A new duality solves a physics mystery
In conventional wisdom, producing a curved space requires
distortions, such as bending or stretching a flat space. A team
of researchers at Purdue University have discovered a new
method to create curved spaces that also solves a mystery in
physics. Without any physical distortions of physical systems,
the team has designed a scheme using non-Hermiticity, which
exists in any systems coupled to environments, to create a
hyperbolic surface and a variety of other prototypical curved
The team recently published their findings in Nature Communications.
Of the members of the team, most work at Purdue University's West
Lafayette campus. Chenwei Lv, graduate student, is the lead author,
and other members of the Purdue team include Prof. Qi Zhou, and
Zhengzheng Zhai, postdoctoral fellow. The co-first author, Prof.
Ren Zhang from Xi'an Jiaotong University, was a visiting scholar
at Purdue when the project was initiated.
In order to understand how this discovery works, first one must
understand the difference between Hermitian and non-Hermitian
systems in physics. Zhou explains it using an example in which
a quantum particle can "hop" between different sites on a lattice.
If the probability for a quantum particle to hop in the right
direction is the same as the probability to hop in the left
direction, then the Hamiltonian is Hermitian. If these two
probabilities are different, the Hamiltonian is non-Hermitian. This
is the reason that Chenwei and Ren Zhang have used arrows with
different sizes and thicknesses to denote the hopping probabilities
in opposite directions in their plot.
He further explains that their work provides an unprecedented
explanation of fundamental non-Hermitian quantum phenomena.
They found that a non-Hermitian Hamiltonian has curved the space
where a quantum particle resides. For instance, a quantum particle
in a lattice with nonreciprocal tunneling is in fact moving on
a curved surface. The ratio of the tunneling amplitudes along one
direction to that in the opposite direction controls how large the
surface is curved. In such curved spaces, all the strange non
Hermitian phenomena, some of which may even appear unphysical,
immediately become natural. It is the finite curvature that requires
orthonormal conditions distinct from their counterparts in flat
spaces. As such, eigenstates would not appear orthogonal if we
used the theoretical formula derived for flat spaces. It is also the
finite curvature that gives rise to the extraordinary non-Hermitian
skin effect that all eigenstates concentrate near one edge of the
Now that the team has published their findings, they anticipate it
spinning off into multiple directions for further study. Physicists
studying curved spaces could implement their apparatuses to address
challenging questions in non-Hermitian physics. Also, physicists
working on non-Hermitian systems could tailor dissipations to
access non-trivial curved spaces that cannot be easily obtained by
conventional means. The Zhou research group will continue to
theoretically explore more connections between non-Hermitian
physics and curved spaces. They also hope to help bridge the gap
between these two physics subjects and bring these two different
communities together with future research.