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Localized eigenstates with enhanced entanglement in quantum Heisenberg spin-glasses

Arun KannawadiDepartment of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA

Auditya SharmaInternational Institute of Physics - Federal University of Rio Grande do Norte, Natal, RN, Brazil and

Tel Aviv University, Tel Aviv, Israel

Arul LakshminarayanDepartment of Physics, Indian Institute of Technology Madras, Chennai 600036

The concurrence versus participation ratio phase diagram of the eigenstates of the quantum infiniterange Heisenberg spin glass shows two distinct separate clouds. We show that the ‘special states’that agglomerate away from the main one, are precisely those that are obtained by ‘promoting’ theeigenstates of a lower number sector of the Hamiltonian to particle added eigenstates of a highernumber sector of the Hamiltonian. We compare the properties of these states with states of similarstructure constructed from GOE random matrices that are easier to understand. In particular,we obtain the scaling behaviour of average entanglement of these special states with system size.By studying a power-law decay Hamiltonian, we see a merger of the main cloud into these specialstates as we go away from the infinite-range model and move towards short-range models. Thiscould indicate that short-range quantum spin glasses are essentially different from the infinite rangemodel.

I. INTRODUCTION

The last decade or so has seen a proliferation of inter-est in understanding quantum condensed matter systemsfrom an entanglement perspective; for a somewhat earlyreview see [1]. The notion of entanglement [2, 3] andhow to measure it has developed through the aid of anumber of works within the general framework of quan-tum information. The condensed matter systems studiedhave concentrated around “clean” systems that displayquantum phase transitions. Exceptions include work onthe von Neumann, or block, entropy in disordered modelswhere it has been shown that the kind of scaling found incorresponding clean critical chains (∼ logL, for a chainof length L) persist and may even be enhanced in thepresence of quenched disorder [4].Another somewhat different but related tack of re-

search has been on disordered systems from the point ofview of quantum chaos and random matrix theories, forexample [5]. Here the applicability of measures conven-tionally used for few-body quantized classically chaoticsystems has been sought to be applied to many-bodysystems without an apparent classical limit, chaotic orotherwise. The rationale being that rather than a puta-tive classical limit, it is the nonintegrable nature of themodels that determine if random matrix theories maybeapplicable. Nonintegrability could naturally be found inclean systems as well and such systems display some sig-natures conventionally attributed to quantum chaos, forexample see [6, 7]. However the well-developed theoriesof random matrices [8, 9] especially the Gaussian ensem-bles, are applicable if there are many-body interactionsthat tend to make the Hamiltonians full matrices ratherthan the typical sparse matrices that arise out of the typ-ically two-body interactions of many-body systems. The

notion of “two-body-random ensembles” and “embeddedensembles” developed mostly within nuclear physics wasprecisely to plug this lacuna [10]. However its applica-bility to condensed matter systems is largely unexplored.It has also been pointed out that enhanced multipartiteentanglement in disordered spin systems is possible thatcan be useful for multiport quantum dense coding [11].

The present work maybe seen in this context as anexploration of entanglement in disordered many bodyspin 1/2 particles. The disorder is via the interactionthat is two-body type and long-ranged. The features inthese systems maybe compared to what is expected fromconventional random matrix theories. Also unlike moststudies related to condensed matter this work looks atnot just ground states but indeed excited states as well.In particular we will study single and two-particle (ormagnon) sectors. There have been many interesting stud-ies related to quantum communication across spin chainswhere such subspaces have played a dominant role, seefor an overview [12]. Being the simplest subspaces wealso concentrate on these, although they may not con-tain the ground state. The kind of systems we study maythen be either classified simply as long-ranged disorderedHeisenberg models or quantum Heisenberg spin-glasses.Although spin glasses have been around for four decades,most studies have focussed on the classical version. SeeTalagrand [13] and references therein. Quantum spinglasses have also naturally been considered, for exam-ple [14]. A reason why quantum spin glasses have re-ceived less attention though is that numerical techniquesto study quantum problems are less developed, as op-posed to the classical world, where Monte Carlo methodsare very advanced [15]. The infinite-range Sherrington-Kirkpatrick spin glass is the prototype model in classicalspin glasses [16], the quantum version of which is the

2

starting point of our study here.

