The two-body random spin ensemble and a new type of quantum phase transition

We study the properties of a two-body random matrix ensemble for distinguishable spins. We require the ensemble to be invariant under the group of local transformations and analyze a parametrization in terms of the group parameters and the remaining parameters associated with the"entangling"part of the interaction. We then specialize to a spin chain with nearest neighbour interactions and numerically find a new type of quantum phase transition related to the strength of a random external field i.e. the time reversal breaking one body interaction term.


Introduction
Eugene Wigner introduced random matrix models about fifty years ago into nuclear physics [1]. The scope of applications has increased over the years [2,3] including fields such as molecular and atomic physics, mesoscopics and field theory. More recently random matrix theory has started to be used in quantum information theory [4,5,6,7,8]. For an introduction to such applications see [9]. There the concept of individual qubits and their interactions becomes important. This implies that we enter the field of two-body random ensembles (TBRE) [10,11], i.e. ensembles of Hamiltonians of n-body systems interacting by two-body forces. While such ensembles have received considerable attention, it was first focussed on fermions and later also included bosons. Yet in quantum information theory the qubits are taken to be distinguishable, and indeed the same holds for spintronics. Interest in both fields has sharply increased recently [12,13].
It is thus very pertinent to formulate and investigate TBRE's for distinguishable qubits. As random matrix ensembles are mainly determined by their symmetry properties this ensemble will be very different from other TBRE's. In particular, as the particles are distinguishable, their interaction can vary from particle pair to particle pair and can indeed be randomly distributed, thus introducing an entirely new aspect. This has the consequence that the topology according to which spins or qubits are distributed or interact will be important, Thus chains, trees and crystals of particles with nearest, second nearest and up to kth order interaction can be represented.
As mentioned above, random matrix ensembles are usually basically defined by the invariance group of their measure and, if that is not enough, some minimal information conditions [14,9] or independence condition [15]. Note that we deal with a symmetry of the ensemble, rather than with a symmetry of individual systems. The two concepts are to some degree complementary, and the former has also been called structural invariance. We propose an adequate definition for such ensembles in a very general framework in terms of independent Gaussian distributed variables. We then give an alternate representation in terms of the invariance group and variables that determine the orbits of the Hamiltonian on the ensemble under the action of the group.
In order to show the relevance of the new ensemble we address the simplest possible topology, namely the chain with nearest neighbour interactions. For this system we focus on the ensemble averaged structure of the ground state and demonstrate the existence of an unusual quantum phase transition [16], which is triggered by breaking of time-reversal invariance (TRI).
Entanglement, a key resource of quantum many-body systems in terms of quantum information, is to large extent related to quantum correlations, localization properties and quantum chaos. Entanglement has also been used as a property, alternative to long-range order in spatial correlation functions, to describe systems undergoing a quantum phase transition [17]. In one-dimensional systems such as quantum spin chains, it was shown [18] that the entanglement entropy of the ground state typically saturates or diverges logarithmically with size when approaching the thermodynamic limit. Furthermore, it has been shown that logarithmic divergence implies quantum criticality.
Interesting results emerge when a spatially homogeneous spin model is replaced by its disordered counterpart, where the spin interactions are taken at random. In this case there is often no physical justification why random interactions should still obey specific restricted forms such as Ising or Heisenberg interactions. In this context, we argue, it is more natural to use two-spin random ensembles (TSRE) for distinguishable particles, specifically choosing quantum spins 1/2, though these ensembles can readily be generalized to arbitrary spin. By construction these ensembles, as given in section 2, are invariant with respect to arbitrary local rotations, which we may view as gauge transformations. Another physical motivation for the definition of such ensembles is the coupling among arbitrary and perhaps mutually independent two level quantum systems which may come from completely different physical contexts such as e.g. twolevel atoms, Josephson junctions and photons.
In section 3 we concentrate on one-dimensional systems or spin chains and present results of numerical calculations, mainly based on density matrix renormalization group (DMRG) [19], in which we investigate entanglement and correlation properties of the ground state, averaged over an ensemble, and the average spectral gap to the first excited state as well as its fluctuations. If we include the interaction with an external random magnetic field, and hence TRI is broken, we find fast decay of correlations, saturation of entanglement entropy, and power law decay g ≈ N −0.4 of the spectral gap g with the system size N while its distribution displays Wigner-type level repulsion. When the strength of external field goes to zero, and time-reversal invariance is restored, we find long range order, logarithmically divergent entanglement entropy, and exponential decay of the spectral gap, while the level repulsion disappears.
We argue that this quantum phase transition is non-conventional from the point of view of established models, since in what we shall call non-critical case we still find slow power law closing of the spectral gap.

