Spontaneous Breaking of $U(1)$ Symmetry in Coupled Complex SYK Models

As shown in [1], two copies of the large $N$ Majorana SYK model can produce spontaneous breaking of a $Z_2$ symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the $U(1) \times U(1)$ symmetry. We also present a tensor counterpart of this coupled model. When the coefficient $\alpha$ of the quartic term lies in a certain range, the coupled large $N$ theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of $\alpha$. We show that the operator $c_{1i}^\dagger c_{2i}$, which is charged under the axial $U(1)$ symmetry, acquires a complex dimension outside of the line of fixed points. We derive the large $N$ Dyson-Schwinger equations and show that, outside the fixed line, this $U(1)$ symmetry is spontaneously broken at low temperatures because this operator acquires an expectation value. We support these findings by exact diagonalizations extrapolated to large $N$.


Introduction and summary
There has been a great deal of interest in the fermionic quantum mechanical models which are exactly solvable in the large N limit because they are dominated by a special class of Feynman diagrams, which are called melonic [2]. Perhaps the simplest such model is the Majorana SYK model consisting of a large number of Majorana fermions with random quartic interactions [3,4]. Quantum mechanical models of this type have non-random tensor counterparts [5,6], which have continuous symmetry groups (for reviews of the melonic models see [7][8][9][10][11][12][13][14]). Both the random and non-random quantum mechanical models are solvable via the same melonic Dyson-Schwinger (DS) equations [4,6,[15][16][17][18], which indicate that the model is nearly conformal. One can obtain richer dynamics when more than one Majorana SYK or tensor models are coupled [1,[19][20][21]. In particular, when two such models are coupled by certain quartic interactions with a coefficient α, one finds a line of fixed points when α is positive, while a gapped Z 2 symmetry breaking phase appears when α is negative [1].
In this paper we make further progress in this direction by obtaining similar coupled models where a U (1) symmetry is broken spontaneously in the large N limit. Our starting point is the complex SYK model [22][23][24][25] (see also the earlier work [26,27]), which has a U (1) global symmetry. When two such models are coupled together by a quartic interaction preserving the U (1) × U (1) symmetry, we find that it is possible to break one of the U (1) symmetries spontaneously. The phase where the U (1) symmetry is broken by a VEV of operator c † 1i c 2i is found for α < 0 and α > 1. In contrast with the breaking of discrete symmetry in the coupled Majorana SYK model [1], there is no gap in the full large N spectrum due to the Nambu-Goldstone phenomenon. It manifests itself in splittings of order 1/N between the lowest states in different charge sectors.
However, some specific charge sectors exhibit gaps of order 1 above the ground state.
We also exhibit a tensor counterpart of the coupled random model (1.1) which consists of two coupled complex tensor models. The basic such model with SU (N ) 2 × O(N ) × U (1) symmetry was introduced in [6], and the two are coupled by an interaction which preserves the SU (N ) 2 × O(N ) × U (1) 2 symmetry. 1 At the special coupling α = 1/4, the U (1) × U (1) symmetry is enhanced to U (2) ∼ 1 The meaning of N in the tensor models is different from that in the SYK models.
U (1) × SU (2), and the Hamiltonian (1.1) may be written compactly as where there is a sum over σ, σ = 1, 2. This is equal to the quartic term in the model of [28], which was argued to provide a description of quantum dots with irregular boundaries. In (1.2) the U (1) is the usual charge symmetry, while the enhanced SU (2) symmetry models the physical spin; we may think of σ as labeling the two spin states, up and down.
Other recently introduced models [29,32,33] include random quartic couplings, as well as the non-random double-trace operator OO † , where O is a "Cooper pair operator" O ∼ c i↑ c i↓ .
The models we study in this paper are somehwat different, and they appear to be the first examples of manifestly melonic theories where the spontaneous breaking of U (1) symmetry can be established through analysis of the exact large N Dyson-Schwinger equations.
The structure of the paper is as follows. In section 2 we introduce some melonic models with U (1) × U (1) symmetry. They include a pair of coupled complex SYK models with Hamiltonian (1.1), as well as the tensor counterpart of this model with Hamiltonian (2.11).
In section 3 we discuss the symmetric saddle point of the large N effective action, as well as fluctuations around it. There is a range 0 ≤ α ≤ 1 where the symmetric saddle point is stable, while outside this fixed line a fermion bilinear operator, c † 1i c 2i , acquires a complex scaling dimension. In section 4 we find a more general solution of the Dyson-Schwinger equations, which contains the off-diagonal Green's function G 12 . It is stable outside the fixed line and indicates that the operator c † 1i c 2i acquires an expectation value. This phase of the theory is characterized by the exponential fall-off of Green's functions at low temperatures. In section 5 we discuss the low-energy effective action in this phase and calculate the compressibility for the broken U (1) degree of freedom. In section 6 we support some of these results by Exact Diagonalizations at accessible values of N . Extrapolating the ground state energies and compressibilities to large N , we obtain good agreement with some of the results obtained using the DS equations. In section 7 we present results for compressibilities at the special value α = 1/4 where the model has U (2) symmetry. Some additional details can be found in the Appendices.
As we were about to submit this paper to arXiv, we noticed the new paper [37]  2 Melonic models with U (1) × U (1) symmetry In this section we introduce some melonic models with quartic Hamiltonians, which possess U (1)×U (1) symmetry. The first model with Hamiltonian (1.1) consists of two copies of complex SYK model with a marginal U (1) × U (1) preserving interaction containing a dimensionless coupling, α. We also formulate its tensor counterpart which has SU (N ) 2 ×O(N )×U (1) 2 symmetry; it has the same Dyson-Schwinger equations as the random model.

