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We consider the forces acting on electrons in magnetic field including the constraints and a condition arising from quantum mechanics. The force is calculated as the electron mass, $m_e$, multiplied by the total time-derivative of the velocity field evaluated using the quantum mechanical many-electron wave function. The velocity field includes a term of the Berry connection from the many-body wave function; thereby, quantum mechanical effects are included. It is shown that additional important forces besides the Lorentz force exist; they include the gradient of the electron velocity field kinetic energy, the gradient of the chemical potential, and the `force' for producing topologically protected loop currents. These additional forces are shown to be important in superconductivity, electric current in metallic wires, and charging of capacitors.

A theory for cuprate superconductivity predicts the existence of nano-sized loop currents called, `` spin-vortex-induced loop currents (SVILCs)''. We calculate magnetic fields produced by them for a model of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi-2212) thin films composed of one surface and two bulk CuO$_2$ bilayers. In this model, bulk CuO$_2$ layers host stable spin-vortices around small polarons formed from doped holes; they give rise to a $U(1)$ gauge field described by the Berry connection from many-body wave functions, and generates the SVILCs. The effect of the gauge field is taken into account by the particle-number conserving Bogoliubov-de Gennes (PNC-BdG) formalism. The magnitude of the calculated magnetic field produced by the SVILCs in the vicinity of the surface ($10a \approx 4$ nm, where $a$ is the lattice constant of the CuO$_2$ plane) is in the order of mT; thus, may be detectable by currently available detection methods. The detection of the SVILCs by the magnetic field measurement may bring about the elucidation of the cuprate superconductivity, and may also lead to their quantum device applications, including qubits.

Recently, a new theory of superconductivity has been put forward that attributes the origin of superconductivity to the appearance of a non-trivial Berry connection from many-electron wave functions. This theory reproduces the major results of the BCS theory with conserving the particle number, and predicts the single-electron supercurrent tunneling across the Josephson junction with keeping the correct Josephson relation. We re-examine the quantization of superconducting qubit circuits by taking into account the above development, and show that the dynamical variables used in the standard theory, the flux nodes relating to the voltage, should be replaced by those relating to the electromagnetic vector potential. The fact that the Josephson junction tunneling allows the single-electron supercurrent tunneling is the reason for the existence of excited single electrons in superconducting qubits with Josephson junctions. We predict that it will be avoided by weakening the coupling between two superconductors in the Josephson junction.

The velocity field composed of the Berry connection from many-body wave functions and electromagnetic vector potential explains the energy-momentum balance during the reversible superconducting-normal phase transition in the presence of an externally applied magnetic field. In this formalism, forces acting on electrons are the Lorentz force and force expressed as the gradient of the kinetic energy. In the stationary situation, they balance; however, an infinitesimal imbalance of them causes a phase boundary shift. In order to explain the energy balance during this phase boundary shift, the electromotive force of the Faraday's magnetic induction type is considered for the Berry connection. This theory assumes that supercurrent exists as a collection of stable quantized loop currents, and the transition from the superconducting to normal phase is due to the loss of their stabilizations through the thermal fluctuation of the winding numbers of the loop currents. We argue that an abrupt change of loop current states with integral quantum numbers should be treated as a quantum transition; then, the direct conversion of the quantized loop currents to the magnetic field occurs; consequently, the Joule heat generation does not occur during the phase transition.

The velocity field composed of the electromagnetic field vector potential and the Berry connection from many-body wave functions explains supercurrent generation, Faraday's law for the electromotive force (EMF) generation, and other EMF generations whose origins are not electromagnetism. An example calculation for the EMF from the Berry connection is performed using a model for the cuprate superconductivity.

Superconductivity is reformulated as a phenomenon in which a stable velocity field is created by a $U(1)$ phase neglected by Dirac in the Schroedinger representation of quantum mechanics. The neglected phase gives rise to a $U(1)$ gauge field expressed as the Berry connection from many-body wave functions. The inclusion of this gauge field transforms the standard particle-number non-conserving formalism of superconductivity to a particle-number conserving one with many results of the former unaltered. In other words, the new formalism indicates that the current standard one is an approximation that effectively takes into account this neglected $U(1)$ gauge field by employing the particle-number non-conserving formalism. Since the standard and new formalisms are physically different, conflicting results are predicted in some cases. We reexamine the Josephson relation and show that a capacitance contribution of the Josephson junction to the $U(1)$ phase is missing in the standard formalism, and inclusion of it indicates that the standard theory actually does one agree with the experiment while the new one does. It is also shown that the dissipative quantum phase transition in Josephson junctions predicted in the standard theory does not exit in the new one in accordance with the recent experimental result.

