Quantum renormalization group approach to geometric phases in spin chains
A relation between geometric phases and criticality of spin chains are studied using the quantum renormalization-group approach. I have shown how the geometric phase evolve as the size of the system becomes large, i.e., the finite size scaling is obtained. The renormalization scheme demonstrates how the first derivative of the geometric phase with respect to the field strength diverges at the critical point and maximum value of the first derivative, and its position, scales with the exponent of the system size.
keywords:Quantum Renormalization Group, Geometric Phase
Email address: email@example.com, firstname.lastname@example.org
Tel: (+98) 241-4152118, Fax: (+98) 241 4244949
Quantum phase transition (QPT) has been one of the most interesting topics in the area of strongly correlated systems Vojta (). At zero temperature, the properties of the ground state may change drastically showing a non-analytic behavior of a physical quantity by reaching the quantum critical point. This can be done by tuning a parameter in the Hamiltonian, for instance, the magnetic field or the amount of disorder. Traditionally such a problem is addressed by resorting to notions such as order-parameter and symmetry breaking i.e., the Landau-Ginzburg paradigm Goldenfeld (). In the last few years a big effort has been devoted to the analysis of QPTs from the Quantum Information perspective Osterloh (); kargarian1 (); kargarian2 (); Jafari1 (); kargarian3 (); Jafari2 (); Langari (), the main tool being the study of different entanglement measures Osborne (). In the view of some difficulties Reuter (), attention has shifted to include other, potentially related, means of characterizing QPTs Zanardi (). One such approach centers around the notion of geometric phase (GP). GP has been offered as a typical mechanism for a quantum system to keep the memory of its evolution in Hilbert space. Such phases were introduced in quantum mechanics by Berry in 1984 Berry (). Since then, geometric phases became objects of theoretical and experimental researches Shapere () uncovering that they are related to a number of important physical phenomena Bohm () such as Aharonov-Bohm Aharonov () and quantum Hall effects Klitzing (). In recent years, this interest is increased due to their applicability in quantum-information processing Zanardi2 (). In other words, GP has become extendable to product states of composite systems since the uncorrelated subsystems pick up independent geometric phase factors. However, GP could be induced by quantum entanglement, if the full state is pure. On the other hand, classical correlations and quantum entanglement can coexist in mixed quantum states, which means the forms of the mixed state of the geometric phases Uhlmann (), applied to the path of the relative states, may contain portions from both types of correlations. Nevertheless, their connection to the quantum phase transitions has been manifested recently in Ref.Carollo (); Zhu (); Hamma (), where it is shown that the geometric phase could be used to investigate the critical properties of the spin chains Carollo (). On the other hand, the critical exponents can be evaluated from the scaling behavior of the geometric phases Zhu (). Therefore, the geometric phase could be considered as a topological test for manifestation of quantum phase transitions Hamma (). These general relation originates from topological property of the geometric phase. It describes the curvature of the Hilbert space and is directly related to the degeneracy property in the quantum systems. The degeneracy in the many-body systems plays crucial role in our understanding of the quantum phase transition. Thus the geometric phase can be considered as another powerful tool for detecting the QPT.
Our main purpose in this work is to hire quantum renormalization group (QRG) Pfeuty1 () to study the evolution of the geometric phase of spin models. To have a concrete discussion, the one dimensional Ising model in transverse field (ITF) is considered by implementing the quantum renormalization group approach kargarian1 (); kargarian2 (); Jafari1 (); Jafari2 (); kargarian3 (); Langari (); miguel1 (); Jafari3 (); miguel2 (). To the best of my knowledge, the GP properties study has only been done for exactly solvable models and this is the first report which addresses how to get GP properties of the models which are not exactly solvable using QRG. I also show that QRG-based investigation of the GP of the models is more convenient and also accurate than that of entanglement (concurrence).
2 Theoretical Model
Consider the ITF model on a periodic chain of sites with Hamiltonian
where and are the exchange coupling and the transverse field, respectively. From the exact solution miguel2 (); Pfeuty2 () it can be seen that a second order phase transition occurs for where the behavior of the order parameter or magnetization is given by for and for .
3 Quantum Renormalization Group
The main idea of the RG method is the mode elimination or thinning of the degrees of freedom followed by an iteration which reduces the number of variables step by step till reaching a fixed point. In Kadanoff’s approach, the first step of the QRG method consists of assembling a set of lattice points into disconnected blocks of sites. In this fashion, the total number of blocks in the whole chain would be . This partitioning of the lattice into blocks induces a decompositioning of the Hamiltonian into two parts: intra-block () and inter-block () Hamiltonians. The block Hamiltonian is a sum of commuting Hamiltonians () acting on individual blocks. The diagonalization of for small is achieved analytically and then intra-block Hamiltonian and inter-block Hamiltonian is projected into the low energy subspace of . Afterwards, the original Hamiltonian is mapped into an effective Hamiltonian () which acts on the renormalized subspace miguel1 (); Jafari3 (); miguel2 ().
