Entanglement Chern Number for an Extensive Partition of a Topological Ground State
Entanglement Chern Number for an Extensive Partition of a Topological Ground State
Takahiro Fukui and Yasuhiro Hatsugai
If an extensive partition in two dimensions yields a gapful entanglement spectrum of the reduced density matrix, the Berry curvature based on the corresponding entanglement eigenfunction defines the Chern number. We propose such an entanglement Chern number as a useful, natural, and calculable topological invariant, which is potentially relevant to various topological ground states. We show that it serves as an alternative topological invariant for time-reversal invariant systems and as a new topological invariant for a weak topological phase of a superlattice Wilson-Dirac model. In principle, the entanglement Chern number can also be effective for interacting systems such as topological insulators in contrast to invariants.
The Chern number, which determines the quantized Hall conductivity in the integer quantum Hall effect as shown by Thouless et al. [TKNN82, Kohmoto85], has become increasingly popular for condensed matter physicists owing to the recent remarkable development of the classification [Zirnbauer96, AltZir97, SRFL08, Kitaev08] of the topological phases of matter. [KanMel05a, KanMel05b, BerZha06, BHZ06, FuKan06, FKM07, MooBal07, QHZ08, Roy09, FuKan07, Fu11, TIreviews] Even in three dimensions, a type of topological insulator can be characterized by the Chern number, called the mirror Chern number, if a system has mirror symmetry.[TFK08] The numerical method of computing the Chern number has also been established [FHS05] and widely applied to various systems with complicated multiband structures. The Chern number is a topological invariant for the bulk, whereas the number of edge states for a system with a boundary gives the same topological invariant. This is well known as the bulk-edge correspondence.[Hatsugai93]
The entanglement spectrum of the reduced density matrix also informs us of the edge states along an artificial boundary introduced by a partition of the system. [RyuHat06, LiHal08] Nowadays, this is widely used to clarify the property of topological insulators. [PHB10, TZV10, HPB11, AHB11, CPSV11, HuaAro12, FGB13] Recently, Hsieh and Fu have introduced the notion of the “bulk entanglement spectrum”. [HsiFu13, HFQ14] By considering an extensive and translationally invariant partition in real space, they demonstrated that a topological ground state intrinsically has a hidden phase transition.
In this work, we study the bulk property of some topological ground states using the wave function of the entanglement Hamiltonian. Namely, we define the Chern number from the Berry curvature based on the entanglement wave function, which is referred to as the entanglement Chern number. For such a Chern number to be well-defined, we consider an extensive partition without a clear boundary between patches of the partition, which makes the entanglement spectrum gapped generically. We then show that this serves as an alternative topological invariant for a time-reversal invariant system. At the same time, it yields a new topological invariant for a system with some spatial structure such as a superlattice system.
To begin with, let us reconsider a typical model of the topological insulator, the Kane-Mele model [KanMel05b], , and exemplify the usefulness of the entanglement Chern number. This is one of models for the quantum spin Hall effect (QSHE), describing the electrons on the honeycomb lattice with spin-orbit couplings. When the Rashba spin-orbit coupling vanishes, the Hamiltonian is decoupled into spin-up and spin-down sectors such that , where is equivalent to the spinless Haldane model for the anomalous Hall effect. [Haldane88] These two sectors are transformed into each other under time reversal, making the model time-reversal invariant. The half-filled ground state of this decoupled model can be characterized by two Chern numbers , being a trivial insulating state when they are , and the QSHE state (or anomalous Hall state from the viewpoint of the Haldane model) when . Although the Rashba spin-orbit coupling does not break time-reversal symmetry, it breaks the spin conservation. Therefore, it is no longer possible to define the set of Chern numbers for the generic Kane-Mele model. What we can know is only the total Chern number , but it is trivial () because of time-reversal symmetry.
If one gives up defining the Chern number in the Brillouin zone, a spin Chern number is available [SWSH06, FukHat07a] using a spin-dependent twisted boundary condition. When the Rashba coupling vanishes, such a spin Chern number corresponds to , as it should be. Even without disorder or interactions, however, we have to always compute it by definition in the coordinate space, which is impossible in the Brillouin zone.
