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Entanglement of a 3D generalization of the Kitaev model on the diamond lattice

Ian Mondragon-Shem, Taylor L. Hughes

Published Jan 1, 2014• 7 citations
OpenAlex

Abstract

Abstract. We study the entanglement properties of a three dimensional generaliza-tion of the Kitaev honeycomb model proposed by Ryu [Phys. Rev. B 79, 075124, (2009)]. The entanglement entropy in this model separates into a contribution from a Z2 gauge field and that of a system of hopping Majorana fermions, similar to what occurs in the Kitaev model. This separation enables the systematic study of the en-tanglement of this 3D interacting bosonic model by using the tools of non-interacting fermions. In this way, we find that the topological entanglement entropy comes ex-clusively from the Z2 gauge field, and that it is the same for all of the phases of the system. There are differences, however, in the entanglement spectrum of the Majorana fermions that distinguish between the topologically distinct phases of the model. We further point out that the effect of introducing vortex lines in the Z2 gauge field will only change the entanglement contribution of the Majorana fermions. We evaluate this contribution to the entanglement which arises due to gapless Majorana modes that are trapped by the vortex lines.

Related papers

Entanglement of a 3D generalization of the Kitaev model on the diamond lattice

Ian Mondragon-Shem, Taylor L. Hughes

We study the entanglement properties of a three dimensional generalization of the Kitaev honeycomb model proposed by Ryu [Phys. Rev. B 79, 075124, (2009)]. The entanglement entropy in this model separates into a contribution from a $Z_2$ gauge field and that of a system of hopping Majorana fermions, similar to what occurs in the Kitaev model. This separation enables the systematic study of the entanglement of this 3D interacting bosonic model by using the tools of non-interacting fermions. In this way, we find that the topological entanglement entropy comes exclusively from the $Z_2$ gauge field, and that it is the same for all of the phases of the system. There are differences, however, in the entanglement spectrum of the Majorana fermions that distinguish between the topologically distinct phases of the model. We further point out that the effect of introducing vortex lines in the $Z_2$ gauge field will only change the entanglement contribution of the Majorana fermions. We evaluate this contribution to the entanglement which arises due to gapless Majorana modes that are trapped by the vortex lines.

arXiv:1407.3328•Journal of Statistical Mechanics Theory and Experiment•Oct 15, 2014•8 citations

Entanglement Entropy and Entanglement Spectrum of the Kitaev Model

Hong Yao, Xiao-Liang Qi

In this letter, we obtain an exact formula for the entanglement entropy of the ground state and all excited states of the Kitaev model. Remarkably, the entanglement entropy can be expressed in a simple separable form S = SG+SF, with SF the entanglement entropy of a free Majorana fermion system and SG that of a Z2 gauge field. The Z2 gauge field part contributes to the universal "topological entanglement entropy" of the ground state while the fermion part is responsible for the nonlocal entanglement carried by the Z2 vortices (visons) in the non-Abelian phase. Our result also enables the calculation of the entire entanglement spectrum and the more general Renyi entropy of the Kitaev model. Based on our results we propose a new quantity to characterize topologically ordered states--the capacity of entanglement, which can distinguish the st ates with and without topologically protected gapless entanglement spectrum.

arXiv:1001.1165•Physical Review Letters•Aug 16, 2010•250 citations

Three-dimensional topological phase on the diamond lattice

Shinsei Ryu

An interacting bosonic model of Kitaev type is proposed on the three-dimensional diamond lattice. Similarly to the two-dimensional Kitaev model on the honeycomb lattice, which exhibits both Abelian and non-Abelian phases, the model has two (``weak'' and ``strong'' pairing) phases. In the weak pairing phase, the auxiliary Majorana hopping problem is in a topological superconducting phase characterized by a nonzero winding number introduced by Schnyder et al. [Phys. Rev. B 78, 195125 (2008)] for the ensemble of Hamiltonians with both particle-hole and time-reversal symmetries. The topological character of the weak pairing phase is protected by a discrete symmetry.

