Simple probability distributions on a Fock-space lattice

We consider some aspects of a standard model employed in studies of many-body localization: interacting spinless fermions with quenched disorder, for non-zero filling fraction, here on $d$-dimensional lattices. The model may be recast as an equivalent tight-binding model on a `Fock-space (FS) lattice' with an extensive local connectivity. In the thermodynamic limit exact results are obtained for the distributions of local FS coordination numbers, FS site-energies, and the density of many-body states. All such distributions are well captured by exact diagonalization on the modest system sizes amenable to numerics. Care is however required in choosing the appropriate variance for the eigenvalue distribution, which has implications for reliable identification of mobility edges.


I. INTRODUCTION
In recent years there has been great interest in the study of highly excited quantum states of disordered, interacting systems, and notably the phenomenon of manybody localization 1 (MBL); for topical reviews see e.g. [2,3]. In particular it is now well appreciated that in interacting systems, localization or its absence can be viewed quite generally as an Anderson localization problem in the Fock or Hilbert space of associated many-body states. This connection was made long ago in the context of the problem of quantum ergodicity in isolated molecules, 4,5 and later in quantum dots. 6 One of the central models 7 extensively studied in MBL is that of interacting spinless fermions with quenched disorder, for non-zero fermion filling fractions. We consider it here, on d-dimensional hypercubic lattices. As detailed in sec. II, the model can be mapped exactly onto a tightbinding model on a 'Fock-space lattice' with an extensive local connectivity, each site of which corresponds to a disordered, interacting many-body state with specified fermion occupancy of all real-space sites, and is thus associated with a Fock-space 'site-energy'; with Fock-space sites connected by the one-electron hopping matrix element of the original Hamiltonian.
While MBL per se is not considered directly in this paper, we believe it is instructive to have some broad understanding of the Fock-space lattice in a statistical sense. To that end we consider the distributions, over both Fock-space sites and (where relevant) disorder realizations, of the local Fock-space coordination numbers, site-energies, and many-body eigenvalues (or density of states). None of these quantities contain information about localization itself, but all reflect 'basic' properties of the system in an obvious sense, and understanding them is a precursor to considering localization (to which we turn in a subsequent work 8 ).
As expected from the central limit theorem all such distributions are normal, with extensive means; the relevant moments are obtained (sec. III) as a function of the bare model parameters, fermion filling fraction and space dimension. A subtlety arising in the choice of variance for the eigenvalue spectrum (and Fock-space siteenergies) is pointed out in sec. IV, with the correct choice enabling a pristine distinction between localized and extended states as a function of energy, as required for a reliable identification of mobility edges which may separate them. Comparison to exact diagonalization shows the distributions considered to be well captured even for the small system sizes practically accessible to numerics in d = 1, 2. Concluding remarks are given in sec. V.

II. MODEL
The Hamiltonian is a standard model of spinless fermions, 7 here considered on a d-dimensional lattice: The hoppings (t) and interactions (V ) are nearest neighbour (NN), with ij denoting distinct NN pairs. The site energies { i } in H W are characterized by quenched random disorder, with a distribution P ( i ) common to all sites (and chosen for convenience to have zero mean, = d P ( ) = 0). The eigenvalues of H are denoted by E n .
The lattice has N sites and contains N e fermions. We are interested in the thermodynamic limit where both N ≡ L d and N e → ∞, holding the filling ν = N e /N fixed and non-vanishing. The coordination number of the lattice is denoted by Z d , with Z d = 2d for hypercubic lattices.
with each occupation number n (I) i = 0 or 1 only, and i n (I) i = N e for any |I . These states are orthonormal, with energies E I under H 0 given by where r I is thus defined; and each of the N H configurations of fermions on the lattice specifies uniquely one such basis state |I . The {|I } may be viewed as forming a lattice of N H sites in state-space, 4 here referred to as a 'FS lattice'. These sites are connected under the hopping term H t (eq. 1), with T IJ = I|H t |J the matrix elements coupling them. Since the hopping is NN only, T IJ = t for T IJ non-vanishing (with T IJ = T JI real symmetric). The total Hamiltonian can thus be written as This has the form of a TBM on the FS lattice; with the number of sites to which any given site I is connected under the hopping defining the local coordination number of the site, denoted Z I . Given the mapping to an effective TBM, the same questions can obviously be asked about this FS lattice as arise for a one-electron disordered TBM. 9 What for example is the disorder-averaged density of many-body states; and are those states of some given energy localized on a vanishingly small fraction of the FS lattice, or delocalized over a finite fraction of it? The mapping also means that some techniques applicable to one-electron problems on a real-space lattice may be extended to encompass the question of MBL in the FS lattice (as we will consider in subsequent work 8 ).

