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arXiv:1407.0084 · Physical Review B

Entanglement across a cubic interface in 3 + 1 dimensions

Trithep Devakul, Rajiv R. P. Singh

Published Aug 20, 2014• 15 citations
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Abstract

We calculate the area, edge, and corner Renyi entanglement entropies in the ground state of the transverse-field Ising model, on a simple-cubic lattice, by high-field and low-field series expansions. We find that while the area term is positive and the line term is negative as required by strong subadditivity, the corner contributions are positive in three dimensions. Analysis of the series suggests that the expansions converge up to the physical critical point from both sides. The leading area-law Renyi entropies match nicely from the high- and low-field expansions at the critical point, forming a sharp cusp there. We calculate the coefficients of the logarithmic divergence associated with the corner entropy and compare them with conformal field theory results with smooth interfaces and find a striking correspondence.

DOI
10.1103/physrevb.90.054415

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We calculate the area, edge, and corner Renyi entanglement entropies in the ground state of the transverse-field Ising model, on a simple-cubic lattice, by high-field and low-field series expansions. We find that while the area term is positive and the line term is negative as required by strong subadditivity, the corner contributions are positive in three dimensions. Analysis of the series suggests that the expansions converge up to the physical critical point from both sides. The leading area-law Renyi entropies match nicely from the high- and low-field expansions at the critical point, forming a sharp cusp there. We calculate the coefficients of the logarithmic divergence associated with the corner entropy and compare them with conformal field theory results with smooth interfaces and find a striking correspondence.
The entanglement entropy of the random transverse-field Ising model is calculated by a numerical implementation of the asymptotically exact strong disorder renormalization group method in 2 d , 3 d and 4 d hypercubic lattices for different shapes of the subregion. We find that the area law is always satisfied, but there are analytic corrections due to E -dimensional edges (1⩽ E ⩽ d − 2). More interesting is the contribution arising from corners, which is logarithmically divergent at the critical point and its prefactor in a given dimension is universal, i.e. , independent of the form of disorder.
We study the bipartite entanglement entropy of the 2D transverse-field Ising model in the thermodynamic limit. Series expansions are developed for the Renyi entropy around both the small-field and large-field limits, allowing the separate calculation of the entanglement associated with lines and corners at the boundary between subsystems. Series extrapolations are used to extract power laws and logarithmic singularities as the quantum critical point is approached, giving access to new universal quantities. In 1D, we find excellent agreement with exact results as well as quantum Monte Carlo simulations. In 2D, we find compelling evidence that the entanglement at a corner is significantly different from a free boson field theory. These results demonstrate the power of the series expansion method for calculating entanglement entropy in interacting systems, a fact that will be particularly useful in searches for exotic quantum criticality in models with and without the sign problem.
We consider a bilayer quantum spin model with anisotropic intralayer exchange couplings. By varying the anisotropy, the quantum critical phenomena changes from XY to Heisenberg to Ising universality class, with two modes, three modes, and one mode, respectively, becoming gapless simultaneously. We use series-expansion methods to calculate the second and third Renyi entanglement entropies when the system is bipartitioned into two parts. Leading area-law terms and subleading entropies associated with corners are separately calculated. We find clear evidence that the logarithmic singularity associated with the corners is universal in each class. Its coefficient along the Ising critical line is in excellent agreement with those obtained previously for the transverse-field Ising model. Our results provide strong evidence for the idea that the universal terms in the entanglement entropy provide an approximate measure of the low energy degrees of freedom in the system.
We calculate the universal contribution to the $\ensuremath{\alpha}$-R\'enyi entropy from a cubic trihedral corner in the boundary of the entangling region in 3+13+13+1 dimensions for a massless free scalar. The universal number ${v}_{\ensuremath{\alpha}}$ is manifest as the coefficient of a scaling term that is logarithmic in the size of the entangling region. Our numerical calculations find that this universal coefficient has both larger magnitude and the opposite sign to that induced by a smooth spherical entangling boundary in 3+13+13+1 dimensions, for which there is a well-known subleading logarithmic scaling. Despite these differences, up to the uncertainty of our finite-size lattice calculations, the functional dependence of the trihedral coefficient ${v}_{\ensuremath{\alpha}}$ on the R\'enyi index $\ensuremath{\alpha}$ is indistinguishable from that for a sphere, which is known analytically for a massless free scalar. We comment on the possible source of this $\ensuremath{\alpha}$ dependence arising from the general structure of (3+1)(3+1)(3+1)-dimensional conformal field theories, and suggest calculations past the free scalar which could further illuminate the general structure of the trihedral divergence in the R\'enyi entropy.
In a D=2+1D=2+1D=2+1 quantum critical system, the entanglement entropy across a boundary with a corner contains a subleading logarithmic scaling term with a universal coefficient. It has been conjectured that this coefficient is, to leading order, proportional to the number of field components NNN in the associated O(N)\text{O}(N)O(N) continuum ${\ensuremath{\phi}}^{4}$ field theory. Using density matrix renormalization group calculations combined with the powerful numerical linked cluster expansion technique, we confirm this scenario for the O(2)\text{O}(2)O(2) Wilson-Fisher fixed point in a striking way, through direct calculation at the quantum critical points of two very different microscopic models. The value of this corner coefficient is, to within our numerical precision, twice the coefficient of the Ising fixed point. Our results add to the growing body of evidence that this universal term in the R\'enyi entanglement entropy reflects the number of low-energy degrees of freedom in a system, even for strongly interacting theories.
Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.