The standard quantum mechanical electronic state calculations for molecules and solids uses the Schroedinger representation where the momentum conjugate to the coordinate $q_r$ is given by $-hbar {partial over {partial q_r}}$. This formalism contains an extra $U(1)$ phase degree-of-freedom. We show that it can be regarded as a Berry phase arising from many-electron interaction, and when it is non-trivial, it gives rise to a current carrying ground state identified as the superconducting ground state. The connection between this superconducting state and the BCS one is presented.

Superconductivity is a phenomenon where electrical current flows without friction. The current standard theory for it is the BCS (Bardeen-Cooper-Schrieffer) theory, which explains it as due to the energy gap formation by the electron-pairing, and the key ingredient for the supercurrent generation is the gauge symmetry breaking brought about by it. It was thought that superconductivity was fully understood by this standard theory; however, the discovery of superconductivity in cuprates in 1986 changed this situation, showing a number of experimental results that contradict the standard theory. It is also notable that the standard theory contradicts in the supecurrent carrier mass in the London moment; the predicted mass is an effective mass, while the experimental value is the free electron mass. The above contradictions suggest the necessity for a fundamental revision for the theory of superconductivity. Here we show that the required revision may be achieved by using the Berry phase formalism. It was developed after the establishment of the BCS theory, and provides a way to detect emergent gauge fields. A self-referencing phase factor on the wave function detected by the Berry phase formalism explains the supercurrent generation in the conventional and cuprate superconductors. It gives rise to a gauge field that enables the gauge symmetry breaking in the standard theory interpretation.

A fundamental revision of superconductivity theory that resolves the supercurrent carrier mass contradiction (the standard theory predicts it to be the effective mass but the London moment measurement indicates it to be the free electron mass) is presented, using a generalized Hatree-Fock approximation that takes into account a Berry connection from many-body wave functions. The new theory explains the pairing energy gap formation accompanying the superconductivity transition in the same manner as the standard theory, yet, provides the free electron carrier mass in accordance with the London moment measurement.

The new superconductivity theory that attributes the $U(1)$ superconductivity phase to a Berry phase arising from many-body wave functions is applied to the circuit quantization for superconducting qubits. The phase-charge duality required for the occurrence of superconductivity in the standard theory becomes irrelevant in the new theory, and the absence of a dissipative quantum phase transition in Josephson junctions is explained. It is shown that a charge-decaying term leads to the compact phase description, and the appearance of Shapiro steps is explained without introducing normal current.

A fundamentally revised version of superconductivity theory has been put forward by the present authors since the standard theory of superconductivity based on the BCS theory cannot explain superconductivity in cuprates discovered in 1986, and reexaminations on several experimental results on the conventional superconductors indicate the necessity for a fundamental revision. The revision is made on the origin of the superconducting phase variable, which is attributed to a Berry connection arising from many-body wave functions. With this revision, the theory can be cast into a particle number conserving formalism. We have developed a method to calculate superconducting states with the Berry connection using the particle number conserving version of the Bogoliubov-de Gennes equations. An example calculation is made for a model originally built for cuprate superconductors.

Although the standard theory of superconductivity based on the BCS theory is a successful one, several experimental results indicate the necessity for a fundamental revision. We argue that the revision is on the origin of the phase variable for superconductivity; this phase appears as a consequence of the electron-pairing in the standard theory, but its origin is a Berry connection arising from many-body wave functions. When this Berry connection is non-trivial, it gives rise to a collective mode that generates supercurrent; this collective mode creates number-changing operators for particles participating in this mode, and these number-changing operators stabilize the superconducting state by exploiting the Cooper instability. In the new theory, the role of the electron-pairing is to stabilize the nontrivial Berry connection; it is not the cause of superconductivity. In BCS superconductors, however, the simultaneous appearance of the nontrivial Berry connection and the electron-pairing occurs. Therefore, the electron-pairing amplitude can be used as an order parameter for the superconducting state. We revisit the Andreev$-$Saint-James reflection and the Josephson effect. They are explained as consequence of the presence of the Berry connection. Bogoliubov quasiparticles are replaced by the particle-number conserving Bogoliubov excitations that describe the transfer of electrons between the collective mod and single particle mode.

Although the standard theory of superconductivity based on the BCS theory is a successful one, there are several experimental results that indicate the necessity for fundamental revisions. One of them is the mass in the London moment. Experiments indicate the mass in the London moment is the free electron mass although the BCS theory and its extension predict it to be an effective mass. We show that this discrepancy is lifted if we install the London's superpotential in the theory, and identify it as the Berry phase arising from the many-body wave functions. Then, the induced current by the applied magnetic field becomes a stable current calculated using the free energy in contrast to the linear response current assumed in the standard theory which yields the Nambu-Goldstone mode. The Nambu-Goldstone mode arising from the breakdown of the global $U(1)$ gauge invariance in the standard theory is replaced by the collective mode arising from the Berry connection. Then, the free electron mass appears in the London moment.