B. Fock space lattice
We have found it helpful to have a concrete picture of the FS lattice, in part to prompt questions about its basic characteristics and their implications. An illustrative example is given in fig. 1, shown for a small 1d open chain of N = 12 sites, N e = 6 fermions (filling ν = 1/2).
Each circle represents a site (associated with a state |I ), and solid lines denote connections between the states under the hopping. The sites are arranged in rows, with row index l. The top row (l = 0) consists of the single state |I = |L = |111111000000 in which all fermions occupy the leftmost real-space sites. This state has a coordination number Z I = 1, since under H t it can connect only to the single state |111110100000 in row l = 1. The latter connects under H t to two further states in row l = 2 (such that its coordination number Z I = 3); and these two states are connected in turn to a total of three states in row l = 3. The process continues in this fashion, with the number of sites/states in successive layers first growing and then decreasing as seen in fig. 1 (reflecting that a fermion cannot move beyond the end site for an open chain); and terminates at the bottom state |I = |R = |000000111111 in which all fermions occupy the rightmost real-space sites (row l = 36 here).
Generally, the total number of rows in the FS lattice is 1 + N e (N − N e ) = 1 + ν(1 − ν)N 2 (as the minimum number of sequential hops required to connect |L to |R is N e (N − N e )). Since the FS lattice is invariant to reading the fermion strings in FS sites from left to right, or vice versa, the number of FS sites in rows l and N e (N −N e )−l are the same. The number of sites in row l can be shown to be the number of integer partitions of l, subject to the restriction that no partition may have more than N e parts and no individual part can be larger than N − N e ; and asymptotically grows exponentially in √ l as l is increased towards the middle rows l ∼ 1 2 ν(1 − ν)N 2 . Although our focus here is on finite filling ν = N e /N in the thermodynamic limit N → ∞, for which the FS lattice clearly 'balloons', we add that in the single-fermion limit N e = 1 it reduces simply to that for the real-space lattice; with N rows, each containing just one state (in which the fermion occupies a particular real-space site).
Several general features of the FS lattice are apparent. First, the hopping T IJ is local in state-space, in the sense that the FS sites {|J } to which any given |I connects under H t necessarily lie in a directly adjacent row, reflecting that one fermion has hopped to a NN site in the physical lattice. Moreover, since the hypercubic realspace lattices considered are bipartite, it follows that the corresponding FS lattice is also bipartite.
Second, physical properties of the system are typically characterized by probability distributions -over FS lattice sites, disorder realizations, or both. An obvious example is the local connectivity under T IJ , reflected in the distribution of coordination numbers Z I over FS sites (which is independent of disorder). In e.g. a 1d chain the minimum Z I for all fillings ν = N e /N is clearly Z I,min = 1 (or 2 for periodic boundary conditions), occurring solely for the apical sites of rows l = 0 and N e (N − N e ) where the fermions are maximally bunched. The maximum Z I by contrast is macroscopically large, Z I,max = 2N e for all fillings ν < 1/2, arising for states in which each fermion is surrounded by at least two empty sites; with Z I,max = 2(N − N e ) for ν > 1/2 (and Z I,max = N − 1 precisely at half-filling).
The FS site energies (eq. 3) are also naturally distributed, over sites and/or disorder realizations. These {E I } are the counterparts, for the equivalent Fock-space TBM eq. 4, of the site-energies { i } in a one-particle TBM. One thus expects their distribution to influence whether many-body states are FS localized or extended; while recognizing that, unlike the (N ) real-space { i } which are independent random variables, the N H FS site-energies {E I } are correlated. A further obvious example is the eigenvalue spectrum of H -or density of states (DoS) -and its distribution over disorder realizations.

III. DISTRIBUTIONS
We seek then the distributions, over FS lattice sites and disorder realizations, of: the FS site-energies {E I }, the eigenvalues of H, the coordination numbers Z I , and r I = ij n (I) i n (I) j of eq. 3 (which determines the interaction contribution to E I ).