Definite-particle states are natural to quantum systemsthat are pure and conserve particle number or total spin.A m−particle state lies in the subspace of the Hilbertspace which is spanned by the basis vectors that havem-number of ‘1’s (or equivalently ‘0’s) when expressedin spin-z basis. In one-particle states (one spin up orone spin down) there is a clear monotonic relation [17–19] between localization, for example as measured by theparticipation ratio, and the inter-spin entanglement asmeasured say by concurrence or tangle: the more the lo-calization, the lesser the entanglement. While this haslong been appreciated, the case of higher particle num-bers or spins is more complicated. There is no such mono-tonic relationship between localization and entanglementeven for two-particle states. To study this in the simpleststatistical context, random definite particle states werestudied using an ensemble that was uniformly distributedin such subspaces [20]. From this it is known for examplethat while the expected entanglement between two spins(or qubits in the language of quantum information) forone particle states having L qubits scales as 1/L and thatof two-particle states scale as 1/L2, in the case of threeor more particles entanglement is practically absent andis (super) exponentially small in L. This is consistentwith such states having larger multipartite entanglementand the fact that entanglement moves away from beinglocally shared. In some sense the “environment” seen insuch cases, by for example two given spins, is too largefor entanglement to remain intact.

Here we make use of concurrence as a measure of inter-spin entanglement, and with the aid of numerical exactdiagonalization and some analytical calculation, pointout and explain a striking class of entangled states thatarise from the consideration of the quantum Heisenberg

spin-glass . Our motivation comes from trying to un-derstand some of these striking features observed in arecent hitherto unreported study [21] of entanglement inquantum spin-glasses . In particular the two-particle sec-tor states separate into two classes, whose entanglementscales very differently with the total number of spins L.Even those with a smaller amount of entanglement arestill larger than those expected of random states. Thusthese highly disordered Hamiltonians are still quite farfrom behaving as random states that are uniformly dis-tributed in the definite particle subspace. Fig. 1 showsup-front two diagrams, one featuring ‘average concur-rence’ and the other ‘particpation ratio’ of the ‘N↑ = 2’sector of the eigenstates of quantum Heisenberg infiniterange spin glass. The illumination of the why-and-howand the consequences of these spikes observed is the coremessage of this paper.

The structure of the paper is as follows. In Sec. II, wedescribe the spin-glass Hamiltonian under study and thequantum properties of the system we are interested in.We also highlight the existence of a special class of eigen-states, which we call as ‘promoted states’. These statesform the subject of this work. In Sec. III, we obtain a few

analytical results for the quantities described in Sec. IIwhen the system is in a ‘promoted’ state, with a numberof assumptions and compare with two specific spin-glassmodels namely the Infinite range and Nearest-Neighbourmodels. In Sec. IV, we show systematically that the ‘pro-moted’ states, that were distinguishable from the rest ofthe eigenstates from an entanglement perspective in thecase of the Infinite range spin-glass model, become indis-tiguishable as the range of interaction between the spinsbecomes smaller. The last section summarizes the con-clusions of our work, and offers an outlook for futurework.

II. FORMULATION OF THE PROBLEM

A. Symmetries of the Spin-glass Hamiltonian

We start by considering a large number (L) of qubits,labelled arbitrarily from 1 to L. The generic quantumHeisenberg model is given by

H =∑

i<j

Jij [σxi σ

xj + σy

i σyj + σz

i σzj ] =

∑

i<j

Jij ~σi. ~σj (1)

where Jij represents the ‘interaction strength’ betweenthe qubits i and j. If all the Jij s are negative(positive),then the system is a ferromagnet(anti-ferromagnet). IfJij s have mixed signs, then the system is said to be ‘frus-trated’ and no long-range order maybe found as in [22],with a few exceptions, see for example [23].A useful way to think of this Hamiltonian is to replace

the ~σi. ~σj with 2Sij − 1, where Sij is the swap operatorthat interchanges the z-component of the spins of qubitsi and j. It is immediately clear that the Hamiltonianconnects only states of the same particle number (num-ber of spin-up qubits). Thus, when expressed in the σzbasis (or Fock state basis), the Hamiltonian takes a blockdiagonal form, where each block is characterized by a def-inite “particle number”N↑, which is nothing but the totalnumber of spins up in the z direction. This is a great sim-plification since we can study only one or a few definiteparticle sectors at a time and the associated Hilbert spaceHN↑

is much smaller and grows only polynomially withL, the number of qubits. This definite particle structureis of course a manifestation of the fact that the systemcan have any of the L + 1 allowed values for the totalz-spin, but once a value is assumed, it is conserved untilacted by a random external magnetic field. Mathemat-ically, it is the consequence of rotational symmetry andin particular, of the fact that σz =

∑

i σzi commutes with

the Hamiltonian. Considering the isotropy of the Hamil-tonian, it also follows that the operators σ± =

∑

i σ±i

too commute with the Hamiltonian. This implies that if|ψ〉 is an eigenstate of H , then σ±|ψ〉 must also be aneigenstate of H with the same eigenvalue. The particleadded state σ+|ψ〉 is referred to as a “promoted” statecorresponding to |ψ〉.

