Topological Boundary Conditions Research Articles

Effects of topological boundary conditions on Bell nonlocality

Effects of topological boundary conditions on Bell nonlocality

On the symmetry TFT of Yang-Mills-Chern-Simons theory

Three-dimensional Yang-Mills-Chern-Simons theory has the peculiar property that its one-form symmetry defects have nontrivial braiding, namely they are charged under the same symmetry they generate, which is then anomalous. This poses a few puzzles in describing the corresponding Symmetry TFT in a four-dimensional bulk. First, the braiding between lines at the boundary seems to be ill-defined when such lines are pulled into the bulk. Second, the Symmetry TFT appears to be too trivial to allow for topological boundary conditions encoding all the different global variants. We show that both of these puzzles can be solved by including endable (tubular) surfaces in the class of bulk topological operators one has to consider. In this way, we are able to reproduce all global variants of the theory, with their symmetries and their anomalies. We check the validity of our proposal also against a top-down holographic realization of the same class of theories.

Real-time vacuum plume flow field reconstruction during lunar landings based on deep learning

In space missions, the vacuum plume generated by rocket engines can negatively impact spacecraft. Therefore, researching the vacuum plume is crucial to guarantee the regular operation of spacecraft. The conventional numerical simulation methodology, the direct simulation Monte Carlo (DSMC) method, is time-consuming and lacks real-time calculation capabilities. Recently, deep learning (DL) methods have emerged in the field of fluid dynamics. In this study, a DL model trained by a convolutional neural network with multiple decoders is introduced to predict the vacuum plume flow field during lunar landings. The network processes shape topology information and boundary conditions as inputs, yielding flow field data including velocity and pressure fields as outputs. Meanwhile, the flow field prediction results under different conditions and training methods are discussed. The results show that the predicted flow field under different lunar surface conditions is in accord with the DSMC results. The maximum mean and standard deviation errors of the data distribution of each flow field do not exceed 9.72% and 9.07%, respectively. Different training methods with flat and inclined lunar surfaces also have an impact on the prediction results. Compared with the DSMC method, the DL method exhibits higher efficiency with a speedup of about four orders of magnitude, indicating that the DL-based flow field reconstruction method has strong application prospects in the real-time computation of vacuum plume flow fields.

Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologies E 1–E 6, E 11, E 12, E 16, and E 18 that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys.

On topology of the moduli space of gapped Hamiltonians for topological phases

The moduli space of gapped Hamiltonians that are in the same topological phase is an intrinsic object that is associated with the topological order. The topology of these moduli spaces has been used recently in the construction of Floquet codes. We propose a systematical program to study the topology of these moduli spaces. In particular, we use effective field theory to study the cohomology classes of these spaces, which includes and generalizes the Berry phase. We discuss several applications for studying phase transitions. We show that a nontrivial family of gapped systems with the same topological order can protect isolated phase transitions in the phase diagram, and we argue that the phase transitions are characterized by screening of topological defects. We argue that the family of gapped systems obeys bulk-boundary correspondence. We show that a family of gapped systems in the bulk with the same topological order can rule out a family of gapped systems on the boundary with the topological order given by the topological boundary condition, constraining phase transitions on the boundary.

Higher central charges and topological boundaries in 2+1-dimensional TQFTs

A 2+1-dimensional topological quantum field theory (TQFT) may or may not admit topological (gapped) boundary conditions. A famous necessary, but not sufficient, condition for the existence of a topological boundary condition is that the chiral central charge c_-c− has to vanish. In this paper, we consider conditions associated with ``higher" central charges, which have been introduced recently in the math literature. In terms of these new obstructions, we identify necessary and sufficient conditions for the existence of a topological boundary in the case of bosonic, Abelian TQFTs, providing an alternative to the identification of a Lagrangian subgroup. Our proof relies on general aspects of gauging generalized global symmetries. For non-Abelian TQFTs, we give a geometric way of studying topological boundary conditions, and explain certain necessary conditions given again in terms of the higher central charges. Along the way, we find a curious duality in the partition functions of Abelian TQFTs, which begs for an explanation via the 3d-3d correspondence.

