Quantum phase transition at non-zero doping in a random t - J model
Henry Shackleton, Alexander Wietek, Antoine Georges, Subir Sachdev
QQuantum phase transition at non-zero doping in a random t - J model Henry Shackleton, Alexander Wietek, Antoine Georges,
2, 3, 4, 5 and Subir Sachdev Department of Physics, Harvard University, Cambridge MA 02138, USA Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010 USA Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France CPHT, CNRS, ´Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France DQMP, Universit´e de Gen`eve, 24 quai Ernest Ansermet, CH-1211 Gen`eve, Suisse
We present exact diagonalization results on finite clusters of a t - J model of spin-1/2 electrons withrandom all-to-all hopping and exchange interactions. We argue that such random models capturequalitatively the strong local correlations needed to describe the cuprates and related compounds,while avoiding lattice space group symmetry breaking orders. The previously known spin glassordered phase in the insulator at doping p = 0 extends to a metallic spin glass phase up to atransition p = p c ≈ /
3. The dynamic spin susceptibility shows signatures of the spectrum ofthe Sachdev-Ye-Kitaev models near p c . We also find signs of the phase transition in the entropy,entanglement entropy and compressibility, all of which exhibit a maximum near p c . The electronenergy distribution function in the metallic phase is consistent with a disordered extension of theLuttinger-volume Fermi surface for p > p c , while this breaks down for p < p c . Two recent experiments [1, 2] have shed new lighton the transformation in the metallic parent stateof the cuprate superconductors near optimal doping,while also highlighting the central theoretical puz-zles. Angle-dependent magnetoresistance measurementsin La . − x Nd . Sr x CuO [1] are compatible with a Lut-tinger volume ‘large’ Fermi surface only at a hole dop-ing p > p c ≈ .
23. Nuclear magnetic resonance inLa − x Sr x CuO [2] in high magnetic fields uncovers glassyantiferromagnetic order for p < p c ≈ .
19. These, andother, observations show that the parent metallic stateof the cuprates exhibits Fermi liquid behavior for p > p c ,and transforms to an enigmatic pseudogap metal withglassy magnetic order for p < p c . Observations also in-dicate that the reshaping of the Fermi surface, and theonset of the pseudogap, for p < p c cannot be explainedby long-range antiferromagnetic order, which sets in at adoping smaller than p c .Here, we present exact diagonalization results on clus-ters of N sites of a t - J model with random and all-to-allhopping and exchange interactions (see (1)). In the ther-modynamic limit N → ∞ , the replica-diagonal saddlepoint of this model, and a related Hubbard model [3],are described by (extended) dynamic mean-field equa-tions in which the disorder self-averages [4]. Moreover,closely related mean-field equations also appear in non-random models in high spatial dimensions [5, 6], indicat-ing that the self-averaging features of the random mod-els properly capture generic aspects of strong correlationphysics. A direct solution of the N = ∞ replica-diagonalsaddle point of the Hubbard model is presented in a sep-arate paper [7], with complementary results which areconsistent with our conclusions below.The insulating model at p = 0 has been studiedpreviously by exact diagonalization [8], and a non-self-averaging spin glass ground state was found. We findsimilar results at p = 0, but with a reduced estimate for the magnitude of the spin glass Edwards-Anderson orderparameter, q . At non-zero p , we find that q decreasesmonotonically, vanishing at a quantum phase transition p c ≈ /
