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Emptiness formation probability for the spin- 1 XX chain with three spin and uniform long-range interactions

Paul C. Lou, J. Y. Lee

Published Jan 1, 2007• 0 citations
Publisher

Abstract

The emptiness formation probability (EFP) that the first n spins of the chain are all aligned downwards is studied for the one-dimensional spin- 1/2 XX chain system with both critical and non-critical regimes with three spin and uniform long-range interactions. By using the emptiness formation probability P(n), the block-block entanglement Sn and the magnetization Mz, the properties of the system are also studied. It is worthy of note \nthat when the quantum phase transition from the phase III to the phase II occurs, the Emptiness Formation Probability P(n) of the system decreases, but the block-block entanglement Sn increases.

DOI
10.5488/cmp.10.3.425

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The emptiness formation probability (EFP) that the first n spins of the chain are all aligned downwards is studied for the one-dimensional spin- 1/2 XX chain system with both critical and non-critical regimes with three spin and uniform long-range interactions. By using the emptiness formation probability P(n), the block-block entanglement Sn and the magnetization Mz, the properties of the system are also studied. It is worthy of note \nthat when the quantum phase transition from the phase III to the phase II occurs, the Emptiness Formation Probability P(n) of the system decreases, but the block-block entanglement Sn increases.
The emptiness formation probability (EFP) that the first n spins of the chain are all aligned downwards is studied for the one-dimensional spin-1 2 XX chain system with both critical and non-critical regimes with three spin and uniform long-range interactions.By using the emptiness formation probability P (n), the block-block entanglement Sn and the magnetization M z , the properties of the system are also studied.It is worthy of note that when the quantum phase transition from the phase III to the phase II occurs, the Emptiness Formation Probability P (n) of the system decreases, but the block-block entanglement Sn increases.
We propose a new correlator in one-dimensional quantum spin chains, the s -emptiness formation probability ( s -EFP). This is a generalization of the emptiness formation probability (EFP), which is the probability that the first n spins of the chain are all aligned downwards. In the s -EFP we let the spins in question be separated by s sites. The usual EFP corresponds to the special case when s = 1. Taking s > 1 allows us to quantify non-local correlations. We express the s -EFP for the anisotropic XY model in a transverse magnetic field, a system with both critical and non-critical regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find that the magnetic field induces an interesting length scale.
We study the Emptiness Formation Probability (EFP) for the spin 1/2 XXZ spin chain. EFP P (n) detects a formation of ferromagnetic string of the length n in the ground state. It is expected that EFP decays in a Gaussian way for large strings P (n) n . Here, we propose the explicit expressions for the rate of Gaussian decay C as well as for the exponent . In order to con rm the validity of our formulae, we employed an ab initio simulation technique of the density-matrix renormalization group to simulate XXZ spin chain of sucient length. Furthermore, we performed Monte-Carlo integration of the Jimbo-Miwa multiple integral for P (n). Those numerical results for P (n) support our formulae fairly de nitely.
We consider the one-dimensional XXX spin 1/2 Heisenberg antiferromagnet at zero temperature and zero magnetic field. We are interested in a probability of a formation of a ferromagnetic string P(n) in the antiferromagnetic ground-state. We call it emptiness formation probability [EFP]. We suggest a new technique for computation of the EFP in the inhomogeneous case. It is based on the quantum Knizhnik-Zamolodchikov equation [qKZ]. We calculate EFP for n≤6 for the inhomogeneous case. The homogeneous limit confirms our hypothesis about the relation of quantum correlations and number theory. We also make a conjecture about a structure of EFP for arbrary n.
We study an asymptotic behaviour of a special correlator known as the emptiness formation probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field. This correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models. For the XY spin chain, the correlator can be expressed as the determinant of a Toeplitz matrix and its asymptotical behaviours for n → ∞ throughout the phase diagram are obtained using known theorems and conjectures on Toeplitz determinants. We find that the decay is exponential everywhere in the phase diagram of the XY model except on the critical lines, i.e. where the spectrum is gapless. In these cases, a power-law prefactor with a universal exponent arises in addition to an exponential or Gaussian decay. The latter Gaussian behaviour holds on the critical line corresponding to the isotropic XY model, while at the critical value of the magnetic field the EFP decays exponentially. At small anisotropy one has a crossover from the Gaussian to the exponential behaviour. We study this crossover using the bosonization approach.
We review some known properties of the "emptiness formation probability" correlation which we have calculated numerically for spin-1/2 XX chains with constant (homogeneous) or alternating (dimerized) nearestneighbor coupling and an external field (in z direction) for arbitrary temperature.The long-distance asymptotic behavior of this correlation is known to be Gaussian at zero temperature and exponential at finite temperature for the homogeneous chain.By simple analytical arguments the exponential behavior at finite temperature extends to the dimerized system.Numerical results for the dimerized chain confirm the exponential decay at finite temperature and show Gaussian decay at zero temperature.