Random Transverse Field Spin-Glass Model on the Cayley tree ...

Random Transverse Field Spin-Glass Model on the Cayley tree : phase transition between the two Many-Body-Localized Phases Cécile Monthus

arXiv:1707.04039v1 [cond-mat.dis-nn] 13 Jul 2017

Institut de Physique Théorique, Université Paris Saclay, CNRS, CEA, 91191 Gif-sur-Yvette, France The quantum Ising model with random couplings and random transverse fields on the Cayley tree is studied by Real-Space-Renormalization in order to construct the whole set of eigenstates. The renormalization rules are analyzed via large deviations. The phase transition between the paramagnetic and the spin-glass Many-Body-Localized phases involves the activated exponent ψ = 1 and the correlation length exponent ν = 1. The spin-glass-ordered cluster containing NSG spins is found to be extremely sparse with respect to the total number N of spins : its size grows only criti logarithmically at the critical point NSG ∝ ln N , and it is sub-extensive NSG ∝ N θ in the finite region of the spin-glass phase where the continuously varying exponent θ remains in the interval 0 < θ < 1.

I.

INTRODUCTION

In the field of Many-Body-Localization (see the recent reviews [1–9] and references therein), one of the important characterization of Many-Body-Localized phases is the existence of an extensive number of Local Integrals of Motion called LIOMs [10–22]. Since these LIOMS are the building blocks of the whole set of eigenstates, it is natural to try to identify them via some real-space renormalization procedure. The Strong Disorder Real-Space RG approach developed by Daniel Fisher [23–25] to construct the ground states of random quantum models (see the review [26]) has been thus generalized into the RSRG-X procedure to construct the whole set of excited eigenstates [27–31] : the idea is that each local renormalization step produces a LIOM that describes the choice between the local energy levels (instead of projecting always onto the lowest energy-level). The RSRG-t procedure developed by Vosk and Altman [32, 33] in order to construct the effective dynamics via the iterative elimination of the degree of freedom oscillating with the highest local eigenfrequency is equivalent to the RSRG-X procedure but gives an interesting different point of view [34]. The RSRG-X is very useful to analyse the long-ranged order of the excited eigenstates and the possible phase transitions between different Many-Body-Localized phases [27]. In the present paper, we wish to analyse the transition between the paramagnetic and the spin-glass Many-Body-Localized phases for the quantum Ising model with random couplings and random transverse fields on the Cayley tree. Since the standard RSRG-X procedure destroys the tree structure and could only be followed numerically, we will instead use an RG procedure that preserves the tree structure in order to obtain some simple analytical approximation : the Pacheco-Fernandez block-RG introduced for the ground state of the one-dimensional chain without disorder [35, 36] or with disorder [37–39] is applied here sequentially [40] around the center of the tree in order to construct the whole set of eigenstates. The paper is organized as follows. In section II, the real-space RG procedure to construct the set of eigenstates of the random quantum Ising model on the Cayley tree is described. Section III is devoted to the large deviation properties of the basic variables that appear in the RG flows. The statistics of the renormalized couplings and of the renormalized transverse field of the center are studied in section IV and in section V respectively in order to characterize the critical properties of the transition between the paramagnetic and spin-glass Many-Body-Localized phases. Our conclusions are summarized in section VI.

2 II.

REAL-SPACE RG PROCEDURE TO CONSTRUCT THE SET OF EIGENSTATES A.

Model

We consider the geometry of a Cayley tree of branching ratio K with L generations around the central spin σ0 . It is convenient to decompose the quantum Ising Hamiltonian in terms of the contributions of the various generations H

=

L X

Hr

r=0

H0

= h0 σ0z

H1

=

K+1 X

(Ji1 σ0x σix1 + hi1 σiz1 )

i1 =1

H2

=

K+1 K X X

(Ji1 ,i2 σix1 σix1 ,i2 + hi1 ,i2 σiz1 ,i2 )

i1 =1 i2 =1

Hr

=

K+1 K X X

..

i1 =1 i2 =1

K X

(Ji1 ,i2 ,..,ir σix1 ,..,ir−1 σix1 ,..,ir + hi1 ,..,ir σiz1 ,..,ir )

