Abstract Fermionic phase space representations are a promising method for studying correlated fermion systems. The fermionic Q-function and P-function have been defined using Gaussian operators of fermion annihilation and creation operators. The resulting phase-space of covariance matrices belongs to the symmetry class D, one of the non-standard symmetry classes. This was originally proposed to study mesoscopic normal-metal-superconducting hybrid structures, which is the type of structure that has led to recent experimental observations of Majorana fermions. Under a unitary transformation, it is possible to express these Gaussian operators using real anti-symmetric matrices and Majorana operators, which are much simpler mathematical objects. We derive differential identities involving Majorana fermion operators and an antisymmetric matrix which are relevant to the derivation of the corresponding Fokker–Planck equations on symmetric space. These enable stochastic simulations either in real or imaginary time. This formalism has direct relevance to the study of fermionic systems in which there are Majorana type excitations, and is an alternative to using expansions involving conventional Fermi operators. The approach is illustrated by showing how a linear coupled Hamiltonian as used to study topological excitations can be transformed to Fokker–Planck and stochastic equation form, including dissipation through particle losses.
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.
The problem of fermion dynamics is studied using the
Q function for fermions. This is a probabilistic phase-space representation, which we express using Majorana operators and a phase-space variable of real antisymmetric matrices. We consider a general Hamiltonian with quadratic and quartic Majorana interactions and arbitrary couplings. Our model includes the Majorana-Hubbard and Fermi-Hubbard Hamiltonians, as well as general quantum field theories of interacting fermions. The relevant coefficients are calculated for a number of both local and nonlocal Hamiltonian models. We also give a detailed example of a four-Majorana interaction. Using the Majorana
Q function we derive a generalized Fokker-Planck equation, with results for the drift and diffusion terms. The diffusion term is shown to be traceless, which has a dynamical interpretation as a forward-backward stochastic process. We prove that an initial pure state has trajectories that remain on a surface of pure states. This approach leads to a model of quantum measurement in terms of an ontology with real vacuum fluctuations.
Wehrl's entropy is the entropy of the probability distribution in the phase space corresponding to the Q representation (antinormal ordering of operators) of a quantum state in terms of coherent states. The Hilbert space of a system of N spins (or N two-level atoms) is of 2 N dimensions. The subspace with maximum total spin N/2 is of N+1 dimensions. All discussions are restricted within this subspace. The probability amplitude for an arbitrary pure state in this subspace is a polynomial of degree N which can be factorised into the product of N linear factors and each root can be identified with one point on the unit sphere. Hence, an arbitrary state in the subspace can be represented geometrically by N unit vectors. The general expression for the Wehrl entropy is obtained as a finite series expansion in terms of some symmetric functions of sin 2 ( omega ij /2), where omega ij is the angle between a pair of unit vectors. The special cases of coherent and almost coherent spin states are then considered.
Understanding the quantum properties of many-body states is important in the fields of both quantum information and condensed-matter physics. For this purpose, we generalize the basic concept of the anticommutator from two operators to the many-body case. Some key properties are given for the many-body anticommutators, and examples are provided for Pauli operators and density matrices. Using these results and techniques from the symmetry group, in a straightforward way, we give the normalization of the completely symmetric states with Majorana's stellar representations. Our developed method will help to pave the way in the study of many-body symmetric systems.
We obtain a positive probability distribution or Q -function for an arbitrary fermionic many-body system. This is different to previous Q -function proposals, which were either restricted to a subspace of the overall Hilbert space, or used Grassmann methods that do not give probabilities. The fermionic Q -function obtained here is constructed using normally ordered Gaussian operators, which include both non-interacting thermal density matrices and BCS states. We prove that the Q -function exists for any density matrix, is real and positive, and has moments that correspond to Fermi operator moments. It is defined on a finite symmetric phase-space equivalent to the space of real, antisymmetric matrices. This has the natural SO (2 M ) symmetry expected for Majorana fermion operators. We show that there is a physical interpretation of the Q -function: it is the relative probability for observing a given Gaussian density matrix. The distribution has a uniform probability across the space at infinite temperature, while for pure states it has a maximum value on the phase-space boundary. The advantage of probabilistic representations is that they can be used for computational sampling without a sign problem.