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Entanglement and Majorana edge states in the Kitaev model

Saptarshi Mandal, Moitri Maiti, Vipin Kerala Varma

We investigate the von Neumann entanglement entropy and Schmidt gap in the vortex-free ground state of the Kitaev model on the honeycomb lattice for square/rectangular and cylindrical subsystems. We find that, for both the subsystems, the free-fermionic contribution to the entanglement entropy ${S}_{E}$ exhibits signatures of the phase transitions between the gapless and gapped phases. However, within the gapless phase, we find that ${S}_{E}$ does not show an expected monotonic behavior as a function of the coupling ${J}_{z}$ between the suitably defined one-dimensional chains for either geometry; moreover, the system generically reaches a point of minimum entanglement within the gapless phase before the entanglement saturates or increases again until the gapped phase is reached. This may be attributed to the onset of gapless modes in the bulk spectrum and the competition between the correlation functions along various bonds. In the gapped phase, on the other hand, ${S}_{E}$ always monotonically varies with ${J}_{z}$ independent of the subregion size or shape. Finally, further confirming the Li-Haldane conjecture, we find that the Schmidt gap $\mathrm{\ensuremath{\Delta}}$ defined from the entanglement spectrum also signals the topological transitions but only if there are corresponding zero-energy Majorana edge states that simultaneously appear or disappear across the transitions. We analytically corroborate some of our results on entanglement entropy, the Schmidt gap, and the bulk-edge correspondence using perturbation theory.

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Classification of gapless Z2 spin liquids in three-dimensional Kitaev models

Kevin M. O'Brien, Maria Hermanns, Simon Trebst

In quantum magnetism, Kitaev's honeycomb model is well known for its gapped and gapless spin liquid ground states in which the elementary spin degrees of freedom fractionalize into Majorana fermions and a ${\mathbb{Z}}_{2}$ gauge field. In this paper, the authors discuss the gapless spin liquids arising in three-dimensional Kitaev models and show that the gapless Majorana fermions form metallic states which, depending on the underlying lattice structure, exhibit Fermi surfaces, nodal lines, or Weyl points.

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Unpaired Majorana modes in the gapped phase of Kitaev's honeycomb model

Olga Petrova, Paula Mellado, Oleg Tchernyshyov

We study the gapped phase of Kitaev's honeycomb model (a ${Z}_{2}$ spin liquid) in the presence of lattice defects. We find that some dislocations and bond defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a ${Z}_{2}$ gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a ${Z}_{2}$ vortex around a dislocation. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes.

