Ground state fidelity in bond-alternative Ising chains with Dzyaloshinskii-Moriya interactions

Ground state fidelity in bond-alternative Ising chains with Dzyaloshinskii-Moriya interactions

Bo Li libo.phy@cqu.edu.cn Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China    Sam Young Cho sycho@cqu.edu.cn Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China    Hong-Lei Wang Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China    Bing-Quan Hu Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China
Abstract

A systematic analysis is performed for quantum phase transitions in a bond-alternative one-dimensional Ising model with a Dzyaloshinskii-Moriya (DM) interaction by using the fidelity of ground state wave functions based on the infinite matrix product states algorithm. For an antiferromagnetic phase, the fidelity per lattice site exhibits a bifurcation, which shows spontaneous symmetry breaking in the system. A critical DM interaction is inversely proportional to an alternating exchange coupling strength for a quantum phase transition. Further, a finite-entanglement scaling of von Neumann entropy with respect to truncation dimensions gives a central charge at the critical point.

pacs:
75.10.Pq, 64.70.Tg, 75.30.Kz

I Introduction

Recent advanced material technologies have made it possible to access low-dimensional quantum systems. Furthermore, material synthesis has offered a great opportunity to explore more intriguing lower-dimensional spin systems rather than well-understood conventional spin systems Verdaguer (). In such a low-dimensional system, for instance, alternating bond interactions and/or less symmetry interactions in spin lattices can be realizable in synthesizing two different magnetic atoms. Of particular importance, therefore, is understanding quantum phase transitions in which one-dimensional spin systems are unlikely found naturally.

Normally, quantum fluctuations in a low-dimensional spin system are stronger than higher dimensional spin systems sachdev (). Quantum phase transitions driven by stronger quantum fluctuations then exhibit more interesting and novel quantum phenomena in low-dimensional spin systems. The effects of alternating bond interactions, especially, have been intensively studied theoretically in spin systems such as antiferromagnetic Heisenberg chains Kolezhuk (); Yamamoto97 (); Dukelsky (); Aplesnin (); Onishi (); Narumi (), Heisenberg chains with next-nearest-neighbor bond alternations Capriotti (); Maeshima (), a tetrameric Heisenberg antiferromagnetic chain Gong (), and two-leg spin ladders Fukui (); Almeida (). A recent experiment has demonstrated a realization of a bond-alternating chain by applying magnetic fields in a spin-1/2 chain antiferromagnet Canevet ().

In this study, we will consider one-dimensional Ising-type spin chains with an alternating exchange coupling. Actually, this bond alternation cannot destroy the antiferromagnetic phase of the uniform bond case but just quantitatively changes the ground state properties originating from a dimerization of the spin lattice. Then, a less symmetric interaction can play a significant role to induce a quantum phase transition. To see a quantum phase transition, we will employ a Dzyaloshinskii-Moriya (DM) interaction Moriya () which results from the spin-orbit coupling.

Based on the ground state fidelity Zhou () with the iMPS presentation iTEBD (), we discuss the quantum criticality in the system. It is shown that a uniform DM interaction can destroy the antiferromagnetic phase, which is a continuous quantum phase transition, and its critical value is inversely proportional to the alternating exchange coupling strength.

Ii Model and numerical method

Let us start with a spin-1/2 Ising chain with antisymmetric anisotropic, and alternative bond interactions on the infinite-size lattice. Our system can be described by the spin Hamiltonian

(1)

where are the spin operators acting on the -th site. The exchange interaction is chosen as and the alternative bond interaction is characterized by the relative strength of exchange coupling for the even and odd lattice sites. To describe an antisymmetric anisotropic exchange coupling between the two spins on the lattice, we employ a uniform DM interaction , that is characterized by the DM vector . For and , Eq. (1) is reduced to the conventional Ising chain Hamiltonian. If and , Eq. (1) can be mapped onto the XXZ spin chain model which has a quantum phase transition from the gapped Neel or antiferromagnetic (AFM) phase to the gapless Luttinger Liquid (LL) phase at the critical point Soltani (). This study will then be focused on the antiferromagnetic exchange interaction , i.e., , and a transverse DM interaction denoting .