Two coupled complex SYK models
Consider two sets of N complex fermions, c σi , where σ = 1, 2 and i = 1, . . . , N : The Hamiltonian coupling them is (1.1), where J ij,kl is the random Gaussian complex tensor with zero mean J ij,kl = 0; it satisfies J ij,kl = J * kl,ij in order for the Hamiltonian to be Hermitian. We also assume anti-symmetry in the first and second pairs of indices: J ij,kl = The corresponding conserved charges are Both Q + and Q − take integer values ranging from −N to N , with the constraint that Q + +Q − is even for N even and odd for N odd. 2 The Hamiltonian also has the Z 4 symmetry which is analogous to the Z 4 symmetry which played an important role in [1]. Another important symmetry is the particle-hole symmetry In order to make the Hamiltonian invariant under this symmetry for general α, we have to add to it certain quadratic and c-number terms which are exhibited in (A.4). 3 Note that, since the random coupling J ij,kl is complex, the U (1) + and U (1) − are on a different footing: the charge conjugation acting on the second flavor c 2i , is not a symmetry of the Hamiltonian (1.1). The U (1) + is the overall charge symmetry, while the "axial" symmetry U (1) − may be thought of as a spatial rotation around the third axis.
We will show that, for α < 0, the U (1) − may be broken spontaneously in the large N limit, but the charge symmetry U (1) + remains unbroken. Holographically, the U (1) − has a simple physical meaning: a holographic state charged under U (1) − corresponds to bulk solutions with an electric field turned on.

Tensor counterpart of the random model
Let us recall that the tensor counterpart of the standard complex SYK model [23,25] is given by the tensor model with Hamiltonian [6,12] h = gψ a 1 b 1 c 1ψ a 2 b 1 c 2 ψ a 1 b 2 c 2 ψ a 2 b 2 c 1 .
Under interchange of the two SU (N ) groups the Hamiltonian changes sign, and we have chosen the coupling term multiplied by α to preserve this discrete symmetry. The U (1)×U (1) symmetry acts analogously to that in the random model, (2.12) The Hamiltonian is also symmetric under the π/2 rotation ψ abc 1 → ψ abc 2 , ψ abc 2 → −ψ abc 1 .
In the tensor model ( are not allowed (in the random model, these operators do not receive ladder corrections). The symmetries do allow correlators of the formψ abc σ ∂ m t ψ abc σ , and their large N scaling dimensions are non-trivial. We will determine their values as functions of α in the next section, and show that one of them is complex for α < 0 and α > 1.

Scaling Dimensions of Fermion Bilinears
First let us study the large N saddle point where so that the U (1) + × U (1) − symmetry is preserved. Next it is reasonable to assume that where G(τ ) is the particle-hole symmetric Green's function, so G(−τ ) = −G(τ ). And we obtain which is the standard SYK Dyson-Schwinger (DS) equations with J = J √ 1 + 8α 2 .
Now we consider the bilinear spectrum at the nearly conformal saddle 3.2. They can be obtained by considering the melonic Bethe-Salpeter equations for the three point functions.
between elementary fermions c † σi , c σ i and a primary operator O ). Note that operators with non-zero U (1) + charge do not receive ladder correction in the large N limit due to J ij,kl being complex.
It is convenient to write down the explicit forms of the primary operators {O 0,0 m, For (0, 0) operators the scaling dimensions are determined by the following matrix: where we define K c as the conformal kernel of a single SYK/tensor model with Majorana fermions: which has eigenvalues in the anti-symmetric and symmetric sectors as g a (h), 3g s (h), with and For the (0, ±2) operators, they have the same anomalous dimensions determined by .
The series of scaling dimensions coming from solving g a (h) = 1 and g s (h) = 1 are the same as those found in a single complex SYK model or the SU (N ) 2 × O(N ) × U (1) tensor model [6].
Thus, for any α = 1 4 , there are two h = 1 modes corresponding to the U (1) × U (1) symmetry.  11) which are proportional to the generators of the U (2) symmetry 1 2 c † si σ a ss c s i .
ln contrast to the coupled Majorana SYK model [1], the large N operator spectrum (3.9) does not exhibit a duality symmetry. A duality (4.7 can be explored at level of DS equations after assuming certain symmetries on the correlators, but fluctuations not obeying such symmetries prevent this duality from being exact. For example, the theory at α = 1 is not equivalent to that at α = 0. For α = 1, we note that the operator O 0,0 2829. Since this lies in the range 1 < h < 3 2 , the conformal solution might not be described by a Schwarzian theory [39]. In fact, O 0,0 1,− has scaling dimension in this range when α > 3 8 .
For α < 0 or α > 1 the nearly conformal phase becomes unstable because the scaling dimension of operators O 0,±2 0 becomes complex. The plot of its imaginary part as a function of α is in fig. 1. We note that it reaches its maximum when α = −1/2. The antisymmetric sector cannot have such an instability for any α since −1/3

General Dyson-Schwinger equations and their numerical solution
In this section we study the DS equations more generally and show that, for α < 0 or α > 1, the solution with lowest free energy breaks the U (1) − symmetry. These equations may be obtained by varying the effective action (2.8). The first series is For the second series we find The equation for Σ 22 (τ 12 ) is obtained from Σ 11 (τ 12 ) by G 11 ↔ G 22 , and that for Σ 21 (τ 12 ) is obtained from Σ 12 (τ 12 ) by G 12 ↔ G 21 . In Appendix C we show how to derive these equations diagrammatically in both the coupled SYK and tensor models.
Combining this with G * 12 (τ ) = G 21 (τ ), we see that G 12 is purely imaginary. Using also (4.3), we find that G 12 (τ ) = G 12 (−τ ). Therefore, similarly to [1], we have to solve for only two functions: an odd real one, G 11 (τ ) = G 22 (τ ), and an even imaginary one, G 12 (τ ). The equations determining these two functions are They are very similar to the equations derived in [1]; the functions of α are somewhat different, but they again demonstrate changes of behavior at α = 0 and 1. The solutions to be a VEV of c 1i ∂ τ c † 2i . However, the latter is unlikely to appear dynamically. Therefore, the interchange symmetry does not appear to be realized. these equations may be obtained similarly to those in [1], and they are plotted in fig. 2.
We note that there is a duality symmetry of (4.6): these equations are invariant under However, this is not a symmetry of the theory even in the large N limit: neither (4.4), nor the bilinear spectrum (3.9) respect it.
Due to the underlying U (1) − symmetry, there is a continuous family of solutions obtained from these ones through the transformation G 12 (τ ) → e iφ G 12 (τ ). If we don't a priori assume the Z 4 symmetry (2.4), we find that the general numerical algorithm typically converges to a solution of this form with some phase φ. We note that such a solution has a modified discrete symmetry c 1j → e −iφ c 2j , c 2j → −e iφ c 1j .
Let us calculate the expectation values of the U (1) × U (1) charges. After introducing a point splitting regulator and writing it follows that Since for the solution in fig.2 G 11 (τ ) = G 22 (τ ) has the symmetry G 11 (τ ) = G 11 (β − τ ), we see that To see this, consider inserting a complete set of states where σ, σ ranges from 1 to 2. In order for the matrix elements to be non-vanishing, |n must have Q + = 1. Using the numerical solutions to DS equations, extrapolated to large βJ, we have plotted in fig. 3 the quantity ∆E from (4.11).
Given the DS solution, we can also calculate the ground state energy via In momentum space this is given by (4.14) We find good agreement between the DS computation of the ground state energy and the exact diagonalization results, as summarized in fig. 11.  Similarly to [1], for fixed α we observe a second-order phase transition from the U (1) symmetric phase to U (1) broken phase. We numerically observe that S/N approaches zero, rather than a finite number, as β → ∞. This may be explained by the U (1) sigma model, where one expects S 0 ∼ log N instead of powers in N.

Charge compressibility and the sigma model
Since the U (1) − symmetry is spontaneously broken for α < 0, we expect the presence of a gapless Goldstone mode. It arise from the degeneracy between ground states in sectors with different values of the charge Q − , which emerges in the large N limit. The expected action for the Goldstone modes is the U (1) sigma model action: where the coefficient K − ∼ O(1) in the large N limit, is the zero-temperature charge compressibility for U (1) − charge.
Let us emphasize that this U (1) − sigma model has a completely different origin from U (1) sigma model arising in the complex SYK model, which was recently discussed in detail in [25]. In the later case, the physics is similar to the conventional SYK model: there is an approximate conformal symmetry in the IR, the Schwartzian effective action, zero- Assuming that in the range 0 < α < 1 the solution is given by standard near-conformal SYK saddle, so there are no anomalous VEVs, we essentially have two non-interacting complex fermions. We then find that we have two sigma-models, for U (1) ± with compressibilities: where K cSY K ≈ 1.04 is the compressibility of a single complex SYK model [25]. The factor of two comes from having two fermions and the square root comes from renormalization of J by non-zero α, (3.2). Let us point out though that at α = 1/4, the U (1) − symmetry is enhanced to SU (2). We will discuss this case separately in section 7.
In the case of spontaneously broken U (1) − symmetry, the physics is different. The solutions of the Dyson-Schwinger equations that we have found for α < 0 do not have a conformal form.
Therefore, there is no approximate reparametrization symmetry or Schwartzian effective action. At zero temperature the entropy is zero, and the U (1) − symmetry is spontaneously broken. So, in the large N limit, the action (5.1) is a conventional Nambu-Goldstone mode action.
On these grounds, we do not expect to have a sigma model for U (1) + symmetry, since it is unbroken. Therefore, the splittings between sectors with different values of Q + should not vanish in the large N limit. This implies that the compressibility K + defined as dQ + /dµ + is zero, so small chemical potential does not generate non-zero charge. We will see this in the large N DS equations momentarily. In the exact diagonalization at finite N , this manifests in the fact that the energy dependence on Q + is not close to quadratic.
Let us return to the U (1) − symmetry and compute the corresponding compressibility K − .
It can be found in three ways: First of all, it is the derivative of the charge with respect to the chemical potential: Secondly, it is related to the grand canonical thermodynamical potential Ω as and finally the action (5.1) can be quantized leading to the spectrum: It would be convenient for us to find K − numerically by introducing a chemical potential into large N Dyson-Schwinger equations and fitting the numerical result for Ω using eq.
(5.4). In fact, to double check our results, we will introduce chemical potentials µ − and µ + for U (1) − and U (1) + and fit Ω with Since U (1) + is unbroken, we expect that K + = K mix = 0. In other words, low energy states have the same charge under U (1) + and the gap to states with different U (1) + charges is big.
The result is presented in Figure 6. We indeed see that K + = K mix = 0.

Results from Exact Diagonalizations
In this section we will study the energy spectra for accessible values of N . We will use the particle-hole symmetric version of the Hamiltonian, given in (A.4). We have generated multiple random samples of the Hamiltonain, which allow us to study various averaged quantities as functions of α and N .

Evidence for Symmetry Breaking
For α < 0 and α > 1, the large N DS equations indicate that U (1) − symmetry is spontaneously broken. In these ranges of α, the absolute ground state appears in the sectors with Q + = 0 and the lowest possible value of |Q − |, which is |Q − | = 0 for even N and |Q − | = 1 for odd N . This means that, for odd N , there are two degenerate ground states, which have Q − = ±1, and their mixture admits an expectation value of operator c † 1i c 2i already at finite N . At any finite even N we cannot see the spontaneous symmetry breaking, but it appears in the large N limit due to the degeneracy of ground states with Q + = 0 and different values of Q − . In fig. 8 we exhibit the spectra in two different charge sectors for N = 10. 5 A characteristic quantity in the broken symmetry phase is the gap between the first excited state and the ground state: such a gap is observed in the sectors with Q + = 0. For example, in the (Q + , Q − ) = (0, 1) sectors we find for α = −1/2 that the average gaps above the ground state are ≈ 0.440, 0.437, 0.473 for N = 7, 9, 11, respectively. These results suggest that the gap is non-vanishing in the large N limit.
Similarly, there is a sizable difference between the ground state energies in sectors with different values of Q + . It is noticeably bigger than the difference between sectors with different values of Q − , which is expected to be of order 1/N . For example, for α = −1/2 and N = 10, we find In the sectors with Q + = 0, we expect the ground state energies to depend quadratically on Q − : where K − is the large N compressibility for the U (1) − degree of freedom. As can be seen in figs. 9 and 10, these quadratic fits work well, and B − (α) is approximately linear in N .   Another important quantity is the leading term in the ground state energy (6.2), A(α, N ), which is expected to grow linearly for large N . In fig. 11 we plot A(α, N ) for α = −0.5 and α = −0.2, and show the fits In fig, 12 we plot E 0 (α) = lim N →∞ E 0 (α)/N for a range of negative α. This shows good agreement with the corresponding calculation using DS equation as a function of α.

Line of Fixed Points
Along the fixed line 0 ≤ α ≤ 1 there is no symmetry breaking, and the large N spectrum is gapless in every charge sector. In fact, for such values of α, near the edge the density of state should behave as Along the fixed line, we expect the gaps to be of order 1/N for excitations of both the Q − and Q + charges, so that both U (1) − and U (1) + compressibilities are well-defined: For α = 0, B − = B + and, therefore, the compressibilities are equal: K + (0) ≈ K − (0) ≈ 2.08. Indeed, for α = 0 the Hamiltonian is simply a sum of two cSYK Hamiltonian with the common J ijkl , so that To compare with our normalizations, K cSYK = K + (0)/2 ≈ 1.04. Thus, our finding for α = 0 is in good agreement with the result K cSYK ≈ 1 from [25].
We have also done fits of the two large N compressibilities for α = 0.5 and 0.7. In these cases, K − is found to be significantly smaller than K + . This is in conflict with the DS calculations which gives equal values. This may be due to the slow convergence of the ED results to the large N limit. The DS formula K + (α) = K + (0)/ √ 1 + 8α 2 predicts the value Let us discuss the low-energy effective action for the U (2) symmetric theory. We expect that instead, of the U (1) + × U (1) − sigma model, we now have SU (2) × U (1) + . The low energy effective action for the SU (2) part is: where U is the matrix from SU (2) group. Previously, we obtained compressibilities from coupling to µ − chemical potential. Let us argue that this calculation does not change. Indeed, corrections ∝ N to the free energy depend on classical properties of this sigma-model, since we have a factor of N in front. Upon introducing a chemical potential to U (1) − subgroup of SU (2) we have to study the following action: Its contribution to the Gibbs potential is again −N K SU(2) µ 2 − /2. We find using the DS equations that However, the low energy spectrum is very different, as it involves quantizing the sigma model. Namely, now the excitations come in SU (2) multiplets with energies given by a quadratic Casimir of SU (2). Namely, for a multiplet with Q − /2 ∈ (−S, −S + 1, . . . , S) the energy is given by: Therefore, we find for large N : (2) S(S + 1) , (7.5) where S is the SU (2) spin. A priori, there are two different compressibilities, but our results indicate that they are equal in the large N limit, K + = K SU (2) . For even N , the unique ground state occurs in the Q + = S = 0. For odd N , there are two ground states: they are SU (2) singlets and have Q + = ±1.
In fig. 14 we show the fits of this dependence for the low-lying ground states. They work well and give approximately equal compressibilities K SU(2) ≈ K + ≈ 1.60. This value is in Figure 14: The dependence of ground state energy at α = 0.25 on SU (2) spin S at Q + = 0, 1, 2. We use the Ansatz E 0 ≈ A + B 2 (Q 2 + + 4S(S + 1)), and plot 1/B against N in the last plot to extrapolate the compressibility K SU(2) ≈ 1.60. good agreement with the formula (5.2) evaluated at α = 1/4.

A Particle-hole symmetry
For the single complex SYK model, the Hamiltonian which respects the particle-hole symmetry c i ↔ c † i , accompanied by J ijkl → J * ijkl , was given in [25] where A denotes total antisymmetrization: To make the Hamiltonian of the coupled model, (1.1), invariant under the full particle-hole symmetry (2.5), we have to add to it similar terms: This can also be written as The quadratic and c-number terms are subleading in N and thus are not important at large N. They can be important at small N, such as in the exact diagonalizations. We note that these terms vanish for α = −1/2, so that the original Hamiltonian (1.1) is automat- , because the DS equations hold up to arbitrary insertion as long as operators are not inserted at τ or 0. In order to not add more contact terms, the operator has to be inserted at ∞.
Therefore δG σσ (τ ) would correspond to a zero mode in the quadratic fluctuation, and the eigenvector dictates the form of the operator. Note in conformal theory, the 3 point functions between primaries are determined up to a constant where h is the scaling dimension of the operator O. In order for the three point function to be non-vanishing, the primary operator O is necessarily bilinear in the elementary fermions, and a O(N ) singlet. Therefore one can use this Ansatz to determine the bilinear operator dimension from the quadratic fluctuation. In the following we are going to omit the integrals over τ 1 , τ 2 for brevity.

D Analytical approximation
If we assume a particular phase such that the Z 4 symmetry 2.4 is preserved, the DS equations can be written using one function G = (G 11 + G 22 + i (G 12 − G 21 )) /2: where parameter k is related to α by: andJ is related to J in the standart formulation with G 11 and α bỹ Evaluating the convolution for τ > 0 yields: A 1 e −µτ + A 2+c e −(2+c)µτ + A 3c e −3cµτ (D.6) The A 1 , A 2+c and A 3c are easily computed functions of a, b, c and µ: A 1 = a 4 c 4(c + 1)µ + a 4 4(c + 1)µ − a 3 bk (c + 1)µ + a 2 b 2 k 2(c + 1)µ + ab 3 µ − 3cµ , (D.7) A 2+c = a 3 bk (c + 1)µ + a 2 b 2 k 2(c + 1)µ , (D.8) Terms e −2µτ , e −3cµτ are subdominant and were not present in the ansatz, so we can safely ignore them. Therefore we have a single equation: Let us assume that c > 4. Then we can again ignore the term e (1+2c)µτ . However, the term e 3µτ has to be zero. Therefore we have two equations: (D. 16) We see that our ansatz is consistent: we managed to eliminate all faster decaying terms.
Moreover, we have 4 unknown variables and only three equations. We will empose one extra condition: a + b = 1 .