(1)

ir =1

In order to avoid degeneracies (such as in the model studied in Ref. [41] where the transverse fields take all the same value hi = h, and the couplings take only two values (±J)), we consider that both the transverse fields hi1 ,i2 ,..,ir and the couplings Ji1 ,i2 ,..,ir are random variables drawn with some continuous distributions. As example, we will focus on the case where the probability distributions of the couplings Ji1 ,i2 ,..,ir and of the random fields hi1 ,i2 ,..,ir are uniformly drawn on [−J, +J] and [−h, +h] respectively θ(−J ≤ Ji1 ,i2 ,..,ir ≤ J) 2J θ(−h ≤ hi1 ,i2 ,..,ir ≤ h) πh (hi1 ,i2 ,..,ir ) = 2h

πJ (Ji1 ,i2 ,..,ir ) =

(2)

In the limit where all couplings Ji1 ,i2 ,..,ir vanish, the model is in a trivial paramagnetic Localized phase, where the LIOMs are the site operators σiz1 ,i2 ,..,ir . In the opposite limit where all fields hi1 ,i2 ,..,ir vanish, the model is in a trivial spin-glass Localized phase, where the LIOMs are the bond operators σix1 ,..,ir−1 σix1 ,..,ir . In the following, we wish to study the phase transition between the paramagnetic and spin-glass Many-Body-Localized phases via some real-space procedure that constructs the LIOMs and thus the set of eigenstates. B.

First RG step

The RSRG-X procedure mentioned in the Introduction can be applied in d > 1, but the changes of the geometry prevents the finding of any analytical description. The renormalization procedure has to be implemented numerically, as was done for the RSRG procedure concerning the ground state in d = 2, 3, 4 [25, 42–51]. Here we wish instead to obtain some analytically solvable RG procedure in order to get more insight into the mechanism of the transition. We have thus chosen to apply sequentially [40] around the center of the tree the idea of the Pacheco-Fernandez elementary step [35–39] in order to keep a simple geometry along the RG flow. More precisely, the first RG step consists in the diagonalization of the Hamiltonian H1 Eq. 1 concerning the center spin and the (K + 1) spins of the first generation H1 =

K+1 X

Ji1 σ0x σix1 + hi1 σiz1

i1 =1

(3)

Since H1 commutes with σ0x , one needs to consider the two possible values σ0x = S0x = ±1, and to diagonalize the (K + 1) remaining effective Hamiltonians involving the single spin σi1 f Hief = Ji1 S0x σix1 + hi1 σiz1 1

(4)

3 The two eigenvalues of Eq. 4 do not depend on the value S0x = ± and read q (τ z ) λi1 i1 = τiz1 Ji21 + h2i1

(5)

where the variable

τiz1 = ±

(6)

labels the choice between the positive or negative energy in Eq. 5. The corresponding eigenvectors depend on the value S0x v  v    u u u u z SxJ z S xJ z τ τ (τ ) u1 u1 0 i1  x 0 i1  x (7) |σi1 = +i + τiz1 sgn(hi1 )t 1 − qi1 |σi1 = −i |λi1 i1 (S0x ) >= t 1 + qi1 2 2 2 2 2 Ji1 + hi1 Ji1 + h2i1 For the whole Hamiltonian H1 of Eq. 3, the 2K+1 energy-levels labelled by the variables (τ1 ...τK+1 ) (τ1 ...τK+1 )

E1

K+1 X

=

τiz1

i1 =1

q Ji21 + h2i1

(8)

x are independent of S0x = ±1. To label this degeneracy, it is thus convenient to introduce the renormalized spin σR0 i1 (S0x = +1)i = |S0x = +1i ⊗K+1 i1 =1 |λi1

x |τ1 ...τK+1 ; σR0 = −1i

i1 (S0x = −1)i = |S0x = −1i ⊗K+1 i1 =1 |λi1

(τ1 ...τK+1 )

The projector onto the energy-level E1 (τ1 ...τK+1 )

P1

(τ z )

x |τ1 ...τK+1 ; σR0 = +1i

(τ z )

(9)

then reads

x x x x = |τ1 ...τK+1 ; σR0 = +1ihτ1 ...τK+1 ; σR0 = +1| + |τ1 ...τK+1 ; σR0 = −1ihτ1 ...τK+1 ; σR0 = −1| (τ1 ...τK+1 )

The projection onto the energy-level E1 obtained from the various contributions

(10)

of the Hamiltonian of Eq 1 concerning the whole tree can be

(τ ...τ ) (τ ...τ ) P1 1 K+1 HP1 1 K+1

=

L X

(τ1 ...τK+1 )

(τ1 ...τK+1 )

Hr P1

(11)

(τ1 ...τK+1 )

(12)

= Hr

(13)

P1

r=0

(τ1 ...τK+1 )

The projection of H1 is simply the energy E1

(τ1 ...τK+1 )

