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The Riemannian metric on the quantum states manifold and its topological characterization: on a simple two-band model

We characterize a generalized quantum geometric tensor to betray topological quantum phase transition of gapped time-reversal invariant simple two-band model by topological numbers. The generalized quantum geometric tensor contains two different measurements. There is the Riemannian metric and Berry curvature, which can derive the z2 number and Euler number to curve the topological properties when the parameter of the Hamiltonian varies. Our results show that the local behavior of the topological properties of the quantum geometric tensor can help us to understand the topological quantum phase transitions. Ⅰ. INTRODUCTION The geometric and topological properties have been more important in quantum physics because of the discovery of Berry phase [1,2] and the topological Chern number [3,4] interpretation for adiabatic pumping and quantized Hall conductance [5,6]. Recently, new topological phase have been found in quantum spin Hall effects and in topological insulators which is described by the z2 numbers [7]. Soon afterwards, people find another quantum numbers to characterize the topological phase which is called Euler number [8]. The Euler number is based on the Gauss-Bonnet theorem on the Riemannian structure established by the real part of the quantum geometric tensor [9, 10] in momentum space. In this paper, we shall briefly introduce a model and its topological characterization. The quantum geometric tensor derives from defining a local U(1) gauge invariant. The real part of the geometric tensor give the Riemannian metric tensor and Riemannian structure of the quantum states manifold. Remarkably, the imaginary part is the curvature form giving rise to a Berry phase. The geometric tensor has recently paid more attention to characterizing the novel collective behaviors of quantum many-body systems in low temperature [11-13]. The quantum geometric tensor which is defined on the manifold of ground state is expected to be better understood of quantum phase transitions [14] in many-body system. And recent studies have shown that quantum geometric tensor unified the fidelity susceptibility [15] can generally analyzed the singularity and scaling behaviors near the quantum critical point. Even both the Berry curvature and metric tensor exhibit singularity around the phase transition points, but neither of them can act as a topological order to characterize a topological phase. So the previous studies on the ground-state Berry phase are mainly interest in the local properties. In this approach [16-18], the partial derivatives of the Berry phase near the critical points and the phase boundaries can be characterized. Recently, people showed that the ground-state Berry phase which is symmetry as local order parameters can study various topological phases. For example, the z2 number characterized the quantum phase transitions gapped time-reversal invariant spin chain systems. The z2 topological number can be generated by a quantized Berry phase of the Bloch state along a loop

over a quarter of the Brillouin zone (Bz) in the two-dimensional parameter space. And the other topological quantum number is Euler number which can characterize the topological phase of a gapped fermionic ground state. The Euler number is based on the Gauss-Bonnet theorem on the Riemannian structure established by the real part of the quantum geometric tensor in momentum space. In this work, we shall introduce the quantum geometric tensor defined in the Bloch states manifold and the topological characterization. We discuss this approach analytically in a two-band simplest model. We show the topological Euler characteristic of the model. Meanwhile, we also obtain a nontrivial z2 number by the integral of the Berry curvature as the imaginary part of the geometric tensor over half of the Brillouin zone. Ⅱ. RIEMANNIAN METRIC IN MOMENTUM SPACE To begin with, we introduce the notion of Riemannian metric [19] and quantum geometric tensor in Bloch momentum space. The Riemannian metric is a meaningful metric tensor on any manifold of quantum states which can be derived from a gauge invariant distance between two Bloch states on the U(1) line bundle. And the manifolds are generated by the action of a lie group on a fixed quantum state. We first consider a family

?? ? s ??of normalized vectors of Hilbert

n

space which based on an n-dimensional parameter s ? ? s1,…, sn ? ?

. Let

and ? , ?

indicted the norm and the scalar product on the Hilbert space [20]. The distance between two close vectors in the family

?? ? s ??

derives a metric in the following way. Writing s1 ? s and

s2 ? s ? ds , we develop the distance

? ? s2 ? ? ? ? s1 ? ? ? ? s ? ds ? ? ? ? s ? ? ?? ? s ? ds ? ? ? ? s ? , ? ? s ? ds ? ? ? ? s ? ?