In the thermodynamic limit of interest we take it as given that all such distributions are Gaussian (as is essentially obvious from the central limit theorem, although can be shown explicitly). We thus focus on first and second moments. As will be seen, all such may be obtained solely from a knowledge of that for r I .
For any quantity O I = I|Ô|I , its average O over both FS sites and disorder realizations is Here ... denotes the disorder average, and TrO I an average over the N H = N C Ne FS sites/states (such that averages over disorder and FS sites commute, (7) such that ν 1 = ν, the filling fraction. Note that ν m ≡ ν m + O(1/N ) in the thermodynamic limit (although eq. 7 holds for any N, N e , and is required below).
Consider first the FS site-energies E I = I|H 0 |I (eq. 3). TrE I is the mean of E I over FS sites for a given disorder realization, given (via eq. 7) by with r ≡ Trr I (as r I is independent of disorder). Since the disorder-averaged = 0, E = TrE I = V r. From eqs. 3,7 Trr I = ν 2 ij = ν 2 Z d N/2, so E is given in the thermodynamic limit by Disorder-induced fluctuations in TrE I are embodied in [TrE I ] 2 , and since the lattice site-energies { i } are independent random variables, i j = δ ij 2 (with 2 ≡ 2 for brevity); so E 2 = Tr(E 2 I ) likewise follows using eqs. 3,7 as (where r 2 = Trr 2 I ). For reasons explained in sec. IV we now define two distinct variances, specifically . These are thus given by with µ 2 r = r 2 − r 2 the variance of r I . We turn now to the distribution of eigenvalues. TrH does not depend on the basis chosen, so TrH = H n E n , and gives the center of gravity of the eigenvalues {E n } for any given disorder realization. Since TrH = TrE I (eq. 4), the disorderaveraged mean eigenvalue E = TrH = E, i.e.
(with E ∝ N e reflecting extensivity). Since [TrH] 2 = [TrE I ] 2 , eq. 10 gives Now consider E 2 = TrH 2 . From eq. 4, I|H 2 |I = E 2 I + J T IJ T JI = E 2 I + t 2 Z I with Z I the coordination number of FS site I. Hence with Z = TrZ I the average coordination number of the FS lattice. In parallel to the FS site energies above, we define again two variances, Physically, µ 2 E gives the disorder-averaged variance of the eigenvalues, relative to their center of gravity TrH = N −1 H n E n for each disorder realization; while µ 2 E0 gives the variance relative to the full mean TrH over both FS sites and disorder. The reasons for introducing these two distinct variances are discussed in sec. IV. Here we simply note that it is µ 2 E (and likewise µ 2 E ) which is of primary relevance.
The variances in eqs. 12,16 thus follow once µ 2 r and the mean FS coordination number Z are known. Obviously neither of the latter depends on disorder. Further, as now shown, the coordination number Z I is simply related to r I , so only the latter need be considered.
Each configuration of fermions on the real-space lattice specifies uniquely one FS basis state |I . With 1 denoting an occupied site and 0 an empty site, the following types of NN pairs arise: 11, 10 and 00. The total number of such pairs in any |I are denoted N (I) mn , and are clearly Using The distribution of FS coordination numbers thus follows directly from a knowledge of that for r I . Eq. 18 is also physically clear: for r I = N (I) 11 = 0, all NN sites to each occupied site in |I are empty, so each of the N e fermions can hop under t to its 2d NNs, whence Z I = 2dN e ; while for r I = N (I) 11 = 0, each additional NN 11-pair clearly 'blocks' one hop for each of the pair of fermions, and thus reduces Z I by 2.
A. Variance of r I From eqs. 12,16 the final step is to determine the second moment r 2 = Tr ij kl n (I) i n (I) j n (I) k n (I) l . Since the sites in ij and kl are not all distinct we partition this sum into terms involving 2 NN sites, 3 adjacent NN sites, and two pairs of distinct NN sites, specifically The middle sum refers to 3 distinct sites, where the site k = i or j, but is a NN to one of them, such that for any given ij there are Z d − 1 sites in the k sum. Hence, 2 The final sum in eq. 19 refers to two pairs of distinct NN sites. Recognizing that the original sum for r 2 contained a total of [ 1 2 Z d N ] 2 terms, while the first pair of terms on the right side of eq. 19 contain respectively 1 2 4 . Using the precise forms for ν m (eq. 7), together with r = 1 2 Z d N ν 2 , then gives µ 2 r = r 2 − r 2 as (20) This is exact under periodic boundary conditions for finite N . In the thermodynamic limit of interest it gives the desired result for the variance of r I , (with leading corrections O(1)).
