We derive the Ward-Takahashi identity and establish the gauge-invariant response theory for open quantum systems described by Lindbladians to show that particle-number conservation is not necessary to satisfy gauge invariance. We construct an observable which can be used to test the gauge invariance in the absence of particle-number conservation. We derive the low-energy collective modes that emerge as a consequence of gauge invariance in open quantum systems, and find that two-body loss induces diffusive modes in dissipative Bardeen-Cooper-Schrieffer (BCS) superconductivity. Possible experimental situations for testing gauge invariance in open quantum systems are also discussed.
We derive the Ward-Takahashi identity and establish the gauge-invariant response theory for open quantum systems described by Lindbladians to show that particle-number conservation is not necessary to satisfy gauge invariance. We construct an observable which can be used to test the gauge invariance in the absence of particle-number conservation. We derive the low-energy collective modes that emerge as a consequence of gauge invariance in open quantum systems, and find that two-body loss induces diffusive modes in dissipative Bardeen-Cooper-Schrieffer (BCS) superconductivity. Possible experimental situations for testing gauge invariance in open quantum systems are also discussed.
We present a diagrammatic theory for determining consistent electromagnetic response functions in strongly correlated fermionic superfluids. While a gauge-invariant electromagnetic response is well understood at the BCS level, a treatment of correlations beyond BCS theory requires extending this theoretical formalism. The challenge in such systems is to maintain gauge invariance, while simultaneously incorporating additional self-energy terms arising from strong correlation effects. Central to our approach is the application of the Ward-Takahashi identity, which introduces collective mode contributions in the response functions and guarantees that the f-sum rule is satisfied. We outline a powerful method, which determines these collective modes in the presence of correlation effects and in a manner compatible with gauge invariance. Since this method is based on fundamental aspects of quantum field theory, the underlying principles are broadly applicable to strongly correlated superfluids. As an illustration of the technique, we apply it to a simple class of theoretical models that contain a frequency-independent order parameter. These models include BCS-BEC crossover theories of the ultracold Fermi gases, along with models specifically associated with the high-Tc cuprates. Finally, as an alternative approach, we contrast with the path integral formalism. Here, the calculation of gauge-invariant response appears more straightforward. However, the collective modes introduced are those of strict BCS theory, without any modification from additional correlations. As the path integral simultaneously addresses electrodynamics and thermodynamics, we emphasize that it should be subjected to a consistency test beyond gauge invariance, namely that of the compressibility sum rule. We show how this sum rule fails in the conventional path integral approach.
In a Lagrange formulation of nonrelativistic quantum mechanics gauge invariance corresponds to a local U(1) symmetry of the Lagrangian density. We review how this gauge symmetry yields the electromagnetic interaction of the free and the locally interacting N-body system and why nonlocal interactions require additional exchange contributions. A nonrelativistic Ward-Takahashi identity is then derived. This relates electromagnetic operators to the full propagator of the particles in question, which may be off their energy shell.
A diagrammatic method of obtaining exact gauge-invariant response functions in strongly correlated Fermi superfluids is implemented for several example condensed matter systems of current interest. These include: topological superfluids, high temperature superconductors, and superfluids with finite center-of-mass momentum pairing known as Fulde-Ferrell superfluids. Much of the literature on these systems has focused on single-particle properties or alternatively has invoked simple approximations to treat response functions. The goal is to show that, for this wide class of topical problems, one can compute exact response functions. This enables assessment of the validity of different physical scenarios and allows a very broad class of experiments to be addressed.,The method developed is based on deriving the full electromagnetic vertex, which satisfies the Ward-Takahashi identity, and determining the collective modes in a manner compatible with the self-consistent gap equation. In the condensed phase of a superfluid and a superconductor, where gauge invariance is spontaneously broken, it is crucial to determine the collective modes from the gap equation in a manner which restores gauge invariance. Our diagrammatic framework provides a very general and powerful method for obtaining these collective modes in a variety of strongly correlated Fermi superfluids. We show that a full electromagnetic vertex satisfying the Ward-Takahashi identity ensures the f-sum rule is satisfied and thus charge is conserved. ,This diagrammatic method is implemented for both normal and superfluid phases. While there are no collective modes in the normal phase, the Ward-Takahashi identity plays a similarly important role. In particular, for the normal phase we study Rashba spin-orbit coupled Fermi gases with intrinsic pairing in the absence and presence of a magnetic field. Exact density and spin response functions are obtained, even in the absence of a spin conservation law, providing signatures of topological order. Another set of examples studied in the normal phase include a broad collection of models of the cuprate pseudogap. For these cases too, the exact response functions are obtained and signatures of each model are identified. A detailed discussion of the superfluid phase is provided by the study of Fulde-Ferrell superfluids. The generally omitted collective modes are computed and the exact gauge-invariant response is obtained. The first complete calculation of the superfluid density, incorporating the collective mode contribution and without any assumptions on the size of the gap, is provided. It is found that the amplitude mode causes the superfluid to become unstable at temperatures much lower than the mean-field transition temperature.,As an alternative to the diagrammatic approach, the functional path integral technique for Fermi superfluids is presented. While this approach is widely studied in the literature, it is of restricted utility, in comparison to the diagrammatic approach presented here, due to the inability to perform the functional integral except in very limited cases. A short coming in the current literature is identified, in that the standard method of deriving gauge-invariant electrodynamics leads to a violation of the compressibility sum rule. It is found that to satisfy this sum rule the amplitude mode of the order parameter must be incorporated. A reformulation of the path integral method for Fermi superfluids is derived, which has both gauge-invariant electrodynamics and thermodynamics compatible with the compressibility sum rule, to all orders of approximation.