The breakthrough was the Z topological invariant introduced by Fu and Kane. [FuKan06] This is, roughly speaking, half the Chern number, i.e., the integration of the Berry curvature over half the Brillouin zone from which the Berry phase along the boundary is subtracted. In this definition, a specific gauge fixing between the wave function of the Kramers pair is needed. [FuKan06] The numerical method of computing the Z invariant was also given, [FukHat07b] which is just a straightforward generalization of the method for the Chern number. If the system has inversion symmetry, the above formula of the Z invariant reduces to the product of the parity of the occupied bands at the time-reversal invariant momenta.[FuKan07]
In what follows, we first propose an alternative invariant for the QSHE, the entanglement spin Chern number. It is similar to the spin Chern number via a twisted boundary condition: Indeed, they are manifestly equivalent when the Rashba coupling vanishes. Moreover, the entanglement spin Chern number can be defined in the Brillouin zone even without the spin conservation. To be concrete, let us regard the spin degrees of freedom as a partition of the system. Then, tracing out one spin sector from the density matrix yields an effective Hamiltonian, called the entanglement Hamiltonian, for the other spin sector. Since the partition in terms of the spin is manifestly extensive and maintains the translational invariance, the entanglement Hamiltonian can be represented in the Brillouin zone, as mentioned above. If the spectrum of such a bulk entanglement Hamiltonian has a gap, we can define a new Chern number different from for . Here, note that the entanglement Hamiltonian thus obtained never has time-reversal symmetry. This implies that a nonzero Chern number can be expected in general. In the classification of the topological phases of matter, symmetry protection has surely been playing a crucial role, [SRFL08, Kitaev08] but symmetry constraints are sometimes too restrictive to define the corresponding topological invariants. Thus, the entanglement Chern number proposed in this paper allows for various possibilities to capture the characteristic feature of symmetry-protected topological phases. The entanglement Chern number so far discussed is useful not only for the QSHE (or more generically symmetry-protected topological states) but also some other systems with some internal or other degrees of freedom. For example, if the system has a spatial structure such as a superlattice, it gives a new topological invariant, as will be discussed in the latter part of this paper.
Let be a many-body ground state of a given noninteracting Hamiltonian , and let and be the subsystems of the total system . The reduced density matrix and the corresponding entanglement Hamiltonian are defined by tracing out such that
where is the normalization constant. In the case of noninteracting fermions, the entanglement Hamiltonian can be written as . Let us next introduce the correlation matrix
where and denote the sites as well as some internal degrees of freedom such as the spin or orbital. When and are restricted in , the correlation matrix may be called , and it is shown as[Peschel03]
Thus, the eigenstates of are those of .
As introduced by Hsieh and Fu, [HsiFu13] we consider the extensive partition with translational symmetry. For example, in the case of the Kane-Mele model, we choose and as the spin-up and spin-down sectors. Then, the Fourier transformation gives
where is the projection operator to the occupied bands expressed by the single-particle multiplet wave functions , and is the projection operator to . Solving the eigenvalue equation for ,
we can define the (entanglement) Chern number as usual. To this end, let us first consider the spectrum . Without the projection operator in Eq. (Entanglement Chern Number for an Extensive Partition of a Topological Ground State), has only two obvious eigenvalues, i.e., and , denoting the occupied and unoccupied states, respectively. The wave functions with the eigenvalue are nothing but those of the ground state for the total system . Because of the projection operator in Eq. (Entanglement Chern Number for an Extensive Partition of a Topological Ground State), the eigenvalues of are not restricted to 1 and 0. Here, we assume that some of them form bands at approximately and others at approximately , and that there is a finite gap between these two bands. Then, their origin is clear: The former are occupied states and the latter are unoccupied states for the subsystem . Therefore, it is natural to choose the entanglement Chern number of the upper bands to characterize the topological property of the ground state. In the examples we study below, in Eq. (Entanglement Chern Number for an Extensive Partition of a Topological Ground State) is a matrix, and the behavior of its spectrum indeed exhibits such a property.[footnote]
To be concrete, we introduce the Berry connection and the curvature , where is the multiplet wave function of the upper bands with , and . The entanglement Chern number is thus defined by
The above procedure is quite easy to carry out numerically using the link and plaquette variables on the discretized Brillouin zone. [FHS05] In this calculation of the (entanglement) Chern number, it does not depend on the gauge of the (entanglement) wave function even for the QSHE with time-reversal symmetry.
Let us now compute the entanglement spin Chern number for the Kane-Mele model. As has already been discussed, this model has two phases, the QSHE phase and trivial insulating phase. They are distinguished by the Z invariant or spin Chern number. Let us choose partitions and as the up-spin and down-spin states, respectively. These partitions are manifestly extensive and translationally invariant. From the numerical calculation,[FHS05] it turns out that the QSHE phase and trivial phase have the entanglement spin Chern numbers and , respectively. Therefore, we conclude that the entanglement spin Chern number can distinguish the phases in time-reversal invariant classes with the conventional Chern number .
We now discuss some details of the numerical calculations. Although the entanglement spin Chern number indeed changes at the transition point between two phases, it is quite difficult to observe the gap closing in the entanglement spectrum, at least, on the discretized lattice of the Brillouin zone that we adopted in our calculation. Furthermore, the entanglement spectrum , and thus the entanglement entropy are similar on both sides of the transition point, even in the vicinity of the transition point. Such behavior of the entanglement spectrum is far from being the conventional one in a partition with a boundary showing the spectral flow of the edge states. This implies that we are studying indeed the bulk entanglement spectrum without boundaries. The change in the (entanglement) Chern number, however, should be due to the gap closing. Therefore, we surmise that it occurs in a singular way like the delta function at a few points on the Brillouin zone, which is generically impossible to observe on the meshes of a discretized Brillouin zone. One reason why the computation of the entanglement spin Chern number needs a rather larger number of meshes (order of meshes), than in the case of the Z invariant (order of meshes), near the transition points may be this singular behavior of the spectrum.