Bulletin of the American Physical Society
APS March Meeting 2011
Volume 56, Number 1
Monday–Friday, March 21–25, 2011; Dallas, Texas
Session T1: Entanglement Spectroscopy |
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Sponsoring Units: DCMP Chair: Ashvin Vishwanath, University of California, Berkeley Room: Ballroom A1 |
Wednesday, March 23, 2011 2:30PM - 3:06PM |
T1.00001: Momentum space entanglement in quantum spin chains Invited Speaker: Daniel Arovas I will discuss work performed in collaboration with R. Thomale and A. Bernevig ({\sl Phys. Rev. Lett.} {\bf 105}, 116805 (2010)) on entanglement spectra in spin chains. Typically, bipartite entanglement entropy and spectra have been studied in the case of spatial partitions, {\it i.e.\/} A denotes the left half of a spin chain, B the right half, $\rho^{\vphantom{\dagger}}_{\rm A}={\rm Tr}^{\vphantom{\dagger}}_{\rm B}|\Psi_0\rangle\,\langle\Psi_0|$ is the reduced density matrix, and ${\rm spec}(\rho_{\rm A})$ is the entanglement spectrum (ES). We find for the $S={1\over 2}$ Heisenberg model that a remarkable structure in the ES is revealed if the partition is performed in momentum space, {\it i.e.\/} A = left-movers and B = right-movers. Further classifying the entanglement eigenstates by total crystal momentum, we observe a universal low-lying portion of the ES with specific multiplicities separated from a higher-lying nonuniversal set of levels by an {\it entanglement gap\/}, similar to what was observed by Li and Haldane ({\sl Phys. Rev. Lett.} {\bf 101}, 010504 (2008)) for the fractional quantum Hall effect. Indeed, the momentum space ES for the Heisenberg chain is understood in terms of the proximity of the Haldane-Shastry model, which corresponds to a fixed point with no nonuniversal corrections, and whose ground state wavefunction is related to that for the $\nu={1\over 2}$ Laughlin state. We further explore the behavior of the ES as one tunes through the spin-Peierls transition in a model with next-nearest- neighbor exchange. We also discuss entanglement gap scaling and applications to other systems. [Preview Abstract] |
Wednesday, March 23, 2011 3:06PM - 3:42PM |
T1.00002: Identifying Topological Order from the Entanglement Spectrum Invited Speaker: F.D.M. Haldane The Schmidt decomposition reveals bipartite entanglement of a quantum state. Calculation of the entanglement entropy reduces it to a single number, which can be studied as a function of the size and shape of the entangled regions. However, this reduction discards additional information contained in the full spectrum of the entanglement, which can be presented as a set of (dimensionless) ``pseudo-energy'' levels spectrum, labeled by quantum numbers such as momentum parallel to the $(d-1)$-dimensional boundary along which the bipartite decomposition of a $d$-dimensional system is made. The nature of the entanglement is revealed by this spectrum, much as the elementary excitations and collective modes characterizes condensed-matter states. (The von Neumann entropy is equivalent to the the thermodynamic entropy of the system of pseudo-energy levels at a particular fictitious ``temperature'' $k_BT = 1$.) The previously-unrecognized importance of the spectrum (as opposed to just its entropy) became immediately apparent when the entanglement spectrum of a 2D fractional Quantum Hall state along a 1D cut was first plotted [1]. The gapless spectrum of the conformal field theory related to the topological order of the FQHE can be recognized, and is the only spectrum in model states like the Laughlin or Moore-Read wavefunctions related to cft. For realistic states, corrections due to collective-mode fluctuations give rise to high-pseudo-energy modes that are separated from the gapless (topological) modes by a finite gap. Previously, it had been believed that the extensive O(L) (``area law'') part of the entanglement entropy of this spectrum was non-universal, and topological order could only be recognized from the O(1) subleading behavior as the length L of the cut was scaled. However, while the ``pseudo-energy'' distribution appears to be non-universal, the distribution of the spectrum {\it as a function of (true) momentum} does not have this drawback, showing that the topological contribution to the O(L) part has a universal character not visible in the numerical value of the entropy itself. In general, (including also systems such as topological or Chern insulators [2]), the signature of topological order is the occurrence of gapless mode in the entanglement spectrum, providing a fingerprint from which this order can be identified. \\[4pt] [1] Hui Li and F. D. M. Haldane, Phys. Rev. Lett. {\bf 101} 246806 (2008).\\[0pt] [2] F. D. M. Haldane, BAPS.2009.MAR.T13.13. [Preview Abstract] |
Wednesday, March 23, 2011 3:42PM - 4:18PM |
T1.00003: Multiplets in the Entanglement Spectrum Invited Speaker: Ari Turner Often, spin chains do not have any long range order, because of quantum mechanical fluctuations. Surprisingly, there can be phase transitions between two such phases, which suggests the existence of a hidden order. In this talk, I demonstrate that the entanglement spectrum can be used to distinguish between these subtly different phases. The central idea is to reduce a one-dimensional chain to a zero-dimensional imaginary system, called the entanglement Hamiltonian. One can then understand the phases of the original spin chain simply by looking at the spectrum of the entanglement Hamiltonian, just as one deduces the properties of atoms from their spectra. The next question is what the physical meaning of the entanglement Hamiltonian is. Properties of the entanglement Hamiltonian are in fact often reflected in physical properties of the ends of a finite chain, such as the appearance of gapless degrees of freedom or surface charge; I will give some examples in higher dimensional systems such as topological insulators as well as one dimension. [Preview Abstract] |
Wednesday, March 23, 2011 4:18PM - 4:54PM |
T1.00004: Interacting Topological Insulators Invited Speaker: Lukasz Fidkowski Topological insulators and superconductors are new phases of matter whose physics is described by non-interacting fermions. They can be understood in terms of the topological ``twisting" of the fermion's phase over the Brillouin zone, and using topology one can come up with a full classification of when such phases can occur. Strangely, this classification fails for some one dimensional systems once higher-order interactions are allowed. In this talk I will use the entanglement spectrum to understand the modified interacting classification, in one dimension. I will also discuss general one dimensional gapped models, and how matrix product states allow us to find their phases. [Preview Abstract] |
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