A Ward-Takahashi identity, as a consequence of gauge invariance and in a form that relates self-energy to the two-particle Bethe-Salpeter scattering kernel, was first derived by Vollhardt and Wölfle for a system of independent particles moving in a random medium. This is generalized to a class of interacting electronic systems in materials with or without random impurities, following a procedure previously used for classical waves transport in disordered media. This class of systems also possesses other symmetry properties such as invariance under time translations and local spin rotations, which imply local conservation laws for energy and spin current. They imply additional Vollhardt-Wölfle type identities. We present non-perturbative derivations of these identities, and consider the constraints they impose on the relationship between the self-energy and the two-particle scattering kernel. PACS numbers: 11.30.-j, 03.80.+r, 72.10.Bg 1 Ward identities are statements of local conservation laws, and reflect the corresponding
Statistical-gauge structures generated by current operators provide a symmetry-based description of forces and hyperforces in closed quantum statistical mechanics, but their status in open quantum dynamics remains largely unexplored. Here we develop a generalized statistical-gauge framework for Markovian open quantum systems governed by Lindblad evolution. The construction formulates statistical shifts in a smeared operator language, thereby avoiding singular local expressions while preserving the force and hyperforce content of the closed theory. Within this framework, the action of the Lindblad generator is analyzed in a jump-resolved representation that separates Hamiltonian transport from dissipative contributions in jump space. The central result is a modular Ward identity that holds for any full-rank quantum state, independently of equilibrium, stationarity, or unitary dynamics. We show that weak covariance is controlled by the full dissipative structure in jump space, rather than by invariance of individual jump operators or by particle-number conservation. This leads to a classification of continuum reservoirs according to density weight and spatial rate profile, and shows that Lindblad dynamics can deform the transported shift algebra through its product defect even when the bath is weakly covariant. Physically, the theory explains why uniform one-body loss or pumping may preserve the statistical-gauge structure, whereas density dephasing, multiparticle loss, and nonuniform bath profiles generically produce covariance-breaking residuals. These residuals provide practical diagnostics for reservoir engineering, approximate master equations, and numerical many-body simulations of open quantum systems.
We show that the gauge-invariant kinetic equation of superconductivity provides an efficient approach to study the electromagnetic response of the gapless Nambu-Goldstone and gapful Higgs modes on an equal footing. We prove that the Fock energy in the kinetic equation is equivalent to the generalized Ward's identity. Hence, the gauge invariance directly leads to the charge conservation. Both linear and second-order optical responses are analytically investigated. The linear response of the Higgs mode vanishes in the long-wave limit, whereas the linear response of the Nambu-Goldstone mode interacts with the long-range Coulomb interaction, causing the original gapless energy spectrum effectively lifted up to the plasma frequency as a result of the Anderson-Higgs mechanism, consistent with previous work. The second-order response exhibits interesting physics. On the one hand, a finite second-order optical response of the Higgs mode is obtained in the long-wave limit. We reveal that this response, which has been experimentally observed, is attributed solely to the drive effect rather than the widely considered Anderson-pump effect. On the other hand, the second-order optical response of the Nambu-Goldstone mode, free from the influence of the long-range Coulomb interaction and hence the Anderson-Higgs mechanism, is predicted. We find that both Anderson-pump and drive effects play important role in this response. A tentative scheme to detect this second-order response is proposed.