3

(a)

0

50

100

150

200

250

300

-50 -40 -30 -20 -10 0 10 20 30 40 50

Par

ticip

atio

n R

atio

(P

R)

Shifted Eigenvalues E-SJ (b)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

-50 -40 -30 -20 -10 0 10 20 30 40 50

Ave

rage

Con

curr

ence

Shifted Eigenvalues E-SJ

FIG. 1. A typical realization of the Hamiltonian for L = 25, N↑ = 2. We plot the participation ratio and average concurrence(averaged over all pairs of spins chosen from the L spins) of all the eigenstates of the infinite range quantum Heisenberg spinglass arranged by their eigenvalues. SJ refers to Σi,jJij , the energy of the all-one state.

For any definite-particle sector of the Hamiltonian, onecan show (readily from the swap operator form) that thestate with equal coefficients for all the basis states is nec-essarily an eigenstate, with eigenvalue SJ :=

∑

i<j Jij .This corresponds to states promoted progressively from

the 0−particle state |0〉⊗Land we will refer to such states

as ‘all-one’ states of the appropriate particle number.

The two models of spin-glass that we use intensivelyin this work are

1. Infinite-range model where Jij ∼ N (0, 1) ∀i < j.

2. Nearest-Neighbour model where Jij ∼ N (0, 1) iffj = i−1 else 0, with a periodic condition such thatj = 0 corresponds to j = L.

Here, N (0, 1) stands for the standard normal distri-bution. Also, we briefly study a one-dimensional powerlaw decay model that contains the above two models asspecial cases.

B. Entanglement & Localization

As stated in the introduction, for 1−particle states,there is a direct relation between the localization of astate and the average entanglement between qubits [17–19] while no definite relation is known to exist in thecase of higher particle states. We will be interested inconcurrence as a measure of bipartite entanglement andparticipation ratio (PR) as a measure of localization.

Concurrence is the entanglement between any two two-level systems in a mixed or pure state. Therefore it is agood measure of entanglement between any two qubits orspin-1/2 particles in a general many-body state. For def-inite particle states, it is known that the reduced density

matrix of any two spins takes the following form [24]:

ρ =

v 0 0 00 w z 00 z∗ x 00 0 0 y

(2)

With ρ of such a form, concurrence has a simple expres-sion [24]:

C(ρ) = max (2(|z| − √vy), 0) . (3)

Appendix describes a simple algorithm to do a fast com-putation of C for Hamiltonians with this symmetry.Next, we recall that the inverse participation ratio is

a basis dependent quantity that is defined as follows. Ifa state |ψ〉 =∑i ai|i〉, where |i〉 are the kets in the com-putational basis, then IPR =

∑

i a4i , from which the

participation ratio PR = 1IPR is immediately obtained.

It is typical to consider the Sz basis for spin-systems, andwe do the same.During our study of entanglement in the spin-glass sys-

tems [21], in addition to the high-PR-high-concurrenceall-one state, several other states with high concurrencewere observed. But these states did not stand out whena similar plot of PR was made. Refer Fig. 1. It is clearthat the eigenstates would clearly separate out into twodistinct clouds in a phase diagram where we plot the av-erage concurrence against PR (+ points in Fig. 2). Thespikes at E−SJ = 0 in both Fig. 1 (a) and (b) correspondto the 2−particle all-one state. While no other significantspikes are found in Fig. 1(a), several such spikes are foundin Fig. 1(b), with some of them being greater than thespike for all-one state. The authors were motivated bythis observation to investigate further, which resulted inthis work.Note that the 2−particle all-one state is not the

state with the highest average concurrence, while the1−particle all-one state, with pairwise concurrence be-tween any two qubits being 2/L, has the maximum pos-sible average pairwise entanglement [17]. Next, we show

4

that the these spikes are in fact the set of all ‘promoted’eigenstates, arising due to the isotropy of the Hamilto-nian.

III. PROMOTED STATES

Recollect that promoted states are of the form σ+ |ψ〉.It is easy to show that the operator σ+σ− leaves thepromoted states invariant, upto a scale factor. We markthose states that have this property in Fig 2 with boxes.Also we include data for random states and ‘promoted’random states.