3. We present several results for thermodynamic,entanglement, and spectral properties across this transi-tion. All our results are consistent with the presence of aself-averaging Fermi liquid state for p > p c ; in particular,we find that the one-particle energy distribution functionis consistent with a disordered analog of the Luttingertheorem [4]. The entropy, entanglement entropy andcompressibility all have maxima near p c . We find thatthe low frequency dynamic spin susceptibility matchesthat of the Sachdev-Ye-Kitaev (SYK) class of models[9, 10] over a significant range of frequencies near p c ;this includes a subleading contribution which arises froma boundary graviton in dual models of two-dimensionalquantum gravity [11–14]. Such spectral features are notpresent in theories that treat the transition at p = p c in aLandau-Ginzburg-Hertz framework for the onset of spinglass order in a Fermi liquid [15, 16]. Random t - J model. We consider the Hamiltonian H = 1 √ N N (cid:88) i (cid:54) = j =1 t ij P c † iα c jα P + 1 √ N N (cid:88) i 3. Using the Lanczosalgorithm, we calculate the spectral function at T = 0, χ (cid:48)(cid:48) ( ω ) = 13 (cid:88) α N (cid:88) i (cid:88) n |(cid:104) ψ n | S αi | ψ (cid:105)| × [ δ ( ω − ( E n − E )) − δ ( ω + ( E n − E ))] , (2)where numerically the delta functions are replaced byGaussians with a small variance. The signature of spinglass order, lim t →∞ N (cid:80) i (cid:104) S i (0) S i ( t ) (cid:105) = q (cid:54) = 0, is re-flected by a qδ ( ω ) contribution to the dynamical struc-ture factor S ( ω ), which is related to the spectral functionat T = 0 by χ (cid:48)(cid:48) ( ω ) = S ( ω ) − S ( − ω ). For a finite systemsize, the exact delta function in S ( ω ) is replaced by a peakat low frequency, whose width approaches 0 in the ther-modynamic limit and whose total spectral weight gives q . Therefore, the spin glass contribution to χ (cid:48)(cid:48) ( ω ) for fi-nite systems is given by a low frequency peak, and wasanalyzed for this model at p = 0 in [8]. Above p c , a dis-ordered Fermi liquid is expected to have a low-frequencybehavior of χ (cid:48)(cid:48) ( ω ) ∼ ω .The spectral function for the random t - J model, calcu-lated using the Lanczos algorithm on an 18-site cluster, isshown for several values of doping in Fig. 1. A prominenthump at low-frequency for dopings p (cid:46) . N analysis of this hump must be performedin order to verify that the hump asymptotes to a deltafunction in the thermodynamic limit. To do this, we firstsubtract off a background contribution to account for therest of the spectral weight. Anticipating SYK behaviornear the critical point at low frequencies, we subtract aspectral weight obtained by rescaling the solution of theSchwinger-Dyson equations of the p = 0 model in thelarge- M limit [9, 14] (we rescale J , while preserving totalspectral weight). This SYK spectral weight has a leadingterm χ (cid:48)(cid:48) ( ω ) ∼ sgn( ω ) as | ω | → T = 0 (which general- . . . . . . ω . . . . . . . ( / n ) χ ( ω ) . . . . . p . . . . . q Large- N Large- Mp = 0 . p = 0 . p = 0 . p = 0 . FIG. 1. The spectral function χ (cid:48)(cid:48) ( ω ) of the random t − J model, averaged over 100 disorder realizations on an 18-sitecluster. At low dopings, a sharp peak at low-frequency atlow doping is indicative of spin glass order. With increas-ing doping, the magnitude of this peak is reduced, and thelow-frequency behavior closely resembles the rescaled spec-tral function of the large M SYK theory [9, 14]. (Inset) Afteran extrapolation to the thermodynamic limit, the integratedweight of the low-frequency peak is non-zero, indicating spinglass order. This weight vanishes near p ≈ . 4. Plotted is theintegrated weight for 8 (cid:54) N (cid:54) 18 (as a gradient from red toblue), and the large- N extrapolation with error bars. izes to tanh ( ω/ T ) at low T ). The next-to-leading SYKterm depends linearly in ω , and arises from the boundarygraviton in the holographic dual [14]. It is important tonote that the exponents of these two leading SYK contri-butions are universal and independent of M . Away fromthe critical point and in the spin glass phase, we findthat the spectral function is described well by a combi-nation of the SYK result and a low-frequency hump. Alarge- N analysis of this low-frequency hump, describedin more detail in the supplementary material, confirmsthat the variance of the hump vanishes in the thermody-namic limit, whereas the spectral weight, shown in Fig. 1,remains non-zero. Our analysis gives a large- N estimateof q ∼ . 02 at p = 0. For larger values of doping, q decreases from its value at p = 0, eventually vanishingat some critical value of doping p c . By linearly extrap-olating the large- N prediction for q to higher dopings,we obtain an estimate of p c = 0 . ± . M critical prediction given inFig. 