Publication: Generalized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies
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We introduce a new class of generalized isotropic Lipkin–Meshkov–Glick models with su(m+1) spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of su(m+1) type. We evaluate in closed form the reduced density matrix of a block of Lspins when the whole system is in its ground state, and study the corresponding von Neumann and Rényi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as a log L when L tends to infinity, where the coefficient a is equal to (m − k)/2 in the ground state phase with k vanishing magnon densities. In particular, our results show that none of these generalized Lipkin–Meshkov–Glick models are critical, since when L-->∞ their Rényi entropy R_q becomes independent of the parameter q. We have also computed the Tsallis entanglement entropy of the ground state of these generalized su(m+1) Lipkin–Meshkov–Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when m-k≥3. Finally, in the su(3) case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of su(3). This is also true in the su(m+1) case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an (m + 1)-simplex in R^m whose vertices are the weights of the fundamental representation of su(m+1).
© 2016 IOP Publishing Ltd and SISSA Medialab srl. This work was supported in part by Spain's MINECO under grant no. FIS2011-22566 and by the Universidad Complutense de Madrid and Banco Santander under grant no. GR3/14-910556. JAC would also like to thank the Madrid township and the 'Residencia de Estudiantes' for their financial support. The authors would also like to thank the anonymous referees of a previous version of this manuscript for their helpful remarks and suggestions.
 Horodecki R, Horodecki P, Horodecki M and Horodecki K 2009 Quantum entanglement Rev. Mod. Phys. 81 865.  Nielsen M A and Chuang I L 2010 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) (10th anniversary edition).  Rényi A 1961 On measures of entropy and information Proc. 4th Berkeley Symp. on Mathematical Statistics and Probability vol 1 ed J Neyman (Berkeley, CA: University of California Press) pp 547–61.  Rényi A 1970 Probability Theory (Amsterdam: North-Holland).  Vidal G, Latorre J I, Rico E and Kitaev A 2003 Entanglement in quantum critical phenomena Phys. Rev. Lett. 90 227902.  Jin B Q and Korepin V E 2004 Quantum spin chain, toeplitz determinants and the Fisher–Hartwig conjecture J. Stat. Phys. 116 79.  Its A R, Jin B Q and Korepin V E 2005 Entanglement in the XY spin chain J. Phys. A: Math. Gen. 38 2975.  Sachdev S 2011 Quantum Phase Transitions 2nd edn (Cambridge: Cambridge University Press).  Eisert J, Cramer M and Plenio M B 2010 Colloquium: area laws for the entanglement entropy Rev. Mod. Phys. 82 277.  Holzhey C, Larsen F and Wilczek F 1994 Geometric and renormalized entropy in conformal field theory 1994 Nucl. Phys. B 424 443.  Calabrese P and Cardy J 2004 Entanglement entropy and quantum field theory J. Stat. Mech. P06002.  Calabrese P and Cardy J 2005 Evolution of entanglement entropy in one-dimensional systems J. Stat. Mech. P04010.  Lipkin H J, Meshkov N and Glick A J 1965 Validity of many-body approximation methods for a solvable model: (I). Exact solutions and perturbation theory Nucl. Phys. 62 188.  Meshkov N, Glick A J and Lipkin H J 1965 Validity of many-body approximation methods for a solvable model: (II). Linearization procedures Nucl. Phys. 62 199.  Glick A J, Lipkin H J and Meshkov N 1965 Validity of many-body approximation methods for a solvable model: (III). Diagram summations Nucl. Phys. 62 211.  Popkov V and Salerno M 2005 Logarithmic divergence of the block entanglement entropy for the ferromagnetic Heisenberg model Phys. Rev. A 71 012301.  Latorre J I, Orús R, Rico E and Vidal J 2005 Entanglement entropy in the Lipkin–Meshkov–Glick model Phys. Rev. A 71 064101.  Haldane F D M 1988 Exact Jastrow–Gutzwiller resonating-valence-bond ground state of the spin-1/2 antiferromagnetic Heisenberg chain with 1/r2 exchange Phys. Rev. Lett. 60 635.  Shastry B S 1988 Exact solution of an Heisenberg antiferromagnetic chain with long-ranged interactions Phys. Rev. Lett. 60 639.  Frahm H 1993 Spectrum of a spin chain with inverse-square exchange J. Phys. A: Math. Gen. 26 L473.  Polychronakos A P 1993 Lattice integrable systems of Haldane–Shastry type Phys. Rev. Lett. 70 2329.  