Quantum critical phase diagram of bond alternating Ising model with Dzyaloshinskii-Moriya interaction: signature of ground state fidelity
We present the zero temperature phase diagram of the bond alternating Ising chain in the presence of Dzyaloshinskii-Moriya interaction. An abrupt change in ground state fidelity is a signature of quantum phase transition. We obtain the renormalization of fidelity in terms of quantum renormalization group without the need to know the ground state. We calculate the fidelity susceptibility and its scaling behavior close to quantum critical point (QCP) to find the critical exponent which governs the divergence of correlation length. The model consists of a long range antiferromagnetic order with nonzero staggered magnetization which is separated from a helical ordered phase at QCP. Our results state that the critical exponent is independent of the bond alternation parameter () while the maximum attainable helical order depends on .
One of the key features of strongly correlated electron systems is their zero temperature behavior where quantum fluctuations have the dominant role to classify different novel phases which are separated by quantum phase transitions vojta (). In this case, ground state (GS) is the sole candidate which receives drastic changes at QCP driven by some external parameters. Identification of QCPs is always a challenging task specially when quantum correlations diminish a single-particle picture. Within last couple of years different measures of quantum entanglement have been proposed to be a new toolkit to detect and categorize QCPs amico2009 (). Recently, ground state fidelity–the overlap of ground state at two slightly different values of coupling constants–has attracted intensive attention as a proper quantity to signal QCP without the need to know the structure of phases close to phase transition zanardi2006 (); Gu2010 ().
An abrupt drop of fidelity in the vicinity of QCP is a consequence of an essential change in the structure of GS which is usually accompanied with a divergence of fidelity susceptibility. Meanwhile, it is rather difficult to obtain the GS of a strongly correlated many body system. However, recent successful attempts to calculate some measures of entanglement kargarian () by quantum renormalization group (QRG) approach qrg () lead to a unified formalism Langari2012 () to obtain fidelity in the thermodynamic limit () without knowing the GS of a large system. It suggests to implement QRG for obtaining the renormalization of fidelity for the bond alternating Ising model with Dzyaloshinskii-Moriya (DM) interaction. DM interaction Dzyaloshinskii (); Moriya () which roots to the spin-orbit coupling is the antisymmetric super-exchange which leads to helical magnetic structures as the most likely candidates to host ferroelectricity kimura2005 ().
In this article we study the quantum critical behavior of the one dimensional bond alternating Ising model with DM interaction in terms of renormalization of fidelity. The model represents an antiferromagnetic (Néel) long range order for small DM interaction while it undergoes via a continuous phase transition to a helical ordered phase. The ground state in the Néel phase is a product state which has essentially zero quantum entanglement and nonzero staggered magnetization while the helical phase poses a correlated quantum ground state with zero staggered magnetization. Within a classical picture the helical phase could be assumed as slightly rotating spins along the direction of chain. In terms of Landau theory of critical phenomena the staggered magnetization could be chosen as the proper order parameter. We have shown that the divergence in fidelity susceptibility (FS) at the QCP is an appropriate signature to find QCP which is more pronounced than the second derivative of ground state energy chen2008 (). The presented scheme makes us to find the critical points and its corresponding exponents more accurately.
Ii Quantum renormalization group
The Hamiltonian of bond alternating Ising chain with DM interaction on a periodic chain of N sites is defined
where is the spin-1/2 operator at site , describes the relative strength of the alternating coupling and shows the nearest-neighboring antiferromagnetic coupling. is the vector of DM interaction which is considered in -direction, i.e. . To apply QRG qrg (), the spin chain is decomposed to three-sites blocks (see Fig.1) where the intra-block Hamiltonian is and the inter-block one is and their sum defines the whole Hamiltonian, (see the bottom part of Fig.1) . In this respect we have , where
and similarly ,
The block Hamiltonian () is diagonalized exactly and the two lowest energy eigenvectors () are kept to construct the embedding operator () to the renormalized Hilbert space, . Here, , represent the renamed states and , respectively in the renormalized space to be considered as the new base kets. and have the following expression (for odd blocks) in the original spin Hilbert space where and represent the eigenvectors of operator at each site,
and is the ground state energy of the block. The presence of bond alternation imposes to consider two types of blocks as depicted in Fig.1, namely even and odd which are the mirror image of each other. For even blocks we find similar eigenstates by replacing , and .
The global embedding operator is the direct product of the embedding operator of each block, which gives the renormalized Hamiltonian by qrg (). The renormalized Hamiltonian is similar to the original one, Eq.1, while the coupling constants are replaced with the renormalized one (denoted with ) as given in the following equations,
ii.1 Fidelity signature
We implement the formalism introduced in Langari2012 () to calculate the ground state fidelity in terms of quantum renormalization group. According to the QRG-flow (Eq.6), does not run within the QRG iterations which hints to study the quantum phase transition by variation of . The fidelity (f) associated to the ground states for a system of size is defined by
where and is a small deviation around . According to the renormalization group approach in which is the global embedding operator and is the ground state of the renormalized Hamiltonian. Thus, fidelity can be written in terms of renormalized ground state, . A straightforward calculation shows that
for both even and odd types of blocks where is the identity operator. Therefore, the first QRG iteration leads to , where fidelity of the original model is expressed in terms of fidelity of the renormalized one, i.e. . It defines the renormalization of fidelity in terms of QRG. The QRG procedure is iterated -times to reach the renormalized system of and the ground state fidelity is expressed by
where has the same expression as given in Eq.II.1 for in which , and are calculated at the -th QRG iteration and is the fidelity of a single block with three sites and -times renormalized couplings.
