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Register a new userTensor chain and constraints in tensor networks
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This paper accompanies with our recent work on quantum error correction (QEC) and entanglement spectrum (ES) in tensor networks (arXiv:1806.05007). We propose a general framework for planar tensor network state with tensor constraints as a model for $AdS_3/CFT_2$ correspondence, which could be viewed as a generalization of hyperinvariant tensor networks recently proposed by Evenbly. We elaborate our proposal on tensor chains in a tensor network by tiling $H^2$ space and provide a diagrammatical description for general multi-tensor constraints in terms of tensor chains, which forms a generalized greedy algorithm. The behavior of tensor chains under the action of greedy algorithm is investigated in detail. In particular, for a given set of tensor constraints, a critically protected (CP) tensor chain can be figured out and evaluated by its average reduced interior angle. We classify tensor networks according to their ability of QEC and the flatness of ES. The corresponding geometric description of critical protection over the hyperbolic space is also given.
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A sort of planar tensor networks with tensor constraints is investigated as a model for holography. We study the greedy algorithm generated by tensor constraints and propose the notion of critical protection (CP) against the action of greedy algorithm. For given tensor constraints, a CP tensor chain can be defined. We further find that the ability of quantum error correction (QEC), the non-flatness of entanglement spectrum (ES) and the correlation function can be quantitatively evaluated by the geometric structure of CP tensor chain. Four classes of tensor networks with different properties of entanglement is discussed. Thanks to tensor constraints and CP, the correlation function is reduced into a bracket of Matrix Production State and the result agrees with the one in conformal field theory.
The SYK model has a wormhole-like solution after averaging over the fermionic coupling in the nearly $AdS_2$ space. Even when the couplings are fixed the contribution of these wormholes continues to exist and new saddle points appear which are interpreted as half-wormholes. In this paper, we will study the fate of these wormholes in a model without quenched disorder namely a tensor model with $O(N)^{q-1}$ gauge symmetry whose correlation function and thermodynamics in the large $N$ limit are the same as that of SYK model. We will restate the factorization problem linked with the wormhole threaded Wilson, operator, in terms of global charges or non-trivial cobordism classes associated with disconnected wormholes. Therefore in order for the partition function to factorize especially at short distances, there must exist certain topological defects which break the global symmetry associated with wormholes and make the theory devoid of global symmetries. We will interpret these wormholes with added topological defects as our half-wormholes. We will also comment on the late time behaviour of the spectral form factor, particularly its leading and sub-leading order contributions coming from higher genus wormholes in the gravitational sector. We also found its underlying connections with the Brownian SYK model, particularly in the plateau region which has constant contributions coming from non-trivial saddle points of holonomy from the wormhole followed by an exponential rising part, where the other non-trivial saddles from half-wormhole dominate and give rise to unusual thermodynamics in the bulk sector due to non-perturbative effects.
We show how the choice of an inflationary state that entangles scalar and tensor fluctuations affects the angular two-point correlation functions of the $T$, $E$, and $B$ modes of the cosmic microwave background. The propagators for a state starting with some general quadratic entanglement are solved exactly, leading to predictions for the primordial scalar-scalar, tensor-tensor, and scalar-tensor power spectra. These power spectra are expressed in terms of general functions that describe the entangling structure of the initial state relative to the standard Bunch-Davies vacuum. We illustrate how such a state would modify the angular correlations in the CMB with a simple example where the initial state is a small perturbation away from the Bunch-Davies state. Because the state breaks some of the rotational symmetries, the angular power spectra no longer need be strictly diagonal.
Recent observations confirm that our universe is flat and consists of a dark energy component with negative pressure. This dark energy is responsible for the recent cosmic acceleration as well as determines the feature of future evolution of the universe. In this paper, we discuss the dark energy of the universe in the framework of scalar-tensor cosmology. In the very early universe, the gravitational scalar field $phi$ plays the roll of the inflaton field and drives the universe to expand exponentially. In this period the field $phi$ acts as a cosmological constant and dominates the energy budget, the equation of state (EoS) is $w=-1$. The universe exits from inflation gracefully and with no reheating. Afterwards, the field $phi$ appears as a cold dark matter and continues to dominate the energy budget, the universe expands according to 2/3 power law, the EoS is $w=0$. Eventually, by the epoch of $zsim O(1)$, the field $phi$ contributes a significant component of dark energy with negative pressure and accellerates the late universe. In the future the universe will expand acceleratedly according to $a(t)sim t^{1.31}$.
Degenerate scalar-tensor theories of gravity extend general relativity by a single degree of freedom, despite their equations of motion being higher than second order. In some cases, this is a mere consequence of a disformal field redefinition carried out in a non-degenerate theory. More generally, this is made possible by the existence of an additional constraint that removes the would-be ghost. It has been noted that this constraint can be thwarted when the coupling to matter involves time derivatives of the metric, which results in a modification of the canonical momenta of the gravitational sector. In this note we expand on this issue by analyzing the precise ways in which the extra degree of freedom may reappear upon minimal coupling to matter. Specifically, we study examples of matter sectors that lead either to a direct loss of the special constraint or to a failure to generate a pair of secondary constraints. We also discuss the recurrence of the extra degree of freedom using the language of disformal transformations in particular for what concerns veiled gravity. On the positive side, we show that the minimal coupling of spinor fields is healthy and does not spoil the additional constraint. We argue that this virtue of spinor fields to preserve the number of degrees of freedom in the presence of higher derivatives is actually very general and can be seen from the level decomposition of Grassmann-valued classical variables.
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