arXiv:1603.05246v1 [cond-mat.str-el] 16 Mar 2016 Physical Review B 94, 035135 (2016) Numerical study of fermion and boson models with infinite-range random interactions Wenbo Fu1 and Subir Sachdev1, 2 1Department of Physics, Harvard University, Cambridge, Massachusetts, 02138, USA 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: March 18, 2016) Abstract We present numerical studies of fermion and boson models with random all-to-all interactions (the SYK models). The high temperature expansion and exact diagonalization of the N -site fermion model are used to compute the entropy density: our results are consistent with the numerical solution of N = 1 saddle point equations, and the presence of a non-zero entropy density in the limit of vanishing temperature. The exact diagonalization results for the fermion Green’s function also appear to converge well to the N = 1 solution. For the hard-core boson model, the exact diagonalization study indicates spin glass order. Some results on the entanglement entropy and the out-of-time-order correlators are also presented. 1 I. INTRODUCTION Fermion and boson models with infinite-range random interactions were studied in the 1990’s and later [1–6] as models of quantum systems with novel non-Fermi liquid or spin glass ground states. More recently, it was proposed that such models are holographically connected to the dynamics of AdS2 horizons of charged black holes [7, 8], and remarkable connections have emerged to topics in quantum chaos and black hole physics [9–15]. The model introduced by Sachdev and Ye [1] was defined on N sites, and each site had particles with M flavors; then the double limit N ! 1, followed by M ! 1. In such a limit, the random interactions depend on 2 indices, each taking N values. Taking the double limit is challenging in numerical studies, and so they have been restricted to M = 2 with increasing values of N [5, 6]. It was found that the ground state for N ! 1 at a fixed M = 2 was almost certainly a spin glass. So a direct numerical test of the more exotic non-Fermi liquid states has not so far been possible. Kitaev [9] has recently introduced an alternative large N limit in which the random interaction depends upon 4 indices, each taking N values; the saddle-point equations in the N ! 1 limit are the same as those in Ref. 1. No separate M ! 1 is required, and this is a significant advantage for numerical study. The present paper will study such Sachdev-Ye-Kitaev (SYK) models by exact diagonalization; some additional results will also be obtained in a high temperature expansion. Our numerical studies will be consistent the fermionic SYK model displaying a non-Fermi liquid state which has extensive entropy, and entanglement entropy, in the zero temperature limit. For the case of the bosonic SYK model, our numerical study indicates spin-glass order: this implies that the analytic study of the large N limit will require replica symmetry breaking [3]. The outline of this paper is as follows. In Section II, we review the large N solution of the SYK model, and present new results on its high temperature expansion. In Section III we present exact diagonalization results for the fermionic SYK model, while the hard-core boson case is considered later in Section V. Section IV contains a few results on out-of-time-order correlators of recent interest. II. LARGE N LIMIT FOR FERMIONS This section will introduce the SYK model for complex fermions, and review its large N limit. We will obtain expressions for the fermion Green’s function and the free energy density. A high temperature expansion for these quantities will appear in Section II B. 2 The Hamiltonian of the SYK model is H = 1 (2N )3/2 XN Jij;k` c†i c†j ckc` i,j,k,`=1 X µ c†i ci i where the Jij;k` are complex Gaussian random couplings with zero mean obeying (1) Jji;k` = Jij;k` , Jij;`k = Jij;k` , Jk`;ij = Ji⇤j;k` |Jij;k`|2 = J 2. (2) The above Hamiltonian can be viewed as a ‘matrix model’ on Fock space, with a dimension which is exponential in N . But notice that there are only order N 4 independent matrix elements, and so Fock space matrix elements are highly correlated. The conserved U(1) density, Q is related to the average fermion number by Q = 1 N XD E c†i ci . i (3) The value of Q can be varied by the chemical potential µ, and ranges between 0 an 1. The solution described below applies for any µ, and so realizes a compressible state. Using the imaginary-time path-integral formalism, the partition function can be written as Z Z = Dc†Dc exp ( S) (4) where where Z S = d⌧ (c†@⌧ c + H), 0 (5) = 1/T is the inverse temperature, and we have already changed the operator c into a Grassman number. In the replica trick, we take n replicas of the system and then take the n ! 0 limit ln Z = lim n!0 1 n (Z n 1) (6) Introducing replicas cia, with a = 1 . . . n, we can average over disorder and obtain the replicated imaginary time (⌧ ) action XZ Sn = ia 0 ✓ d⌧ c†ia @ @⌧ ◆ µ cia J2 XZ 4N 3 ab 0 X4 d⌧ d⌧ 0 c†ia(⌧ )cib(⌧ 0) ; i (7) (here we neglect normal-ordering corrections which vanish as N ! 1). Then the partition function can be written as ZY Zn = Dc†aDca exp ( Sn) a (8) 3 If we ignore the time-derivative term in Eq. (7), notice that the action has a SU(N ) gauge invariance under cia ! Uij(⌧ )cja. And indeed, in the low energy limit leading to Eq. (20), the time-derivative term can be neglected. However, we cannot drop the time-derivative term at the present early stage, as it plays a role in selecting the manner in which the SU(N ) gauge invariance is ‘broken’ in the low energy limit. In passing, we note that this phenomenon appears to be analogous to that described in the holographic study of non-Fermi liquids by DeWolfe et al. [16]: there, the bulk fermion representing the low energy theory is also argued to acquire the color degeneracy of the boundary fermions due to an almost broken gauge invariance. As in Ref. [16], we expect the bulk degrees of freedom of gravitational duals to the SYK model to carry a density of order N [10]. Following the earlier derivation [1], we decouple the interaction by two successive HubbardStratonovich transformations. First, we introduce the real field Qab(⌧, ⌧ 0) obeying Qab(⌧, ⌧ 0) = Qba(⌧ 0, ⌧ ). (9) The equation above is required because the action is invariant under the reparameterization a $ b, ⌧ $ ⌧ 0. In terms of this field XZ Sn = ia 0 d⌧ c†ia ✓ @ @⌧ 1 2N ◆ µ cia + Qab(⌧, ⌧ 0) XZ ab 0 X ( d⌧ d⌧ 0 N 4J 2 [Qab(⌧, 2) c†ia(⌧ )cib(⌧ 0) . ⌧ 0 )]2 i (10) A second decoupling with the complex field Pab(⌧, ⌧ 0) obeying Pab(⌧, ⌧ 0) = Pb⇤a(⌧ 0, ⌧ ) (11) yields XZ Sn = ia 0 ✓ d⌧ c†ia @ @⌧ ◆ XZ µ cia + ab 0 ( d⌧ d⌧ 0 N 4J 2 [Qab(⌧, ⌧ 0)]2 + N 2 Qab(⌧, ⌧ 0) |Pab(⌧, ⌧ 0)|2 ) X Qab(⌧, ⌧ 0)Pba(⌧ 0, ⌧ ) c†ia(⌧ )cib(⌧ 0) (12) i Now we study the saddle point of this action in the large N limit. Using Z [d ] F[ ] (x) =0 (13) Taking F = exp ( Sn) and = Pba we obtain Pab(⌧, ⌧ 0) = 1 N hc†ia(⌧ )cib(⌧ 0)i (14) 4 Note that we have combined N 2 Qab|Pab|2 and N 2 Qba|Pba|2 as one term. Similarly, taking derivative with respect to Qab, we have Qab(⌧, ⌧ 0) = J 2|Pab(⌧, ⌧ 0)|2. (15) If we only consider diagonal solution in the replica space (non spin-glass state), we can define the self energy: ⌃(⌧, ⌧ 0) = Q(⌧, ⌧ 0)P (⌧ 0, ⌧ ), (16) and the Green’s function Then we have G(⌧, ⌧ 0) = hT⌧ c(⌧ )c†(⌧ 0)i. P (⌧, ⌧ 0) = G(⌧ 0, ⌧ ), (17) (18) and the saddle point solution becomes G(i!n) = 1 i!n + µ ⌃(i!n) ⌃(⌧ ) = J2G2(⌧ )G( ⌧ ) (19) The above equation shows a re-parameterization symmetry at low temperature if we ignore the i!n term [9, 10]. At zero temperature, the low energy Green’s function is found to be[1, 10] G(z) = C e i(⇡/4+✓) p z , Im(z) > 0, |z| ⌧ J, T = 0 (20) where C is a positive number, and ⇡/4 < ✓ < ⇡/4 characterizes the particle-hole asymmetry. A full numerical solution for Eq. (19) at zero temperature was also obtained in Ref. 1, and is shown p in Fig. 1. We can see the 1/ z behavior at low energy. However, it is not possible to work entirely within this low energy scaling limit to obtain other low temperature properties: the i!n term is needed to properly regularize the ultraviolet, and select among the many possible solutions of the low-energy equations [4, 10]. A. Free energy and thermal entropy The free energy is defined to be F = 1 ln Zeff where Zeff has only one replica. So Z Zeff = Dc†Dc exp ( S), 5 (21) (22) 1.2 1 0.8 p !J I m(G) 0.6 0.4 0.2 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 !/J FIG. 1. Figure, adapted from Ref 1, showing the imaginary part of Green’s function multiplied by p! as a function of ! at particle-hole symmetric point ✓ = 0. Our definition of the Green’s function, Eq. (17), di↵ers by a sign from Ref. [1]. with XZ S= i0 ✓ d⌧ c†i @ @⌧ ◆Z µ ci + 0 ( d⌧ d⌧ 0 N 4J 2 [Q(⌧, ⌧ 0)]2 + N 2 Q(⌧, ⌧ 0) |P (⌧, ⌧ 0)|2 ) X Q(⌧, ⌧ 0)P (⌧ 0, ⌧ ) c†i (⌧ )ci(⌧ 0) i (23) For the free energy density F/N , we can just drop the site index i to give the single site action, substituting the Green’s function and self energy Z S= 0 ✓◆ d⌧ d⌧ 0c† Z + 0 (⌧ ) d⌧ d⌧ @ @(⌧ (⌧ 01 4J 2 ⌧ 0) [⌃(⌧, ⌧ µ (⌧ 0)/G(⌧, ⌧ 0) ⌧ 0)]2 + ⌃(⌧, ⌧ 0) c(⌧ 0) 1 2 ⌃(⌧ , ⌧ 0)G(⌧ 0, ⌧ ) ) (24) After integrating out the fermion field S= Tr ln [(@⌧ µ) (⌧ Z ⌧ 0) + ⌃(⌧, ⌧ 0)] + 0 ( d⌧ d⌧ 0 1 4J 2 [⌃(⌧, ⌧ 0)/G(⌧, ⌧ 0)]2 ) 1 2 ⌃(⌧ , ⌧ 0)G(⌧ 0, ⌧ ) (25) 6 To verify this result, we can vary with respect to ⌃(⌧, ⌧ 0) and G(⌧, ⌧ 0), also using the fact that ⌃(⌧, ⌧ 0) = ⌃⇤(⌧ 0, ⌧ ), G(⌧, ⌧ 0) = G⇤(⌧ 0, ⌧ ), to obtain the equations of motions as before. In the large N limit, we can substitute in the classical solution, and then free energy density is F N = T X ln ( n G(i!n)) Z 0 d⌧ 3 4 ⌃(⌧ )G( ⌧) (26) The thermal entropy density can be obtained by S N = 1 @F N @T (27) B. High temperature expansion Now we present a solution of Eqs. (19) by a high temperature expansion (HTE). Equivalently, this can be viewed as an expansion in powers of J. We will limit ourselves to the simpler particle-hole symmetric case, Q = 1/2, for which both G and ⌃ are odd functions of !n. We start with the high temperature limit G0(i!n) = 1 i!n (28) and then expand both G and ⌃ in powers of J2: G = G0 + G1 + · · · and ⌃ = ⌃0 + ⌃1 + · · · . The successive terms can be easily obtained by iteratively expanding both equations in Eq. (19), and repeatedly performing Fourier transforms between frequency and time. ⌃1(i!n) = J2 1 4i!n G1(i!n) = J2 1 4(i!n)3 ⌃2(i!n) = J4 3 16(i!n)3 G2(i!n) = J4 1 4(i!n)5 ⌃3(i!n) = J6 15  32(i!n)5 + 3 128T 2(i!n)3 G3(i!n) = J6 37  64(i!n)7 + 3 128T 2(i!n)5 ⌃4(i!n) = J8 561  256(i!n)7 + 75 512T 4(i!n)5 1 256T 4(i!n)3 G4(i!n) = J 8 5 2(i!n)9 + 81 512T 2(i!n)7 1 256T 4(i!n)5 (29) 7 The free energy density can be written in terms of G(i!n) and ⌃(i!n) F N =T X ln ( n G(i!n)) 3T 4 X ⌃(i!n)G(i!n) n (30) We also need to regularize the above free energy by subtracting and adding back the