In the standard theory of superconductivity, the origin of superconductivity is the electron-pairing. The induced current by a magnetic field is calculated by the linear response to the vector potential, and the supercurrent is identified as the dissipationless flow of the paired-electrons, while single electrons flow with dissipation. This supercurrent description suffers from the following serious problems: 1) it contradicts the reversible superconducting-normal phase transition in a magnetic field observed in type I superconductors; 2) the gauge invariance of the supercurrent induced by a magnetic field requires the breakdown of the global $U(1)$ gauge invariance, or the non-conservation of the particle number; 3) the explanation of the ac Josephson effect is based on the boundary condition that is different from the real experimental one. We will show that above problems are resolved if the supercurrent is attributed to the collective mode arising from the Berry connection for many-body wave functions. The problem 1) is resolved by attributing the appearance and disappearance of the supercurrent to the abrupt appearance and disappearance of topologically-protected loop currents produced by the Berry connection; the problem 2) is resolved by assigning the non-conserved number to that for the particle number participating in the collective mode produced by the Berry connection; and the problem 3) is resolved by identifying the relevant phase in the Josephson effect is that arising from the Berry connection, and using the modified Bogoliubov transformation that conserves the particle number.

We show that the Rashba spin-orbit interaction causes spin-twisting itinerant motion of electrons in metals and realizes the quantized cyclotron motion of conduction electrons without an external magnetic field. From the view point of the Berry connection, the cause of this {quantized} motion is the appearance of a non-trivial Berry connection ${\bf A}^{\rm fic}=-{\hbar \over {2e}}\nabla \chi$ ($\chi$ is an angular variable with period $2\pi$ that generates $\pi$ flux (in the units of $\hbar=1, e=1,c=1$) inside the nodal singularities of the wave function (a "Dirac string") along the centers of spin-twisting. Since it has been shown in our previous work[Ref.1]that the collective mode of $\nabla \chi$ is stabilized by the electron-pairing and generates supercurrent, the $\pi$-flux Dirac string created by the spin-twisting itinerant motion will be stabilized by the electron-pairing and produce supercurrent.

We show that two phenomena of superfluidity, superfluidity of weakly interacting bosons and superconductivity of the BCS model, are unified using the collective mode arising from the Berry connection for many-body wave functions. The superfluidity is attributed to the presence of this mode, which is stabilized by the interaction between particles that causes fluctuations of the number of particles participating in it. It is suggested that the existence of this collective mode and its stabilization is more fundamental to the occurrence of superconductivity than the electron-pair formation.

We revisit the supercurrent generation mechanism for superconductors whose superconducting transition temperature are explained by the BCS theory, motivated by the reexamination of the ac Josephson effect [J. Supercond. Nov. Magn. (2015) 28:61] that indicates the electromagnetic vector potential ${\bf A}^{\rm em}$ couples to each electron in the pairing electrons, separately, as $e{\bf A}^{\rm em}$, not together as $2e{\bf A}^{\rm em}$. To satisfy the above finding, we argue that the origin of the supercurrent generation is the emergence of Dirac strings with $\pi$ flux inside. It appears if the Rashba spin-orbit interaction is added to the BCS model due to its stabilization of the spin-twisting itinerant motion of electrons. The $\pi$-flux Dirac string generates the cyclotron motion without external magnetic field, and produces topologically protected loop current. This can be also attributed to the emergence of the $U(1)$ instanton of the Berry connection ${\bf A}^{\rm fic}=-{ \hbar \over {2e}} \nabla \chi$, ${\bf \varphi}^{\rm fic}={ \hbar \over {2e}} \partial_t \chi$, where $\chi$ is an angular variable of period $2\pi$. The phase of the macroscopic wave function for the Ginzburg-Landau theory or the phase of the pair potential of the Bogoliubov-de Gennes equations is identified as $\chi$. If it is treated as a phenomenological parameter, the Ginzburg-Landau theory or the Bogoliubov-de Gennes equations can be used without modification. However, the new origin requires the Rashba interaction for the occurrence of superconductivity. This may explain the fact that ideal metals like sodium does not show superconductivity since the screening of the electric field is efficient in such materials, suppressing the internal electric field too weak to occur superconductivity.

Superconductivity is a phenomena where an external-feeding current flows through the system without voltage drop. This indicates the existence of an energy minimum under the current feeding boundary condition. Although it is believed that superconductivity is explained by the BCS theory that attributes the origin of superconductivity to the electron pair formation, such an energy minimum has never been obtained by it. We have found such a minimum in the system exhibiting spin twisting itinerant motion of electrons with the Rashba spin-orbit interaction, but without referring to the electron pairing. The supercurrent is shown to have the London equation form that exhibits the flux quantum $h/2e$. This supercurrent generation mechanism may be the one actually occurring in real superconductors instead of the one explained by the BCS theory.

