# Entanglement negativity in free-fermion systems: An overlap matrix approach

###### Abstract

In this paper, we calculate the entanglement negativity in free-fermion systems by use of the overlap matrices. For a tripartite system, if the ground state can be factored into triples of modes, we show that the partially transposed reduced density matrix can be factorized and the entanglement negativity has a simple form. However, the factorability of the ground state in a tripartite system does not hold in general. In this situation, the partially transposed reduced density matrix can be expressed in terms of the Kronecker product of matrices. We explicitly compute the entanglement negativity for the Su-Schrieffer-Heeger model, the integer Quantum Hall state, and a homogeneous one-dimensional chain. We find that the entanglement negativity for the integer quantum Hall states shows an area law behavior. For the entanglement negativity of two adjacent intervals in a homogeneous one-dimensional gas, we find agreement with the conformal field theory. Our method provides a numerically feasible way to study the entanglement negativity in free-fermion systems.

## I Introduction

The study of quantum entanglement provides a powerful tool to analyze the many-body states in condensed matter physicsKitaev (); Levin (); Calabrese0 (); Amico (); Eisert (); Vidal0 (); Calabrese00 (); Calabrese000 (); Benatti (); Benatti2 (). The most celebrated measure of entanglement is given by the entanglement entropy (von Neumann entropy) of the reduced density matrix of the subsystem . This quantity measures the entanglement between two complementary subsystems and , when the total system is in a pure state. One remarkably common result is an area law behavior of the entanglement entropy in gapped systems, which grows proportionally with the boundary between two subsystemsEisert (). An important exception is the one-dimensional systems at criticality, where the entanglement entropy has a logarithmic scaling behaviorCalabrese0 (); Vidal0 (); Eisert () and have been understood by use of a conformal field theory (CFT) approachCalabrese000 (); Calabrese0 ().

However, using the entanglement entropy as a probe to diagonalize the entanglement for a mixed state is a difficult task. It requires a heavy optimization process. Hence, an alternative measure of entanglement for a mixed state has been proposed—the entanglement negativityEisert1 (); Vidal (); Plenio (). The advantage of using entanglement negativity is that it only requires linear algebraic computations. The entanglement negativity is defined as follows. Suppose a tripartite system is divided into , , and subsystems. The ground state after tracing out the degrees of freedom in a subsystem is a general mixed state. The entanglement negativity, which quantifies the entanglement between and , is obtained by partially transposing (with respect to the degrees of freedom in ) the reduced density matrix , which is denoted as . Then the entanglement negativity between subsystems and is defined as

(1) |

where the trace norm represents the sum of all absolute values of the eigenvalues of . Recently, the entanglement negativity has been extensively studied in numerous many-body systems, e.g., harmonic oscillators in one dimensionAudenaert (); Marcovitch (); Ferraro (); Cavalcanti (); Anders (); Anders1 (), quantum spin chainsWichterich2 (); Wichterich3 (); Bayat (); Bayat2 (); Bayat3 (); Santos (), systems with topological orders Castelnovo (); Lee (), and (2+1) dimensional Chern-Simons theoriesEN_CS (). A CFT approachCalabrese (); Calabrese2 () and many useful numerical methods are also developed, including tree tensor techniquesCalabrese5 (), Monte Carlo simulationsAlba (); Chung (), and rational interpolationsNobili (). Further studies on non-equilibrium situationsEisler (); Coser (); Hoogeveen (); Wen () and finite temperatureCalabrese4 (); Eisler () refine our understanding of the entanglement negativity.

