Quantum systems with short-range interactions are known to respect an area law for the entanglement entropy: The von Neumann entropy $S$ associated to a bipartition scales with the boundary $p$ between the two parts. Here we study the case in which the boundary is a fractal. We consider the topologically ordered phase of the toric code with a magnetic field. When the field vanishes it is possible to analytically compute the entanglement entropy for both regular and fractal bipartitions ( $A,B$ ) of the system and this yields an upper bound for the entire topological phase. When the $A\ensuremath{-}B$ boundary is regular we have $S/p=1$ for large $p$ . When the boundary is a fractal of the Hausdorff dimension $D$ , we show that the entanglement between the two parts scales as $S/p=\ensuremath{\gamma}\ensuremath{\leqslant}1/D$, and $\ensuremath{\gamma}$ depends on the fractal considered.Quantum systems with short-range interactions are known to respect an area law for the entanglement entropy: The von Neumann entropy $S$ associated to a bipartition scales with the boundary $p$ between the two parts. Here we study the case in which the boundary is a fractal. We consider the topologically ordered phase of the toric code with a magnetic field. When the field vanishes it is possible to analytically compute the entanglement entropy for both regular and fractal bipartitions ($A,B$) of the system and this yields an upper bound for the entire topological phase. When the $A\ensuremath{-}B$ boundary is regular we have $S/p=1$ for large $p$. When the boundary is a fractal of the Hausdorff dimension $D$, we show that the entanglement between the two parts scales as $S/p=\ensuremath{\gamma}\ensuremath{\leqslant}1/D$, and $\ensuremath{\gamma}$ depends on the fractal considered.

arXiv:0903.4444Physical Review AJan 27, 20103 citations

Generalized boundary condition, free energies of six-vertex models and fractal dimension

Kazuhiko Minami

The structures of the configuration space of the six-vertex models with various boundaries and boundary conditions are investigated, and it is derived that the free energies depend on the boundary conditions, and that they are classified by the fractal structures. The "n-equivalences" of the boundary conditions are defined with a property that the models with n-equivalent boundary conditions result in the identical free energy. The configurations which satisfy the six-vertex restriction are classified, through the n-equivalences, into sets of configurations called islands. It is derived that each island shows a fractal structure when it is mapped to the real axis. Each free energy is expressed by the weighted fractal dimension (multi-fractal dimension) of the island. The fractal dimension of the island has a strict relation with the maximum eigenvalue of the corresponding block element of the transfer matrix. It is also found that the fractal dimensions of islands take the absolute maximum when the boundary is "alternative"; in this case, the free energy equals to the solution obtained by Lieb and Sutherland.The structures of the configuration space of the six-vertex models with various boundaries and boundary conditions are investigated, and it is derived that the free energies depend on the boundary conditions, and that they are classified by the fractal structures. The "n-equivalences" of the boundary conditions are defined with a property that the models with n-equivalent boundary conditions result in the identical free energy. The configurations which satisfy the six-vertex restriction are classified, through the n-equivalences, into sets of configurations called islands. It is derived that each island shows a fractal structure when it is mapped to the real axis. Each free energy is expressed by the weighted fractal dimension (multi-fractal dimension) of the island. The fractal dimension of the island has a strict relation with the maximum eigenvalue of the corresponding block element of the transfer matrix. It is also found that the fractal dimensions of islands take the absolute maximum when the boundary is "alternative"; in this case, the free energy equals to the solution obtained by Lieb and Sutherland.

arXiv:cond-mat/0401485Jan 25, 20040 citations

Quantum mechanics in fractal geometry

Johanna L. Miller

Physics TodayNov 19, 20180 citations

Quantum fractals in boxes

Michael Berry

A quantum wave with probability density , confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state . For almost all positions , the graph of the evolution of P is a fractal curve with dimension . For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension . When D = 1, there are, in addition to these generic time and space fractals, infinitely many special `quantum revival' times when P is piecewise constant, and infinitely many special spacetime slices for which the dimension of P is 5/4. If the surface of the box is a fractal with dimension , simple arguments suggest that the dimension of the time fractal is , and that of the space fractal is .A quantum wave with probability density , confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state . For almost all positions , the graph of the evolution of P is a fractal curve with dimension . For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension . When D = 1, there are, in addition to these generic time and space fractals, infinitely many special `quantum revival' times when P is piecewise constant, and infinitely many special spacetime slices for which the dimension of P is 5/4. If the surface of the box is a fractal with dimension , simple arguments suggest that the dimension of the time fractal is , and that of the space fractal is .

Journal of Physics A Mathematical and GeneralOct 21, 1996193 citations

Quantum systems with short-range interactions are known to respect an area law for the entanglement entropy: The von Neumann entropy

S

associated to a bipartition scales with the boundary

p

between the two parts. Here we study the case in which the boundary is a fractal. We consider the topologically ordered phase of the toric code with a magnetic field. When the field vanishes it is possible to analytically compute the entanglement entropy for both regular and fractal bipartitions (

A,B

) of the system and this yields an upper bound for the entire topological phase. When the $A\ensuremath{-}B$ boundary is regular we have

S/p=1

for large

p

. When the boundary is a fractal of the Hausdorff dimension

D

, we show that the entanglement between the two parts scales as $S/p=\ensuremath{\gamma}\ensuremath{\leqslant}1/D$, and $\ensuremath{\gamma}$ depends on the fractal considered.