Polychronakos A P 1994 Exact spectrum of spin chain with inverse-square exchange Nucl. Phys. B 419 553.  Frahm H and Inozemtsev V I 1994 New family of solvable 1D Heisenberg models J. Phys. A: Math. Gen. 27 L801.  Gebhard F and Ruckenstein A E 1992 Exact results for a Hubbard chain with long-range hopping Phys. Rev. Lett. 68 244.  Finkel F and González-López A 2005 Global properties of the spectrum of the Haldane–Shastry spin chain Phys. Rev. B 72 174411.  Haldane F D M, Ha Z N C, Talstra J C, Bernard D and Pasquier V 1992 Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory Phys. Rev. Lett. 69 2021.  Haldane F D M 1991 Fractional statistics in arbitrary dimensions: a generalization of the Pauli principle Phys. Rev. Lett. 67 937.  Bernard D, Gaudin M, Haldane F D M and Pasquier V 1993 Yang–Baxter equation in long-range interacting systems J. Phys. A: Math. Gen. 26 5219.  Greiter M 2009 Statistical phases, momentum spacings for one-dimensional anyons Phys. Rev. B 79 064409.  Popkov V, Salerno M and Schütz G 2005 Entangling power of permutation-invariant quantum states Phys. Rev. A 72 032327.  Affleck I 1985 Critical behavior of two-dimensional systems with continuous symmetries Phys. Rev. Lett. 55 1355.  Schoutens K 1994 Yangian symmetry in conformal field theory Phys. Lett. B 331 335.  Bekenstein J D 1973 Black holes and entropy Phys. Rev. D 7 2333.  Bekenstein J D 1974 Generalized second law of thermodynamics in black-hole physics Phys. Rev. D 9 3292.  Hawking S W 1974 Black hole explosions? Nature 248 30.  Hawking S W 1976 Black holes and thermodynamics Phys. Rev. D 13 191.  Caruso F and Tsallis C 2008 Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics Phys. Rev. E 78 021102.  Tsallis C 1988 Possible generalization of Boltzmann–Gibbs statistics J. Stat. Phys. 52 479.  Tsallis C 2009 Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World (Berlin: Springer).  Botet R, Jullien R and Pfeuty P 1982 Size scaling for infinitely coordinated systems Phys. Rev. Lett. 49 478.  Botet R and Jullien R 1983 Large-size critical behavior of infinitely coordinated systems 1983 Phys. Rev. B 28 3955.  Ribeiro P, Vidal J and Mosseri R 2007 Thermodynamical limit of the Lipkin–Meshkov–Glick model Phys. Rev. Lett. 99 050402.  Ribeiro P, Vidal J and Mosseri R 2008 Exact spectrum of the Lipkin–Meshkov–Glick model in the thermodynamic limit and finite-size corrections Phys. Rev. E 78 021106.  Barthel T, Dusuel S and Vidal J 2006 Entanglement entropy beyond the free case Phys. Rev. Lett. 97 220402.  Orus R, Dusuel S and Vidal J 2008 Equivalence of critical scaling laws for many-body entanglement in the Lipkin–Meshkov–Glick model Phys. Rev. Lett. 101 25701.  Wilms J, Vidal J, Verstraete F and Dusuel S 2012 Finite-temperature mutual information in a simple phase transition 2012 J. Stat. Mech. P01023.  Barba J C, Finkel F, González-López A and Rodríguez M A 2010 Inozemtsev’s hyperbolic spin model and its related spin chain Nucl. Phys. B 839 499.  Castro-Alvaredo O A and Doyon B 2011 Permutation operators, entanglement entropy, and the XXZ spin chain in the limit J. Stat. Mech. P02001.  Castro-Alvaredo O A and Doyon B 2012 Entanglement entropy of highly degenerate states, fractal dimensions Phys. Rev. Lett. 108 120401.  Castro-Alvaredo O A and Doyon B 2013 Entanglement in permutation symmetric states and fractal dimensions, and geometric quantum mechanics J. Stat. Mech. P02016.  Latorre J I, Rico E and Vidal J 2004 Ground state entanglement in quantum spin chains Quantum Inf. Comput. 4 48.  Ercolessi E, Evangelisti S and Ravanini F 2010 Exact entanglement entropy of the XYZ model and its sine-Gordon limit Phys. Lett. A 374 2101.  Wolf M M 2006 Violation of the entropic area law for fermions Phys. Rev. Lett. 96 010404.  Barthel T, Chung M C and Schollwöck U 2006 Entanglement scaling in critical two-dimensional fermionic and bosonic systems Phys. Rev. A 74 022329.  Cadarso A, Sanz M, Wolf M M, Cirac J I and Pérez-García D 2013 Entanglement, fractional magnetization, and long-range interactions Phys. Rev. B 87 035114.  Tempesta P 2015 Formal groups and Z-entropies arXiv:1507.07436 [math-ph] Preprint.  Tempesta P 2016 Beyond the Shannon–Khinchin formulation: the composability axiom and the universal-group entropy Ann. Phys. 365 180.  Feller W 1971 An Introduction to Probability Theory and its Applications vol 2 3rd edn (New York: Wiley).