In this work, we demonstrate a systematical way to compute the entanglement negativity in free-fermion systems by using the overlap matrices. For the prior works on entanglement negativity in free fermion systemsEisler2 (); Coser2 (); Coser3 (); Eisler3 (); Coser4 (), the quantities people calculated are the moments of partially transposed reduced denity matrix , with . In other words, the entanglement negativity, which is obtained from the trace norm , has not been computed in the prior works due to reasons of technical difficulty. In our work, we will give an explicit calculation of by using the overlap matrices, and therefore the entanglement negativity is obtained. The overlap matrices are built up from the single-particle states corresponding to the lowest (occupied) energy levels of the systemIsrael (); Calabrese6 (); Calabrese3 (); Calabrese7 (); Ossipov (); Drut ():

(2) |

where the integration is restricted to the subsystems or in arbitrary dimension . We show that for a tripartite system, when the overlap matrices , , and can be simultaneously diagonalized, the ground state of fermions can be factored into triples of modes. In this situation, the partially transposed reduced density matrix can be factorized and the entanglement negativity can be obtained from the spectra of the overlap matrices. On the other hand, when the simultaneous diagonalizability of the overlap matrices does not hold, the factorability of the ground state in a tripartite system is failed. In this situation, the partially transposed reduced density matrix can only be expressed in terms of the Kronecker product of matrices. Then the entanglement negativity is directly obtained from the partially transposed reduced density matrix. The overlap matrix approach provides a systematical way to calculate the entanglement negativity in any free-fermion systems, which can be either gapped topological phases or critical systems. This approach is also applicable for disordered systems and can be extended to higher dimensions.

The paper is organized as follows: In Sec. II, we review the entanglement negativity for a pure state. In Sec. III.1, we obtain the entanglement negativity for a mixed state when the ground state can be factored into triples of modes in a tripartite system. We find that the entanglement negativity for the integer quantum Hall state satisfies an area law. In Sec. III.2, we derive the entanglement negativity for a general mixed state. We compute the entanglement negativity for free fermions on a ring and show the entanglement negativity is in agreement with CFT. In Sec. IV, we conclude our results.

## Ii Entanglement negativity in free-fermion systems for pure states

Let us start with the Hamiltonian for free fermions

(3) |

where is the fermion operator with corresponding energy , and is the number of bands. The ground state is

(4) |

where is the total number of particles in the system. We consider a bipartite system that the system is partitioned into and parts and the single-particle wave function can be partitioned by the projection operator . Following Ref. [Israel, ], we introduce an overlap matrix,

(5) |

where and is the orthogonal projection operator on Hilbert space , such that . Since , one can find a unitary matrix that simultaneously diagonalizes and , such that and . The single-particle Hilbert space is decomposed into two parts . The orthonormal modes at and are

(6) |

Let is the subspace of the single particle Hilbert space . Since is unitary, we have . The ground state in Eq. (4) can be rotated by this unitary matrix . By using

(7) |

the ground state under the rotation can be written as (up to a global phase)

(8) |

where .

We can compute the entanglement negativity directly from the partially transposed density matrix . The density matrix from the ground state under occupation basis in Eq. (4) is

(9) |

where the basis is .

The partially transposed density matrix is

(10) |

where the basis is .

Notice that the eigenvalues of the Kronecker product of two matrices are , where and are the eigenvalues of and , respectively. The eigenvalues of are the product of one of the eigenvalues of each . The set of eigenvalues of is with being the label of the elements in the set. Hence the entanglement is

(11) |

Examples of the entanglement negativity for pure states are shown in App. A.

## Iii Entanglement negativity in free-fermion systems for mixed states

### iii.1 The ground state in a tripartite system can be factored into triples of modes

Now we generalize the bipartite case to a tripartite system. The overlap matrices , , and are constructed from Eq. (5), with the projected single-particle wave function , , and . If the overlap matrices can be simultaneously diagonalized by a unitary matrix , we can find the orthonormal modes at the subsystems

(12) | ||||

Since the single-particle Hilbert space does note change under unitary transformation, i.e., , and

(13) |

the ground state can be factored into triples of modes

(14) |

where with .

The reduced density matrix after tracing part is

(15) |

where the basis of the reduced density is .

The partially transposed reduced density matrix after transposing the is

(16) |

The set of eigenvalues of is with being the label of the elements in the set. The entanglement negativity is

(17) |

Notice that if the probability for all the eigenstates in subsystem is zero, which can be realized by taking the dimension of Hilbert space of to be zero, the entanglement negativity becomes the case of pure states,

(18) |

where is the Hibert space of region .

