Showing papers in "Journal of Mathematical Physics in 2019"
TL;DR: In this paper, the dissipative dynamics of contact Hamiltonian systems are interpreted as a Legendrian submanifold and a coisotropic reduction theorem similar to the one in symplectic mechanics is proved.
Abstract: In this paper, we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We show how the dissipative dynamics can be interpreted as a Legendrian submanifold, and also prove a coisotropic reduction theorem similar to the one in symplectic mechanics; as a consequence, we get a method to reduce the dynamics of contact Hamiltonian systems.In this paper, we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We show how the dissipative dynamics can be interpreted as a Legendrian submanifold, and also prove a coisotropic reduction theorem similar to the one in symplectic mechanics; as a consequence, we get a method to reduce the dynamics of contact Hamiltonian systems.
102 citations
TL;DR: In this article, the authors considered the low temperature quantum theory of a charged black hole of zero temperature horizon radius Rh in a spacetime which is asymptotically AdSD (D > 3) far from the horizon.
Abstract: We consider the low temperature quantum theory of a charged black hole of zero temperature horizon radius Rh in a spacetime which is asymptotically AdSD (D > 3) far from the horizon. At temperatures T ≪ 1/Rh, the near-horizon geometry is AdS2, and the black hole is described by a universal 0+1 dimensional effective quantum theory of time diffeomorphisms with a Schwarzian action and a phase mode conjugate to the U(1) charge. We obtain this universal 0+1 dimensional effective theory starting from the full D-dimensional Einstein-Maxwell theory, while keeping quantitative track of the couplings. The couplings of the effective theory are found to be in agreement with those expected from the thermodynamics of the D-dimensional black hole.
102 citations
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TL;DR: In this article, an SU(2) theory with an odd number of fermion multiplets in the spin 1/2 representation of the gauge group, and more generally in representations of spin 2r+1/2, is inconsistent.
Abstract: A familiar anomaly affects SU(2) gauge theory in four dimensions: a theory with an odd number of fermion multiplets in the spin 1/2 representation of the gauge group, and more generally in representations of spin 2r + 1/2, is inconsistent. We describe here a more subtle anomaly that can affect SU(2) gauge theory in four dimensions under the condition that fermions transform with half-integer spin under SU(2) and bosons transform with integer spin. Such a theory, formulated in a way that requires no choice of spin structure, and with an odd number of fermion multiplets in representations of spin 4r + 3/2, is inconsistent. The theory is consistent if one picks a spin or spinc structure. Under Higgsing to U(1), the new SU(2) anomaly reduces to a known anomaly of “all-fermion electrodynamics.” Like that theory, an SU(2) theory with an odd number of fermion multiplets in representations of spin 4r + 3/2 can provide a boundary state for a five-dimensional gapped theory whose partition function on a closed five-manifold Y is (−1)∫Yw2w3. All statements have analogs with SU(2) replaced by Sp(2N). There is also an analog in five dimensions.
99 citations
TL;DR: In this article, a general-purpose quantum algorithm for ground states of quantum Hamiltonians from a given trial state is proposed, which is based on techniques recently developed in the context of solving the quantum linear system problem.
Abstract: We propose a general-purpose quantum algorithm for preparing ground states of quantum Hamiltonians from a given trial state. The algorithm is based on techniques recently developed in the context of solving the quantum linear system problem. We show that, compared to algorithms based on phase estimation, the runtime of our algorithm is exponentially better as a function of the allowed error, and at least quadratically better as a function of the overlap with the trial state. We also show that our algorithm requires fewer ancilla qubits than existing algorithms, making it attractive for early applications of small quantum computers. Additionally, it can be used to determine an unknown ground energy faster than with phase estimation if a very high precision is required.
91 citations
TL;DR: In this article, a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems is introduced.
Abstract: Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustrationfree models satisfying a local topological quantum order condition, which we present in a sequel to this paper.Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frust...
89 citations
TL;DR: In this paper, a higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth is considered and it is shown that strong logistic damping can prevent blow-up in the higher dimensions.
Abstract: This paper concerns a higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth. It is shown that the strong logistic damping can prevent blow-up in the higher dimensions.