The embedded ensemble of spin Hamiltonians with random two-body interactions
In this section we define the two-spin random ensembles of Hamiltonians for systems with N distinguishable spins or qubits with at most two body interactions and describe its basic (invariance) properties. If we do not allow all spins to interact, we have to define which ones do. The simplest case will be a chain with nearest neighbour interactions, but in general we need a graph, whose vertices correspond to spins and whose edges correspond to two-body interactions. We proceed to formalize this.
Let G = (V, E) be an undirected graph with a finite set of N vertices V and a set of M edges E ⊂ V × V. In addition, let λ : V → R + and µ : E → R + be two nonnegative functions defined on the sets of vertices and edges, respectively. To such a graph we assign a 2 N dimensional Hilbert space of N spins or qubits, placed at its vertices H G = ⊗ j∈V C 2 ≡ C 2 N , and a set of N Pauli operators σ α j : H G → H G , α ∈ {1, 2, 3}, j ∈ V satisfying SO(3) commutation relations [σ α j , σ β k ] = iε αβγ σ γ j δ j,k . We also use the notation σ j = (σ 1 j , σ 2 j , σ 3 j ). Let A (j,k) ∈ R 3×3 , (j, k) ∈ E, be a set of M random real 3×3 matrices, and b (j) ∈ R 3 , j ∈ V, be a set of N random 3 dimensional real vectors. The TSRE then consists of the random Hamiltonians The above defined functions on the edges and vertices of the graph are used to determine the average strength of the corresponding terms in the Hamiltonian. The distribution of random two-body interaction matrices (for short also bond matrices) A (j,k) and the random external field vectors b (j) shall be uniquely determined by requiring the following two conditions: maximum local invariance and maximum independence expressed formally as: (i) An ensemble of Hamiltonians (1) should be invariant with respect to an arbitrary local SO(3) transformation, namely where O j ∈ SO(3) j , j ∈ V, meaning that the choice of local coordinate system is arbitrary for each spin/qubit. Obviously, (2) preserves the canonical commutation relations for the Pauli operators. Then it follows immediately that the joint probability distributions of {A (j,k) , b (j) } should be invariant with respect to transformations where O, O ′ are arbitrary independent SO(3) rotations for each j, k.
(ii) The matrix elements of the tensors A (j,k) α,β and of the vectors b (j) α should be independent random variables.
Following arguments similar to those presented in [15] it is straightforward to show that, in order to satisfy conditions (i) and (ii) above for pre-determined but general strengths of bonds µ(j, k) and external fields λ(j), A (j,k) α,β and b (j) α should be Gaussian independent random variables of zero mean and equal variance, which are uniquely specified in terms of the correlators where • denotes an ensemble average. We abbreviate the ensemble defined in this way by TSRE(G, µ, λ) noting that it depends on the graph and the strength functions µ and λ.
Let us now describe some other elementary properties of TSRE. We have seen that each member H of TSRE(G, µ, λ) can be parametrized (1) by 9M + 3N independent random parameters. However, we know for the classical ensembles, that a parametrization in terms of the structural invariance group, i.e. the invariance group of the ensemble and the remaining parameters is very useful. Similarly in the present case for an arbitrary set of local SO(3) rotations O j , the transformation of the parameters preserves the spectrum and all entanglement properties of H. In fact, the transformation (4) can be considered as a gauge transformation since, composed with the local canonical transformation (2), it preserves the Hamiltonian H (1) exactly. Two Hamiltonians, specified by and where is a standard canonical singular value decomposition (SVD) of the original bond matrix (1,2) . This symmetrization is unique provided that singular values of all SVD's (6) are non-degenerate which is the case for a generic member H. Thus the number of parameters specifying the bond matrices is 3 + 6(M − 1) = 6M − 3 and in addition to 3N external field parameters this gives K = 6M − 3 + 3N independent parameters. We recover the original set of parameters if we add the 3N parameters of the group of local rotations. Second, we consider the case of a ring graph with N vertices and M = N bonds, which is obtained from the previous case by specifying the periodic boundary condition N + 1 ≡ N. We see that a general H as given in (1) can now be symmetrized only if the additional topological condition R := R N · · · R 2 R 1 = ½ is satisfied. Now all but one bond matrix can be symmetrized, for example the last one may in addition be multiplied by a topological rotation A (N,1) ′ = A (N,1) ′ symmetric R leading to three additional parameters. Therefore in the case of a ring graph we have K = 6M + 3N independent parameters; again adding the 3N = 3M parameters of the local rotations we obtain the full set of parameters.
We can now consider the case of a general (connected) graph. From the previous examples it is evident that the only crucial additional parameter is the number L of primitive cycles, i.e. such cycles which cannot be decomposed into other primitive cycles. It is clear that each primitive cycle adds 3 additional topological parameters (or one SO(3) topological R matrix) to the 6M − 3 + 3N parameters which we would have for the case of a tree graph. Therefore we have Counting the cycle-contributions and taking the primitivity criterion into account again we obtain the the total number of parameters by adding those of the local rotations. The above considerations hold for the case of general bond and vertex strength functions µ and λ. If, however, these functions are degenerate, or even constant, i.e. the average interaction strength and field strength do not depend on edges/vertices of the graph, then the structural invariance group of the TSRE may be even larger. In the latter case this group is obtained as a semi-direct product of a discrete symmetry group of the graph G and the gauge group of local rotations, the latter being the normal subgroup. We hope that the considerable invariance properties of the TSRE will prove useful in a future analytical treatment of its properties.

