Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2002.02066 · Physical review. D/Physical review. D.

Hagedorn temperature in large N Majorana quantum mechanics

G. Gaitan, Igor R. Klebanov, Kiryl Pakrouski, P. N. Pallegar, Fedor K. Popov

Published Jun 2, 2020• 10 citations
arXivHTMLPDF

Abstract

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 $gN=\ensuremath{\lambda}$ fixed, where $g$ multiplies the quartic Hamiltonian. We introduce a simple model with $O(N)\ifmmode\times\else\texttimes\fi{}SO(4)$ 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 $O(N)\ifmmode\times\else\texttimes\fi{}O(2{)}^{2}$, and the Hamiltonian reduces to that studied in [I. R. Klebanov et al., Phys. Rev. D 97, 106023 (2018)]. 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 ${e}^{\ensuremath{-}|E|/\ensuremath{\lambda}}$ for a wide range of energies. This gives rise to critical behavior as the temperature approaches the Hagedorn temperature ${T}_{\mathrm{H}}=\ensuremath{\lambda}$. In the formal large $N$ limit, the specific heat blows up as $({T}_{H}\ensuremath{-}T{)}^{\ensuremath{-}2}$, which implies that ${T}_{H}$ 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 $O(N{)}^{2}\ifmmode\times\else\texttimes\fi{}O(2)$ 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.

DOI
10.1103/physrevd.101.126002

Related papers

Fermionic matrix models and bosonization

Matías N. Sempé, Guillermo A. Silva

We explore different limits of exactly solvable vector and matrix fermionic quantum mechanical models with quartic interactions at finite temperature. The models preserve a $U(1)\ifmmode\times\else\texttimes\fi{}SU(N)\ifmmode\times\else\texttimes\fi{}SU(L)$ symmetry at the classical level and we analyze them through bosonization techniques introducing scalar (singlet) and matrix (nonsinglet) bosonic fields. The bosonic path integral representations in the vector limits $(N,1)$ and $(1,L)$ are matched to fermionic Fock space Hamiltonians expressed in terms of quadratic Casimirs and some additional terms involving the Cartan subalgebra, which makes explicit the symmetries preserved by scalar and matrix bosonizations at the quantum level. For the case of nonsinglet bosonization we find an equivalence between the vector model and the Polychronakos-Frahm spin model. Using this relation, we compute the free energy. Finally, we compute the eigenvalue distribution in the large $N$, $L$ limit with $\ensuremath{\alpha}=\frac{L}{N}$ fixed. The model displays a third order phase transition as we vary the temperature which, in the $\ensuremath{\alpha}\ensuremath{\gg}1$ limit, can be characterized analytically. We conclude finding the critical curve in the parameter space were the eigenvalue distribution transitions from single to double cut.

arXiv:2112.14270•Physical review. D/Physical review. D.•Nov 10, 2022•0 citations

Spectra of eigenstates in fermionic tensor quantum mechanics

Igor R. Klebanov, Alexey Milekhin, Fedor K. Popov +1 more

We study the $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}O({N}_{3})$ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks ${N}_{i}$ are all equal, this model has a large $N$ limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of group invariant states for any set of ${N}_{i}$. It is non-vanishing only when each ${N}_{i}$ is even. For equal ranks the number of singlets exhibits rapid growth with $N$: it jumps from 36 in the $O(4{)}^{3}$ model to 595 354 780 in the $O(6{)}^{3}$ model. We derive bounds on the values of energy, which show that they scale at most as ${N}^{3}$ in the large $N$ limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order $1/N$. For ${N}_{3}=1$ the tensor model reduces to $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})$ fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with $SU({N}_{1})\ifmmode\times\else\texttimes\fi{}SU({N}_{2})\ifmmode\times\else\texttimes\fi{}U(1)$ symmetry. Finally, we study the ${N}_{3}=2$ case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}U(1)$. All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard 't Hooft large $N$ limits where the ground state energies are of order ${N}^{2}$, while the energy gaps are of order 1.

arXiv:1802.10263•Physical review. D/Physical review. D.•May 31, 2018•51 citations