In this paper, to implement QRG, the Hamiltonian is divided into two-site blocks
and the remaining part of the Hamiltonian is included in the inter-block part
where refers to the -component of the Pauli matrix at site of the block labeled by . The Hamiltonian of each block () is diagonalized exactly and the projection operator
is constructed from the two lowest eigenstates in which is the ground state and is the first excited state. In this respect the effective Hamiltonian
is matched to the original one (Eq.(1)) replacing the couplings with the following renormalized coupling constants.
4 Geometric Phase and Renormalization Group Application
To investigate the geometric phase in systems, a new family of Hamiltonians are introduced that can be described by applying a rotation of around the direction to each spin Carollo (), i.e.,
The critical behavior is independent of as the spectrum of the system is independent Zhu (). The geometric phase of the ground state, accumulated by varying the angle from to , is described by
here is the ground state of Carollo ().
The eigenvalues of the Hamiltonian will not affected by this unitary transformation. So the eigenvectors of new Hamiltonians can be obtained by acting the rotation operator on the eigenvectors of the former Hamiltonian (). In other words, where and are the eigenvectors of and , respectively. However, the projection operators of new Hamiltonian (() and the unrotated Hamiltonian (Eq. (2)) are related by
On the other hand, the ground state of the renormalized chain will be related to that of the original one by the transformation . It is straightforward to show that the geometric phase in the renormalized chain is described by
where is geometric phase at the step of RG and is defined by . The expression for is similar to where the coupling constants should be replaced by the renormalized ones at the corresponding RG iteration (). In this approach, geometric phase at each iteration of RG is connected to its value after a RG iteration by Eq. (6). This will be continued till reaching a controllable fixed point where the value of the geometric phase could be obtained Jafari1 (); Jafari3 ().
5 Numerical Results: Scaling properties of Geometric Phase
In this section the numerical results of the model would be discussed. The evolution of under RG steps versus is presented in Fig. 1. In the RG step the expression given in Eq. (6) is evaluated at the renormalized coupling given by the iteration of given in Eq. (3). The zero RG step means a bare two-site model, while in the first RG step the effective two-site model represents a four-site chain. Generally, in the step of RG, a chain of sites is represented effectively by the two sites with renormalized couplings. All curves in Fig. (1) have a kink at the critical point, , for large systems. At the critical point, correlation length is infinite and fluctuations occur on all length scales which means that the system is scale-invariant. The non-analytic behavior is a feature of second-order quantum phase transition. It is also accompanied by a scaling behavior since the correlation length diverges and there is no characteristic length in the system at the critical point.
Zhu has verified that the GP of ground state in the XY model in the transverse field obeys scaling behavior in the vicinity of a quantum phase transition Zhu (). In particular he has shown that the geometric phase is non-analytical and its derivative with respect to the magnetic field diverges at the critical point. As it is previously stated, a large system, i.e. , can be effectively describe by two sites with the renormalized coupling in the RG step. The first derivative of GP is analyzed as a function of magnetic field at different RG steps which manifest the size of the system. In Fig. (2) the derivative of GP with respect to the coupling constant is presented which, shows a singular behavior at the critical point as the size of system becomes large. This singular behavior is the result of the kink in GP at (Fig. (1)).
It is found that the position of the maximum of () tends towards the critical point like
The exponent is directly related to the correlation length exponent () close to the critical point. The correlation length exponent gives the behavior of the correlation length in the vicinity of , i.e., . Under the RG transformation of Eq. (3), the correlation length scales in the th RG step as , which immediately leads to an expression for in terms of and . Dividing the last equation into gives rise to , which implies that , since at the critical point. We should note that the scaling of in the the position of (Fig. (3)), comes from the divergence of the correlation length near the critical point. In the large system size limit, when approaching to the critical point the correlation length almost covers the size of the system which results in the following scaling form
To obtain the finite-size scaling behavior of , we look for a scaling function when all graphs tend to collapse on each other under RG evolution which results in a large system. This is also a manifestation of the existences of the finite size scaling for the GP. Fig. (5) shows the plot of versus . The lower curves, which are for large system sizes, clearly show that all plots fall on each other.
The similar scaling behaviors as well as their relation to correlation length exponent have been reported in our previous works kargarian1 (); Jafari2 (), where the static properties of the ground state entanglement and low energy state dynamics of entanglement of ITF model by RG method were studided. These facts strongly imply the important relation between quantum entanglement and geometric phase, and provides a possible understanding of entanglement from the topological structure of the systems. This point can be understood by noting that both of the mentioned methods are connected to the correlation functions, and also are connected directly to each other by the inequality Cui ().