We have plotted fidelity (Eq.9) versus in Fig.2-(left) for different chain length (), and at . By definition, fidelity is bounded by and an abrupt drop is a signature of quantum phase transition. We observe a sharp drop in Fig.2-(left) for which manifests that the ground state has encountered an essential change. The deep of drop is enhanced as the system size is increased which justifies an unfailing drop in the thermodynamic limit. This signature of quantum phase transition is more pronounced in the fidelity susceptibility (FS) which is the leading nonzero term in the expansion of fidelity and shows the change rate of fidelity, i.e. . Thus, FS is obtained by the following relation
Fig.2-(right) presents versus for various system sizes, and at . A maximum appears in which is increased by the size of system representing a divergence in the thermodynamic limit. The position of is exactly at the point where fidelity receives a drop.
ii.2 Scaling analysis
It has been shown Gu2010 () that the fidelity susceptibility at the quantum critical point obeys a scaling relation. The scaling analysis for finite system size () states
where is the quantum critical point, is the position of maximum in FS and is the critical exponent governing the divergence of correlation length. The analysis of data of Fig.2-(right) is presented as an inset to this figure. It clearly verifies that the scaling relation Eq.11 is satisfied with and for . Moreover, we have obtained similar behavior for different values of (not presented here) where the critical point is found to be a function of bond alternating parameter, while is the same for all values of . In contrast to what stated in Ref.Hao2010 () the critical exponent which we got does not depend on . It means that the whole phase boundary for belongs to the same universality class of a second order phase transition.
Iii Summary and discussions
We have implemented the quantum renormalization group approach to study the zero temperature phase diagram of bond alternating Ising chain in the presence of DM interaction. To get a self similar Hamiltonian two types of blocks with 3-sites have been considered which leads to the QRG-flow equations of Eq.6. The QRG-flow tells that does not vary within QRG procedure while is renormalized. To get the phase diagram of the model we have calculated the renormalization of ground state fidelity which has been developed recently Langari2012 ().
Fidelity as a geometric quantity zanardi2007 () shows how much the ground state encounters an essential change by slightly moving in the parameter space. Therefore, a sharp drop of fidelity versus a control parameter is a signature of quantum phase transition. The renormalization of fidelity obtained in Eq.9 in addition to QRG-flow, Eq.6 give the fidelity of our model for very large system sizes without the need to get the ground state. A clear drop of fidelity versus in Fig.2-(left) and consequently a maximum in the fidelity susceptibility, Fig.2-(right), verifies the existence of a quantum phase transition at . We have applied the finite size scaling on the susceptibility data presented in the inset of Fig.2-(right) for and generally got the critical phase boundary . The phase boundary which separates the antiferromagnetic (AF) Néel order from a helical order is shown by the blue line in Fig.3-(right). However, our scaling analysis gives a single value for the correlation length exponent (independent of ) for the whole phase boundary which is in contrast to Hao2010 (). Although the presence of bond alternation breaks the translation invariance of the Hamiltonian it does not change the symmetry of the ground state which has already been spontaneously broken due to antiferromagnetic long range order. A comparison of our results with Jafari2008 () concludes that the bond alternation does not change the universality class of the model as far as .
We calculate the staggered magnetization (SM) and helical order parameter which is presented in Fig.3-(left). Staggered magnetization is defined by
which can be expressed in terms of the renormalized ground state by replacing . We use the projection of spin operators into the renormalized Hilbert space which finally leads to
where is the staggered magnetization of the renormalized chain. A large number of iterations of Eq.13 give the staggered magnetization in the thermodynamic limit. Similarly, the helical order parameter () is defined by the following relation Jafari2008 ()
which can be calculated using the QRG-flow. In this respect, is expressed in terms of the helical order parameter in the renormalized model () by the following relation
The above equation is iterated by replacing the couplings with the renormalized ones to reach the stable thermodynamic limit. We have plotted both staggered magnetization and helical order parameter versus in Fig.3-(left) for three values of and . For , is at its saturated value and . The onset of nonzero induces a helical order on the spins in the presence of antiferromagnetic order. The increase of reduces the antiferromagnetic order and enhances (helical order). Exactly at the quantum critical point , vanishes and remains zero for while saturates to a finite amount which is less than its maximum attainable value. Similar behavior has been observed for all while the saturated value of is slightly decreased by increase of .
To complete the phase diagram let us concentrate on the axis (Fig.3-(right)). At the model is simply an antiferromagnetic Ising chain with Néel order and the ground state energy is . For , there are two types of couplings and which are positive and induce the previous Néel order and ground state energy. Exactly at , the weak interaction becomes zero and the spin model decomposes to independent pairs of antiferromagnetically coupled spins. The ground state energy is still while the ground state is exponentially degenerate, namely . The degeneracy arises from the pairs which are decoupled. For instance, , and are examples of different configurations which is possible for the ground state. The entropy (S) is proportional to which leads to . This high amount of entropy at is a signature of a phase transition which is actually of first order. Meanwhile, for one of the interactions becomes ferromagnetic, and the other is still antiferromagnetic which totally leads to . Thus, the derivative of with respect to receives a discontinuity at . The ground state for is either or . The first order tri-critical point is represented by the filled black circle in Fig.3-(right) and the ferro-antiferromagnetic ground state is denoted by the red line for .
Acknowledgments We would like to thank A. T. Rezakhani for fruitful discussions. This work was supported in part by Sharif University of Technology’s Center of Excellence in Complex Systems and Condensed Matter and the Office of Vice-President for Research. A. L. acknowledges partial support from the Alexander von Humboldt Foundation and Max-Planck-Institut für Physik komplexer Systeme (Dresden-Germany).
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