Experimental analysis and transient numerical simulation of a large diameter pulsating heat pipe in microgravity conditions

A multi-parametric transient numerical simulation of the start-up of a large diameter Pulsating Heat Pipe (PHP) specially designed for future experiments on the International Space Station (ISS) are compared to the results obtained during a parabolic flight campaign supported by the European Space Agency. Since the channel diameter is larger than the capillary limit in normal gravity, such a device behaves as a loop thermosyphon on ground and as a PHP in weightless conditions; therefore, the microgravity environment is mandatory for pulsating mode. Because of a short duration of microgravity during a parabolic flight, the data concerns only the transient start-up behavior of the device. One of the most comprehensive models in the literature, namely the in-house 1-D transient code CASCO (French acronym for Code Avancé de Simulation du Caloduc Oscillant: Advanced PHP Simulation Code in English), has been configured in terms of geometry, topology, material properties and thermal boundary conditions to model the experimental device. The comparison between numerical and experimental results is performed simultaneously on the temporal evolution of multiple parameters: tube wall temperature, pressure and, wherever possible, velocity of liquid plugs, their length and temperature distribution within them. The simulation results agree with the experiment for different input powers. Temperatures are predicted with a maximum deviation of 7%. Pressure variation trend is qualitatively captured as well as the liquid plug velocity, length and temperature distribution. The model also shows the ability of capturing the instant when the fluid pressure begins to oscillate after the heat load is supplied, which is a fundamental information for the correct design of the engineering model that will be tested on the ISS. We also reveal the existence of strong liquid temperature gradients near the ends of liquid plugs both experimentally and by simulation. Finally, a theoretical prediction of the stable functioning of a large diameter PHP in microgravity is given. Results show that the system provided with an input power of 185W should be able to reach the steady state after 1min and maintain a stable operation from then on.

Prediction of Toric Code Topological Order from Rydberg Blockade

The physical realization of Z2 topological order as encountered in the paradigmatic toric code has proven to be an elusive goal. We predict that this phase of matter can be realized in a two-dimensional array of Rydberg atoms placed on the ruby lattice, at specific values of the Rydberg blockade radius. First, we show that the blockade model—also known as a “PXP” model—realizes a monomer-dimer model on the kagome lattice with a single-site kinetic term. This model can be interpreted as a Z2 gauge theory whose dynamics is generated by monomer fluctuations. We obtain its phase diagram using the numerical density matrix renormalization group method and find a topological quantum liquid (TQL) as evidenced by multiple measures including (i) a continuous transition between two featureless phases, (ii) a topological entanglement entropy of ln2 as measured in various geometries, (iii) degenerate topological ground states, and (iv) the expected modular matrix from ground state overlap. Next, we show that the TQL persists upon including realistic, algebraically decaying van der Waals interactions V(r)∼1/r6 for a choice of lattice parameters. Moreover, we can directly access topological loop operators, including the Fredenhagen-Marcu order parameter. We show how these can be measured experimentally using a dynamic protocol, providing a “smoking gun” experimental signature of the TQL phase. Finally, we show how to trap an emergent anyon and realize different topological boundary conditions, and we discuss the implications for exploring fault-tolerant quantum memories.12 MoreReceived 24 November 2020Accepted 19 May 2021DOI:https://doi.org/10.1103/PhysRevX.11.031005Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasTopological phases of matterTopological quantum computingPhysical SystemsRydberg atoms & moleculesCondensed Matter, Materials & Applied Physics

Interfaces and the extended Hilbert space of Chern-Simons theory

The low energy effective field theories of (2 + 1) dimensional topological phases of matter provide powerful avenues for investigating entanglement in their ground states. In [1] the entanglement between distinct Abelian topological phases was investigated through Abelian Chern-Simons theories equipped with a set of topological boundary conditions (TBCs). In the present paper we extend the notion of a TBC to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases. These boundary conditions furnish a defining relation for the extended Hilbert space of the quantum theory and allow the calculation of entanglement directly in the gauge theory. Because we allow for trivial interfaces, this includes a generic construction of the extended Hilbert space in any (compact) Chern-Simons theory quantized on a Riemann surface. Additionally, this provides a constructive and principled definition for the Hilbert space of effective ground states of gapped phases of matter glued along gapped interfaces. Lastly, we describe a generalized notion of surgery, adding a powerful tool from topological field theory to the gapped interface toolbox.

Exact solution of a topological spin ring with an impurity**Project supported by the National Natural Science Foundation of China (Grant No. 11664001).

The spin-1/2 Heisenberg chain coupled to a spin-S impurity moment with anti-periodic boundary condition is studied via the off-diagonal Bethe ansatz method. The twisted boundary breaks the U(1) symmetry of the system, which leads to that the spin ring with impurity can not be solved by the conventional Bethe ansatz methods. By combining the properties of the R-matrix, the transfer matrix, and the quantum determinant, we derive the T–Q relation and the corresponding Bethe ansatz equations. The residual magnetizations of the ground states and the impurity specific heat are investigated. It is found that the residual magnetizations in this model strongly depend on the constraint of the topological boundary condition, the inhomogeneity of the impurity comparing with the hosts could depress the impurity specific heat in the thermodynamic limit. This method can be expand to other integrable impurity models without U(1) symmetry.