free particle part F N = X T ln 2 + T [ln ( G(i!n)) n The series expansion of the entropy density is ln ( i!n)] 3T 4 X ⌃(i!n)G(i!n) n (31) S N = ln 2 1 64 J2 T2 + 1 512 J4 T4 11 36864 J6 T6 + 599 11796480 J8 T8 + ··· (32) Next, we describe our numerical solution of Eq. (19) at non-zero temperature. We used a Fourier transform (FT) to iterate between the two equations, until we obtained a convergent solution. For faster convergence, we started at high temperature, and used the above high temperature expansion as the initial form. Then we decreased temperature to get the full temperature dependence. We compare the large N exact numerical result with the high temperature expansion in Fig. 2. At high temperatures, all methods converge to ln 2 as expected. The HTE results fit the exact numerics quite well for T /J > 0.6, but are no longer accurate at lower T . The exact numerics shows a finite entropy density in the limit of vanishing temperature, with a value consistent with earlier analytic results [4][10]. 8 S/N 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 large N exact HTE O(T-8) HTE O(T-6) 0.2 0.4 0.6 0.8 T/J 1 FIG. 2. Entropy computation from exact large N EOM and HTE: at hight temperature, all approaches the infinite temperature limit S/N = ln 2. HTE result fit the exact result quite well for T /J > 0.6. . III. EXACT DIAGONALIZATION FOR FERMIONS We now test the validity of the large N results by comparing with an exact diagonalization (ED) computation at finite N . For the numerical setup, it was useful to employ the Jordan- Wigner transformation to map the Hamiltonian to a spin model YY ci = i z j , c†i = + i z j j 0.5. Although there is a clear trend that a larger system size gives rise to larger thermal entropy at low temperature, we cannot obtain a finite zero temperature entropy for any finite N . This is due to the fact that the non- zero zero temperature entropy is obtained by taking the large N limit first then taking the zero temperature limit. 13 "S 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 1/N FIG. 6. The di↵erence of integrated thermal entropy between ED at di↵erent N and large N result. The di↵erence goes to 0 as 1/N approaches 0. . As in Fig. 4, we estimate the deviation from the large N theory by defining Z S = dT |SED(T )/N SN=1(T )/N1|, (45) and plot the result in Fig. 6. The finite size correction goes to 0 as 1/N goes to zero. C. Entanglement entropy Finally, we compute the entanglement entropy in the ground state, obtained by choosing a subsystem A of N sites, and tracing over the remaining sites; the results are in Fig. 7. For NA < N/2, we find that SEE is proportional to NA, thus obeying the volume law, and so even the ground state obeys eigenstate thermalization [14]. We would expect that SEE/NA equals the zero temperature limit of the entropy density S/N [17]. However, our value of SEE/NA appears closer to ln 2 (see Fig. 7) than the value of S/N as T ! 0. Given the small di↵erence between ln 2 = 0.69 and S/N (T ! 0) = 0.464848..., we expect this is a finite-size discrepancy. 14 SEE 4 y=ln(2)x 3.5 N=8 N=12 3 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 NA FIG. 7. Entanglement entropy for the ground state. We divide the system into two subsystems, A and B, we trace out part B and calculate the entropy for the reduced density matrix ⇢A. The x-axis is the size of subsystem A. IV. OUT-OF-TIME-ORDER CORRELATIONS AND SCRAMBLING One of the interesting properties of the SYK model is that it exhibits quantum chaos [9, 13]. The quantum chaos can be quantified in terms of an out-of-time-ordered correlator