75 citations
TL;DR: In this article, the generalized hydrodynamics for the integrable Toda system were derived from the elastic and factorized scattering of Toda particles using Boltzmann statistics.
Abstract: We obtain the exact generalized hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and factorized scattering of Toda particles, we obtain the generalized free energy and exact average currents and write down the Euler hydrodynamic equations. This is written both as a continuity equation for the density of asymptotic momenta and in terms of normal modes. This is based on the classical thermodynamic Bethe ansatz (TBA), with a single quasiparticle type of Boltzmann statistics. By explicitly connecting chain and gas conserved densities and currents, we then derive the thermodynamics and hydrodynamics of the chain. As the gas and chain have different notions of length, they have different hydrodynamics, and, in particular, the velocities of normal modes differ. We also give a derivation of the classical TBA equations for the gas thermodynamics from the factorized scattering of Toda particles.
69 citations
TL;DR: In this paper, a basis for rational gl(N) spin chains in an arbitrary rectangular representation (SA) that factorises the Bethe vectors into products of Slater determinants in Baxter Q-functions was proposed.
Abstract: We propose a basis for rational gl(N) spin chains in an arbitrary rectangular representation (SA) that factorises the Bethe vectors into products of Slater determinants in Baxter Q-functions. This basis is constructed by repeated action of fused transfer matrices on a suitable reference state. We prove that it diagonalises the so-called B-operator; hence, the operatorial roots of the latter are the separated variables. The spectrum of the separated variables is also explicitly computed, and it turns out to be labeled by Gelfand-Tsetlin patterns. Our approach utilises a special choice of the spin chain twist which substantially simplifies derivations.We propose a basis for rational gl(N) spin chains in an arbitrary rectangular representation (SA) that factorises the Bethe vectors into products of Slater determinants in Baxter Q-functions. This basis is constructed by repeated action of fused transfer matrices on a suitable reference state. We prove that it diagonalises the so-called B-operator; hence, the operatorial roots of the latter are the separated variables. The spectrum of the separated variables is also explicitly computed, and it turns out to be labeled by Gelfand-Tsetlin patterns. Our approach utilises a special choice of the spin chain twist which substantially simplifies derivations.
64 citations
TL;DR: In this paper, the authors studied fermionic topological phases using the technique of fermion condensation, which can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions and condensing pairs of physical and emergent fermanions.
Abstract: We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases that 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 the resulting fermionic fusion categories, which we call “m-type” and “q-type” objects. The endomorphism algebras of q-type objects are complex Clifford algebras, and they have no analogs in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations arising from the condensed theories. We prove a series of results relating data in fermionic theories to data in their parent bosonic theories; for example, if C is a modular tensor category containing a fermion, then the tube category constructed from the condensed theory satisfies T u b e ( C / ψ ) ≅ C × ( C / ψ ) . We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the SO(3)6 theory, and the 1 2 E 6 theory and compute the quasiparticle excitation spectrum in each of the condensed theories.
58 citations
TL;DR: In this article, the authors introduce quantum generalizations of the inequalities (TC1) and (TC2), making use of appropriate quantum versions of the Wasserstein distances, one recently defined by Carlen and Maas and the other defined by us.
Abstract: Quantum functional inequalities (e.g., the logarithmic Sobolev and Poincare inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these inequalities are related to each other via a so-called transportation cost inequality of order 2 (TC2). The latter inequality relies on the notion of a metric on the set of probability distributions called the Wasserstein distance of order 2. (TC2) in turn implies a transportation cost inequality of order 1 (TC1). In this paper, we introduce quantum generalizations of the inequalities (TC1) and (TC2), making use of appropriate quantum versions of the Wasserstein distances, one recently defined by Carlen and Maas and the other defined by us. We establish that these inequalities are related to each other, and to the quantum modified logarithmic Sobolev- and Poincare inequalities, as in the classical case. We also show that these inequalities imply certain concentration-type results for the invariant state of the underlying semigroup. We consider the example of the depolarizing semigroup to derive concentration inequalities for any finite dimensional full-rank quantum state. These inequalities are then applied to derive upper bounds on the error probabilities occurring in the setting of finite blocklength quantum parameter estimation.