The Hamiltonian in Eq. (1) is actually invariant under the transformation with for -th site and for ()-th site. Our model Hamiltonian then possesses a symmetry generated by the transformation . The ground state of the system may undergo a spontaneous symmetry breaking which gives rise to a quantum phase transition between an ordered phase and a disordered phase.

For a quantum spin system with a finite lattice site, its wave function with the periodic boundary condition can be expressed in the matrix product state (MPS) representation Verstrate () as , where is a site-dependent matrix with the truncation dimension of the local Hilbert space at the -th site, is a basis of the local Hilbert space at the -th site, and the physical index takes value with the dimension of the local Hilbert space. This MPS representation for a finite lattice system can be extended to describe an infinite lattice system. To do this, for an infinite lattice, one may replace the matrix with iTEBD (), where is a three-index tensor and is a diagonal matrix at the -th site, which is called the canonical infinite matrix product state (iMPS) representation.

If system Hamiltonian is translational invariant for an infinite lattice, for instance, our system Hamiltonian describe by Eq. (1) has a two-site translational invariance, the two-site translational invariance allows us to reexpress the Hamiltonian as , where is the nearest-neighbor two-body Hamiltonian density. In such a case, one can introduce a two-site translational invariant iMPS representation, i.e., for the even (odd) sites A (B), only two three-index tensors and two diagonal matrices can be applied in representing a system wave function:

(2)

Note that, actually, for an infinite lattice sites, the diagonal elements of the matrix are the normalized Schmidt decomposition coefficients of the bipartition between the semi-infinite chains and .

In order to find a ground state of our system in the iMPS representation, the infinite time-evolving block decimation (iTEBD) algorithm introduced by Vidal iTEBD () is employed. For a given initial state and the Hamiltonian , a ground-state wave function can be yield by the imaginary time evolution for a large enough . To realize the imaginary time evolution operation numerically, the imaginary time is divided into the time slices and a sequence of the time slice evolution gates approximates the continuous time evolution. Meanwhile, the time evolution operator for is expanded to a product of the evolution operators acting on two successive and sites through the Suzuki-Trotter decomposition Suzuki (). After absorbing a time-slice evolution gate, in order to recover the iMPS representation, a singular value decomposition of a matrix is performed, which contracted from , , one , two and the evolution operators. Then only the largest singular values are retained. This procedure yields the new tensors , and that are used to update the tensors for all the sites. As a result, the translational invariance under two-site shifts is recovered. Repeating the above procedure until the ground-state energy converges yields the ground-state wave function of the system in the iMPS representation.

Iii Fidelity per lattice site and bifurcations

One can define a fidelity from a ground state wavefunction . A fidelity per lattice site (FLS) Zhou () can be defined as

(3)

where is the system size. Remarkably, FLS is well defined and plays a similar role to an order parameter although trivially becomes zero in the thermodynamic limit when approaches infinity. The FLS satisfies the properties inherited from fidelity : (i) normalization , (ii) symmetry , and (iii) range .

Figure 1: (Color online) Fidelity per lattice site as a function of for various truncation dimensions with . For higher truncation dimension, a bifurcation point moves to saturate to its critical value. Inset: Extrapolation of bifurcation point . From the numerical fitting function with and , the critical point is estimated as .

On adapting the transfer matrix approach iPEPS-fidelity (), the iMPS representation of the ground-state wave-functions allows us to calculate the fidelity per lattice site (FLS) . Let us choose as a reference state for the FLS . For , in Fig. 1, the FLS is plotted as a function of the DM interaction parameter for various values of the truncation dimensions with a randomly chosen initial state in the iMPS representation. Figure 1 shows a singular behavior of the FLS , which indicates that there occurs a quantum phase transition across the singular point. A bifurcation behavior of the FLS is also shown when the interaction parameter becomes smaller than its characteristic singular value that can be called a ‘bifurcation point’ . The bifurcation points depend on the truncation dimension , i.e., . As the truncation dimension increases, the bifurcation occurs starting at the lower value of . For , the bifurcation point at which a bifurcation starts to occur can be extrapolated. In the inset of Fig. 1, we use an extrapolation function , characterized by the coefficients , , and being a positive real number, which guarantees that becomes a finite value. The numerical fitting gives , , and . From the extrapolation, the bifurcation point is shown to saturate to which can be regarded as a critical point Bifur (); Dai (). Actually, the critical point also corresponds to the pinch point of the FLS in the thermodynamics limit, i.e., . Consequently, a FLS bifurcation point plays the role of a pseudo phase transition point for a given finite truncation dimension in the MPS representation. In addition, the continuous function behavior of the FLS across the bifurcation point implies that a continuous quantum phase transition occurs at the critical point Zhou ().