Properties of ground states of the TSRE on a 1D chain
In this section we shall only consider the simplest case of a TSRE on a 1D chain of N vertices. In addition, we consider the most symmetric case of constant strength functions, say µ(j, j + 1) ≡ 1 and λ(j) ≡ λ = const. Such an ensemble of random spin chain Hamiltonians shall be designated as TSRE (N, λ) where we explicitly assume open boundary conditions; we shall, however, also consider a ring graph with periodic boundary conditions in which case we shall stress this separately. In particular we shall be interested in the zero temperature (ground state) properties of TSRE(N, λ). We note that due to the large co-dimension of bond strength space the standard perturbative renormalization group of decimating the strongest bonds [20] would not work and we have at this point to rely on a brute numerical investigation. Still, it turns out that most zero temperature properties of TSRE(N, λ) can be efficiently simulated using White's density matrix renormalization group (DMRG) finitesize algorithm [19] by which spin chains of sizes up to N = 80 could at present be achieved. One should not forget that all numerical estimates of ensemble average or expectation value of some physical quantity A, which will be designated as A , require averaging over many, say N r , realizations from TSRE(N, λ), such that the statistical error estimated as σ A ∼ ( A 2 − A 2 )/N r is sufficiently small. Due to the lack of translational invariance the implementation of the DMRG is non-trivial and was only done for a chain with open boundaries.
We note that for λ = 0, any H as defined in (8), or even more generally in (1), commutes with the following anti-unitary time-reversal operation so, for odd N, all eigenvalues of H have to be doubly degenerate (Kramer's degeneracy [15]). However, as this represents more a technical than conceptual problem for the ground state (or ground plane) properties of TSRE(N, λ) we shall in the following restrict ourselves to the case of even N.