In the following, we consider the entanglement negativity between and regions of two examples in which the overlap matrices can be simultaneously diagonalized. The first one is the fully dimerized Su-Schrieffer-Heeger (SSH) model and the second one is the integer quantum Hall state (IQHS).

#### iii.1.1 Dimerized Su-Schrieffer-Heeger (SSH) model

The Su-Schrieffer-Heeger model is first introduced in Ref. [Su79, ], which describes one-dimensional polyacetylene. The Hamiltonian of a dimerized SSH model is

(19) |

where is the hopping amplitude and is the fermion operator. The ground state for this dimerized SSH model is

(20) |

where .

We consider an infinite chain that the system is divided into , , and as shown in Fig. 1. The state is localized at orbital on -th site and at orbital on -th site. This localized state is the eigenstate of three overlap matrices, , , and . For the geometry shown in Fig. 1(a), the corresponding eigenvalues of and are,

(21) |

(22) |

For the geometry shown in Fig.1(b), the corresponding eigenvalues of and are,

(23) |

(24) |

##### Adjacent intervals

##### Disjointed intervals

From Eqs. (17), (23), and (24), the entanglement negativity between and [see Fig. 1(b)] after tracing out is zero. We can understand the vanishing of the entanglement negativity from the following picture. All the single-particle states are localized on the dimerized bonds that no single-particle states can spread out on both and . Hence there is no entanglement between and .

#### iii.1.2 Integer Quantum Hall state (IQHS)

We consider the IQHS on a cylinder with the periodic boundary condition along direction. The entanglement entropy/spectrum analysis on IQHS on a cylinder has been studied in Refs. [Ivan, ; Chang, ]. The single-particle state in the lowest Landau level with the Landau gauge is

(26) |

where

(27) |

The particles are restricted in the region and the number of particles is (we set the magnetic length ). The system is divided into , , and . For the case of two adjacent intervals and . We have , , and along axis [Fig. 2(a)]. From Refs. [Ivan, ; Chang, ], the corresponding probabilities for a particle at regions , and are

(28) |

For the case of two disjointed intervals and . We have , , and along axis [Fig. 2(b)]. The corresponding probabilities for a particle at regions , , and are

(29) |

For simplicity, we set . From Eq. (17), the entanglement negativity between and after tracing out is

(30) |

We can also compute the slope of the entanglement negativity by taking the continuous limit

(31) |

##### Adjacent intervals

The entanglement negativity is linearly proportional to [Fig. 3 (a)], i.e., an area law behavior. For , the dimension of Hilbert space of region is zero that leads to the vanishing of the entanglement negativity. In the limit that , the entanglement negativity increases monotonically as increasing. Notice that when , all the particles are restricted in the subsystem . Hence the entanglement negativity goes to zero. The slope of the entanglement computed from Eq. (31) is shown in Fig. 3 (b). The plateau of is around , which indicates the system approaches to the case of a pure state when . We demonstrate the case of a pure state in the IQHS in the Appendix. A.2.

##### Disjointed intervals

In Fig. 4 (a), the entanglement negativity also shows an area law behavior as the case of adjacent intervals. For , the dimension of Hilbert space of region is zero. Hence the entanglement negativity is exact the same as the entanglement negativity for a pure state. As is increasing, the entanglement negativity decreases. This indicates that and become less entangled when the distance between these two intervals becomes larger. The slope of the entanglement computed from Eq. (31) is shown in Fig. 4 (b). When , the system can be described by a pure state and . Since the decay length in the system is proportional to , and become less entangled when the distance between two disjointed intervals is greater then . We find that monotonically decreases as is increasing.

