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arXiv:2012.07991 · Journal of High Energy Physics

Rotons in anyon superfluids

Yi-Hsien Du, Umang Mehta, Dam Thanh Son

Published Mar 1, 2021• 3 citations
arXivHTMLPDF

Abstract

Abstract We consider the problem of calculating the excitation spectrum of a gas of nonrelativistic anyons. When the anyons have statistics close to fermionic and the statistical angle has the form θ = π(1 − $$ \frac{1}{k} $$ 1 k ) where k is a large integer, the problem can be solved by employing the method of bosonization, which maps the problem to that of an infinite number of bosonic excitations coupled to a U(1) Chern-Simons gauge field. The spectrum consists of a Goldstone boson branch and a large number of massive branches, each having roton minima and maxima. The dispersion curves asymptote to the Landau levels at large momentum.

DOI
10.1007/jhep03(2021)101

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Abstract We consider the problem of calculating the excitation spectrum of a gas of nonrelativistic anyons. When the anyons have statistics close to fermionic and the statistical angle has the form θ = π(1 − 1k\frac{1}{k}k1​ 1 k ) where k is a large integer, the problem can be solved by employing the method of bosonization, which maps the problem to that of an infinite number of bosonic excitations coupled to a U(1) Chern-Simons gauge field. The spectrum consists of a Goldstone boson branch and a large number of massive branches, each having roton minima and maxima. The dispersion curves asymptote to the Landau levels at large momentum.
We consider the problem of calculating the excitation spectrum of a gas of nonrelativistic anyons. When the anyons have statistics close to fermionic and the statistical angle has the form θ = π(1 - 1/k) where k is a large integer, the problem can be solved by employing the method of bosonization, which maps the problem to that of an infinite number of bosonic excitations coupled to a U(1) Chern-Simons gauge field. The spectrum consists of a Goldstone boson branch and a large number of massive branches, each having roton minima and maxima. The dispersion curves asymptote to the Landau levels at large momentum.
I consider a gas of “free” anyons with statistical paremeter δ on a two dimensional lattice. Using a recently derived Jordan-Wigner transformation, I map this problem onto a gas of fermions on a lattice coupled to a Chern-Simons gauge theory with coupling [Formula: see text]. I show that if [Formula: see text] and the density [Formula: see text], with r and q integers, the system is a superfluid. If q is even and the system is half filled the state may be either a superfluid or a Quantum Hall System depending on the dynamics. Similar conclusions apply for other values of ρ and δ. The dynamical stability of the Fetter-Hanna-Laughlin goldstone mode is insured by the topological invariance of the quantized Hall conductance of the fermion problem. This leads to the conclusion that anyon gases are generally superfluids or quantum Hall systems.
We prove, working in the Hamiltonian formalism, that a U(1) Chern-Simons theory coupled to fermions on a lattice can be mapped exactly onto a theory of interacting lattice anyons. This map does not involve any singular gauge transformations, and is everywhere well defined. We also prove that, when the statistics parameter is an odd integer so that the anyons are bosons, the ground state, which consists of a condensate of bound pairs of flux tubes and fermions, breaks phase invariance. The ensuing long range order implies that the system is an unconventional superfluid.
I consider a gas of ``free'' anyons with statistical parameter \ensuremath{\delta}, with hard cores, on a two-dimensional square lattice. Using a recently derived Jordan-Wigner transformation, I map this problem onto a gas of fermions on the same lattice coupled to a Chern-Simons gauge theory with coupling \ensuremath{\theta}=1/2\ensuremath{\delta}. At the semiclassical level, the system is found to be equivalent to a gas of fermions, with the same density, in an average effective magnetic field \ensuremath{\rho}/\ensuremath{\theta}. I consider the case in which an integer number of the Landau bands of the saddle-point problem are completely filled. If \ensuremath{\delta}=\ensuremath{\pi}/m and the density \ensuremath{\rho}=r/q, with m, r, and q integers, the system is a superfluid, provided that q is larger than twice the largest common factor of m and r. If q is even and the system is half filled, the state may be either a superfluid or a quantum Hall phase. For all other values of \ensuremath{\rho} and \ensuremath{\delta}, compatible with integer filling of the Landau bands, the system is in a quantum Hall phase. The dynamical stability of the superfluid state is ensured by the topological invariance of the quantized Hall conductance of the fermion problem. I find a close analogy between anyon superconductivity and the Schwinger mechanism. The effective Lagrangian for the low-energy modes coupled to the electromagnetic field is derived. The energies of fermion and flux states are logarithmically divergent, but finite for the anyon state. The system has flux quantization, a zero-temperature Hall effect with a quantized Hall conductance, Meissner effect, charged vortices, screening with induced magnetic fields for static charges, and different masses for the longitudinal and transverse components of the electromagnetic field.
The ground state of an ideal gas of anyons with statistical angle θ∈(0,π)θ\in(0,π)θ∈(0,π) is a long-standing open problem, with renewed interest from the recent observations of fractional Chern insulators. In this work we construct variational wavefunctions that build in the exact anyon statistics and combine theory-guided composite-boson, composite-fermion and parton wavefunctions with neural-network correlations on the sphere. For spinless and SU(2) semions we confirm the paired anyon superfluid, well below the best crystal states. At θ=π/3θ=π/3θ=π/3 liquids and crystals are closer in energy for the spinless case. For θ=π/3θ=π/3θ=π/3 anyons with SU(3) flavor, we find two candidates. One is a skyrmion Hall crystal, in which the anyons form an integer quantum Hall state in the effective magnetic field from the spin texture. The other is a chiral color superconductor of six-anyon clusters, the color-flavor-locked pairing descendant of a holon metal with SU(3) color gauge symmetry and SU(3) flavor symmetry; its wavefunction is a hyperdeterminant rather than a product of parton determinants. Within our variational ansatz the color superconductor has better energy than both crystals and the color-flavor locked phases with U(1)2^22 and Z3_33​ gauge groups. For twisted MoTe2_22​ this suggests that the reentrant integer quantum anomalous Hall phase is the skyrmion Hall crystal of the doped anyons, and that the observed superconductivity may be the color superconductor, which can undergo a transition to an f+iff+iff+if BCS superconductor with a jump of the chiral central charge by 3.
We revisit the long-standing problem of the anyon gas. We study the solvable near-bosonic limit of a large number of anyons NNN at fixed N/kN/kN/k, where 1/k1/k1/k characterizes the statistical interaction. In this limit, the ground state is a crystal with order N/kN/kN/k lattice sites. The statistical interactions, mediated by a long-range gauge field, are completely screened within the crystalline phase. The resulting state is a conventional crystal without superfluidity. We characterize this anyon crystal by determining its unit-cell size, ground-state energy, acoustic excitations, angular momentum crossings, and other properties. Our results indicate a quantum phase transition between the crystal at k>kck>k_ck>kc​ and a superfluid at $k