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arXiv:1911.02451 · Physical review. D/Physical review. D.

Continuous quantum phase transition in the fermionic mass solutions of the Nambu–Jona-Lasinio model

Alireza Beygi, S. P. Klevansky, Richard H. Lemmer

Published Feb 13, 2020• 1 citation
arXivHTMLPDF

Abstract

Recently, quantum simulators have been constructed to investigate experimentally the most prominent theoretical four-point many-body system described by the Hubbard model. By varying the coupling strength of the four-point interaction in relation to the kinetic term, it is possible to analyze the phase structure of the model. This intriguing fact leads us to ask the question as to whether similar Hamiltonians with four-point interactions can also be studied as a function of their four-point coupling strength. In this paper, we reexamine the Nambu--Jona-Lasinio model, regarding it generally beyond the context of quantum chromodynamics. Essentially, it is a model in which particle-antiparticle pairing leads to a Bardeen-Cooper-Schrieffer-like condensate, with the result that chiral symmetry is broken dynamically in the strong-coupling regime, where $G{\mathrm{\ensuremath{\Lambda}}}^{2}$ is larger than a critical value, i.e., $G{\mathrm{\ensuremath{\Lambda}}}^{2}>{G}_{c}{\mathrm{\ensuremath{\Lambda}}}^{2}$. To study the behavior of the system, it is necessary to move from this regime to a hypothetical regime of weak coupling, altering the coupling strength of the interaction arbitrarily. In order to do this, the gap equation must be regarded as complex and its Riemann surface structure must be known. We do this and obtain a continuous quantum phase transition characterized by the development of a complex order parameter (the dynamically generated mass) from the second sheet of the Riemann surface associated with the gap equation, as we move into the weak-coupling regime. The power-law behavior of the order parameter in the vicinity of the phase transition point is demonstrated to be independent of the choice of the regularization scheme with the critical exponent as $\ensuremath{\beta}\ensuremath{\approx}0.55$. At the same time, the isovector pseudoscalar modes retain their feature as Goldstone modes and still have zero mass, while the isoscalar scalar meson follows the behavior of the order parameter and gains a width. Energetically, this mode is not favored over the normal, uncondensed mode but would have to be accessed through an excitation process.

