11institutetext: Department of Physics, Anyang Normal University, Anyang,
455000, China, 11email<EMAIL_ADDRESS>
# Pairwise Entanglement and Geometric Phase in High Dimensional Free-Fermion
Lattice Systems

H. T. Cui  Y. F. Zhang

(Received: / Revised version: )

###### Abstract

The pairwise entanglement, measured by concurrence and geometric phase in high
dimensional free-fermion lattice systems have been studied in this paper. When
the system stays at the ground state, their derivatives with the external
parameter show the singularity closed to the phase transition points, and can
be used to detect the phase transition in this model. Furthermore our studies
show for the free-fermion model that both concurrence and geometric phase show
the intimate connection with the correlation functions. The possible
connection between concurrence and geometric phase has been also discussed.

###### pacs:

03.65.Vf Phases: geometric; dynamic or topological; 03.65.Ud Entanglement and
quantum nonlocality; and 05.70.Fh Phase transitions: general studies

##  1 introduction

The understanding of quantum many-body effects based on the fundamentals of
quantum mechanics, has been raising greatly because of the rapid development
in quantum information theoryafov07 . Encouraged by the suggestion of
Preskillpreskill , the connection between the quantum entanglement and quantum
phase transition has been demonstrated first in 1D spin-$1/2$ $XY$ modeoo02 ,
and then was extended to more other spin-chain systems and fermion systems
(see Ref afov07  for a review). Furthermore the decoherence of a simple
quantum systems coupled with the quantum critical environment has been shown
the significant features closed to the critical points ycw06 ; quan .
Regarding these findings, the fidelity between the states across the
transition point has also been introduced to mark the happening of the phase
transitions zanardi . These intricate connections between quantum entanglement
and phase transition in many-body systems have sponsored great effort devoted
to the understanding of many-body effects from quantum information pointafov07
. In general quantum entanglement as a special correlation, is believed to
play an essential role for the many-body effects since it is well accepted
that the non-trivial correlation is at the root of many-body effects. Although
the ambiguity existsyang , quantum entanglement provides us a brand-new
perspective into quantum many-body effects. However the exact physical meaning
of quantum entanglement in many body systems remains unclearvedral07 .
Although the entanglement witnesses has been constructed in some many-body
systemswvb05 , a general and physical understanding of quantum entanglement in
many-body systems is still absent.

On the other hand, the geometric phase, which was first studied systemically
by Berryberry  and had been researched extensively in the past 20 yearsgp ,
recently has also been shown the intimate connection to quantum phase
transitionscp05 ; zhu ; hamma ; cui06 ; plc06 ; cui08 ; hkh08 (or see a recent
review Ref.zhu08 ). This general relation roots at the topological property of
the geometric phase, which depicts the curvature of the Hilbert space, and
especially has direct relation to the property of the degeneracy in quantum
systems. The degeneracy in the many-body systems is critical in our
understanding of the quantum phase transition sachdev . Thus the geometric
phase is another powerful tool for detecting the quantum phase transitions.
Moreover recently geometric phase has been utilized to distinguish different
topological phases in quantum Hall systemsshen , in which the traditional
phase transition theory based on the symmetry-broken theory is not in
functionSenthil .

Hence it is very interesting to discuss the possible connection between
entanglement and geometric phase, since both issues show the similar
sensitivity to the happening of quantum phase transition. Recently the
connection between the entanglement entropy and geometric phase has first been
discussed with a special model in strongly correlated systems; the geometric
phase induced by the twist operator imposed on the filled Fermi sphere, was
shown to present a lower bound for the entanglement entropyrh06 . This
interesting result implies the important relation between quantum entanglement
and geometric phase, and provides an possible understanding of entanglement
from the topological structure of the systems. In another way the two-particle
entanglement was also importantoo02 . Especially in spin-chain systems two-
particle entanglement is more popular and general because of the interaction
between spins, and furthermore the quantum information transferring based on
spin systems are generally dependent on the entanglement between two
particlesss05 . So it is a tempting issue to extend this discussion to the
universal two-particle entanglement situation.

For this purpose the pairwise entanglement and geometric phase are studied
systemically in this paper. Our discussion focuses on nearest-neighbor
entanglement in the ground state in free-Fermion lattice systems because of
the availability of the exact results. By our own knowledge, this paper first
presents the exact results of entanglement and geometric phase in higher
dimensional systems. In Sec.2 the model will be provided, and the entanglement
measured by Wootter’s concurrence is calculated by introducing pseudospin
operators. Furthermore the geometric phase is obtained by imposing a globe
rotation, and its relation with concurrence are also discussed generally. In
Sec.3, we discussed respectively the concurrence and geometric phase in 2D and
3D cases. Finally, the conclusion is presented in Sec.4.

