2. Department of Applied Physics, Xi'an Jiaotong University, Xi'an 710049, China ;
3. China Aviation Optical-Electrical Technology Co. Ltd, Electro-Optical Equipment Division, Luoyang 471000, Henan, China
2. 西安交通大学 理学院, 陕西 西安 710049;
3. 中航光电科技股份有限公司 光电设备事业部, 河南 洛阳 471000
Last decades, characterization of the quantum phase transition of the low-dimensional systems has attracted a considerable amount of attention in quantum many-body systems[1]. Compared to the order parameter, quantum entanglement have been applied to analysis of the quantum phase transitions, because it is a necessary resource of the quantum computation[2], and is at the heart of connections between quantum information theory and traditional quantum many-body theory.
Actually, spin-1 models have richer phase diagrams and show more complex physical phenomena. In 1983, Haldane proposed a famous conjecture[3], i.e., the half-integer spin antiferromagnetic Heisenberg chain is gapless, and the integer spin chain has an energy gap between the ground state and the first excited state. There are qualitative differences between integer and half-integer spin quantum antiferromagnetic chains based on the large-S expansion[4-6]. In the spin-1 Heisenberg chain, a hidden Z2×Z2 symmetry is fully broken. It induces a non-local string order in the Haldane phase[5]. Moreover, an energy gap appears between a spin-singlet ground state and a spin-triplet excited state. If a bond-alternating effect exists in the Heisenberg chain, the system is structural dimerised. Such an one-dimensional dimerised Heisenberg model is able to exhibit plentiful quantum critical phenomena[6-9]. However, it has not been investigated clearly yet, especially for the non-local string order.
To investigate the quantum critical phenomena of the quantum many-body system, the key step is calculating the ground state wavefunction of the system. However, the difficulty is that, with increase the size of the systems, the dimension of the ground state is exponentially increasing in the Hilbert space. Recently, significant progress has been made in numerical studies based on tensor network (TN) representations for the investigation of quantum phase transitions, such as the infinite matrix product states (iMPS) with the time-evolving block decimation (iTEBD) algorithm for one-dimensional systems[10] and the projected entangled-pair states (PEPSs) with the variational tensor network (VTN) algorithm for more than one spatial dimension systems[11-12]. These methods offer a new perspective for quantum entanglement, order parameter, and fidelity, provide a deeper understanding on characterizing critical phenomena in finite and infinite spin-lattice systems. This numerical method provides a convenient way to study the many-body systems for quantum phase transitions.
In this paper, the spin-1 bond-alternating Heisenberg model in one spatial dimension is investigated. By using the iMPS approach, the ground states are obtained in the infinite lattice system. As a measurement of the quantum entanglement, the singular peaks of the odd-bond and even-bond von Neumann entropies show that the phase transition from the Haldane phase to the dimerised phase is estimated at δc=0.26, which support to the DMRG result. However, in the whole parameter range, the odd-bond and even-bond von Neumann entropies have different values. It indicates that, due to the bond-alternating effect, the dimerization also exists in the Haldane phase. Moreover, to investigate the topological Haldane phase, the non-local string order parameter is calculated. It is shown that the string order parameter is able to characterize a topologically ordered phases and capture the quantum phase transitions.
1 Numerical method: iMPS approachFor the spin-1 systems, the Jordan-Wigner transformation cannot apply any more. Thus, recently, the iTEBD method in the iMPS representation is used to investigate the quantum many-body systems[13]. In an infinite one-dimensional lattice system, an iMPS state[14] with a two-site translational invariance Hamiltonian can be written as
$ \left| {\left. \mathit{\Psi } \right\rangle = \sum\limits_{\left\{ S \right\}} {\sum\limits_{\left\{ {l, r} \right\}} { \cdots {\lambda _A}{\mathit{\Gamma }_A}{\lambda _B}{\mathit{\Gamma }_B} \cdots } } } \right| \cdots {S_{i-1}}{S_i}{S_{1 + 1}}\left. \cdots \right\rangle, $ | (1) |
where |Si〉 denotes a basis of the local Hilbert space at the site i, l and r are the left and right bond indices, respectively. The diagonal matrix λ is the Schmidt decomposition coefficients of the bipartition between the semi-infinite chains L and R, and Γ is a three index tensor with bond indices taking the value 1, …, χ, where χ is the truncation dimension in the iMPS representation. The physical indices Si take the value 1, …, d at the site i.
Assuming that the model Hamiltonian H describes the nearest-neighbor interactions, i.e., the Hamiltonian can be expressed by
$ \left| {\mathit{\Psi }\left( \tau \right)} \right\rangle = \frac{{\exp \left[{-H\tau } \right]\left| {\mathit{\Psi }\left( 0 \right)} \right\rangle }}{{\left\| {\exp \left[{-H\tau } \right]\left| {\mathit{\Psi }\left( 0 \right)} \right\rangle } \right\|}}, $ | (2) |
if τ is large enough, it yields a good approximation to the ground state wave function. During this procedure, the Suzuki-Trotter decomposition[15] can be used to reduce the imaginary time operator to a product of two-site evolution operators that only act on two successive sites. After a singular value decomposition step, all the tensors λA, λB, ΓA and ΓB are updated. Then, repeating the procedure until the groundstate energy per site converges, an approximation groundstate wavefunction is generated in the TN representation.