56 citations
TL;DR: In this article, the authors analyzed the topological properties of systems of Dirac equations in the presence of heterogeneities to model transport in topological insulators by means of indices of Fredholm operators.
Abstract: We analyze the topological properties of systems of Dirac equations in the presence of heterogeneities to model transport in topological insulators. The topology is described by means of indices of Fredholm operators. We describe bulk and interface topological invariants first for two-dimensional materials, which find practical applications, and then in arbitrary dimensions. In the two-dimensional setting, we relate the interface invariant to a physical observable describing asymmetric current along the interface.We analyze the topological properties of systems of Dirac equations in the presence of heterogeneities to model transport in topological insulators. The topology is described by means of indices of Fredholm operators. We describe bulk and interface topological invariants first for two-dimensional materials, which find practical applications, and then in arbitrary dimensions. In the two-dimensional setting, we relate the interface invariant to a physical observable describing asymmetric current along the interface.
TL;DR: In this paper, the authors study the initial value problem for the integrable nonlocal nonlinear Schrodinger (NNLS) equation with decaying boundary conditions, and they adapt the nonlinear steepestdecent method to the study of the Riemann-Hilbert problem associated with the NNLS equation.
Abstract: We study the initial value problem for the integrable nonlocal nonlinear Schrodinger (NNLS) equation iqt(x,t)+qxx(x,t)+2σq2(x,t) q¯ (−x,t)=0 with decaying (as x → ±∞) boundary conditions. The main aim is to describe the long-time behavior of the solution of this problem. To do this, we adapt the nonlinear steepest-decent method to the study of the Riemann-Hilbert problem associated with the NNLS equation. Our main result is that, in contrast to the local NLS equation, where the main asymptotic term (in the solitonless case) decays to 0 as O(t−1/2) along any ray x/t = const, the power decay rate in the case of the NNLS depends, in general, on x/t and can be expressed in terms of the spectral functions associated with the initial data.
TL;DR: In this article, the stability of free fermion Hamiltonians with gapped and local Hamiltonians was shown to be robust to weak local perturbations, assuming several conditions.
Abstract: Recent results have shown the stability of frustration-free Hamiltonians to weak local perturbations, assuming several conditions. In this paper, we prove the stability of free fermion Hamiltonians which are gapped and local. These free fermion Hamiltonians are not necessarily frustration-free, but we are able to adapt previous work to prove stability. The key idea is to add an additional copy of the system to cancel topological obstructions. We comment on applications to quantization of Hall conductance in such systems.Recent results have shown the stability of frustration-free Hamiltonians to weak local perturbations, assuming several conditions. In this paper, we prove the stability of free fermion Hamiltonians which are gapped and local. These free fermion Hamiltonians are not necessarily frustration-free, but we are able to adapt previous work to prove stability. The key idea is to add an additional copy of the system to cancel topological obstructions. We comment on applications to quantization of Hall conductance in such systems.
TL;DR: In this paper, the authors studied finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation on Z2 with independent and identically distributed exponential weights on the vertices.
Abstract: Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper, we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on Z2 with independent and identically distributed exponential weights on the vertices. Fix two points v1 = (0, 0) and v2 = (0, k2/3) for some k > 0, and consider the maximal paths Γ1 and Γ2 starting at v1 and v2, respectively, to the point (n, n) for n ≫ k. Our object of study is the point of coalescence, i.e., the point v ∈ Γ1 ∩ Γ2 with smallest |v|1. We establish that the distance to coalescence |v|1 scales as k, by showing the upper tail bound P(|v|1>Rk)≤R−c for some c > 0. We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction (1, 1) starting from v3 = (−k2/3, k2/3) and v4 = (k2/3, −k2/3), we establish the optimal tail estimate P(|v|1>Rk)≍R−2/3, for the point of coalescence v. This answers a question left open by Pimentel [Ann. Probab. 44(5), 3187–3206 (2016)] who proved the corresponding lower bound.