##  2 Model

The Hamiltonian for spinless fermions in lattice systems reads

$H=\sum_{\mathbf{ij}}^{L}c_{\mathbf{i}}^{\dagger}A_{\mathbf{ij}}c_{\mathbf{j}}+\frac{1}{2}(c_{\mathbf{i}}^{\dagger}B_{\mathbf{ij}}c_{\mathbf{j}}^{\dagger}+\text{h.c.}),$
(1)

in which $c_{\mathbf{i}}^{(\dagger)}$ is fermion annihilation(creation)
operator and $L$ is the total number of lattice sites. The hermitity of $H$
imposes that matrix $A$ is Hermit and $B$ is an anti-symmetry matrix. The
configuration of lattice does not matter for Eq. (1) since our discussion
focuses on the general case and available exact results. This model obviously
is solvable exactly and can be transformed into the free Bogoliubov fermionic
model. So it is also called free-fermion model. By Jordan-Wigner
transformationjw28  one can convert the spin-chain systems into spinless
fermions systems, in which the physical properties can be readily determined.
Therefore an alternative approach is necessary by which one can treat solvable
fermion systems of arbitrary size. The model Eq. (1) serves this purpose.

Without the loss of generality we assume $A$ and $B$ to be reallsm61 . An
important property of Eq. (1) is

$[H,\prod_{\mathbf{i}}^{L}(1-2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}})]=0.$ (2)

This symmetry would greatly simplify the consequent calculation of the reduced
density matrix for two fermions. One can diagonalize Eq. (1) by introducing
linear transformation with real $g_{\mathbf{ki}}$ and $h_{\mathbf{ki}}$lsm61

$\eta_{\mathbf{k}}=\frac{1}{\sqrt{L}}\sum_{\mathbf{i}}^{L}g_{\mathbf{ki}}c_{\mathbf{i}}+h_{\mathbf{ki}}c_{\mathbf{i}}^{\dagger},$
(3)

in which the normalization factor $1/\sqrt{L}$ have been included to ensure
the convergency under the thermodynamic limit. After some algebra, the
Hamiltonian Eq. (1) becomes

$H=\sum_{\mathbf{k}}\Lambda_{\mathbf{k}}\eta_{\mathbf{k}}^{\dagger}\eta_{\mathbf{k}}+\text{const}.$
(4)

in which $\Lambda_{\mathbf{k}}^{2}$ is the common eigenvalue of the matrices
$(A-B)(A+B)$ and $(A+B)(A-B)$ with the corresponding eigenvectors
$\phi_{\mathbf{ki}}=g_{\mathbf{ki}}+h_{\mathbf{ki}}$ and
$\psi_{\mathbf{ki}}=g_{\mathbf{ki}}-h_{\mathbf{ki}}$ respectively (see
Ref.lsm61  for details). The ground state is defined as $|g\rangle$, which
satisfies the relation

$\eta_{\mathbf{k}}|g\rangle=0$ (5)

With respect to fermi operator $\eta_{\mathbf{k}}$, one has relations

$\displaystyle\frac{1}{L}\sum_{\mathbf{i}}g_{\mathbf{ki}}g_{\mathbf{k^{\prime}i}}+h_{\mathbf{ki}}h_{\mathbf{k^{\prime}i}}$
$\displaystyle=$ $\displaystyle\delta^{(3)}_{\mathbf{k^{\prime}k}}$
$\displaystyle\frac{1}{L}\sum_{\mathbf{i}}g_{\mathbf{ki}}h_{\mathbf{k^{\prime}i}}+h_{\mathbf{ki}}g_{\mathbf{k^{\prime}i}}$
$\displaystyle=$ $\displaystyle 0$ (6)

Furthermore the requirement that $\\{\phi_{k},\forall k\\}$ and
$\\{\psi_{k},\forall k\\}$ be normalized and complete, reinforce the relations
lsm61

$\displaystyle\frac{1}{L}\sum_{\mathbf{k}}g_{\mathbf{ki}}g_{\mathbf{kj}}+h_{\mathbf{ki}}h_{\mathbf{kj}}$
$\displaystyle=$ $\displaystyle\delta_{\mathbf{ij}}$
$\displaystyle\frac{1}{L}\sum_{\mathbf{k}}g_{\mathbf{ki}}h_{\mathbf{kj}}+h_{\mathbf{ki}}g_{\mathbf{kj}}$
$\displaystyle=$ $\displaystyle 0$ (7)

With the help of these formula above, one obtains

$c_{\mathbf{i}}=\frac{1}{\sqrt{L}}\sum_{\mathbf{k}}g_{\mathbf{ki}}\eta_{\mathbf{k}}+h_{\mathbf{ki}}\eta_{\mathbf{k}}^{\dagger},$
(8)

which would benefit our calculation for the correlation functions.