Let us simply recap the essential results arising in the thermodynamic limit. Eq. 13 gives the mean energies, E = E = V r = V νdN e . For the variances, eqs. 12,16 give and likewise all being a sum of independent contributions from disorder, interactions and (for the eigenvalues) hopping. The mean FS coordination number follows directly from eq. 18 as Z = Z d N e − 2r, so from eq. 9 for r while the variance µ 2 Z = 4µ 2 r (from eq. 18), with µ 2 r given by eq. 21. All these quantities follow directly on specifying the filling fraction ν = N e /N , the space dimension d of the real-space lattice, and the number of fermions N e . Note also that µ 2 E and µ 2 E are invariant under a particle-hole transformation ν ↔ 1 − ν (since Distribution p(N, Ne; Z) of coordination numbers Z for the FS lattice arising for the 1d chain, shown for N = 16, Ne = 8; and compared to the normal distribution (solid line) of mean Z (eq. 24) and variance µ 2 Z = 4µ 2 r (eq. 21) appropriate to the thermodynamic limit. Inset: same comparison for N = 100, Ne = 50.
As an indication of the size-dependence of this distribution, Fig. 2 shows p(N, N e ; Z) vs Z for a half-filled N = 16-site system (as typically employed in exact diagonalization studies of MBL, and with N H = 12870 FS sites). The Gaussian distribution appropriate to the thermodynamic limit is shown for comparison. Even for N = 16 this is quite well approached, despite the modest range of available Z's; and by N = 100 ( fig. 2 inset) the discrete and normal distributions are barely distinguishable. Analogous comparison for the eigenvalue distribution is given below.

IV. ENERGY VARIANCES
We have introduced two distinct variances for the eigenvalues (eqs. 16), and the FS site-energies (eq. 12). Here we explain why, and why it is that µ 2 E and µ 2 E are the physically relevant variances; focusing in the following on the eigenvalues (the same considerations apply to the FS site-energies). The reason is a little subtle, does not usually arise in considering one-body localization (1BL)although it may do, as explained -and has implications for the identification of mobility edges.
Consider first the 1BL case of a single fermion, 10 with filling ν = 1/N . Here, trivially, Z ≡ Z d is the coordination number of the physical lattice and r = 0 = µ 2 r (since N e = 1). In this case (eq. 16), µ 2 E ≡ µ 2 E0 (= 2 + t 2 Z d ) coincide in the thermodynamic limit, so which of the two one considers is immaterial. This reflects the fact (eq. 14) that for 1BL [TrH] 2 = 2 /N vanishes as N → ∞, i.e. that TrH = N −1 H n E n -the center of gravity of the eigenvalue distribution for any given disorder realization -does not fluctuate from realization to realization. For a macroscopic system, each realization will then yield the same eigenvalue spectrum (self-averaging). By itself, that spectrum does not of course contain information about whether states are localized (L) or extended (E). But states of any given energy are either L or E with probability unity over an ensemble of disorder realizations; so the fact that the same spectrum is obtained for all disorder realizations ensures a pristine distinction between L and E states as a function of energy, and hence an unambiguous identification of mobility edges separating them.
The situation above is not however ubiquitous, even for 1BL. To illustrate this, consider the Hamiltonian N a constant which is irrelevant in the following). As for the H of eq. 1, this Hamiltonian is widely studied in MBL, since in d = 1 it maps directly to a random XXZ model under a Jordan-Wigner transformation. H is regarded as being equivalent to H; and indeed, aside from the trivial disorder-independent constant (c), the two Hamiltonians differ only by a constant: However C depends on the disorder realization, C ≡ C({ i }); with a vanishing disorder-averaged mean C = 0, but a non-zero variance C 2 = 2 N proportional to system size N . For H one has TrH = (ν − 1 2 ) i i + V r = (ν − 1 2 )C + V r, with the variances µ 2 E and µ 2 E0 given by Note that µ 2 E = Tr([H − TrH ] 2 ) for H is identical to that arising for H (eq. 16a), so is unaffected by C ≡ C({ i }); by contrast, µ 2 E0 = Tr([H − TrH ] 2 ) differs from that for H (eq. 16b).