2 2

(2.1)

up to second order:

? ? s ? ds ? ? ? ? s ? ? ? i , j ? ? i? , ? j? ? dsi ds j ? ? i ?

2

? ?

? ? ? ?si ?

(2.2)

Separating the hermitian product into the real and the imaginary parts

(?i?, ? j?) ? ? i , j ? i? i , j

Obviously,

(2.3) (2.4)

? i, j ? s ? ? ? j ,i ? s ? ,?i , j ? s ? ? ?? j ,i ? s ?

2

Thus, its imaginary parts, canceled out in the quantum distance. So (2.2) reads:

? ? s ? ds ? ? ? ? s ? ? ?i , j ? i , j ? s ?dsi ds j

The quantities

(2.5)

? i , j ? s ? so defined the components of the metric tensor. However, this tensor is

meaningless as a metric tensor on a manifold of quantum states in ordinary quantum mechanics. Therefore, so long as the phase of a vector state is invisible, the physical states are standed for manifold of the Hilbert space and the two vectors

? ? s ? and ?? ? s ?

? ? ? s ? ? ei? ? s ?? ? s ?

(2.6)

Define the same point on the manifold. Calculating the two metric tensors, they should be identical but is not true. From (2.3) the tensor r ’ with components

? i , j? ? s ? ? Re ?i?? , ? j??

?

? is

different from r. And more precisely, we have to make the metric tensor invariant under the gauge transformation (2.6). Then, we calculate a meaningful metric tensor as following

gi , j ? s ? ? ?? ? s ? ds ? ? ? ? s ? , ? ? s ? ds ? ? ? ? s ? ? ? ? ? s ? , ? ? s ? ds ? ? ? ? s ?

(2.7)

And this metric tensor is not the only one which can be invariant under the transformation. There we introduce a more general one-quantum geometric tensor. Its expression is

Qij ? ? i? ?1 ? ? ?

???

j

(2.8)

This quantity is Hermitian, i.e., Qi , j ? Qj ,i* . Both its real and imaginary parts have a relevant physical meaning. The real part is a Riemannian metric tensor which defines the distance between the two nearby vectors in Hilbert space as we deduce above, i.e. gi , j ? ?Qi , j . The Hilbert space one between the corresponding ground-state is the meaning of this distance function. We now consider the imaginary parts Fi , j ? ?2Im Qi , j . For ? ? j? is a purely

imaginary, one finds Im Qi , j ? Im ? i? ? j? ? ? i? ? j? ? ? j? ? i? . Thus, its imaginary part is nothing but Berry curvature. Recently, some papers have shown that the ground-state Berry curvature exist some singularity and scaling behave near the quantum critical point. Soon, they indicated the derivative of the ground state Berry phase shows a scaling behavior near the quantum critical point of spin chain systems. This fact points out that the critical points are related to the divergence of Berry curvature in the thermodynamic limit. What’s more, the significance thing is the singularity behavior of the Berry curvature near the quantum critical point can be detailed analyzed by the framework of quantum phase transitions. This physical mechanism is based on the fact that the different phases are independent by the adiabatic evolution of the ground state. And the ground-state Berry phase is accumulated by a cyclic parallel transport of the state which is generated by the quantum adiabatic approximation. On the other hand, some previous literatures are interested in studying the local properties. Those studies can only portray the position of phase boundaries. Now, we have noted that some symmetry of ground-state Berry phase as local order parameters can discuss various topological phases. III.ANALYTICAL CHERN NUMBER IN TWO-BAND MODEL As an example, here let us consider a two-band model which is one of the simplest models. We study this system to exhibit the topological nontrivial states. This model was first showed by Qi( ) which is a two-dimensional lattice model on a two torus T 2 . The Hamiltonian H ? k ? can be generally given by H ? k ? ? ? ? k ? I 2?2 ?