###  2.1 Concurrence

The concurrence, first introduced by Wootterswootters  for the measure of two-
qubit entanglement, is defined as

$c=\max\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\},$ (9)

in which $\lambda_{i}(i=1,2,3,4)$ are the square roots of eigenvalues of
matrix $R=\rho(\sigma^{y}\otimes\sigma^{y})\rho(\sigma^{y}\otimes\sigma^{y})$
with decreasing order. Then the critical step is to determine the two-body
reduced density operator $\rho$. The reduced density operator
$\rho_{\mathbf{ij}}$ for two spin-half particles labeled $\mathbf{i,j}$ can be
written generally as,

$\rho_{\mathbf{ij}}=\text{tr}_{\mathbf{ij}}\rho=\frac{1}{4}\sum_{\alpha,\beta=0}^{4}p_{\alpha,\beta}\sigma^{\alpha}_{\mathbf{i}}\otimes\sigma^{\beta}_{\mathbf{j}},$
(10)

in which $\rho$ is the density matrix for the whole system and $\sigma^{0}$ is
the $2\times 2$ unity matrix and $\sigma^{\alpha}(\alpha=1,2,3)$ are the Pauli
operators $\sigma^{x},\sigma^{y},\sigma^{z}$, which also the generators of
$SU(2)$ group.
$p_{\alpha\beta}=\text{tr}[\sigma^{\alpha}_{\mathbf{i}}\sigma^{\beta}_{\mathbf{j}}\rho_{\mathbf{ij}}]=\langle\sigma^{\alpha}_{\mathbf{j}}\sigma^{\beta}_{\mathbf{j}}\rangle$
is the correlation function. With the symmetry Eq. (2), one can verify that
only $p_{00},p_{03},p_{30},p_{11},p_{22},p_{33},p_{12},p_{21}$ are not
vanishing. After some efforts, one obtain

$c=\max\\{0,c_{I},c_{II}\\},$ (11)

in which

$\displaystyle c_{I}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\sqrt{(p_{11}+p_{22})^{2}+(p_{12}-p_{21})^{2}}$
$\displaystyle-\sqrt{(1+p_{33})^{2}-(p_{30}+p_{03})^{2}}]$ $\displaystyle
c_{II}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[|p_{11}-p_{22}|-\sqrt{(1-p_{33})^{2}-(p_{30}-p_{03})^{2}}].$
(12)

In order to obtain the reduced density operator for two fermions, it is
crucial to construct $SU(2)$ algebra for the fermions in lattice systems. In
1D case, the Jordan-Wigner (JW) transformation is availablejw28 ; cp05 ; zhu ;
lsm61 . For higher dimension cases the JW-like transformation has been
constructed by different methodsjw . However the transformation is very
complex and the calculation is difficult. Hence instead of a general
calculation, we focus on the nearest neighbor two lattices in this paper. In
this situation, the $SU(2)$ algebra can be readily constructed

$\displaystyle\sigma_{\mathbf{i}}^{+}=(\sigma_{\mathbf{i}}^{x}+i\sigma_{\mathbf{i}}^{y})/2=c^{\dagger}_{\mathbf{i}}$
$\displaystyle\sigma_{\mathbf{i}}^{-}=(\sigma_{\mathbf{i}}^{x}-i\sigma_{\mathbf{i}}^{y})/2=c_{\mathbf{i}}$
$\displaystyle\sigma_{\mathbf{i}}^{z}=2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}-1$
$\displaystyle\sigma_{\mathbf{i}+1}^{+}=(\sigma_{\mathbf{i}+1}^{x}+i\sigma_{\mathbf{i}+1}^{y})/2=(2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}-1)c^{\dagger}_{\mathbf{i}+1}$
$\displaystyle\sigma_{\mathbf{i}+1}^{-}=(\sigma_{\mathbf{i}+1}^{x}-i\sigma_{\mathbf{i}+1}^{y})/2=(2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}-1)c_{\mathbf{i}+1}$
$\displaystyle\sigma_{\mathbf{i}+1}^{z}=2c_{\mathbf{i}+1}^{\dagger}c_{\mathbf{i}+1}-1$
(13)

in which $\mathbf{i}+1$ denotes the nearest neighbor lattice for site
$\mathbf{i}$. This point can be explained as the following. The difficulty for
the JW transformation in higher dimension case comes from the absence of a
natural ordering of particles. However when one focuses on the nearest
neighbored particle, this difficulty does not appear since for a definite
direction the nearest neighbor particle is unique (for non-nearest neighbored
case one have to consider the effect from the other particles). Then the
correlation functions for the ground state are in this case

