• DOI:10.1103/PhysRevB.80.113103
• Corpus ID: 118670024

Critical exponents of one-dimensional quantum critical models by means of MERA tensor network

@inproceedings{Montangero2008CriticalEO,
  title={Critical exponents of one-dimensional quantum critical models by means of MERA tensor network},
  author={Simone Montangero and Matteo Rizzi and Vittorio Giovannetti and Rosario Fazio},
  year={2008}
}

• S. Montangero, M. Rizzi, +1 author R. Fazio
• Published 8 October 2008
• Physics

Critical phenomena are ubiquitous in science ranging from condensed matter to biological or economic systems. They are associated to scale invariance, a diverging correlation lenght, and the various correlation functions decay as power law. A key issue in the study of of critical systems is the computation of the critical exponents, i.e. the exponents which govern the decay of the correlations. The power of methods like the renormalization group stems from its capability to compute (analitically or numerically) the critical exponents. In the field of quantum many-body systems important progresses have been made recently in one-dimension where a number of tensor network approximations [1, 2, 3, 4, 5, 6, 7, 8] of the many-body wave-function allowing an efficient computation of all the relevant observables. The Multi-scale Entanglement Renormalization Ansatz (MERA) introduced by Vidal [8] is particularly appealing since embeds scale invariance typical of critical systems. A method to relate the critical exponents in the MERA tensor network has been very recently put forward by some of us [9] by relating them to the eigenvalues of the MERA transfer matrix. In the present work we want to apply this method for the computation of the critical exponents of two cornerstone models in quantum statistical mechanics, the Ising and the XXZ model. To this end we provide an algorithm for optimizing the MERA tensor network in an infinite system and then, by employing the results given in [9] we compute the critical exponents. The paper is organized as follows. In the next section we briefly review the optimization of the MERA tensor network and how to compute the critical exponent. We then proceed with the comparison with the Ising and XXZ model defined in Section III. In Section IV we discuss the results and compare them with the exact known values. The last Section is devoted to the conclusions.