P1

by construction

(τ1 ...τK+1 )

H1 P1

= E1

while the projection of Hr is unchanged for r ≥ 3 (τ1 ...τK+1 )

P1

(τ1 ...τK+1 )

Hr P1

The projection of H0 (τ1 ...τK+1 )

P1

(τ1 ...τK+1 )

H0 P1

(τ1 ...τK+1 ) z (τ1 ...τK+1 ) σ0 P1 s ! K+1 Y h2i1 (τ ...τ ) z (τ ...τ ) P1 1 K+1 σR0 P1 1 K+1 2 + h2 J i i 1 1 i1 =1

= h0 P1 = h0

z gives the renormalized transverse field hR 0 associated to the renormalized spin operator σR0 s K+1 Y h2i1 R h0 = h0 Ji21 + h2i1 i =1 1

(14)

(15)

4 The projection of H2 (τ1 ...τK+1 )

P1

(τ1 ...τK+1 )

H2 P1

=

K+1 K X X

i1 =1 i2 =1

=

K+1 K X X

(τ ...τ ) (τ ...τ ) Ji1 ,i2 (P1 1 K+1 σix1 P1 1 K+1 )σix1 ,i2 + hi1 ,i2 σiz1 ,i2 

τiz1 Ji1



(τ ...τ ) x (τ ...τ ) Ji1 ,i2 q (P1 1 K+1 σR0 P1 1 K+1 )σix1 ,i2 + hi1 ,i2 σiz1 ,i2  (16) 2 2 Ji1 + hi1 i1 =1 i2 =1

x gives the renormalized coupling between the operators σR0 and σix1 ,i2

τ z Ji1 JiR1 ,i2 = Ji1 ,i2 q i1 Ji21 + h2i1 C.

(17)

RG rules

The iteration of the above procedure yields the following RG rules after r RG steps. The renormalized transverse r z field hR 0 associated to the renormalized spin operator σRr 0 evolves according to (Eq 15) s K K+1 K Y Y Y h2i1 ,..,ir Rr Rn−1 .. h0 = h0 (18) r−1 h2i1 ,..,ir + [JiR1 ,..,ir ]2 i =1 i =1 i =1 1

2

r

x x reads (Eq 17) while the renormalized coupling between the operators σR r 0 and σi ,i ,..,i 1 2 r+1 r−1

r JiR1 ,..,ir+1

τiz1 ,..,ir JiR1 ,..,ir

= Ji1 ,...,ir+1 q r−1 h2i1 ,..,ir + [JiR1 ,..,ir ]2

D.

(19)

Solution of the RG rules

The RG rule of Eq. 19 for the couplings involve only the initial transverse fields and not the renormalized transversed fields, so that it can be solved independently. The sign r

sgn(JiR1 ,..,ir+1 )

r−1

= τiz1 ,..,ir sgn(Ji1 ,...,ir+1 )sgn(JiR1 ,..,ir ) = τiz1 ,..,ir τiz1 ,..,ir−1 ...τiz1 sgn(Ji1 ,...,ir+1 )sgn(Ji1 ,...,ir )...sgn(Ji1 )

(20)

is simply the product of all the couplings J and of all the variablesτ z along the path between the sites 0 and (i1 , .., ir ). The absolute value reads (Eq. 19) r

|JiR1 ,..,ir+1 |

= |Ji1 ,...,ir+1 |Ci1 ,..,ir

(21)

where Ci1 ,..,ir

#− 21 r r X Y h2i1 ,..,ik ≡ 1+ J2 m=1 k=m i1 ,..,ik " #− 21 h2i1 ,..,ir h2i1 ,..,ir−1 ..h2i1 ,i2 h2i1 h2i1 ,..,ir h2i1 ,..,ir−1 h2i1 ,..,ir = 1+ 2 + .. + 2 + 2 Ji1 ,..,ir Ji1 ,..,ir Ji21 ,..,ir−1 Ji1 ,..,ir Ji21 ,..,ir−1 ..Ji21 ,i2 Ji21 "

(22)

involves in the denominator a so-called Kesten random variable [52–56] that has been much studied in relation with the surface magnetization in the ground-state of the one-dimensional chain [26, 40, 57].

5 This solution for the renormalized couplings can be plugged into the RG flow of Eq. 18 for the renormalized transverse field to obtain   Rr K K+1 K X X X   1 h0  r ln  .. = ln r−1   R 2 Ji ,..,ir 2 h0 i1 =1 i2 =1 ir =1 1 + h21 Ci1 ,..,ir−1 i1 ,..,ir

= in terms of the Kesten variables of Eq. 22. E.

