We test a new approach to bosonization in relativistic field theories and many-body systems, based on the use of fermionic composites as integration variables in the Berezin integral defining the partition function of the system. The method appears promising since at zeroth order it correctly describes the propagators of the composites, which can be evaluated in a number of significant cases. Still to be established is a general procedure for deriving the free action of the composites starting from the one of the constituents. To shed light on this problem and to explore further features of the method we study a simplified version of the BCS model. In this simple case the action of the composites can indeed be obtained: whether this result can be generalized it remains hewever to be seen. Yet an interesting property of the wave operators appearing in the free actions of bilinear composites already emerges from the simple problem we have treated: here the wave operators do not describe any time evolution, even though they generate the right propagators. This outcome relates to the basic properties of the integrals over the even elements of a Grassmann algebra where the composites live, which entails that the propagators are no longer the inverse of the wave operators.
We prove that the Gärtner–Ellis generating function of probability distributions associated with KMS states of weakly interacting fermions on the lattice can be written as the limit of logarithms of Gaussian Berezin integrals. The covariances of the Gaussian integrals are shown to have a uniform Pfaffian bound and to be summable in general cases of interest, including systems that are not translation invariant. The Berezin integral representation can thus be used to obtain convergent expansions of the generating function in terms of powers of its parameter. The derivation and analysis of the expansions of logarithms of Berezin integrals are the subject of the second part of the present work. Such technical results are also useful, for instance, in the context of quantum information theory, in the computation of relative entropy densities associated with fermionic Gibbs states, and in the theory of quantum normal fluctuations for weakly interacting fermion systems.
Understanding the structure of operators that commute with
k identical replicas of unitary ensembles, also known as their
k-commutants, is an important problem in quantum many-body physics with deep implications for the late-time behavior of physical quantities such as correlation functions and entanglement entropies under unitary evolution. In this work, we study the
k-commutants of free-fermion unitary systems, which are heuristically known to contain
SO(k) and
SU(k) groups without and with particle number conservation respectively, with formal derivations of projectors onto these commutants appearing only very recently. We establish a complementary perspective by highlighting a larger
O(2k) replica symmetry (or
SU(2k) respectively) that the
k-commutant transforms irreducibly under, which leads to a simple geometric understanding of the commutant in terms of coherent states parametrized by a Grassmannian manifold. We derive this structure by mapping the
k-commutant to the ground state of effective ferromagnetic Heisenberg models, analogous to the ones that appear in the noisy circuit literature, which we solve exactly using standard representation theory methods. Further, we show that the Grassmannian manifold of the
k-commutant is exactly the manifold of fermionic Gaussian states on
2k sites, which reveals a duality between real space and replica space in free-fermion systems. This geometric understanding also provides a compact projection formula onto the
k-commutant, based on the resolution of identity for coherent states, which can prove advantageous in analytical calculations of averaged non-linear functionals of Gaussian states, as we demonstrate using some examples for the entanglement entropies. In all, this work provides a geometric perspective on the
k-commutant of free-fermions that naturally connects to problems in quantum many-body physics.
Abstract In this work, the perturbative aspects of quantum mechanics (QM) and quantum field theory (QFT), to a large extent, are studied with functional (path or field) integrals and functional techniques. This physics textbook thus begins with a discussion of algebraic properties of Gaussian measures, and Gaussian expectation values for a finite number of variables. The important role of Gaussian measures is not unrelated to the central limit theorem of probabilities, although the interesting physics is generally hidden in essential deviations from Gaussian distributions. A few algebraic identities about Gaussian expectation values, in particular Wick's theorem are recalled. Integrals over some type of formally complex conjugate variables, directly relevant for boson systems are defined. Fermion systems require the introduction of Grassmann or exterior algebras, and the corresponding generalization of the notions of differentiation and integration. Both for complex and Grassmann integrals, Gaussian integrals, and Gaussian expectation values are calculated, and generalized Wick's theorems proven. The concepts of generating functions and Legendre transformation are recalled.
The methods that have been described in previous chapters can be used to study general quantum Bose systems. They are based in a direct way on the introduction of a generating function of symmetric wave functions of bosons. In the case of fermion systems, however, one faces the problem that fermion wave functions, or fermion correlation functions (or Green functions) are antisymmetric with respect to the exchange of a fermion pair. Thus, the construction of generating functions requires the introduction of an antisymmetric or Grassmann algebra of ‘classical functions’. It is then possible to generalize to Grassmann algebras the notions of derivatives and integrals, yielding quite parallel formalisms for bosons and fermions, in particular, to define a path integral for fermion systems, analogous to the holomorphic path integral for bosons. This chapter discusses differentiation and integration in Grassmann algebras, Gaussian integrals and perturbative expansion, partition function, and quantum Fermi gas.