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arXiv:1105.0853 · Physical Review A

Statistical mechanics of the cluster Ising model

Pietro Smacchia, Luigi Amico A, Paolo Facchi A, Rosario Fazio, Giuseppe Florio, Saverio Pascazio, Vlatko Vedral

Published Aug 2, 2011• 140 citations
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Abstract

We study a Hamiltonian system describing a three-spin-$1/2$ clusterlike interaction competing with an Ising-like antiferromagnetic interaction. We compute free energy, spin-correlation functions, and entanglement both in the ground and in thermal states. The model undergoes a quantum phase transition between an Ising phase with a nonvanishing magnetization and a cluster phase characterized by a string order. Any two-spin entanglement is found to vanish in both quantum phases because of a nontrivial correlation pattern. Nevertheless, the residual multipartite entanglement is maximal in the cluster phase and dependent on the magnetization in the Ising phase. We study the block entropy at the critical point and calculate the central charge of the system, showing that the criticality of the system is beyond the Ising universality class.

DOI
10.1103/physreva.84.022304

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We investigate the cluster Ising chain with an additional nearest-neighbor interaction using tensor-network methods and a weak-coupling field theory. Without the interaction, the Jordan-Wigner transformation decomposes the model into a triplet of identical Majorana chains related by an exact O(3)O(3)O(3) flavor symmetry. Their common mass vanishes at the SU(2)2SU(2)_2SU(2)2​ Wess-Zumino-Novikov-Witten critical point with central charge c=3/2c = 3/2c=3/2 separating the symmetry-protected topological cluster phase from a ferromagnet. The interaction reduces O(3)O(3)O(3) to its cyclic subgroup C3C_3C3​, splitting the triplet into a singlet and a doublet whose gaps close separately, producing a critical bifurcation into Ising (c=1/2c = 1/2c=1/2) and Gaussian (c=1c = 1c=1) critical lines. A second ferromagnetic phase opens between these critical lines for repulsive interactions and a disordered phase for attractive ones. The Gaussian line then separates two Landau-incompatible ferromagnets, realizing a deconfined quantum critical line with emergent O(2)O(2)O(2) symmetry, along which our independently extracted exponents vary continuously yet satisfy the parameter-free relation β=(2ν−1)/4β= (2ν- 1)/4β=(2ν−1)/4 of the eight-vertex weak universality class. At stronger repulsion this line opens into an extended gapless floating phase with incommensurate algebraic correlations, entered through Berezinskii-Kosterlitz-Thouless transitions. All of these phases and the transitions between them are captured by the weak-coupling theory. Beyond its regime of validity, our simulations reveal a translation-symmetry-breaking antiferromagnet reached through first-order transitions.
We study the ground-state properties of a spin-1/2 model on a chain containing four-spin Ising-like interactions in the presence of both transverse and longitudinal magnetic fields. We use entanglement entropy and finite-size scaling methods to obtain the phase diagrams of the model. Our numerical calculations reveal a rich variety of phases and the existence of multicritical points in the system. We identify phases with both ferromagnetic and antiferromagnetic orderings. We also find periodically modulated orderings formed by a cluster of like spins followed by another cluster of opposite like spins. The quantum phases in the model are found to be separated by either first- or second-order transition lines.
We study a cluster Ising model with non-Hermitian external field which can be exactly solved in the language of free fermions. By investigating the second derivative of energy density and fidelity, the possible new critical points are tentatively located. The string order parameter and staggered magnetization are then detected to reveal emergent phases of brand new characteristics. To categorize the exotic phases and phase transitions induced by non-Hermiticity, we calculate the variation mode of the spin-correlation function as well as of the string correlation function, which characterize the emergent phases and critical points with different patterns of decay and critical exponents. With the help of the string order parameter and staggered magnetization, we find that there are four phases after introducing the non-Hermiticity---the cluster phase, the gapless phase, the paramagnetic phase, and the antiferromagnetic phase. A phase diagram is then presented to graphically illustrate, based on two ``Kosterlitz-Thouless-like'' phase transitions and an Ising phase transition, respectively, the generation of three critical lines as non-Hermitian strength increases. Our theoretical work is expected to be realized in the experiment of ultracold atoms, pushing for progress in exploring novel phases and phase transitions.
Abstract The quantum Gaussian phase transition is investigated for the infinite spin-1 bond-alternative Heisenberg model in one spatial dimension. By using a tensor network representation with an infinite matrix product state approach, the ground state energy, bipartite entanglement entropy, non-local string order, and fidelity per lattice site are calculated to characterize the phase transition. At the quantum phase transition point, the scaling behavior of various physical observables with respect to the finite truncation dimension are discussed for the ground state wavefunctions. In addition, the central charge is extracted from the finite entanglement entropies and the finite correlation lengths. Furthermore, the various critical exponents of the string order are calculated. The characteristic critical exponents and the central charge determine the universality class of the phase transition.