$\displaystyle p_{00}$ $\displaystyle=$ $\displaystyle
1,p_{30}=1-\frac{2}{L}\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2};p_{03}=1-\frac{2}{L}\sum_{\mathbf{k}}h_{\mathbf{k(i+1)}}^{2};$
$\displaystyle p_{11}$ $\displaystyle=$
$\displaystyle\frac{2}{L}\sum_{\mathbf{k}}(h_{\mathbf{ki}}-g_{\mathbf{ki}})(h_{\mathbf{k(i+1)}}+g_{\mathbf{k(i+1)}});$
$\displaystyle p_{22}$ $\displaystyle=$
$\displaystyle\frac{2}{L}\sum_{\mathbf{k}}(h_{\mathbf{ki}}+g_{\mathbf{ki}})(h_{\mathbf{k(i+1)}}-g_{\mathbf{k(i+1)}})$
$\displaystyle p_{33}$ $\displaystyle=$
$\displaystyle(1-\frac{2}{L}\sum_{\mathbf{k}}h^{2}_{\mathbf{ki}})(1-\frac{2}{L}\sum_{\mathbf{k}}h^{2}_{\mathbf{k(i+1)}})$
$\displaystyle+\frac{4}{L^{2}}\sum_{\mathbf{k,k^{\prime}}}h_{\mathbf{ki}}h_{\mathbf{k(i+1)}}g_{\mathbf{k^{\prime}i}}g_{\mathbf{k^{\prime}(i+1)}}-h_{\mathbf{ki}}g_{\mathbf{ki}}h_{\mathbf{k^{\prime}(i+1)}}g_{\mathbf{k^{\prime}(i+1)}}$
$\displaystyle p_{12}$ $\displaystyle=$ $\displaystyle p_{21}=0$ (14)

###  2.2 Geometric Phase

Following the method in Refs.cp05 ; zhu , one can introduce a globe rotation
$R(\phi)=\exp[i\phi\sum_{i}c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}]$ to obtain
the geometric phase(GP). Then we have Hamiltonian with parameter $\phi$

$H(\phi)=\sum_{\mathbf{ij}}^{L}c_{\mathbf{i}}^{\dagger}A_{ij}c_{\mathbf{j}}+\frac{1}{2}(c_{\mathbf{i}}^{\dagger}B_{\mathbf{ij}}c_{\mathbf{j}}^{\dagger}e^{2i\phi}+\text{h.c.}),$
(15)

and the ground state becomes $|g(\phi)\rangle=R(\phi)|g\rangle$. GP is defined
as berry

$\displaystyle\gamma_{g}$ $\displaystyle=$ $\displaystyle-i\int d\phi\langle
g(\phi)|\frac{\partial}{\partial\phi}|(\phi)\rangle$ (16) $\displaystyle=$
$\displaystyle\frac{\phi}{L}\sum_{\mathbf{i}}\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2}$

Regarding to Eq.(15), one only require $\phi=\pi$ for a cycle evolution. Hence
one has
$\gamma_{g}=\frac{\pi}{L}\sum_{i}\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2}=\frac{1}{L}\sum_{\mathbf{i}}\gamma_{g\mathbf{i}}$.

###  2.3 GP vs. Concurrence

At a glance of Eq.(2.1) and Eq.(16), GP and concurrence both are related
directly to correlation functions. Hence it is tempting to find the relation
between the two quantities, which would benefit to the understanding of the
physical meaning of concurrence.

According to Eqs.(2.1) and (2.1), the following inequality can be obtained
(see Appendix for details of calculations)

$\displaystyle c_{I}$ $\displaystyle\leq$
$\displaystyle\frac{1}{L\pi}(\gamma_{g\mathbf{i}}+\gamma_{g(\mathbf{i+1})})-\sqrt{(1+p_{33})^{2}-(p_{30}+p_{03})^{2}}$
$\displaystyle c_{II}$ $\displaystyle\leq$ $\displaystyle
1+\frac{1}{L\pi}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})-\frac{1}{2L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})^{2}$
(17)

For the first inequality, a much tighter bound is difficult to find. While if
the average of $c_{II}$ over all site $\mathbf{i}$ is considered, $c_{II}\leq
1-\frac{1}{2L^{3}\pi^{2}}\sum_{i}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})^{2}$.
Fortunately in the following examples $c_{I}$ is always negative. Although the
existence of this defect, in our own points, the relation between GP and
concurrence have been displayed genuinely from the inequality above.

##  3 GP and Concurrence in Higher Dimensional $XY$ model

The previous section presents the general discussion of GP and concurrence in
free fermion lattice system Eq.(1). In this section a concrete model would be
checked explicitly, of which the Hamiltonian is

$H=\sum_{\langle\mathbf{i,j}\rangle}[c_{\mathbf{i}}^{\dagger}c_{\mathbf{j}}-\gamma(c_{\mathbf{i}}^{\dagger}c_{\mathbf{j}}^{\dagger}+\text{h.c.})]-2\lambda\sum_{\mathbf{i}}c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}},$
(18)

in which $\langle\mathbf{i,j}\rangle$ denotes the nearest-neighbor lattice
sites and $c_{\mathbf{i}}$ is fermion operator. This Hamiltonian, first
introduced in Ref.li06 , depicts the hopping and pairing between nearest-
neighbor sites in hypercubic lattice systems, in which $\lambda$ is the
chemical potential and $\gamma$ is the pairing potential. Eq.(18) could be
considered as a $d$-dimensional generalization of 1D XY model. However for
$d>1$ case, this model shows different phase features li06 .