To summarize, the idea of renormalization group (RG) to study the geometric phase of Ising model in transverse field is implemented. In order to explore the critical behavior of the ITF model the evolution of geometric phase through the renormalization of the lattice were examined. In this respect I have shown that the RG procedure can be implemented to obtain the GP of a system and its finite size scaling in terms of the effective Hamiltonian which is described by the renormalized coupling constants. The phase transition becomes significant which shows a diverging behavior in the first derivative of the geometric phase. This divergence of GP are accompanied by a scaling behavior near the critical point where the size of the system becomes large. The scaling behavior characterizes how the critical point of the model is touched as the system size is increased. It is also shown that the non-analytic behavior of GP is originated from the correlation length exponent in the vicinity of the critical point. This shows that the behavior of the GP near the critical point is directly connected to the quantum critical properties of the model. We get the properties of GP for a large system dealing with a small block which make it possible to get analytic results. However, the numerical results of QRG show that the application of QRG to manifest the GP properties, is quantitatively more accurate than its application on quantum information resource kargarian1 (); kargarian2 (); Jafari1 (); Jafari2 (); kargarian3 ().
The author would like to thank S. N. S. Reihani, A. Akbari and A. Langari for reading the manuscript, fruitful discussions and comments.
- (1) M. Vojta, Rep. Prog. Phys. 66, (2003) 2069 and references therein.
- (2) N. Goldenfeld, Lectures on phase transitiosn and the renormalization group, Westview Press, Boulder, 1992.
- (3) A. Osterloh, Luigi Amico, G. Falci and Rosario Fazio, Nature 416, 608 (2002).
- (4) M. Kargarian, R. Jafari and A. Langari, Phys. Rev. A 76, 060304(R)(2007).
- (5) M. Kargarian, R. Jafari and A. Langari, Phys. Rev. A 77, 032346 (2008).
- (6) R. Jafari, M. Kargarian, A. Langari, and M. Siahatgar, Phys. Rev. B 78, 214414 (2008).
- (7) M. Kargarian, R. Jafari and A. Langari, Phys. Rev. A 79, 042319 (2009).
- (8) R. Jafari, Phys. Rev. A 82, 052317 (2010).
- (9) A. Langari, A. T. Rezakhani, New J. Phys. 14, 053014 (2012); N. Amiri, A. Langari, physica status solidi (b). 250, 417 (2013).
- (10) T.J. Osborne, M.A. Nielsen, Phys. Rev. A. 66, 032110 (2002); G. Vidal, J.I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003); Y. Chen, P. Zanardi, Z. D. Wang, F. C. Zhang, New J. Phys. 8, 97 (2006); L.-A. Wu, M.S. Sarandy, D. A. Lidar, Phys. Rev. Lett. 93, 250404 (2004)
- (11) M.E. Reuter, M.J. Hartmann, M.B. Plenio, Proc. Roy. Soc. Lond. A 463, 1271 (2007).
- (12) P. Zanardi and N. Paunkovic, Phys. Rev. E 74, 031123 (2006); A. T. Rezakhani, P. Zanardi, Phys. Rev. A 73, 012107 (2006); 73, 052117 (2006).
- (13) M. V. Berry, Proc. R. Soc. London 392, 45 (1984).
- (14) A. Shapere and F. Wilczek (Eds.), Geometric Phases in Physics, World Scientific, Singapore, 1989.
- (15) A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, and J. Zwanziger, The Geometric Phase in Quantum Systems, Springer, Berlin, 2003.
- (16) Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
- (17) K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
- (18) P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999); J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Nature (London) 403, 869 (2000).
- (19) A. Uhlmann, Rep. Math. Phys. 24, 229 (1986); E. Sjoqvist, A.K. Pati, A. Ekert, J.S. Anandan, M. Ericsson, D.K.L. Oi, V. Vedral, Phys. Rev. Lett. 85 2845 (2000).
- (20) A. Carollo, and J. K. Pachos, Phys. Rev. Lett. 95, 157203(2005).
- (21) S. L. Zhu, Phys. Rev. Lett. 96, 077206(2006).
- (22) A. Hamma, e-print: quant-ph/0602091.
- (23) P. Pfeuty, R. Jullian, K. L. Penson, in Real-Space Renormalization, edited by T. W. Burkhardt and J. M. J. van Leeuwen (Springer, Berlin, 1982), Chap. 5.
- (24) M. A. Martin-Delgado and G. Sierra, Int. J. Mod. Phys. A 11, 3145 (1996).
- (25) R. Jafari and A. Langari, Phys. Rev. B 76, 014412 (2007); Physica A 364, 213 (2006); A. Langari, Phys. Rev. B 69, 100402(R) (2004).
- (26) M. A. Martin-Delgado and G. Sierra, Phys. Rev. Lett. 76, 1146 (1996).
- (27) P. Pfeuty, ANNALS of Physics, 57, 79 (1970).
- (28) H.T. Cui, Y.F. Zhang, Eur. Phys. J. D, 51, 393 (2009).