hA(t)B(0)A(t)B(0)i (OTOC) obtained from the cross terms in h[A(t), B(0)]2i [11]. The exponential decay in the OTOC results in an exponential growth of h[A(t), B(0)]2i at short times, and the latter was connected to analogous behavior in classical chaos. In particular, Ref. 11, established a rigorous bound, 2⇡/ , for the decay rate, L, of the OTOC, and the Majorana SYK model is expected [9] to saturate this bound in the strong-coupling limit J 1. In the opposite perturbative limit, J ⌧ 1, one expects L ⇠ J. Ref. 12 performed a ED calculation of the OTOC on the Majorana SYK model in the infinite temperature limit, = 0. Here we will perform a similar calculation on the complex SYK model, and also obtain results at large J. We define our renormalized OTOC by OTOC = hA(t)B(0)A(t)B(0)i + hB(0)A(t)B(0)A(t)i 2hAAihBBi . (46) We choose the Hermitian Majorana operators A = c1 + c†1, B = c2 + c†2. The negative sign gives a positive initial value for OTOC. At infinite temperature, the result is shown in Fig. 8. We observe the fast scrambling e↵ect from the quick decay of OTOC, and the early time decay 15 OTOC rate L is proportional to J as expected. Similar behavior is found in the Majorana SYK model [12]. At finite temperature, although we can perform the computation in the strong coupling limit 1.2 1 J=1 J=2 0.8 0.6 0.4 0.2 0 -0.2 0 2 4 6 8 10 t FIG. 8. OTOC as a function of time at infinite temperature with di↵erent interaction strength J = 1 and J = 2. Here the total system size N = 7. . J 1, because of finite size e↵ects, we do not get the predicted decay rate L = 2⇡/ . And the OTOC only has a weak dependence on even in the strong coupling limit as shown in Fig. 9. Theoretically [11], in the large N and strong coupling limit, 1 OTOC ⇠ ( J/N )e(2⇡/ )t. Fig. 9 does not display a large change in the exponent, and the pre-factor di↵erence is also small. It is clearly that of our small system sizes, J is the most relevant energy scale that controlling the chaos. 16 OTOC 1.2 T/J=1 T/J=0.1 1 0.8 0.6 0.4 0.2 0 -0.2 0 2 4 6 8 10 t FIG. 9. OTOC as a function of time at di↵erent temperature with interaction strength J = 1. Here the total system size N = 7. . V. SYK MODEL FOR BOSONS Now we consider a ‘cousin’ of the present model: SYK model for hardcore bosons. The bosonic case was also considered in the early work [1–4] but with a large number of bosons on each site. It was found that over most of the parameter regime the ground state had spin glass order. We will find evidence of similar behavior here. The Hamiltonian will be quite similar as Eq. (1), except that because of the Bose statistics now the coe cients obey Jji;kl = Jij;kl , Jij;lk = Jij;kl , Jkl;ij = Ji⇤j;kl (47) Hardcore boson satisfies [bi, bj] = 0 for i 6= j and {bi, b†i } = 1. Also to make particle-hole symmetry (40) hold, we only consider pair hoping between di↵erent sites, i. e. site indices i, j, k, l are all di↵erent, and we drop the normal order correction terms. The spin formalism in ED will be even simpler, as we do not need to attach a Jordan-Wigner string of z: bi = i , b†i = + i (48) We can define a similar Green’s function for bosons: GB(t) = i✓(t)h{b(t), b†(0)}i (49) 17 We identify the infinite time limit of GB as the Edward-Anderson order parameter qEA, which can characterize long-time memory of spin-glass: qEA = lim t!1 GB (t) (50) Then qEA 6= 0 indicates p that GB (!) ⇠ (!). This is quite di↵erent from the fermionic case, where we have GF (z) ⇠ 1/ z; this inverse square-root behavior also holds in the bosonic case without spin glass order [1]. Fig. 10 is our result from ED, with a comparison between GB with GF . It is evident that the behavior of GB is qualitatively di↵erent from GF , and so an inverse square-root behavior is ruled out. Instead, we can clearly see that, as system size gets larger, GB’s peak value increases much faster than the GF ’s peak value. This supports the presence of spin glass order. Im(G)J 18 15 16 10 14 5 0 -0.2 12 10 8 6 4 2 0 -1 0 0.2 -0.5 0 !