1. At dopings well above p = 0 . 4, we find the spec-tral function to be largely independent of system size. Nogap at low frequency is visible, and χ (cid:48)(cid:48) ( ω ) ∼ ω behaviorconsistent with Fermi liquid predictions is clear. We willprovide a more rigorous verification of the Fermi liquidphase at higher dopings via Luttinger’s theorem later inthe paper. . 00 0 . 25 0 . 50 0 . 75 1 . T . . . . . . . C / N (a) p = 0 p = 1 / p = 1 / p = 1 / p = 1 / . 00 0 . 25 0 . 50 0 . 75 1 . T . . . . . . . γ / N (b) p = 0 p = 1 / p = 1 / p = 1 / p = 1 / . 00 0 . 25 0 . 50 0 . 75 1 . p . . . . . . γ / N T = 0 . N = 12 N = 16 N = 18 0 . 00 0 . 25 0 . 50 0 . 75 1 . p . . . . . . S / N (d) T = 0 . T = 0 . T = 0 . T = 1 . T = 5 . 00 0 . 00 0 . 25 0 . 50 0 . 75 1 . T . . . . . S fi t p a r a m e t e r s (e) ˜ p ˜ sλ / λ / FIG. 2. Thermodynamics of the random t - J model for system sizes N = 12 , , 18, indicated by increasing opacity. (a) Thespecific heat C as a function of temperature for various values of doping. (b) The linear-in- T coefficient of specific heat, γ = C/T , for various dopings as a function of temperature, and (c) for T = 0 . 05 as a function of doping. (d) The thermalentropy S as a function of doping for various temperatures. Black dots show the ansatz Eq. 4 at optimal fitting parameters. (e)Estimates of the parameters in Eq. 4. (cid:101) p corresponds to the doping value with maximal entropy, (cid:101) s corresponds to the maximalentropy density. Thermodynamics and Entanglement. We investigate the specific heat and thermal entropygiven by, C = ∂E∂T , and S = log( Z ) + ET , (3)where Z denotes the canonical partition function, and E = (cid:104) H (cid:105) the internal energy. Results for system sizes N = 12 , , 18 are shown in Fig. 2. To obtain theresults on system sizes N = 16 , 18 we employed ther-mal pure quantum (TPQ) states [24, 25] as described inRefs. [26–28] similar to the finite-temperature Lanczosmethod [29, 30] (see [4] for details). For each set of ran-dom couplings we sampled R = 5 TPQ states, cf. [27].Error estimates have been obtained from 1000 , (400 , N = 12 , (16 , p = 0 exhibits in Fig. 2(a) ex-hibits a broad maximum at T ≈ . 25, in agreement withprevious results [8]. At small values of doping p (cid:46) / T ≈ . 25 while we observean increase of the specific heat at higher temperatures.The maximum is gradually shifted towards a higher value T ≈ . 50 for dopings from p = 1 / p = 1 / 2. Atlow temperatures we observe that the specific heat is ap-proximately linear in temperature, with a maximal slopeattained between dopings p = 0 . 20 and p = 0 . 40. Thelinear-in- T coefficient of the specific heat, γ = C/T , isshown in Fig. 2(b). We observe an increase of γ whenlowering the temperature for all values of doping. Weshow γ at temperature T = 0 . 05 as a function of dopingin Fig. 2(c) for N = 12 , , 18. At this temperature, themaximum is attained at p ≈ . 25. However, we find thatthis maximum is dependent on the temperature. At tem-peratures below T = 0 . 05 sample fluctuations become toolarge for a reliable estimate of the maximum. We notethat a divergence of the γ coefficient has been reported at the pseudogap quantum critical point in cuprate su-perconductors [31].The thermal entropy for different temperatures and N = 12 , , 18 is shown in Fig. 2(d). Again we observemaxima at dopings between p = 0 . 20 and p = 0 . 40 de-pending on temperature. For T → ∞ the maximum isattained exactly at p = 1 / N → ∞ in the canonicalensemble. The ansatz, S/N = (cid:40) − K | p − (cid:101) p | λ + (cid:101) s for p (cid:54) (cid:101) p − K | p − (cid:101) p | λ + (cid:101) s for p > (cid:101) p , (4)is found to describe the entropy data with considerableprecision. A comparison between the ansatz (black cir-cles) and the ED data is shown in Fig. 2(d). The parame-ters (cid:101) p , (cid:101) s , λ , λ , and K are fitted for dopings p ∈ [0 , . (cid:101) p is increasing when lowering the tem-perature below T = 0 . 25. At T = 0 . 05 and N = 18 weobtain an estimate, (cid:101) p ≈ . ± . 025 [from S ( T = 0 . . (5)This value is consistent with the maximum of γ , observedin Fig. 2(c). However, we find that both increasing thesystem size and lowering temperature increases our esti-mate of the critical doping (cid:101) p when estimated as above.At temperatures below T = 0 . 