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arXiv:1508.03646 · Physical Review Letters

Universal Finite-Size Scaling around Topological Quantum Phase Transitions

Tobias Gulden, Michael Janas, Yuting Wang, Alex Kamenev

Published Jan 14, 2016• 23 citations
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Abstract

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.

DOI
10.1103/physrevlett.116.026402

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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.
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.
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.
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.
Topological phase transitions constitute a new class of quantum critical phenomena. They cannot be described within the usual framework of the Landau theory since, in general, the different phases cannot be distinguished by an order parameter, neither can they be related to different symmetries. In most cases, however, one can identify a diverging length at these topological transitions. This allows us to describe them using a scaling approach and to introduce a set of critical exponents that characterize their universality class. Here we consider some relevant models of quantum topological transitions associated with well-defined critical exponents that are related by a quantum hyperscaling relation. We extend to these models a finite-size scaling approach based on techniques for calculating the Casimir force in electromagnetism. This procedure allows us to obtain universal Casimir amplitudes at their quantum critical points. Our results verify the validity of finite-size scaling in these systems and confirm the values of the critical exponents obtained previously.
The correlation functions related to topological phase transitions in inversion-symmetric lattice models described by $2\ifmmode\times\else\texttimes\fi{}2$ Dirac Hamiltonians are discussed. In one dimension, the correlation function measures the charge-polarization correlation between Wannier states at different positions, while in two dimensions it measures the itinerant-circulation correlation between Wannier states. The correlation function is nonzero in both the topologically trivial and nontrivial states, and allows us to extract a correlation length that diverges at topological phase transitions. The correlation length and the curvature function that defines the topological invariants are shown to have universal critical exponents, allowing the notion of universality classes to be introduced. Particularly in two dimensions, the universality class is determined by the orbital symmetry of the Dirac model. The scaling laws that constrain the critical exponents are revealed, and are predicted to be satisfied even in interacting systems, as demonstrated in an interacting topological Kondo insulator.
We analyze the finite-size corrections to entanglement in quantum critical systems. By using conformal symmetry and density functional theory, we discuss the structure of the finite-size contributions to a general measure of ground state entanglement, which are ruled by the central charge of the underlying conformal field theory. More generally, we show that all conformal towers formed by an infinite number of excited states (as the size of the system $L\ensuremath{\rightarrow}\ensuremath{\infty}$) exhibit a unique pattern of entanglement, which differ only at leading order (1∕L)2{(1∕L)}^{2}(1∕L)2. In this case, entanglement is also shown to obey a universal structure, given by the anomalous dimensions of the primary operators of the theory. As an illustration, we discuss the behavior of pairwise entanglement for the eigenspectrum of the spin-1∕21∕21∕2 XXZXXZXXZ chain with an arbitrary length LLL for both periodic and twisted boundary conditions.