We study the classical two-dimensional RP^{2} and Heisenberg models, using the tensor-network renormalization (TNR) method. The determination of the phase diagram of these models has been challenging and controversial due to the very large correlation lengths at low temperatures. The finite-size spectrum of the transfer matrix obtained by TNR is useful in identifying the conformal field theory describing a possible critical point. Our results indicate that the ultraviolet fixed point for the Heisenberg model and the ferromagnetic RP^{2} model in the zero-temperature limit corresponds to a conformal field theory with central charge c=2, in agreement with two independent would-be Nambu-Goldstone modes. On the other hand, the ultraviolet fixed point in the zero-temperature limit for the antiferromagnetic Lebwohl-Lasher model, which is a variant of the RP^{2} model, seems to have a larger central charge. This is consistent with c=4 expected from the effective SO(5) symmetry. At T>0, the convergence of the spectrum is not good in both the Heisenberg and ferromagnetic RP^{2} models. Moreover, there seems to be no appropriate candidate of conformal field theory matching the spectrum, which shows the effective central charge c∼1.9. These suggest that both models have a single disordered phase at finite temperatures, although the ferromagnetic RP^{2} model exhibits a strong crossover at the temperature where the dissociation of Z_{2} vortices has been reported.
We study the classical two-dimensional RP^{2} and Heisenberg models, using the tensor-network renormalization (TNR) method. The determination of the phase diagram of these models has been challenging and controversial due to the very large correlation lengths at low temperatures. The finite-size spectrum of the transfer matrix obtained by TNR is useful in identifying the conformal field theory describing a possible critical point. Our results indicate that the ultraviolet fixed point for the Heisenberg model and the ferromagnetic RP^{2} model in the zero-temperature limit corresponds to a conformal field theory with central charge c=2, in agreement with two independent would-be Nambu-Goldstone modes. On the other hand, the ultraviolet fixed point in the zero-temperature limit for the antiferromagnetic Lebwohl-Lasher model, which is a variant of the RP^{2} model, seems to have a larger central charge. This is consistent with c=4 expected from the effective SO(5) symmetry. At T>0, the convergence of the spectrum is not good in both the Heisenberg and ferromagnetic RP^{2} models. Moreover, there seems to be no appropriate candidate of conformal field theory matching the spectrum, which shows the effective central charge c∼1.9. These suggest that both models have a single disordered phase at finite temperatures, although the ferromagnetic RP^{2} model exhibits a strong crossover at the temperature where the dissociation of Z_{2} vortices has been reported.
Tensor-network methods are used to perform a comparative study of the two-dimensional classical Heisenberg and RP^{2} models. We demonstrate that uniform matrix product states (MPSs) with explicit SO(3) symmetry can probe correlation lengths up to O(10^{3}) sites accurately, and we study the scaling of entanglement entropy and universal features of MPS entanglement spectra. For the Heisenberg model, we find no signs of a finite-temperature phase transition, supporting the scenario of asymptotic freedom. For the RP^{2} model we observe an abrupt onset of scaling behavior, consistent with hints of a finite-temperature phase transition reported in previous studies. A careful analysis of the softening of the correlation length divergence, the scaling of the entanglement entropy, and the MPS entanglement spectra shows that our results are inconsistent with true criticality, but are rather in agreement with the scenario of a crossover to a pseudocritical region which exhibits strong signatures of nematic quasi-long-range order at length scales below the true correlation length. Our results reveal a fundamental difference in scaling behavior between the Heisenberg and RP^{2} models: Whereas the emergence of scaling in the former shifts to zero temperature if the bond dimension is increased, it occurs at a finite bond-dimension independent crossover temperature in the latter.