05 estimates are found tobe unreliable due to sample fluctuations.To access the limit T → S vN ( A ) = − Tr[ ρ A log ρ A ] . Here, ρ A = Tr B ( | ψ (cid:105) (cid:104) ψ | ) is the reduced density matrixof the ground state | ψ (cid:105) on a subsystem A . B denotesthe complement of A . Results for S vN ( A ) for subsystemsizes M = 1 , , , N = 10 , , M = 1)and two-site ( M = 2) entanglement entropy are well con-verged as a function of total system size N . Analogouslyto the thermal entropy, we find the ansatz in Eq. 4 toalso be an excellent fit for the entanglement entropy. Thevalue of (cid:101) p only weakly depends on the subsystem size M and the total system size N . For a N = 16 site clusterand M = 4 we estimate, (cid:101) p ≈ . ± . 024 [from S vN ( A )] , (6)in agreement with our estimate obtained from the ther-mal entropy in Eq. 5.Finally, we investigate the charge susceptibility (com-pressibility), χ c = ∂n∂µ = (cid:18) ∂ e∂n (cid:19) − = (cid:18) ∂ e∂p (cid:19) − , (7)computed by taking the inverse of the second deriva-tive of the internal state energy density e = E/N w.r.t.doping p . Here, the chemical potential is given by µ = ∂e/∂n . Results for different temperatures at N = 18are shown in Fig. 3(b). At temperatures T = 0 and T = 0 . p = 1 / 3. Weobserve a shoulder-like feature at lower doping. At highertemperatures T = 0 . T = 0 . p ≈ . 2. We notice, that thisshift matches the shift of (cid:101) p in the thermal entropy shownin Fig. 2(b,c). We note that the occurrence of a maxi-mum in the compressibility, specific heat coefficient andlocal entanglement entropy has been recently discussedin cluster-DMFT studies of the Hubbard model withoutrandomness in relation to the pseudogap and Mott criti-cal points [34–37]. Luttinger’s theorem. Having found strong signaturesof a spin glass phase persisting from half filling up to p c ≈ / 3, we now provide evidence of a Fermi liquid phaseat higher values of doping, which vanishes at a criticalvalue of doping near the onset of spin glass order. Toverify the presence of a Fermi liquid phase, we introducethe one-particle energy distribution function , N ( (cid:15) ) = 1 N (cid:88) λ δ ( (cid:15) − (cid:15) λ ) (cid:88) ijσ (cid:104) λ | i (cid:105) (cid:104) c † iσ c jσ (cid:105) (cid:104) j | λ (cid:105) (8)where | λ (cid:105) are the single-particle non-interacting eigen-states with energy (cid:15) λ , obtained by diagonalizing the hop-ping matrix t ij . This quantity is analogous to the parti-cle occupation number in momentum space, n ( k ), com-monly used in systems with translational invariance. Fora non-interacting system with fixed particle number n ,the averaged quantity N ( (cid:15) ) converges to D ( (cid:15) ) θ ( (cid:15) − (cid:15) F ),where D ( (cid:15) ) is the single-particle density of states and (cid:15) F . 00 0 . 25 0 . 50 0 . 75 1 . p S v N (a) M = 1 M = 2 M = 3 M = 4 . 00 0 . 25 0 . 50 0 . 75 1 . p . . . . . χ c (b) T = 0 . T = 0 . T = 0 . T = 0 . FIG. 3. (a) The ground state entanglement entropy S vN ofsubsystems of size M . Results are compared for total systemsize N = 10 , , 16, shown as increasing opacity. The maximaare attained at values close to p = 1 / 3, indicated by the graydashed line. Black dots show the ansatz Eq. 4 at optimalfitting parameters. (b) Charge susceptibility χ c for differenttemperatures at N = 18. The low-temperature maximum atdoping p = 1 / p ≈ . − (cid:15) . . . . . N ( (cid:15) ) / ( n D ( (cid:15) )) (a) p = 0 . p = 0 . p = 0 . p = 0 . 12 0 . . p − . − . − . − . . ∆ (cid:15) F (b) Free fermionRandom tJ FIG. 4. (a) At high values of doping, the one-particle en-ergy distribution function drops sharply near the energy levelpredicted by Luttinger’s theorem (marked by crosses). Atlower values of doping, this function becomes more broad-ened, suggesting a breakdown of Luttinger’s theorem. (b) Acomparison of the Fermi energy given by Luttinger’s theoremand the numerically-computed value given by the inflectionpoint of the one-particle energy distribution function. For a16 site cluster, the two show good agreement up to a criticalvalue between 6 / 16 = 0 . 38 and 7 / 16 = 0 . 