Corner Contributions Research Articles

Entanglement Entropy and Deconfined Criticality: Emergent SO(5) Symmetry and Proper Lattice Bipartition.

We study the Rényi entanglement entropy (EE) of the two-dimensional J-Q model, the emblematic quantum spin model of deconfined criticality at the phase transition between antiferromagnetic and valence-bond-solid ground states. State-of-the-art quantum MonteCarlo calculations of the EE reveal critical corner contributions that scale logarithmically with the system size, with a coefficient in remarkable agreement with the form expected from a large-N conformal field theory with SO(N=5) symmetry. However, details of the bipartition of the lattice are crucial in order to observe this behavior. If the subsystem for the reduced density matrix does not properly accommodate valence-bond fluctuations, logarithmic contributions appear even for cornerless bipartitions. We here use a 45° tilted cut on the square lattice. Beyond supporting an SO(5) deconfined quantum critical point, our results for both the regular and tilted cuts demonstrate important microscopic aspects of the EE that are not captured by conformal field theory.

Universal Features of Entanglement Entropy in the Honeycomb Hubbard Model.

The entanglement entropy is a unique probe to reveal universal features of strongly interacting many-body systems. In two or more dimensions these features are subtle, and detecting them numerically requires extreme precision, a notoriously difficult task. This is especially challenging in models of interacting fermions, where many such universal features have yet to be observed. In this Letter we tackle this challenge by introducing a new method to compute the Rényi entanglement entropy in auxiliary-field quantum MonteCarlo simulations, where we treat the entangling region itself as a stochastic variable. We demonstrate the efficiency of this method by extracting, for the first time, universal subleading logarithmic terms in a two-dimensional model of interacting fermions, focusing on the half-filled honeycomb Hubbard model at T=0. We detect the universal corner contribution due to gapless fermions throughout the Dirac semi-metal phase and at the Gross-Neveu-Yukawa critical point, where the latter shows a pronounced enhancement depending on the type of entangling cut. Finally, we observe the universal Goldstone mode contribution in the antiferromagnetic Mott insulating phase.

C-functions in flows across dimensions

We explore the notion of c-functions in renormalization group flows between theories in different spacetime dimensions. We discuss functions connecting central charges of the UV and IR fixed point theories on the one hand, and functions which are monotonic along the flow on the other. First, using the geometric properties of the holographic dual RG flows across dimensions and the constraints from the null energy condition, we construct a monotonic holographic c-function and thereby establish a holographic c-theorem across dimensions. Second, we use entanglement entropies for two different types of entangling regions in a field theory along the RG flow across dimensions to construct candidate c-functions which satisfy one of the two criteria but not both. In due process we also discuss an interesting connection between corner contributions to the entanglement entropy and the topology of the compact internal space. As concrete examples for both approaches, we holographically study twisted compactifications of 4d mathcal{N} = 4 SYM and compactifications of 6d mathcal{N} = (2, 0) theories.

Reparametrization dependence and holographic complexity of black holes

We refine the calculation of holographic complexity of black holes in the complexity equals action approach by applying the recently introduced criterion that the action of any causal diamond in static vacuum regions must vanish identically. This criterion fixes empty anti--de Sitter (AdS) spacetime as the reference state with vanishing complexity and renders holographic complexity explicitly finite in all the cases we consider. The cases considered here include the Reissner-Nordstr\"om-AdS black hole, the rotating BTZ black hole, the Kerr-AdS black hole, and AdS-Vaidya spacetime. The criterion is equivalent to imposing that the corner contributions vanish. Contrary to earlier results, we find that the generalized Lloyd bound always holds in the Reissner-Nordstr\"om-AdS and BTZ cases.

Universal signatures of Dirac fermions in entanglement and charge fluctuations

We investigate the entanglement entropy (EE) and charge fluctuations in models where the low energy physics is governed by massless Dirac fermions. We focus on the response to flux insertion which, for the EE, is widely assumed to be universal, \emph{i.e.}, independent of the microscopic details. We provide an analytical derivation of the EE and charge fluctuations for the seminal example of graphene, using the dimensional reduction of its tight-binding model to the one-dimensional Su-Schrieffer-Heeger model. Our asymptotic expression for the EE matches the conformal field theory prediction. We show that the charge variance has the same asymptotic behavior, up to a constant prefactor. To check the validity of universality arguments, we numerically consider several models, with different geometries and number of Dirac cones, and either for strictly two-dimensional models or for gapless surface mode of three-dimensional topological insulators. We also show that the flux response does not depend on the entangling surface geometry as long as it encloses the flux. Finally we consider the universal corner contributions to the EE. We show that in the presence of corners, the Kitaev-Preskill subtraction scheme provides non-universal, geometry dependent results.