??

3

?1

d? ? k ? ? ? ,

where I 2?2 is the 2 ? 2 identity matrix and σα are the three Pauli matrix. The eigenvalues is readily to be obtained as E? ? k ? ? ? ? k ? ?

T

??

3

?1

d? 2 ? k ? . The eigenvectors are given by

T

? ?? ? ?? ? ? ?? ? ? cos , ei? sin ? and ?? ? ? ? sin , ei? cos ? , where 2 2? 2 2? ? ?

? ? arccos d3 ? k ? / d12 ? k ? ? d 2 2 ? k ? ? d32 ? k ? and ? ? d1 ? k ? / d12 (k ) ? d 2 2 ? k ? . In this

model, the coefficients are written as ? ? k ? ? 0, d1 ? sin kx , d2 ? sin k y and

d3 ? m ? cos kx ? cos k y

As we all known, the properties of the ground state can be insensitive to local perturbations and the system can undergo a topological phase transition which is beyond the Landau’s second order phase transitions paradigm and local order parameters. And all of these are expressed as the Chern number. Now we analyze the Chern number of this model when the parameter m is changed. The topological invariant on the U(1) line bundle of all occupied bands is the first Chern number. And the Chern number can be calculated as Ch1 ?

1 4?

? ? d? ? ?

kx

??? d ? dxdy (2.9) d ky

?

Through the qualities we show above, we can calculate the topological invariant

if ? 2 ? m ? 0 ?1 ? Ch1 ? ?1 if 0 ? m ? 2 , ?0 otherwise ?

(2.10)

which is the first Chern number of the induced U(1) line bundle. The phase diagram is plotted in Fig1.

1.0 Chern number

0.5

3

2

1

1

2

3 m

0.5

1.0

Fig.1(Color online) The first Chern number Ch1 of the induced U(1) line bundle as a function of m which can curve the topological characters.

IV.THE EULER NUMBER IN THIS MODEL In the thermodynamic limit, the base manifold is the two-dimensional momentum space

k ? ? k x , k y ? . Then substituting ?? into Eq(2.8). Then we can obtain the quantum geometric

tensor. We have Qxy ? ? kx ?? k y ? ? ? kx ?? k y ? sin 2 ? / 4 ? i sin ? ? kx ?? k y ? ? ? k y ?? kx? / 4 . And by using the relation: g xy ? ?Qxy , we have g xy ? ? k x ?? k y ? ? ? k x ?? k y ? sin 2 ? / 4 . What’s more, the corresponding Berry curvature can be obtained by using Fxy ? ?2Qxy . We have

?

?

?

?

?

?

Fxy ? i sin ? ? kx ?? k y ? ? ? k y ?? kx? / 2 . Here we can also calculate the Ricci scalar R=8 by the

standard calculation. We know that the system is time reversal invariant, so Berry curvature Fxy is odd with x. As a consequence, the first Chern number as the integral of the Berry curvature Fxy in Brillouin zone is equivalent to zero. Obviously, the Chern number is not the one which can depict the topological phase as a appropriate topological number. However, it has been noted in recent work that a nontrivial Z2 number can be obtained do the same work as Chern number. We can derive the Berry curvature Fxy of the Bloch state. The z2 number can be calculated as a quantized Berry phase along a loop over the quarter of the Brillouin zone,

?

?

z2 ?

1

? /2

?

?

o

?1 , ? dy ? Fxy dx ? ??1, 0 ?0 , ?

?

if m ? 0 if m ? 0 if m ? 0

(2.11)

The phase diagram is plotted in Fig2.

