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arXiv:1210.1225 · Physical Review B

Monte Carlo study of a U(1)×U(1) loop model with modular invariance

Scott Geraedts, Olexei I. Motrunich

Published Dec 19, 2012• 9 citations
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Abstract

We study a $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ system in (2+1) dimensions with long-range interactions and mutual statistics. The model has the same form after the application of operations from the modular group, a property which we call modular invariance. Using the modular invariance of the model, we propose a possible phase diagram. We obtain a sign-free reformulation of the model and study it in Monte Carlo. This study confirms our proposed phase diagram. We use the modular invariance to analytically determine the current-current correlation functions and conductivities in all the phases in the diagram, as well as at special ``fixed'' points which are unchanged by an operation from the modular group. We numerically determine the order of the phase transitions, and find segments of second-order transitions. For the statistical interaction parameter $\ensuremath{\theta}=\ensuremath{\pi}$, these second-order transitions are evidence of a critical loop phase obtained when both loops are trying to condense simultaneously. We also measure the critical exponents of the second-order transitions.

DOI
10.1103/physrevb.86.245121

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We study a U(1)\ifmmode×\else\texttimes\fiU(1)U(1)\ifmmode\times\else\texttimes\fi{}U(1)U(1)\ifmmode×\else\texttimes\fiU(1) system in (2+1) dimensions with long-range interactions and mutual statistics. The model has the same form after the application of operations from the modular group, a property which we call modular invariance. Using the modular invariance of the model, we propose a possible phase diagram. We obtain a sign-free reformulation of the model and study it in Monte Carlo. This study confirms our proposed phase diagram. We use the modular invariance to analytically determine the current-current correlation functions and conductivities in all the phases in the diagram, as well as at special ``fixed'' points which are unchanged by an operation from the modular group. We numerically determine the order of the phase transitions, and find segments of second-order transitions. For the statistical interaction parameter \ensuremathθ=\ensuremathπ\ensuremath{\theta}=\ensuremath{\pi}\ensuremathθ=\ensuremathπ, these second-order transitions are evidence of a critical loop phase obtained when both loops are trying to condense simultaneously. We also measure the critical exponents of the second-order transitions.
We study a U(1)×U(1) system in (2+1) dimensions with long-range interactions and mutual statistics. The model has the same form after the application of operations from the modular group, a property which we call modular invariance. Using the modular invariance of the model, we propose a possible phase diagram. We obtain a sign-free reformulation of the model and study it in Monte Carlo. This study confirms our proposed phase diagram. We use the modular invariance to analytically determine the current-current correlation functions and conductivities in all the phases in the diagram, as well as at special “fixed” points which are unchanged by an operation from the modular group. We numerically determine the order of the phase transitions, and find segments of second-order transitions. For the statistical interaction parameter θ=π, these second-order transitions are evidence of a critical loop phase obtained when both loops are trying to condense simultaneously. We also measure the critical exponents of the second-order transitions.
We study a $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ system with two species of loops with mutual $\ensuremath{\pi}$ statistics in (2 + 1) dimensions. We are able to reformulate the model in a way that can be studied by Monte Carlo and we determine the phase diagram. In addition to a phase with no loops, we find two phases with only one species of loop proliferated. The model has a self-dual line, a segment of which separates these two phases. Everywhere on the segment, we find the transition to be first-order, signifying that the two loop systems behave as immiscible fluids when they are both trying to condense. Moving further along the self-dual line, we find a phase where both loops proliferate, but they are only of even strength and, therefore, avoid the statistical interactions. We study another model, which does not have this phase, and also find first-order behavior on the self-dual segment.
We study a lattice model of interacting loops in three dimensions with a 1/r21/{r}^{2}1/r2 interaction. Using Monte Carlo methods, we have found that the phase diagram contains a line of second-order phase transitions between a phase where the loops are gapped and a phase where they proliferate. The correlation length exponent and critical conductivity vary continuously along this line. Our model is exactly self-dual at a special point on the critical line, which allows us to calculate the critical conductivity exactly at this point.
We study a $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ system with short-range interactions and mutual $\ensuremath{\theta}=2\ensuremath{\pi}/3$ statistics in (2+1) dimensions. We are able to reformulate the model to eliminate the sign problem and perform a Monte Carlo study. We find a phase diagram containing a phase with only small loops and two phases with one species of proliferated loop. We also find a phase where both species of loop condense, but without any gapless modes. Lastly, when the energy cost of loops becomes small, we find a phase that is a condensate of bound states, each made up of three particles of one species and a vortex of the other. We define several exact reformulations of the model that allow us to precisely describe each phase in terms of gapped excitations. We propose field-theoretic descriptions of the phases and phase transitions, which are particularly interesting on the ``self-dual'' line where both species have identical interactions. We also define irreducible responses useful for describing the phases.
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I will discuss a family of quantum loop models in 2+1 spacetime dimensions with marginally long-ranged and statistical interactions mediated by a U(1) gauge field, both purely in 2+1 dimensions and on a surface in a 3+1 dimensional bulk system.  In the absence of fractional spin, these theories have been shown to be self-dual under particle-vortex duality and shifts of the statistical angle of the loops by 2π, which form a subgroup of the modular group. We show that careful consideration of fractional spin in these theories completely breaks their statistical periodicity and describe how this occurs, resolving a disagreement with the conformal field theories they appear to approach at criticality. We show explicitly that incorporation of fractional spin leads to loop model dualities which parallel the recent web of 2+1 dimensional field theory dualities, providing a nontrivial check on its validity.