Hagedorn Temperature in Large $N$ Majorana Quantum Mechanics

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 $gN=\lambda$ fixed, where $g$ multiplies the quartic Hamiltonian. We introduce a simple model with $O(N)\times SO(4)$ 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 $O(N)\times O(2)^2$, and the Hamiltonian reduces to that studied in arXiv:1802.10263. 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 $e^{-|E|/\lambda}$ for a wide range of energies. This gives rise to critical behavior as the temperature approaches the Hagedorn temperature $T_{\rm H} = \lambda$. In the formal large $N$ limit, the specific heat blows up as $(T_H- T)^{-2}$, 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 $O(N)^2\times O(2)$ symmetry, which was studied in arXiv:1802.10263. In contrast with the vector models, the density of states is smooth and nearly Gaussian near the middle of the spectrum.


Introduction and summary
Strongly interacting fermionic systems describe some of the most challenging and interesting problems in physics. For example, one of the big open questions in condensed matter physics is the microscopic description of the various phases observed in the high-temperature superconducting materials. Models relevant in this context [2][3][4] include the Hubbard [5,6] and t-J models [7]. The Hamiltonians of these models include the quadratic hopping terms for fermions on a lattice, as well as approximately local quartic interaction terms. The analysis of such models often begins with treating a quartic interaction term as a small perturbation. In the cases when such an expansion is not possible, for example, the fractional quantum Hall effect, one typically has to resort to numerical calculations.
Fortunately, there are also fermionic systems which can be solved analytically in the strongly interacting regime, when the number of degrees of freedom is sent to infinity. Such large N systems include the Sachdev-Ye-Kitaev (SYK) models [8][9][10][11][12][13] (see also the earlier work [14,15]). The SYK models have been studied extensively in the recent years; for reviews and recent progress, see [16][17][18].
The simplest of them, the so-called Majorana SYK model [9,13], has the Hamiltonian H = J ijkl ψ i ψ j ψ k ψ l , which describes a large number N SYK of Majorana fermions ψ i (we assume summation over repeated indices throughout this work). They have random quartic couplings J ijkl with appropriately chosen variance. A remarkable feature of this model is that, in the limit where N SYK → ∞, it becomes nearly conformal at low energies. The low-lying spectrum exhibits gaps which are exponentially small in N SYK . In further work, models consisting of coupled pairs of Majorana SYK models [19][20][21], as well as the SYK chain models [22,23], have produced a host of dynamical phenomena which include gapped phases and spontaneous symmetry breaking. In addition to the terms quartic in fermions, they can include quadratic terms which describe hopping between different SYK sites.
Another class of solvable large N fermionic models are those with degrees of freedom transforming as tensors under continuous symmetry groups [24,25] (for reviews, see [26,27]).
A simple example [25] is the O(N ) 3 symmetric quantum mechanics for N 3 Majorana fermions ψ abc . In these tensor models the interaction is disorder-free, so the standard rules of quantum mechanics apply. Interestingly, the large N limit is similar to that in the SYK model because in both classes of models the perturbative expansion is dominated by the "melonic" Feynman diagrams, which can be summed [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. Since the Hubbard and t-J models do not have any random couplings, the disorder-free tensor models may be viewed as their generalization, and it is interesting to investigate if they can incorporate some interesting physical effects in a solvable setting. One possibility is to interpret the three indices of the tensor ψ abc , where a, b, c = 1, . . . , N , as labeling the sites of a 3-dimensional cubic lattice [43]. Then the tensor models may perhaps be interpreted as non-local versions of the Hubbard model.
It is also natural to generalize the Majorana tensor model of [25] to the cases where the indices have different ranges: a = 1, . . . [1,44] (see also [37,45]). The traceless Hamiltonian of this model is [1,25] where {ψ abc , ψ a b c } = δ aa δ bb δ cc . If the ranks N i are sent to infinity with fixed ratios, then the perturbation theory is dominated by the melonic graphs. However, it is also interesting to consider the cases where one or two of the N i are not sent to infinity. Such models with The O(N ) 2 × O(2) model, which may be viewed as a complex fermionic matrix model [1], has the 't Hooft large N limit where all the planar diagrams contribute (similar fermionic matrix models were studied in [46,47]).
In this paper we will carry out further analysis of the fermionic vector and matrix models.
