# Douglas Lundholm's research

**Spectral theory of quantum systems with exotic symmetries**

## Non technical background and introduction

The theory of quantum mechanics (QM) lies at the foundation of much of modern technology, from microelectronics to lasers, and the set of mathematical tools adopted and developed for the study of quantum systems, here briefly summarized under the term*spectral theory*, has also turned out to be very useful in many other areas of pure and applied mathematics. Several decades after the initial development and establishment of QM in the 1920s, certain unconventional symmetries entered the context, first only as exotic logical possibilities, but later realized to be of relevance for predicted and/or observed physical phenomena. One such possibility is

*supersymmetry*, whereby the two standard classes of particle types realized in nature - bosons and fermions - are intertwined via an extended symmetry of space and time. This symmetry underlies some of the serious modern attempts to reconcile QM with Einstein's theory of gravitation. In a mathematically rigorous approach to this problem, a particular class of QM models called

*supersymmetric matrix models*has been proposed, and in particular the spectral theoretic question of existence and structure of ground states, i.e. states of lowest energy in these models, is crucial for their relevance as models for unified interactions. Another relatively recent and exotic class of symmetries is the notion of alternative (

*intermediate, or fractional*)

*particle statistics*in lower dimensions, which opens up for the possibility of having types of particles different from the usual bosons and fermions. Although such particles are not believed to exist as fundamental particles, they have been found useful as effective models in condensed matter physics. However, with these new types of symmetries come new types of challenges for mathematicians and physicists, and it is often the case that standard methods of spectral theory are found inapplicable in these contexts. The aim of my research project is therefore to develop new mathematical tools and techniques to study the spectrum, structure and geometry of quantum mechanical systems with such exotic symmetries - such as supersymmetry or intermediate and fractional statistics - as well as other models with similar features arising in mathematical physics. The project is divided into two main parts which have independent interest from the point of view of applications within physics, but with many unifying elements from the point of view of spectral theory.

## Intermediate particle statistics and anyons

The possibility of an extension of the conventional quantum mechanics of identical particles to intermediate (fractional) statistics was discovered in the 1970s (see e.g. [A1,A2] for more recent reviews). It applies only for particles in spaces of dimension strictly less than three and, while the one-dimensional case was soon found to be related to certain exactly solvable models (the Lieb-Liniger [A3] and Calogero-Sutherland [A4,A5] models), the two-dimensional case, with the corresponding particles of intermediate statistics called anyons, was under intensive investigation in the late 80s and early 90s, in particular motivated by connections to newly found phenomena in condensed matter physics such as the fractional quantum Hall effect and certain types of superconductors. However, the new anyon particle statistics posed serious difficulties compared to the usual bosons and fermions (which are comparatively well understood), and the spectral theory of quantum systems of many anyons remained mostly undeveloped until very recently. Among earlier results we can mention the rigorous mathematical papers [A6,A7] and the harmonic oscillator spectrum [A8,A9]. Today anyons are a hot topic in physics because of their potential for applications in quantum computing [A10].In joint work with J. P. Solovej we have studied the many-particle quantum mechanics of anyons, modeled conveniently using magnetic interactions of topological type. We have proved certain Hardy and Lieb-Thirring inequalities for anyons in [A11], thereby extending several standard tools in many-particle spectral theory to this context, and as an application we have obtained the first set of rigorous non-trivial bounds for the ground state energy of the ideal anyon gas. Our results depend sensitively on the value of a statistics parameter which labels the type of anyons, and therefore a natural question arising from this work is whether the lower bounds we have found are in some sense optimal, implying that two classes of types of anyons behave very differently. We have also applied some of our new techniques to identical particles in one dimension, resulting in new Lieb-Thirring type bounds for the kinetic energy in terms of the density [A12]. A summary of these results and some physical applications are given in [A13]. Apart from these results, a new approach to a mean-field description of anyons has been initiated in joint work with N. Rougerie [A14].

These models of intermediate statistics have in common that they can be modeled using bosons together with a statistical interaction. Recently we have, together with Solovej and F. Portmann, extended our study to more general interacting quantum Bose gases, proving Lieb-Thirring type inequalities for a large class of such systems in [A15]. Furthermore, together with Portmann and P. T. Nam, we have considered generalizations to kinetic energy operators of fractional type (relevant for relativistic systems), Hardy-Lieb-Thirring inequalities, and more general interpolation inequalities [A16].

Some ideas for student projects related to this topic:

1. Consider extensions to non-abelian anyons (a type of anyons appearing in some recent approaches to quantum computing, see also [A17] for an initial study).

2. Intermediate Calogero-Sutherland statistics (presently our methods are limited to 'superfermions').

3. Analytical and numerical studies of small systems of anyons (see [A18]).

4. Many-particle Lieb-Thirring and related inequalities

## Supersymmetric matrix models

Due to relevance in quantum gravity (M-theory), as well as membrane theory, and reduced Yang-Mills gauge theory, there has been considerable effort during the last twenty years to determine the existence and structure of ground states of supersymmetric matrix models (SMM). A brief review of the subject and its difficulties is given in [M1]. Of central relevance is the fact that, although the scalar (bosonic) version of the model has a purely discrete energy spectrum [M2], the spectrum of the SMM covers the whole positive real axis [M3] (making the investigation of ground states much more complicated), as well as the BFSS conjecture [M4,M5] regarding the existence and uniqueness of ground states and their physical relevance. It is conjectured that, for the SMM, parametrized by a dimension d=2,3,5, or 9, and matrix size N >= 2, there is a unique normalizable zero-energy ground state for d=9 and all N, while for d=2,3,5 there is no such state for any N. A count on the number of ground states, by means of the Witten index, was first computed in [M6,M7], although not entirely satisfactorily from a mathematical perspective because of the technical complications due to the continuous spectrum of the operator, and because no information is obtained on the structure of ground states. It has only been proven rigorously in [M8] that the simplest (d=2,N=2) of the SMM does not possess a normalizable ground state (in agreement with the conjecture), and an asymptotic analysis [M9] of the models, for N=2, also agrees with the conjecture. Except for the asymptotic description, one has (until recently, in approach 4 below) not been able to extract much information on the actual structure of the conjectured ground state.Four new approaches to this problem have been initiated and discussed in [M1] (some of these are joint work with V. Bach, J. Hoppe and M. Trzetrzelewski):

1. Weighted Hilbert spaces, negative spectrum, and index theory (see [M10,M11]).

2. Averaging with respect to symmetries (see [M12]).

3. Deformation techniques (see [M13]).

4. Construction by recursive methods (see [M14,M15,M16,M17,M18]).

Some ideas for student projects related to this topic:

1. Computing the weighted index for simpler models (cp. Section 4.3 in [M1]).

2. Averaging of eigenstates of a truncated operator (cp. [M12]).

## Other projects

I also have an interest in more general mathematical tools and techniques and their applications in mathematical physics. Together with L. Svensson I am writing lecture notes [O1] on a general approach to the theory of Clifford algebras, and related to this there are several interesting research questions. I have also considered some applications of Clifford algebraic techniques to many-particle quantum mechanics [O2].Some ideas for student projects:

1. Norm functions in Clifford algebras (see Section 6.7 in [O1])

2. Spinors and Fierz identities

## References

[A1] | J. Myrheim, Anyons; in Topological aspects of low dimensional systems (Les Houches, 1998), pp. 265-413, EDP Sci., Les Ulis, 1999 [DOI] | |

[A2] | A. Khare, Fractional Statistics and Quantum Theory, (World Scientific, Singapore, Second Edition 2005) [DOI] | |

[A3] | E. H. Lieb, W. Liniger, Phys. Rev. 130 (1963) 1605-1616 [DOI] | |

[A4] | F. Calogero, J. Math. Phys. 10 (1969), 2191-2200 [DOI] | |

[A5] | B. Sutherland, J. Math. Phys. 12 (1971), 246-256 [DOI] | |

[A6] | G. A. Baker et. al., Commun. Math. Phys. 153 (1993) 277-295 [DOI] | |

[A7] | G. Dell'Antonio, R. Figari, A. Teta, Lett. Math. Phys. 40 (1997) 235-256 [DOI] | |