0

0.01

0.02

0.03

0.04

0.05

0 20 40 60 80 100 120 140 160

Ave

rage

Con

curr

ence

Participation ratio (PR)

Spin-glass 2-particle statesSpin-glass promoted 2-particle states

Random 2-particle statesRandom promoted 2-particle states

FIG. 2. Scatter plot the average concurrence vs. the partic-ipation ratio (PR)of all the eigenstates of the N↑ = 2 sectorin infinite range quantum Heisenberg spin glass for L = 25.Data for random states and random-promoted states are in-cluded.

It is known [20] that concurrence of a random2−particle state (N↑ = 2) is given by 16

L2π3/2 , which withL = 25 gives ∼ 0.005. We infer that the eigenstates ofthese Hamiltonians tend to have higher concurrence andare more localized than random states (circles). Clearly,the distribution of points for the eigenstates arising fromthe Hamiltonian is very different from what is expectedfor random states. This immediately suggests that en-tanglement properties of random states, such as scalingbehavior [20] may not be found for eigenstates of spinsystems, at least those with two-body interactions [21].

The most striking feature in the infinite-range modelis that the eigenstates cluster into two separate clouds.From Fig. 2, we see that it is the promoted states, identi-fied and labelled as boxes (over + points), that stand outand form a separate cloud. These promoted states alsotend to be distinct when we consider 2-qubit and 3-qubitentanglement in 3−particle sector as we verified (figurenot included).

0.5

0.55 0.566

0.6

0.65

0.7

0.75

0.8

0.85

10 20 30 40 50 60 70 80 90 100 110 120

Pro

babi

lity

of n

on-z

ero

conc

urre

nce

No. of qubits L

Random PromoInfinite Range

Nearest-Neighbour

0.5

0.55 0.566

0.6

0.65

0.7

0.75

0.8

0.85

10 20 30 40 50 60 70 80 90 100 110 120

Pro

babi

lity

of n

on-z

ero

conc

urre

nce

No. of qubits L

FIG. 3. P (C > 0) plotted against L. The dashed lines arethe fits taking into account of these errorbars and the dotted-dashed lines are fits without taking the errorbars into acc-count. Points with L < 8 are not considered for finding fitparameters.

0.01

0.1

10 20 30 40 50 60 70 80 90 100 110 120

Ave

rage

Con

curr

ence

No. of qubits L

Nearest-NeighbourInfinite Range

Random PromoAnalytical2-particle

Upper bound

FIG. 4. Scaling behaviour of the promoted states comparedagainst the scaling behaviour expected for random 2−particlestates, which is 16

L2π3/2 and random promoted 2−particle

states and its estimate. ‘Analytical’ refers to curve 0.465/L.The solid line corresponds to the upper bound given inSec. III B. The fit parameters are given in Table III. They-axis is in logscale for better visibility.

A. Random Promoted States

The correlations in the matrix elements and the struc-ture of the Hamiltonian make any analytical study of thespin-glass eigenstates complicated. In order to under-stand why the ‘promoted’ nature makes states special,we study random promoted states which are amenableto an analytical approach. From the orthogonality con-dition applied with respect to the ‘all-one’ eigenstate, forall other eigenstates, the coefficients in the standard basismust add to zero. We will enforce this property exten-sively on the random promoted states to simplify our ex-

5

pressions. Random 1−particle states, by definition, spanthe surface of the unit sphere in the associated Hilbertspace uniformly. In addition, we impose the constraintthat the states must lie in the hyperplane orthogonal tothe all-one state, to mimic the promoted states arising inspin-glass systems. Thus, if we denote an unnormalizedrandom 1−particle state as

|ψ1p〉 =∑

i

a′i |i〉 , (4)

where |i〉 refers to the 1−particle basis state where onlythe qubit at i is in |1〉 state and the rest in |0〉, thenthe a′i s come from a j.p.d.f containing δ (

∑

i a′i) but we

shall still consider them as i.i.d variables for all practicalpurposes. Isotropy in the Hilbert space is achieved whentheir marginals happen to be the standard normal distri-bution N (0, 1), see for example [25]. The normalizationis assumed to only set the scale of the coefficients and thenormalized coefficients can continued to be treated inde-pendent of each other [20]. Although we restrict ourselvesto the subspace that satisfies

∑

i ai = 0 for theoreticalanalyses that follow, in practice, we do not include thiscondition in the Monte Carlo simulations.The action of the ‘promotion-operator’, σ+, on a

1−particle state is given by

∑

k

ak |k〉 σ+

−−→∑

i<j

aij |ij〉 , where aij ∼ ai + aj . (5)