The Hamiltonian can be diagonalized by introducing the $d$-dimensional Fourier
transformation with periodic boundary condition in momentum space li06

$H=\sum_{\mathbf{k}}2t_{\mathbf{k}}c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}}-i\Delta_{\mathbf{k}}(c_{\mathbf{k}}^{\dagger}c_{-\mathbf{k}}^{\dagger}-\text{h.c.}),$
(19)

in which $t_{\mathbf{k}}=\sum_{\alpha=1}^{d}\cos k_{\alpha}-\lambda$ and
$\Delta_{\mathbf{k}}=\gamma\sum_{\alpha=1}^{d}\sin k_{\alpha}$. With the help
of Bogoliubov transformation, one obtains

$H=\sum_{\mathbf{k}}2\Lambda_{\mathbf{k}}\eta_{\mathbf{k}}^{\dagger}\eta_{\mathbf{k}}+\text{const}.$
(20)

in which
$\Lambda_{\mathbf{k}}=\sqrt{t_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2}}$.
Based on the degeneracy of the eigenenergy $\Lambda_{\mathbf{k}}=0$, the phase
diagram can be determined clearlyli06 ; When $d=2$, the phases diagram should
be identified as two different situations; for $\gamma=0$, the degeneracy of
the ground state occurs when $\lambda\in[0,2]$, whereas the gap above the
ground state is non-vanishing for $\lambda>2$. However for $\gamma\neq 0$
three different phases can be identified as $\lambda=0$, $\lambda\in(0,2]$ and
$\lambda>2$. The first two phases correspond to case that the energy gap above
the ground state vanishes, whereas not for $\lambda>2$. One should note that
$\lambda=0$ means a well-defined Fermi surface with $k_{x}=k_{y}\pm\pi$, whose
symmetry is lowered by the presence of $\lambda$ term. For $d=3$ two phases
can be identified as $\lambda\in[0,3]$ with the vanishing energy gap above the
ground state and $\lambda>3$ with a non-vanishing energy gap above ground
state. In a word the critical points can be identified as
$\lambda_{c}=d(d=1,2,3)$ for any anisotropy of $\gamma$, and $\lambda=0$ for
$d=2$ with $\gamma\neq 0$. One should note that since the $\gamma^{2}$
dependence of $\Lambda_{\mathbf{k}}$, the sign of $\gamma$ does not matter.
Hence the plots below are only for positive $\gamma$.

The correlation functions between nearest-neighbor lattice sites would play a
dominant role in the transition between different phases because of the
nearest-neighbor interaction, similar to the case in XY model oo02 . Then it
is expected that the pairwise entanglement is significant in this model. In
the following, concurrence for the nearest-neighbor sites of ground state is
calculated for $d=2,3$ respectively. The geometric phase of ground state is
also calculated by imposing a globe rotation $R(\phi)$. our calculation shows
that both quantities show interesting singularity closed to the boundary of
different phases.

###  3.1 Concurrence

For $d>1$ case, the nearest-neighbor lattice sites appear in different
directions. In order to eliminate the dependence of orientations, the
calculation of correlation functions Eqs.(2.1) is implemented by averaging in
all directions. With the transformation Eq.(2.1), one can determine under the
thermodynamic limit

$\displaystyle p_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{d(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha}^{d}dk_{\alpha}(\Delta_{k}\sum_{\alpha=1}^{d}\sin
k_{\alpha}-t_{k}\sum_{\alpha=1}^{d}\cos k_{\alpha})/\Lambda_{k}$
$\displaystyle p_{22}$ $\displaystyle=$
$\displaystyle-\frac{1}{d(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha}^{d}dk_{\alpha}(\Delta_{k}\sum_{\alpha=1}^{d}\sin
k_{\alpha}+t_{k}\sum_{\alpha=1}^{d}\cos k_{\alpha})/\Lambda_{k}$
$\displaystyle p_{12}$ $\displaystyle=$ $\displaystyle p_{21}=0$
$\displaystyle p_{03}$ $\displaystyle=$ $\displaystyle
p_{30}=p_{3}=\frac{1}{(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha}^{d}dk_{\alpha}\frac{t_{k}}{\Lambda_{k}}$
$\displaystyle p_{33}$ $\displaystyle=$ $\displaystyle
p_{3}^{2}-(\frac{p_{11}+p_{22}}{2})^{2}+(\frac{p_{11}-p_{22}}{2})^{2}$ (21)

$d=2$ Our calculation shows that $c_{I}$ is negative. So in Fig. 1, only
$c_{II}$ and its derivative with $\lambda$ are numerically illustrated. In
order to avoid the ambiguity because of the cutoff in the definition of
concurrence, the derivative of $c_{II}$ with $\lambda$ is depicted in all
region whether $c_{II}$ positive or notyang . Obviously the singularity for
$\partial c_{II}/\partial\lambda$ can be found at the point $\lambda=0,2$
respectively, which are consistent with our knowledge about phase transitions.

