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arXiv:1911.01512 · Physical review. B./Physical review. B

Universal finite-size scaling around tricriticality between topologically ordered, symmetry-protected topological, and trivial phases

Ke Wang, Tigran Sedrakyan

Published Jan 10, 2020• 7 citations
arXivHTMLPDF

Abstract

Quantum one-dimensional criticality with conformal invariance is known to be characterized by conformal field theory with the inherent central charge ($c$), and universal finite-size amplitudes. Here, the authors demonstrate that around a quantum criticality with $c$=\textonehalf{} and two Majorana boundary modes, the symmetry-breaking effect modifies the finite-size amplitudes from a universal constant to be a universal function of an emergent scale. This finding provides a new perspective to study the interplay of topology and quantum criticality.

DOI
10.1103/physrevb.101.035410

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Quantum one-dimensional criticality with conformal invariance is known to be characterized by conformal field theory with the inherent central charge (ccc), and universal finite-size amplitudes. Here, the authors demonstrate that around a quantum criticality with ccc=\textonehalf{} and two Majorana boundary modes, the symmetry-breaking effect modifies the finite-size amplitudes from a universal constant to be a universal function of an emergent scale. This finding provides a new perspective to study the interplay of topology and quantum criticality.
The critical point of a topological phase transition is described by a conformal field theory (CFT), where the finite-size corrections to the ground state energy are uniquely related to its central charge. We study the finite-size scaling of the energy of non-Hermitian Su-Schrieffer-Heeger (SSH) model with parity and time-reversal symmetry (PT\mathcal{PT}PT) symmetry. We find that under open boundary condition (OBC), the energy scaling E(L)∼c/LE(L)\sim c/LE(L)∼c/L reveals a negative central charge c=−2c=-2c=−2 at the non-Hermitian critical point, indicative of a non-unitary CFT. Furthermore, we discover a universal scaling function capturing the flow of a system from Dirac CFT with c=1c=1c=1 to a non-unitary CFT with c=−2c=-2c=−2. The scaling function demonstrates distinct behaviors at topologically non-trivial and trivial sides of critical points. Notably, within the realm of topological criticality, the scaling function exhibits an universal rise-dip-rise pattern, manifesting a characteristic singularity inherent in the non-Hermitian topological critical points. The analytic expression of the scaling function has been derived and is in good agreement with the numerical results.
The critical point of a topological phase transition is described by a conformal field theory, where finite-size corrections to energy are uniquely related to its central charge. We investigate the finite-size scaling away from criticality and find a scaling function, which discriminates between phases with different topological indices. This function appears to be universal for all five Altland-Zirnbauer symmetry classes with nontrivial topology in one spatial dimension. We obtain an analytic form of the scaling function and compare it with numerical results.
We consider a finite-size scaling function across a topological phase transition in one-dimensional models. For models of noninteracting fermions it was shown to be universal for all topological symmetry classes and markedly asymmetric between trivial and topological sides of the transition [T. Gulden, M. Janas, Y. Wang, and A. Kamenev, Phys. Rev. Lett. 116, 026402 (2016)]. Here we verify its universality for the topological transition between dimerized and Haldane phases in bilinear-biquadratic spin-1 chain. To this end we perform high-accuracy variational matrix product state simulations. We show that the scaling function, expressed in terms of $L/\ensuremath{\xi}$, where LLL is the chain length and $\ensuremath{\xi}$ is the correlation length, coincides with that of three species of noninteracting massive Majorana fermions. The latter is known to be a proper description of the conformal critical theory with central charge c=3/2c=3/2c=3/2. We have shown that it still holds away from the conformal point, including the finite-size corrections. We have also observed peculiar differences between even- and odd-size chains, which may be fully accounted for by residual interactions of the edge states.
We introduce topological invariants for gapless systems and study the associated boundary phenomena.More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants establishing the notion of symmetry-enriched quantum criticality.The charges of nonlocal scaling operators, or more generally, of symmetry defects, are topological and imply the presence of localized edge modes.We primarily focus on the 1 þ 1d case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors.An example of the exponential case is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase.An example of the algebraic case arises between the gapped Ising and cluster phases: This symmetry-enriched Ising CFT has an edge mode with finite-size splitting scaling as 1=L 14 .In addition to such new cases, our formalism unifies various examples previously studied in the literature.Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs.This raises the question of classification, to which we give a partial answer-including a complete characterization of symmetry-enriched 1 þ 1d Ising CFTs.Nontrivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched 2 þ 1d CFT without gapped sectors.
We show that for conformally invariant two-dimensional systems, the amplitude of the finite-size corrections to the free energy of an infinitely long strip of width L at criticality is linearly related to the conformal anomaly number c, for various boundary conditions. The result is confirmed by renormalization-group arguments and numerical calculations. It is also related to the magnitude of the Casimir effect in an interacting one-dimensional field theory, and to the low-temperature specific heat in quantum chains.
Under an appropriate symmetric bulk bipartition in a one-dimensional symmetry protected topological phase with the Affleck-Kennedy-Lieb-Tasaki matrix product state wave function for the odd integer spin chains, a bulk critical entanglement spectrum can be obtained, describing the excitation spectrum of the critical point separating the topological phase from the trivial state with the same symmetry. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry-breaking phase transitions. Recently, the framework of matrix product states for topological phases with Majorana fermions/parafermions has been established. Here we first generalize these fixed-point matrix product states with the zero correlation length to the more generic ground-state wave functions with a finite correlation length for the general one-dimensional interacting Majorana fermion/parafermion systems. Then we employ the previous method to decode quantum criticality to the interacting Majorana fermion/parafermion matrix product states. The obtained quantum critical spectra are described by the conformal field theories with central charge $c\ensuremath{\le}1$, characterizing the quantum critical theories separating the fermionic/parafermionic topological phase from the trivial phases with the same symmetry.