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.
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.
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.
The Hohenberg-Mermin-Wagner theorem states that there is no spontaneous breaking of continuous internal symmetries in spatial dimensions d≤2 at finite temperature. At zero temperature, the quantum-to-classical mapping further implies the absence of such symmetry breaking in one dimension, which is also known as Coleman's theorem in the context of relativistic quantum field theories. One route to violate this "folklore" is requiring an order parameter to commute with a Hamiltonian, as in the classic example of the Heisenberg ferromagnet and its variations. However, a systematic understanding has been lacking. In this Letter, we propose a family of one-dimensional models that display spontaneous breaking of a U(1) symmetry at zero temperature, where the order parameter does not commute with the Hamiltonian. While our models can be deformed continuously within the same phase, there exist symmetry-preserving perturbations that render the observed symmetry breaking fragile. We argue that a more general condition for this behavior is that the Hamiltonian is frustration-free.
Abstract The Mermin-Wagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with short-range interactions at spatial dimension D ≤ 2. For long-range interactions with a power-law form (1/ r α ), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2 D . However, the situation for α ∈ (2, 4) at D = 2 is not covered by the theorem. To address this, we conduct large-scale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of S U (2) symmetry in the ferromagnetic Heisenberg model with 1/ r α -form long-range interactions at D = 2. We determine critical exponents through finite-size analysis for α < 3 (above the upper critical dimension with Gaussian fixed point) and 3 ≤ α < 4 (below the upper critical dimension with non-Gaussian fixed point). These results reveal new critical behaviors in 2D long-range Heisenberg models, encouraging further experimental studies of quantum materials with long-range interactions beyond the Mermin-Wagner theorem’s scope.
The Hohenberg-Mermin-Wagner theorem motivates the expectation that continuous symmetry remains unbroken in one-dimensional quantum ground states. Here, contrary to this expectation, we demonstrate the spontaneous breaking of U(1) spin-rotation symmetry and the emergence of an interacting quantum critical point in lattice spin-1/2 and spin-1 chains. Using subsystem Binder scaling, we locate the critical point and determine the correlation-length critical exponent as ν=0.60±0.02, distinct from the Gaussian value 1/2 but consistent with the perturbative renormalization-group value. We further estimate the dynamical critical exponent to be z=1.986±0.005 from the lowest neutral excitation gap and z=1.955±0.021 from the longitudinal spin correlations. The results z<2 are incompatible with microscopic realizations of this critical theory by frustration-free models, implying a mechanism for continuous symmetry breaking in one-dimensional quantum ground states distinct from frustration freeness.
In local theories with short-range interactions, continuous symmetries cannot be spontaneously broken in two spatial dimensions at nonzero temperature. 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-N expansion, we study a (2+1)-dimensional quantum field theory in which a complex scalar is coupled to the critical 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) amplitude. Integrating out the gapped fluctuations yields an effective theory for the compact phase mode at distances large compared to 1/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 N and is fixed entirely by the fixed point. This renders vortex operators irrelevant for sufficiently large N. We further show that the vortex fugacity is exponentially small in N.
The basic problem how spontaneous symmetry breaks is studied rigorously for local fields in a finite domain. Another basic problem is to investigate whether or not there exist lower dimensions (of the domain to which a field H is applied) below which no spontaneous symmetry breaking (SSB) occurs. This is studied using scaling, renormalization group and mean-field theory for partial fields in an infinite domain. Our main results are the following: (i) The total magnetization is proportional to the square of the spontaneous magnetization when a field is applied only to a finite region (an extension of Yang's result [Phys. Rev. 85 (1952) 808], M = N m 2 s for a fixed single spin). A general formulation is given to calculate the proportionality coefficient which is shown to vanish as the field goes to zero. This indicates no SSB in this situation. (ii) On the other hand, if we apply partial fields to a d '-dimensional region and make them vanish, we find that SSB occurs, at least, for d '> d ' c =β/ν. In the region 0< d '≤ d ' c , the situation is subtle, as is exemplified by an exactly soluble model with such a field on one-side boundary as depends on the system-size.