We present a possible explanation for a recently observed magnetic field induced charge order in cuprate superconductors (Edkins et al.~arXiv:1802.04673 [cond-mat.supr-con]). We argue that it arises from the reorganization of spin-vortex-induced loop current (SVILC) pattern due to supercurrent-flow caused by the magnetic field. The reorganization is from the most stable tiling of $4a times 6a$ spin-vortex quartets ($a$ is the lattice constant in the CuO$_2$ plane, and a spin-vortex quartet is stable of unit of spin-vortices that contains four holes, four spin-vortices, and four SVILCs) to that of $4a times 8a$ spin-vortex quartets. The consequence of this reorganization will lead to the enhancement of $8a$ charge order, and reduction of $6a$ charge order. The former is observed in the experiment, but the latter is not confirmed, so far. However, it may be confirmed if the experimental result is carefully reexamined.

We derive the AC Josephson frequency using the gauge invariant equations of motion with including the battery-contact effect. The frequency is given as a sum of a contribution from an Aharonov-Bohm phase that arises when charged-partices pass through an electric field in the insulator and that from a chemical potential difference arising form the battery contact; each of them contributes $|q|V/h$ to the frequency where $q$ is the charge, thus, the sum is $2|q|V/h$. The observed Josepshon frequency, $2|e|V/h$, hence, means that the charge on the tunneling particles is $e$. A variety of derivations so far miss one of the above two contributions; the original derivation misses the first contribution due to the lack of inclusion of the electric field in the insulator. The present result indicates that the phase of the order-parameter does not arise from the number fluctuations of Cooper pairs. We present an alternative origin for it; we argue that a Berry phase that arises from spin-vortices is the origin., Comment: 24 pages, 6 figures

Persistent current generation by small polarons embedded in a spin-density wave background in a two dimensional lattice is theoretically studied. When the embedded small polarons become cores of merons and antimerons (vortices in spin configuration with winding numbers +1 and -1, respectively), a spin Berry phase arises from the spiral spin texture of the merons and antimerons, and generates a spin current around the polarons. Electric current is, however, absent due to the exact cancellation of the up- and down-spin electron currents. By addition of an imaginary magnetic flux that negates the fictitious magnetic flux from the spin Berry phase, the state with merons and antimerons is stabilized and non-zero electric current appears; this imaginary flux becomes a real one when side-by-side bipolaronic cores of merons and antimerons are formed, where the real flux arises spontaneously from the wave function for the motion of the system as a whole; thus, the bipolarons serve as pumps for the persistent electric current. It is also shown that a Nernst signal stems from thermal diffusion of the bipolarons. Implications of the present results to anomalous properties of cuprate superconductors, including the enhanced Nernst signal, Drude peak, and Fermi arc are also discussed. The present result suggests the appearance of the phase variable or ``Nambu-Goldstone'' mode for the superconductivity without the breakdown of the global U(1) gauge invariance in the cuprates., Comment: 26 pages, 9 figures

We show that a stable spontaneous current appears in a system of a spin density wave order with merons, vortices in the spin configuration with winding number +1. A meron in the spin density wave order modifies the boundary condition for eigenfunctions around it to a sign-change one. As a consequence, two types of stable current states, one with a clockwise circulation and the other with a counterclockwise one, arise. When a magnetic field is present, one produces a diamagnetic current is chosen. A collection of such currents results in a large diamagnetic current; and if the meron density is sufficiently large, a perfect diamagnetism is realized. The stability of this diamagnetic current is attributed to the topological nature of the merons, and as long as the distribution of the merons remains the same the current will persist even in a macroscopic system.

The spin-charge-orbital complex structures of manganites are studied using topological concepts. The key quantity is the "winding number" w associated with the Berry-phase connection of an e_g electron parallel-transported through Jahn-Teller centers, along zigzag one-dimensional paths in an antiferromagnetic environment of t_{2g} spins. From these concepts, it is shown that the "bi-stripe" and "Wigner-crystal" states observed experimentally have different w's. Predictions for the spin structure of the charge-ordered states for heavily doped manganites are discussed., Comment: 5 pages, Revtex, with 3 figures embedded in the text. Submitted to Phys. Rev. Lett

Electronic states of an $e_g$ electron are calculated in the system composed of two MnO$_6$ octahedra with the inclusion of the Berry phase acquired by parallel transport. Based on this calculation, a comment is made on the controversy between ``Wigner-crystal'' and ``paired-stripe'' models for the the insulating charge-ordered manganese oxides., Comment: 4 pages, 2 figures