DOI
10.1103/physrevd.101.036005

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In this paper we study a (1+11+11+1)-dimensional version of the famous Nambu--Jona-Lasinio model of quantum chromodynamics (QCD2) both at zero and at finite baryon density. We use nonperturbative techniques (non-Abelian bosonization and the truncated conformal spectrum approach). When the baryon chemical potential, $\ensuremath{\mu}$, is zero, we describe the formation of fermion three-quark (nucleons and $\mathrm{\ensuremath{\Delta}}$ baryons) and boson (two-quark mesons, six-quark deuterons) bound states. We also study at $\ensuremath{\mu}=0$ the formation of a topologically nontrivial phase. When the chemical potential exceeds the critical value and a finite baryon density appears, the model has a rich phase diagram which includes phases with a density wave and superfluid quasi-long-range (QLR) order, as well as a phase of a baryon Tomonaga-Luttinger liquid (strange metal). The QLR order results in either a condensation of scalar mesons (the density wave) or six-quark bound states (deuterons).
We present quantum Monte Carlo simulations for the chiral Heisenberg Gross-Neveu-Yukawa quantum phase transition of relativistic fermions with N=4 Dirac spinor components subject to a repulsive, local four fermion interaction in (2+1)D. Here we employ a two-dimensional lattice Hamiltonian with a single, spin-degenerate Dirac cone, which exactly reproduces a linear energy-momentum relation for all finite size lattice momenta in the absence of interactions. This allows us to significantly reduce finite size corrections compared to the widely studied honeycomb and π-flux lattices. A Hubbard term dynamically generates a mass beyond a critical coupling of U_{c}=6.76(1) as the system acquires antiferromagnetic order and SU(2) spin rotational symmetry is spontaneously broken. At the quantum phase transition, we extract a self-consistent set of critical exponents ν=0.98(1), η_{ϕ}=0.53(1), η_{ψ}=0.18(1), and β=0.75(1). We provide evidence for the continuous degradation of the quasiparticle weight of the fermionic excitations as the critical point is approached from the semimetallic phase. Finally, we study the effective "speed of light" of the low-energy relativistic description, which depends on the interaction U, but is expected to be regular across the quantum phase transition. We illustrate that the strongly coupled bosonic and fermionic excitations share a common velocity at the critical point.
directions. By tuning coupling strengths along distinct directions, the system exhibits different symmetries, ranging from a discrete Z2 to a continuous U(1) symmetry. The anisotropic coupled-top model displays a discrete Z2 symmetry, and the symmetry breaking induced by strong coupling drives a quantum phase transition from a disordered paramagnetic phase to an ordered ferromagnetic or antiferromagnetic phase. In particular, the isotropic coupled-top model possesses a continuous U(1) symmetry, whose breaking gives rise to the Goldstone mode. The phase boundary can be well captured by the mean-field approach, characterized by the distinct behaviors of the order parameter. Higher-order quantum effects beyond the mean-field contribution can be achieved by mapping the large spins to bosonic operators via the Holstein-Primakoff transformation. For the anisotropic coupled-top model with Z2 symmetry, the energy gap closes, and both quantum fluctuations and entanglement entropy diverge near the critical point, signaling the onset of second-order quantum phase transitions. Strikingly, when U(1) symmetry is broken, the energy gap vanishes beyond the critical point, yielding a novel critical exponent of 1, rather than 1/2 for Z2 symmetry breaking. The rich symmetry structure of the coupled-top model underpins its role as a paradigmatic model for studying quantum phase transitions and exploring associated physical phenomena.
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Spontaneous symmetry breaking occurs in a physical system whenever the ground state does not share the symmetry of the underlying theory, e.g., the Hamiltonian. This mechanism gives rise to massless Nambu-Goldstone modes and massive Anderson-Higgs modes. These modes provide a fundamental understanding of matter in the Universe and appear as collective phase or amplitude excitations of an order parameter in a many-body system. The amplitude excitation plays a crucial role in determining the critical exponents governing universal nonequilibrium dynamics in the Kibble-Zurek mechanism (KZM). Here, we characterize the amplitude excitations in a spin-1 condensate and measure the energy gap for different phases of the quantum phase transition. At the quantum critical point of the transition, finite-size effects lead to a nonzero gap. Our measurements are consistent with this prediction, and furthermore, we demonstrate an adiabatic quench through the phase transition, which is forbidden at the mean field level. This work paves the way toward generating entanglement through an adiabatic phase transition.
We characterize, by means of large-scale fermion quantum Monte Carlo simulations, metallic and deconfined quantum phase transitions in a bilayer honeycomb model in terms of their quantum critical and finite-temperature properties. The model features three distinct phases at zero temperature as a function of interaction strength. At weak interaction, a fully symmetric Dirac semimetal state is realized. At intermediate and strong interaction, respectively, two long-range-ordered phases, each breaking different symmetries are stabilized. The ordered phases feature partial and full gap openings in the fermion spectrum, respectively. We clarify the symmetries of the different zero-temperature phases and the symmetry-breaking patterns across the two quantum phase transitions between them. The first transition between the disordered and long-range-ordered semimetallic phases has previously been argued to be described by the (2+1)(2+1)(2+1)-dimensional Gross-Neveu-SO(3) field theory. By performing simulations with an improved symmetric Trotter decomposition, we further substantiate this claim by computing the critical exponents $1/\ensuremath{\nu}$, ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, and ${\ensuremath{\eta}}_{\ensuremath{\psi}}$, which turn out to be consistent with the field-theoretical expectation within numerical and analytical uncertainties. The second transition between the two long-range-ordered phases has previously been proposed as a possible instance of a metallic deconfined quantum critical point. We further develop this scenario by analyzing the spectral functions in the single-particle, particle-hole, and particle-particle channels. Our results indicate gapless excitations with a unique velocity, supporting the emergence of Lorentz symmetry at criticality. We also compute the finite-temperature phase boundaries of the ordered states above the fully gapped state at large interaction. The phase boundary vanishes smoothly in the vicinity of the putative metallic deconfined quantum critical point, in agreement with the expectation for a continuous or weakly first-order transition.