$d=3$ Similar to the case of $d=2$, our calculation shows $c_{I}<0$. Only
$c_{II}$ and its derivative with $\lambda$ are numerically displayed in Fig.2.
Different from the case of $d=2$, no singularity of the first derivative of
$c_{II}$ with $\lambda$ is found at $\lambda=3$. While a cusp appears at
$\lambda=1$. A further calculation demonstrates that the second derivative of
$c_{II}$ is divergent genuinely at exact $\lambda=3$, as shown in Figs.2(c).
which means the phase transition at this points. Furthermore our numerical
calculations show that $\partial^{2}c_{II}/\partial\lambda^{2}$ is finite at
$\lambda=1$, as shown in Figs.2(b). Hence one cannot attribute this feature to
the phase transition. The similar feature has been found in the previous
studies oo02 ; yang ; gu . However the underlying physical reason is unclear
in general. But this special feature is not unique for concurrence; van Hove
singularity in solid state physics displays the similar feature, which is
because of the vanishing of the moment-gradient of the energy. Although we
cannot established the direct relation between these two issues because of the
bad definition of the moment-gradient of the energy when degeneracy happening,
we affirm that this feature is not an accident and the underlying physical
reason is still to be found.

In a word the discussion above first demonstrates the exact connection between
concurrence and quantum phase transitions in high-dimensional many body
systems. However a question is still open; what the physical interpretation of
concurrence is in many-body systems. In this study, we includes the negative
part of $c_{II}$ to identify the phase diagram in free-fermion systems. In
general, it is believed that the negative $c_{II}$ means no entanglement
between two particles and then include no any useful information about state.
But from the discussion one can note that the omission of the negative part of
$c_{II}$ would lead to incorrect results. Moreover, for $\gamma=0$, our
calculations show that $c_{I},c_{II}$ always are zero, and so one cannot
obtain any the phase transition information from pair wise entanglement in
this case. Further discussions will be presented in the final part of this
paper.

###  3.2 Geometric Phase

Geometric phase manifests the structure of Hilbert in the system and has
intimate relation to the degeneracy. GP, defined in Eq. (16) by imposing a
globe rotation $R(\phi)$ on ground state $|g\rangle$ is calculated in this
section. After some algebra, one obtains

$\gamma_{g}=\frac{\pi}{2(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha=1}^{d}dk_{\alpha}(1-\frac{t_{k}}{\Lambda_{k}}).$
(22)

$d=2$ In Fig.3, $\gamma_{g}$ and its derivative with $\lambda$ are displayed
explicitly. Obviously one notes that $\partial\gamma_{g}/\partial\lambda$
shows the singularity closed to $\lambda=0,2$, which are exactly the phase
transition points of Hamiltonian Eq.(18). An interesting observation is that
closed to these points, both GP and concurrence $c_{II}$ show the similar
behaviors.

$d=3$ GP and its derivative are plotted explicitly in Fig.(4). One should note
that there is a platform below $\lambda=1$ for
$\partial\gamma_{g}/\partial\lambda$, as shown in Fig.4(a), but a further
calculation shows that $\partial^{2}\gamma_{g}/\partial\lambda^{2}$ is
continued (Fig.4(b)) and $\partial\gamma_{g}/\partial\lambda$ has no
divergency at this point. This phenomena is very similar to the case of
concurrence (see Fig.2(b, c)). As expected,
$\partial^{2}\gamma_{g}/\partial\lambda^{2}$ is divergent at exact
$\lambda=3$, which means a phase transition happens at this point. Together
with respect of the case of $d=2$, it makes us a suspect that GP and
concurrence in our model have the same physical origination.

Furthermore for $\gamma=0$, GP fails to mark the phase transition too. This is
similar to the case of concurrence, but has different physical reason. The
further discussion is presented in the next section.

##  4 Discussion and Conclusions

The pairwise entanglement and geometric phase for ground state in
$d$-dimensional ($d=2,3$) free-fermion lattice systems are discussed in this
paper. By imposing the transformation Eq.(2.1), the reduce two-body density
matrix for the nearest neighbor particles can be determined exactly for any
dimension, and the concurrence is also calculated explicitly. Furthermore
geometric phase for ground state, obtained by introducing a globe rotation
$R(\phi)$, has also been calculated. Given the known results for XY model oo02
; cp05 ; zhu , our calculations show again that both GP and concurrence
display intimate connection with the phase transitions. Moreover an inequality
relation between concurrence and geometric phase is also presented in Eq.
(2.3). The similar scaling behaviors at the transition point $\lambda=3$ has
also been shown in Figs. 5. These facts strongly mean the intimate connection
between the two items. This point can be understand by noting that both of
them are connected to the correlation functions, as shown in Eqs. (2.1) and
(16).

An interesting point in our study is that in order to obtain all information
of phase diagram in model Eq.(18), the negative part of $c_{II}$ has to be
included to avoiding the confusion because of the mathematical cutoff in the
definition of concurrenceyang . In general, it is well accepted that the
negative part of $c_{II}$ gives no any information of quantum pairwise
entanglement, and then is considered to be meaningless. However, in our
calculation, the negative part of $c_{II}$ appears as an indispensable
consideration to obtain the correct phase information. This point means that
the pairwise entanglement does not provide the all information about the
system since the two-body reduced density operator throw away much
information.