TL;DR: In this article, a 3 × 3 Riemann-Hilbert problem was formulated to solve the Cauchy problem for the Sasa-Satsuma equation on the line.
Abstract: We formulate a 3 × 3 Riemann–Hilbert problem to solve the Cauchy problem for the Sasa–Satsuma equation on the line, which allows us to give a representation for the solution of the Sasa–Satsuma equation. We then apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Sasa–Satsuma equation.
TL;DR: In this paper, the authors partially integrate a system of rectangular matrix Riccati equations which describe the synchronization behavior of a nonlinear complex system of N globally connected oscillators, and show by numerical example that complete synchronization can occur even for the mixed case.
Abstract: We partially integrate a system of rectangular matrix Riccati equations which describe the synchronization behavior of a nonlinear complex system of N globally connected oscillators. The equations take a restricted form in which the time-dependent matrix coefficients are independent of the node. We use linear fractional transformations to perform the partial integration, resulting in a system of reduced size which is independent of N, generalizing the well-known Watanabe-Strogatz reduction for the Kuramoto model. For square matrices, the resulting constants of motion are related to the eigenvalues of matrix cross ratios, which we show satisfy various properties such as symmetry relations. For square matrices, the variables can be regarded as elements of a classical Lie group, not necessarily compact, satisfying the matrix Riccati equations. Trajectories lie either within or on the boundary of a classical domain, and we show by numerical example that complete synchronization can occur even for the mixed case. Provided that certain unitarity conditions are satisfied, we extend the definition of cross ratios to rectangular matrix systems and show that again the eigenvalues are conserved. Special cases are models with real vector unknowns for which trajectories lie on the unit sphere in higher dimensions, with well-known synchronization behavior, and models with complex vector wavefunctions that describe synchronization in quantum systems, possibly infinite-dimensional.We partially integrate a system of rectangular matrix Riccati equations which describe the synchronization behavior of a nonlinear complex system of N globally connected oscillators. The equations take a restricted form in which the time-dependent matrix coefficients are independent of the node. We use linear fractional transformations to perform the partial integration, resulting in a system of reduced size which is independent of N, generalizing the well-known Watanabe-Strogatz reduction for the Kuramoto model. For square matrices, the resulting constants of motion are related to the eigenvalues of matrix cross ratios, which we show satisfy various properties such as symmetry relations. For square matrices, the variables can be regarded as elements of a classical Lie group, not necessarily compact, satisfying the matrix Riccati equations. Trajectories lie either within or on the boundary of a classical domain, and we show by numerical example that complete synchronization can occur even for the mixed ca...
TL;DR: In this paper, the Verlinde formula for fusion rules in the Weyl vertex algebra was constructed, and a result that relates irreducible weight modules for the Wey vertex algebra to the affine Lie superalgebra gl(1|1)^ was given.
Abstract: In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way, we confirm the conjecture on fusion rules based on the Verlinde formula. We explicitly construct intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra gl(1|1)^.
TL;DR: In this paper, the Ricci tensor has been shown to have the form of a perfect fluid in all the O(n)-invariant subspaces provided by Gray's decomposition of the gradient of the R tensor.
Abstract: We give new necessary and sufficient conditions on the Weyl tensor for generalized Robertson-Walker (GRW) space-times to be perfect-fluid space-times. For GRW space-times, we determine the form of the Ricci tensor in all the O(n)-invariant subspaces provided by Gray’s decomposition of the gradient of the Ricci tensor. In all but one, the Ricci tensor is Einstein or has the form of perfect fluid. We discuss the corresponding equations of state that result from the Einstein equation in dimension 4, where perfect-fluid GRW space-times are Robertson-Walker.
TL;DR: In this article, the existence of infinitely many solutions for a fractional Kirchhoff-Schrodinger-Poisson system under certain assumptions was shown to be true for the subcritical case and the symmetric mountain pass theorem established by Kajikiya for the critical case.