We return to this below, but first consider again the 1BL limit, where ν = 1/N . In this case, unlike that for H considered above, µ 2 E and µ 2 E0 (eqs. 26) no longer coincide: µ 2 E = 2 + t 2 Z d (as arises also for H), while µ 2 E0 = 1 4 2 N + t 2 Z d grows with system size N . This reflects the fact that the center of gravity of the eigenvalue distribution (TrH ) fluctuates from realization to realization, [TrH ] 2 = 1 4 2 N = 1 4 C 2 growing with N (in contrast to [TrH] 2 = 2 /N ). The implications of this are clear -for a macroscopic system each disorder realization no longer yields the same eigenvalue spectrum as a function of energy. Rather, any two disorder realizations will yield identical copies of the spectrum, but energetically displaced/offset from each other by the difference in the C ≡ C({ i })'s for the two realizations. This merely reflects the fact that for any given disorder , and compared to the Gaussian thermodynamic limit result eq. 27 with variance µ 2 E of eq. 16a (black line). The corresponding 'fully-averaged' spectrum of variance µ 2 E 0 discussed in text is also shown (blue). Inset: D(ω) vs ω/t (red) for an N = 4 × 4 square lattice (with same parameters as for the d = 1 case), again compared to the Gaussian eq. 27 with variance µ 2 E (black line).
realization, H ≡ H − 1 2 C and H have exactly the same eigenstates, which hence have the same L or E character; but their eigenvalues, while in 1:1 correspondence, are each shifted by the disorder-dependent C({ i }). The cure is obvious: to restore the sharp distinction between L and E states as a function of energy, as required for unambiguous identification of mobility edges, one needs only to eliminate these realization-dependent offsets; thereby referring all energies to a common origin independent of disorder (specifically TrH = 0 = TrH ), with eigenvalue fluctuations treated relative to their center of gravity for each disorder realization. It is of course precisely this which is captured by µ 2 E = Tr([H −TrH ] 2 ) (= Tr([H − TrH] 2 ) ), hence our focus on it rather than on µ 2 E0 . The 1BL situation just described for the case of H is in fact the norm when considering MBL (where the filling ν = N e /N is strictly non-vanishing in the thermodynamic limit). Here, whether H or H is considered, the center of gravity of the eigenvalue distribution fluctuates with disorder realization, with [TrH] 2 = ν 2 2 N + V 2 r 2 (eq. 14) and [TrH ] 2 = [ν − 1 2 ] 2 2 N + V 2 r 2 each inevitably ∝ N . The resolution of the disorderinduced offsets is precisely the same as for the 1BL example above: all energies are referred to a common origin (E = V r = TrH = TrH ), with eigenvalue fluctuations treated relative to their center of gravity (TrH or TrH ) for each disorder realization, as embodied in µ 2 E . With this, the (normalized) eigenvalue spectrum is given by the Gaussian Note again that µ 2 E is the same for both Hamiltonians H and H (eqs. 16a,26a), whence so too are their spectra D(ω). Indeed this is readily seen to be true for any Hamiltonian with a site-energy term i i (n i − ζ) with ζ an arbitrary disorder-independent constant, encompassing H and H as particular cases. It is eq. 27 we refer to in the following as the eigenvalue spectrum/DoS.
The previous considerations are salutary. If one does not account for disorder-induced energy offsets as above, and instead averages the eigenvalue distribution willy nilly over all disorder realizations, then if a sharp distinction between L or E states as a function of energy occurs, it will be lost (as above). The resultant averaged distribution in this case is again Gaussian with the same mean E = V r, but now with a variance µ 2 E0 = Tr([H − TrH ] 2 ) (with µ 2 E0 differing for H and H , eqs. 16b,26b respectively, and µ 2 E0 = µ 2 E for either H or H ).