In particular, we study the large N densities of states ρ and analyze the resulting temperature dependence of the specific heat. In the matrix model case, the density of states is smooth and nearly Gaussian, which is a rather familiar behavior. In the large N vector models, we instead find a surprise: for a wide range of energies we find log ρ ≈ −|E|/λ plus slowly varying terms. The approximately exponential growth of the density of states, discussed long ago in the context of hadronic physics and string theory [48,49], leads to interesting behavior as the temperature approaches the Hagedorn temperature, T H = λ. In the Majorana vector models we indeed find critical behavior as the temperature is tuned to λ, with a sharp peak in the specific heat. In the formal large N limit, the specific heat blows up as ( This means that T H is the limiting temperature, and it is impossible to heat the system above it. However, at any finite N , no matter how large, the specific heat does not blow up, so it is possible to reach arbitrarily large temperatures. Thus, our model provides a demonstration of how the finite N effects can smooth the Hagedorn transition. In section 2, we study the O(N ) × O(2) 2 symmetric vector model. We find that the density of states exhibits exponential growth in a large range of energies, and match this with analytical results. In section 3 we study a related vector model, where the symmetry is enhanced to O(N ) × SO (4). In this case, we obtain simple closed-form expressions for the large N density of states, free energy, and specific heat. In section 4, we consider the fermionic matrix model with O(N ) 2 × O(2) symmetry and find that the spectrum now exhibits a nearly Gaussian distribution for sufficiently large N . In appendix A we study the structure of the Hilbert space of the above models, and derive the Cauchy identities from simple physical arguments. results of the appendix, has a simple decomposition in the irreducible representations of the where [µ] G stands for a representation of the group G described by the Young Tableaux µ.
In the Hilbert space of our model, the Young Tableaux of SO(N ) contains at most 2 columns and N/2 rows. In terms of fermions ψ aI , the Hamiltonian (1.1) may be rewritten as and can be used to split the lie algebra so(4) into the direct sum of the two su(2) algebras, which we have labeled by + and −, as follows: It is easy to see that both sets K + i and K − i comprise an SU (2) algebra, and thus the representations of the two SU (2) groups with spins Q + /2 and Q − /2, respectively, fully determine the representation of the SO(4) group. One can derive the following algebraic relation: where we have used that K + i 2 is the quadratic Casimir operator and we know its value in each of the representations of SU (2). It is also interesting to notice that from (2.4) we have This allows one to rewrite the Hamiltonian only in terms of the SO(4) representations. If we have a representation with SU (2) spins (Q + /2, Q − /2), then all eigenvectors with definite K ± 1 are the eigenvalues of Hamiltonian with energies The degeneracy of such a state is determined by the dimension of the corresponding SO(N ) representation. Because we know the structure of the Hilbert space (2.1), we can determine the complete structure of the spectrum. If we have a SO(N ) representation with a Young tableaux µ consisting of two columns of the length µ 1 ≥ µ 2 ≥ 0, the corresponding rep- From this one can see that each set of pairs of non-negative integers (Q + , Q − ) whose sum is constrained to take values N, N − 2, N − 4, . . . appears once. This formula allows us to study the density of states in the vicinity of the ground state and of E = 0.
The ground state (E 0 = −gN (N + 2)) corresponds to the choice of Q + = 0, Q − = N , thus q + ≡ 0 and the spectrum in its vicinity has the form, The states immediately above the ground state are labeled by q − and the gap between them is of the order g ∼ λ N . The next excited states correspond to the choice Q + > 0. The gap between such states and the ground state is of the order ∆E ∼ gN ∼ λ and is finite in the large N limit, but the dimension of the representation is of the order dim ∼ N Q + and diverges in the large N limit. Immediately above the ground state (δE ∼ λ, Q + = 0) the density of states may be approximated as On the other hand, near E = 0, the logarithm of the density of states exhibits an unusual cusp-like behavior shown in figure 1. Another remarkable feature is its approximately linear behavior for a large range of energies.
For |E|/λ of order 1, the dominant contributions come from the states with large charges In this regime we can apply the Stirling approximation to the factorials in To obtain the density of states in the large N limit, we introduce the variables This may be evaluated if we first perform the integrals over T ± = t 2 ± : The formula (2.13) is in good agreement with the numerical results (see figure 1). Expanding ρ(x) near x = 0 we see that which exhibits a singularity at x = 0: ρ (0) diverges, signaling a breakdown of the Gaussian approximation of the density of states. We also note that, for x 1, ρ(x) ∼ |x| 1 2 e −|x| . We can present an argument for why the density of states is not Gaussian near the origin.
