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Entanglement in Fractional Quantum Hall states Emil Bergholtz1, Andreas Läuchli1, Masud Haque1 and Juha Suorsa2 1 Max Planck Institute for the Physics of Complex Systems, Dresden, Germany 2 Department of Physics, University of Oslo, Norway Entanglement spectrum Introduction |<L|C>|=0.995 Entanglement in CM systems The description of condensed matter (CM) phases using entanglement measures, borrowed from the field of quantum information (QI) theory, has led to an explosive growth of interdisciplinary work. However, there are only a few cases where entanglement concepts provide physical information that is not obtainable through more conventional quantities, such as correlation functions. A prominent example where these new concepts are indeed valuable is ... Bz The QH system: The quantum Hall (QH) system - cold electrons in two dimensions in a perpendicular magnetic field - is a striking example of a system where unexpected phenomena emerge at low energies due to subtle correlations [1,2]. Emergent CFT structure in Coulomb GS |ψ! = A |ψi ! e i ⊗ B |ψi !, |ψiA !, |ψiB ! where |ψ! is the ground state and " Vy ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! • Only experimentally realized system |<L|C>|=0.985 L1=10 L1=12 L1=14 Laughlin Coulomb 4 label states in the two partitions. 0 L1=6 -4 analyze the ES of Laughlin states on the torus and show that it is arranged in towers, each of which is generated by modes of two spatially separated chiral edges. This structure is present for all torus circumferences, which allows for a microscopic identification of the prominent features of the spectrum by perturbing around the thin torus limit. [6] + + + + + + + + + + + + + + + + + + + |<L|C>|=0.986 8 Recently it was shown [5] that the structure of the ES is related to the edge theory 5 of FQH states and provides a promising route to identifying topological order. We... Ix −ξi /2 |<L|C>|=0.987 12 The entanglement spectrum (ES), {ξi }, is defined in terms of the Schmidt decomposition ! |<L|C>|=0.978 L1=8 0 4 -4 !KA 0 4 -4 !KA 0 4 -4 0 !KA 4 -4 !KA 0 4 !KA Comparison of the ES (and overlaps) between the Laughlin wave function and the Coulomb ground state for various L1. The ES become very similar for sufficiently large L1. The appearance of "generic levels" in the Coulomb state lead to a tentative notion of "Entanglement gap". FIG. 1: Sketch of the Hall experiment. The 2DEG is exposed to a strong perpendicular magnetic field Bz . A current Ix is ES as two chiral edges passed through the sample along the x-direction, and the resulting transverse voltage Vy measured for varying values of the 2 1 1 B! -D ! 3 2 1 20 1 ! C -A -3 -2 -1 0 1 2 3 investigate two probes of bi-partite entanglement - the FIG. 2: Sketch of the (integer) quantum Hall effect. The Hall resistance as function of the magnetic field is quantized, i.e. A-B exhibits and plateaux. von (The result predicted by the classical Hall effect corresponds entanglement spectrum (ES) Neumann entropy - in to a straight line through the centres 0 of the plateaux.) The longitudinal resistivity is zero except at transitions between plateaux. Courtesy of D.R. Leadley, Warwick fractional quantum Hall (FQH) states on the torus [7-8]. The torus University 1997. geometry provides new natural continuos parameter; its (bulk) to excitations. The integer effect was discovered in 1980 by von Klitzing et al[15]; two years later, using even circumference. We exploit this 20 !(NA=6) 2 "K B-A ! A -B ! 10 ! 0 20 cleaner samples, Tsui and collaborators reported the discovery of the fractional effect[16] at ν = 1/3. Since then, with • Disentangle the ES originating from two spatially separated cuts. • Obtain a physical understanding of the ES starting from the the fabrication of ever-higher mobility samples, a large number of fractions have been observed[17]. The quantization 10 10 of the Hall resistance turned out to be extremely exact (to at least ten parts in a billion), which has led to the introduction of a new standard of resistance, with the so-called von Klitzing constant RK = h/e2 , roughly equal to B-B microscopic Hamiltonian. 