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arXiv:1411.4236

Semiclassical time crystal in the chiral Gross-Neveu model

MICHAEL F. THIES

Published Nov 16, 2014• 4 citations
arXivHTMLPDF

Abstract

In the limit of a large number of flavors, the ground state of the chiral Gross-Neveu model at finite fermion density exhibits a spatially periodic mean field in the form of the chiral spiral, thereby breaking translational invariance. Here we show that the ground state of the same model at finite fermion current density gives rise to a mean field which is periodic in time, a temporal chiral spiral. Since the current density is the same as axial charge density in two dimensions and axial charge is conserved, this may serve as an example of a time crystal. More specifically, as mean field theory is invoked in the large N limit, we are dealing with a semiclassical time crystal.

DOI
10.48550/arxiv.1411.4236

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In the limit of a large number of flavors, the ground state of the chiral Gross-Neveu model at finite fermion density exhibits a spatially periodic mean field in the form of the chiral spiral, thereby breaking translational invariance. Here we show that the ground state of the same model at finite fermion current density gives rise to a mean field which is periodic in time, a temporal chiral spiral. Since the current density is the same as axial charge density in two dimensions and axial charge is conserved, this may serve as an example of a time crystal. More specifically, as mean field theory is invoked in the large N limit, we are dealing with a semiclassical time crystal.
There is substantial evidence that the ground state of the 2D chiral Gross-Neveu model, in the presence of a U(1) fermion number chemical potential μ and in the large N limit, is given by a "chiral spiral" phase, namely an inhomogeneous phase with a chiral condensate having a spatially periodic phase. We show that the chiral spiral configuration persists at finite N and T=0 for any μ>0. Our analysis is based on nonabelian bosonization, that relates the model to a U(N)_{1} Wess-Zumino-Witten model deformed by current-current interactions. In this description, the appearance of the inhomogeneous phase is surprisingly simple. We also rederive the phase diagram of the large N chiral Gross-Neveu model via a direct diagrammatic computation, finding agreement with previous results in the literature.
We explore the finite-density phase diagram of the single-flavour Gross-Neveu-Wilson (GNW) model in (1+1) dimensions using matrix product state (MPS) simulations. At zero temperature and along the symmetry line of the phase diagram, we find a sequence of inhomogeneous ground states that arise through a real-space version of the mechanism of Hilbert-space fragmentation. For weak interactions, doping the symmetry-protected topological (SPT) phase of the GNW model leads to localised charges or holes at periodic arrangements of immobile topological defects separating the fragmented subchains: a topological crystal. Increasing the interactions, we observe a transition into a parity-broken phase with a pseudoscalar condensate displaying a modulated periodic pattern. This soliton lattice is a sequence of topological charges corresponding to anti-kinks, which also bind the doped fermions at their respective centres. Out of this symmetry line, we show that quasi-spiral profiles appear with a characteristic wavevector set by the density k = 2\pi\rho k = 2 π ρ , providing non-perturbative evidence for chiral spirals beyond the large- N N limit. These results demonstrate that various exotic inhomogeneous phases can arise in lattice field theories, and motivate the use of quantum simulators to confirm such QCD-inspired phenomena in future experiments.
Abstract We study the phase diagram of the (1 + 1)-dimensional Gross-Neveu model with both gx2(ψ‾ψ)2 {g}_x^2{\left(\overline{\psi}\psi \right)}^2 gx2​(ψ​ψ)2 g x 2 ψ ¯ ψ 2 and gy2(ψ‾iγ5ψ)2 {g}_y^2{\left(\overline{\psi}i{\gamma}_5\psi \right)}^2 gy2​(ψ​iγ5​ψ)2 g y 2 ψ ¯ i γ 5 ψ 2 interaction terms on a spatial lattice. The continuous chiral symmetry, which is present in the continuum model when gx2=gy2 {g}_x^2={g}_y^2 gx2​=gy2​ g x 2 = g y 2 , has a mixed ’t Hooft anomaly with the charge conservation symmetry, which guarantees the existence of a massless mode. However, the same ’t Hooft anomaly also implies that the continuous chiral symmetry is broken explicitly in our lattice model. Nevertheless, from numerical matrix product state simulations we find that for certain parameters of the lattice model, the continuous chiral symmetry reemerges in the infrared fixed point theory, even at strong coupling. We argue that, in order to understand this phenomenon, it is crucial to go beyond mean-field theory (or, equivalently, beyond the leading order term in a 1/N expansion). Interestingly, on the lattice, the chiral Gross-Neveu model appears at a Landau-forbidden second order phase transition separating two distinct and unrelated symmetry-breaking orders. We point out the crucial role of two different ’t Hooft anomalies or Lieb-Schultz-Mattis obstructions for this Landau-forbidden phase transition to occur.
The two-dimensional Gross-Neveu model is anticipated to undergo a crystalline phase transition at high baryon charge densities. This conclusion is drawn from the mean-field approximation, which closely resembles models of Peierls instability. We demonstrate that this transition indeed occurs when both the rank of the symmetry group and the dimension of the particle representation contributing to the baryon density are large (the large N limit). We derive this result through the exact solution of the model, developing the large N limit of the Bethe ansatz. Our analytical construction of the large-N solution of the Bethe ansatz equations aligns perfectly with the periodic (finite-gap) solution of the Korteweg-de Vries (KdV) of the mean-field analysis.
A time-dependent projection technique is used to treat the initial-value problem for self-interacting fermionic fields. On the basis of the general dynamics of the fields, we derive formal equations of kinetic-type for the set of one-body dynamical variables. A nonperturbative mean-field expansion can be written for these equations. We treat this expansion in lowest order, which corresponds to the Gaussian mean-field approximation, for a uniform system described by the chiral Gross-Neveu Hamiltonian. Standard stationary features of the model, such as dynamical mass generation due to chiral symmetry breaking and a phenomenon analogous to dimensional transmutation, are reobtained in this context. The mean-field time evolution of nonequilibrium initial states is discussed.