We introduce the concept of fermionic matrix product operators, and show that they provide a natural representation of fermionic fusion tensor categories. This allows for the classification of two dimensional fermionic topological phases in terms of matrix product operator algebras. Using this approach we give a classification of fermionic symmetry protected topological phases with respect to a group $G$ in terms of three cohomology groups: $H^1(G,\mathbb{Z}_2)$, describing which matrix product operators are of Majorana type, $H^2(G,\mathbb{Z}_2)$, describing the fermionic nature of the fusion tensors that arise when two matrix product operators are multiplied, and the supercohomolgy group $\bar{H}^3(G,U(1))$ which corresponds to the associator that changes the order of fusion. We also generalize the tensor network description of the string-net ground states to the fermionic setting, yielding simple representations of a class that includes the fermionic toric code.
We study fermionic matrix product operator algebras and identify the associated algebraic data. Using this algebraic data we construct fermionic tensor network states in two dimensions that have non-trivial symmetry-protected or intrinsic topological order. The tensor network states allow us to relate physical properties of the topological phases to the underlying algebraic data. We illustrate this by calculating defect properties and modular matrices of supercohomology phases. Our formalism also captures Majorana defects as we show explicitly for a class of $\mathbb{Z}_2$ symmetry-protected a…
We construct a topological invariant that classifies density matrices of symmetry-protected topological orders in two-dimensional fermionic systems. As it is constructed out of the previously introduced Uhlmann phase, we refer to it as the topological Uhlmann number ${\rm n}_{\rm U}$. With it, we study thermal topological phases in several two-dimensional models of topological insulators and superconductors, computing phase diagrams where the temperature $T$ is on an equal footing with the coupling constants in the Hamiltonian. Moreover, we find novel thermal-topological transitions between tw…
We study a chain of ferromagnetic nano-particles or ferromagnetic molecule/atoms on a substrate of fully gapped superconductors. We find that under quite realistic conditions, the fermion-number-parity symmetry $Z_2^f$ can spontaneously break. In other words, such a chain can realize a 1+1D fermionic topologically ordered state and the corresponding two-fold topological degeneracy on an open chain. Such a topological degeneracy becomes the so called Majorana zero mode in the non-interacting limit. More specifically, we find that $Z_2^f$ symmetry breaking or fermionic 1+1D topological order can…
We study the effect of interactions on 2D fermionic symmetry-protected topological (SPT) phases using the recently proposed braiding statistics approach. We focus on a simple class of examples: superconductors with a Z2 Ising symmetry. Although these systems are classified by Z in the noninteracting limit, our results suggest that the classification collapses to Z8 in the presence of interactions -- consistent with previous work that analyzed the stability of the edge. Specifically, we show that there are at least 8 different types of Ising superconductors that cannot be adiabatically connecte…
The Z_2 topological order in Z_2 spin liquid and in exactly soluble Kitaev toric code model is the simplest topological order for 2+1D bosonic systems. More general 2+1D bosonic topologically ordered states can be constructed via exact soluble string-net models. However, the most important topologically ordered phases of matter are arguably the fermionic fractional quantum Hall states. Topological phases of matter for fermion systems are strictly richer than their bosonic counterparts because locality has different meanings for the two kinds of systems. In this paper, we describe a simple ferm…
We present a fully many-body formulation of topological invariants for various topological phases of fermions protected by antiunitary symmetry, which does not refer to single particle wave functions. For example, we construct the many-body $\mathbb{Z}_2$ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele $\mathbb{Z}_2$ invariant written in terms of single-particle Bloch wave functions. We show that an important ingredient for the construction of the many-body topological invariants is a fermioni…
Tensor network states, and in particular projected entangled pair states, play an important role in the description of strongly correlated quantum lattice systems. They do not only serve as variational states in numerical simulation methods, but also provide a framework for classifying phases of quantum matter and capture notions of topological order in a stringent and rigorous language. The rapid development in this field for spin models and bosonic systems has not yet been mirrored by an analogous development for fermionic models. In this work, we introduce a framework of tensor networks hav…
Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料国际中心）；C. N. Yang Institute for Theoretical Physics, Stony Brook University（石溪大学陈省身理论物理研究所）
There can exist topological obstructions to continuously deforming a gapped Hamiltonian for free fermions into a trivial form without closing the gap. These topological obstructions are closely related to obstructions to the existence of exponentially localized Wannier functions. We show that by taking two copies of a gapped, free fermionic system with complex conjugate Hamiltonians, it is always possible to overcome these obstructions. This allows us to write the ground state in matrix product form using Grassman-valued bond variables, and show insensitivity of the ground state density matrix…
We analyse in detail the effect of non-trivial band topology on the area law behaviour of the entanglement entropy in Kitaev's honeycomb model. By mapping the translationally invariant 2D spin model into 1D fermionic subsystems, we identify those subsystems responsible for universal entanglement contributions in the gapped phases and those responsible for critical entanglement scaling in the gapless phases. For the gapped phases we analytically show how the topological edge states contribute to the entanglement entropy and provide a universal lower bound for it. For the gapless semi-metallic p…
