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Showing papers in "Communications in Mathematical Physics in 2017"


Journal ArticleDOI
Daniel Harlow1
TL;DR: In this article, it was shown that a version of the quantum-corrected Ryu-Takayanagi formula holds in any quantum error-correcting code, with the final statement expressed in the language of operatoralgebra quantum error correction.
Abstract: I argue that a version of the quantum-corrected Ryu–Takayanagi formula holds in any quantum error-correcting code. I present this result as a series of theorems of increasing generality, with the final statement expressed in the language of operator-algebra quantum error correction. In AdS/CFT this gives a “purely boundary” interpretation of the formula. I also extend a recent theorem, which established entanglement-wedge reconstruction in AdS/CFT, when interpreted as a subsystem code, to the more general, and I argue more physical, case of subalgebra codes. For completeness, I include a self-contained presentation of the theory of von Neumann algebras on finite-dimensional Hilbert spaces, as well as the algebraic definition of entropy. The results confirm a close relationship between bulk gauge transformations, edge-modes/soft-hair on black holes, and the Ryu–Takayanagi formula. They also suggest a new perspective on the homology constraint, which basically is to get rid of it in a way that preserves the validity of the formula, but which removes any tension with the linearity of quantum mechanics. Moreover, they suggest a boundary interpretation of the “bit threads” recently introduced by Freedman and Headrick.

339 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered the Fermi-Hubbard model with periodic driving at high frequency and showed that up to a quasi-exponential time, the system barely absorbs energy.
Abstract: Prethermalization refers to the transient phenomenon where a system thermalizes according to a Hamiltonian that is not the generator of its evolution. We provide here a rigorous framework for quantum spin systems where prethermalization is exhibited for very long times. First, we consider quantum spin systems under periodic driving at high frequency $${ u}$$ . We prove that up to a quasi-exponential time $${\tau_* \sim {\rm e}^{c \frac{ u}{\log^3 u}}}$$ , the system barely absorbs energy. Instead, there is an effective local Hamiltonian $${\widehat D}$$ that governs the time evolution up to $${\tau_*}$$ , and hence this effective Hamiltonian is a conserved quantity up to $${\tau_*}$$ . Next, we consider systems without driving, but with a separation of energy scales in the Hamiltonian. A prime example is the Fermi–Hubbard model where the interaction U is much larger than the hopping J. Also here we prove the emergence of an effective conserved quantity, different from the Hamiltonian, up to a time $${\tau_*}$$ that is (almost) exponential in $${U/J}$$ .

259 citations


Journal ArticleDOI
TL;DR: In this paper, a flow is defined as a divergenceless norm-bounded vector field, or equivalently a set of Planck-thickness "bit threads", which represent entanglement between points on the boundary, and naturally implement the holographic principle.
Abstract: The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a region in a holographic theory to the area of a corresponding bulk minimal surface. Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface. Instead, we invoke the notion of a "flow", defined as a divergenceless norm-bounded vector field, or equivalently a set of Planck-thickness "bit threads". The entanglement entropy of a boundary region is given by the maximum flux out of it of any flow, or equivalently the maximum number of bit threads that can emanate from it. The threads thus represent entanglement between points on the boundary, and naturally implement the holographic principle. As we explain, this new picture clarifies several conceptual puzzles surrounding the RT formula. We give flow-based proofs of strong subadditivity and related properties; unlike the ones based on minimal surfaces, these proofs correspond in a transparent manner to the properties' information-theoretic meanings. We also briefly discuss certain technical advantages that the flows oer over minimal surfaces. In a mathematical appendix, we review the max flow-min cut theorem on networks and on Riemannian manifolds, and prove in the network case that the set of max flows varies Lipshitz continuously in the network parameters.

199 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proved that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge, using the eigenvector moment flow, a multiparticle random walk in a random environment.
Abstract: We prove that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries. The proof relies on analyzing the eigenvector flow under the Dyson Brownian motion. The key new ideas are: (1) the introduction of the eigenvector moment flow, a multi-particle random walk in a random environment, (2) an effective estimate on the regularity of this flow based on maximum principle and (3) optimal finite speed of propagation holds for the eigenvector moment flow with very high probability.

