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arXiv:1906.06052 · Physical review. B./Physical review. B

Quantum phase transition and criticality in quasi-one-dimensional spinless Dirac fermions

Yasuhiro Tada

Published Sep 20, 2019• 4 citations
arXivHTMLPDF

Abstract

We study the quantum criticality of spinless fermions on a quasi-one-dimensional $\ensuremath{\pi}$-flux square lattice in cylinder geometry, by using the infinite density matrix renormalization group and Abelian bosonization. For a series of cylinder circumferences ${L}_{y}=4n+2=2,6,...$ with a periodic boundary condition, there are quantum phase transitions from gapped Dirac fermion states to charge density wave (CDW) states. We find that the quantum phase transitions for such circumferences are continuous and belong to the ($1+1$)-dimensional Ising universality class. On the other hand, when ${L}_{y}=4n=4,8,...$, there are gapless Dirac fermions at the noninteracting point and the phase transition to the CDW state is Gaussian. Both of these criticalities are described in a unified way by bosonization. We clarify their intimate relationship and demonstrate that a central charge $c=1/2$ Ising transition line arises as a critical state of an emergent Majorana fermion from the $c=2$ Gaussian transition point.

DOI
10.1103/physrevb.100.125145

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We study the quantum criticality of spinless fermions on a quasi-one-dimensional \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux square lattice in cylinder geometry, by using the infinite density matrix renormalization group and Abelian bosonization. For a series of cylinder circumferences Ly=4n+2=2,6,...{L}_{y}=4n+2=2,6,...Ly​=4n+2=2,6,... with a periodic boundary condition, there are quantum phase transitions from gapped Dirac fermion states to charge density wave (CDW) states. We find that the quantum phase transitions for such circumferences are continuous and belong to the (1+11+11+1)-dimensional Ising universality class. On the other hand, when Ly=4n=4,8,...{L}_{y}=4n=4,8,...Ly​=4n=4,8,..., there are gapless Dirac fermions at the noninteracting point and the phase transition to the CDW state is Gaussian. Both of these criticalities are described in a unified way by bosonization. We clarify their intimate relationship and demonstrate that a central charge c=1/2c=1/2c=1/2 Ising transition line arises as a critical state of an emergent Majorana fermion from the c=2c=2c=2 Gaussian transition point.
We study orbital diamagnetism in (2+1)(2+1)(2+1)-dimensional Dirac electrons with a short-range interaction at zero temperature. We introduce orbital magnetic fields into spinless Dirac electrons on the π\piπ-flux square lattice, and analyze them by using infinite density matrix renormalization group. It is found that the diamagnetism remains intact when the short-range interaction VVV is weak, while it is monotonically suppressed for larger interactions. Around the quantum critical point (QCP) of a charge density wave phase transition, we find a scaling behavior of the ground state energy density characteristic of the chiral Ising universality class. This leads to universal orbital diamagnetism in correlated Dirac electrons around the QCP, which may be regarded as a quantum, magnetic analogue of critical Casimir effect which has been widely studied for classical phase transitions.
Quantum critical phenomena may be qualitatively different when massless Dirac fermions are present at criticality. Using our recently-discovered fermion-sign-free Majorana quantum Monte Carlo method introduced by us in (Li et al 2015 Phys. Rev. B 91 241117 ), we investigate the quantum critical phenomena of spinless Dirac fermions at their charge-density-wave phase transitions on the honeycomb lattice having sites with largest L = 24. By finite-size scaling, we accurately obtain critical exponents of this so-called Gross–Neveu chiral-Ising universality class of two (two-component) Dirac fermions in 2+1D: , , and , which are qualitatively different from the mean-field results but are reasonably close to the ones obtained from renormalization group calculations.
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Abstract We study orbital diamagnetism at zero temperature in (2 + 1)-dimensional Dirac fermions with a short-range interaction which exhibits a quantum phase transition to a charge density wave (CDW) phase. We introduce orbital magnetic fields into spinless Dirac fermions on the π-flux square lattice, and analyze them by using infinite density matrix renormalization group. It is found that the diamagnetism remains intact in the Dirac semimetal regime, while it is monotonically suppressed in the CDW regime. Around the quantum critical point of the CDW phase transition, we find a scaling behavior of the diamagnetism characteristic of the chiral Ising universality class. Besides, the scaling analysis implies that the robust orbital diamagnetism at weak magnetic fields in a Dirac semimetal regime would hold not only in our model but also in other interacting Dirac fermion systems as long as scaling regions are wide enough. The scaling behavior may also be regarded as a quantum, magnetic analogue of the critical Casimir effect which has been widely studied for classical phase transitions.
Spinless fermions on a honeycomb lattice provide a minimal realization of lattice Dirac fermions. Repulsive interactions between nearest neighbors drive a quantum phase transition from a Dirac semimetal to a charge-density-wave state through a fermionic quantum critical point, where the coupling of the Ising order parameter to the Dirac fermions at low energy drastically affects the quantum critical behavior. Encouraged by a recent discovery (Huffman and Chandrasekharan 2014 Phys. Rev. B 89 111101 ) of the absence of the fermion sign problem in this model, we study the fermionic quantum critical point using the continuous-time quantum Monte Carlo method with a worm-sampling technique. We estimate the transition point with the critical exponents and . Compatible results for the transition point are also obtained with infinite projected entangled-pair states.
We present an unbiased numerical density-matrix renormalization group study of the one-dimensional Bose-Hubbard model supplemented by nearest-neighbor Coulomb interaction and bond dimerization. It places the emphasis on the determination of the ground-state phase diagram and shows that, besides dimerized Mott and density-wave insulating phases, an intermediate symmetry-protected topological Haldane insulator emerges at weak Coulomb interactions for filling factor one, which disappears, however, when the dimerization becomes too large. Analyzing the critical behavior of the model, we prove that the phase boundaries of the Haldane phase to Mott insulator and density-wave states belong to the Gaussian and Ising universality classes with central charges c=1c=1c=1 and c=1/2c=1/2c=1/2, respectively, and merge in a tricritical point. Interestingly we can demonstrate a direct Ising quantum phase transition between the dimerized Mott and density-wave phases above the tricritical point. The corresponding transition line terminates at a critical end point that belongs to the universality class of the dilute Ising model with c=7/10c=7/10c=7/10. At even stronger Coulomb interactions the transition becomes first order.