charge-fermion interaction”. Such information is important since it is well established by the ARPES data that the d-wave pairing scenario is realized in the copper based superconductors.

Also, in my opinion the comment that the “The explicit 2D momentum dependence of the superconducting gap was not considered in this simplified picture.” should also be in some form included somewhere in the paper for the sake of clarity

Reply 1)

1) The author is grateful to the Referee for pointing out the missing explanations, that now are added in the text (marked with bold font) as follows:

 

1. Concerning the d-wave vs s-wave symmetry of the order parameter in the case of spin-/charge-fermion couplings it is added after Eq. (26):

 

This would, in turn, lead to the absence of the factor $\sigma\bar{\sigma}=-1$ in the Eq. (\ref{DOs}). Hence, in order to keep $U_f <0$ (the driving force of the Q-ball transition) one has to compensate for this sign change. For this, it is necessary to change the sign of the Green's functions product $\overline{F}_{\sigma,\bar{\sigma}}(\omega,{\bf{p}}){F}_{\bar{\sigma},\sigma}(\omega-\Omega,{\bf{p}}-{\bf{Q_{DW}}})$ in Eq. (\ref{DOs}). Then, allowing for the structure of the Gor'kov's anomalous Green's function in Eq. (\ref{F}) one concludes, that in order to change the sign of the Green's functions product one has to change relation between the signs of superconducting order parameters in the points connected by the 'nesting' wave vector $Q_{CDW}$. Hence, in case of CDW-mediated pairing \cite{Seibold, Bianconi (1994)} the 'nesting' wave vector should couple points with the same sign of superconducting order parameter corresponding to the s-wave coupling \cite{Chubukov (2003)}: $\Sigma _{2p - Q_{CDW},\sigma }  =  \Sigma _{2p,\sigma }$. Such choice then changes the sign of the product $\overline{F}_{\sigma,\bar{\sigma}}(\omega,{\bf{p}}){F}_{\bar{\sigma},\sigma}(\omega-\Omega,{\bf{p}}-{\bf{Q_{DW}}})$ in the free energy integral in Eq. (\ref{DOs}) just compensating for the absence of the factor $\sigma\bar{\sigma}=-1$, and, hence, keeping intact the major condition : $U_f<0$.

 

 

2. Concerning the explicit 2D momentum dependence of the superconducting gap it is added now the following text after  Eq. (23):

Hence, the superconducting (pseudo)gap g_0 in the spectrum of fermions populating the Q-balls arises under such scheme only in the vicinities of the antinodal points. A problem of creation of the Fermi arcs is, in principle, treatable in the presented Q-ball gas picture, but demands elaboration of the self consistency equation for momenta $p$ also outside the antinodal points. An explicit 2D momentum dependence of the superconducting gap is not considered in the simplified picture used in the present work and will be considered elsewhere. The consequence of the latter approximation is discussed below in the subsection "The size of the Cooper-pair function" below.

 3. and in subsection 6.1 it is added after Eq. (49):

 

Hence, in our rough approximation for the anomalous self-energy $\Sigma_2$ being non-zero only in the vicinity of the 'antinodal' points in Brillouin zone the 'shape' of the Cooper-pair is 'starfish' like: it is characterised by the length scale $v_f/g_0$ in the 'antinodal' direction, and by the Q-ball radius in the 'nodal' direction, in qualitative correspondence with experiment \cite{Li}. Substituting $g_0\sim T^*\sim \mu_0$ according to Eq. (\ref{cross}), one estimates the characteristic size scale along the 'antinodal' direction: $v_f/g_0\sim v_f/\mu_0\gg\xi\sim s/\mu_0$, where $\xi$ is correlation length of the spin-/charge fluctuation inside the Q-ball, see Eq. (\ref{Eu}). Power law decrease of the wave-function in Eq. (ref{psi})leading to formal divergence of its size has its origin in the crude ansatz for the anomalous self-energy $\Sigma_2$ momentum dependence in Eq. (\ref{freeint}) mentioned after Eq. (\ref{pgspect}). 

 

 

 

 

 

2) Referee comment:

Since g~U than I believe that it would be convenient to comment in the text on the comparison between the value of g (or the range of values) taken in the calculations and the U value which is appropriate for the cuprates - the value of U for the single band Hubbard model as applied to the description of cuprates is believed to be 4-6 eV.

Reply 2)

1. The important issue concerning the value of the constant g used in the present work is now explicitly discussed in the text after Eq. (16):

 

Actually, spin-fermion coupling action (\ref{f}) could be obtained after decoupling of the on-site inter-fermion repulsive U-term: U$n_{i,\uparrow} n_{i,\downarrow}$ in the Hubbard Hamiltonian  via auxiliary Hubbard-Stratonovich  field $M(r,\tau)$, see e.g.  \cite{schulz, Mukhin (2018)}. Then, before renormalisations, the constant g in Eq. (\ref{Eu}) is formally inferred from the term ${\mu _0 ^2 }{|M|}^2$ and Eq. (\ref{f}): $g\sim$ U$\mu_0^2\sim 10^{-2}$eV$^3$, where the last estimate is based on the the value of U$\sim 4-6$ eV for the single band Hubbard model \cite{scalapino, spalek}, and on results of neutron measurements, $\mu_0\sim 100-200$meV, of the spin-wave excitations in doped high-T$_c$ cuprates \cite{tranq}.

 

 

3) Referee comment:

One additional remark. Namely I believe that it would be beneficial for the reader to provide a broader perspective in the introduction when it comes to the theoretical concepts related with the theoretical description of the cuprates which have been analyzed over the years. For example the mechanism based on the antiferromagnetic spin fluctuations has been analyzed in the review article D. J. Scalapino. Rev. Mod. Phys. 84 (2012), 1383. Also, general information about the theoretical description of the cuprates is provided in the review article M. Ogata, H. Fukuyama, Rep. Prog. Phys. 71 (2008), 036501 and relatively recent article studying the pairing mechanism within the paradigm of strong electronic correlations which also could be useful is J. Spałek et al., Phys Rev. B 95, 024506 (2017).

Reply 3)

1. All the corresponding references are added in the Introduction :

This mechanism of local/Cooper pairing differs from the usual Frohlich mechanism of exchange with infinitesimal lattice/charge/spin quasiparticles in e.g. phonon- or spin-fermion coupling models considered for high-T$_c$ cuprates \cite{Chubukov (2003)}and underlying $t-J-U$ Hubbard models reviewed in \cite{scalapino, fukuyama, spalek}.

 

2. and also are used in the text concerning the evaluation of coupling g after Eq. (16), where reference to John Tranquada’s work is added as well:

 

Then, before renormalisations, the constant g in Eq. (\ref{Eu}) is formally inferred from the term ${\mu _0 ^2 }{|M|}^2$ and Eq. (\ref{f}): $g\sim$ U$\mu_0^2\sim 10^{-2}$eV$^3$, where the last estimate is based on the the value of U$\sim 4-6$ eV for the single band Hubbard model \cite{scalapino, spalek}, and on results of neutron measurements, $\mu_0\sim 100-200$meV, of the spin-wave excitations in doped high-T$_c$ cuprates \cite{tranq}.