arXiv:1307.1668•Physical Review B•Oct 14, 2013•21 citations
Abstract. We study the entanglement properties of a three dimensional generaliza-tion of the Kitaev honeycomb model proposed by Ryu [Phys. Rev. B 79, 075124, (2009)]. The entanglement entropy in this model separates into a contribution from a Z2 gauge field and that of a system of hopping Majorana fermions, similar to what occurs in the Kitaev model. This separation enables the systematic study of the en-tanglement of this 3D interacting bosonic model by using the tools of non-interacting fermions. In this way, we find that the topological entanglement entropy comes ex-clusively from the Z2 gauge field, and that it is the same for all of the phases of the system. There are differences, however, in the entanglement spectrum of the Majorana fermions that distinguish between the topologically distinct phases of the model. We further point out that the effect of introducing vortex lines in the Z2 gauge field will only change the entanglement contribution of the Majorana fermions. We evaluate this contribution to the entanglement which arises due to gapless Majorana modes that are trapped by the vortex lines.
We study the entanglement properties of a three dimensional generalization of the Kitaev honeycomb model proposed by Ryu [Phys. Rev. B 79, 075124, (2009)]. The entanglement entropy in this model separates into a contribution from a Z2Z_2Z2​ gauge field and that of a system of hopping Majorana fermions, similar to what occurs in the Kitaev model. This separation enables the systematic study of the entanglement of this 3D interacting bosonic model by using the tools of non-interacting fermions. In this way, we find that the topological entanglement entropy comes exclusively from the Z2Z_2Z2​ gauge field, and that it is the same for all of the phases of the system. There are differences, however, in the entanglement spectrum of the Majorana fermions that distinguish between the topologically distinct phases of the model. We further point out that the effect of introducing vortex lines in the Z2Z_2Z2​ gauge field will only change the entanglement contribution of the Majorana fermions. We evaluate this contribution to the entanglement which arises due to gapless Majorana modes that are trapped by the vortex lines.
In this letter, we obtain an exact formula for the entanglement entropy of the ground state and all excited states of the Kitaev model. Remarkably, the entanglement entropy can be expressed in a simple separable form S = SG+SF, with SF the entanglement entropy of a free Majorana fermion system and SG that of a Z2 gauge field. The Z2 gauge field part contributes to the universal "topological entanglement entropy" of the ground state while the fermion part is responsible for the nonlocal entanglement carried by the Z2 vortices (visons) in the non-Abelian phase. Our result also enables the calculation of the entire entanglement spectrum and the more general Renyi entropy of the Kitaev model. Based on our results we propose a new quantity to characterize topologically ordered states--the capacity of entanglement, which can distinguish the st ates with and without topologically protected gapless entanglement spectrum.
An interacting bosonic model of Kitaev type is proposed on the three-dimensional diamond lattice. Similarly to the two-dimensional Kitaev model on the honeycomb lattice, which exhibits both Abelian and non-Abelian phases, the model has two (``weak'' and ``strong'' pairing) phases. In the weak pairing phase, the auxiliary Majorana hopping problem is in a topological superconducting phase characterized by a nonzero winding number introduced by Schnyder et al. [Phys. Rev. B 78, 195125 (2008)] for the ensemble of Hamiltonians with both particle-hole and time-reversal symmetries. The topological character of the weak pairing phase is protected by a discrete symmetry.
We investigate the von Neumann entanglement entropy and Schmidt gap in the vortex-free ground state of the Kitaev model on the honeycomb lattice for square/rectangular and cylindrical subsystems. We find that, for both the subsystems, the free-fermionic contribution to the entanglement entropy SE{S}_{E}SE​ exhibits signatures of the phase transitions between the gapless and gapped phases. However, within the gapless phase, we find that SE{S}_{E}SE​ does not show an expected monotonic behavior as a function of the coupling Jz{J}_{z}Jz​ between the suitably defined one-dimensional chains for either geometry; moreover, the system generically reaches a point of minimum entanglement within the gapless phase before the entanglement saturates or increases again until the gapped phase is reached. This may be attributed to the onset of gapless modes in the bulk spectrum and the competition between the correlation functions along various bonds. In the gapped phase, on the other hand, SE{S}_{E}SE​ always monotonically varies with Jz{J}_{z}Jz​ independent of the subregion size or shape. Finally, further confirming the Li-Haldane conjecture, we find that the Schmidt gap $\mathrm{\ensuremath{\Delta}}$ defined from the entanglement spectrum also signals the topological transitions but only if there are corresponding zero-energy Majorana edge states that simultaneously appear or disappear across the transitions. We analytically corroborate some of our results on entanglement entropy, the Schmidt gap, and the bulk-edge correspondence using perturbation theory.
In quantum magnetism, Kitaev's honeycomb model is well known for its gapped and gapless spin liquid ground states in which the elementary spin degrees of freedom fractionalize into Majorana fermions and a Z2{\mathbb{Z}}_{2}Z2​ gauge field. In this paper, the authors discuss the gapless spin liquids arising in three-dimensional Kitaev models and show that the gapless Majorana fermions form metallic states which, depending on the underlying lattice structure, exhibit Fermi surfaces, nodal lines, or Weyl points.
We study the gapped phase of Kitaev's honeycomb model (a Z2{Z}_{2}Z2​ spin liquid) in the presence of lattice defects. We find that some dislocations and bond defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a Z2{Z}_{2}Z2​ gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a Z2{Z}_{2}Z2​ vortex around a dislocation. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes.