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arXiv:1608.01817 · Physical Review A

First-order phase transitions in spinor Bose gases and frustrated magnets

Thibault Debelhoir, N. Dupuis

Published Nov 23, 2016• 7 citations
arXivHTMLPDF

Abstract

We show that phase transitions in spin-1 Bose gases and stacked triangular Heisenberg antiferromagnets---an example of frustrated magnets with competing interactions---are described by the same Landau-Ginzburg-Wilson Hamiltonian with $\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ symmetry. In agreement with previous nonperturbative-renormalization-group studies of the three-dimensional $\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ model, we find that the transition from the normal phase to the superfluid ferromagnetic phase in a spin-1 Bose gas is weakly first order and shows pseudoscaling behavior. The (nonuniversal) pseudoscaling exponent $\ensuremath{\nu}$ is fully determined by the scattering lengths ${a}_{0}$ and ${a}_{2}$. We provide estimates of $\ensuremath{\nu}$ in $^{87}\mathrm{Rb},^{41}\mathrm{K}$, and $^{7}\mathrm{Li}$ atom gases which can be tested experimentally. We argue that pseudoscaling comes from either a crossover phenomenon due to proximity of the O(6) Wilson-Fisher fixed point $(^{87}\mathrm{Rb}$ and $^{41}\mathrm{K})$ or the existence of two unphysical fixed points (with complex coordinates) which slow down the RG flow $(^{7}\mathrm{Li})$. These unphysical fixed points are a remnant of the chiral and antichiral fixed points that exist in the $\mathrm{O}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ model when $N$ is larger than ${N}_{c}\ensuremath{\simeq}5.3$ (the transition being then second order and controlled by the chiral fixed point). Finally, we discuss a $\mathrm{O}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ lattice model and show that our results, even though we find the transition to be first order, are compatible with Monte Carlo simulations yielding an apparent second-order transition.