128 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated the structure of certain protected operator algebras that arise in three-dimensional superconformal field theories, and they found that these algebra can be understood as a quantization of (either of) the half-BPS chiral ring(s).
Abstract: We investigate the structure of certain protected operator algebras that arise in three-dimensional \({\mathcal{N}=4}\) superconformal field theories. We find that these algebras can be understood as a quantization of (either of) the half-BPS chiral ring(s). An important feature of this quantization is that it has a preferred basis in which the structure constants of the quantum algebra are equal to the OPE coefficients of the underlying superconformal theory. We identify several nontrivial conditions that the quantum algebra must satisfy in this basis. We consider examples of theories for which the moduli space of vacua is either the minimal nilpotent orbit of a simple Lie algebra or a Kleinian singularity. For minimal nilpotent orbits, the quantum algebras (and their preferred bases) can be uniquely determined. These algebras are related to higher spin algebras. For Kleinian singularities the algebras can be characterized abstractly—they are spherical subalgebras of symplectic reflection algebras—but the preferred basis is not easily determined. We find evidence in these examples that for a given choice of quantum algebra (defined up to a certain gauge equivalence), there is at most one choice of canonical basis. We conjecture that this is the case for general \({\mathcal{N}=4}\) SCFTs.

125 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that with high probability the maximum of the characteristic polynomial on the unit circle of a random unitary matrix sampled from the Haar measure grows in the range Θ(CN/n log n) √ 3/4 for some random variable C.
Abstract: It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a \({N\times N}\) random unitary matrix sampled from the Haar measure grows like \({CN/({\rm log} N)^{3/4}}\) for some random variable C. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the range \({[N^{1 - \varepsilon},N^{1 + \varepsilon}]}\), for arbitrarily small \({\varepsilon}\).

120 citations


Journal ArticleDOI
TL;DR: In this article, the authors prove remainder terms for the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability for the commutative case.
Abstract: We prove several trace inequalities that extend the Golden–Thompson and the Araki–Lieb–Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb’s triple matrix inequality. As an example application of our four matrix extension of the Golden–Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities.

116 citations


Journal ArticleDOI
TL;DR: In this paper, a construction for the quantum-corrected Coulomb branch of a general 3D gauge theory with N = 4 supersymmetry is proposed in terms of local coordinates associated with an abelianized theory.
Abstract: We propose a construction for the quantum-corrected Coulomb branch of a general 3d gauge theory withN = 4 supersymmetry, in terms of local coordinates associated with an abelianized theory. In a xed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkahler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations.

114 citations


Journal ArticleDOI
TL;DR: In this paper, an unramified extension of the p-adic numbers replaces Euclidean space as the boundary and a version of the Bruhat-Tits tree replaces the bulk.
Abstract: We construct a p-adic analog to AdS/CFT, where an unramified extension of the p-adic numbers replaces Euclidean space as the boundary and a version of the Bruhat–Tits tree replaces the bulk. Correlation functions are computed in the simple case of a single massive scalar in the bulk, with results that are strikingly similar to ordinary holographic correlation functions when expressed in terms of local zeta functions. We give some brief discussion of the geometry of p-adic chordal distance and of Wilson loops. Our presentation includes an introduction to p-adic numbers.

101 citations


Journal ArticleDOI
TL;DR: In this article, the existence of ground states with prescribed mass for the focusing nonlinear Schrodinger equation with L 2-critical power nonlinearity on noncompact quantum graphs was investigated.
Abstract: We investigate the existence of ground states with prescribed mass for the focusing nonlinear Schrodinger equation with L 2-critical power nonlinearity on noncompact quantum graphs. We prove that, unlike the case of the real line, for certain classes of graphs there exist ground states with negative energy for a whole interval of masses. A key role is played by a thorough analysis of Gagliardo–Nirenberg inequalities and by estimates of the optimal constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs.

