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arXiv:2304.10390 · Physical review. B./Physical review. B

Critical properties of the Majorana chain with competing interactions

Natalia Chepiga

Published Aug 11, 2023• 5 citations
arXivHTMLPDF

Abstract

We explore critical properties of a chain of interacting Majorana fermions, particles that are their own antiparticles. We study the combined effect of two competing interaction terms of the shortest possible range and show this results in a very rich phase diagram with nine different phases, five of which are critical. In addition, we report a wide variety of quantum phase transitions: the tri-critical Ising lines; the Lifshitz critical line characterized by the dynamical critical exponent $z=3$; two Kosterlitz-Thouless transitions; and an exotic first-order transition between the floating and the gapped phases. However, the most surprising result is the emergence of the commensurate line at which the floating phases collapse into direct transition. We provide numerical evidences that the resulting multicritical point belongs to the universality class of the eight-vertex model. Implications in the context of supersymmetric properties of the Majorana chain are briefly discussed.

DOI
10.1103/physrevb.108.054509

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We explore critical properties of a chain of interacting Majorana fermions, particles that are their own antiparticles. We study the combined effect of two competing interaction terms of the shortest possible range and show this results in a very rich phase diagram with nine different phases, five of which are critical. In addition, we report a wide variety of quantum phase transitions: the tri-critical Ising lines; the Lifshitz critical line characterized by the dynamical critical exponent z=3z=3z=3; two Kosterlitz-Thouless transitions; and an exotic first-order transition between the floating and the gapped phases. However, the most surprising result is the emergence of the commensurate line at which the floating phases collapse into direct transition. We provide numerical evidences that the resulting multicritical point belongs to the universality class of the eight-vertex model. Implications in the context of supersymmetric properties of the Majorana chain are briefly discussed.
We study Majorana chain with the shortest possible interaction term and in the presence of hopping alternation. When formulated in terms of spins the model corresponds to the transverse field Ising model with nearest-neighbor transverse and next-nearest-neighbor longitudinal repulsion. The phase diagram obtained with extensive DMRG simulations is very rich and contains six phases. Four gapped phases include paramagnetic, period-2 with broken translation symmetry, \mathbb{Z}_2 ℤ 2 with broken parity symmetry and the period-2- \mathbb{Z}_2 ℤ 2 phase with both symmetries broken. In addition there are two floating phases: gapless and critical Luttinger liquid with incommensurate correlations, and with an additional spontaneously broken \mathbb{Z}_2 ℤ 2 symmetry in one of them. By analyzing an extended phase diagram we demonstrate that, in contrast with a common belief, the Luttinger liquid phase along the self-dual critical line terminates at a weaker interaction strength than the end point of the Ising critical line that we find to be in the tri-critical Ising universality class. We also show that none of these two points is a Lifshitz point terminating the incommensurability. In addition, we analyzed topological properties through Majorana zero modes emergent in the two topological phases, with and without incommensurability. In the weak interaction regime, a self-consistent mean-field treatment provides a remarkable accuracy for the description of the spectral pairing and the parity switches induced by the interaction.
We investigate the cluster Ising chain with an additional nearest-neighbor interaction using tensor-network methods and a weak-coupling field theory. Without the interaction, the Jordan-Wigner transformation decomposes the model into a triplet of identical Majorana chains related by an exact O(3)O(3)O(3) flavor symmetry. Their common mass vanishes at the SU(2)2SU(2)_2SU(2)2​ Wess-Zumino-Novikov-Witten critical point with central charge c=3/2c = 3/2c=3/2 separating the symmetry-protected topological cluster phase from a ferromagnet. The interaction reduces O(3)O(3)O(3) to its cyclic subgroup C3C_3C3​, splitting the triplet into a singlet and a doublet whose gaps close separately, producing a critical bifurcation into Ising (c=1/2c = 1/2c=1/2) and Gaussian (c=1c = 1c=1) critical lines. A second ferromagnetic phase opens between these critical lines for repulsive interactions and a disordered phase for attractive ones. The Gaussian line then separates two Landau-incompatible ferromagnets, realizing a deconfined quantum critical line with emergent O(2)O(2)O(2) symmetry, along which our independently extracted exponents vary continuously yet satisfy the parameter-free relation β=(2ν−1)/4β= (2ν- 1)/4β=(2ν−1)/4 of the eight-vertex weak universality class. At stronger repulsion this line opens into an extended gapless floating phase with incommensurate algebraic correlations, entered through Berezinskii-Kosterlitz-Thouless transitions. All of these phases and the transitions between them are captured by the weak-coupling theory. Beyond its regime of validity, our simulations reveal a translation-symmetry-breaking antiferromagnet reached through first-order transitions.
The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the Bethe ansatz, density matrix renormalization group (DMRG), and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher-dimensional cases. Recently, an interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve four sites. We present a thorough analysis of the phase diagram of this model in one dimension using field-theory/RG and DMRG methods. It includes a gapped supersymmetric region and a gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first-order transition, three critical points occur: tricritical Ising, Lifshitz, and a generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1, and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.
We introduce a two-ladder lattice model with interacting Majorana fermions that could be realized on the surfaces of a topological insulator film. We study this model with a combination of analytical and numerical techniques, and we find a phase diagram that features both gapless and gapped phases as well as interesting phase transitions including a quantum critical point in the tricritical Ising (TCI) universality class. The latter occurs at an intermediate-coupling strength at a meeting point of a first-order transition line and an Ising critical line, and it is known to be described by a superconformal field theory with central charge c=710c=\frac{7}{10}c=107​. We discuss the experimental feasibility of constructing the model and tuning parameters to the vicinity of the TCI point where signatures of the elusive supersymmetry can be observed.
We analyze a series of interacting Majorana Fermion chains with finite range pair interactions with coupling strength ggg that all exhibit a tri-critical point that separates an Ising critical phase from a supersymmetric gapped phase. We first notice that the interacting models exhibit an even-odd asymmetry depending on the number of sites, δδδ, over which the interaction ranges. The even case exhibits competing order, thereby making it numerically untractable while the odd case exhibits an exactly solvable point at g=−0.5g=-0.5g=−0.5 where the entanglement entropy vanishes. By introducing a swirling geometrical twist, we map our 1D δδδ-range chains to a series of 2D δ/2δ/2δ/2-width models. Our new 2D models possess a unique helical boundary condition, constructed from 1D chains with the end of one connected to the start of another. We propose that the phase transition in the 1D system can be understood as a finite-system size transition in 2D. That is, the gc−δg_c-δgc​−δ behavior is controlled by a 2D tri-critical universality class at δ→∞δ\to\inftyδ→∞ limit and is predicted by finite-size scaling theory.
We show that including pairing and repulsion into the description of one-dimensional spinless fermions, as in the domain wall theory of commensurate melting or the interacting Kitaev chain, leads, for strong enough repulsion, to a line of critical points in the eight-vertex universality class terminating floating phases with emergent U(1)U(1)U(1) symmetry. For nearest-neighbor repulsion and pairing, the variation of the critical exponents along the line that can be extracted from Baxter's exact solution of the XYZXYZXYZ chain at ${J}_{x}=\ensuremath{-}{J}_{z}$ is fully confirmed by extensive density matrix renormalization group (DMRG) simulations of the entire phase diagram, and the qualitative features of the phase diagram are shown to be independent of the precise form of the interactions.