Entanglement entropy and the Berry phase in the solid state

Ryu, Shinsei; Hatsugai, Yasuhiro

doi:10.1103/physrevb.73.245115

Cited by 181 publications

(261 citation statements)

References 46 publications

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“…We finally mention the interesting connection between the block entropy and the Berry phase in lattice models of fermions recently discussed in (Ryu and Hatsugai, 2006).…”

Section: Fermi Systemsmentioning

confidence: 99%

Entanglement in many-body systems

Amico¹,

et al. 2008

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The recent interest in aspects common to quantum information and condensed matter has prompted a flory of activity at the border of these disciplines that were far distant untill few years ago. Numerous interesting questions have been addressed so far. Here we review an important part of this field, the properties of the entanglement in many-body systems. We discuss the zero and finite temperature properties of entanglement in interacting spin, fermion and boson model systems. Both bipartite and multipartite entanglement will be considered. In equilibrium we show how entanglement is tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium we discuss how to generate and manipulate entangled states by means of many-body Hamiltonians.

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“…We finally mention the interesting connection between the block entropy and the Berry phase in lattice models of fermions recently discussed in (Ryu and Hatsugai, 2006).…”

Section: Fermi Systemsmentioning

confidence: 99%

Entanglement in many-body systems

Amico¹,

et al. 2008

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“…The single particle entanglement spectrum is closely related to the entanglement entropy 42,[44][45][46] by…”

Section: Single Particle Entanglement Spectrummentioning

confidence: 99%

Entanglement spectrum classification ofCn-invariant noninteracting topological insulators in two dimensions

2013

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We study the single particle entanglement spectrum in 2D topological insulators which possess n-fold rotation symmetry. By defining a series of special choices of subsystems on which the entanglement is calculated, or real space cuts, we find that the number of protected in-gap states for each type of these real space cuts is a quantum number indexing (if any) non-trivial topology in these insulators. We explicitly show the number of protected in-gap states is determined by a Z n -index, (z1, ..., zn), where zm is the number of occupied states that transform according to m-th one-dimensional representation of the Cn point group. We find that the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues [1/n, 1 − 1/n]. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the Zm quantum numbers. Furthermore, we show that in a homogeneous system, the Z n index can be determined through an evaluation of the eigenvalues of point group symmetry operators at all high-symmetry points in the Brillouin zone. When disordered n-fold rotationally symmetric systems are considered, we find that the number of protected in-gap states is identical to that in the clean limit as long as the disorder preserves the underlying point group symmetry and does not close the bulk insulating gap.The study of novel topological phases of matter has become one of the most active fields in condensed matter physics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] . Chronologically speaking, the first of these phases to be experimentally realized is the integer quantum Hall (IQH) state, which is a striking departure from the traditional theory of conductivity due to its quantized Hall conductance and chiral edge modes 21 . Shortly after its experimental discovery, Thouless demonstrated that the special properties of the IQH state came down to its non-trivial bandstructure topology 22 . For the IQH state, one can prove that the quantized Hall conductance is equivalent to a momentum space integral of the Berry curvature, which is a non-zero integer (known as the Chern number in topology), multiplied by the conductance quantum e 2 /h. Beyond this, Haldane demonstrated that the IQH state may be further generalized to a system which does not require an external overall magnetic flux, yet still possesses a non-zero Chern number and quantized Hall conductance. This system is generally referred to as a Chern insulator 23 .Further as systems which preserve time-reversal invariance (TRI) pose an interesting problem as their Chern number vanishes in the presence of non-trivial topology. This necessitates the definition of a new quantum number capable of distinguishing between trivial and non-trivial topological states in systems which possess TRI. In 2D 1-3 and 3D 7,24 , one can define a Z 2 -number, which here we call γ 0 , as a quantity which is uniquely determined by the matrix representations of the time-reversal symmetry operator at all time-reversal inv...

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“…Entanglement properties also play an important role in condensed matter physics, such as phase transitions (Osterloh, et al 2002;Osborne & Nielsen 2002) and macroscopic properties of solids (Ghosh, et al 2003;Vedral 2004). Extensive research has been undertaken to understand quantum entanglement for spin chains, correlated electrons, interacting bosons as well as other models, see Amico, et al (2007), Audenaert, et al (2002), Fan & Korepin (2008), Katsura, et al (2007b), Fan, et al (2007), Arnesen, et al (2001), Korepin (2004), Verstraete, et al (2004a, b), Campos Venuti, et al (2006), Jin & Korepin (2004), Vedral (2004), Latorre, et al (2004aLatorre, et al ( , b, 2005, Orus (2005), Orus & Latorre (2004), Pachos & Plenio (2004), Plenio et al (2004), Fan & Lloyd (2005), Chen, et al (2004), Zanardi & Rasetti (1999), Popkov & Salerno (2004), Keating & Mezzadri (2004), Gu, et al (2003Gu, et al ( , 2004, , , Holzhey, et al (1994), Calabrese & Cardy (2004), Levin & Wen (2006), Kitaev & Preskill (2006), Ryu & Hatsugai (2006), Hirano & Hatsugai (2007) for reviews and references. Characteristic functions of quantum entanglement, such as von Neumann entropy and Renyi entropy, are obtained and discussed through studying reduced density matrices of subsystems (Fan, et al 2004;…”

Section: Introductionmentioning

confidence: 99%

Entanglement and Density Matrix of a Block of Spins in AKLT Model

Katsura

Hirano

et al. 2008

J Stat Phys

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We study a 1-dimensional AKLT spin chain, consisting of spins S in the bulk and S/2 at both ends. The unique ground state of this AKLT model is described by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a contiguous block of bulk spins in this ground state. It is shown that the density matrix is a projector onto a subspace of dimension (S + 1)2 . This subspace is described by nonzero eigenvalues and corresponding eigenvectors of the density matrix. We prove that for large block the von Neumann entropy coincides with Renyi entropy and is equal to ln (S + 1) 2 .

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Entanglement entropy and the Berry phase in the solid state

Cited by 181 publications

References 46 publications

Entanglement in many-body systems

Entanglement in many-body systems

Entanglement spectrum classification ofCn-invariant noninteracting topological insulators in two dimensions

Entanglement and Density Matrix of a Block of Spins in AKLT Model

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