Spectrum of Majorana Quantum Mechanics with O(4)3 Symmetry

Kiryl Pakrouski, Igor R. Klebanov, Fedor K. Popov +1 more

We study the quantum mechanics of three-index Majorana fermions ψ^{abc} governed by a quartic Hamiltonian with O(N)^{3} symmetry. Similarly to the Sachdev-Ye-Kitaev model, this tensor model has a solvable large-N limit dominated by the melonic diagrams. For N=4 the total number of states is 2^{32}, but they naturally break up into distinct sectors according to the charges under the U(1)×U(1) Cartan subgroup of one of the O(4) groups. The biggest sector has vanishing charges and contains over 165 million states. Using a Lanczos algorithm, we determine the spectrum of the low-lying states in this and other sectors. We find that the absolute ground state is nondegenerate. If the SO(4)^{3} symmetry is gauged, it is known from earlier work that the model has 36 states and a residual discrete symmetry. We study the discrete symmetry group in detail; it gives rise to degeneracies of some of the gauge singlet energies. We find all the gauge singlet energies numerically and use the results to propose exact analytic expressions for them.

arXiv:1808.07455•Physical Review Letters•Jan 10, 2019•27 citations

Notes on melonic O(N)q−1 tensor models

Sayantan Choudhury, Anshuman Dey, Indranil Halder +3 more

Abstract It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group O(N) q−1 agrees with the large N limit of the SYK model. In these notes we investigate aspects of the dynamics of the O(N) q−1 theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has $$ \left(q-1\right)\frac{N^2}{2} $$ q − 1 N 2 2 new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like E ln E (i.e. faster than Hagedorn) up to energies of order N 2. The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order 1/ln N. We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in J/m.

arXiv:1707.09352•Journal of High Energy Physics•Jun 1, 2018•92 citations

Majorana scars as group singlets

Zimo Sun, Fedor K. Popov, Igor R. Klebanov +1 more

In some quantum many-body systems, the Hilbert space breaks up into a large ergodic sector and a much smaller scar subspace. It has been suggested [K. Pakrouski et al., Phys. Rev. Lett. 125, 230602 (2020)] that the two sectors may be distinguished by their transformation properties under a large group whose rank grows with the system size (it is not a symmetry of the Hamiltonian). The quantum many-body scars are invariant under this group, while all other states are not. Here we apply this idea to lattice systems containing $M$ Majorana fermions per site. The Hilbert space for $N$ sites may be decomposed under the action of the $\text{SO}(N)\ifmmode\times\else\texttimes\fi{}\text{SO}(M)$ group, and the scars are the $\text{SO}(N)$ singlets. For any even $M$ there are two families of scars. One of them, which we call the $\ensuremath{\eta}$ states, is symmetric under the group $\mathrm{O}(N)$. The other, the $\ensuremath{\zeta}$ states, has the $\text{SO}(N)$ invariance. For $M=4$, where our construction reduces to spin-$1/2$ fermions on a lattice with local interactions, the former family are the $N+1\phantom{\rule{4pt}{0ex}}\ensuremath{\eta}$-pairing states, while the latter are the $N+1$ states of maximum spin. We generalize this construction to $M>4$. For $M=6$ we exhibit explicit formulas for the scar states and use them to calculate the bipartite entanglement entropy analytically. For large $N$, it grows logarithmically with the subsystem size. We present a general argument that any group-invariant scars should have the entanglement entropy that is parametrically smaller than that of typical states. The energies of the scars we find are not equidistant in general but can be made so by choosing Hamiltonian parameters. For $M>6$ we find that with local Hamiltonians the scars typically have certain degeneracies. The scar spectrum can be made ergodic by adding a nonlocal interaction term. We derive the dimension of each scar family and show the scars could have a large contribution to the density of states for small $N$.

arXiv:2212.11914•Physical Review Research•Dec 6, 2023•14 citations

Quantum critical scaling and the Gross-Neveu model in 2+1 dimensions

Hassan Chamati, N. S. Tonchev

The quantum critical behavior of the (2 + 1)-dimensional Gross-Neveu model in the vicinity of its zero-temperature critical point is considered.The model is known to be renormalisable in the large-N limit, which offers the possibility to obtain expressions for various thermodynamic functions in closed form.We have used the concept of finite-size scaling to extract information about the leading temperature behavior of the free energy and the mass term, defined by the fermionic condensate and determined the crossover lines in the coupling (g)-temperature (T ) plane.These are given by T ∼ |g -gc|, where gc denotes the critical coupling at zero temperature.According to our analysis no spontaneous symmetry breaking survives at finite temperature.We have found that the leading temperature behavior of the fermionic condensate is proportional to the temperature with the critical amplitude √ 5 3 π.The scaling function of the singular part of the free energy is found to exhibit a maximum at ln 2 2π corresponding to one of the crossover lines.The critical amplitude of the singular part of the free energy is given by the universal number 1 3 1 2π ζ(3) -Cl2 π 3 = -0.274543 . .., where ζ(z) and Cl2(z) are the Riemann zeta and Clausen's functions, respectively.Interpreted in terms of the thermodynamic Casimir effect, this result implies an attractive Casimir "force".This study is expected to be useful in shedding light on a broader class of four fermionic models.