96 citations


Journal ArticleDOI
TL;DR: In this paper, the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties was proved for nonlinearities which are L 2 -supercritical.
Abstract: We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are L 2-supercritical; in particular, we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the phase transition of the nearest-neighbor ferromagnetic q-state Potts model is continuous for q ≥ 2,3,4.
Abstract: This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic q-state Potts model on $${\mathbb{Z}^2}$$ is continuous for $${q \in \{2,3,4\}}$$ , in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions. The proof uses the random-cluster model with cluster-weight $${q \ge 1}$$ (note that q is not necessarily an integer) and is based on two ingredients: The result has important consequences toward the study of the scaling limit of the random-cluster model with $${q \in [1,4]}$$ . It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. Let us mention that the result should be instrumental in the study of critical exponents as well.

Journal ArticleDOI
TL;DR: In this paper, the authors gave a polynomial time algorithm for finding low energy states for 1D Hamiltonians acting on a chain of nparticles. But the existence of a succinct classical description and area laws were not rigorously proved before this work.
Abstract: One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of nparticles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n^(O(log n)) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is O(nM(n)), where M(n) is the time required to multiply two n × n matrices.

Journal ArticleDOI
TL;DR: In this paper, a variation of the notion of topological phase reflecting metric structure of the position space is introduced, which contains not only periodic and non-periodic systems with symmetries in Kitaev's periodic table but also topological crystalline insulators.
Abstract: In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev’s periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant K-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane–Mele $${\mathbb{Z}_2}$$ -invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse Mayer-Vietoris exact sequence. As a new example, we study the index of reflection-invariant systems.

Journal ArticleDOI
TL;DR: In this paper, the authors study complex Chern-Simons theory on a Seifert manifold M_3 by embedding it into string theory and show that it is equivalent to a topologically twisted supersymmetric theory and its partition function can naturally regularize by turning on a mass parameter.
Abstract: We study complex Chern–Simons theory on a Seifert manifold M_3 by embedding it into string theory. We show that complex Chern–Simons theory on M_3 is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between (1) the Verlinde algebra, (2) quantum cohomology of the Grassmannian, (3) Chern–Simons theory on Σ×S^1 and (4) index of a spin^c Dirac operator on the moduli space of flat connections to a new set of relations between (1) the “equivariant Verlinde algebra” for a complex group, (2) the equivariant quantum K-theory of the vortex moduli space, (3) complex Chern–Simons theory on Σ×S^1 and (4) the equivariant index of a spin^c Dirac operator on the moduli space of Higgs bundles.

Journal ArticleDOI
TL;DR: In this paper, the exact strong converse exponent of classical-quantum channel coding, for every rate above the Holevo capacity, has been determined, which is an exact analogue of Arimoto's, given as a transform of the Renyi capacities with parameters $${\alpha > 1}$$¯¯.
Abstract: We determine the exact strong converse exponent of classical-quantum channel coding, for every rate above the Holevo capacity. Our form of the exponent is an exact analogue of Arimoto’s, given as a transform of the Renyi capacities with parameters $${\alpha > 1}$$ . It is important to note that, unlike in the classical case, there are many inequivalent ways to define the Renyi divergence of states, and hence the Renyi capacities of channels. Our exponent is in terms of the Renyi capacities corresponding to a version of the Renyi divergences that has been introduced recently in Muller-Lennert et al. (J Math Phys 54(12):122203, 2013. arXiv:1306.3142 ), and Wilde et al. (Commun Math Phys 331(2):593–622, 2014. arXiv:1306.1586 ). Our result adds to the growing body of evidence that this new version is the natural definition for the purposes of strong converse problems.

Journal ArticleDOI
TL;DR: In this paper, a strong-converse bound on the private capacity of a quantum channel assisted by unlimited two-way classical communication was established, based on the max-relative entropy of entanglement and its proof uses a new inequality for the sandwiched Renyi divergences.
Abstract: We find a strong-converse bound on the private capacity of a quantum channel assisted by unlimited two-way classical communication. The bound is based on the max-relative entropy of entanglement and its proof uses a new inequality for the sandwiched Renyi divergences based on complex interpolation techniques. We provide explicit examples of quantum channels where our bound improves upon both the transposition bound (on the quantum capacity assisted by classical communication) and the bound based on the squashed entanglement. As an application, we study a repeater version of the private capacity assisted by classical communication and provide an example of a quantum channel with high private capacity but negligible private repeater capacity.

