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Europhysics Letters (EPL)

Momentum space Z 2 number, quantized Berry phase and the quantum phase transitions in spin chain systems

Yu-Quan Ma, Zhaoxian Yu, Deng‐Shan Wang, Bing-Hao Xie, Xiang‐Gui Li

Published Dec 1, 2012• 7 citations
Publisher

Abstract

We characterize the quantum phase transitions of gapped time-reversal invariant spin chain systems by nontrivial Z 2 numbers derived from a quantized Berry phase in a (1 + 1)-dimensional Bloch momentum space. We study this approach analytically in a general two-band model and give a concrete example in a transverse field XY spin chain system. In our scheme, an extra local gauge transformation is performed to the spin system by a time-dependent twist operator, which endows the Hamiltonian of the system with the topology of a torus without changing its energy spectrum. The Z 2 number is obtained by a loop integral of the Berry gauge potential along a quarter of the Brillouin zone. We show that the different phases of the XY spin chain system are distinguished by the Z 2 topological order in momentum space.

DOI
10.1209/0295-5075/100/60001

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We characterize the quantum phase transitions of gapped time-reversal invariant spin chain systems by nontrivial Z 2 numbers derived from a quantized Berry phase in a (1 + 1)-dimensional Bloch momentum space. We study this approach analytically in a general two-band model and give a concrete example in a transverse field XY spin chain system. In our scheme, an extra local gauge transformation is performed to the spin system by a time-dependent twist operator, which endows the Hamiltonian of the system with the topology of a torus without changing its energy spectrum. The Z 2 number is obtained by a loop integral of the Berry gauge potential along a quarter of the Brillouin zone. We show that the different phases of the XY spin chain system are distinguished by the Z 2 topological order in momentum space.
We propose a topological Euler number to characterize nontrivial topological phases of gapped fermionic systems, which originates from the Gauss-Bonnet theorem on the Riemannian structure of Bloch states established by the real part of the quantum geometric tensor in momentum space. Meanwhile, the imaginary part of the geometric tensor corresponds to the Berry curvature which leads to the Chern number characterization. We discuss the topological numbers induced by the geometric tensor analytically in a general two-band model. As an example, we show that the zero-temperature phase diagram of a transverse field XY spin chain can be distinguished by the Euler characteristic number of the Bloch states manifold in a -dimensional Bloch momentum space.
We show that the Z_{N} Berry phase (Berry phase quantized into 2π/N) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size. The Z_{N} Berry phase is defined in a N-1-dimensional parameter space of local gauge twists, which we call the "synthetic Brillouin zone," and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase. We demonstrate the usefulness of the Z_{N} Berry phase by studying two 1D models of bosons, SU(3) and SU(4) Affleck-Kennedy-Lieb-Tasaki models, where topological phase transitions are captured by Z_{3} and Z_{4} Berry phases, respectively. We find that the exact quantization of the Z_{N} Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone.
Recently, by using quantized Berry phases, a prescription for a local characterization of gapped topological insulators has been given (Hatsugai 2006 Preprint cond-mat/0603230). It requires that the ground state is gapped and is invariant under some anti-unitary operation. A spin liquid which is realized as a unique ground state of the Heisenberg spin system with frustrations is a typical target system, since pairwise exchange couplings are always time-reversal invariants even with frustrations. As for a generic Heisenberg model with a finite excitation gap, we locally modify the Hamiltonian by a continuous SU (2) twist only at a specific link and define the Berry connection by the derivative. Then the Berry phase evaluated by the entire many-spin wavefunction is used to define the local topological order parameter at the link. We numerically apply this scheme for several spin liquids and show its physical validity. For example, it implies that the Haldane phase of the S = 1 chains is characterized by uniform π quantized Berry phases.
Recently by using quantized Berry phases, a prescription for a local characterization of gapped topological insulators is given [1].One requires the ground state is gapped and is invariant under some anti-unitary operation.A spin liquid which is realized as a unique ground state of the Heisenberg spin system with frustrations is a typical target system, since pairwise exchange couplings are always time-reversal invariants even with frustrations.As for a generic Heisenberg model with a finite excitation gap, we locally modify the Hamiltonian by a continuous SU (2) twist only at a specific link and define the Berry connection by the derivative.Then the Berry phase evaluated by the entire many-spin wavefunction is used to define the local topological order parameter at the link.We numerically apply this scheme for several spin liquids and show its physical validity.
We characterize several phases of gapped spin systems by local order parameters defined by quantized Berry phases [Y. Hatsugai, J. Phys. Soc. Jpn. 75, 123601 (2006)]. This characterization is topologically stable against any small perturbation as long as the energy gap remains finite. The models we pick up are S=1,2S=1,2S=1,2 dimerized Heisenberg chains and S=2S=2S=2 Heisenberg chains with uniaxial single-ion-type anisotropy. Analytically, we also evaluate the topological local order parameters for the generalized Affleck-Kennedy-Lieb-Tasaki model. The relation between the present Berry phases and the fractionalization in the integer spin chains are discussed as well.
We investigate the properties of antiferromagnetic spin-SSS ladders with the help of local Berry phases defined by imposing a twist on one or a few local bonds. In gapped systems with time-reversal symmetry, these Berry phases are quantized, hence able in principle to characterize different phases. In the case of a fully frustrated ladder where the total spin on a rung is a conserved quantity that changes abruptly upon increasing the rung coupling, we show that two Berry phases are relevant to detect such phase transitions: the rung Berry phase defined by imposing a twist on one rung coupling, and the twist Berry phase defined by twisting the boundary conditions along the legs. In the case of nonfrustrated ladders, we have followed the fate of both Berry phases when interpolating between standard ladders and dimerized spin chains, with the surprising conclusion that, at least far enough from dimerized chains, they define different domains in parameter space. A careful investigation of the spin gap and of edge states shows that a change of twist Berry phase is associated with a quantum phase transition at which the bulk gap closes, and at which, with appropriate boundary conditions, edge states appear or disappear, while a change of rung Berry phase is not necessarily associated with a quantum phase transition. The difference is particularly acute for regular ladders, in which the twist Berry phase does not change at all upon increasing the rung coupling from zero to infinity while the rung Berry phase changes 2S2S2S times. By analogy with the fully frustrated ladder, these changes are interpreted as crossovers between domains in which the rungs are in different states of total spin from 0 in the strong rung limit to 2S2S2S in the weak rung limit. This interpretation is further supported by the observation that these crossovers turn into real phase transitions as a function of rung coupling if one rung is strongly ferromagnetic, or equivalently if one rung is replaced by a spin 2S2S2S impurity.