25812.8 ohms, as the fundamental unit. entanglement entropy, and extract its • Determine the scaling of the 0 Physically, the number ν in (1) corresponds to the Landau level filling fraction at the center of the corresponding -18 -12 topological part, to an accuracy superior totheprevious studies [8] on plateau[18]. In other words, IQHE occurs when an integer number of Landau levels is filled, while the FQHE the sphere. 3 B A x 6 A x BL B x 1ν 0=0p/q, 1 1 0q 0 0 1 0 0 0 0 1 0 ν =: p/(2mp Hamiltonian Hee =+ 1) Vkm c†n+m c†n+k cn+m+k cn 1.2 No kinetic energy! 0 100 = 102 , ν = 1/3 • Following Ref. [8] we divide the lattice sites into two disjoint groups A and B. This is very convenient for numerical studies and amounts to a similar, but not identical, to a spatial partitioning. 1 • Fractional quantum Hall states have degenerate ground states on the torus geometry. We label the ground states by their corresponding thin torus (or Tao-Thouless, TT) ground states [9] which are adiabatically connected to the bulk ground states. For example, for 1/3 filling there are three degenerate states, which correspond to the following TT configurations ! ! ! ! 100100100!100100100100100100!100100100 ! ! ! ! 010010010!010010010010010010!010010010 ! ! ! ! 001001001!001001001001001001 !001001001 #$ % " 2 A • Thinking about the entanglement in terms of the FQH states as dressed TT states turn out to be very convenient as we explain below. 3 L1=12 (a) 6 9 12 15 2 lA 0.1 k+m e! = ±e/q L1=10 0.6 (all ee-terms that preserve position of CM) L1=10 1 (b) Ns=18 1 Ns=24 Ns=24 0.0 ΔS 0.0 L1=8 Ns=12 Ns=30 1 Ns=30 (c) Ns=36 00 0.0 k-m 1......1 • Exact mapping of a single Landau level! • The matrix elements k,m depend on the real space interaction ν =V1/3 and on the torus dimensions, in particular the lattice constant 2π/L1 0.8 3 L1=14 wave function 1 wave function 2 wave function 3 averaged S(lA) 0 5 10 15 20 0 3 6 L1 9 12 We... 0 15 a) SA in different degenerate sectors for the 1/3 Laughlin state, and their average for fixed L1. b) Difference between degeneracy-averaged SA and largest individualsector SA, as a function of L1. They differ significantly only for intermediate L1. c) Degeneracy-averaged SA versus lA, for different L 1. -0.5 -Ln(3) -Ln(3) -Ln(5) -1.5 ν=1/3, Laughlin (a) ν=1/5, Laughlin (c) 0 -0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 4 2 -Ln(3) -1.5 -Ln(3) -2 -Ln(5) 0 -2 -4 -2.5 -3 ν=1/3, Coulomb (b) -3.5 0 3 6 L1 18 0 6 12 L1 0 a) Chiral edge levels identified from ES, as a function of torus thickness L1. b) `Aspect ratio' of the diamond formed by the four lowest ES levels. 9 L1 12 15 18 0 ν=1/5, Coulomb (d) 3 Ns=12 4 Ns=18 Ns=24 3 Ns=30 2 1 Ns=36 5 Ns=39 4 0.2 0 Ns=18 3 (c) Ns=24 Ns=30 2 0 5 10 L1 Ns=36 1 (a) 0.1 Ns=12 15 -0.1 -0.2 ν=1/3, Laughlin ν=1/3, Coulomb Ns=36 (Laughlin) (a) 0 0 (b) 0.6 0.5 0.5 0.4 0.3 0.2 0.1 0.4 0 0 2 4 6 8 10 12 14 16 18 20 L1 0.3 0.2 0.1 0 0 (b) 2 4 6 8 10 12 14 16 18 20 L1 Conclusions and outlook 0 -1 12 Degeneracy-averaged SA (a) and its derivative (b) for the 1/3 Laughlin state as a function of L1. The inset shows results for the Coulomb interaction. Extraction of the topological part of SA -2 6 Exploit the possibility of continuously varying L=2L1 in the torus setting to obtain significantly improved control of the scaling form of SA. We find that the scaling regime is indeed reached (at least for simple FQH states) in a sizable window of numerically accessible parameter space. lA -1 0 0.5 Sphere 5 Conclusion: Degeneracy-averaged SA have much less finite size oscillations. Moreover, lA should be chosen large (close to lB). A way of determining the topological part, −2γ , is to fit to the scaling for each L1 as is done here. The sought value should be extracted at very large L1. Torus 1 Entanglement scaling in action 1.4 n k>m Vk,m B There are several sources of finite sizes corrections that has to be considered; such as finite torus dimensions (L1 and L2) and finite block size (lA). Moreover the degenerate states do in general lead to different SA. 1 10 for n large enough block boundaries of total length L [3,4]. For the torus setup we have n=2 and L=2L1. The constant term is related to the topological field theory and thus contains information of the universality class of the state. The leading term gives the entropy density and is of relevance for the simulation cost of the state. 