In Fig. 1, the bifurcation occurs for , which is captured in the iMPS representation with a randomly chosen initial state. In fact, the bifurcation behavior of the FLS means that there are two possible ground states for while there is a single ground state for . Such a property of the ground states can be understood by the symmetry of the Hamiltonian from the invariant transformation . In the thermodynamic limit, there are two possible ground states satisfying , that is, or . Once a spontaneous symmetry breaking happens, the system can choose one of two possible ground states or , which indicate a broken symmetry phase. For symmetric phase, the system has a single ground state, being a linear combination of two possible ground state, which should satisfy the transformation invariance. In Fig. 1, then, the FLS is plotted from the two fidelities, i.e., (upper lines) and (lower lines). Then, the bifurcation point is the transition point between the symmetry phase and the broken-symmetry phase.

Iv Phase diagram

From the iMPS with the numerical extrapolation of the bifurcation points, in Fig. 2, we draw the ground-state phase diagram in the interaction parameter plane. Below the phase boundary (red solid line) the system is in an antiferromagnetic phase while above the boundary the system is in a disordered phase. The phase diagram shows that the is inversely proportional to the . A best fitting function (dotted line) of the critical points is given by with a single parameter . The characteristic phase boundary can be understood as follows: If and , the lattice sites are dimerised and the Hamiltonian in Eq. (1) has a two-site translational invariance due to the alternating bond coupling . As assumed, for the antiferromagnetic exchange interaction , i.e., for , the Ising chain with the alternating bond coupling is in an antiferromagnetic state even though the lattice sites are strongly dimerised. If , the antisymmetric anisotropic DM interaction can destroy the antiferromagnetic order originating from the antiferromagnetic exchange interaction . The antiferromagnetic order may be destroyed more easily by the uniform antisymmetric anisotropic DM interaction for the dimerised lattice sites than for the Ising chain without the alternating bond coupling because the antiferromagnetic correlation between the sites becomes weaker due to the dimerised lattice sites. Then, to destroy the antiferromagnetic order, a stronger dimerisation of the lattice sites (bigger ) requires a much weaker uniform antisymmetric anisotropic interaction (much smaller ). Consequently, the phase boundary separating the antiferromagnetic phase and a disordered phase might have a inversely proportional relation between and .

Figure 2: (Color online) Ground state phase diagram in the plane of the DM interaction and the alternating bond strengths. From the iMPS, the phase boundary shows that the is inversely proportional to the . A best fitting is denoted by the dotted line. This is in sharp contrast with the results from the real space renormolaztion group (RSRG) method XiangHao2010 ().

Very recently, a real space renormalization group (RSRG) approach XiangHao2010 () has been applied in the same model and has shown that a symmetric phase boundary is given by (blue dashed line in Fig. 2). Compared with the RSRG approach, our results from the iMPS show that there is a quite significant discrepancy in the phase boundary line because, contrast to the iMPS, the RSRG is an approximate method based on low-energy states, which implies that it does not capture properly a contribution from relevant higher energy states. Then, the ground state phase diagram from our fidelity approach based on the iMPS is more reliable and accurate.