#### iii.1.3 Discussion

For the IQHS, as the distance is larger than the magnetic length , the entanglement negativity vanishes dramatically. It may be straightforwardly understood in the following. As studied in Ref. [Qi2011, ], for a (2+1) dimensional topological quantum state which possesses chiral edge states, such as the IQHS we discussed here, the reduced density matrix of a spatial region can be expressed as the density matrix of the corresponding edge states. In other words, the entanglement is mainly contributed by the boundary states which are localized at the interface. Therefore, for the configuration in Fig. 2, the reduced density matrix can be expressed as

(32) |

where represents the density matrix corresponding to the boundary states located at the interface between region and region . In this case, one has

(33) |

which leads to a vanishing entanglement negativity. Our conclusion also applies to a general topological ordered states such as the fractional quantum hall effect. The topological flux threading through the cylinder in Fig. 2 can be in a superposition of different topological sectors, and the reduced density matrix has the expression

(34) |

where is the probability corresponding to the topological sector . Again, the state is explicitly separable, and therefore the entanglement negativity is zero. Note that the above conclusion was also obtained in Ref. [Lee, ] based on the solvable toric code mode, but here we give an argument which holds for a general topological ordered state in (2+1)D. A more strict proof based on the boundary state picture can be found in Ref. [Wen2015, ]. What ia interesting, most recently, the mutual information between and in Fig. 2 for a (2+1) dimensional topological ordered state was studied Qi2015 (). It was found that

(35) |

which is nonzero. One may understand the difference between the entanglement negativity and the mutual information in the following way. As pointed out in Ref. [Qi2015, ], the nonzero contribution in Eq. (35) comes from the “classical” correlation between extended objects. It is known that the mutual information contains both classical and quantum correlations, while the entanglement negativity measures the completely quantum part of correlationsQiEmail (); Castelnovo (). Therefore, one may have a vanishing entanglement negativity while the mutual information is finite.

### iii.2 The ground state in a tripartite system cannot be factored into triples of modes

For the case where the overlap matrices cannot be simultaneous diagonalized in a tripartite system, the ground state cannot be factorized. Although the ground state cannot be factorized, we can express the ground state in terms of the fermion operators in each subsystem. The procedure of constructing the ground state is the following. We start with the bipartite ground state

(36) |

The orthonormal modes at are the set . Next, we use these orthonormal modes in subsystem to construct the overlap matrices and ,

(37) |

where is the orthogonal projection operator on Hilbert space . In other words, the tripartite system is built from partitioning the system into and parts first, and partitioning the subsystem into and parts. Since is the orthonormal set, we have . We can simultaneously diagonalize and by a unitary matrix , such that and . The orthonormal modes in and are

(38) |

Now the ground state can be expressed as

(39) |

where with . The second line in the above equation comes from

(40) |

In terms of occupation basis, the ground state can be expressed as

(41) |

where indicates that there are electrons in subsystem and indicates that there are electrons in subsystem . We can construct the partially transposed reduced density matrix from the occupational basis [see detail derivations in App. B]. The final expression of is

(42) |

where are four elemental matrices (see Eq. (59) in App. B) as the building blocks of . Thus, the entanglement negativity can be obtained by taking the trace norm, .

#### iii.2.1 Free fermions on a 1D ring

We consider free fermions on a periodic one-dimensional chain. The normalized single-particle eigenstate is

(43) |

with energy . The overlap matrix can be constructed from these eigenstates from Eq. (5) and the partially transposed reduced density matrix can be obtained from Eq. (62).

Due to the limitation of numerical calculation, we consider few fermions (up to five) on a long chain (). For simplicity, two intervals and have equal length and the total length of the ring is .

##### Adjacent intervals

The configuration of adjacent intervals is shown in Fig. 5(a). For fixed number of particles, the entanglement negativity of adjacent intervals depends on the single parameter . The entanglement negativity increases monotonically as the function of [Fig. 6 (a)]. In the CFT calculationCalabrese (); Calabrese2 (), . In order to compare with the CFT calculation and get rid of the constant part in the entanglement negativity, we plot for . in Fig. 6 (b). The entanglement negativity after the subtraction of agrees with the CFT prediction.

##### Disjointed intervals

The entanglement negativity of disjointed intervals for harmonic chains has been studied in Refs. [Wichterich2, ; Marcovitch, ; Calabrese, ]. Here, we consider a symmetric configuration of disjointed intervals as shown in Fig. 5(b). For fixed number of particles, the entanglement negativity of disjointed intervals depends on the single parameter , where is the distance between two intervals. The entanglement negativity decreases monotonically as the function of [Fig. 7].

