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arXiv:2406.08865 · Journal of High Energy Physics

Quantum phase transition of (1+1)-dimensional O(3) nonlinear sigma model at finite density with tensor renormalization group

Xiao Luo, Y. Kuramashi

Published Nov 26, 2024• 1 citation
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Abstract

A bstract We study the quantum phase transition of the (1+1)-dimensional O(3) nonlinear sigma model at finite density using the tensor renormalization group method. This model suffers from the sign problem, which has prevented us from investigating the properties of the phase transition. We investigate the properties of the phase transition by changing the chemical potential μ at a fixed coupling of β . We determine the transition point μ c and the critical exponent ν from the μ dependence of the number density in the thermodynamic limit. The dynamical critical exponent z is also extracted from the scaling behavior of the temporal correlation length as a function of μ .

DOI
10.1007/jhep11(2024)144

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A bstract We study the quantum phase transition of the (1+1)-dimensional O(3) nonlinear sigma model at finite density using the tensor renormalization group method. This model suffers from the sign problem, which has prevented us from investigating the properties of the phase transition. We investigate the properties of the phase transition by changing the chemical potential μ at a fixed coupling of β . We determine the transition point μ c and the critical exponent ν from the μ dependence of the number density in the thermodynamic limit. The dynamical critical exponent z is also extracted from the scaling behavior of the temporal correlation length as a function of μ .
We study the quantum phase transition of the (1+1)-dimensional O(3) nonlinear sigma model at finite density using the tensor renormalization group method. This model suffers from the sign problem, which has prevented us from investigating the properties of the phase transition. We investigate the properties of the phase transition by changing the chemical potential mu at a fixed coupling of beta. We determine the transition point mu c and the critical exponent nu from the mu dependence of the number density in the thermodynamic limit. The dynamical critical exponent z is also extracted from the scaling behavior of the temporal correlation length as a function of mu.
We study (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method with the infinite limit of the bond dimension Dcut→∞D_{\rm cut}\rightarrow \inftyDcut​→∞. At the vanishing chemical potential μ=0μ=0μ=0, we investigate the von Neumann and Rényi types of entanglement entropies. The central charge is determined to be c=1.97(9)c=1.97(9)c=1.97(9) by using the asymptotic scaling properties of the entropies. We also examine the consistency between two entropies. In the finite density region with μ≠0μ\ne 0μ=0, where this model suffers from the sign problem in the standard Monte Carlo approach, we investigate the properties of the quantum phase transition. We determine the transition point μcμ_{\rm c}μc​ and the critical exponent of the correlation length ννν from the μμμ dependence of the number density in the thermodynamic limit. The dynamical critical exponent zzz is also extracted from the scaling behavior of the temporal correlation length as a function of μμμ. This is the first successful calculation of the dynamical critical exponent with the TRG method.
We study (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method with the infinite limit of the bond dimension Dcut→∞D_{\rm cut}\rightarrow \inftyDcut​→∞. At the vanishing chemical potential μ=0\mu=0μ=0, we investigate the von Neumann and R\'enyi types of entanglement entropies. The central charge is determined to be c=1.97(9)c=1.97(9)c=1.97(9) by using the asymptotic scaling properties of the entropies. We also examine the consistency between two entropies. In the finite density region with μ≠0\mu\ne 0μ=0, where this model suffers from the sign problem in the standard Monte Carlo approach, we investigate the properties of the quantum phase transition. We determine the transition point μc\mu_{\rm c}μc​ and the critical exponent of the correlation length ν\nuν from the μ\muμ dependence of the number density in the thermodynamic limit. The dynamical critical exponent zzz is also extracted from the scaling behavior of the temporal correlation length as a function of μ\muμ. This is the first successful calculation of the dynamical critical exponent with the TRG method.
We investigate the metal-insulator transition of the (1+11+11+1)-dimensional Hubbard model in the path-integral formalism with the tensor renormalization group method. The critical chemical potential ${\ensuremath{\mu}}_{\mathrm{c}}$ and the critical exponent $\ensuremath{\nu}$ are determined from the $\ensuremath{\mu}$ dependence of the electron density in the thermodynamic limit. Our results for ${\ensuremath{\mu}}_{\mathrm{c}}$ and $\ensuremath{\nu}$ show consistency with an exact solution based on the Bethe ansatz. Our encouraging results indicate the applicability of the tensor renormalization group method to the analysis of higher-dimensional Hubbard models.
We discuss the thermodynamics of the O(3) nonlinear sigma model in 1+11+11+1 dimensions at nonzero chemical potential (equivalent to a magnetic field). In its conventional field theory representation the model suffers from a sign problem. By dualizing the model, we are able to fully access the nonzero density regime of an asymptotically free theory with dynamical mass gap at arbitrary chemical potential values. We find a quantum phase transition at zero temperature where as a function of the chemical potential the density assumes a nonzero value. Measuring the spin stiffness we present evidence for a corresponding dynamical critical exponent zzz close to 2. The low energy O(3) model is conjectured to be described by a massive boson triplet with repulsive interactions. We confirm the universal square-root behavior expected for such a system at low density (and temperature) and compare our data to the results of Bethe Ansatz solutions of the relativistic and nonrelativistic one-dimensional Bose gas. We also comment on a potential Berezinskii-Kosterlitz-Thouless transition at nonzero density.
We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equations of motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d=2+ɛ expansion about its lower critical dimension d_{lc}=2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d_{c}=4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε=4-d expansion near d_{c}. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z_{Γ}=d-2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values z_{c}=d/2 in an intermediate crossover region.