Geometric formulation of the covariant phase space methods with boundaries

We analyze in full-detail the geometric structure of the covariant phase space (CPS) of any local field theory defined over a space-time with boundary. To this end, we introduce a new frame: the "relative bicomplex framework". It is the result of merging an extended version of the "relative framework" (initially developed in the context of algebraic topology by R.~Bott and L.W.~Tu in the 1980s to deal with boundaries) and the variational bicomplex framework (the differential geometric arena for the variational calculus). The relative bicomplex framework is the natural one to deal with field theories with boundary contributions, including corner contributions. In fact, we prove a formal equivalence between the relative version of a theory with boundary and the non-relative version of the same theory with no boundary. With these tools at hand, we endow the space of solutions of the theory with a (pre)symplectic structure canonically associated with the action and which, in general, has boundary contributions. We also study the symmetries of the theory and construct, for a large group of them, their Noether currents, and charges. Moreover, we completely characterize the arbitrariness (or lack thereof for fiber bundles with contractible fibers) of these constructions. This clarifies many misconceptions about the role of the boundary terms in the CPS description of a field theory. Finally, we provide what we call the CPS-algorithm to construct the aforementioned (pre)symplectic structure and apply it to some relevant examples.

On the corner contributions to the heat coefficients of geodesic polygons

Soit 𝒪 une orbisurface riemannienne compacte. Nous calculons des formules pour la contribution des singularités coniques de 𝒪 au coefficient de t 2 du développement asymptotique de la trace du noyau de la chaleur de 𝒪, les contributions de t 0 et t 1 étant connues. Comme application, nous calculons le coefficient de t 2 de la contribution d’un angle intérieur de la forme γ=π/k dans un polygone géodésique sur une surface au développement asymptotique du noyau de la chaleur de Dirichlet du polygone, sous une hypothèse locale de symétrie près du sommet correspondant. La principale nouveauté ici est la détermination de la façon dont le Laplacien de la courbure de Gauss au sommet en question entre dans le coefficient de t 2 . Nous terminons par une conjecture concernant la contribution analogue d’un angle γ arbitraire dans un polygone géodésique.

Thermodynamics of a higher-order topological insulator

We investigate the order of the topological quantum phase transition in a two dimensional quadrupolar topological insulator within a thermodynamic approach. Using numerical methods, we separate the bulk, edge and corner contributions to the grand potential and detect different phase transitions in the topological phase diagram. The transitions from the quadrupolar to the trivial or to the dipolar phases are well captured by the thermodynamic potential. On the other hand, we have to resort to a grand potential based on the Wannier bands to describe the transition from the trivial to the dipolar phase. The critical exponents and the order of the phase transitions are determined and discussed in the light of the Josephson hyperscaling relation.

Gravity edges modes and Hayward term

We argue that corner contributions in gravity action (Hayward term) capture the essence of gravity edge modes, which lead to gravitational area entropies, such as the black hole entropy and holographic entanglement entropy. We explain how the Hayward term and the corresponding edge modes in gravity are explained by holography from two different viewpoints. One is an extension of AdS/CFT to general spacetimes and the other is the AdS/BCFT formulation. In the final part, we explore how gravity edge modes and its entropy show up in string theory by considering open strings stuck to a Rindler horizon.

Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks

The concept of universality is a cornerstone of theories of critical phenomena. It is very well understood in most systems, especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free energies and response functions is less investigated and less well understood. In particular, the question arises of how universality is maintained in correction-to-scaling in systems of the same universality class but with very different corner geometries. Two-dimensional geometries deliver the simplest such examples that can be constructed with and without corners. To investigate how the presence and absence of corners manifest universality, we analyze the spanning tree generating function on two different finite systems, namely the cobweb and fan networks. The corner free energies of these configurations have stimulated significant interest precisely because of expectations regarding their universal properties and we address how this can be delivered given that the finite-size cobweb has no corners while the fan has four. To answer, we appeal to the Ivashkevich–Izmailian–Hu approach which unifies the generating functions of distinct networks in terms of a single partition function with twisted boundary conditions. This unified approach shows that the contributions to the individual corner free energies of the fan network sum to zero so that it precisely matches that of the web. It therefore also matches conformal theory (in which the central charge is found to be ) and finite-size scaling predictions. Correspondence in each case with results established by alternative means for both networks verifies the soundness of the Ivashkevich–Izmailian–Hu algorithm. Its broad range of usefulness is demonstrated by its application to hitherto unsolved problems—namely the exact asymptotic expansions of the logarithms of the generating functions and the conformal partition functions for fan and cobweb geometries. We also investigate strip geometries, again confirming the predictions of conformal field theory. Thus, the resolution of a universality puzzle demonstrates the power of the algorithm and opens up new applications in the future.