Journal ArticleDOI
TL;DR: In this paper, the pseudolocal charges and their involvement in time evolutions and in the thermalization process of arbitrary states with strong enough clustering properties were rigorously studied, and it was shown that the densities of pseudoline charges form a Hilbert space, with inner product determined by thermodynamic susceptibilities.
Abstract: Recently, it was understood that modified concepts of locality played an important role in the study of extended quantum systems out of equilibrium, in particular in so-called generalized Gibbs ensembles. In this paper, we rigorously study pseudolocal charges and their involvement in time evolutions and in the thermalization process of arbitrary states with strong enough clustering properties. We show that the densities of pseudolocal charges form a Hilbert space, with inner product determined by thermodynamic susceptibilities. Using this, we define the family of pseudolocal states, which are determined by pseudolocal charges. This family includes thermal Gibbs states at high enough temperatures, as well as (a precise definition of) generalized Gibbs ensembles. We prove that the family of pseudolocal states is preserved by finite time evolution, and that, under certain conditions, the stationary state emerging at infinite time is a generalized Gibbs ensemble with respect to the evolution dynamics. If the evolution dynamics does not admit any conserved pseudolocal charges other than the evolution Hamiltonian, we show that any stationary pseudolocal state with respect to these dynamics is a thermal Gibbs state, and that Gibbs thermalization occurs. The framework is that of translation-invariant states on hypercubic quantum lattices of any dimensionality (including quantum chains) and finite-range Hamiltonians, and does not involve integrability.

Journal ArticleDOI
TL;DR: In this article, the authors analyzed the convergence rate of the local statistics of Dyson Brownian motion to the GOE/GUE for short times t = o(1) with deterministic initial data V. They showed that if the density of states of V is bounded both above and away from 0 down to scales in a small interval of size, then the local statistic coincide with the GUE near the energy of V after time t.
Abstract: We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times t = o(1) with deterministic initial data V. Our main result states that if the density of states of V is bounded both above and away from 0 down to scales $${\ell \ll t}$$ in a small interval of size $${G \gg t}$$ around an energy $${E_0}$$ , then the local statistics coincide with the GOE/GUE near the energy $${E_0}$$ after time t. Our methods are partly based on the idea of coupling two Dyson Brownian motions from Bourgade et al. (Commun Pure Appl Math, 2016), the parabolic regularity result of Erdős and Yau (J Eur Math Soc 17(8):1927–2036, 2015), and the eigenvalue rigidity results of Lee and Schnelli (J Math Phys 54(10):103504, 2013).

Journal ArticleDOI
TL;DR: In this paper, the Schrodinger equation with smooth potentials and magnetic type terms with controlled growth at infinity was studied and it was shown that if the perturbation belongs to a class of unbounded symbols including smooth potential, then the system is reducible.
Abstract: We study the Schrodinger equation on $${\mathbb{R}}$$ with a potential behaving as $${x^{2l}}$$ at infinity, $${l \in [1, + \infty)}$$ and with a small time quasiperiodic perturbation. We prove that if the perturbation belongs to a class of unbounded symbols including smooth potentials and magnetic type terms with controlled growth at infinity, then the system is reducible.

Journal ArticleDOI
TL;DR: Schweyer et al. as discussed by the authors considered the energy critical semilinear heat equation and gave a complete classification of the flow near the ground state solitary wave in dimension (d \ge 7) and without radial symmetry assumption.
Abstract: We consider the energy critical semilinear heat equation $$\partial_tu = \Delta u + |u|^{\frac{4}{d-2}}u, \quad x \in \mathbb{R}^d$$ and give a complete classification of the flow near the ground state solitary wave $$Q(x) = \frac{1}{\left(1+\frac{|x|^2}{d(d-2)}\right)^{\frac{d-2}{2}}}$$ in dimension $${d \ge 7}$$ , in the energy critical topology and without radial symmetry assumption. Given an initial data $${Q + \varepsilon_0}$$ with $${\| abla \varepsilon_0\|_{L^2} \ll 1}$$ , the solution either blows up in the ODE type I regime, or dissipates, and these two open sets are separated by a codimension one set of solutions asymptotically attracted by the solitary wave. In particular, non self similar type II blow up is ruled out in dimension $${d \ge 7}$$ near the solitary wave even though it is known to occur in smaller dimensions (Schweyer, J Funct Anal 263(12):3922–3983, 2012). Our proof is based on sole energy estimates deeply connected to Martel et al. (Acta Math 212(1):59–140, 2014) and draws a route map for the classification of the flow near the solitary wave in the energy critical setting. A by-product of our method is the classification of minimal elements around Q belonging to the unstable manifold.