6 9 12 L1 15 • In the ideal case (1/3 Laughlin) our estimates are accurate to about 2-3% which is much better than earlier results (10-30% at best). • For other states the performance is worse albeit still better than previous results. 18 -6 -8 SA(L1) - L1 dSA/dL1 2π/L1 L1 , L 2 !! y 1.5 12 18 24 30 36 SA = αL − nγ + O(1/L), L2 y 2 The entanglement entropy of topologically ordered states in two dimensions is expected to fulfill an ”area law” and scale as A L2 y 2.5 ξ x SA(lA) y ν = p/q 18 (b) 20 Finite size considerations 01 ν = 1/2, 1/4, 3/8, . . . SA(lA) ν = 1/3 ν≤1 A x y BB L1 B 1 SA(L1) - L1 dSA/dL1 e = e/(2m + 1) 12 30 KA ν=1 ! Partitioning: 0 SA = −tr[ρA lnρA ] 2 4 10 ν= , , ... 5 11 21 1..0.....0..1 ↔ 0..1.....1..0 -6 The von Neumann entropy measure of entanglement, SA, measure the entanglement between a block (A) and the rest (B) of a many-particle system in a pure state |ψ! in terms of the reduced density matrix ρA = trB |ψ!"ψ| as .... 1 ν= 2m + 1 N-particle states: -12 Entanglement entropy ν = 2/5, 4/11, 10/21, . . . A -18 0 Δk=0 (1) Δk=1 (1) Δk=2 (2) Δk=3 (3) Δk=4 (5) dSA/dL1 2 18 KA SA(L1) - L1 dSA/dL1 1 12 3 (a) SA(L1) (! = 1) Note: The y-position is given by the momentum in the x-direction! ν = 1/(2m + 1) k= 6 SA(L1) - L1 dSA/dL1 1 0 A!-A! dSA/dL1 A Landau level (LL) is 1D! With periodic b.c. using Landau gauge this is explicit: 1-particle ψk ∼ e 1 e ν =states: 1/(2m + 1) = 1/3, 1/5, . . . -6 B! -B! Entanglement spectrum of the 12-particle 1/3 Laughlin state for L1 =10. Left panels show the symmetric cut and right panels show one of the asymmetric cuts (see above). The blue squares represent numerically obtained data. The assigned edge modes are labeled by green dots while the combinations of those edges are marked by red crosses. The script letters are microscopic identifiers for the two edges combining to form each tower. The inset shows a CFT tower formed by two ideal (linear dispersion) chiral edges and the degeneracies of each such level. Setup and partitioning 2π 2π 2 ik L x −(y−k L ) /2 A-A Diamond Aspect Ratio 10 2 S(C)-S(L) We... 3 3 2 1 0 precision, be reproduced from two chiral edges with split degeneracies. • There is a lot of non-trivial structure such as identical edges appearing in several sectors. • The ES structure is present for all L1, enabling us to understand many features by studying the problem close to the TT limit: The TT state determines the position of the lowest ES level, and the positions and energetics of the other prominent towers can be understood by studying the leading quantum fluctuations. SA(L1) !(NA=5) 20 [001001001001001001]A !(NA=5) [010010010010010010]A • The ES is arranged in towers. • Each tower can, to a remarkably high !(NA=6) exhibiting topological order. magnetic field. • Extremely rich system: precision and universality, fractional charge and statistics, topological q-bits (?), ... • Ideal setting for applying ideas from quantum information/entanglement theory [3-5]. • Investigating the ES on the torus gave us the opportunity to study the interplay of two chiral edges, and to reach a microscopic understanding thereof [6]. • The continuous variation of the block boundary significantly improves on the performance of numerically determining the entanglement scaling of FQH states [7]. • Similar studies of non-abelian states are under way... References 1. D.C. Tsui, H.L. Störmer, and A.C. Gossard, Phys. Rev. Lett. 48, 1599 (1982). 2. R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 3. A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006). 4. M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006). 5. H. Li and F.D.M. Haldane, Phys. Rev. Lett. 101, 010504 (2008). 6. A.M. Läuchli, E.J. Bergholtz, J. Suorsa and M. Haque, Phys. Rev. Lett. 104, 156404 (2010). 7. A.M. Läuchli, E.J. Bergholtz and M. Haque, New J. Phys. 12, 075004 (2010). 8. M. Haque, O. Zozulya, and K. Schoutens, Phys. Rev. Lett. 98, 060401 (2007). 9. See eg E.J. Bergholtz and A. Karlhede, Phys. Rev. B 77, 155308 (2008).