arXiv:1112.4853•Europhysics Letters (EPL)•Aug 1, 2011•4 citations
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 NNN limit is taken keeping gN=\ensuremathλgN=\ensuremath{\lambda}gN=\ensuremathλ fixed, where ggg multiplies the quartic Hamiltonian. We introduce a simple model with O(N)\ifmmode×\else\texttimes\fiSO(4)O(N)\ifmmode\times\else\texttimes\fi{}SO(4)O(N)\ifmmode×\else\texttimes\fiSO(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)\ifmmode×\else\texttimes\fiO(2)2O(N)\ifmmode\times\else\texttimes\fi{}O(2{)}^{2}O(N)\ifmmode×\else\texttimes\fiO(2)2, and the Hamiltonian reduces to that studied in [I. R. Klebanov et al., Phys. Rev. D 97, 106023 (2018)]. We find analytic expressions for the large NNN density of states and free energy. In both vector models, the large NNN density of states varies approximately as e\ensuremath−∣E∣/\ensuremathλ{e}^{\ensuremath{-}|E|/\ensuremath{\lambda}}e\ensuremath−∣E∣/\ensuremathλ for a wide range of energies. This gives rise to critical behavior as the temperature approaches the Hagedorn temperature TH=\ensuremathλ{T}_{\mathrm{H}}=\ensuremath{\lambda}TH​=\ensuremathλ. In the formal large NNN limit, the specific heat blows up as (TH\ensuremath−T)\ensuremath−2({T}_{H}\ensuremath{-}T{)}^{\ensuremath{-}2}(TH​\ensuremath−T)\ensuremath−2, which implies that TH{T}_{H}TH​ is the limiting temperature. However, at any finite NNN, it is possible to reach arbitrarily large temperatures. Thus, the finite NNN effects smooth out the Hagedorn transition. We also study models of matrix type, which have two O(N)O(N)O(N) symmetry groups with large rank. An example is provided by the Majorana matrix model with O(N)2\ifmmode×\else\texttimes\fiO(2)O(N{)}^{2}\ifmmode\times\else\texttimes\fi{}O(2)O(N)2\ifmmode×\else\texttimes\fiO(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.
We explore different limits of exactly solvable vector and matrix fermionic quantum mechanical models with quartic interactions at finite temperature. The models preserve a $U(1)\ifmmode\times\else\texttimes\fi{}SU(N)\ifmmode\times\else\texttimes\fi{}SU(L)$ symmetry at the classical level and we analyze them through bosonization techniques introducing scalar (singlet) and matrix (nonsinglet) bosonic fields. The bosonic path integral representations in the vector limits (N,1)(N,1)(N,1) and (1,L)(1,L)(1,L) are matched to fermionic Fock space Hamiltonians expressed in terms of quadratic Casimirs and some additional terms involving the Cartan subalgebra, which makes explicit the symmetries preserved by scalar and matrix bosonizations at the quantum level. For the case of nonsinglet bosonization we find an equivalence between the vector model and the Polychronakos-Frahm spin model. Using this relation, we compute the free energy. Finally, we compute the eigenvalue distribution in the large NNN, LLL limit with $\ensuremath{\alpha}=\frac{L}{N}$ fixed. The model displays a third order phase transition as we vary the temperature which, in the $\ensuremath{\alpha}\ensuremath{\gg}1$ limit, can be characterized analytically. We conclude finding the critical curve in the parameter space were the eigenvalue distribution transitions from single to double cut.
We study the $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}O({N}_{3})$ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks Ni{N}_{i}Ni​ are all equal, this model has a large NNN limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of group invariant states for any set of Ni{N}_{i}Ni​. It is non-vanishing only when each Ni{N}_{i}Ni​ is even. For equal ranks the number of singlets exhibits rapid growth with NNN: it jumps from 36 in the O(4)3O(4{)}^{3}O(4)3 model to 595 354 780 in the O(6)3O(6{)}^{3}O(6)3 model. We derive bounds on the values of energy, which show that they scale at most as N3{N}^{3}N3 in the large NNN limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order 1/N1/N1/N. For N3=1{N}_{3}=1N3​=1 the tensor model reduces to $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})$ fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with $SU({N}_{1})\ifmmode\times\else\texttimes\fi{}SU({N}_{2})\ifmmode\times\else\texttimes\fi{}U(1)$ symmetry. Finally, we study the N3=2{N}_{3}=2N3​=2 case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}U(1)$. All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard 't Hooft large NNN limits where the ground state energies are of order N2{N}^{2}N2, while the energy gaps are of order 1.
