Emergent Geometry from Entanglement Structure
355
with A = (u + , v − ) and A c = (−∞, u) ∪ (v, ∞), by choosing
J (x, y) =
1
(x − y) 2 .
(13)
This equation indicates that J (x, y) is the two-point correlator, on the complex
plane, of a current operator J, whose integration along segments, as in Eq. (12), is
invariant under reparametrization. J (x, y)dxdy represents the amount of entanglement between the intervals (x, x + dx) and (y, y + dy). This field theory realization
leads to think of entanglement as a flow among the parts of the system, in analogy
to the flow of energy that is characterized by the stress tensor. Moreover, using the
construction described in Ref. [26] for entanglement Hamiltonians in CFT, one can
show that Eq. (12) reproduces the values of S A , for the space-time geometries Σ,
that are conformally equivalent to an annulus. In these cases J (x, y) is given by
the two-point correlator J (x, y) = =J(x) J(y) Σ . Notice that in the conformal field
theory systems the representation is exact only when A is an interval, but not in
general.
5 Conclusion
To conclude, in this work, we aimed to unveil the geometry revealed from the
entanglement properties of any pure quantum state through the elements of a
generalized adjacency matrix, such that the entropy values of any bipartition of the
state can be approximated as a weighted sum of all the links connecting the sites
across that bipartition. We reported certain examples, where the optimal geometry
emerged from the entanglement structure, turned out to be completely different
from that suggested by the parent Hamiltonian of the system. Subsequently, we
showed that our formalism provided a natural route to compute the entanglement
contour, introduced earlier for the non-interacting models, which essentially helped
us to extend the concept for interacting models as well. Finally, we showed that for
conformal invariant systems, a more insightful explanation of the elements of such
generalized adjacency matrices can be obtained in terms of a two-point correlator
of an entanglement current operator.
Acknowledgments GS would like to thank William Witczak-Krempa for the invitation to
participate in the Quantum Theory and Symmetry XI conference held in Montreal in July
2019. We acknowledge financial support from the grants PGC2018-095862-B-C21, QUITEMAD+
S2013/ICE-2801, SEV-2016-0597 of the “Centro de Excelencia Severo Ochoa” Programme and
the CSIC Research Platform on Quantum Technologies PTI-001.
355
with A = (u + , v − ) and A c = (−∞, u) ∪ (v, ∞), by choosing
J (x, y) =
1
(x − y) 2 .
(13)
This equation indicates that J (x, y) is the two-point correlator, on the complex
plane, of a current operator J, whose integration along segments, as in Eq. (12), is
invariant under reparametrization. J (x, y)dxdy represents the amount of entanglement between the intervals (x, x + dx) and (y, y + dy). This field theory realization
leads to think of entanglement as a flow among the parts of the system, in analogy
to the flow of energy that is characterized by the stress tensor. Moreover, using the
construction described in Ref. [26] for entanglement Hamiltonians in CFT, one can
show that Eq. (12) reproduces the values of S A , for the space-time geometries Σ,
that are conformally equivalent to an annulus. In these cases J (x, y) is given by
the two-point correlator J (x, y) = =J(x) J(y) Σ . Notice that in the conformal field
theory systems the representation is exact only when A is an interval, but not in
general.
5 Conclusion
To conclude, in this work, we aimed to unveil the geometry revealed from the
entanglement properties of any pure quantum state through the elements of a
generalized adjacency matrix, such that the entropy values of any bipartition of the
state can be approximated as a weighted sum of all the links connecting the sites
across that bipartition. We reported certain examples, where the optimal geometry
emerged from the entanglement structure, turned out to be completely different
from that suggested by the parent Hamiltonian of the system. Subsequently, we
showed that our formalism provided a natural route to compute the entanglement
contour, introduced earlier for the non-interacting models, which essentially helped
us to extend the concept for interacting models as well. Finally, we showed that for
conformal invariant systems, a more insightful explanation of the elements of such
generalized adjacency matrices can be obtained in terms of a two-point correlator
of an entanglement current operator.
Acknowledgments GS would like to thank William Witczak-Krempa for the invitation to
participate in the Quantum Theory and Symmetry XI conference held in Montreal in July
2019. We acknowledge financial support from the grants PGC2018-095862-B-C21, QUITEMAD+
S2013/ICE-2801, SEV-2016-0597 of the “Centro de Excelencia Severo Ochoa” Programme and
the CSIC Research Platform on Quantum Technologies PTI-001.