44, in contrast withthe same quantity computed for free fermions which agreewell for all values of doping. is the Fermi energy, defined by: D ( (cid:15) ) = 1 N (cid:88) λ δ ( (cid:15) − (cid:15) λ ) , n = 2 (cid:90) (cid:15) F −∞ d (cid:15) D ( (cid:15) ) . (9)For the interacting system, we show in the supplementalmaterial [4] that, because the random couplings are allto all, N ( (cid:15) ) displays self-averaging properties in the ther-modynamic limit N → ∞ . In this limit, the signatureof Luttinger’s theorem is a discontinuity of N ( (cid:15) ) at thenon-interacting value of (cid:15) F defined in Eq. (9).In Fig. 4, we plot the quantity N ( (cid:15) ) /D ( (cid:15) ), averagedover 1000 realizations on a 16-site cluster. The densityof states D ( (cid:15) ) is a semicircle distribution in the large- N limit; however, in order to account for finite-size correc-tions to this distribution, we instead use the numericallycalculated value of D ( (cid:15) ) obtained from our data. Al-though the drop in particle occupation at the Fermi en-ergy is substantially broadened due to interactions andfinite-size effects, the location of the inflection point stillreliably tracks the location of the Fermi energy predictedby Luttinger’s theorem at high values of doping as shownin Fig. 4. The effects of the infinite-strength Hubbard re-pulsion becomes stronger at lower values of doping, even-tually causing a breakdown of Luttinger’s theorem at acritical doping 0 . < p c < . 44, which is also the loca-tion where spin glass order appears to emerge. Discussion and Conclusion. Our numerical resultsdemonstrate a transition in the random all-to-all t - J model from a spin glass to a disordered Fermi liquidat a critical value of doping. The near-critical behaviorhas similarities to the criticality of SYK models, con-sistent with recent theoretical proposals [21] and nu-merical results on related systems [3]. We find a near-critical dynamic spin susceptibility which is consistentwith the SYK behavior χ (cid:48)(cid:48) ( ω ) ∼ sgn( ω ) [1 − g | ω | + . . . ]over a significant frequency regime; the g term is a uni-versal boundary “graviton” contribution. This is the firstappearance of such features in a doped spin-1/2 SU(2)model. SYK criticality also predicts an extensive zerotemperature entropy: we do find a maximum in the en-tropy near the critical point, but our finite-size data doesnot allow us to identify if there is an extensive contribu-tion. However, we note that for SU( M = 2) models, SYKcriticality is pre-empted at small enough T by a spin glassinstability [7, 18], and so the extensive T = 0 entropy isnot ultimately expected. We also find a maximum inthe entanglement entropy, specific heat coefficient, andcompressibility near criticality.An interesting observation is that the breakdown ofLuttinger’s theorem coming from high doping, as well asthe vanishing of spin glass order from low doping, occursnear p = 0 . 4, which differs from the maxima in the ther-modynamic and entanglement entropy closer to p = 0 . t - J model withfinite Hubbard repulsion [7] also give evidence of SYKcriticality occurring at a lower value of doping than thespin glass/Fermi liquid transition. These observationsare consistent with the spin glass instability of SYK crit-icality for finite M [18] noted above. Understanding thenature of this separation, and the very low T at whichthe spin- instability of SYK criticality appears, remainopen questions to be explored. Acknowledgements. We thank P. Dumitrescu, O. Par-collet, M. Rozenberg and N. Wentzell for valuable dis-cussions. 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Tremblay, Entanglement and Classical Correla-tions at the Doping-Driven Mott Transition in the Two-Dimensional Hubbard Model, PRX Quantum , 020310(2020), arXiv:2007.00562 [cond-mat.str-el]. upplement toQuantum phase transition at non-zero doping in a random t - J model Henry Shackleton, Alexander Wietek, Antoine Georges, 2, 3, 4, 5 and Subir Sachdev Department of Physics, Harvard University, Cambridge MA 02138, USA Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010 USA Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France CPHT, CNRS, ´Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France DQMP, Universit´e de Gen`eve, 24 quai Ernest Ansermet, CH-1211 Gen`eve, Suisse SPIN GLASS ANALYSIS As described in the main