/J Boson N=12 Boson N=16 Fermion N=12 Fermion N=16 0.5 1 FIG. 10. Imaginary part of Green’s function for hardcore boson and fermion model. The peak near the center gets much higher in the boson model when system size gets larger. The inset figure is zoomed in near ! = 0. Unlike the fermionic case, P 2 = 1 for all N in the bosonic model. We can apply similar symmetry argument as in Ref. [14]: for the half-filled sector (only in even N cases), the level statistics obeys the Wigner-Dyson distribution of Gaussian orthogonal random matrix ensembles, while in other filling sectors, it obeys distribution of Gaussian unitary random matrix ensembles. Our thermal entropy results for bosons are similar to the fermionic results: although the entropy eventually approaches 0 at zero temperature, there is still a trend of a larger low temperature entropy residue as the system size gets larger. 18 We have also computed the entanglement entropy for the ground state of the hardcore boson SYK model. It still satisfies volume law, and the entanglement entropy density is still quite close to ln 2. Finally results for the OTOC are qualitatively similar to the fermionic results. VI. CONCLUSIONS We have presented exact diagonalization results on the fermionic SYK model. The trends in the computed Green’s functions, high temperature expansion, entropy density, and entanglement entropy all support the conclusion that the large N limit approaches the compressible non-Fermi liquid state obtained in the earlier N = 1 analysis. Note that the entropy density approaches a non-zero value in the limit T ! 0 taken after the N ! 1, and so the ground state itself exhibits eigenstate thermalization. This conclusion is also supported by the volume-law behavior of the entanglement entropy. The original model of Ref. 1pwas argued [4] to have an instability to spin glass order at temperatures exponentially small in N ; the consonance between large N theory and our finite N numerics indicates that the model in (1) (with a random interaction with 4 indices [9]) does not have such an instability. For the SYK model for hard-core bosons, our results for the single-particle Green’s function were very di↵erent, and indicate the presence of spin glass order. Similar quantum spin glass states were examined in random models of bosons in Refs. 3 and 4. ACKNOWLEDGMENTS We thank Kartiek Agarwal, Shiang Fang, Tarun Grover, Yingfei Gu, Steve Gubser, Steve Shenker, and Yi-Zhuang You for valuable discussions. This research was supported by the NSF under Grant DMR-1360789 and and MURI grant W911NF-14- 1-0003 from ARO. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. [1] S. Sachdev and J. Ye, “Gapless spin-fluid ground state in a random quantum Heisenberg magnet,” Phys. Rev. Lett. 70, 3339 (1993), cond-mat/9212030. [2] O. Parcollet and A. Georges, “Non-Fermi-liquid regime of a doped Mott insulator,” Phys. Rev. B 59, 5341 (1999), cond-mat/9806119. 19 [3] A. Georges, O. Parcollet, and S. Sachdev, “Mean Field Theory of a Quantum Heisenberg Spin Glass,” Phys. Rev. Lett. 85, 840 (2000), cond-mat/9909239. [4] A. Georges, O. 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[17] We thank Tarun Grover and Yi-Zhuang You for pointing this out to us. 20