Improving the nonlocal advantage of quantum coherence in the Heisenberg model by topological boundary conditions

The control of quantum correlations in a physical system is of practical importance. Nonlocal advantage of quantum coherence (NAQC) is a type of quantum correlation defined on the basis of quantum coherence. We present a comparative investigation of NAQC in the three-spin Heisenberg model with periodic boundary conditions (PBCs) and topological boundary conditions (TBCs). The results show that while there is no NAQC in the absence of an external magnetic field, one can create a considerable NAQC by applying a nonuniform transverse magnetic field to the three spins. In particular, the NAQC is significantly improved for the model with TBCs compared to that with PBCs, and the regions of nonvanishing NAQC can also be extended. The model considered with TBC can therefore serve as a potential candidate for creating NAQC.

Transient basin as indicator of tectonic expressions in bedrock landscape: Approach based on MATLAB geomorphic tool (Transient-profiler)

The tectonic and erosional processes sculpture the Earth's surface, producing a landscape that preserves the evolutionary history of rock uplift and incision. The bedrock channels act as an efficient gradation agent where the river profile adjusts to changing tectonic, climatic, or topological (e.g., river capture) boundary conditions, and develops a transient basin. A suite of slope-break knickpoints at comparable elevations in the channel profiles of the transient basin act as important markers. We developed a set of comprehensive and interactive MATLAB tools, called a Transient-profiler, to extract the geomorphic indices from the digital elevation model for a basin characterization and landscape evolution period. The paleo-channel reconstruction from the relict reach in the three studied transient drainage basins (Palar, Umiam, and Dibang) suggest a varying surface uplift from 514, 1246, and 1981 m with a respective relief gain of 158%, 230%, and 95%, respectively. We found that the horizontal knickpoint retreat rate is faster in the Himalayan thrust system as compared to the Shillong pop-up system, whereas the vertical knickpoint retreat rate is faster in the latter.

Elastic colloidal monopoles and reconfigurable self-assembly in liquid crystals.

Monopole-like electrostatic interactions are ubiquitous in biology1 and condensed matter2-4, but they are often screened by counter-ions and cannot be switched from attractive to repulsive. In colloidal science, where the main goal is to develop colloidal particles2,3 that mimic and exceed the diversity and length scales of atomic and molecular assembly, electrostatically charged particles cannot change the sign of their surface charge or transform from monopoles to higher-order multipoles4. In liquid-crystal colloids5-7, elastic interactions between particles arise to minimize the free energy associated with elastic distortions in the long-range alignment of rod-like molecules around the particles5. In dipolar6,8, quadrupolar8-12 and hexadecapolar13 nematic colloids, the symmetries of such elastic distortions mimic both electrostatic multipoles14 and the outermost occupied electron shells of atoms7,15,16. Electric and magnetic switching17,18, spontaneous transformations19 and optical control20 of elastic multipoles, as well as their interactions with topological defects and surface boundary conditions, have been demonstrated in such colloids21-23. However, it has long been understood5,24 that elastic monopoles should relax to uniform or higher-order multipole states because of the elastic torques that they induce5,7. Here we develop nematic colloids with strong elastic monopole moments and with elastic torques balanced by the optical torques induced by ambient light. We demonstrate the monopole-to-quadrupole reconfiguration of these colloidal particles by unstructured light, which resembles the driving of atoms between the ground state and various excited states. We show that the sign of the elastic monopoles can be switched, and that like-charged monopoles attract whereas oppositely charged ones repel, unlike in electrostatics14. We also demonstrate the out-of-equilibrium dynamic assembly of these colloidal particles. This diverse and surprising behaviour is explained using a model that considers the balance of the optical and elastic torques that are responsible for the excited-state elastic monopoles and may lead to light-powered active-matter systems and self-assembled nanomachines.

Protecting quantum correlations of the XXZ model by topological boundary conditions

The differences between the XXZ model with topological and periodical boundary conditions were compared by studying their entanglement, quantum discord, and critical temperature above which the entanglement vanishes. It shows that the different boundary conditions mainly affect bipartite quantum correlations of the boundary spins rather than that of other spin pairs. The topological boundary spins can protect entanglement and discord against strong magnetic fields while the periodical boundary spins can protect them against nonuniform magnetic fields. Compared with the periodical XXZ model, the critical temperature is significantly improved for the topological XXZ model. The topological XXZ model also allows us to improve significantly its critical temperature by increasing the strength of magnetic field, which is not feasible for the periodical XXZ model. It is therefore more promising for preparing entangled states at high temperature in the topological XXZ model.