[A8] | C. Chou, Phys. Lett. A 155 (1991) 245-251 [DOI] | |

[A9] | R. Chitra, D. Sen, Phys. Rev. B 46, (1992) 10923-10930 [DOI] | |

[A10] | C. Nayak et. al., Rev. Mod. Phys. 80 (2008) 1083-1159 [DOI] | |

[A11] | D. Lundholm, J. P. Solovej, Commun. Math. Phys. 322 (2013) 883-908 [DOI] | |

[A12] | D. Lundholm, J. P. Solovej, Ann. Henri Poincaré 15 (2014) 1061-1107 [DOI] | |

[A13] | D. Lundholm, J. P. Solovej, Phys. Rev. A 88 (2013) 062106 [DOI] | |

[A14] | D. Lundholm, N. Rougerie, 2015 [arXiv] | |

[A15] | D. Lundholm, F. Portmann, J. P. Solovej, Commun. Math. Phys. 335 (2015) 1019-1056 [DOI] | |

[A16] | D. Lundholm, P. T. Nam, F. Portmann, Arch. Rational Mech. Anal. [DOI] | |

[A17] | O. Weinberger, B.Sc. thesis [DiVA] | |

[A18] | D. Lundholm, IHES/P/13/25 [IHES] | |

[M1] | D. Lundholm, Ph.D. thesis, KTH, 2010 [URN] | |

[M2] | B. Simon, Ann. Phys. 146 (1983) 209-220 [DOI] | |

[M3] | B. de Wit, W. Lüscher, H. Nicolai, Nucl. Phys. B320 (1989) 135-159. [DOI] | |

[M4] | E. Witten, Nucl. Phys. B460 (1996) 335-350 [DOI] | |

[M5] | T. Banks, W. Fischler, S. Shenker, L. Susskind, Phys. Rev. D55 (1997) 5112 [DOI] | |

[M6] | P. Yi, Nucl. Phys. B505 (1997) 307-318 [DOI] | |

[M7] | S. Sethi, M. Stern, Commun. Math. Phys. 194 (1998) 675-705 [DOI] | |

[M8] | J. Fröhlich, J. Hoppe, Commun. Math. Phys. 191 (1998) 613-626 [DOI] | |

[M9] | J. Fröhlich, G. M. Graf, D. Hasler, J. Hoppe, S.-T. Yau, Nucl. Phys. B567 (2000) 231-248 [DOI] | |

[M10] | D. Lundholm, J. Math. Phys. 49 (2008) 062101 [DOI] | |

[M11] | D. Lundholm, Lett. Math. Phys. 92 (2010), 125-141 [DOI] | |

[M12] | J. Hoppe, D. Lundholm, M. Trzetrzelewski, J. Math. Phys. 50, 043510 (2009) [DOI] | |

[M13] | J. Hoppe, D. Lundholm, M. Trzetrzelewski, Ann. Henri Poincaré 10 (2009), 339-356 [DOI] | |

[M14] | V. Bach, J. Hoppe, D. Lundholm, Documenta Math. 13 (2008) 103-116 [DOI] | |

[M15] | J. Hoppe, D. Lundholm, M. Trzetrzelewski, Nucl. Phys. B 817 (2009) 155-166 [DOI] | |

[M16] | M. Hynek, M. Trzetrzelewski, Nucl. Phys. B 838 (2010) 413-421 [DOI] | |

[M17] | Y. Michishita, JHEP 1009 (2010) 075; J. Math. Phys. 51 (2010) 122309; Prog. Theor. Phys. Suppl. 188 (2011) 75-82 | |

[M18] | Y. Michishita, M. Trzetrzelewski, Nucl. Phys. B 868 (2013) 539-553 [DOI] | |

[O1] | D. Lundholm, L. Svensson, 2009 [arXiv] | |

[O2] | D. Lundholm, J. Phys. A: Math. Theor. 48 (2015) 175203 [DOI] |

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