DOI
10.1103/physreva.94.053623

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We show that phase transitions in spin-1 Bose gases and stacked triangular Heisenberg antiferromagnets---an example of frustrated magnets with competing interactions---are described by the same Landau-Ginzburg-Wilson Hamiltonian with O(3)\ifmmode×\else\texttimes\fiO(2)\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)O(3)\ifmmode×\else\texttimes\fiO(2) symmetry. In agreement with previous nonperturbative-renormalization-group studies of the three-dimensional O(3)\ifmmode×\else\texttimes\fiO(2)\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)O(3)\ifmmode×\else\texttimes\fiO(2) model, we find that the transition from the normal phase to the superfluid ferromagnetic phase in a spin-1 Bose gas is weakly first order and shows pseudoscaling behavior. The (nonuniversal) pseudoscaling exponent \ensuremathν\ensuremath{\nu}\ensuremathν is fully determined by the scattering lengths a0{a}_{0}a0​ and a2{a}_{2}a2​. We provide estimates of \ensuremathν\ensuremath{\nu}\ensuremathν in 87Rb,41K^{87}\mathrm{Rb},^{41}\mathrm{K}87Rb,41K, and 7Li^{7}\mathrm{Li}7Li atom gases which can be tested experimentally. We argue that pseudoscaling comes from either a crossover phenomenon due to proximity of the O(6) Wilson-Fisher fixed point (87Rb(^{87}\mathrm{Rb}(87Rb and 41K)^{41}\mathrm{K})41K) or the existence of two unphysical fixed points (with complex coordinates) which slow down the RG flow (7Li)(^{7}\mathrm{Li})(7Li). These unphysical fixed points are a remnant of the chiral and antichiral fixed points that exist in the O(N)\ifmmode×\else\texttimes\fiO(2)\mathrm{O}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)O(N)\ifmmode×\else\texttimes\fiO(2) model when NNN is larger than Nc\ensuremath≃5.3{N}_{c}\ensuremath{\simeq}5.3Nc​\ensuremath≃5.3 (the transition being then second order and controlled by the chiral fixed point). Finally, we discuss a O(2)\ifmmode×\else\texttimes\fiO(2)\mathrm{O}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)O(2)\ifmmode×\else\texttimes\fiO(2) lattice model and show that our results, even though we find the transition to be first order, are compatible with Monte Carlo simulations yielding an apparent second-order transition.
Although there is a broad consensus on the fact that critical behavior in stacked triangular Heisenberg antiferromagnets---an example of frustrated magnets with competing interactions---is described by a Landau-Ginzburg-Wilson Hamiltonian with $\text{O}(3)\ifmmode\times\else\texttimes\fi{}\text{O}(2)$ symmetry, the nature of the phase transition in three dimensions is still debated. We show that spin-one Bose gases provide us with a simulator of the $\text{O}(3)\ifmmode\times\else\texttimes\fi{}\text{O}(2)$ model. Using a renormalization-group approach, we argue that the transition is weakly first order and shows pseudoscaling behavior, and give estimates of the pseudocritical exponent $\ensuremath{\nu}$ in 87Rb^{87}\mathrm{Rb}87Rb, 41K^{41}\mathrm{K}41K, and 7Li{}^{7}\mathrm{Li}7Li atom gases which can be tested experimentally.
Using Monte Carlo and renormalization group methods, we investigate systems with critical behavior described by two order parameters: continuous (vector) and discrete (scalar). We consider two models of classical three-dimensional Heisenberg magnets with different numbers of spin components N = 1,…,4: the model on a cubic lattice with an additional competing antiferromagnetic exchange interaction in a layer and the model on a body-centered lattice with two competing antiferromagnetic interactions. In both models, we observe a first-order transition for all values of N. In the case where competing exchanges are approximately equal, the first order of a transition is close to the second order, and pseudoscaling behavior is observed with critical exponents differing from those of the O(N) model. In the case N = 2, the critical exponents are consistent with the well-known indices of the class of magnets with a noncollinear spin ordering. We also give a possible explanation of the observed pseudoscaling in the framework of the renormalization group analysis.
In recent years the quantum simulation of condensed-matter physics problems has resulted from ex- citing experimental progress in the realm of ultracold atoms and molecules in optical lattices. In this thesis we analyze theoretically a spinor Bose gas loaded into a three-dimensional cubic optical lattice. In order to account for different superfluid phases of spin-1 bosons with a linear Zeeman effect, we work out a Ginzburg-Landau theory for the underlying spin-1 Bose-Hubbard model. To this end we add artificial symmetry- breaking currents to the spin-1 Bose-Hubbard Hamiltonian in order to break the global U (1) symmetry. With this we determine a diagrammatic expansion of the grand-canonical free energy up to fourth order in the symmetry-breaking currents and up to the leading non-trivial order in the hopping strength which is of first order. As a cross-check we demonstrate that the resulting grand- canonical free energy allows to recover the mean-field theory. Applying a Legendre transfor- mation to the grand-canonical free energy, where the symmetry-breaking currents are transformed to order parameters, we obtain the effective Ginzburg-Landau action. With this we calculate in detail at zero temperature the Mott insulator-superfluid quantum phase boundary as well as condensate and particle number density in the superfluid phase. We find that both mean-field and Ginzburg-Landau theory yield the same quantum phase transition between the Mott insulator and superfluid phases, but the range of validity of the mean-field theory turns out to be smaller than that of the Ginzburg-Landau theory. Due to this finding we expect that the Ginzburg-Landau theory gives better results for the superfluid phase and, thus, we restrict ourselves to extremize only the effective Ginzburg-Landau action with respect to the order parameters. With- out external magnetic field the superfluid phase is a polar (ferromagnetic) state for anti-ferromagnetic (ferromagnetic) interactions, i.e. only the hyperfine spin 0 (1) is macroscopically occupied, in accor- dance with previous mean-field results. On the other hand, in the presence of the external magnetic field for ferromagnetic interaction, the superfluid phase does not change as the minimization of the energy implies the maximal spin value. However, when an anti-ferromagnetic interaction competes with the linear Zeeman effect, we can distinguish various ferromagnetic and anti-ferromagnetic su- perfluid phases within the range of validity of the Ginzburg-Landau theory. Increasing the external magnetic field yields a breaking of spin singlet pairs and a subsequent alignment of spins, thus anti- ferromagnetic phases decrease until only a ferromagnetic superfluid phase prevails. In addition, we find that the superfluid-Mott insulator phase transition is always of second order for both ferromag- netic and anti- ferromagnetic interactions. However, the transitions between different superfluid phases for an anti-ferromagnetic interaction can be both of first and second order depending on whether the respective macroscopic occupation of hyperfine spin states changes discontinuously or continuously. The established Ginzburg-Landau theory for spin-1 bosons in optical lattices will certainly be the basis for many further applications as, for instance, time-of-flight absorption pictures or collective excitations, which are of experimental importance.
