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Entanglement in the quantum Ising model

Geoffrey Grimmett, Tobias J. Osborne, Petra F. Scudo, G. R. Grimmett, Tobias J. Osborne, P. F. Scudo

Published Jan 1, 2007• 28 citations
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Abstract

We study the asymptotic scaling of the entanglement of a block of spins for the ground state of the one-dimensional quantum Ising model with transverse field. When the field is sufficiently strong, the entanglement grows at most logarithmically in the number of spins. The proof utilises a transformation to a model of classical probability called the continuum random-cluster model, and is based on a property of the latter model termed ratio weak-mixing. Our proof applies equally to a large class of disordered interactions. 1 The quantum Ising model The quantum Ising model in a transverse magnetic field is one of the most famous examples of exactly solvable one-dimensional quantum models. The solution was first given by Pfeuty in [26], based on earlier works by Lieb, Schultz, and Mattis [18] and by McCoy [21]. The diagonalisation of the Hamiltonian and the determination of the energy eigenstates is based on methods developed by Jordan and Wigner [16] in the theory of second quantisation of fermion fields, and by Bogoliubov [7] in the theory of superconductivity. This model exhibits a second-order phase transition in the ground state when the temperature of the system is zero. The existence of the phase transition and

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We study the asymptotic scaling of the entanglement of a block of spins for the ground state of the one-dimensional quantum Ising model with transverse field. When the field is sufficiently strong, the entanglement grows at most logarithmically in the number of spins. The proof utilises a transformation to a model of classical probability called the continuum random-cluster model, and is based on a property of the latter model termed ratio weak-mixing. Our proof applies equally to a large class of disordered interactions. 1 The quantum Ising model The quantum Ising model in a transverse magnetic field is one of the most famous examples of exactly solvable one-dimensional quantum models. The solution was first given by Pfeuty in [26], based on earlier works by Lieb, Schultz, and Mattis [18] and by McCoy [21]. The diagonalisation of the Hamiltonian and the determination of the energy eigenstates is based on methods developed by Jordan and Wigner [16] in the theory of second quantisation of fermion fields, and by Bogoliubov [7] in the theory of superconductivity. This model exhibits a second-order phase transition in the ground state when the temperature of the system is zero. The existence of the phase transition and
A rigorous proof is presented of the boundedness of the entanglement entropy of a block of spins for the ground state of the one-dimensional quantum Ising model with sufficiently strong transverse field. This is proved by a refinement of the stochastic geometric arguments in the earlier work by Grimmett et al. (J Stat Phys 131:305–339, 2008). The proof utilises a transformation to a model of classical probability called the continuum random-cluster model. Our method of proof is fairly robust, and applies also to certain disordered systems. © 2019, The Author(s).
Abstract A rigorous proof is presented of the boundedness of the entanglement entropy of a block of spins for the ground state of the one-dimensional quantum Ising model with sufficiently strong transverse field. This is proved by a refinement of the stochastic geometric arguments in the earlier work by Grimmett et al. (J Stat Phys 131:305–339, 2008). The proof utilises a transformation to a model of classical probability called the continuum random-cluster model. Our method of proof is fairly robust, and applies also to certain disordered systems.
The quantum phase transition (QPT) of the one-dimensional (1D) quantum compass model in a transverse magnetic field is studied in this paper. An exact solution is obtained by using an extended Jordan and Wigner transformation to the pseudo-spin operators. The fidelity susceptibility, the concurrence, the block-block entanglement entropy, and the pseudo-spin correlation functions are calculated with antiperiodic boundary conditions. The QPT driven by the transverse field only emerges at zero field and is of the second-order. Several critical exponents obtained by finite size scaling analysis are the same as those in the 1D transverse field Ising model, suggesting the same universality class. A logarithmic divergence of the entanglement entropy of a block at the quantum critical point is also observed. From the calculated coefficient connected to the central charge of the conformal field theory, it is suggested that the block entanglement depends crucially on the detailed topological structure of a system.
The random Ising model in a transverse field in one dimension with Hamiltonian H=- Sigma ( Gamma i S i z +J i S i x S i+1 x ) is studied at T=O from a real-space renormalisation group block method which preserves duality transformations. The ground state magnetisation and the ground state energy are determined for random distributions P(J)(P( Gamma ))=N J J NJ-1 /J 0 N(J) for 0<J<J 0 and P(J)=0 for J>J O . A new critical behaviour corresponding to a new fixed point takes place in the presence of disorder. The crossover exponent describing the departure from the pure system behaviour is calculated. The second derivative of the ground state energy delta 2 E/ delta Gamma 2 which diverges logarithmically for the pure system is rounded in the presence of disorder but a sharp transition field still exists where the magnetisation goes to zero with an exponent beta about twice as large as the pure system exponent. Comparison is made with the analytical results of McCoy and Wu (1969) for the classical equivalent random-striped Ising 2D model.
The quantum phase transition (QPT) of the one-dimensional (1D) quantum compass model in a transverse magnetic field is studied in this paper. An exact solution is obtained by using an extended Jordan and Wigner transformation to the pseudospin operators. The fidelity susceptibility, the concurrence, the block-block entanglement entropy, and the pseudospin correlation functions are calculated with antiperiodic boundary conditions. The QPT driven by the transverse-field only emerges at zero field and is of the second order. Several critical exponents obtained by finite-size scaling analysis are the same as those in the 1D transverse-field Ising model, suggesting the same universality class. A logarithmic divergence of the entanglement entropy of a block at the quantum critical point is also observed. From the calculated coefficient connected to the central charge of the conformal field theory, it is suggested that the block entanglement depends crucially on the detailed topological structure of a system.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.