2 Spin-1 bond-alternating HeisenbergTo investigate the dimerization effect on quantum phase transitions in many-body systems, we consider the bond-alternating antiferromagnetic spin-1 Heisenberg chain in one spatial dimension[16]. The Hamiltonian is written as
$ H = J\sum\limits_{i =-\infty }^\infty {\left( {1-{{\left( {-1} \right)}^i}\delta } \right)} {S_i}{S_{i + 1}}, $ | (3) |
where Si is the spin-1 operator at the lattice site i, and δ∈[0, 1] is the bond-alternating strength parameter. J denotes the antiferromagnetic spin-exchange interaction between the nearest-neighbor spins. If δ=0, the Hamiltonian in Eq. (3) is reduced to the conventional Heisenberg model. This model is in a gaped topological phase, i.e., Haldane phase. If δ=1, the Hamiltonian in Eq. (3) is fully dimerised. Then, a quantum phase transition appears.
In 1996, A. Kitazawa[16] gave the phase diagram of the XXZ model, where Δ is the anisotropic exchange interaction of the z-component. Actually, the bond-alternating antiferromagnetic spin-1 Heisenberg model is reduced to the case of Δ=1 in the the XXZ model. From the phase diagram in Fig. 1, it is seen that there is a Gauss phase transition at
Quantum entanglement has been shown to be a universal tool to characterize the quantum phase transitions[20]. Meanwhile, the TN representation is convenient to exact the entanglement. For MPS state in a canonical form, the von Neumann entropy S can be directly evaluated by the elements of the diagonal matrix λ that is the Schmidt decomposition coefficients of the bipartition between the semi-infinite chains L(-∞, …, i) and R(i+1, …, ∞). Then, the MPS state in Eq. (1) can be written in the form as
$ \left| \mathit{\Psi } \right\rangle = \sum\limits_{\alpha = 1}^\chi {{\lambda _\alpha }} \left| {\mathit{\Psi }_\alpha ^L} \right\rangle \left| {\mathit{\Psi }_\alpha ^R} \right\rangle, $ | (4) |
where
$ S =- {\rm{Tr}}\left[{{\rho _L}\log {\rho _L}} \right] = - {\rm{Tr}}\left[{{\rho _R}\log {\rho _R}} \right], $ | (5) |
where ρL=TrRρ and ρR=TrLρ are the reduced density matrices of the subsystems L and R, respectively.
As a measurement of the quantum entanglement, the von Neumann entropies are calculated in Figs. 2(a) and (b) for different bipartitions, i.e., odd-bond entropy Sodd and even-bond entropy Seven. It is shown that, for the truncation dimension χ=8, 16, and 32, there are two singular points exist in Sodd and Seven. It indicates that there are two phase transitions. Then, a pseudo phase exists between the Haldane phase and the dimerization phase. However, as increase of χ, the two singular points become close to each other. When χ=40, they are merged together, and only one singular point corresponding to a quantum phase transition point exists at δc=0.26. Such a behavior remind us, when we use the iMPS algorithm to calculate the many-body systems, a necessary requirement is the truncation dimension should be big enough to obtain the proper physics. To guarantee the numerical accuracy, in this paper, we calculate the numerical data based on χ=50 throughout. Actually, when the truncation dimension χ increases from 40 to 50, the singular point of the von Neumann entropies almost do not changed. Then, the phase transition point is able to estimate at δc=0.26 from iMPS method. This result is matched with the DMRG method[21].
Furthermore, the odd-bond and even-bond von Neumann entropies have different values in the whole parameter range. It implies that, due to the bond-alternating effect in the system, the dimerization also exists in the Haldane phase. The von Neumann entropies for the odd bonds are always larger than those for the even bonds. It implies that the entanglement of odd bonds is bigger than even bonds. Comparing the entanglement entropies between the odd-bond Sodd and even-bond Seven for δ>δc, with increasing δ, one can see that the value of Sodd is converged, and Seven is gradually reduced to zero. When δ=1, it indicates that the system is fully dimerised in the odd bond.
3.2 Non-local string order parametersIn the Haldane phase, due to the fully broken Z2×Z2 hidden symmetry, a non-local string order was firstly introduced by den Nijs and Rommelse[7] and later by Tasaki to clarify the nature of spin-1 Heisenberg chain. The string order parameter is defined as
$ {O_{{\rm{String}}}} = \mathop {\lim }\limits_{\left| {i- j} \right| \to \infty }- \left\langle {{S_i}\exp \left[{{\rm{i\pi }}\sum\limits_{k = i + 1}^{j-1} {{S_k}} } \right]{S_j}} \right\rangle, $ | (6) |
where |i-j| denotes the lattice distance between site i and j. Actually, the iMPS representation approach can directly calculate the non-local order parameter in an infinite-size system rather than the exploration of a finite-size system.
As shown in Eq.(6), the string order is a saturated value of the string correlation for a large lattice distance. Actually, the iMPS representation approach can directly calculate the non-local order parameter in an infinite-size system rather than the exploration of a finite-size system. In Fig. 3, the non-local string order parameter OString is plotted as a function of the bond-alternating strength parameter δ for the truncation dimension χ=50. It is shown that the string order has finite value for δc < 0.26, which characterizes a topologically ordered phase, i.e., the gapful Haldane phase. One can obtain the phase transition point at δc=0.26 corresponding to the Haldane-dimerization phase. The phase transition point is matched with that from the von Neumann entropies.
4 ConclusionsWe have investigated the one-dimensional bond-alternating Heisenberg chain. Employing the iMPS representation with the iTEBD approach in the infinite lattice system, the ground state bipartite entanglement measure, i.e., von Neumann entropies, and order parameters are calculated. It is shown that, the phase boundary can be determined from the von Neumann entropy and the non-local string order parameter. It indicates that, varying the bond-alternating effect parameter from zero to one, there is a quantum phase transition from the Haldane phase to the dimerised phase. The phase transition point is able to estimate at δc=0.26 from iMPS method. The results support the DMRG approach. Hence, the iMPS approach would be widely used for capturing the quantum phase transitions.
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