A scaling theory of conserved currents is studied at critical points with critical charge fluctuations. It is shown that the conserved current has no anomalous dimension even for anisotropic critical points. Many universal amplitudes are identified. One of the universal amplitudes at the critical points is ${\mathrm{lim}}_{(\mathit{k},\mathrm{\ensuremath{\omega}})\ensuremath{\rightarrow}0\mathrm{\ensuremath{\omega}}}^{2\mathrm{\ensuremath{-}}\mathit{d}}$${\mathit{k}}^{(\mathit{z}\mathrm{\ensuremath{-}}\mathit{d})}$ $^{(2\mathrm{\ensuremath{-}}\mathit{d})}\mathrm{\ensuremath{\sigma}}$(\ensuremath{\omega}) ${]}^{\mathit{z}}$ [\ensuremath{\kappa}(k)${]}^{2\mathrm{\ensuremath{-}}\mathit{d}}$, which reduces to the universal conductance in 2+1 dimensions. We find that the ratio \ensuremath{\surd}\ensuremath{\kappa}\ensuremath{\chi} /\ensuremath{\sigma} always approaches a constant as we approach the critical point. We also determine exponents of many scaling relations, using the scaling properties of the current.A scaling theory of conserved currents is studied at critical points with critical charge fluctuations. It is shown that the conserved current has no anomalous dimension even for anisotropic critical points. Many universal amplitudes are identified. One of the universal amplitudes at the critical points is ${\mathrm{lim}}_{(\mathit{k},\mathrm{\ensuremath{\omega}})\ensuremath{\rightarrow}0\mathrm{\ensuremath{\omega}}}^{2\mathrm{\ensuremath{-}}\mathit{d}}$${\mathit{k}}^{(\mathit{z}\mathrm{\ensuremath{-}}\mathit{d})}$ $^{(2\mathrm{\ensuremath{-}}\mathit{d})}\mathrm{\ensuremath{\sigma}}$(\ensuremath{\omega})${]}^{\mathit{z}}$[\ensuremath{\kappa}(k)${]}^{2\mathrm{\ensuremath{-}}\mathit{d}}$, which reduces to the universal conductance in 2+1 dimensions. We find that the ratio \ensuremath{\surd}\ensuremath{\kappa}\ensuremath{\chi} /\ensuremath{\sigma} always approaches a constant as we approach the critical point. We also determine exponents of many scaling relations, using the scaling properties of the current.

Physical review. B, Condensed matterAug 1, 199225 citations

Theory of Finite-Entanglement Scaling at One-Dimensional Quantum Critical Points

Frank Pollmann, Subroto Mukerjee, Ari M. Turner +1 more

Studies of entanglement in many-particle systems suggest that most quantum critical ground states have infinitely more entanglement than noncritical states. Standard algorithms for one-dimensional 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. Finite-entanglement scaling in one-dimensional systems is governed not by the scaling dimension of an operator but by the "central charge" of the critical point. An important ingredient is the universal distribution of density-matrix eigenvalues at a critical point [P. Calabrese and A. Lefevre, Phys. Rev. A 78, 032329 (2008)10.1103/PhysRevA.78.032329]. The parameter-free theory is checked against numerical scaling at several quantum critical points.Studies of entanglement in many-particle systems suggest that most quantum critical ground states have infinitely more entanglement than noncritical states. Standard algorithms for one-dimensional 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. Finite-entanglement scaling in one-dimensional systems is governed not by the scaling dimension of an operator but by the "central charge" of the critical point. An important ingredient is the universal distribution of density-matrix eigenvalues at a critical point [P. Calabrese and A. Lefevre, Phys. Rev. A 78, 032329 (2008)10.1103/PhysRevA.78.032329]. The parameter-free theory is checked against numerical scaling at several quantum critical points.

arXiv:0812.2903Physical Review LettersJun 26, 2009364 citations

Phase Transitions and Critical Points

James G. Glimm, Arthur Jaffe

Springer eBooksJan 1, 19814 citations

Scaling in driven dynamics starting in the vicinity of a quantum critical point

Shuai Yin, Chung-Yu Lo, Pochung Chen

Driven dynamics across a quantum critical point is usually described by the Kibble-Zurek scaling. Although the original Kibble-Zurek scaling requires an adiabatic initial state, it has been shown that scaling behaviors exist even when the driven dynamics is triggered from a thermal equilibrium state exactly at the critical point, in spite of the breakdown of the initial adiabaticity. In this paper, we show that the existence of the scaling behavior can be generalized to the case of the initial state being a thermal equilibrium state near the critical point. We propose a scaling theory in which the initial parameters are included as additional scaling variables due to the breakdown of the initial adiabaticity. In particular, we demonstrate that for the driven critical dynamics in a closed system, the nontrivial thermal effects are closely related to the initial distance to the critical point. We numerically confirm the scaling theory by simulating the real-time dynamics of the one-dimensional quantum Ising model at both zero and finite temperatures.Driven dynamics across a quantum critical point is usually described by the Kibble-Zurek scaling. Although the original Kibble-Zurek scaling requires an adiabatic initial state, it has been shown that scaling behaviors exist even when the driven dynamics is triggered from a thermal equilibrium state exactly at the critical point, in spite of the breakdown of the initial adiabaticity. In this paper, we show that the existence of the scaling behavior can be generalized to the case of the initial state being a thermal equilibrium state near the critical point. We propose a scaling theory in which the initial parameters are included as additional scaling variables due to the breakdown of the initial adiabaticity. In particular, we demonstrate that for the driven critical dynamics in a closed system, the nontrivial thermal effects are closely related to the initial distance to the critical point. We numerically confirm the scaling theory by simulating the real-time dynamics of the one-dimensional quantum Ising model at both zero and finite temperatures.

arXiv:1607.01840Physical review. B./Physical review. BAug 25, 201611 citations

Phase Transitions and Critical Points

James G. Glimm, Arthur Jaffe

Springer eBooksJan 1, 19874 citations

{]}^{\mathit{z}}

[\ensuremath{\kappa}(k)${]}^{2\mathrm{\ensuremath{-}}\mathit{d}}$, which reduces to the universal conductance in 2+1 dimensions. We find that the ratio \ensuremath{\surd}\ensuremath{\kappa}\ensuremath{\chi} /\ensuremath{\sigma} always approaches a constant as we approach the critical point. We also determine exponents of many scaling relations, using the scaling properties of the current.