We apply the charge pumping argument to fermionic tensor network representations of d-dimensional topological insulators (TIs) to obtain tensor network states for (d+1)-dimensional TIs. We exemplify the method by constructing a two-dimensional projected entangled pair states (PEPS)for a Chern insulator starting from a matrix product state (MPS) in d=1 describing pumping in the Su-Schrieffer-Heeger (SSH) model. In extending the argument to second-order TIs, we build a three-dimensional TNS for a chiral hinge TI from a PEPS in d=2 for the obstructed atomic insulator (OAI) of the quadrupole model…
Topological phases of matter possess intricate correlation patterns typically probed by entanglement entropies or entanglement spectra. In this work, we propose an alternative approach to assessing topologically induced edge states in free and interacting fermionic systems. We do so by focussing on the fermionic covariance matrix. This matrix is often tractable either analytically or numerically and it precisely captures the relevant correlations of the system. By invoking the concept of monogamy of entanglement we show that highly entangled states supported across a system bi-partition are la…
We develop the formalism of fermionic matrix product states (fMPS) and show how irreducible fMPS fall in two different classes, related to the different types of simple $\mathbb{Z}_2$ graded algebras, which are physically distinguished by the absence or presence of Majorana edge modes. The local structure of fMPS with Majorana edge modes also implies that there is always a two-fold degeneracy in the entanglement spectrum. Using the fMPS formalism we make explicit the correspondence between the $\mathbb{Z}_8$ classification of time-reversal invariant spinless superconductors and the modulo 8 pe…
The string-net approach by Levin and Wen, and the local unitary transformation approach by Chen, Gu, and Wen, provide ways to classify topological orders with gappable edge in 2D bosonic systems. The two approaches reveal that the mathematical framework for 2+1D bosonic topological order with gappable edge is closely related to unitary fusion category theory. In this paper, we generalize these systematic descriptions of topological orders to 2D fermion systems. We find a classification of 2+1D fermionic topological orders with gappable edge in terms of the following set of data $(N^{ij}_k, F^{…
We construct exactly solved commuting projector Hamiltonian lattice models for all known 2+1d fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group $G_f = G \times \mathbb{Z}_2^f$, where $G$ is finite and $\mathbb{Z}_2^f$ is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1d. A natural ingredient in our construction is a discrete form of the spin structure of the 2d spatial surface $M$ on which our model is defined, namely a `Kasteleyn' orientati…
A topological measure characterizing symmetry-protected topological phases in one-dimensional open fermionic systems is proposed. It is built upon the kinematic approach to the geometric phase of mixed states and facilitates the extension of the notion of topological phases from zero-temperature to nonzero-temperature cases. In contrast to a previous finding that topological properties may not survive above a certain critical temperature, we find that topological properties of open systems, in the sense of the measure suggested here, can persist at any finite temperature and disappear only in …
Under an appropriate symmetric bulk bipartition in a one-dimensional symmetry protected topological phase with the Affleck-Kennedy-Lieb-Tasaki matrix product state wave function for the odd integer spin chains, a bulk critical entanglement spectrum can be obtained, describing the excitation spectrum of the critical point separating the topological phase from the trivial phase with the same symmetry. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry breaking phase transitions. Recently, the framework of matrix product states for topological phases with Ma…
Recently, there is a considerable study on gapped symmetric phases of bosons that do not break any symmetry. Even without symmetry breaking, the bosons can still be in many exotic new states of matter, such as symmetry-protected trivial (SPT) phases which are short-range entangled and symmetry-enriched topological (SET) phases which are long-range entangled. It is well-known that non-interacting fermionic topological insulators are SPT states protected by time-reversal symmetry and U(1) fermion number conservation symmetry. In this paper, we construct three-dimensional exotic phases of bosons …
We present a scheme to implement a Fermi-Hubbard-like model in ultracold atoms in optical lattices and analyze the topological features of its ground state. In particular, we show that the ground state for appropriate parameters has a large overlap with a lattice version of the bosonic Laughlin state at filling factor one half. The scheme utilizes laser assisted and normal tunnelling in a checkerboard optical lattice. The requirements on temperature, interactions, and hopping strengths are similar to those needed to observe the Néel antiferromagnetic ordering in the standard Fermi-Hubbard mode…
We investigate topological phase transitions driven by interaction and identify a novel topological Mott insulator state in one-dimensional fermionic optical superlattices through numerical density matrix renormalization group (DMRG) method. Remarkably, the low-energy edge excitations change from spin-1/2 fermionic single-particle modes to spin-1 bosonic collective modes across the phase transition. Due to spin-charge separation, the low-energy theory is governed by an effective spin superexchange model, whereas the charge degree of freedom is fully gapped out. Such topological Mott state can …
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory.…
Finite-temperature quantum topological order in three dimensions
Nanjing University（南京大学）；Yale University（耶鲁大学）；Stanford University（斯坦福大学）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；King’s College London（伦敦国王学院）；Perimeter Institute for Theoretical Physics（理论物理珀米特研究所）