Interface contributions to topological entanglement in abelian Chern-Simons theory

We study the entanglement entropy between (possibly distinct) topological phases across an interface using an Abelian Chern-Simons description with topological boundary conditions (TBCs) at the interface. From a microscopic point of view, these TBCs correspond to turning on particular gapping interactions between the edge modes across the interface. However, in studying entanglement in the continuum Chern-Simons description, we must confront the problem of non-factorization of the Hilbert space, which is a standard property of gauge theories. We carefully define the entanglement entropy by using an extended Hilbert space construction directly in the continuum theory. We show how a given TBC isolates a corresponding gauge invariant state in the extended Hilbert space, and hence compute the resulting entanglement entropy. We find that the sub-leading correction to the area law remains universal, but depends on the choice of topological boundary conditions. This agrees with the microscopic calculation of [1]. Additionally, we provide a replica path integral calculation for the entropy. In the case when the topological phases across the interface are taken to be identical, our construction gives a novel explanation of the equivalence between the left-right entanglement of (1+1)d Ishibashi states and the spatial entanglement of (2+1)d topological phases.

Predictions of first passage times in sparse discrete fracture networks using graph-based reductions.

We present a graph-based methodology to reduce the computational cost of obtaining first passage times through sparse fracture networks. We derive graph representations of generic three-dimensional discrete fracture networks (DFNs) using the DFN topology and flow boundary conditions. Subgraphs corresponding to the union of the k shortest paths between the inflow and outflow boundaries are identified and transport on their equivalent subnetworks is compared to transport through the full network. The number of paths included in the subgraphs is based on the scaling behavior of the number of edges in the graph with the number of shortest paths. First passage times through the subnetworks are in good agreement with those obtained in the full network, both for individual realizations and in distribution. Accurate estimates of first passage times are obtained with an order of magnitude reduction of CPU time and mesh size using the proposed method.

Unoriented 3d TFTs

This paper generalizes two facts about oriented 3d TFTs to the unoriented case. On one hand, it is known that oriented 3d TFTs having a topological boundary condition admit a state-sum construction known as the Turaev-Viro construction. This is related to the string-net construction of fermionic phases of matter. We show how Turaev-Viro construction can be generalized to unoriented 3d TFTs. On the other hand, it is known that the “fermionic” versions of oriented TFTs, known as Spin-TFTs, can be constructed in terms of “shadow” TFTs which are ordinary oriented TFTs with an anomalous ℤ2 1-form symmetry. We generalize this correspondence to Pin+-TFTs by showing that they can be constructed in terms of ordinary unoriented TFTs with anomalous ℤ2 1-form symmetry having a mixed anomaly with time-reversal symmetry. The corresponding Pin+-TFT does not have any anomaly for time-reversal symmetry however and hence it can be unambiguously defined on a non-orientable manifold. In case a Pin+-TFT admits a topological boundary condition, one can combine the above two statements to obtain a Turaev-Viro-like construction of Pin+-TFTs. As an application of these ideas, we construct a large class of Pin+-SPT phases.

Off-Diagonal Bethe Ansatz and Exact Solution of a Topological Spin Ring

A general method is proposed for constructing the Bethe ansatz equations of integrable models without U(1) symmetry. As an example, the exact spectrum of the XXZ spin ring with a Möbius-like topological boundary condition is derived by constructing a modified T-Q relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodromy matrix. With the exact solution, the elementary excitations of the topological XX spin ring are discussed in detail. It is found that the excitation spectrum indeed shows a nontrivial topological nature.

Bicategories for Boundary Conditions and for Surface Defects in 3-d TFT

We analyze topological boundary conditions and topological surface defects in three-dimensional topological field theories of Reshetikhin-Turaev type based on arbitrary modular tensor categories. Boundary conditions are described by central functors that lift to trivializations in the Witt group of modular tensor categories. The bicategory of boundary conditions can be described through the bicategory of module categories over any such trivialization. A similar description is obtained for topological surface defects. Using string diagrams for bicategories we also establish a precise relation between special symmetric Frobenius algebras and Wilson lines involving special defects. We compare our results with previous work of Kapustin-Saulina and of Kitaev-Kong on boundary conditions and surface defects in abelian Chern-Simons theories and in Turaev-Viro type TFTs, respectively.

Surface states of topological insulators

We develop an effective bulk model with a topological boundary condition to study the surface states of topological insulators. We find that the Dirac point energy, the band curvature and the spin texture of surface states are crystal face-dependent. For a given face on a sphere, the Dirac point energy is determined by the bulk physics that breaks p-h symmetry in the surface normal direction and is tunable by surface potentials that preserve T symmetry. Constant energy contours near the Dirac point are ellipses with spin textures that are helical on the S/N pole, collapsed to one dimension on any side face, and tilted out-of-plane otherwise. Our findings identify a route to engineering the Dirac point physics on the surfaces of real materials.