Fig2(color online)The z2 number as a function of m, derived from a quantized Berry phase along a loop over the quarter of the Brillouin zone. Here, the z2 number is derived by the Berry curvature Fxy which is the imaginary part of the geometric tensor. This number reflect the topological properties of the genus of Bloch states manifold in a (1+1)-dimensional momentum space. The more important is that there is another

topological invariant-Euler number. It derived from the Gauss-Bonnet theorem on the 2D closed manifold established by the Riemannian metric g for the Bloch state. In two dimensions, the Euler number can be calculated using the metric g as follows [8]

??

1 4?

??

n

n

d12 ? k ? +d 2 2 ? k ? +d32 ? k ?dk x dk y

(2.12)

where the ? is Ricci scalar curvature which is mentioned above. Here we can give an intuitional

n

picture to illustrate this result as Fig2.

Fig2(color online) The Euler number as a function of the m which is a characterization of the phase diagram . In this case, the first Chern number is not a useful number to carve the topological characterization because of the time reversal invariance. So we have the z2 number and Euler number to express the model’s characterization. V.CONCLUSION In summary, we have characterized an adiabatic origin for the generalized quantum geometric tensor and shown the topological characterization of the simple two-band model. In addition, we introduced two different numbers based on an intuitive physical picture when parameter varies. The two different numbers which can be unified in the generalized quantum geometric tensor as its symmetric and antisymmetric parts are z2 number and Euler number. We use the Riemannian metric on the quantum states manifold to calculate the topological numbers and to understand the singular behavior of the quantum phase transitions. REFERENCES [1] BERRY M.V., Proc. R. Soc. London, Ser. A, 392 (1984) 45 [2] M. V. Berry: The Quantum Phase, Five Years After. Phys. Rev. Lett1988 [3]B. Simon, Phys. Rev.Lett.51, 2167(1983) [4] Q. Niu, D. J. Thouless, and Y.S. Wu, Phys. Rev. B 31, 3372(1985)

[5] R. B. Laughlin, Phys. Rev. B 23, 5632(1981) [6] Q. Niu and D. J. Thouless, J. Phys. A 17, 2453(1984) [7] YU-QUAN MA,ZHAO-XIAN YU,DENG-SHAN WANG,BING-HAO XIE and XIANG-GUI LI: Momentum space Z2 number, quantized Berry phase and the quantum phase transitions in spin chain systems.EPL,100(2012)60001 [8] Yu-Quan Ma, Shi-jian Gu, Shu Chen, Heng Fan, and Wu-Ming Liu: The Euler Number of Bloch States Manifold and the Quantum Phases in Gapped Fermionic Systems.EPL A79,022116(2009) [9] Ran Cheng: Quantum Geometric Tensor (Fubini-Study Metric) in Simple Quantum System: A pedagogical Introduction [10] Lorenzo Campos Venuti and Paolo Zanardi: Quantum Critical Scaling of the Geometric Tensors. PRL99, 095701(2007) [11] R. Resta, Phys. Rev. Lett. 95, 196805 (2005) [12] Y. Q. Ma, S. Chen, H. fan, and W. M. Liu, Phys. Rev. B 81, 245129(2010) [13] X. G. Wen: Quantum Field Theory of Many-Body Systems (Oxford University, New York, 2004) [14] L. Campos Venuti and P. Zanardi, Phys. Rev.Lett.99, 095701(2007) [15] W. L. You, Y. W. Li and S. J. Gu, Phys. Rev. E 76, 022101(2007) [16] S. Yang, S. J. Gu, C. P. Sun, and H. Q. Lin, Phys. Rev. A 78,012304(2008) [17] A. Hamma, W. Zhang, S. Hass, and D. A. Lidar, Phys. Rev. A 79,032302(2009) [18] D. F. Abasto, A. Hamma, and P. Zanardi, Phys. Rev. A 78, 010301(2008) [19] J. P. Provost and G. Vallee: Riemannian Structure on Manifolds of Quantum. Math.Phys.76, 289-301(1980) [20]Kibble, T.W.B.: Commun.Math.Phys.65.189-201(1979)

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