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Volumn 80, Issue 18, 2009, Pages

Shannon and entanglement entropies of one- and two-dimensional critical wave functions

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EID: 73249141480     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.80.184421     Document Type: Article
Times cited : (198)

References (73)
  • 4
    • 2442560099 scopus 로고    scopus 로고
    • 10.1103/PhysRevLett.92.096402
    • V. E. Korepin, Phys. Rev. Lett. 92, 096402 (2004). 10.1103/PhysRevLett. 92.096402
    • (2004) Phys. Rev. Lett. , vol.92 , pp. 096402
    • Korepin, V.E.1
  • 6
    • 4644275078 scopus 로고    scopus 로고
    • 10.1016/j.physletb.2004.08.072
    • H. Casini and M. Huerta, Phys. Lett. B 600, 142 (2004). 10.1016/j.physletb.2004.08.072
    • (2004) Phys. Lett. B , vol.600 , pp. 142
    • Casini, H.1    Huerta, M.2
  • 9
    • 33645151438 scopus 로고    scopus 로고
    • 10.1103/PhysRevLett.96.110404
    • A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006). 10.1103/PhysRevLett.96.110404
    • (2006) Phys. Rev. Lett. , vol.96 , pp. 110404
    • Kitaev, A.1    Preskill, J.2
  • 10
    • 33645161396 scopus 로고    scopus 로고
    • 10.1103/PhysRevLett.96.110405
    • M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006). 10.1103/PhysRevLett.96.110405
    • (2006) Phys. Rev. Lett. , vol.96 , pp. 110405
    • Levin, M.1    Wen, X.-G.2
  • 13
    • 48449107047 scopus 로고    scopus 로고
    • 10.1103/PhysRevB.78.035320
    • B. A. Friedman and G. C. Levine, Phys. Rev. B 78, 035320 (2008). 10.1103/PhysRevB.78.035320
    • (2008) Phys. Rev. B , vol.78 , pp. 035320
    • Friedman, B.A.1    Levine, G.C.2
  • 16
    • 34347407227 scopus 로고    scopus 로고
    • 10.1103/PhysRevB.75.214407
    • S. Furukawa and G. Misguich, Phys. Rev. B 75, 214407 (2007). 10.1103/PhysRevB.75.214407
    • (2007) Phys. Rev. B , vol.75 , pp. 214407
    • Furukawa, S.1    Misguich, G.2
  • 17
    • 36049024363 scopus 로고    scopus 로고
    • 10.1103/PhysRevB.76.174416
    • C. Castelnovo and C. Chamon, Phys. Rev. B 76, 174416 (2007). 10.1103/PhysRevB.76.174416
    • (2007) Phys. Rev. B , vol.76 , pp. 174416
    • Castelnovo, C.1    Chamon, C.2
  • 21
  • 22
    • 73249153072 scopus 로고    scopus 로고
    • A measure of phase space which is invariant under a chaotic transformation can sometimes be represented by a probability distribution on the configurations of a spin chain.
    • A measure of phase space which is invariant under a chaotic transformation can sometimes be represented by a probability distribution on the configurations of a spin chain.
  • 24
  • 25
    • 33746922351 scopus 로고    scopus 로고
    • 10.1103/PhysRevLett.97.050404
    • E. Fradkin and J. E. Moore, Phys. Rev. Lett. 97, 050404 (2006). 10.1103/PhysRevLett.97.050404
    • (2006) Phys. Rev. Lett. , vol.97 , pp. 050404
    • Fradkin, E.1    Moore, J.E.2
  • 28
    • 73249144691 scopus 로고    scopus 로고
    • Here we take a standard notation used in statistical mechanics and field theory, where R=1 for a free fermion and R=2 for a SU(2)-symmetric case. In condensed matter physics, a common notation is Rcm =R/ 4π, and the coupling constant K=1/ R2 known as TLL parameter is also widely used.
    • Here we take a standard notation used in statistical mechanics and field theory, where R=1 for a free fermion and R=2 for a SU(2)-symmetric case. In condensed matter physics, a common notation is Rcm =R/ 4π, and the coupling constant K=1/ R2 known as TLL parameter is also widely used.