text, the spectral function of the random t - J model near half filling has a peak at lowfrequency, suggesting spin glass order. To establish this rigorously, one must show that the variance of the peak goesto zero in the thermodynamic limit while the integrated spectral weight remains non-zero, indicating delta function-like behavior. We isolate the low-frequency peak by subtracting off a background contribution, given by the large- M solution of the SY model. We then fit the remaining low-frequency peak to the function χ low ( ω ) = ωC exp (cid:20) − ω σ (cid:21) . (1)In Fig. 1, we show the extrapolation of Γ to the thermodynamic limit for several values of doping up to p = 1 / q . . 00 0 . 05 0 . 10 0 . 15 0 . 20 0 . /N . . . . . . Γ p = 0 p = 1 / p = 1 / FIG. 1. At low dopings, the low-frequency peak in the spectral function can be isolated and fit to Eq. 1. In the thermodynamiclimit, we confirm that the variance Γ vanishes up to p = 1 / 3. Due to a prominent even/odd particle effect at half filling, weonly extrapolate Γ at half-filling for even system sizes. THERMAL PURE QUANTUM STATES The computation of thermodynamic quantities in the main text has been performed using thermal pure quantumstates [1, 2] together with the Lanczos algorithm. This allowed us to reach system sizes beyond the reach of full exactdiagonalization. This approach is closely related to the finite-temperature Lanczos method [3, 4]. We will now brieflyexplain the method. The trace of any operator H can be evaluated by taking random average values,Tr( A ) = D h r | A | r i , (2)where | r i is a normalized random vector, h r | r i = 1, with normal distributed coefficients, h m | r i ∼ N (0 , D denotes the dimension of the Hilbert space. Here, {| m i} m =1 ,...D denotes an arbitrary orthonormal basis of the Hilbertspace and · · · denotes averaging over random realizations of | r i . Hence, a thermal expectation value of an observable O can be written as as, hOi = 1 Z Tr(e − βH O ) = h β |O| β ih β | β i , (3)where Z = Tr(e − βH ) denotes the partition function and we define the so-called thermal pure quantum (TPQ) state [1,2] at inverse temperature β = 1 /T , | β i = e − βH/ | r i . (4)This way, thermal expectation values can be evaluated efficiently using the Lanczos algorithm, whereas the exactcomputation of the trace of an exponential requires full diagonalization. In the main text we present data for thespecific heat, internal energy and entropy, which are all computed from expectation values of powers of the Hamiltonianwith TPQ states of the form h β | H k | β i . Using the Lanczos algorithm this quantity is efficiently approximated by, h β | H k | β i ≈ e † e − β T n T kn e − β T n e , (5)where e = (1 , , . . . , † and T n denotes the tridiagonal matrix of the Lanczos algorithm after n steps. The convergenceis typically exponentially fast, such that results can be attained up to machine precision. For a more detaileddescription of the method we refer the reader to Ref. [5]. We notice in Eq. (5), that once the Lanczos algorithm hasbeen applied to compute the tridiagonal matrix, results can be derived at all temperatures simultaneously withoutrerunning the expensive Lanczos algorithm.Instead of one single computation as done for evaluating a trace, using TPQ states requires us to to perform randomsampling with multiple vectors | r i and compute error estimates. Since expectation values of the form Eq. (3) arenon-linear in | r i , we perform jackknife resampling [6] of the data. Interestingly, larger system sizes typically requireless random realizations | r i to obtain comparable errorbars. Refs. [1, 2] give a mathematical proof, that for a constantfree energy density, the variance of the estimate in Eq. (3) is exponentially small in the system size. In the main textwe typically average over R = 5 random realizations of the TPQ states. SELF-AVERAGING, ELECTRON DISTRIBUTIONS, AND THE LUTTINGER THEOREMSelf-averaging from the cavity method In this section, we establish that, in the thermodynamic limit N → ∞ , some local observables have self-averaging properties in this fully connected random model. This means that, when considered for a given site, they convergewith probability one to their average over samples. We also establish the