This dissertation brings together ideas from quantum information theory and condensed matter physics to study many-particle systems of fermions. We revisit the structure of the Hilbert space and operator algebras in Fermi systems which provides a natural platform to discuss the notion of entanglement and local operations. We generalize the partial transpose, which is a well-known operation to diagnose entanglement in density matrices of qubits and bosonic systems, to fermionic systems. Our idea was inspired by the observation that time-reversal acts as transposition on density matrices; hence, we looked for a way to apply partial time-reversal to density matrices of fermions. We present a comprehensive set of benchmarks on our proposed definition of fermionic partial transpose and explain the fundamental differences between the bosonic and fermionic partial transpose. We use this new framework and the associated entanglement measure, logarithmic negativity, to study the entanglement content of free fermions with an arbitrary shape of Femi surface in all dimensions as well as topological insulators and superconductors. In particular, we show how thermal fluctuations destroy the quantum coherence of the ground state as temperature is increased. Furthermore, we report the discovery of a surprising connection between topological invariants and the partial transpose in time-reversal symmetric topological insulators and superconductors. In short, we find that partition functions on non-orientable spacetime manifolds such as the Klein bottle or real-projective plane can be obtained by combining untransposed and partially transposed density matrix. This relation turns out to be general and can be used to study various other topological phases protected by anti-unitary symmetries. Similar to the partition functions, the quantities we introduce are complex numbers and their complex phase is a topological invariant. These topological invariants can be regarded as order parameters of topological phases in the sense that they admit discrete values and can only change by jumping from one value to another as we transition from one topological phase to another.
This dissertation brings together ideas from quantum information theory and condensed matter physics to study many-particle systems of fermions. We revisit the structure of the Hilbert space and operator algebras in Fermi systems which provides a natural platform to discuss the notion of entanglement and local operations. We generalize the partial transpose, which is a well-known operation to diagnose entanglement in density matrices of qubits and bosonic systems, to fermionic systems. Our idea was inspired by the observation that time-reversal acts as transposition on density matrices; hence, we looked for a way to apply partial time-reversal to density matrices of fermions. We present a comprehensive set of benchmarks on our proposed definition of fermionic partial transpose and explain the fundamental differences between the bosonic and fermionic partial transpose. We use this new framework and the associated entanglement measure, logarithmic negativity, to study the entanglement content of free fermions with an arbitrary shape of Femi surface in all dimensions as well as topological insulators and superconductors. In particular, we show how thermal fluctuations destroy the quantum coherence of the ground state as temperature is increased. Furthermore, we report the discovery of a surprising connection between topological invariants and the partial transpose in time-reversal symmetric topological insulators and superconductors. In short, we find that partition functions on non-orientable spacetime manifolds such as the Klein bottle or real-projective plane can be obtained by combining untransposed and partially transposed density matrix. This relation turns out to be general and can be used to study various other topological phases protected by anti-unitary symmetries. Similar to the partition functions, the quantities we introduce are complex numbers and their complex phase is a topological invariant. These topological invariants can be regarded as order parameters of topological phases in the sense that they admit discrete values and can only change by jumping from one value to another as we transition from one topological phase to another.
Here, the authors introduce a set of quantities to diagnose symmetry-protected topological phases of fermions protected by antiunitary symmetries. These quantities, which can be written as expectation values of nonlocal operators, effectively simulate the partition function on nonorientable spacetime manifolds in the operator formalism. The important observation is that the proposed quantities are complex valued for topological states, where the complex phase is the many-body topological invariant. Numerous symmetry classes and dimensionalities are studied in detail. The key ingredient of all quantities is the ``fermionic partial transpose''. For example, the Z2 topological invariant for time-reversal symmetric topological insulators in two spatial dimensions is given.
We define and compute many-body topological invariants of interacting fermionic symmetry-protected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry. The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a nonlocal operation, "partial" reflection or transpose. As an application of our scheme, we study the Z_{8} and Z_{16} classification of topological superconductors in one and three dimensions.
Quantum systems are known to exhibit several types of exotic behavior beyond classical physics, including quantum entanglement, which describes a web of nonlocal correlations among the constituents of the system. Despite numerous advances in understanding and quantifying the bipartite entanglement entropy of a pure state, where the whole system is described by a wave function, less is understood about quantum entanglement of a mixed state, where the system is described by a density matrix. The partial transpose of the density matrix, in which one takes the transpose only for a subsystem, and its corresponding entanglement measure -- called the (logarithmic) negativity -- have been introduced as an effective probe for the quantum entanglement in bosonic mixed states. In this work, the authors present a scheme to compute an analog of the entanglement negativity in the fermionic mixed states using a partial time-reversal transformation. Various examples are investigated. In particular, it is shown that the partial time reversal is an intrinsically fermionic construction, in that it can capture the formation of the edge Majorana fermions, while the partial transpose obtained from the bosonic partial transpose through the Jordan-Wigner transformation fails.
Abstract We study quantized non-local order parameters, constructed by using partial time-reversal and partial reflection, for fermionic topological phases of matter in one spatial dimension protected by an orientation reversing symmetry, using topological quantum field theories (TQFTs). By formulating the order parameters in the Hilbert space of state sum TQFT, we establish the connection between the quantized non-local order parameters and the underlying field theory, clarifying the nature of the order parameters as topological invariants. We also formulate several entanglement measures including the entanglement negativity on state sum spin TQFT, and describe the exact correspondence of the entanglement measures to path integrals on a closed surface equipped with a specific spin structure.