1 2

K+1 K X X

i1 =1 i2 =1

..

K X

ir =1

ln 1 − Ci21 ,..,ir−1 ,ir

(23)

Solution at lowest order in the couplings

For the one-dimensional chain, the location of paramagnetic/spin-glass phase transition of the quantum Ising model is know to occur at ln |Ji | = ln |hi | [24, 26, 58]. For the Cayley tree with branching ratio K > 1, the transition is thus expected to occur in the region ln |Ji | < ln |hi |

(24)

where the couplings are typically smaller than the transverse fields. To analyse the RG rules in this region, it is convenient to introduce the products Ji1 ,..,ir Ji1 ,..,ir−1 ..Ji1 ,i2 Ji1 Pi1 ,..,ir ≡ (25) hi ,..,i hi ,..,i ..hi ,i hi r

1

r−1

1

1

2

1

In the region of Eq. 24, the Kesten variable of the denominator in Eq. 22 is dominated by the last term, while it is convenient to keep the term unity to maintain the important bound Ci1 ,..,ir ≤ 1, so that we make the following approximation at lowest order in the couplings #− 21 " Pi ,..,i 1 = q 1 r Ci1 ,..,ir ≃ 1 + 2 (26) Pi1 ,..,ir 1 + P2 i1 ,..,ir

Then the absolute values of the renormalized couplings of Eq 21 become r

|JiR1 ,..,ir+1 |

Pi ,..,i = |Ji1 ,...,ir+1 | q 1 r 1 + Pi21 ,..,ir

(27)

q For the ground state, the result |Ji1 ,...,in+1 |Pi1 ,..,ir (i.e. without the denominator 1 + Pi21 ,..,ir ) that involves the product of all couplings in the numerator and all the transverse fields in the denominator has been obtained in the paramagnetic phase via various approaches including the Cavity-Mean-Field approach [59–61], the Strong Disorder RG framework when only sites are decimated [62] or simply perturbation theory in the couplings [63]. The approximation of Eq 26 yields that the RG flow of Eq. 23 for the renormalized transverse field becomes Rr K K+1 K 1 X X X h0 ≃ − (28) ln 1 + Pi21 ,..,ir−1 ,ir ln .. r−1 R 2 i =1 i =1 i =1 h0 1

r

2

To analyse the statistical properties of the RG flows Eq 27 and Eq 28, one needs first to characterize the large deviation properties of the products of Eq. 25. III.

LARGE DEVIATION ANALYSIS

In this section, we describe the statistical properties of the product of Eq. 25 with the simplified notation r Y Ji1 ,..,ik P (r) = hi ,..,i 1 k k=1 Ji1 ,..,ik where r represents the number of random variables hi ,..,i in this product. 1

k

(29)

6 A.

Typical behavior

The logarithm of Eq. 29 reduces to a sum of random variables ln P (r)

≃

r X

(ln |Ji1 ,..,ik | − ln |hi1 ,..,ik |)

(30)

k=1

The Central Limit Theorem thus yields the following typical behavior for large r √ ln P (r) ≃ −ra0 + ru

(31)

r→+∞

where a0 = (ln |hi | − ln |Ji |)

(32)

is positive a0 > 0 in the region under study (Eq. 24) and governs the typical exponential decay of P (r), while u is a Gaussian random variable. For the one-dimensional chain, only this typical behavior is relevant, but here on the Cayley tree of branching ratio K > 1 where the number of sites at distance r grows exponentially as K r with the distance r, one needs to analyze the large deviations properties. B.

Large deviations

In the field of large deviations (see the review [64] and references therein), one is interested into the exponentially small probability to see an exponential decay with some coefficient a different from the typical value a0 of Eq. 32 Prob(P (r) ∝ e−ar )

∝

r→+∞

e−rI(a)

(33)

where the rate function I(a) vanishes at the typical value I(a0 ) = 0

(34)

and is strictly positive otherwise I(a 6= n0 ) > 0. The standard way to evaluate the rate function I(a) [64] is to consider the generalized moments that display the following exponential behavior !r 2q |J | i P 2q (r) = = erλ(q) (35) |hi |2q where λ(q)

= ln

|Ji |2q |hi |2q

!

