## CQuIC Seminars

### Undecidability of the Spectral Gap in One Dimension

#### Presented by Johannes Bausch, University of Cambridge

The spectral gap problem - determining whether the energy

spectrum of a system has an energy gap above ground state, or if there

is a continuous range of low-energy excitations - pervades quantum

many-body physics. Recently, this important problem was shown to be

undecidable for quantum systems in two (or more) spatial dimensions: it

is provably impossible to determine in general whether a system is

gapped or gapless, a result which has many unexpected consequences for

the physics of such systems. However, there are many indications that

one dimensional systems are simpler than their higher-dimensional

counterparts: for example, they cannot have thermal phase transitions or

topological order, and there exist highly-effective numerical algorithms

such as DMRG for gapped 1D systems, exploiting the fact that such

systems obey an entropy area-law. Furthermore, the spectral gap

undecidability construction crucially relied on aperiodic tilings, which

are easily seen to be impossible in 1D.

So does the spectral gap problem become decidable in 1D? In this paper

we prove this is not the case, by constructing a family of 1D spin

chains with translationally-invariant nearest neighbour interactions

with undecidable spectral gap. This not only proves that the spectral

gap of 1D systems is just as intractable, but also predicts the

existence of qualitatively new types of complex physics in 1D spin

chains. In particular, it implies there are 1D systems with constant

spectral gap and unique classical ground state for all systems sizes up

to an uncomputably large size, whereupon they switch to a gapless

behaviour with dense spectrum.

**3:30 pm, Thursday, February 14, 2019
Room 190, Physics & Astronomy
Northeast corner of Lomas and Yale, Albuquerque, New Mexico**

Individuals with disabilities who need an auxiliary aid or service to attend or participate in P&A events should contact Sandra Ortiz (phone: 505-277-5900, email: sportiz@unm.edu) well in advance to ensure your needs are accomodated. Event handouts can be provided in alternative accessible formats upon request. Please contact Mrs. Ortiz if you need written information in an alternative format.

*A schedule of talks within the Department of Physics and Astronomy is available on the P&A web site at http://physics.unm.edu/pandaweb/events/index.php*