As for the geometric phase, defined in Eq. (16), it is obvious that
$\gamma_{g}$ can tell us the happening of phase transition at the point, where
$\gamma_{g}$ display some kinds of singularity. However it cannot
distinguished the degenerate region from the nondegenerate, as shown in Figs.
3 and 4. Recently GP imposing by the twist operator in many-body systems is
introduced as an order parameter to distinguish the phases cui08 ; hkh08 . For
the free-fermion lattice system, this GP have also calculated and shows the
intimate connection with the vanishing of energy gap above the ground state.
However the boundary between the two different phases becomes obscure with the
increment of dimensionality in that discussion cui08 , and moreover it cannot
distinguish the phase transition not come from the degeneracy of the ground-
state energy. While the geometric phase imposing by the globe rotation
$R(\phi)$ clearly demonstrate the existence of this kind of phase transition,
as shown in Fig.3, whether originated from the degeneracy or not. In fact this
point can be understood by noting the intimate relation between $\gamma_{g}$
and correlation functions. It maybe hint that one has to find different
methods for different many-body systems to identify the phase diagram.

Although the intimate relationship of concurrence and GP with phase
transitions in the model Eq.(18), a exceptional happens when $\gamma=0$, in
which $c_{I},c_{II}$ are zero and GP is a constant independent of $\lambda$.
From Eq.(18), $\gamma=0$ means the hopping of particles is dominant, and the
position of particle becomes meaningless. Since the calculation of concurrence
depend on the relative position of lattice site, the pairwise entanglement is
disappearing. However one could introduce the spatial entanglement to detect
the phase transition in this casehav07 . For GP, $\gamma=0$ means the
emergency of new symmetry. One can find
$[\sum_{\mathbf{i}}c^{\dagger}_{\mathbf{i}}c_{\mathbf{i}},H]=0$ in this case,
which leads to the failure of $R(\phi)$ for construction of nontrivial GP.

Finally we try to transfer two viewpoints in this paper. One is that
concurrence and geometric phase can be used to mark the phase transition in
many-body systems since both of them are intimately connected to the
correlation functions. The other is that concurrence and the geometric phase
are connected directly by the inequality Eq. (2.3). Then it is interesting to
extend this relation to multipartite entanglement in the future works, which
would be helpful to establish the physical understanding of entanglement.

###### Acknowledgements.

The author (Cui) would appreciate the help from Dr. Kai Niu (DLUT) and Dr.
Chengwu Zhang (NJU) in the numerical calculations and permission of the usage
of their powerful computers. We also thank greatly the enlightening discussion
with Dr. Chong Li (DLUT). Especially we thank the first referee for his/her
important hint for the van Hove singularity. This work is supported by the
Special Foundation of Theoretical Physics of NSF in China, Grant No.
10747159\.

## APPENDIX

For the first inequality, one should note

$\displaystyle|p_{11}+p_{22}|$ (23) $\displaystyle=$
$\displaystyle\frac{4}{L}|\sum_{\mathbf{k}}h_{\mathbf{ki}}h_{\mathbf{k(i+1)}}|\leq\frac{4}{L}\sum_{\mathbf{k}}|h_{\mathbf{ki}}h_{\mathbf{k(i+1)}}|$
$\displaystyle\leq$
$\displaystyle\frac{2}{L}\sum_{\mathbf{k}}(h^{2}_{\mathbf{ki}}+h^{2}_{\mathbf{k(i+1)}})=\frac{2}{L\pi}(\gamma_{g\mathbf{i}}+\gamma_{g(\mathbf{i+1})}).$

From inequality $\sqrt{x^{2}-y^{2}}\geq|x|-|y|(|x|>|y|)$, one reduces

$\sqrt{(1+p_{33})^{2}-(p_{30}+p_{03})^{2}}]\geq|1+p_{33}|-|p_{30}+p_{03}|.$
(24)

Then one obtains

$c_{I}\leq\frac{1}{L\pi}(\gamma_{g\mathbf{i}}+\gamma_{g(\mathbf{i+1})})+\frac{1}{2}(|p_{30}+p_{03}|-|1+p_{33}|).$
(25)

However a much tighter bound is difficult to decide because of the complexity
of $p_{33}$.