## Iv Discussion and conclusion

In this paper, we systematically derive the entanglement negativity in free-fermion systems by using the overlap matrices. For a pure state, the entanglement negativity is obtained from a bipartite system and the ground state can always be factored into pairs of modes. For a mixed state, the factorability of the ground state in a tripartite system does not always hold. In the case that the ground state can still be factored into triples of modes, the entanglement negativity has a simple form. We show two examples in this case: the dimerized SSH model and the IQHS. The entanglement negativity for the former is when two intervals are adjacent. The entanglement negativity for the latter shows an area law behavior. For both examples, we observe that the entanglement negativity vanishes when the separation between two intervals is greater than the correlation length of the systems. We argue the vanish of the entanglement negativity holds for a general topological ordered state in D. A proof based on the edge theory approach as well as the bulk theory calculation is given in Refs. [Wen2015, ] and Ref. [EN_CS, ], respectively. On the other hand, if the ground state cannot be factored into triples of modes, we show that the partially transposed reduced density matrix can be expressed in terms of the Kronecker product of matrices. In this case, we compute the entanglement negativity for free fermions on a one-dimensional ring. We find that the entanglement negativity of two adjacent intervals agrees with the CFT prediction.

We would like to end the conclusion by mentioning a few of future problems.

It is interesting to study the non-equilibrium dynamics of entanglement negativity for a free-fermion system. In particular, as discussed in Ref. [Wen, ], for the time evolution of entanglement negativity after a local quench, the result is not universal for some region , i.e., may depend on the concrete lattice models. It is interesting to check the behavior of for a critical free fermion chain, and compare it with the results of other lattice models Eisler ().

In our current work, we mainly focus on the free-fermion systems at zero temperature. It is desirable to generalize our method to the finite temperature case, and compare with the CFT resultsCalabrese4 ().

One can easily apply our methods to free-fermion systems with disorder. Studying the quantum phase transitions of disordered systems by using the entanglement negativity is needed for a future investigation.

It is interesting to extend our method to the study of interacting fermionic systems, such as fractional quantum hall systems. Although there is no entanglement negativity for two disjoint intervals, it is still interesting to study the entanglement negativity for two adjacent intervals, where we may have a nonvanishing contribution from the long-range entanglement.

## V Acknowledgement

The authors thank Shinsei Ryu for his encouragement and discussions during this work. PYC is supported by the Rutgers Center for Materials Theory group postdoc grant.

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## Appendix A Entanglement negativity for pure states

Let us consider a bipartite system that the ground state can be Schmidt decomposed as

(44) |

where is the coefficient that in between zero and one and the system is bipartite into A and B parts. The density matrix and the reduced density matrix are defined as

(45) |

One can define a partial transpose of the density matrix as

(46) |

The trace norm of is defined by

(47) |

where is the eigenvalue of .

The (logarithmic) entanglement negativity is defined as

(48) |

### a.1 Su-Schrieffer-Heeger (SSH) model

Let us consider an one-dimensional tight binding Hamiltonian (SSH model) under periodic boundary condition

where and is the fermion operator. The configuration of a bipartite ring is shown in Fig. 8. We consider the lengths of and are equal. The overlap matrix has two eigenstates with eigenvalue when . On the other hand, there is no eigenstates of with eigenvalue when . For , the set of the eigenvalues of is , that leads to . For , the set of the entanglement spectrum is . This implies . At critical point, , the scaling function of the entanglement negativity is proportional to with being the length of the system. The set of points in Fig. 8(c) fits . From the CFT predictionCalabrese (), the entanglement negativity is . In our consideration, , that leads to . One can read off the central charge .

### a.2 Integer Quantum Hall state (IQHS)

We consider the integer quantum Hall state (IQHS) on a cylinder. We choose the periodic boundary condition along direction. Hence is a good quantum number. The single-particle state in the lowest Landau level with the Landau gauge is

(50) |

where