This article is devoted to the study of the critical properties of classical XY and Heisenberg frustrated magnets in three dimensions. We first analyze the experimental and numerical situations. We show that the unusual behaviors encountered in these systems, typically nonuniversal scaling, are hardly compatible with the hypothesis of a second order phase transition. Moreover, the fact that the scaling laws are significantly violated and that the anomalous dimension is negative in many cases provides strong indications that the transitions in frustrated magnets are most probably of very weak first order. We then review the various perturbative and early nonperturbative approaches used to investigate these systems and argue that none of them provides a completely satisfactory description of the three-dimensional critical behavior. We then recall the principles of the nonperturbative approach---the effective average action method---that we have used to investigate the physics of frustrated magnets and show how it enables to clarify most of the problems encountered in the previous theoretical descriptions of frustrated magnets. First, we get an explanation of the long-standing mismatch between different perturbative approaches which consists in a nonperturbative mechanism of annihilation of fixed points between two and three dimensions. Secondly, we get a coherent picture of the physics of frustrated magnets in agreement with the numerical and experimental results. The central feature that emerges from our approach is the existence of scaling behaviors without fixed or pseudofixed point and that relies on a slowing down of the renormalization group flow in a whole region in the coupling constants space. This phenomenon allow us to explain the occurrence of generic weak first order behaviors and to understand the absence of universality in the critical behavior of frustrated magnets.
The nature of the phase transition of antiferromagnetic vector spin models on the stacked triangular lattice (STA) has been an issue of debate. In order to account for the coplanar non-collinear order in the ground state one needs two vector fields in a Ginzburg–Landau–Wilson Hamiltonian. The field theoretical renormalization group analysis shows that there is a threshold value Nc for the number of spin component N, and for N > Nc the transition is of second order belonging to a new ‘‘chiral’’ universality class, while for N < Nc the chiral fixed points ceases to exist in the real parameter space and the transition is of first order. According to a recent estimate based on a Monte Carlo renormalization group study, 3 < Nc < 8. 2) Therefore a first order transition is expected for the physically relevant XY and Heisenberg spin systems. The phase transition of the XY or Heisenberg STA is of very weak first order and is said to be of ‘‘almost second order’’ with large (but finite) correlation lengths due to a slow velocity region in the renormalization group flow. Monte Carlo simulations on systems with moderate size appear to show second order transitions with pseudo-critical behavior. Strong first order transitions were found for variants of the model on which a constraint of local rigidity was imposed which still belong to the same universality class as the original STA. Recently the first order nature of the phase transition in XY antiferromagnet on the stacked triangular lattice (XY-STA model) was demonstrated directly by canonical Monte Carlo simulations on large systems, in which the energy probability distribution shows a double peak structure near the transition temperature. The purpose of the present paper is to show the first order transition of the XY-STA model by applying a novel technique: a microcanonical method. To distinguish a weak first order transition from a second order one can be difficult, especially if the energy jump, the characteristic of a first order transition, is tiny as is the case for frustrated spin systems. Limited to accessible system sizes one may analyze the different power law behavior of the maximum of specific heat or susceptibility, with an exponent equal to three for a first order transition and less than three for a second order transition. The obvious solution to increase the size of the system runs into difficulties for the canonical simulations (at constant temperature) because of the growing autocorrelation times. Also the histogram method is then restricted to smaller intervals of temperature shrinking as the inverse power of the system size. In this short note we will show that the microcanonical simulations (at constant energy) can help to solve the problems encountered in canonical simulations. The method has been tested for strong first order transitions and also for second order ones. Before presenting the method and the results we have to describe the model we study. It is given by the usual Hamiltonian
The recent experimental condensation of ultracold atoms in a triangular optical lattice with a negative effective tunneling parameter paves the way for the study of frustrated systems in a controlled environment. Here, we explore the critical behavior of the chiral phase transition in such a frustrated lattice in three dimensions. We represent the low-energy action of the lattice system as a two-component Bose gas corresponding to the two minima of the dispersion. The contact repulsion between the bosons separates into intra- and intercomponent interactions, referred to as V0{V}_{0}V0​ and V12{V}_{12}V12​, respectively. We first employ a Huang-Yang-Luttinger approximation of the free energy. For V12/V0=2{V}_{12}/{V}_{0}=2V12​/V0​=2, which corresponds to the bare interaction, this approach suggests a first-order phase transition, at which both the U(1)\mathrm{U}(1)U(1) symmetry of condensation and the Z2{\mathbb{Z}}_{2}Z2​ symmetry of the emergent chiral order are broken simultaneously. Furthermore, we perform a renormalization-group calculation at one-loop order. We demonstrate that the coupling regime $0<{V}_{12}/{V}_{0}\ensuremath{\le}1$ shares the critical behavior of the Heisenberg fixed point at V12/V0=1{V}_{12}/{V}_{0}=1V12​/V0​=1. For V12/V0>1{V}_{12}/{V}_{0}>1V12​/V0​>1 we show that V0{V}_{0}V0​ flows to a negative value, while V12{V}_{12}V12​ increases and remains positive. This results in a breakdown of the effective quartic-field theory due to a cubic anisotropy and, again, suggests a discontinuous phase transition.