  • 29
    • 35549006412 scopus 로고
    • 10.1063/1.1703773;
    • F. J. Dyson, J. Math. Phys. 3, 140 (1962) 10.1063/1.1703773
    • (1962) J. Math. Phys. , vol.3 , pp. 140
    • Dyson, F.J.1
  • 30
    • 0141913605 scopus 로고
    • 10.1063/1.1703774;
    • F. J. Dyson, J. Math. Phys. 3, 157 (1962) 10.1063/1.1703774
    • (1962) J. Math. Phys. , vol.3 , pp. 157
    • Dyson, F.J.1
  • 31
    • 0242436583 scopus 로고
    • 10.1063/1.1703775
    • F. J. Dyson, J. Math. Phys. 3, 166 (1962). 10.1063/1.1703775
    • (1962) J. Math. Phys. , vol.3 , pp. 166
    • Dyson, F.J.1
  • 33
    • 0000680872 scopus 로고
    • 10.1063/1.1664820
    • F. Calogero, J. Math. Phys. 10, 2191 (1969). 10.1063/1.1664820
    • (1969) J. Math. Phys. , vol.10 , pp. 2191
    • Calogero, F.1
  • 34
    • 0000612877 scopus 로고
    • 10.1063/1.1665584
    • B. Sutherland, J. Math. Phys. 12, 246 (1971). 10.1063/1.1665584
    • (1971) J. Math. Phys. , vol.12 , pp. 246
    • Sutherland, B.1
  • 37
    • 17144436460 scopus 로고    scopus 로고
    • 10.1103/PhysRevLett.86.1881
    • R. Moessner and S. Sondhi, Phys. Rev. Lett. 86, 1881 (2001). 10.1103/PhysRevLett.86.1881
    • (2001) Phys. Rev. Lett. , vol.86 , pp. 1881
    • Moessner, R.1    Sondhi, S.2
  • 41
    • 73249148737 scopus 로고    scopus 로고
    • It is interesting to notice that ρΩ (Ω=A,B) is block diagonal in the spin basis, each block being labeled by a spin configuration i at the boundary. Block number i is a card EiΩ ×card EiΩ matrix and has only one nonzero eigenvalue pi, the corresponding eigenvector being | R KiΩ □. In the special case where all the energies EΩ (ω,i) vanish, each state | RK iΩ □ is an equal-amplitude superposition of spin configurations. Then, each block of ρΩ is particularly simple since all its matrix elements are identical. The block labeled by the boundary configuration i, of size card EiΩ ×card EiΩ, has all its entries equal to pi /card EiΩ and a trace equal to pi.
    • It is interesting to notice that ρΩ (Ω=A,B) is block diagonal in the spin basis, each block being labeled by a spin configuration i at the boundary. Block number i is a card EiΩ ×card EiΩ matrix and has only one nonzero eigenvalue pi, the corresponding eigenvector being | R KiΩ □. In the special case where all the energies EΩ (ω,i) vanish, each state | RK iΩ □ is an equal-amplitude superposition of spin configurations. Then, each block of ρΩ is particularly simple since all its matrix elements are identical. The block labeled by the boundary configuration i, of size card EiΩ ×card EiΩ, has all its entries equal to pi /card EiΩ and a trace equal to pi.
  • 42
    • 73249119389 scopus 로고    scopus 로고
    • Here the factor 2 is inserted because it will be convenient in Sec. when identifying E (i) as a Coulomb gas model. One may also think that the interactions occurring in A and B make a contribution E (i) each and that the total contribution is twice of it.
    • Here the factor 2 is inserted because it will be convenient in Sec. when identifying E (i) as a Coulomb gas model. One may also think that the interactions occurring in A and B make a contribution E (i) each and that the total contribution is twice of it.
  • 43
    • 46949084674 scopus 로고    scopus 로고
    • 10.1103/PhysRevLett.101.010504
    • H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008). 10.1103/PhysRevLett.101.010504
    • (2008) Phys. Rev. Lett. , vol.101 , pp. 010504
    • Li, H.1    Haldane, F.D.M.2
  • 44
    • 73249147509 scopus 로고    scopus 로고
    • KITP Conference, February
    • F. D. M. Haldane, Talk at KITP Conference, February 2009 (unpublished), http://online.kitp.ucsb.edu/online/lowdim-c09/haldane/
    • (2009)
    • Haldane, F.D.M.1
  • 45
    • 73249121494 scopus 로고    scopus 로고
    • The particle coordinates are restricted to be multiple of 2π/L ground-state wave function of the Calogero-Sutherland model.
    • The particle coordinates are restricted to be multiple of 2π/L ground-state wave function of the Calogero-Sutherland model.
