TY - JOUR

T1 - Hagedorn temperature in large N Majorana quantum mechanics

AU - Gaitan, G.

AU - Klebanov, I. R.

AU - Pakrouski, K.

AU - Pallegar, P. N.

AU - Popov, F. K.

N1 - Funding Information:
This research was supported in part by the U.S. NSF under Grants No. PHY-1620059 and PHY-1914860. K. P. was also supported by the Swiss National Science Foundation through the Early Postdoc.Mobility Grant No. P2EZP2_172168, and by DOE Grant No. DE-SC0002140. Some of the results presented here are from Gabriel Gaitan’s Princeton University Senior Thesis (May 2019). I. R. K. is grateful to the Kavli Institute for Theoretical Physics at UC, Santa Barbara and the organizers of the program “Chaos and Order: From strongly correlated systems to black holes” for the hospitality and stimulating atmosphere during some of his work on this project. His research at K. I. T. P. was supported in part by the National Science Foundation under Grant No. NSF PH-1748958. We are grateful to S. Sondhi, P. Werner and C. Xu for useful discussions, and to A. Milekhin and G. Tarnopolsky for valuable discussions and comments on a draft of this paper. F. K. P. is grateful to A. Morozov, D. Vasiliev, S. Anokhina, S. Shakirov, A. Sleptsov, K. Aleshkin and Y. Kononov for useful discussions and comments on the representation theory. F. K. P. is grateful to the Mainz Institute for Theoretical Physics (MITP) of the DFG Cluster of Excellence PRISMA+ (Project ID 39083149), for its hospitality and its partial support. P. N. P. is grateful to the Yukawa Institute for Theoretical Physics (YITP) for its hospitality and partial support.
Publisher Copyright:
© 2020 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the "https://creativecommons.org/licenses/by/4.0/" Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP

PY - 2020/6/15

Y1 - 2020/6/15

N2 - We discuss two types of quantum mechanical models that couple large numbers of Majorana fermions and have orthogonal symmetry groups. In models of vector type, only one of the symmetry groups has a large rank. The large N limit is taken keeping gN=λ fixed, where g multiplies the quartic Hamiltonian. We introduce a simple model with O(N)×SO(4) symmetry, whose energies are expressed in terms of the quadratic Casimirs of the symmetry groups. This model may be deformed so that the symmetry is O(N)×O(2)2, and the Hamiltonian reduces to that studied in [I. R. Klebanov et al., Phys. Rev. D 97, 106023 (2018)PRVDAQ2470-001010.1103/PhysRevD.97.106023]. We find analytic expressions for the large N density of states and free energy. In both vector models, the large N density of states varies approximately as e-|E|/λ for a wide range of energies. This gives rise to critical behavior as the temperature approaches the Hagedorn temperature TH=λ. In the formal large N limit, the specific heat blows up as (TH-T)-2, which implies that TH is the limiting temperature. However, at any finite N, it is possible to reach arbitrarily large temperatures. Thus, the finite N effects smooth out the Hagedorn transition. We also study models of matrix type, which have two O(N) symmetry groups with large rank. An example is provided by the Majorana matrix model with O(N)2×O(2) symmetry, which was studied in Klebanov et al. In contrast with the vector models, the density of states is smooth and nearly Gaussian near the middle of the spectrum.

AB - We discuss two types of quantum mechanical models that couple large numbers of Majorana fermions and have orthogonal symmetry groups. In models of vector type, only one of the symmetry groups has a large rank. The large N limit is taken keeping gN=λ fixed, where g multiplies the quartic Hamiltonian. We introduce a simple model with O(N)×SO(4) symmetry, whose energies are expressed in terms of the quadratic Casimirs of the symmetry groups. This model may be deformed so that the symmetry is O(N)×O(2)2, and the Hamiltonian reduces to that studied in [I. R. Klebanov et al., Phys. Rev. D 97, 106023 (2018)PRVDAQ2470-001010.1103/PhysRevD.97.106023]. We find analytic expressions for the large N density of states and free energy. In both vector models, the large N density of states varies approximately as e-|E|/λ for a wide range of energies. This gives rise to critical behavior as the temperature approaches the Hagedorn temperature TH=λ. In the formal large N limit, the specific heat blows up as (TH-T)-2, which implies that TH is the limiting temperature. However, at any finite N, it is possible to reach arbitrarily large temperatures. Thus, the finite N effects smooth out the Hagedorn transition. We also study models of matrix type, which have two O(N) symmetry groups with large rank. An example is provided by the Majorana matrix model with O(N)2×O(2) symmetry, which was studied in Klebanov et al. In contrast with the vector models, the density of states is smooth and nearly Gaussian near the middle of the spectrum.

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U2 - 10.1103/PhysRevD.101.126002

DO - 10.1103/PhysRevD.101.126002

M3 - Article

AN - SCOPUS:85087007079

SN - 2470-0010

VL - 101

JO - Physical Review D

JF - Physical Review D

IS - 12

M1 - 126002

ER -