Studies of entanglement in many-particle systems suggest that most quantum critical ground states have infinitely more entanglement than non-critical states. Standard algorithms for onedimensional many-particle systems construct model states with limited entanglement, which are a worse approximation to quantum critical states than to others. We give a quantitative theory of previously observed scaling behavior resulting from finite entanglement at quantum criticality: the scaling theory of finite entanglement is only superficially similar to finite-size scaling, and has a different physical origin. We find that finite-entanglement scaling is governed not by the scaling dimension of an operator but by the "central charge" of the critical point, which counts its universal degrees of freedom. An important ingredient is the recently obtained universal distribution of density-matrix eigenvalues at a critical point [1]. The parameter-free theory is checked against numerical scaling at several quantum critical points. A system in its ground state can undergo a secondorder (continuous) quantum phase transition as a control parameter is varied through a critical value. As at a thermal phase transition, the critical point has correlations over long length scales. These correlations are described by properties of the critical point that are "universal", i.e., independent of microscopic details. The entanglement entropy is a measure of the quantum-mechanical nature of correlations and in many cases is also universal [2,3,4,5,6,7,8,9]. The entanglement entropy of a pure state of a bipartite system AB is defined aswhere ρ A (ρ B ) is the reduced density matrix of subsystem A(B). A nonzero entanglement entropy reflects that in quantum mechanics, a "complete description" of a system (i.e., a pure state) does not imply a complete description of subsystems. We would like to understand a consequence of diverging entanglement at quantum criticality: any approach constructing states with a limited amount of entanglement will show universal, systematic errors in describing quantum critical states. It was shown previously by Tagliacozzo et al. [10] in a numerical study of two one-dimensional critical models that finite entanglement leads to scaling behavior like that induced by other perturbations of a critical point: it introduces a finite correlation length ξ ∼ χ κ , where χ (defined below) is related to how much entanglement is retained and κ is the finiteentanglement scaling exponent. That work gave convincing evidence for this behavior in the two models studied but did not attempt to explain its origin or develop a theory predicting κ. As the retained entanglement χ increases, ξ → ∞ and criticality is restored.The main result of this Letter is a theory for this behavior for conformally invariant critical points in one dimension. The theory predicts that κ, unlike other scaling exponents, is determined by the central charge of the critical point. It leads to a specific formula for this dependence and also explains the observed sca...
We show that a molecular junction can give large values of the thermoelectric figure of merit ZT , and so could be used as a solid state energy conversion device that operates close to the Carnot efficiency. The mechanism is similar to the Mahan-Sofo model for bulk thermoelectrics -the Lorenz number goes to zero violating the Wiedemann-Franz law while the thermopower remains non-zero. The molecular state through which charge is transported must be weakly coupled to the leads, and the energy level of the state must be of order kBT away from the Fermi energy of the leads. In practice, the figure of merit is limited by the phonon thermal conductance; we show that the largest possible ZT ∼ (G ph th ) −1/2 , whereG ph th is the phonon thermal conductance divided by the thermal conductance quantum.
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