Exact finite-size corrections in the dimer model on a planar square lattice

We consider the dimer model on the rectangular lattice with free boundary conditions. We derive exact expressions for the coefficients in the asymptotic expansion of the free energy in terms of the elliptic theta functions () and the elliptic integral of second kind (E), up to 22nd order. Surprisingly we find that ratio of the coefficients f p in the free energy expansion for strip () and square () geometries in the limit of large p tends to 1/2. Furthermore, we predict that the ratio of the coefficients f p in the free energy expansion for rectangular () for aspect ratio to the coefficients of the free energy for square geometries, multiplied by , that is , is also equal to 1/2 in the limit of . With these results, one can find the asymptotic behavior of the from that of the , whose asymptotic behavior is derived explicitly here. We find that the corner contribution to the free energy for the dimer model on rectangular lattices with free boundary conditions is equal to zero and explain that result in the framework of conformal field theory, with two consistent values of the central charge, namely, c = −2 for the construction of a conformal field theory using a mapping of spanning trees and c = 1 for the height function description. We also derive a simple exact expression for the free energy of open strips of arbitrary width.

Probing trihedral corner entanglement for Dirac fermions

We investigate the universal information contained in the Renyi entanglement entropies for a free massless Dirac fermion in three spatial dimensions. Using numerical calculations on the lattice, we examine the case where the entangling boundary contains trihedral corners. The entropy contribution arising from these corners grows logarithmically in the entangled subsystem's size with a universal coefficient. Our numerical results provide evidence that this logarithmic coefficient has a simple structure determined by two universal functions characterizing the underlying critical theory and the geometry of the corner. This form is similar to that of the analogous coefficient appearing for smooth entangling surfaces. Furthermore, our results support the idea that one of the geometric factors in the corner coefficient is topological in nature and related to the Euler characteristic of the boundary, in direct analogy to the case of the smooth surface. We discuss implications, including the possibility that one can use trihedral corner contributions to the Renyi entropy to determine both of the universal central charges of the underlying critical theory.

Modeling Temporal Interaction Dynamics in Organizational Settings

Most workplace phenomena take place in dynamic social settings and emerge over time, and scholars have repeatedly called for more research into the temporal dynamics of organizational behavior. One reason for this persistent research gap could be that organizational scholars are not aware of the methodological advances that are available today for modeling temporal interactions and detecting behavioral patterns that emerge over time. To facilitate such awareness, this Methods Corner contribution provides a hands-on tutorial for capturing and quantifying temporal behavioral patterns and for leveraging rich interaction data in organizational settings. We provide an overview of different approaches and methodologies for examining temporal interaction patterns, along with detailed information about the type of data that needs to be gathered in order to apply each method as well as the analytical steps (and available software options) involved in each method. Specifically, we discuss and illustrate lag sequential analysis, pattern analysis, statistical discourse analysis, and visualization methods for identifying temporal patterns in interaction data. We also provide key takeaways for integrating these methods more firmly in the field of organizational research and for moving interaction analytical research forward.

Extreme Wave-Induced Oscillation in Paradip Port Under the Resonance Conditions

A mathematical model is constructed to analyze the long wave-induced oscillation in Paradip Port, Odisha, India under the resonance conditions to avert any extreme wave hazards. Boundary element method (BEM) with corner contribution is utilized to solve the Helmholtz equation under the partial reflection boundary conditions. Furthermore, convergence analysis is also performed for the boundary element scheme with uniform and non-uniform discretization of the boundary. The numerical scheme is also validated with analytic approximation and existing studies based on harbor resonance. Then, the amplification factor is estimated at six key record stations in the Paradip Port with multidirectional incident waves and resonance modes are also estimated at the boundary of the port. Ocean surface wave field is predicted in the interior of Paradip Port for the different directional incident wave at various resonance modes. Moreover, the safe locations in the port have been identified for loading and unloading of moored ship with different resonance modes and directional incident waves.

Entanglement in Lifshitz-type quantum field theories

We study different aspects of quantum entanglement and its measures, including entanglement entropy in the vacuum state of a certain Lifshitz free scalar theory. We present simple intuitive arguments based on “non-local” effects of this theory that the scaling of entanglement entropy depends on the dynamical exponent as a characteristic parameter of the theory. The scaling is such that in the massless theory for small entangling regions it leads to area law in the Lorentzian limit and volume law in the z → ∞ limit. We present strong numerical evidences in (1+1) and (2+1)-dimensions in support of this behavior. In (2 + 1)-dimensions we also study some shape dependent aspects of entanglement. We argue that in the massless limit corner contributions are no more additive for large enough dynamical exponent due to non-local effects of Lifshitz theories. We also comment on possible holographic duals of such theories based on the sign of tripartite information.