The spin can be described in the star product formalism by extending the bosonic Moyal product in the fermionic sector. The fermionic star product is then the Clifford product of geometric algebra and it is possible to formulate the fermionic star product formalism in analogy to the bosonic star product formalism. For the fermionic star product description of spin, one can then establish the relation to other approaches that describe spin with fermionic variables, i.e. the operator formalism and the path integral formalism. It is shown that the fermionic star product formalism and the fermioni…
In this paper, we apply the K-theory scheme of classifying the topological insulators/superconductors to classify the topological classes of the massive multi-flavor fermions in anti-de Sitter (AdS) space. In the context of AdS/CFT correspondence, the multi-flavor fermionic mass matrix is dual to the pattern of operator mixing in the boundary conformal field theory (CFT). Thus, our results classify the possible patterns of operator mixings among fermionic operators in the holographic CFT.
Quantum theory sets a limit on the minimum time required to transform from an initial state to a target state. It is known as quantum speed limit time. quantum speed limit time can be used to determine the rate of quantum evolution for closed and open quantum systems. Given that in the real world we are dealing with open quantum systems, the study of quantum speed limit time for such systems has particular importance. In this work we consider the topological qubit realized by two Majorana modes. We consider the case in which the topological qubit is influenced by fermionic and bosonic environm…
We propose a topological Euler number to characterize nontrivial topological phases of gapped fermionic systems, which originates from the Gauss-Bonnet theorem on the Riemannian structure of Bloch states established by the real part of the quantum geometric tensor in momentum space. Meanwhile, the imaginary part of the geometric tensor corresponds to the Berry curvature which leads to the Chern number characterization. We discuss the topological numbers induced by the geometric tensor analytically in a general two-band model. As an example, we show that the zero-temperature phase diagram of a …
The topological Anderson and Mott insulators are two phases that have so far been separately and widely explored beyond topological band insulators. Here we combine the two seemingly different topological phases into a system of spin-1/2 interacting fermionic atoms in a disordered optical lattice. We find that the topological Anderson and Mott insulators in the noninteracting and clean limits can be adiabatically connected without gap closing in the phase diagram of our model. Lying between the two phases, we uncover a disordered correlated topological insulator, which is induced from a trivia…
We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in fermionic theories, which we call "m-type" and "q-type" particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have n…
Two-dimensional ($p_{x}+ip_{y}$) superfluids/superconductors offer a playground for studying intriguing physics such as quantum teleportation, non-Abelian statistics, and topological quantum computation. Creating such a superfluid in cold fermionic atom optical traps using p-wave Feshbach resonance is turning out to be challenging. Here we propose a method to create a $p_{x}+ip_{y}$ superfluid directly from an s-wave interaction making use of a topological Berry phase, which can be artificially generated. We discuss ways to detect the spontaneous Hall mass current, which acts as a diagnostic f…
Deformation quantization is a powerful tool for quantizing theories with bosonic and fermionic degrees of freedom. The star products involved generate the mathematical structures which have recently been used in attempts to analyze the algebraic properties of quantum field theory. In the context of quantum mechanics they provide a canonical quantization procedure for systems with either bosonic of fermionic degrees of freedom. We illustrate this procedure for a number a physical examples, including bosonic, fermionic and supersymmetric oscillators. We show how non-relativistic and relativistic…
Quantum nonlinear optics is a quickly growing field with large technological promise, at the same time involving complex and novel many-body phenomena. In the usual scenario, optical nonlinearities originate from the interactions between polaritons, which are hybrid quasi-particles mixing matter and light degrees of freedom. Here we introduce a type of polariton which is intrinsically nonlinear and emerges as the natural quasi-particle in presence quantum degenerate fermionic matter. It is a composite object made of a fermion trapped inside an optical soliton forming a topological defect in a …
We provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups $G_f$ and general values of the chiral central charge $c_-$. Here $G_f$ is a central extension of a bosonic symmetry group $G_b$ by fermion parity, $(-1)^F$, specified by a second cohomology class $[ω_2] \in \mathcal{H}^2(G_b, \mathbb{Z}_2)$. Our approach proceeds by gauging fermion parity and classifying the resulting $G_b$ symmetry-enriched topological orders while keeping track of certain additional data and constraints. We perfor…
We study multi-qubit quantum channels that can be represented as a product of one-mode fermionic attenuation channels. An explicit formula for the classical capacity $C_1$ and for the minimum output entropy $S_{min}$ of these channels is proposed. We compute $S_{min}$ analytically for any number of qubits under assumption that the minimum is achieved on a Gaussian input. Apart from that, a simple numerical method for evaluating $S_{min}$ is developed. The method is applicable to any channels that are sufficiently noisy. For fermionic product channels the proposed formula for $S_{min}$ agrees w…
We propose to use the recently predicted two-dimensional `weak-pairing' $p_x + ip_y$ superfluid state of fermionic cold atoms as a platform for topological quantum computation. In the core of a vortex, this state supports a zero-energy Majorana mode, which moves to finite energy in the corresponding topologically trivial `strong-pairing' state. By braiding vortices in the `weak-pairing' state, unitary quantum gates can be applied to the Hilbert space of Majorana zero-modes. For read-out of the topological qubits, we propose realistic schemes suitable for atomic superfluids.