Abstract: In this paper, we study the existence of infinitely many solutions for a fractional Kirchhoff–Schrodinger–Poisson system. Based on variational methods, especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by Kajikiya for the critical case, we obtain infinitely many solutions for the system under certain assumptions. The novelties of this article lie in the appearance of the possibly degenerate Kirchhoff function and weak assumptions on the nonlinear term which are quite mild.In this paper, we study the existence of infinitely many solutions for a fractional Kirchhoff–Schrodinger–Poisson system. Based on variational methods, especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by Kajikiya for the critical case, we obtain infinitely many solutions for the system under certain assumptions. The novelties of this article lie in the appearance of the possibly degenerate Kirchhoff function and weak assumptions on the nonlinear term which are quite mild.
TL;DR: All polynomial tau-functions of the BKP, DKP, and MDKP hierarches are constructed.
Abstract: We construct all polynomial tau-functions of the BKP, DKP, and MDKP hierarches.We construct all polynomial tau-functions of the BKP, DKP, and MDKP hierarches.
TL;DR: In this paper, the authors considered both the entanglement assisted and unassisted cases for the point-to-point quantum channel, the quantum multiple access channel, quantum channel with a state, and the quantum broadcast channel.
Abstract: We study the problem of transmission of classical messages through a quantum channel in several network scenarios in the one-shot setting. We consider both the entanglement assisted and unassisted cases for the point to point quantum channel, the quantum multiple-access channel, the quantum channel with a state, and the quantum broadcast channel. We show that it is possible to near-optimally characterize the amount of communication that can be transmitted in these scenarios, using the position-based decoding strategy introduced in a prior study (A. Anshu, R. Jain, and N. Warsi, https://ieee.org/document/8399830). In the process, we provide a short and elementary proof of the converse for entanglement-assisted quantum channel coding in terms of the quantum hypothesis testing divergence [obtained earlier in the work of W. Matthews and S. Wehner, IEEE Trans. Inf. Theory 60, 7317–7329 (2014)]. Our proof has the additional utility that it naturally extends to various network scenarios mentioned above. Furthermore, none of our achievability results require a simultaneous decoding strategy, existence of which is an important open question in quantum Shannon theory.
TL;DR: In this article, the existence of a positive solution with minimal energy for a planar Schrodinger-Poisson system, where the nonlinearity is a continuous function with exponential critical growth, was proved using a minimization technique on the Nehari manifold.
Abstract: In this paper, we prove the existence of a positive solution with minimal energy for a planar Schrodinger-Poisson system, where the nonlinearity is a continuous function with exponential critical growth. We find this solution using a minimization technique on the Nehari manifold.In this paper, we prove the existence of a positive solution with minimal energy for a planar Schrodinger-Poisson system, where the nonlinearity is a continuous function with exponential critical growth. We find this solution using a minimization technique on the Nehari manifold.
TL;DR: In this paper, the authors studied various direct and inverse spectral problems for the one-dimensional Schrodinger equation with distributional potential and boundary conditions containing the eigenvalue parameter.
Abstract: We study various direct and inverse spectral problems for the one-dimensional Schrodinger equation with distributional potential and boundary conditions containing the eigenvalue parameter.
TL;DR: In this paper, a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case, is presented, which works in any lattice dimension, for any number of internal degrees of freedom, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.
Abstract: Describing a particle in an external electromagnetic field is a basic task of quantum mechanics. The standard scheme for this is known as “minimal coupling” and consists of replacing the momentum operators in the Hamiltonian by the modified ones with an added vector potential. In lattice systems, it is not so clear how to do this because there is no continuous translation symmetry, and hence, there are no momenta. Moreover, when time is also discrete, as in quantum walk systems, there is no Hamiltonian, but only a unitary step operator. We present a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case. In particular, we show how to implement minimal coupling in a way that automatically guarantees unitary dynamics. The scheme works in any lattice dimension, for any number of internal degrees of freedom, for walks that allow jumps to a finite neighbourhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.Describing a particle in an external electromagnetic field is a basic task of quantum mechanics. The standard scheme for this is known as “minimal coupling” and consists of replacing the momentum operators in the Hamiltonian by the modified ones with an added vector potential. In lattice systems, it is not so clear how to do this because there is no continuous translation symmetry, and hence, there are no momenta. Moreover, when time is also discrete, as in quantum walk systems, there is no Hamiltonian, but only a unitary step operator. We present a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case. In particular, we show how to implement minimal coupling in a way that automatically guarantees unitary dynamics. The scheme works in any lattice dimension, for any number of internal degrees of freedom, for walks that allow jumps to a finite neighbourhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions t...