Entanglement negativity in a two dimensional harmonic lattice: area law and corner contributions

We study the logarithmic negativity and the moments of the partial transpose in the ground state of a two dimensional massless harmonic square lattice with nearest neighbour interactions for various configurations of adjacent domains. At leading order for large domains, the logarithmic negativity and the logarithm of the ratio between the generic moment of the partial transpose and the moment of the reduced density matrix at the same order satisfy an area law in terms of the length of the curve shared by the adjacent regions. We give numerical evidence that the coefficient of the area law term in these quantities is related to the coefficient of the area law term in the Rényi entropies. Whenever the curve shared by the adjacent domains contains vertices, a subleading logarithmic term occurs in these quantities and the numerical values of the corner function for some pairs of angles are obtained. In the special case of vertices corresponding to explementary angles, we provide numerical evidence that the corner function of the logarithmic negativity is given by the corner function of the Rényi entropy of order 1/2.

Universal corner contributions to entanglement negativity

It has been realised that corners in entangling surfaces can induce new universal contributions to the entanglement entropy and Rényi entropy. In this paper we study universal corner contributions to entanglement negativity in three- and four-dimensional CFTs using both field theory and holographic techniques. We focus on the quantity χ defined by the ratio of the universal part of the entanglement negativity over that of the entanglement entropy, which may characterise the amount of distillable entanglement. We find that for most of the examples χ takes bigger values for singular entangling regions, which may suggest increase in distillable entanglement. However, there also exist counterexamples where distillable entanglement decreases for singular surfaces. We also explore the behaviour of χ as the coupling varies and observe that for singular entangling surfaces, the amount of distillable entanglement is mostly largest for free theories, while counterexample exists for free Dirac fermion in three dimensions. For holographic CFTs described by higher derivative gravity, χ may increase or decrease, depending on the sign of the relevant parameters. Our results may reveal a more profound connection between geometry and distillable entanglement.

Holographic mutual information for singular surfaces

We study corner contributions to holographic mutual information for entangling regions composed of a set of disjoint sectors of a single infinite circle in three-dimensional conformal field theories. In spite of the UV divergence of holographic mutual information, it exhibits a first order phase transition. We show that tripartite information is also divergent for disjoint sectors, which is in contrast with the well-known feature of tripartite information being finite even when entangling regions share boundaries. We also verify the locality of corner effects by studying mutual information between regions separated by a sharp annular region. Possible extensions to higher dimensions and hyperscaling violating geometries is also considered for disjoint sectors.

Corner contributions to holographic entanglement entropy in non-conformal backgrounds

We study corner contributions to holographic entanglement entropy in non-conformal backgrounds: a kink for D2-branes as well as a cone and two different types of crease for D4-branes. Unlike 2+1-dimensional CFTs, the corner contribution to the holographic entanglement entropy of D2-branes exhibits a power law behaviour rather than a logarithmic term. However, the logarithmic term emerges in the holographic entanglement entropy of D4-branes. We identify the logarithmic term for a cone in D4-brane background as the universal contribution under appropriate limits and compare it with other physical quantities.

Universal corner entanglement from twist operators

The entanglement entropy in three-dimensional conformal field theories (CFTs) receives a logarithmic contribution characterized by a regulator-independent function $a(\theta)$ when the entangling surface contains a sharp corner with opening angle $\theta$. In the limit of a smooth surface ($\theta\rightarrow\pi$), this corner contribution vanishes as $a(\theta)=\sigma\,(\theta-\pi)^2$. In arXiv:1505.04804, we provided evidence for the conjecture that for any $d=3$ CFT, this corner coefficient $\sigma$ is determined by $C_T$, the coefficient appearing in the two-point function of the stress tensor. Here, we argue that this is a particular instance of a much more general relation connecting the analogous corner coefficient $\sigma_n$ appearing in the $n$th R\'enyi entropy and the scaling dimension $h_n$ of the corresponding twist operator. In particular, we find the simple relation $h_n/\sigma_n=(n-1)\pi$. We show how it reduces to our previous result as $n\rightarrow 1$, and explicitly check its validity for free scalars and fermions. With this new relation, we show that as $n\rightarrow 0$, $\sigma_n$ yields the coefficient of the thermal entropy, $c_S$. We also reveal a surprising duality relating the corner coefficients of the scalar and the fermion. Further, we use our result to predict $\sigma_n$ for holographic CFTs dual to four-dimensional Einstein gravity. Our findings generalize to other dimensions, and we emphasize the connection to the interval R\'enyi entropies of $d=2$ CFTs.