Second order quantum phase transitions, with well-known features such as long-range entanglement, symmetry breaking, and gap closing, exhibit quantum enhancement for sensing at criticality. However, it is unclear which of these features are responsible for this enhancement. To address this issue, we investigate phase transitions in free-fermionic topological systems that exhibit neither symmetry-breaking nor long-range entanglement. We analytically demonstrate that quantum enhanced sensing is possible using topological edge states near the phase boundary. Remarkably, such enhancement also endu…
The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers. Motivated by long standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here, we establish the existence of an intrin…
We discuss the topological phase transition of the spin-$\frac{1}{2}$ fermionic Haldane model with repulsive on-site interaction. We show that the Berry curvature of the topological Hamiltonian, the first Chern number, and the topological phase transition point can be extracted from the single-particle density matrix for this interacting system. Furthermore, we design a tomography scheme for the single-particle density matrix of interacting fermionic two-band models in experimental realizations with cold atoms in optical lattices.
We investigate the properties of the Lieb lattice, i.e a face-centered square lattice, subjected to external gauge fields. We show that an Abelian gauge field leads to a peculiar quantum Hall effect, which is a consequence of the single Dirac cone and the flat band characterizing the energy spectrum. Then we explore the effects of an intrinsic spin-orbit term - a non-Abelian gauge field - and demonstrate the occurrence of the quantum spin Hall effect in this model. Besides, we obtain the relativistic Hamiltonian describing the Lieb lattice at low energy and derive the Landau levels in the pres…
In this paper we study entropy production and transport of heat and matter in an 1D bipartite fermionic chain. Moreover, we input a phenomenological dephasing noise in order to show its effect on the dynamics of the system. Specially, it is shown the importance of such noise in the emergence of the entropy production. For the fluxes, analytical solutions are obtained using both Boltzmann and Fermi-Dirac distribuition. Finally, we show that a particular fluctuation theorem for energy and matter exchange is obeyed even in the presence of such noise.
We study matrix product unitary operators (MPUs) for fermionic one-dimensional (1D) chains. In stark contrast with the case of 1D qudit systems, we show that (i) fermionic MPUs do not necessarily feature a strict causal cone and (ii) not all fermionic Quantum Cellular Automata (QCA) can be represented as fermionic MPUs. We then introduce a natural generalization of the latter, obtained by allowing for an additional operator acting on their auxiliary space. We characterize a family of such generalized MPUs that are locality-preserving, and show that, up to appending inert ancillary fermionic de…
We study the error correcting properties of Majorana Surface Codes (MSC), topological quantum codes constructed out of interacting Majorana fermions, which can be used to store quantum information and perform quantum computation. These quantum memories suffer from purely "fermionic" errors, such as quasiparticle poisoning (QP), that have no analog in conventional platforms with bosonic qubits. In physical realizations where QP dominates, we show that errors can be corrected provided that the poisoning rate is below a threshold of $\sim11\%$. When QP is highly suppressed and fermionic bilinear …
The dynamics of quantum field theories on bounded domains requires the introduction of boundary conditions on the quantum fields. We address the problem from a very general perspective by using charge conservation as a fundamental principle for scalar and fermionic quantum field theories. Unitarity arises as a consequence of the choice of charge preserving boundary conditions. This provides a powerful framework for the analysis of global geometrical and topological properties of the space of physical boundary conditions. Boundary conditions which allow the existence of edge states can only ari…