Determining whether an additional local symmetry affects the universality class of a sta-tistical model is an important issue in the theory of critical phenomena. A basic example is provided by the RP N −1 model, in which N -component spin variables at each lattice site interact through an Hamiltonian invariant under global O(N ) rotations and local spin re-versals. The local symmetry makes the difference with the usual O(N ) model and amounts to the head-tail symmetry characteristic of liquid crystals. In three dimensions, the weak first order transition observed in numerical simulations of the ferromagnetic model is con-sistent with the mean field scenario. On the other hand, in the two-dimensional case –the one we focus on in this thesis – fluctuations are stronger and minimize the reliability of mean field predictions, as illustrated by the phase transition of the three-state Potts model, which becomes continuous on planar lattices. For the RP N −1 model, the absence of spontaneous breaking of continuous symmetry in two dimensions generically suggests that criticality is limited to zero temperature, and Monte Carlo studies for T → 0 showed a fast growth of the correlation length which made particularly hard to reach the asymp-totic limit and draw conclusions. On the other hand, the possibility of finite temperature topological transitions similar to the Berezinskii-Kosterlitz-Thouless (BKT) one – which should definitely occur for RP 1 ∼ O(2) – and mediated by ”disclination” defects has also been debated in numerical studies. While two-dimensional criticality has allowed for an impressive amount of exact solutions thanks to lattice integrability and conformal field theory, models with local symmetries traditionally remained outside the range of application of these methods. In this thesis, however, we show how the renormalization group fixed points of the RP N −1 model and of its complex generalization, the CP N −1 model, can be accessed in an exact way. This is achieved in the scale invariant scattering framework which, as we review in the introductory part of the thesis, implements in the basis of particle excitations the infinite-dimensional conformal symmetry characteristic of critical points in two dimensions and has provided in the last few years new results for pure and disordered systems. In the last part of the thesis, we will exploit the generality of the scale invariant scattering method to progress with another long standing problem of two-dimensional criticality, namely that of spin clusters in Potts correlated percolation. It has been known for long time that the problem can be addressed considering two coupled Potts models, but the need to consider the number of states of one of these models as a continuous variable has severely limited the analytical or numerical study of the critical points. Also here, the ability of the scattering method to enforce conformal invariance for the internal symmetry characteristic of the universality class will allow us to obtain exact equations for the critical points and to determine their solutions in the relevant limits.
We present an extensive study of the AB2 polymeric Hubbard chains with experimental motivation in inorganic and organic compounds. In the half-filled strong-coupling limit, these systems can be mapped onto a quantum antiferromagnetic spin-1/2AB2 Heisenberg model that presents a zero-temperature quantum phase transition to a ferrimagnetic ground state. We derive the spin-wave modes with the same correct ferromagnetic dispersion relation calculated through two different techniques. We also build an Euclidean integral formulation in the coherent state representation which naturally gives rise to Wess-Zumino terms of topological origin. The low-energy infrared behavior of the AB2 chains is properly described through a renormalization group analysis by the semiclassical fixed point of a one-dimensional quantum nonrelativistic nonlinear $\ensuremath{\sigma}$ model with critical dynamical exponent z=2. Their low-temperature behavior is shown to be the same of a quantum ferromagnetic spin-1/2 Heisenberg chain, as evidenced by the temperature dependence of the correlation length, susceptibility, and specific heat. On the other hand, a quantum critical fixed point is also found, with the same zero-temperature critical behavior of a classical Heisenberg system in D=d+z=3 dimensions. Its low-temperature critical properties are related to the latter through the crossover exponent $\ensuremath{\varphi}=z{\ensuremath{\nu}}_{3},$ where ${\ensuremath{\nu}}_{3}$ is the correlation length exponent of the D=3 classical Heisenberg model. Sufficiently strong quantum critical fluctuations destroy the ordered ground state of the quantum z=2 nonlinear $\ensuremath{\sigma}$ model (but not that of the quantum AB2 Heisenberg spin systems, in agreement with a theorem by Lieb), giving rise to a quantum disordered phase at zero temperature. These results are supported by a connection between the quantum AB2 Heisenberg and generalized quantum rotor models.
We propose a general procedure for extracting the running coupling constants of the underlying field theory of a given classical statistical model on a two-dimensional lattice, combining tensor network renormalization (TNR) and the finite-size scaling theory of conformal field theory. By tracking the coupling constants at each scale, we are able to visualize the renormalization group flow and demonstrate it with the classical Ising and three-state Potts models. Furthermore, utilizing this methodology, we reveal the limitations due to finite bond dimension D on TNR applied to critical systems. We find that a finite correlation length is imposed by the finite bond dimension in TNR, and it can be attributed to an emergent relevant perturbation that respects the symmetries of the system. The correlation length shows the same power-law dependence on D as the ``finite entanglement scaling'' of the matrix product states.
The classical two-dimensional Heisenberg model revisited: An SU(2)-symmetric tensor network study
The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the O(3) O ( 3 ) non-linear sigma model in 1+1 1 + 1 dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in 3+1 3 + 1 dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an SU(2) S U ( 2 ) symmetry in our two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to \chi_E^\text{eff} \sim 1500 χ E eff ∼ 1500 , a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the literature, namely a finite- T T transition and asymptotic freedom, though with a slight preference for the second.
The classical two-dimensional Heisenberg model revisited: An SU(2)-symmetric tensor network study
The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the O(3) non-linear sigma model in 1+1 dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in 3+1 dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an SU(2) symmetry in our two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to χEeff∼1500, a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the literature, namely a finite-T transition and asymptotic freedom, though with a slight preference for the second.