connection to extended dynamical mean-fieldequations (EDMFT) that allow for a direct study of the model in the thermodynamic limit. We do not consider thespin-glass phase in this section.For the sake of generality, we consider the finite- U version of the model, the t - J limit corresponding to U = ∞ .The model is defined on a fully connected lattice of N sites by the Hamiltonian: H = − X ij,σ = ↑ , ↓ t ij c † iσ c jσ + U X i n i ↑ n i ↓ − X i We can use the eigenstates of the one-particle non-interacting problem ( U = J ij = 0) as a basis set to represent anysingle-particle correlation function of the interacting problem. These states are defined by, for a given sample t ij :ˆ t | λ i = ε λ | λ i , i . e . X j t ij h j | λ i = ε λ h i | λ i (16)The Fock space of the many-body problem is constructed as the number occupancy states |{ n λ }i and is a full basisfor the many-body problem. The single-particle DOS of the non-interacting system reads: D ij ( ε ) = 1 N X λ δ ( ε − ε λ ) h i | λ i h λ | j i (17)In the N → ∞ limit, D ii converges (and self-averages) to the semi-circular DOS D ∞ defined above.Consider now the interacting system, for a given sample t ij , J ij and finite N . We define the one-electron Green’sfunction in the usual way G ij ( τ − τ ) ≡ −h T c † i ( τ ) c j ( τ ) i , but it can actually be viewed as a one-body operator ˆ G thatwe can look at in any basis set, for example in the eigenstate basis: G λλ ( iω n ) = h λ | ˆ G | λ i = X ij h λ | i i G ij ( iω n ) h j | λ i (18)Note that for a given sample and finite N , this is not diagonal in λ . Correspondingly, a self-energy σ ij can be definedas (we are careful to use a different notation here, since this is for a given sample and finite N ):ˆ G − ≡ iω n + µ − ˆ t − ˆ σ , in site basis : [ G − ] ij = ( iω n + µ ) δ ij − t ij − σ ij (19)Things get simpler in the infinite-volume limit. The off-diagonal components of the self-energy σ i = j vanish, andthe diagonal ones self-average and converge to the local self-energy defined above: σ ii → Σ. Hence, the expression ofthe Green’s function becomes: [ G − ] ij = [ iω n + µ − Σ( iω n )] δ ij − t ij , ( N → ∞ ) (20)Note that off-diagonal components of the Green’s functions do not self-average . They are individually of typical order1 / √ N , but we have to take them into account when calculating the kinetic energy for example, since we sum overall bonds. Given (20), the Green’s function for N = ∞ now acquires a simple diagonal representation in the basis ofeigenstates of ˆ t : G ij ( iω n ) = X λ h i | λ i iω n + µ − Σ( iω n ) − ε λ h λ | j i (21)It is convenient to define the (sample independent) Green’s function for a given energy ε in the semi-circular ‘band’as: G ( iω n , ε ) ≡ iω n + µ − Σ( iω n ) − ε (22)which is the natural quantity we would routinely look at in the EDMFT framework. The connection between thisand the Green’s function for a given sample, in the infinite size limit N = ∞ , is given by: G ij ( iω n ) = Z dε D ij ( ε ) G ( iω n , ε ) , ( N = ∞ ) (23)Let us now consider (for a given sample and any N ) the one-body distribution function: N λ = h ˆ n λ i = X ij h λ | i i X σ h c † iσ c jσ i h j | λ i (24)In the non-interacting case, the ground-state is a Slater determinant of the λ states, and hence at T = 0 N λ = 1 forall filled states and 0 for empty states. We can more conveniently look at it by filtering in energy and define: N ( ε ) = 1 N X λ δ ( ε − ε λ ) N λ = 1 N X λ δ ( ε − ε λ ) X ijσ h λ | i i h c † iσ c jσ i h j | λ i (25)which can also be written: N ( ε ) = X ijσ D ij ( ε ) h c † iσ c jσ i (26)obeying (with n the electron density): Z dε N ( ε ) = N e N = n (27)We can also sample average and consider N ( ε ).Now we establish the connection, in the N = ∞ limit, between this distribution function and what we wouldnaturally calculate in the EDMFT context, which is the number distribution function as a function of the single-particle energy: N ( ε ) = 2 G ( τ = 0 − , ε ) = 2 1 β X n e iω n + iω n + µ − Σ( iω n ) − ε (28)The factor of 2 is for the sum over spin. We note that: Z dεD ∞ ( ε ) N ( ε ) = n (29)Using (23), we obtain: N ( ε ) = 1 N X λ δ ( ε − ε λ ) N ( ε λ ) = D ∞ ( ε ) N ( ε ) , ( N = ∞ ) (30) Luttinger’s theorem. 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