  • 46
    • 4243741619 scopus 로고
    • 10.1103/PhysRevLett.75.890
    • Y.-S. Wu and Y. Yu, Phys. Rev. Lett. 75, 890 (1995). 10.1103/PhysRevLett. 75.890
    • (1995) Phys. Rev. Lett. , vol.75 , pp. 890
    • Wu, Y.-S.1    Yu, Y.2
  • 47
    • 0004815079 scopus 로고
    • 10.1103/PhysRevLett.60.635
    • F. D. M. Haldane, Phys. Rev. Lett. 60, 635 (1988). 10.1103/PhysRevLett. 60.635
    • (1988) Phys. Rev. Lett. , vol.60 , pp. 635
    • Haldane, F.D.M.1
  • 48
    • 11744320978 scopus 로고
    • 10.1103/PhysRevLett.60.639
    • B. S. Shastry, Phys. Rev. Lett. 60, 639 (1988). 10.1103/PhysRevLett.60. 639
    • (1988) Phys. Rev. Lett. , vol.60 , pp. 639
    • Shastry, B.S.1
  • 49
    • 27144512211 scopus 로고
    • 10.1016/0550-3213(77)90280-2
    • W. Nahm, Nucl. Phys. B 124, 121 (1977). 10.1016/0550-3213(77)90280-2
    • (1977) Nucl. Phys. B , vol.124 , pp. 121
    • Nahm, W.1
  • 50
    • 0003826278 scopus 로고    scopus 로고
    • Cambridge University Press, Cambridge
    • J. Polchinski, String Theory (Cambridge University Press, Cambridge, 1998).
    • (1998) String Theory
    • Polchinski, J.1
  • 51
  • 52
    • 4244099951 scopus 로고
    • 10.1103/PhysRevLett.51.1506
    • A. Schmid, Phys. Rev. Lett. 51, 1506 (1983). 10.1103/PhysRevLett.51.1506
    • (1983) Phys. Rev. Lett. , vol.51 , pp. 1506
    • Schmid, A.1
  • 61
    • 0037268624 scopus 로고    scopus 로고
    • 10.1016/S0003-4916(02)00018-0
    • A. Y. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003). 10.1016/S0003-4916(02) 00018-0
    • (2003) Ann. Phys. (N.Y.) , vol.303 , pp. 2
    • Kitaev, A.Y.1
  • 64
    • 73249134212 scopus 로고    scopus 로고
    • One can recall that at a certain value of R, the sine potential for φ becomes relevant and φ is locked at a constant value, leading to a crystallization. Conversely, a crystal state can be regarded as a φ -locked state.
    • One can recall that at a certain value of R, the sine potential for φ becomes relevant and φ is locked at a constant value, leading to a crystallization. Conversely, a crystal state can be regarded as a φ -locked state.
  • 65
    • 73249118756 scopus 로고    scopus 로고
    • Note that for the dimer models on the square and hexagonal lattices, the radius of φ is given by R=1, not by R=2 used by Hsu
    • Note that for the dimer models on the square and hexagonal lattices, the radius of φ is given by R=1, not by R=2 used by Hsu (Ref.).
  • 67
    • 0001546262 scopus 로고
    • 10.1088/0305-4470/15/11/011;
    • H. W. J. Blöte and H. J. Hilhorst, J. Phys. A 15, L631 (1982) 10.1088/0305-4470/15/11/011
    • (1982) J. Phys. A , vol.15 , pp. 631
    • Blöte, H.W.J.1    Hilhorst, H.J.2
  • 69
    • 5544298947 scopus 로고
    • 10.1103/PhysRevB.52.6628;
    • J. Kondev and C. L. Henley, Phys. Rev. B 52, 6628 (1995) 10.1103/PhysRevB.52.6628
    • (1995) Phys. Rev. B , vol.52 , pp. 6628
    • Kondev, J.1    Henley, C.L.2
  • 70
    • 0030574904 scopus 로고    scopus 로고
    • 10.1016/0550-3213(96)00064-8;
    • J. Kondev and C. L. Henley, Nucl. Phys. B 464, 540 (1996) 10.1016/0550-3213(96)00064-8
    • (1996) Nucl. Phys. B , vol.464 , pp. 540
    • Kondev, J.1    Henley, C.L.2
  • 71
    • 0009059263 scopus 로고    scopus 로고
    • 10.1103/PhysRevB.55.14935;
    • C. Zeng and C. L. Henley, Phys. Rev. B 55, 14935 (1997) 10.1103/PhysRevB.55.14935
    • (1997) Phys. Rev. B , vol.55 , pp. 14935
    • Zeng, C.1    Henley, C.L.2


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