The high temperature expansion of the free energy is: (2.15) The quantity on the right-hand side of (2.15) may be computed with the use of Feynman diagrams. For vector models, the "cactus" or "snail" diagrams, shown in figure 2, typically dominate in the large N limit [26,51]. However, in our problem they vanish due to the Majorana nature of the variables. Therefore, for any connected part, the trace begins with the subleading term It is easy to see that C 1 comes from the necklace diagrams in figure 2, which give where the factor of 1 k comes from the symmetries of the necklace diagrams. These necklace diagrams may be interpreted as trajectories of a particle propagating in one dimension.
Introducing the 't-Hooft coupling λ = gN and taking the large N limit while keeping λ finite, we calculate the free energy, (2.18) The inverse Laplace transformation with respect to β yields the density of states log ρ(E) ∼ a − |E| λ . From this one can derive that the distribution must have a Laplace-like form, and this agrees with the numerical results.
Let us review the physical effects of the approximately exponential behavior of ρ. In the canonical ensemble, the partition function as a function of inverse temperature β is  energies is The presence of the denominator produces a logarithmic term in the free energy, but it is cut off by the numerator before it diverges. It follows that the specific heat C = −T ∂ 2 F/∂T 2 may be approximated by where δẼ goes to infinity in the large N limit and the second term vanishes. Thus, for large enough N , there should be a clear peak in the specific heat. This simple analytic argument for the existence of a peak is supported by the numerical plots of specific heat shown in figure   3. For any finite N , the height of the peak in C is finite, so that it is possible to heat the system up to any temperature. However, in the formal large N limit, the specific heat blows  This is shown in figure 4.
Let us study this function more precisely, follow the procedure used in the previous section. We include the contributions of representations where Q ± ∼ √ N , and introduce variables x ± = Q ± / √ N . The energy is then given by E = λ x 2 + − x 2 − . Using the Stirling approximation for the factorials in (3.1), we find that the density of states is This integral can be evaluated in closed form: where K 1 is the modified Bessel function, and the normalization is such that ρ integrates to the total number of states, 2 2N . Plotting  shows that ρ (0) diverges. The reasons for this unusual behavior in the large N limit were discussed in the previous section. We also note that ρ ∼ |x| 1/2 e −|x| for |x| 1.
The approximation (3.3) can be used to get the large N limit of the free energy: up to an additive term linear in T . The specific heat diverges at the Hagedorn temperature Note that this is of order N 0 for T < T H , as usual for the Hagedorn transition. For a finite N , the divergence is cut off, but the peak is prominent; see figure 5.
We can write the Hamiltonian (1.2) in terms of the complex Dirac fermions by introducing the following operators: The Hamiltonian may be rewritten as where Q is the U (1) charge. This implies that the SU (N ) invariant states with Q = 0 must be in the spin N/2 representation of SU (2). Therefore, there are N + 1 such states. There are also two SU (N ) × SU (2) invariant states, which have Q = ±N . Thus, the total number of SU (N ) invariant states is N + 3.
We can generalize such a model to the case of O(N ) × SO(2M ) with the Hamiltonian The spectrum consists of half-integers running from E = − N 2 + q and the degeneracy deg(E) = N q corresponds to the representation of the fully antisymmetric tensors.

Fermionic matrix models
In this section we study the fermionic matrix models with O(N 1 )×O(N 2 )×O(2) symmetry [1]. They contain 2N 1 N 2 Majorana fermions that are coupled by the Hamiltonian The direct numerical diagonalization of this Hamiltonian is hampered by the exponential growth of the dimension of Hilbert space as 2 N 1 N 2 . For N 1 = N 2 = 6 it is ≈ 7 · 10 10 , while for N 1 = N 2 = 8 it is ≈ 2 · 10 18 states. For the former we were able to carry out Lanczos diagonalization giving the wave functions and energies of the lowest few states.
Fortunately, the Hamiltonian (4.1) may be expressed in terms of the U (1) charge Q, the Casimir operators of the SO(N i ) symmetry groups, as well as of the SU (N 1 ) group which acts on the spectrum [1]: This analytical expression allows us to proceed to higher values of N i . In general, all the energy eigenvalues are integers in units of g, but finding their degeneracies requires some  where λ = gN is the 't-Hooft coupling, which is held fixed as N → ∞. We find nice agreement, which is shown for N 1 = N 2 = 8 and N 1 = N 2 = 10 in figure 6 and for To demonstrate the validity of this approximation, let us compute Therefore we can use standard Feynman techniques with the propagator ψ ab ψ a b = 1 2 δ aa δ bb and H as an interaction vertex. Since H has the form of a single-trace operator in the large N limit, this product is dominated by the planar diagrams and moreover by the disconnected parts. From this point of view one can see that Then one can invert (4.4) and get that ρ(E) is the Gaussian distribution The second moment, σ 2 E , is easy to compute using the diagrammatic technique: To get the higher order corrections to the distribution function, we can continue calculating the energy moments, or we can instead simply compute the free energy and perform the inverse Laplace transformation to get the energy distribution. To be more precise, the free energy is defined as (4.8) This gives us a formula to compute F (β) as a sum of the connected diagrams with H as an interaction vertex β n tr (H n ) con = β 2 tr H 2 con + β 4 tr (H n ) con + . . .  The spectrum for N 1 = N 2 = 9. As one can see it has the same features as for N 1 = N 2 = 8 and N 1 = N 2 = 10, but there is no separation between the even and the odd energy sectors. It could indicate that this difference has a purely group theoretic explanation.