These differences are clearly evident in finite-size calculations. We illustrate them in fig. 3, considering the Hamiltonian H (eq. 1) for the d = 1 chain, with a standard box distribution P ( ) = 1 W θ( 1 2 W − | |) for the siteenergy distribution (and choosing W/t = 12, V /t = 2). Results are shown for N = 16 sites at half-filling ν = 1 2 , generated from 200 disorder realizations. The resultant eigenvalue spectrum is shown [red], 11 and compared (solid line) to the Gaussian D(ω) eq. 27 with variance µ 2 E (eq. 16a). The latter is seen to be excellently captured, even for N = 16. The fully-averaged spectrum is also shown [blue], and likewise compared to a Gaussian of variance µ 2 E0 (eq. 16b), which similarly captures it well. The two spectra are visibly distinct (with µ 2 E0 > µ 2 E ), as expected from the considerations above.
The inset to fig. 3 also shows the numerical DoS for a d = 2-dimensional square lattice (dimensionality d entering both the mean eigenvalue E, eq. 13, and the variance eq. 16a via the interaction and hopping terms, V 2 µ 2 r and t 2 Z). Parameters considered are otherwise the same as those for d = 1 in the main figure, and comparison is made to the Gaussian D(ω) eq. 27 with variance eq. 16a. Despite the relatively meagre N = 4×4 real-space lattice, this again captures the numerics very well.
Finally, note that the DoS eq. 27 is obviously Ndependent, in two ways: via E ∝ N (which is trivially dealt with by referring energies relative to the band center ω = E), and because its standard deviation µ E ∝ √ N . As such, it is natural to rescale energies asω = (ω − E)/µ E , such that the DoSD(ω) (normalized to unity overω) is a standard normal distribution, We have considered a canonical model employed in studies of MBL: a disordered system of interacting spinless fermions, here on a d-dimensional lattice with N ≡ L d sites and N e fermions, for non-vanishing filling ν = N e /N . The model can be cast as an equivalent tightbinding model (eq. 4) on a locally connected Fock-space lattice of dimension N H ∝ e cN , the sites of which correspond to the many-particle states of the system in the absence of hopping. As such, precisely the same questions may be asked as for a conventional one-body TBM, including about the density of many-body states, and whether those states are FS localized or extended. We have barely touched on the latter question here; but, as a precursor to it, have considered the distributions -over FS lattice sites and/or disorder realizations as appropriate -of the system's eigenvalues, the FS site energies, and the local FS coordination numbers. All such are normally distributed, with variances in the thermodynamic limit that are readily determined, and found to be well captured by exact diagonalization on the small system sizes of up to N = 16 sites typically used in numerical work.
Two aspects of the results above may warrant final brief comment. First, from the discussion in sec. IV of the eigenvalue spectrum D(ω), eq. 27 (orD(ω), eq. 28), all but an exponentially small fraction of states lie in a ' √ N scaling window' about the band center ω = E, i.e. on energy scales set by µ E ∝ √ N . It is worth considering what implications this might have for many-body mobility edges (ME). Above a certain non-zero critical disorder W = W c , states at the band center -and by presumption all states -are MBL. What then happens as W is reduced below W c ? Without prejudice there would seem to be two distinct possibilities: either essentially all states become extended as W is decreased just below W c ; or not. Were the former to occur, we have nothing to say about it. But if the latter arises, one expects many-body MEs at [ω − E] = ω mob± to open up continuously about the band center, separating manybody localized from extended states; as indeed detailed numerical work finds. [12][13][14][15] In this case we simply point out that -by virtue of the µ E ∝ √ N scaling of the eigenvalue spectrum -mobility edge trajectories (as MEs move further into the band with decreasing W ) must likewise scale with √ N , and not with N . This does not of course preclude a subsequent crossover to MEs scaling with N itself, but by definition the latter can occur only when ω mob lies O(N ) away from the band center; and as such lies deep in the tails of the eigenvalue spectrum, where the fraction of states is exponentially small.
Second, note that a typical local coordination number for the FS lattice at non-zero filling is its average, Z ∝ N (eq. 24), which thus grows unboundedly in the thermodynamic limit. This is of course radically different from the one-body case in any finite dimension d (although the limit of infinite coordination number is familiar in the different context of dynamical mean-field theory 16 in d = ∞). A theory of localization in Fock-space must thus be able to explain how the occurrence of a divergent coordination number is in effect mitigated, such that an MBL transition exists in the thermodynamic limit N → ∞. We will suggest one way to do so in subsequent work. 8