TL;DR: Brittin et al. as discussed by the authors showed that the zero-field Ising model has no Fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay.
Abstract: We study the complex zeros of the partition function of the Ising model, viewed as a polynomial in the “interaction parameter”; these are known as Fisher zeros in light of their introduction by Fisher in 1965 [Fisher, M. E., “The nature of critical points,” in Lecture notes in Theoretical Physics, edited by Brittin, W. E. (University of Colorado Press, 1965), Vol. 7c]. While the zeros of the partition function as a polynomial in the “field” parameter have been extensively studied since the classical work of Lee and Yang, comparatively little is known about Fisher zeros for general graphs. Our main result shows that the zero-field Ising model has no Fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay. In addition to shedding light on Fisher zeros themselves, this result also establishes a formal connection between two distinct notions of phase transition for the Ising model: the absence of complex zeros (analyticity of the free energy density or the normalized logarithm of the partition function) and decay of correlations with distance. We also discuss the consequences of our result for efficient deterministic approximation of the partition function. Our proof relies heavily on algorithmic techniques, notably Weitz’s self-avoiding walk tree, and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics.We study the complex zeros of the partition function of the Ising model, viewed as a polynomial in the “interaction parameter”; these are known as Fisher zeros in light of their introduction by Fisher in 1965 [Fisher, M. E., “The nature of critical points,” in Lecture notes in Theoretical Physics, edited by Brittin, W. E. (University of Colorado Press, 1965), Vol. 7c]. While the zeros of the partition function as a polynomial in the “field” parameter have been extensively studied since the classical work of Lee and Yang, comparatively little is known about Fisher zeros for general graphs. Our main result shows that the zero-field Ising model has no Fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay. In addition to shedding light on Fisher zeros themselves, this result also establishes a formal connection between two distinct notions of phase transition for the Ising model: the absence of complex zeros (analyticity of the free energy density...
TL;DR: In this paper, the authors extend a criterion originally developed for periodic systems by Knabe and Gosset-Mozgunov to systems with a boundary and show that if the spectral gaps at linear system size n exceed an explicit threshold of order n−3/2, then the whole system is gapped.
Abstract: In quantum many-body systems, the existence of a spectral gap above the ground state has far-reaching consequences. In this paper, we discuss “finite-size” criteria for having a spectral gap in frustration-free spin systems and their applications. We extend a criterion that was originally developed for periodic systems by Knabe and Gosset-Mozgunov to systems with a boundary. Our finite-size criterion says that if the spectral gaps at linear system size n exceed an explicit threshold of order n−3/2, then the whole system is gapped. The criterion takes into account both “bulk gaps” and “edge gaps” of the finite system in a precise way. The n−3/2 scaling is robust: it holds in 1D and 2D systems on arbitrary lattices and with arbitrary finite-range interactions. One application of our results is to give a rigorous foundation to the folklore that 2D frustration-free models cannot host chiral edge modes (whose finite-size spectral gap would scale like n−1).
TL;DR: In this article, the authors investigated the local and global optimality of the triangular, square, simple cubic, face-centered-cubic (fcc), cubic, fcc, and bcc lattices and the hexagonal-close-packing (hcp) structure for a potential energy per point generated by a Morse potential with parameters (α, r0).