Journal ArticleDOI
TL;DR: This work proposes that 2+1D topological/SPT orders with a fixed finite symmetry E are classified by the unitary modular tensor categories C over E and proves that the set E has a natural structure of a finite abelian group.
Abstract: A finite bosonic or fermionic symmetry can be described uniquely by a symmetric fusion category $${\mathcal{E}}$$ . In this work, we propose that 2+1D topological/SPT orders with a fixed finite symmetry $${\mathcal{E}}$$ are classified, up to $${E_8}$$ quantum Hall states, by the unitary modular tensor categories $${\mathcal{C}}$$ over $${\mathcal{E}}$$ and the modular extensions of each $${\mathcal{C}}$$ . In the case $${\mathcal{C}=\mathcal{E}}$$ , we prove that the set $${\mathcal{M}_{ext}(\mathcal{E})}$$ of all modular extensions of $${\mathcal{E}}$$ has a natural structure of a finite abelian group. We also prove that the set $${\mathcal{M}_{ext}(\mathcal{C})}$$ of all modular extensions of $${\mathcal{E}}$$ , if not empty, is equipped with a natural $${\mathcal{M}_{ext}(\mathcal{C})}$$ -action that is free and transitive. Namely, the set $${\mathcal{M}_{ext}(\mathcal{C})}$$ is an $${\mathcal{M}_{ext}(\mathcal{E})}$$ -torsor. As special cases, we explain in detail how the group $${\mathcal{M}_{ext}(\mathcal{E})}$$ recovers the well-known group-cohomology classification of the 2+1D bosonic SPT orders and Kitaev’s 16 fold ways. We also discuss briefly the behavior of the group $${\mathcal{M}_{ext}(\mathcal{E})}$$ under the symmetry-breaking processes and its relation to Witt groups.

Journal ArticleDOI
TL;DR: In this paper, the joint distribution of two last-passage times at positions ordered in a time-like direction was studied in the zero temperature Brownian semi-discrete directed polymer.
Abstract: In the zero temperature Brownian semi-discrete directed polymer we study the joint distribution of two last-passage times at positions ordered in the time-like direction. This is the situation when we have the slow de-correlation phenomenon. We compute the limiting joint distribution function in a scaling limit. This limiting distribution is given by an expansion in determinants that is not a Fredholm expansion. A somewhat similar looking formula was derived non-rigorously in a related model by Dotsenko.

Journal ArticleDOI
TL;DR: In this article, a quantum algorithm for systems of (possibly inhomogeneous) linear ODEs with constant coefficients is presented, which produces a quantum state that is proportional to the solution at a desired final time.
Abstract: We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the complexity of several problems related to the Transverse field Ising Model (TIM) and showed that the LHP for TIM on degree-3 graphs is equivalent modulo polynomial reductions to the local Hamiltonian problem for general k-local stoquastic Hamiltonians.
Abstract: We study complexity of several problems related to the Transverse field Ising Model (TIM). First, we consider the problem of estimating the ground state energy known as the Local Hamiltonian Problem (LHP). It is shown that the LHP for TIM on degree-3 graphs is equivalent modulo polynomial reductions to the LHP for general k-local ‘stoquastic’ Hamiltonians with any constant \({k \ge 2}\). This result implies that estimating the ground state energy of TIM on degree-3 graphs is a complete problem for the complexity class \({\mathsf{StoqMA}}\) —an extension of the classical class \({\mathsf{MA}}\). As a corollary, we complete the complexity classification of 2-local Hamiltonians with a fixed set of interactions proposed recently by Cubitt and Montanaro. Secondly, we study quantum annealing algorithms for finding ground states of classical spin Hamiltonians associated with hard optimization problems. We prove that the quantum annealing with TIM Hamiltonians is equivalent modulo polynomial reductions to the quantum annealing with a certain subclass of k-local stoquastic Hamiltonians. This subclass includes all Hamiltonians representable as a sum of a k-local diagonal Hamiltonian and a 2-local stoquastic Hamiltonian.