Here, |ij〉 refers to the 2−particle basis state where thequbits at i and j are in |1〉 state and rest in |0〉. TheIPR of the promoted 2−particle state can be expressedin terms of the IPR of the corresponding 1−particle statewhen

∑

i ai = 0:

∑

i<j

a4ij =1

(L− 2)2

(

(L− 8)∑

i

a4i + 3

)

, (6)

where aij = (ai + aj)/√L− 2, are the normalized co-

efficients. In finding the normalization, the condition∑

i ai = 0 is used, which is the case for eigenstates ofspin-glass Hamiltonian. In a more general case, the nor-malization depends on the coefficients themselves. Weobserve in passing the somewhat amusing fact that what-ever maybe the 1−particle state, when L = 8 the pro-moted 2−particle state has an IPR of exactly 1/12, pro-vided the coefficients sum to zero.For random N dimensional states, the average IPR

〈I〉 ∼ 3/N [26]. Using this in Eq. 6 for random2−particle promoted states, the average IPR

〈Ipromoted 2p〉 ∼6

L2, (7)

same as what we expect for “genuine” 2−particle randomstates, as the dimensionality of such a space is ∼ L2/2,and is confirmed by Fig. 2. Thus as far as localization is

concerned there is no difference, typically, between pro-moted and genuine 2−particle states. We find a verydifferent situation regarding quantum correlations suchas entanglement to which we now turn.The elements of the two-spin reduced density matrix

that are involved in the entanglement between them, asquantified by concurrence, are z, v, y (refer to Eqs. 2 and3). For 2−particle states, when ρ is the density matrix ofspins at positions 1 and 2 (which we consider for simplic-ity and without any loss of generality), these elementsare

y = a212, z =

L∑

k=3

a2ka1k, v =

L∑

k,l=3k<l

a2kl. (8)

Note that we are considering real state ensembles. Wemay expect that while all the three quantities are ran-dom variables, y being a single term has more fluctuationthan the other two which are sums. For generic random2−particle states 〈|z|2〉 ∼ 4/L3 and 〈|z|〉2 = (2/π)〈|z|2〉[20].However for 2−particle states promoted from

1−particle states obeying∑

i ai = 0, it is straightfor-ward to show that, up to the leading order,

y ≈ (a1+a2)2/L, z ≈ (1+La1a2)/L, v ≈ 1. (9)

Therefore although z appears as a sum of order L num-ber of terms, it simplifies for promoted states to this sim-ple form which implies that both |z| and √

y are of thesame order of magnitude, namely 1/L. This follows since

ai ∼ 1/√L. As v = O(1), the concurrence in promoted

2−particle states is always in a fine balance between thetwo competing terms |z| and √

y. In contrast, for generic2−particle states, the order of

√y is 1/L which is much

larger than the order of |z| which is 1/L3/2, resulting inthe probability that the concurrence is nonzero decreas-ing as 1/

√L [20].

The probability that the concurrence is nonzero forpromoted 2−particle states is now estimated. This isthe same as P (z2 > y), which using Eq. (9) results in the

following where xi =√Lai are independently distributed

according to N (0, 1).

P (C > 0) = P [(1− x21)(1− x22) > 0] =

erf2(

1√2

)

+ erfc2(

1√2

)

≈ 0.566.(10)

Note that as v < 1 in reality, the above can be expectedto underestimate the actual probability. It is also worthrecounting that random 1−particle states have a proba-bility 1 that the concurrence is nonzero.Fig. 3 shows how P (C > 0) varies with L for differ-

ent systems. We fit a model of the form p + q/Lr tothe data points from Monte Carlo simulations for ran-dom promoted states (without enforcing

∑

i ai = 0), andthe values for the parameters obtained are tabulated inTable I.

6

Parameter Estimate

p 0.564± 0.002

q 0.426± 0.032

r 0.754± 0.042

TABLE I. Best fit curve p + q/Lr to P (C > 0) vs. L forrandom promoted 2−particle states. Fits have been made forL ≥ 8.

The probability of non-zero concurrence approachesfrom above the theoretical asymptotic value of 0.566slower than 1/L. In sharp contrast, the probability ofnon-zero concurrence increases with L for the spin-glasssystems, with the nearest-neighbour model increasingfaster than the infinite-range model. The P (C > 0)curve is described an exponential curve of the formp − q exp (−L/r). The functional forms are purely datadriven and are not motivated by any physical argument.

Parameter Infinite range Nearest-Neighbour

p 0.660 ± 0.004 0.834 ± 0.003

q 0.106 ± 0.003 0.402 ± 0.004

r 59.565 ± 5.365 21.891 ± 0.585

TABLE II. Best fit curves p−q exp (−L/r) to P (C > 0) vs. Lfor promoted 2−particle states of Infinite range and Nearest-Neighbour spin-glass models. Fits have been made for L ≥ 8.