We study the quantum mechanics of three-index Majorana fermions ψ^{abc} governed by a quartic Hamiltonian with O(N)^{3} symmetry. Similarly to the Sachdev-Ye-Kitaev model, this tensor model has a solvable large-N limit dominated by the melonic diagrams. For N=4 the total number of states is 2^{32}, but they naturally break up into distinct sectors according to the charges under the U(1)×U(1) Cartan subgroup of one of the O(4) groups. The biggest sector has vanishing charges and contains over 165 million states. Using a Lanczos algorithm, we determine the spectrum of the low-lying states in this and other sectors. We find that the absolute ground state is nondegenerate. If the SO(4)^{3} symmetry is gauged, it is known from earlier work that the model has 36 states and a residual discrete symmetry. We study the discrete symmetry group in detail; it gives rise to degeneracies of some of the gauge singlet energies. We find all the gauge singlet energies numerically and use the results to propose exact analytic expressions for them.
Abstract It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group O(N) q−1 agrees with the large N limit of the SYK model. In these notes we investigate aspects of the dynamics of the O(N) q−1 theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has (q−1)N22 \left(q-1\right)\frac{N^2}{2} (q−1)2N2​ q − 1 N 2 2 new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like E ln E (i.e. faster than Hagedorn) up to energies of order N 2. The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order 1/ln N. We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in J/m.
In some quantum many-body systems, the Hilbert space breaks up into a large ergodic sector and a much smaller scar subspace. It has been suggested [K. Pakrouski et al., Phys. Rev. Lett. 125, 230602 (2020)] that the two sectors may be distinguished by their transformation properties under a large group whose rank grows with the system size (it is not a symmetry of the Hamiltonian). The quantum many-body scars are invariant under this group, while all other states are not. Here we apply this idea to lattice systems containing MMM Majorana fermions per site. The Hilbert space for NNN sites may be decomposed under the action of the $\text{SO}(N)\ifmmode\times\else\texttimes\fi{}\text{SO}(M)$ group, and the scars are the SO(N)\text{SO}(N)SO(N) singlets. For any even MMM there are two families of scars. One of them, which we call the $\ensuremath{\eta}$ states, is symmetric under the group O(N)\mathrm{O}(N)O(N). The other, the $\ensuremath{\zeta}$ states, has the SO(N)\text{SO}(N)SO(N) invariance. For M=4M=4M=4, where our construction reduces to spin-1/21/21/2 fermions on a lattice with local interactions, the former family are the $N+1\phantom{\rule{4pt}{0ex}}\ensuremath{\eta}$-pairing states, while the latter are the N+1N+1N+1 states of maximum spin. We generalize this construction to M>4M>4M>4. For M=6M=6M=6 we exhibit explicit formulas for the scar states and use them to calculate the bipartite entanglement entropy analytically. For large NNN, it grows logarithmically with the subsystem size. We present a general argument that any group-invariant scars should have the entanglement entropy that is parametrically smaller than that of typical states. The energies of the scars we find are not equidistant in general but can be made so by choosing Hamiltonian parameters. For M>6M>6M>6 we find that with local Hamiltonians the scars typically have certain degeneracies. The scar spectrum can be made ergodic by adding a nonlocal interaction term. We derive the dimension of each scar family and show the scars could have a large contribution to the density of states for small NNN.
The quantum critical behavior of the (2 + 1)-dimensional Gross-Neveu model in the vicinity of its zero-temperature critical point is considered.The model is known to be renormalisable in the large-N limit, which offers the possibility to obtain expressions for various thermodynamic functions in closed form.We have used the concept of finite-size scaling to extract information about the leading temperature behavior of the free energy and the mass term, defined by the fermionic condensate and determined the crossover lines in the coupling (g)-temperature (T ) plane.These are given by T ∼ |g -gc|, where gc denotes the critical coupling at zero temperature.According to our analysis no spontaneous symmetry breaking survives at finite temperature.We have found that the leading temperature behavior of the fermionic condensate is proportional to the temperature with the critical amplitude √ 5 3 π.The scaling function of the singular part of the free energy is found to exhibit a maximum at ln 2 2π corresponding to one of the crossover lines.The critical amplitude of the singular part of the free energy is given by the universal number 1 3 1 2π ζ(3) -Cl2 π 3 = -0.274543 . .., where ζ(z) and Cl2(z) are the Riemann zeta and Clausen's functions, respectively.Interpreted in terms of the thermodynamic Casimir effect, this result implies an attractive Casimir "force".This study is expected to be useful in shedding light on a broader class of four fermionic models.