Condensed matter physics studies how many quantum particles, when brought together, conspire to give rise to collective behaviours that are far richer than those of the individual particles themselves. This is the physics of emergence: the idea that a large system can have properties that are not obvious from its microscopic building blocks alone. The work presented in this thesis explores how to characterise the properties of such many-particle states from three interconnected points of view: topology, quantum transport, and quantum information. The common idea behind these themes is that the properties of quantum matter may be accessed directly from quantum states themselves, without necessarily referring to an underlying Hamiltonian, circuit, or protocol used to generate them. Symmetry-protected topological phases of matter arise from the impossibility of continuously deforming certain quantum states into others whilst preserving their symmetries. In fermionic quantum matter, the phases resulting from such impossibility are those of topological insulators and superconductors. While having an insulating bulk, these materials possess metallic boundary states robust to perturbations preserving the protecting symmetries of their phase. Mathematically, the topological nature of these materials is characterised by the calculation of topological invariants, integer or half-integer numbers distinguishing between different topological phases. These invariants have been traditionally formulated for crystalline quantum matter, where one may use momentum space to perform mathematical calculations. This thesis focuses instead on characterising the topological phases of noncrystalline matter, which includes disorder and amorphicity. We develop a series of mathematical tools, known as local topological markers, which reformulate traditional topological invariants in real space and are therefore applicable even in the absence of crystallinity. Local topological markers are formulated in terms of the one-particle density matrix, a correlation function that only requires the knowledge of the quantum state in question to be calculated. Thus, the framework for topological characterisation we discuss, which also extends to certain higher-order topological phases, is purely state-based: it does not need to specify any parent Hamiltonian nor protocol used to prepare the quantum state to be characterised. Contrary to the classical picture of electricity where electrons behave as solid spheres drifting through a wire, the electronic transport through a nanostructure is shaped by quantum effects like tunnelling and interference. Quantum transport studies how these effects determine the conducting properties of mesoscopic samples---those that are neither microscopic nor fully macroscopic, but lie in an intermediate regime. In this thesis, we use quantum transport to study topological insulator nanowires, where the interplay between topology, deviations from crystallinity, and symmetry constraints shape electronic transmission. The characteristic feature of topological nanowires is the appearance of a perfectly transmitted mode when a certain magnetic flux is threaded through the wire's cross section. Varying the magnetic flux interpolates from perfect transmission to situations where electronic interference is enhanced, suppressing transmission. This is seen in the behaviour of conductance, which oscillates as a function of magnetic flux in what is known as Aharanov-Bohm oscillations. The mechanism originating the perfectly transmitted mode in a topological nanowire relies on time-reversal symmetry appearing at specific values of the magnetic flux. In amorphous nanowires, where time-reversal symmetry may be broken at any magnetic flux, the perfectly transmitted mode loses its protection, and the characteristic conductance oscillations of a topological nanowire may be lost. We study how the perfectly transmitted mode may be protected even when time-reversal symmetry is broken, and relate its loss to a topological phase transition driven by amorphicity. Entanglement is a distinctive feature of condensed matter systems: the possibility that quantum particles may be correlated in a way that has no classical analogue. Although inherently quantum-mechanical, entanglement alone is not a sufficient ingredient for quantum computation to outperform classical computers. The reason is that there exists a special class of quantum states, stabiliser states, that may be highly entangled whilst admitting an efficient classical representation---calculations involving stabiliser states are tractable in modern-day computers. The deviation of a quantum state from the set of stabiliser states goes by the name of quantum nonstabiliserness or magic, and, in the same way as entanglement, it is not a guarantee of quantum complexity on its own. In this thesis, we develop a method to characterise quantum states that combines entanglement and nonstabiliserness. To do so, we use the framework of the information lattice, whichsystematically decomposes the correlations in a quantum state in terms of their scale---the spatial extent of correlations. Through the information lattice, we distinguish different contributions to the nonstabiliserness content of a quantum state, which may have either a single-particle origin or come from intrinsic many-particle correlations. This framework, which is purely state-based, provides a first step to a fully scale-resolved description of nonstabiliserness and contributes to the characterisation of quantum states in terms of their complexity.
Entanglement of mixed quantum states can be quantified using the partial transpose and its corresponding entanglement measure, the logarithmic negativity. Recently, the notion of partial transpose has been extended to systems of anyons, which are exotic quasiparticles whose exchange statistics go beyond the bosonic and fermionic cases. Studying the fundamental properties of this anyonic partial transpose, we first reveal that when applied to the special case of fermionic systems, it can be reduced to the fermionic partial transpose or its twisted variant depending on whether or not a boundary Majorana fermion is present. Focusing on ground state properties, we find that the anyonic partial transpose captures both the correct entanglement scaling for gapless systems, as predicted by conformal field theory, and the phase transition between a topologically trivial and a nontrivial phase. For non-Abelian anyons and the bipartition geometry, we find a rich multiplet structure in the eigenvalues of the partial transpose, the so-called negativity spectrum, and reveal the possibility of defining both a charge- and an imbalance-resolved negativity.