Journal ArticleDOI
TL;DR: In this article, the authors proved the quantization of the Hall conductivity for general weakly interacting gapped fermionic systems on two-dimensional periodic lattices, based on fermion cluster expansion techniques combined with lattice Ward identities.
Abstract: We prove the quantization of the Hall conductivity for general weakly interacting gapped fermionic systems on two-dimensional periodic lattices. The proof is based on fermionic cluster expansion techniques combined with lattice Ward identities, and on a reconstruction theorem that allows us to compute the Kubo conductivity as the analytic continuation of its imaginary time counterpart.

Journal ArticleDOI
TL;DR: In this article, the authors show that the ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problems, and they also show that when there is no external field, the location of the ground states energy is chaotic under small perturbations of the disorder.
Abstract: We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an application, we obtain explicit formulas for the limiting ground state energy in the replica symmetry, one level of replica symmetry breaking and full replica symmetry breaking phases at zero temperature. In addition, our approach leads to new results on disorder chaos in spherical mixed even p-spin models. In particular, we prove that when there is no external field, the location of the ground state energy is chaotic under small perturbations of the disorder. We also establish that in the spherical mixed even p-spin model, the ground state energy superconcentrates in the absence of external field, while it obeys a central limit theorem if the external field is present.

Journal ArticleDOI
TL;DR: In this article, a unified treatment of two distinct blue-shift instabilities for the scalar wave equation on a fixed Kerr black hole background is provided, and the results make essential use of the scattering theory developed in Dafermos, Rodnianski and Shlapentokh-Rothman (2014).
Abstract: In this paper, we provide an elementary, unified treatment of two distinct blue-shift instabilities for the scalar wave equation on a fixed Kerr black hole background: the celebrated blue-shift at the Cauchy horizon (familiar from the strong cosmic censorship conjecture) and the time-reversed red-shift at the event horizon (relevant in classical scattering theory). Our first theorem concerns the latter and constructs solutions to the wave equation on Kerr spacetimes such that the radiation field along the future event horizon vanishes and the radiation field along future null infinity decays at an arbitrarily fast polynomial rate, yet, the local energy of the solution is infinite near any point on the future event horizon. Our second theorem constructs solutions to the wave equation on rotating Kerr spacetimes such that the radiation field along the past event horizon (extended into the black hole) vanishes and the radiation field along past null infinity decays at an arbitrarily fast polynomial rate, yet, the local energy of the solution is infinite near any point on the Cauchy horizon. The results make essential use of the scattering theory developed in Dafermos, Rodnianski and Shlapentokh-Rothman (A scattering theory for the wave equation on Kerr black hole exteriors (2014). arXiv:1412.8379) and exploit directly the time-translation invariance of the scattering map and the non-triviality of the transmission map.

Journal ArticleDOI
TL;DR: In this paper, Carleson's problem regarding convergence for the Schrodinger equation in dimensions in which time tends to zero was considered, and it was shown that the solution converges almost everywhere with respect to the Hausdorff measure to its initial data.
Abstract: We consider Carleson’s problem regarding convergence for the Schrodinger equation in dimensions $${d\ge 2}$$ . We show that if the solution converges almost everywhere with respect to $${\alpha}$$ -Hausdorff measure to its initial datum as time tends to zero, for all data $${H^{s}(\mathbb{R}^{d})}$$ , then $${s\ge \frac{d}{2(d+2)}(d+1-\alpha)}$$ . This strengthens and generalises results of Bourgain and Dahlberg–Kenig.

Journal ArticleDOI
TL;DR: In this article, an a priori bound for the dynamic Euclidean local-in-time solution model on the torus which is independent of the initial condition is established.
Abstract: We prove an a priori bound for the dynamic $${\Phi^4_3}$$ model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows one to construct invariant measures via the Krylov–Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean $${\Phi^4_3}$$ field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities.