# Two indices Sachdev-Ye-Kitaev model

###### Abstract

We study the original Sachdev-Ye (SY) model in its Majorana fermion representation which can be called the two indices Sachdev-Ye-Kitaev (SYK) model. Its advantage over the original SY model in the complex fermion representation is that it need no local constraints, so a expansion can be more easily performed. Its advantage over the 4 indices SYK model is that it has only two site indices instead of four indices , so it may fit the bulk string theory better. By performing a expansion at , we show that a quantum spin liquid (QSL) state remains stable at a finite . The corrections are exactly marginal, so the system remains conformably invariant at any finite . The 4-point out of time correlation ( OTOC ) shows quantum chaos neither at at any finite , nor at at any finite . By looking at the replica off-diagonal channel, we find there is a quantum spin glass (QSG) instability at an exponentially suppressed temperature in . We work out a criterion for the two large numbers and to satisfy so that the QSG instability may be avoided. We speculate that at any finite , the quantum chaos appears at the order of , which is the subleading order in the expansion. When the quantum fluctuations at any finite are considered, from a general reparametrization symmetry breaking point of view, we argue that the effective action should still be described by the Schwarzian one, the OTOC shows maximal quantum chaos. This work may motivate future works to study the possible new gravity dual of the 2 indices SYK model.

1. Introduction. Sachdev-Ye(SY) SY () studied the random Heisenberg model with infinite-range interactions:

(1) |

where the random bond satisfies the Gaussian distribution .

In order to achieve some analytical results, SY generalized the to by introducing complex fermions : subject to the local constraint , then Eq.1 becomes:

(2) |

In the ( number of sites ) limit, followed by a limit in Eq.2, SY found a gapless conformably invariant quantum spin liquid (QSL) ground state. At zero temperature , the QSL has an extensive GPS entropy SY3 (); subir1 (); subir2 () in the limit followed by which is equal to the Bekenstein-Hawking (BH) entropy in Einstein gravity subir1 (); subir2 (); subir3 ().

In a series of talks in 2015, Kitaev Kittalk () simplified Sachdev-Ye model Eq.2 to an infinite range four-indices Majorana fermion interacting model, each has species:

(3) |

where also satisfies the Gaussian distribution with . By showing its possible maximal chaotic behaviour matching the feat of the quantum black holes, Kitaev suggested that the dimensional SYK model may have a gravity dual in asymptotic space. This speculation sparked great interests from both quantum gravity/string theory Pol (); Mald (); Gross (); sff1 (); liu1 (); liu2 (); superSYK (); randomsusy1 (); randomsusy2 (); u1zero (); tensor1 (); tensor2 (); tensor3 () and condensed matter/AMO community CSYKnum (); Rcft (); MBLSPT (); tran1 (); tran2 (); highSYK1 (); highSYK3 (); highcSYK (); longtime1 (); longtime2 (); rev (). Especially, Maldacena and Stanford did a systematic expansion Mald () on the SYK model. In the large limit, it leads to the same gapless QSL ground state as that in the SY model. If dropping the irrelevant time derivative term , the saddle point equation ( and also the action ) has the time re-parametrization invariance , however, the saddle point solution spontaneously breaks it to , leading to ”zero mode ” or Goldstone mode, while the irrelevant time derivative term explicitly breaks the re-parametrization symmetry and lifts the Goldstone mode to a pseudo-Goldstone mode whose quantum fluctuations can be described by the Schwarzian action in terms of re-parametrization. From the Schwarzian, at the order , they evaluated the 4 point out of time ordered correlation (OTOC) function at early times and extracted the Lyapunov exponent at a small finite temperature . It is maximally chaotic and saturates the upper bound of some classes of quantum systems bound1 (); bound2 (); bound3 (). This feat precisely matches that of quantum black holes in the Einstein gravity which are the fastest quantum information scramblers in the universe, therefore confirmed the Kitaev’s claim that the SYK model maybe dual to black holes in asymptotically , which is, in fact, nearly conformably invariant/nearly with a scalar dilaton ( ).

In this paper, we study the original Sachdev-Ye model in its Majorana fermion representation which can be called the two indices Sachdev-Ye-Kitaev model. Its advantage over the original SY model in the fermion representation Eq.2 is that it need no local constraints. Its advantage over the 4 indices SYK model Eq.3 is that it has only two site indices instead of four indices , so it may fit the bulk string theory better Pol () . After the limit was taken, the expansion can be easily performed due to the absence of the constraint. It may also be easily generalized to short-range interaction in any space dimension than the four indices SYK. By performing a expansion, we show that a quantum spin liquid (QSL) state remains stable at a finite . The corrections are exactly marginal, so they only change the values of the zero temperature entropy, the coefficient of the linear specific heat and the overall constants of all the 2 and 4 point correlation functions. The system remains conformably invariant at any finite . The 4-point out of time correlation ( OTOC ) shows quantum chaos neither at at any values of , nor at at any values of . Quantum chaos may only show up at a finite and a finite which can be explored by a expansion, followed by a expansion. The two large numbers and play very different roles, needs to be a large number to have a time window for quantum chaotic behaviours, therefore have a gravity dual in , however, also needs to be large enough to avoid the QSG phase, the expansion can only be used as a tool to evaluate the conformably invariant 2- or 4-point functions or thermodynamic quantities at any finite . By looking at the replica off-diagonal channel, we find there is a quantum spin glass instability at an exponentially suppressed temperature . We show that the QSG instability may be washed away when by the finite size effects at a finite . We argue that when the quantum fluctuations at any finite are considered and if the QSG can be avoided, the effective action may still be described by the Schwarzian, the OTOC still show maximal quantum chaos. We expect the results achieved here also apply to the SY model which maybe called two indices complex fermion SYK, so it may also have a gravity dual in space, if QSG can be avoided. This work may inspire other works to study the possible new gravity dual of the 2 indices Majorana or Complex SYK model.

2. Two indices SYK model. Here we introduce a new class of SY model which can be named as two indices SYK model. Because , there are two different ways to go to larger groups, one is generalize to as originally done by SY in Eq.2. Here we take a different route, generalize to . One can write a quantum spin in terms of Majorana fermions at each site :

(4) |

where is the generators of the group om (). The Majorana fermions satisfy the Clifford algebra . For , . The total spin square . Setting leads to the total spin . Its main advantages over the original SY model Eq.2 is that there are no constraints here.

Using the Majorana fermion representation for a quantum spin has a long history: several authors including the author used it to solve multi-channel Kondo problems kondoye12345 (); stevekondo (), Kitaev kit () and many others steveQSL () used a different version ( namely used 4 Majorana fermions by imposing an constraint ) to solve the quantum spin liquid (QSL) phase in a honeycomb lattice. Several authors used it to study QSL phases in anisotropic triangular lattices major1 (); major2 (). This could be the first time to use it to solve a random quantum spin system. However, there are several tricky features by using the Majorana fermions representation of quantum spins which were noticed before major1 (); major2 (): there is a gauge degree of freedom in Eq.4, which played crucial roles in any description of QSL states. The Hilbert space of spin quantum spin is , each spin is represented by 3 Majorana fermions in the case, each Majorana fermion has quantum dimension , so the Hilbert space of of them is enlarged to where takes the integer part of . The extra dimension is due to the gauge degree of freedoms. We suspect that the physical consequences of this extra degree of freedoms may increase the quantum fluctuations over the original quantum spins, therefore may favor quantum spin liquids over ordered states compared to the original quantum spin models.

Substituting Eq.4 to Eq.1 leads to the two indices SYK model written as SYK/2:

(5) |

which, just like the SYK model Eq.3, also contains 4 Majorana fermions, but with only 2 site indices and additional index . As argued below, may have several advantages over the original fermionic SY models in the representation and the four indices SYK models.

To solve the original fermions SY model Eq.2, one need to take first, then followed by . One must also introduce a Lagrangian multiplier to enforce the local constraint in Eq.2 which becomes a global constraint at . Fixing at , the leads to the gapless QSL. As said in the introduction, all these main difficulties of the original SY model were ingeniously circumvented by Kitaev by replacing the fermions with Majorana fermions, the two indices by 4 indices , double large , large limit by just one large limit. This is a significant improvement over the original SY model both analytically and numerically. Here, we still keep the 2 indices , replacing the fermions by 3 Majorana fermions to keep the spin algebra at , then extend the to , still use the expansion, followed by a expansion.

One of the biggest advantages of this group over the group is the absence of a Lagrangian multiplier to enforce the local constraint in Eq.2. This make the following expansion much easier to perform than that of . The advantage over the 4 indices SYK model Kittalk () is that here we still stick to the two indices . As argued in Pol (), two index coupling may fit better with a bulk string theory.

3. The mean field solution at followed by the . By using replica where is the number of replicas, doing quenched average over the Gaussian distribution and introducing the Hubbard-Stratonovich (HS) field , different sites are decoupled:

(6) |

where is the single site partition function:

(7) | |||||

In the limit, we get the following saddle-point equation for the field:

(8) |

We assume that in both quantum spin liquid and quantum spin glass phase which is obviously anti-symmetric in and .

In a quantum spin-glass, which is the Edward-Anderson (EA) order parameter SY (); rotor1 (); rotor2 (). For its replica off-diagonal component which is independent of . If the replica symmetry is not broken in the QSG phase, then .

Introducing a second HS field , one can transform into the following form:

(9) |

where is the single-site and single-component partition function:

(10) | |||||

In the limit, we reach the saddle-point equation for the two-point function:

(11) |

In the limit, Eq.8 becomes:

(12) |

From Eq.10, one can identify the system’s self-energy:

(13) |

and reach the following self-consistent equation:

(14) |

where the matrix inversion is taken in the replica space.

Obviously, due to the fermions can not condense, so for . So the fermion Green function only has the replica diagonal saddle point solution, there is no QSG at . So in the following, we focus on the quantum spin-liquid phase. The possible instability to the QSG order will be discussed in the conclusion section. Then the self-consistent equations 13,14 for a single replica take the identical form as the SY model in the representation SY (); SY3 () and the SYK model in the limit Kittalk (); Pol (); Mald (); Gross (). So if dropping the irrelevant term in Eq.14, the saddle point equations 13,14 have parametrization invariance spin () under :

(15) |

where with .

The conformably invariant solution at a long time was found to be ( after replacing in Kittalk (); Pol (); Mald (); Gross () by ):

(16) |

which breaks the parametrization symmetry in Eq.15 down to the .

4. expansion at . Fixing at , the saddle point Eq.8 still holds. However, the saddle point Eq.11 suffers quantum fluctuations. In performing the expansion at a fixed , it is convenient to use the self-energy to replace , then Eq.9 becomes jacob ():

(17) |

where should be taken as a fixed external potential. Obviously taking the saddle point recovers Eq.13, 14. At a finite , one can write:

(18) |

In principle, when performing the expansion, one need to keep the saddle point Eq.8 and solve it self-consistently order by order rotor1 () in . Fortunately, to evaluate N-point correlation functions at the order of , one can simply ignore the self-consistency Eq.8 and set ( However, as to be shown later, this is not true in evaluating corrections to the free energy ).

In the following, we will ignore the replica off-diagonal fluctuations, so only focus on the replica diagonal ones ( so we will drop the replica index ). Substituting Eq.18 into Eq.17, one can see that the linear term vanishes, the quadratic term becomes:

(19) | |||||

It is instructive to see that the first term ( or the first term in Eq.17 ) coming from the combination of two HS fields and is diagonal in space, the Green function appears in the denominator, in the long time limit which diverges linearly naive (). However, the second term ( or the second term in Eq.17 ) coming from the integrations of the Majorana fermion bubbles ( Fig.1 ) is off-diagonal in and space ( but they become diagonal in the imaginary frequency space ), the Green functions appear in the numerator, in the long time limit, the product decay to zero in the long time limit. However, due to the completely different dependencies of the two terms on the Green function, the first term dominates over the second, so remains positive definite. It shows the stability of the QSL phase at least to the order of . We expect it to be stable to all orders of . It is also easy to see the in Eq.16 factors out, points to the conformably invariant form of in the long time limit. In fact, the first term is invariant under the following scale transformation: , one knew , then if one assumes , then it indicates which takes the same scaling form as the saddle point . Similarly, one can check the second term is invariant under the same scale transformation: , so Eq.19 indicates which will be confirmed by a direct Feymann diagram calculation in Fig.2a and Eq.21. In fact, one can check that the next order ( the sixth order ) term is also invariant under the same scale transformation.

In fact, one may make Eq.19 physically more transparent by defining , then Eq.19 can be re-written as:

(20) | |||||

where is identical to the kernel of the ladder diagram of the 4-point function in the SYK model Mald (); kernel (); absolute (). The eigenvalues and eigen-functions of the Kernel have been worked out in Mald () using the conformal invariance. By taking into account the replacement in Eq.16, one can see that kernel has a positive eigenvalue for the continuous conformal weight , negative eigenvalue for the discrete conformal weight . In both cases, Eq.20 is positive definite.

We also solve Eq.13,14 numerically just like in SY () which recover the conformally invariant solution only in the long time limit, then plug them into Eq.19 to show it remains positive definite when using the complete solutions.

5. corrections to two and four point correlation functions. Because of the conformably invariant form of Eq.19, we expect the propagator takes also conformably invariant form naive () . Its contribution to self-energy at the order of was shown in Fig.2a:

(21) | |||||

which takes the same scaling form as the saddle point at in Eq.14. This indicates that the conformal invariance is kept at least to order of . For example, it may change the coefficient in Eq.16, but not the function form such as the decay exponent . We expect the conformal invariance is kept to all orders in . In a sharp contrast, in the quantum rotor model, the corrections is more subleading to the result .

Its contribution to the function in Eq.12 ( which is equal to the spin-spin correlation function ) at the order of was shown in Fig.2b:

(22) |

which takes the same scaling form as that at in Eq.12. It confirms the conformal invariance at least to order of .

In contrast to the SYK model which shows quantum chaos at the order , here we fix at the limit and perform a expansion, so in evaluating the OTOC Eq.23, we need to take the same site index , but different component ( no sum over ):

(23) |

which is essentially the extension of the function to 4 different times.

When it is analytically continued to to compute the OTOC in real time with . One may extract the Lyapunov exponent . Unfortunately, just like in Eq.22, it is still conformably invariant in the long time limit. Just like SYK, it may not show any quantum chaos at and any . In order to study possible quantum chaos and evaluate the Lyapunov exponent , one may need to study the effects, then followed by the expansion which be discussed below.

6. corrections to the Free energy, zero temperature entropy and specific heat. From Eq.6,9,10, we can evaluate the free energy per site and per spin component at both and can be solely expressed in terms of the Green function:

(24) |

which can be used to evaluate the zero temperature entropy and specific heat,

If plugging the conformally invariant solution Eq.16 into Eq.24, one can see the first term just vanishes after regularizing the ultra-violet divergent integral properly, the second term leads to the zero temperature entropy which was evaluated in the SY and the SYK model SY3 (); Kittalk (); Mald ():

(25) | |||||

where is the Catalan’s constant.

In order to evaluate the coefficient of the linear specific heat at low temperature, one need to consider the correction to the conformally invariant solution Eq.16. Then the first term in Eq.24 does not vanish anymore, so it will also contribute under such a correction.

In principle, using Eqs.19, one can evaluate the corrections to the free energy Eq.24. However, as alerted earlier, in contrast to evaluate the correction to the point correlation functions, to get all the possible corrections to the free energy, one need also include the correction to in Eq.22. Because all these corrections are exactly marginal, so they will change the zero temperature entropy at to become dependent. It will not change the linear specific heat behaviour, but will make also dependent.

7. Instability to the QSG phase at and a finite : So far, we only focused on the QSL phase, also ignored the replica off-diagonal fluctuations in the QSL phase. It would be interesting to study if there is an instability to QSG order. For SY model, it was argued in SY3 () that the QSG still emerges as the true ground state at any finite below an exponentially suppressed temperatureSY3 () . This is a non-perturbative effects which maybe inaccessible to any orders in the expansion.

To look at the QSG instability, as said below Eq.8, in the QSG phase, the replica off-diagonal component which is independent of . If the replica symmetry is not broken, then . So we split Eq.6 into the replica off-diagonal part and diagonal part:

(26) | |||||

where the means the replica diagonal part. Then the in Eq.7 need to be replaced by:

(27) | |||||

After performing the cumulant expansion in Eq.27, we can collect all the replica off-diagonal part into:

(28) | |||||

where is taking the average over the replica diagonal part of the single site/single component partition function in Eq.10. Finally, we reach:

(29) |

whose structure may be contrasted to Eq.19.

Substituting the conformably invariant solution Eq.16 into Eq.29 and regularizing the integral properly by introducing the natural dimensionless short-time ( or high-energy ) cut-off , one reach

(30) |

which leads to the QSG instability temperature

(31) |

which shows this QSG instability is non-perturbative and in-accessible to any orders in the expansion.

As shown above, the expansion preserves the conformally invariant form Eq.16, so it will not change the exponential form Eq.31 except it may modify the coefficient in Eq.31. Of course, due to the fermions can not condense, So the fermion Green function only has the replica diagonal saddle point solution, there is no QSG at . In fact, the QSG instability is non-perturbative, in-accessible to expansion to any orders.

8. expansion at , the Schwarzian action at both finite and .

In this work, we showed that at a fixed , the correction still keeps the long time conformal or reparametrization invariance under in Eq.15. So the result still applies to case. However, it is not known if it applies to the Heisenberg model due to the extra Majorana fermions and the associated gauge field. Then what would be the crucial quantum fluctuations effects from the effects ? Here£¬ we will derive the effective action at a finite , but with . We find that it did not show any quantum chaos in this limit. We did not expect it to be. Because we only expect quantum chaos show up at at any finite , which maybe explored in the expansion followed by a expansion.

We expect the quantum chaos happen only in the spin singlet channel, so we ignore the quantum fluctuations in spin symmetric and anti-symmetric channels spin (). We also ignore the quantum fluctuations in the replica-off diagonal channel. So we still assume that in both quantum spin liquid and quantum spin glass phase which is obviously anti-symmetric in and . Then Eq.6 is simplified to:

(32) |

where the single site partition function is:

(33) | |||||

where is the single-site and single-component partition function:

(34) | |||||

where is the self-energy.

In the limit, takes the saddle-point value:

(35) |

Then Eq.33 is simplified to

(36) | |||||

and Eq.32 is simplified to

(37) | |||||

In Eq.37, the and appears in the combination , so also implies limit. As alerted earlier above Eq.32, this is due to the fact that we have dropped the quantum spin fluctuations spin (). The physical limit should be first, then followed by to keep instead of the other way around, so the order of limit may not commute. So we expect the quantum chaos may only happen in this physical limit instead of in the un-physical one. Both and need to be finite and to detect the possible quantum chaos.

To be instructive, one may still take the saddle point of Eq.37 which recovers Eq.12. Now we may substitute spin ()

(38) |

into Eq.37 and expand it to the quadratic order in . The zero-th order just leads to Eq.24. The first order vanishes due to the saddle point Eq.12. The quadratic order becomes:

(39) | |||||

where . It is twice of the kernel in the expansion in Eq.20. By using the results achieved in the expansion below Eq.20, one can see Eq.39 is positive definite instead of having a zero mode three (). Just like the expansion at presented in the previous sections, it only leads to conformably invariant OTOC Eq.23 instead of an exponential growth spin (), so at the leading order of the expansion. As argued above, it is not expected to appear at anyway.