The average concurrence of random promoted2−particle states may also be estimated as

〈C〉 ≈ 2

∫

∏i(1−x2

i )>0

(|z| − √vy)e−(x2

1+x22)/2dx1dx2

≈ 0.465

L,

(11)

where xi =√Lai and Eq. 9 and the assumption of

independent marginals have been used. The final resultwas obtained by setting v = 1 and factoring out the Ldependence and the L-independent integral was evalu-ated numerically to obtain 0.465. The 1/L behaviouris to be compared with generic 2−particle states thathave an average concurrence ∼ 1/L2 [20], and that forgeneric random 1−particle states which goes as ∼ 1/L[17]. The promoted states have a smaller entanglementthan 1−particle states, however they are much largerthan what maybe expected for generic 2−particle states.Interestingly, for the promoted 0−particle state i.e. the

all-one state in the 1−particle (N↑=1) sector, the con-currence between any two spins is 2/L and in 2−particle(N↑=2) sector, it is

C =2(

L2

)

(

L− 2−√

(L2 − 5L+ 6)

2

)

, (12)

which also scales as 1/L.

Fig. 4 shows how in our Monte Carlo simulationsthe average concurrence scales with the size for dif-ferent systems. The exponent is smaller for the pro-moted 2−particle eigenstates when compared to pro-moted 2−particle random states. We fit a model 〈C〉 =b/La to the points (L ≥ 8) and obtain parameters dis-played in Table III.

Model a b

Infinite Range 1.063 ± 0.008 0.832 ± 0.023

Nearest-Neighbour 0.758 ± 0.003 0.570 ± 0.006

Random Promo 1.138 ± 0.004 0.900 ± 0.014

TABLE III. Best fit curve b/La to 〈C〉 vs L. Fits have beenmade for L ≥ 8.

Thus the promoted random 2−particle states retainthe larger entanglement present in 1−particle stateswhile being delocalized like 2−particle states. It is inter-esting that this feature of random states is present intactin the eigenstates of quantum spin glass Hamiltonianswith long range interactions.

B. Heuristic understanding of the deviations

From Fig 4, it is clear that the infinite range spin-glassHamiltonian and nearest-neighbour (NN) Hamiltoniandiffer from each other and from the random promotedstates. The average entanglement in the spin-glass eigen-states is higher than the entanglement in random states,which can be related to the higher probability of non-zeroconcurrence. We can understand this as follows: In thecase of 1−particle sector, eigenstates of the Hamiltonianare typically more localized than random states and as aresult, less entangled. When promoted to the 2−particlesector, these states tend to be more entangled than therandom promoted states.Random states may be viewed as the ensemble of eigen-

states of random matrices that belong to the GaussianOrthogonal Ensemble (GOE). GOE matrices are sym-metric, with off-diagonal terms being i.i.d and drawnfrom N (0, 1) and the diagonal terms drawn from N (0, 2),where as for the spin-glass Hamiltonians under consider-ation, the variance of the diagonal terms is much largerwhen compared to that of the off-diagonal terms. Thevariance of the diagonal terms in the 1−particle sectorof the spin-glass Hamiltonian matrices are ∼ L2/2 timesbigger than that of the non-zero off-diagonal elements.As a result, the diagonal elements end up being typi-cally much larger in magnitude when compared to theoff-diagonal terms and hence are ‘close’ to being a di-agonal matrix, with the Nearest-Neighbour model being‘closer’ because of its sparse structure.For a diagonal matrix, the basis vectors are its eigen-

vectors (in the non-degenerate case). Thus, we expect theenergy eigenstates of the spin-glass systems to be typi-cally localized. As a toy problem, consider a 1−particle

7

basis state i.e. where only one of the coefficients is non-zero and hence 1 (upto a sign). The condition

∑

i ai = 0is no longer applicable and the corresponding promotedstate is given by aij = (ai + aj)/

√L− 1.

We can factor out the state of the qubit with up spin,implying that that particular qubit is not entangled tothe remaining L − 1 qubits, that are in the maximallyentangled (on average) all-one 1−particle state. Givensuch a promoted localized state, the probability of non-zero concurrence is

(

L−12

)

/(

L2

)

∼ 1 and the concurrencebetween any two pair of qubits, when not zero is 2/(L−1).Thus, the average concurrence 〈C〉 = 2

L−1L−2L . This is

plotted as the solid line in Fig. 4However, in the Nearest-Neighbour model, for large

values of L, the diagonal elements tend to assume theeigenvalue corresponding to the all-one state. This leadsto a high degeneracy and as a result, we have a highdimensional eigen sub-space corresponding to the eigen-value

∑

i<j Jij . Since the quantities of interest are basisdependent, it is no longer meaningful to discuss aboutthe average concurrence and participation ratio of theeigenstates for large values of L.