This integral can be calculated with the use of general diagrammatic technique, where iE is the source for the energy, tr(H 2 ) con is the propagator, and tr (H 4 ) con and the higher correlators are the vertices. By using these procedures we can compute the connected contribution.
It is easy to compute the leading contributions to the connected trace of H 4 , tr H 4 con. = tr H 4 − 3 tr H 2 2 con. = 8g 4 N 6 . (4.11) After that we can restore log ρ(E) = N 2 log 2 − 1 2 (4.12) Comparing this expression with the numerical data we find a nice agreement between these two formulas.
Let us note the splitting between the even and the odd energies, which is seen in figure   6 but absent in figure 7. These two sets of energies are distinguished by the value of The trace of this operator counts the difference between the number of these branches. The trace of this operator over the whole space can be computed via the representation theory and is equal to tr P C = 2 2N 2 −N +1 .
We can study the thermodynamic properties of the matrix model in a similar fashion as in the case of the vector models. The behavior of the system would be analogous to a system of the spins in an external magnetic field. The partition function is 14) and the heat capacity C is This behavior is nicely captured by the numerical results shown in figure 8. Note that the peak near T peak ∼ g ∼ λ N is due to the discreteness of the spectrum; it may be seen if we consider the contributions coming only from the ground state and the first excited state.
These relations respect the larger symmetry group U (N 1 ) a ×U (N 2 ) b , and could be considered as symmetries of the Hilbert space, in contrast to the Hamiltonian (4.1) which does not respect these symmetries. We can now try to decompose the Hilbert space in terms of the representations of these unitary groups using the character theory [52]. We notice that the generator of the U (N 1 ) a and U (N 2 ) b groups could be rewritten in the following form where T A,B aa are hermitian matrices and can be considered as elements of the u(N i ) algebra.
Then the operators J A,B T are the corresponding representations of these elements of the u(N i ) algebra. Hence, a general element of the U a (N 1 ) × U b (N 2 ) group, acting on the Hilbert space, Therefore we can compute the trace of this operator in the Hilbert space, and it is equal to the following: This gives the following formula for the character 2 cos One can see that this integral has the correct normalization, because if x a = y b = 0 we restore the dimension of the space and χ H = 2 N 1 N 2 as it should be. We can decompose this product in terms of the Schur polynomials, which are the characters of the irreducible representations of U (N i ). Fortunately, this problem is easily solved with the use of the dual where the λ is the Young Tableaux and λ T is the transpose. Therefore the Hilbert space has a very simple decomposition in terms of the U (N i ) groups. To each Young tableaux λ ⊂ (N N 2 1 ) with no more than N 1 columns and N 2 rows we assign only one U a (N 1 ) representation; this  [54] and he obtained the following result [55], where It is interesting to notice that if, instead of complex fermions c ab , we considered Majorana fermions ψ ab , we can compute the partition function to get the following character, i=a,j=b e ixa + e −ixa + e iy b + e −iy b . (A.10) We can deduce this structure heuristically. Note that, because of the Fermi-nature of each state λ of the O(2n) representation, we must include the correspondence λ ⊂ ((N 1 /2) (N 2 /2) ).
One can compute the dimension of all of these representations and find that it is equal to the full Hilbert space. This gives a new dual Cauchy identity for orthogonal Schur polynomials, λ⊂(n m ) o λ (x)o ((N 1 /2) (N 2 /2) /λ) (y) = i,j x i + x −1 i + y j + y −1 j .
(A. 16) Substituting these into the character of the Hilbert space we get As one can see, the representation of the one-dimensional fermions gives a very powerful tool for proving famous combinatorial equalities. It would be interesting to expand these ideas for other groups, say Sp(N ), and to generalize it for the case of MacDonald polynomials [53].