For the second inequality, it can be obtained easily by observing

$\displaystyle p_{33}\leq
1-\frac{1}{L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}^{2}-\gamma_{g(\mathbf{i+1})}^{2}),$
(26)

in which we have used the relation $2ab\leq a^{2}+b^{2}$. Then $1-p_{33}$ is
non-negative and

$\displaystyle c_{II}$ $\displaystyle=$
$\displaystyle\frac{2}{L}\sum_{\mathbf{k}}|h_{\mathbf{ki}}g_{\mathbf{k(i+1)}}|+\frac{p_{33}-1}{2}$
(27) $\displaystyle\leq$
$\displaystyle\frac{1}{L}\sum_{\mathbf{k}}(h^{2}_{\mathbf{ki}}+g^{2}_{\mathbf{k(i+1)}})-\frac{1}{2L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}-\gamma_{g(\mathbf{i+1})})^{2}$
$\displaystyle\leq$ $\displaystyle
1+\frac{1}{L\pi}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})-\frac{1}{2L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}-\gamma_{g(\mathbf{i+1})})^{2}$

in which
$1/L\sum_{\mathbf{k}}g_{\mathbf{ki}}^{2}=1-1/L\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2}$
is used.

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\begin{overpic}[width=170.71652pt]{1.eps} \put(-5.0,40.0){\large$c_{II}$}
\put(50.0,0.0){\large$\lambda$}\put(97.0,50.0){\large\begin{rotate}{-90.0}$\partial
c_{II}/\partial\lambda$\end{rotate}} \end{overpic} Figure 1:  $c_{II}$
($\bigcirc$) and its derivative with $\lambda$ ($\triangle$) vs. $\lambda$
when $d=2$. We have chosen $\gamma=1$ for this plot.

\begin{overpic}[width=170.71652pt]{2a.eps}
\put(15.0,60.0){(a)}\put(-5.0,40.0){\large$c_{II}$}
\put(45.0,0.0){\large$\lambda$}
\put(94.0,50.0){\large\begin{rotate}{-90.0}$\partial
c_{II}/\partial\lambda$\end{rotate}} \end{overpic}

\begin{overpic}[width=128.0374pt]{2b.eps} \put(18.0,78.0){(b)}
\put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}c_{II}/\partial\lambda^{2}$\end{rotate}}
\put(40.0,0.0){\large$\lambda$}
\end{overpic}\begin{overpic}[width=136.5733pt]{2c.eps}
\put(22.0,78.0){(c)}\put(40.0,0.0){\large$\lambda$} \end{overpic}

Figure 2:  $c_{II}$ ($\bigcirc$) and its derivative with $\lambda$
($\triangle$) (a) vs. $\lambda$ when $d=3$. We have chosen $\gamma=1$ for this
plot. The second derivative of $c_{II}$ with $\lambda$ are also displayed in
this plot and focus the points closed to $\lambda=1$ (b) and $\lambda=3$ (c).
\begin{overpic}[width=170.71652pt]{3.eps}
\put(0.0,35.0){\large\begin{rotate}{90.0}$\gamma_{g}/\pi$\end{rotate}}
\put(50.0,0.0){\large$\lambda$}\put(98.0,53.0){\large\begin{rotate}{-90.0}$\partial\gamma_{g}/\pi\partial\lambda$\end{rotate}}
\end{overpic} Figure 3: $\gamma_{g}$ ($\bigcirc$) and its derivative with
$\lambda$ ($\triangle$) vs. $\lambda$ when $d=2$. We have chosen $\gamma=1$
for this plot.

\begin{overpic}[width=170.71652pt]{4a.eps}
\put(13.0,60.0){(a)}\put(0.0,35.0){\large\begin{rotate}{90.0}$\gamma_{g}/\pi$\end{rotate}}
\put(50.0,0.0){\large$\lambda$}
\put(93.0,52.0){\large\begin{rotate}{-90.0}$\partial\gamma_{g}/\pi\partial\lambda$\end{rotate}}
\end{overpic}\begin{overpic}[width=113.81102pt]{4b.eps} \put(13.0,77.0){(b)}
\put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}\gamma_{g}/\pi\partial\lambda^{2}$\end{rotate}}
\put(35.0,0.0){\large$\lambda$}
\end{overpic}\begin{overpic}[width=130.88284pt]{4c.eps}
\put(22.0,78.0){(c)}\put(42.0,0.0){\large$\lambda$} \end{overpic}

Figure 4: $\gamma_{g}$ ($\bigcirc$) and its derivative with $\lambda$
($\triangle$) (a) vs. $\lambda$ when $d=3$. We have chosen $\gamma=1$ for this
plot. The second derivative‘ of $\gamma_{g}$ with $\lambda$ are also displayed
in this plot and focus the points closed to $\lambda=1$ (b) and $\lambda=3$
(c).

\begin{overpic}[width=125.19194pt]{5a.eps} \put(55.0,78.0){(a)}
\put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}c_{II}\partial\lambda^{2}$\end{rotate}}
\put(50.0,0.0){\large$\log|\lambda-\lambda_{c}|/\lambda_{c}$}
\end{overpic}\begin{overpic}[width=130.88284pt]{5b.eps}
\put(57.0,78.0){(b)}\put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}\gamma_{g}/\pi\partial\lambda^{2}$\end{rotate}}
\end{overpic}

Figure 5: The scaling of GP and concurrence for 3D case closed to the critical
point $\lambda_{c}=3$. We have chosen $\gamma=1$ for this plot.