IV. THE ONE-DIMENSIONAL POWER-LAW

DECAY MODEL

To study the differences between the infinite-range andnearest-neighbour models, let us introduce a notion ofdistance that have been absent so far. Consider a familyof Hamiltonians with the spins being put on a closedchain and where the interactions fall with distance as apower law, with perdioic boundary conditions imposed.The distance between the ith and jth spins is taken to bethe length of the chord between the two sites, when allthe sites are put on a circle [27] given by

rij =L

πsin[π

L(i − j)

]

. (13)

The Jijs defined in Eq. 1 obey N (0, 1/rσij). We ob-tain the infinite range Heisenberg spin-glass when σ = 0and recover the 1-dimensional nearest neighbour modelasymptotically as σ → ∞.The cloud of non-promoted eigenstates has a tendency

to merge with that of promoted states as we increase σas shown in Fig 5. In the study of the same model withclassical spins [27], the regime σ = (0, 0.5) was identi-fied as the infinite-range universality class, the regimeσ = (0.5, 0.67) as the mean-field universality class andσ = (0.67, 1) as short-range. So, in that case anythinggreater than σ = 1 is already in the super-short rangeregime, and the system should already have characteris-tics of the nearest-neighbor model. But in the quantumversion of the model, even at σ = 2, a faint separation ofthe clouds can still be discerned, and only at σ ∼ 2.5, thefigure is for all practical purposes, is same as what we ob-tain for nearest neighbour spin chain. Here we note that

it is the lower cloud that merges into the cloud of pro-moted states. The position of the promoted eigenstatesin the Concurrence-PR plot is relatively stable. Thus, theproperties of these states are relatively robust and mustnot be too sensitive to our assumption of the model. Thiscould be advantageous in some eventual quantum infor-mation application, since defects in physical realizationof spin glass wouldn’t affect these states much.There are two interesting aspects that come out from

the phase diagram for large σ (short-range). One is thatapparently there are no distinct clouds. The second isthat inspite of the absence of a clear separation betweenclouds, the ‘special eigenstates’ continue to occupy thehigher concurrence - higher PR region. A purely numer-ical approach can perhaps never really tell whether themerger is a true merger or if the clouds continue to exist,but only get arbitrarily close.

V. CONCLUSIONS AND FUTURE WORK

We have discovered that in the general spin-glassHamiltonian, a special class of eigenstates form a dis-tinct cloud in the concurrence-PR phase diagram. Thesespecial eigenstates, which we show to be ‘promoted-eigenstates’, display signfincantly higher concurrencecompared with the rest of the eigenstates. As a first stepto understand the peculiarities of ‘promoted-eigenstates’,we have studied the properties of random promotedstates by an analytical approach. We find that the ran-dom promoted states are quite different from the pro-moted eigenstates of the Hamiltonian, which we atributeto the additional structures in the Hamiltonian. Whilewe understand some deviations qualitatively, a lot ofthem still remain a mystery.By considering a power-law decay model in 1-d, we

have shown that the ‘regular eigenstates’ tend to mergewith the ‘special eigenstates’ as we move further and fur-ther away from the infinite-range case and move towardsthe short-range model. This is a significant observa-tion because it could indicate that there is somethinginherently deferent between the infinite-range and short-range models in the quantum version of the model. Ifone speculates that there is some direct connection be-tween the quantum and classical version of the Heisen-berg spin glass, this could indicate that perhaps the RSBpicture which is valid for the SK model, may not beapplicable for the short-range models. The applicationof a uniform magnetic field to the quantum spin-glass, in spite of breaking the symmetry with regard to thethe σ± operators, preserves the set of eigenstates of thezero-field Hamiltonian, and therefore the promoted eigen-states continue to exist. However, if the applied externalmagnetic field is random over different sites, there wouldbe no special promoted eigenstates. Some of our prelim-inary checks on the infinite-range Hamiltonian suggestthat in fact, the clouds remain in tact in the presenceof a uniform field, whereas they disappear with random