Distribution of the spectral gap
Let |0 , |1 , represent the ground state, and the first excited state, of H, with eigenenergies E 0 , and E 1 , respectively. It is well known that the crucial quantity which determines the rate of relaxation of zero-temperature quantum dynamics is the spectral To clarify the difference with respect to TRI, we plot the normalized gap distribution dP/dg whereg = g/ g for both cases, together with the theoretical level spacing distributions for the Gaussian unitary ensemble (GUE) of random matrices [21] and for an uncorrelated Poissonian spectrum (Fig. 1). For the non-TRI case we choose λ = 1 and observe, to our accuracy, good agreement with the GUE case for two chain sizes N = 10 and N = 16. Our results suggest that the GUE-like gap distribution, exhibiting level repulsion, also holds in the thermodynamic limit N → ∞; numerical results for odd N give the same results. In the case of λ = 0 the level repulsion between |0 and |1 gradually vanishes as we approach the thermodynamic limit, although no conclusive statement can be made about the limiting distribution. For odd N, the ground state is degenerate for λ = 0 and the present analysis does not apply.

Size scaling of the spectral gap
Being interested in the thermodynamic limit, it is an important issue to understand how g(N) scales with N. The theory of quantum criticality [16] states that g remains finite in the thermodynamic limit for non-critical systems, and rapidly converges to zero, as N → ∞ for critical systems.
In Fig.2 we plot the ensemble averaged spectral gap g versus N for different values of the field strength λ. We find a clear indication that in the non-TRI case the spectral gap exhibits universal asymptotic power law scaling whereas in the TRI case, λ = 0, the asymptotic decay of the gap is faster than a power law, perhaps exponential g ∼ exp(−ξN), with ξ ≈ 0.07 ± 0.02. According to the standard theory [16] both cases, λ = 0 and λ = 0, should be classified as quantum critical, however as we shall see later, the case of slow power-law decaying average gap (10) has many-features of non-critical systems, such as finite correlation length and finite (saturated) entanglement entropy. Therefore we shall, at least for the purposes of the present paper, name the case λ = 0 as random non-critical (RNC) and the case λ = 0 as random critical (RC). We note that the results for odd and even N are in agreement in the RNC case. Also results for the case with periodic boundary conditions up to N = 20 show no significant difference from the results in Fig. 2 for any λ. which measures the entanglement in the ground state between two equal halves of the chain. It has been suggested in non-random systems [18] that for critical cases S ∝ log 2 N whereas in non-critical cases S saturates in the thermodynamic limit. Indeed, as shown in Fig.3, we find for the TSRE(N, λ) that S saturates to a constant finite S ∞ (λ) = lim N →∞ S(N, λ) for the RNC case λ = 0, while in the RC case it grows logarithmically We also note an interesting even-odd-N/2 effect which slowly diminishes as we approach the thermodynamic limit. As pointed out in Ref. [22] such an effect is induced by open boundary conditions. For periodic boundary conditions the entanglement entropy for RNC case is twice as large as in the TSRE (N, λ) with open boundaries. This further confirms the conjecture that only short-range correlations around the boundary between the two halves contribute to the entanglement. We note that our result is essentially different from results for other models, which can be obtained by perturbative real space renormalization group [20], for example for the disordered critical Heisenberg chain [23], where c = (ln 2)/3 and is in general model dependent [24].
The fact that the entanglement is reduced in the RNC case with local disorder can be explained by chaotic behavior [25] signalized by the level repulsion in the gap distribution. A similar effect can be observed in localization properties where hopping of excitations induced by inter-particle interactions is diminished by introducing local disorder [26,27]. This effect of increased localization is useful for successful quantum computing and has important consequences for transport properties such as conductivity [28].

Correlation functions
The most direct probe of criticality is perhaps to investigate of long-range order and (space) correlation functions. In order to do this we compute the ensemble averaged fluctuation of the spin-spin correlation function between two vertices Note that we have to consider average fluctuations of the spin-spin correlation function instead of the correlation function itself, since the latter have to vanish due to the local  gauge invariance properties of the TSRE. Because of the local invariance of the TSRE it is enough to consider a single type of correlation function, as the RHS of (13) does not depend on indices α, β if j = k; in fact in numerical computations we average over α, β in order to improve statistics. We consider the Hamiltonian (8) with periodic boundary conditions. This allows us to average the fluctuation of the correlation over the chain and hence C(r) = 1 N i C(i, i + r). We expect that the results would be qualitatively the same for the model with open boundaries and sufficiently large N, but we obtain better statistics in this way. Fig. 4 shows the averaged correlation function fluctuation C(r) for a few choices of the control parameter λ and of the chain lengths N = 16, 20, 24. The results for chains of different lengths coincide for small distances r whereas for larger r finite-size effects are noticeable. For sufficiently large chains it can be conjectured that the fluctuations (or effective correlations) asymptotically decay in the RNC case as C(r) ≍ C 0 2 −r/ξ with a finite correlation length ξ whereas the decay for the RC case is slower than exponential, perhaps a power-law, which indicates long-range order ξ = ∞.

