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arXiv:2412.09459 · Physical Review Letters

Temperature-Resistant Order in 2 + 1 Dimensions

Zohar Komargodski, Fedor K. Popov

Published Jul 24, 2025• 3 citations
arXivHTMLPDF

Abstract

High temperatures are typically thought to increase disorder. Here we examine this idea in quantum field theory in 2+1 dimensions. For this sake we explore a novel class of tractable models, consisting of nearly-mean-field scalars interacting with critical scalars. We identify UV-complete, local, unitary models in this class and show that symmetry breaking Z_{2}→∅ occurs at any temperature in some regions of the phase diagram. This phenomenon, previously observed in models with fractional dimensions, or in the strict planar limits, or with nonlocal interactions, is now exhibited in a local, unitary 2+1 dimensional model with a finite number of fields.

DOI
10.1103/yb7d-6tvc

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High temperatures are typically thought to increase disorder. Here we examine this idea in quantum field theory in 2+1 dimensions. For this sake we explore a novel class of tractable models, consisting of nearly-mean-field scalars interacting with critical scalars. We identify UV-complete, local, unitary models in this class and show that symmetry breaking Z_{2}→∅ occurs at any temperature in some regions of the phase diagram. This phenomenon, previously observed in models with fractional dimensions, or in the strict planar limits, or with nonlocal interactions, is now exhibited in a local, unitary 2+1 dimensional model with a finite number of fields.
Spontaneous symmetry breaking can persist at all temperatures in certain biconical O(N)×Z_{2} vector models when the underlying field theories are ultraviolet complete. So far, the existence of such theories has been established in fractional dimensions for local but nonunitary models or in 2+1 dimensions but for nonlocal models. Here, we study local models at zero and finite temperature directly in 2+1 dimensions employing functional methods. At zero temperature, we establish that our approach describes the quantum critical behavior with good accuracy for all N≥2. We then exhibit the mechanism of discrete symmetry breaking from O(N)×Z_{2}→O(N) for increasing temperature near the biconical critical point when N is finite but large. We calculate the corresponding finite-temperature phase diagram and further show that the Hohenberg-Mermin-Wagner theorem is fully respected within this approach, i.e., symmetry breaking only occurs in the Z_{2} sector. Finally, we determine the critical N above which this phenomenon can be observed to be N_{c}≈15.
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.
The combined effect of disorder and symmetry-breaking fields on the two-dimensional XY\mathrm{XY}XY model is examined. The study includes disorder in the interaction among spins in the form of random phase shifts as well as disorder in the local orientation of the field. The phase diagrams are determined and the properties of the various phases and phase transitions are calculated. We use a renormalization-group approach in the Coulomb gas representation of the model. Our results differ from those obtained for special cases in previous works. In particular, we find a changed topology of the phase diagram that is composed of phases with long-range order, quasi-long-range order, and short-range order. The discrepancies can be ascribed to a breakdown of the fugacity expansion in the Coulomb gas representation. Implications for physical systems such as planar Josephson junctions and the faceting of crystal surfaces are discussed.
Temperature-driven first-order phase transitions that involve symmetry breaking are converted to second order by the introduction of infinitesimal quenched bond randomness in spatial dimensions d≤2 or d≤4, respectively, for systems of n=1 or n≥ (R18)2 component microscopic degrees of freedom. Even strongly first-order transitions undergo this conversion to second order! Above these dimensions, this phenomenon still occurs, but requires a threshold amount of bond randomness. For example, under bond randomness, the phase transitions of q-state Potts models are second order for all q in d≤2. If no symmetry breaking is involved, temperature-driven first-order phase transitions are eliminated under the above conditions. Another consequence is that bond randomness drastically alters multicritical phase diagrams. Tricritical points and critical endpoints are entirely eliminated (d≤2) or depressed in temperature (d≳2). Similarly, bicritical phase diagrams are converted (d≤2) to reentrant-disorder-line or decoupled-tetracritical phase diagrams. These quenched-fluctuation-induced second-order transitions (a diametric opposite to the previously known annealed-fluctuation-induced first-order transitions) should lead to a multitude of new universality classes of criticality, including many experimentally accessible cases.
We study the zero-temperature phase diagram of the disordered spin-111 Bose-Hubbard model in a two-dimensional square lattice. To this aim, we use a mean-field Gutzwiller ansatz and a probabilistic mean-field perturbation theory. The spin interaction induces two different regimes, corresponding to a ferromagnetic and antiferromagnetic order. In the ferromagnetic case, the introduction of disorder reproduces analogous features of the disordered scalar Bose-Hubbard model, consisting in the formation of a Bose glass phase between Mott insulator lobes. In the antiferromagnetic regime, the phase diagram differs more from the scalar case. Disorder in the chemical potential can lead to the disappearance of Mott insulator lobes with an odd-integer filling factor and, for sufficiently strong spin coupling, to Bose glass of singlets between even-filling Mott insulator lobes. Disorder in the spinor coupling parameter results in the appearance of a Bose glass phase only between the nnn and the n+1n+1n+1 lobes for nnn odd. Disorder in the scalar Hubbard interaction inhibits Mott insulator regions for occupation larger than a critical value.
In local theories with short-range interactions, continuous symmetries cannot be spontaneously broken in two spatial dimensions at nonzero temperature. U(1)\mathrm{U}(1)U(1)-symmetric models instead admit a Berezinskii--Kosterlitz--Thouless (BKT) phase with quasi-long-range order and nonzero phase stiffness, which is normally destroyed at high temperatures by vortex proliferation. Here, we show that a BKT phase, and with it superfluidity, can persist to arbitrarily high temperatures. Using the large-NNN expansion, we study a (2+1)(2+1)(2+1)-dimensional quantum field theory in which a complex scalar is coupled to the critical O(N)\mathrm{O}(N)O(N) model. At the biconical fixed point, where the inter-sector coupling is negative, the thermal effective potential has a circle of minima with nonzero U(1)\mathrm{U}(1)U(1) amplitude. Integrating out the gapped fluctuations yields an effective theory for the compact phase mode at distances large compared to 1/T1/T1/T. Since temperature is the only scale at the fixed point, the phase stiffness grows linearly with temperature, with a proportionality factor that scales linearly with NNN and is fixed entirely by the fixed point. This renders vortex operators irrelevant for sufficiently large NNN. We further show that the vortex fugacity is exponentially small in NNN.