Abstract: We investigate the local and global optimality of the triangular, square, simple cubic, face-centered-cubic (fcc) and body-centered-cubic (bcc) lattices and the hexagonal-close-packing (hcp) structure for a potential energy per point generated by a Morse potential with parameters (α, r0). In dimension 2 and for α large enough, the optimality of the triangular lattice is shown at fixed densities belonging to an explicit interval, using a method based on lattice theta function properties. Furthermore, this energy per point is numerically studied among all two-dimensional Bravais lattices with respect to their density. The behavior of the minimizer, when the density varies, matches with the one that has been already observed for the Lennard-Jones potential, confirming a conjecture we have previously stated for differences of completely monotone functions. Furthermore, in dimension 3, the local minimality of the cubic, fcc, and bcc lattices is checked, showing several interesting similarities with the Lennard-Jones potential case. We also show that the square, triangular, cubic, fcc, and bcc lattices are the only Bravais lattices in dimensions 2 and 3 being critical points of a large class of lattice energies (including the one studied in this paper) in some open intervals of densities as we observe for the Lennard-Jones and the Morse potential lattice energies. More surprisingly, in the Morse potential case, we numerically found a transition of the global minimizer from bcc, fcc to hcp, as α increases, that we partially and heuristically explain from the lattice theta function properties. Thus, it allows us to state a conjecture about the global minimizer of the Morse lattice energy with respect to the value of α. Finally, we compare the values of α found experimentally for metals and rare-gas crystals with the expected lattice ground-state structure given by our numerical investigation/conjecture. Only in a few cases does the known ground-state crystal structure match the minimizer we find for the expected value of α. Our conclusion is that the pairwise interaction model with Morse potential and fixed α is not adapted to describe metals and rare-gas crystals if we want to take into consideration that the lattice structure we find in nature is the ground-state of the associated potential energy.We investigate the local and global optimality of the triangular, square, simple cubic, face-centered-cubic (fcc) and body-centered-cubic (bcc) lattices and the hexagonal-close-packing (hcp) structure for a potential energy per point generated by a Morse potential with parameters (α, r0). In dimension 2 and for α large enough, the optimality of the triangular lattice is shown at fixed densities belonging to an explicit interval, using a method based on lattice theta function properties. Furthermore, this energy per point is numerically studied among all two-dimensional Bravais lattices with respect to their density. The behavior of the minimizer, when the density varies, matches with the one that has been already observed for the Lennard-Jones potential, confirming a conjecture we have previously stated for differences of completely monotone functions. Furthermore, in dimension 3, the local minimality of the cubic, fcc, and bcc lattices is checked, showing several interesting similarities with the Lennard...
TL;DR: In this paper, a generalization of a method developed in the context of Yang-Baxter maps was proposed, based on the invariants of symmetry groups of the lattice equations.
Abstract: A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on Z3 is investigated. Our approach is a generalization of a method developed in the context of Yang–Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case–by–case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations.A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on Z3 is investigated. Our approach is a generalization of a method developed in the context of Yang–Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case–by–case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations.
TL;DR: In this paper, the existence of multiple weak solutions for a nonlinear nonlocal elliptic partial differential equation involving a singularity and a power nonlinearity was proved for the case where Ω is an open bounded domain in RN with smooth boundary.
Abstract: In this paper, we prove the existence of multiple weak solutions for a nonlinear nonlocal elliptic partial differential equation involving a singularity and a power nonlinearity, which is given as (−Δp)su=λuγ+uq;u>0 in Ω with zero Dirichlet boundary condition. Here, Ω is an open bounded domain in RN with smooth boundary, N > ps, s ∈ (0, 1), λ > 0, 0 < γ < 1, 1 < p < ∞, and p−1
TL;DR: It is proved that any incompatibility witness can be implemented as a state discrimination task in which some intermediate classical information is obtained before completing the task, and implies that any incompatible pair of channels gives an advantage over compatible pairs in some such state Discrimination task.
Abstract: We introduce the notion of incompatibility witness for quantum channels, defined as an affine functional that is non-negative on all pairs of compatible channels and strictly negative on some incompatible pair. This notion extends the recent definition of incompatibility witnesses for quantum measurements. We utilize the general framework of channels acting on arbitrary finite-dimensional von Neumann algebras, thus allowing us to investigate incompatibility witnesses on measurement-measurement, measurement-channel, and channel-channel pairs. We prove that any incompatibility witness can be implemented as a state discrimination task in which some intermediate classical information is obtained before completing the task. This implies that any incompatible pair of channels gives an advantage over compatible pairs in some such state discrimination task.