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FIG. 5. Scatter plot of the average concurrence vs. participation ratio PR of all the N↑ = 2 eigenstates of the 1-d power-lawdecay Heisenberg spin glass for L = 25, with periodic boundary conditions for Nsamp = 50 samples of disorder. We show datafor σ = 0 (infinite-ranged), 0.5, 1, 2, 2.5, and ∞ (nearest-neighbour). Notice that σ = 2.5 looks for all practical purposes likethe strict ‘nearest-neighbour’. Every eigenstate is marked as +. Promoted eigenstates also marked as boxes.

fields. In a recent work [28], it was shown that the socalled Almeida-Thouless line of phase transitions existsfor the infinite-range classical vector spin glass under theapplication of random fields. It would therefore be ex-citing to investigate if and what consequence the dis-appearance of these special promoted eigenstates has tothe Almeida-Thouless line in quantum Heisenberg spinglasses. The AT line lies at the heart of the RSB-versus-droplet-picture debate in classical spin-glasses , and itwould be a significant avenue of research to study the

same with quantum spins. Another future work [21] ofinterest for us involves the study of the impact of thesepromoted states in inhibiting the transition from power-law to exponential scaling as reported in Vijayaraghavanet al [20].

We end this paper by suggesting an experimental tech-nique to attain these promoted m−particle states forapplications that may required localized yet highly en-tangled states. Starting from the 0−particle eigenstate,one can adiabatically flip any one qubit and the system

9

would be in the all-one state of the 1−particle sector.Now holding the particle number constant, we can ‘heatup’ or ‘cool down’ the system so that it is in any other1−particle state. Now again, we could adibatically flipone of the qubits and obtain a promoted 2−particle state.By repeating the process of adibatic flipping and heat-ing/cooling, we can obtain a promoted m−particle state.

ACKNOWLEDGMENTS

We thank Dr. Subrahmanyam for useful discussions.The Hamiltonian matrices are diagonalized using Eigen

package [29]. Curve fitting to find optimal parame-ters, including the errorbars, were done using SciPy’scurve fit routine.

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Appendix: Algorithm for concurrence

Computing the average concurrence of a given stateturns out to be computationally the most expensive partof our code1. Here we give a brief description of the al-gorithm we used to compute the essential elements ofthe reduced-density-matrix-of-any-two-spins ρ en-routeto computing the concurrence for systems with definite-particle-symmetry for any particle number.Let us say we are interested in the concurrence between

sites i, j of some particular eigenstate |ψ〉 =∑

k ak |k〉,where |k〉 are the basis states. For a given pair i, j, wefirst initialize all the elements of the reduced density ma-trix (which is real in our case) to 0 and then loop over allk. For each state |k〉, we first find the states of the i andj sites. If both i and j have up(down) spins we add a2kto v(y). The off-diagonal element is a bit more compli-cated and requires another loop and a search algorithmwithin, but not difficult. If i has an up(down) spin andj has a down(up) spin, we do the following. We find allstates in the basis which have a down(up) spin at i and anup(down) spin at j, but are exactly identical to the state|k〉 at every other site. Let us call such states |l〉, whichthus have coefficients al in the eigenstate |ψ〉 of interest.We just add a2k to w(x) and add sum of the products

1 https://github.com/arunkannawadi/spinglass

10

akal for all l to z. Although, w and x are not requiredto compute concurrence, they may be required for othermeasures of entanglement, such as log-negativity or theVon-Neumann entropy that measures how the qubits iand j are entangled to rest of the system. Having thuscomputed v,y, and z, the concurrence is trivially obtainedby the formula of Connor and Wootters [24].For the 2−particle states, this algorithm loops over all

possible pairs of spins (∼ O(L2)), over all basis vectors

(∼ O(L2)) and for (L − 2) cases out of(

L2

)

, it compares

the kth basis with all other basis, which goes asO(L2). Inshort, calculating the average concurrence for any given2−particle state is O(L5). A full analysis of a 2−particlestates will involve diagonalization of N↑=2 sector of theHamiltonian which is O(L6). But it would be wise to

obtain the eigenstates ofN↑=1 sector of the Hamiltonian,which is O(L3), and then ‘promote’ them to 2−particlestates. One might also consider computing concurrencebetween randomly selected pairs of spins instead of allspins.

The most compact way to represent the basis vectorsis to denote them by the positions of up spins, as denotedby |ij〉. But we quickly realized that generalizing it to anarbitrary definite-particle state is tedious. Instead, thebasis are represented by integers and the binary stringsof the basis vectors are encoded in bit representation ofintegers. In order to be able to use large values of L, the128 bit integers in C++ Boost2 libraries were used. Thisis generic and any m−particle basis vectors can be easilycreated.

2 http://www.boost.org