V Quantum entanglement and phase transition

In order to understand more clearly the quantum phase transition in our system, let us consider quantum entanglement that can also detect a quantum phase transition ent_review (). To quantify the quantum entanglement, we employ the von Neumann entropy, which is a good measure of bipartite entanglement between two subsystems of a pure state Bennett (), because our ground states are in a pure state. Then, the spin chain can be partitioned into the two parts denoted by the left semi-infinite chain and the right semi-infinite chain . The von Neumann entropy is defined as in terms of the reduced density matrix or of the subsystems and . In the iMPS representation, the von Neumann entropy for the semi-infinite chains or becomes

(4)

where ’s are diagonal elements of the matrix that could be directly obtained in the iMPS algorithm. This is because, when one partitions the two semi-infinite chains and , one gets the Schmidt decomposition . From the spectral decomposition, are actually eigenvalues of the reduced density matrices for the two semi-infinite chains and . In our two-site translational invariant iMPS representation, there are two Schmidt coefficient matrices and that describe two possible ways of the partitions, i.e., one is on the odd sites, the other is on the even sites. From the and , one can obtain the two von Neumann entropies depending on the odd- or even-site partitions.

Figure 3: (Color online) The von Neumann entropy between left and right halves of a chain as a function of for and . Both the von Neumann entropies for odd and even bonds show a singularity at .

In Fig. 3, we plot the von Neumann entropy as a function of the DM interaction strength for even ( site) and odd (() site) bonds with . The von Neumann entropy for the odd bonds is always larger than those for the even bonds because, by the definition, the odd-site exchange interaction is stronger that the even-site exchange interaction . Furthermore, it is shown that both the entropies for the even and odd bonds have a singularity at the same value of the DM interaction strength . Note that the singularities of the entropies occur at the critical point . This result then shows clearly that both the FLS and the von Neumann entropy give the same phase transition point. As a consequence, in fact, the von Neumann entropy gives the same phase diagram from the FLS in Fig. 2.

As discussed, for the antiferromagnetic state of our system, there are two possible ground states that are connected by the unitary transformation . From the bifurcation of FLS, then, one might expect a bifurcation in the von Neumann entropy too. However, contrary to the FLS , in Fig. 3, no bifurcation is seen in the von Neumann entropy for the antiferromagnetic state even though the initial state is randomly chosen in the iMPS representation. The reason for the absence of bifurcation in the von Neumann entropy is why the singular values in Eq. (4) do not depend on the unitary transformation because the unitary transformation acts only on a single site of the spin lattice in the iMPS representation.

Vi Central charge and universality class

At a critical point, characteristic singular behaviors of thermodynamics system properties depend only on few features such as dimensionality and symmetry, which can be classified by the concept of universality classes. Especially, the central charge can be used for the classification of universality classes Cardy (); Calabrese (). Owing to implement the iMPS representation, we can obtain a central charge and a so-called finite-entanglement scaling exponent numerically via the unique behaviors of the correlation length and the von Neumann entropy with respect to the truncation dimension at a critical point Corrlen (); GVidal (), i.e.,

(5)
(6)
Figure 4: (Color online) (a) Correlation length as a function of the truncation dimension at the critical point. The power curve fitting yields and . (b) Scaling of the von Neumann entropy with the truncation dimension at the critical point. For from (a), the linear fitting yields the central charge . Here, the alternating bond strength is chosen as .

In Fig. 4, the correlation length and the von Neumann entropy as a function of the truncation dimension at the critical point for . Here, the truncation dimensions are taken as , and . It is shown that both the correlation length and the von Neumann entropy diverges as the truncation dimension increases. From a power-law fitting on the correlation length , we have and . As shown in Fig. 4 (b), our numerical result demonstrates a linear scaling behavior, which gives a central charge with . Our central charge is close to the exact value . Consequently, the quantum phase transition in our system is in the same universality class as the quantum transverse field Ising model.

Vii Summary

Quantum phase transitions have been investigated in the Ising chain with the Dzyaloshinskii-Moriya interaction as well as the alternating bond-coupling. The FLS and its bifurcation have clearly shown a characteristic singular point as a signature of the quantum phase transition and behaves as a continuous function, which shows a continuous phase transition occurring at the critical point. The phase diagram was obtained from the FLS and the von Neumann entropy. With a finite-entanglement scaling of the von Neumann entropy with respect to the truncation dimension in the iMPS representation, a central charge was estimated to be , which shows that the system is in the same universality class with the quantum transverse field Ising model.

Acknowledgements.
We thank Huan-Qiang Zhou for helpful discussions. This work was supported by the Fundamental Research Funds for the Central Universities (Project No. CDJZR10100027). SYC acknowledges the support from the NSFC under Grant No.10874252.

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