Correlation length and the entanglement entropy saturation value
In Fig. 4 we observe that the degree of localization depends on the control parameter λ and the correlation function C(r) decays on larger scales as we approach the critical point λ = 0 which results in a larger correlation length ξ. Eventually, the correlation length becomes infinite at the critical point λ = 0. In Fig. 5  ξ Figure 5. The correlation length ξ as a function of the control parameter λ. Open and full symbols designate correlation length obtained from the best of C(r) with c exp(−r/ξ) on sets r ∈ {4, . . . , 8} and r ∈ {5, . . . , 7}, respectively, all for N = 24. The inset demonstrates ∝ − log 2 λ scaling. Please note that last point at λ = 0.1 is rather inaccurate due to insufficiently large system size N , so it is not used in a linear fit (full line) in the inset, for which other points with λ ≤ 1 have been used.
of the correlation length ξ on the control parameter λ as obtained by exponential fit of C(r) for a finite size N = 24. Unlike conventional phase transition as in e.g. [17], the correlation length seems to diverge logarithmically as ξ(λ) ∼ −ξ 0 log 2 λ + const with ξ 0 = 0.26 (indicated in the inset of fig.5), even though algebraic scaling cannot be entirely excluded with the numerical data that are available at present. Large correlation length has strong effect on entanglement. Long range correlations demand longer chains for the entanglement entropy to saturate whereas the saturation value itself also grows when the critical point is approached. In Fig. 6 we plot the entanglement entropy saturation value for various values of parameter λ and observe similar behavior as for the correlation length. However, unlike the correlation length the quantity which diverges logarithmically when λ → 0 is not the entanglement entropy S but its exponential. In fact, numerical data for small λ show good agreement with 2 kS ∝ − log 2 λ + const, where k = 4.0 (indicated in the inset of fig.6). Note that 2 S is to  Figure 6. Saturated entanglement entropy S ∞ as a function of the control parameter λ. The inset demonstrates ∝ log 2 log 2 (λ ⋆ /λ) scaling, where λ * = 4, which is indicated by a straight line obtained from a best fit to points with λ < 1.
a good approximation proportional to the effective rank, or the Schmidt number of the ground state χ ǫ [18,29], which denotes the number of eigenvalues of the reduced density matrix needed to describe the state of the system up to an error ǫ. In fact, the effective rank χ ǫ , rather than the entanglement entropy, is the decisive indicator of simulability by the DMRG method [30] and, we believe, also a relevant quantity in the description of a quantum phase transition.

Conclusions
In the present paper we have defined a two-body random matrix ensemble of independent spin Hamiltonians which are invariant under local SO(3) transformations and described them in a framework of undirected graphs. As the simplest example, we have studied a chain with nearest-neighbour interactions in a random external field and observed a non-conventional phase transition when the external field is switched off. The system is always critical in conventional terminology as it has a vanishing gap in the thermodynamic limit in all cases studied. Yet we have shown that, in the presence of a random external field breaking time-reversal invariance, the locally disordered system has many properties of non-critical systems such as finite correlation length and finite bipartite entanglement entropy in the thermodynamic limit, whereas the gap decay obeys a universal power law dependence. The transition towards the critical point with vanishing of the external field exhibits logarithmic divergence for the correlation length and the effective rank of the ground state. We have no explanation for the logarithmic behavior in the quantum phase transition. The model proposed is much richer than the example discussed. Thus we expect, that higher connectivity of the graph will yield very different results, but even an exploration of high temperature behaviour for the chain seems very worthwhile. In view of the large structural invariance group of the ensemble in the case of site independent average coupling and external fields we hope, that some analytic results can be obtained for this ensemble.