(36)

can be explicitly computed from the probability distribution of the couplings Ji and of the random fields hi (see the example below). The evaluation of Eq. 35 via the saddle-point approximation P 2q (r)

≃

Z

+∞

dae

−rI(a) −ar2q

e

=e

r

max(−I(a) − 2qa) a

(37)

0

yields λq in terms of the saddle-point aq λ(q) 0

= −I(aq ) − 2qaq = I ′ (aq ) + 2q

(38)

I(a) 0

= −λ(qa ) − 2aqa = λ′ (qa ) + 2a

(39)

The reciprocal Legendre transform yields

7 C.

Explicit example with the two box distributions of Eq. 2

Let us now focus on the example where the probability distributions of the couplings and of the random fields are the two box distributions of parameters J and h respectively (Eq 2). In the region h > J, the typical decay of the renormalized couplings is governed by (Eq. 32) a0 =

h

Z

0

dhi ln hi − h

Z

0

J

dJi ln Ji = ln J

h >0 J

(40)

The generalized moments of Eq. 35 converge only in the region −1 < 2q < 1 and Eq. 36 becomes e

λ(q)

|Ji |2q = = |hi |2q

J

Z

0

dJi 2q J J i

Z

h

0

1 dhi −2q h = h i 1 − 4q 2

2q 1 J = e−2qa0 h 1 − 4q 2

(41)

so that the function λ(q) and its derivative read in terms of the typical value a0 = −2qa0 − ln(1 − 4q 2 ) 8q = −2a0 + 1 − 4q 2

λ(q) λ′ (q)

(42)

The second equation of the system 39 0

= 2a + λ′ (qa ) = 2(a − a0 ) +

8qa 1 − 4qa2

(43)

leads to the following second-order equation for qa = qa2 −

0

qa 1 − a − a0 4

(44)

The appropriate solution qa that tends to qa → 0 when a → a0 reads qa =

2(1 +

a −a p0 1 + (a0 − a)2 )

(45)

The rate function given by the first equation of the system 39 reads I(a)

IV.

4qa ) = −λ(qa ) − 2aqa = 2qa (a0 − a) + ln(1 − 4qa2 ) = 2qa (a0 − a) + ln( a0 − a ! p 1 + 1 + (a0 − a)2 (a0 − a)2 p = − ln 2 1 + 1 + (a0 − a)2

(46)

STATISTICAL PROPERTIES OF THE RENORMALIZED COUPLINGS

In this section, we focus on the absolute values of the renormalized couplings given by Eq 27 r

|JiR1 ,..,ir+1 |

A.

Pi ,..,i = |Ji1 ,...,ir+1 | q 1 r 1 + Pi21 ,..,ir

(47)

Location of the critical point

On the Cayley tree where the number of points at distance r grows exponentially as K r , the number of products P (r) displaying the decay P (r) ∝ e−ar reads (Eq. 33) N (P (r) ∝ e−ar )

∝

r→+∞

K r e−rI(a) = er(ln K−I(a)) θ(amin ≤ a ≤ amax )

(48)

8 where the minimum value amin and the maximal value amax are respectively smaller and bigger than the typical value amin < a0 < amax and satisfy I(amin ) = ln K = I(amax )

(49)

so that they occur only on a finite number O(1) of branches, while the typical value a0 where I(a0 ) = 0 occur on an extensive O(K n ) number of branches, From Eq 47, it is clear that the renormalized coupling J(r) inherits the exponential decay of P (r) of Eq. 48 as long as a > 0, while the region a ≤ 0 produces finite renormalized couplings O(1) so that the critical point corresponds to amin = 0 i.e. in terms of the large deviation function I(a) ( Eq. 49) I criti (0) = ln K

(50)

For the special case of the box distribution of Eq 2, Eq 46 yields the following explicit condition in terms of the control parameter a0 = ln Jh ! p 2 ) (acriti )2 1 + 1 + (acriti 0 p 0 − 0 = ln K (51) 2 1 + 1 + (acriti )2 0 Paramagnetic phase for amin > 0

B.

In the paramagnetic phase amin > 0, all K r renormalized couplings decay exponentially (Eq 48) N (J(r) ∝ e−ar ) C.

∝

r→+∞

er(ln K−I(a)) θ(amin ≤ a ≤ amax )

(52)

Spin-Glass phase for amin < 0

In the spin-glass phase amin < 0, the K r renormalized couplings can be split into two groups : the number of finite couplings grows exponentially in r as Z 0 daer(ln K−I(a)) ≃ er(ln K−I(0)) = er(I(amin )−I(0)) (53) N (J(r) ∝ O(1)) ∝ r→+∞

amin

while the other branches are still characterized by exponential decays with exponents a > 0 N (J(r) ∝ e−ar )

∝

r→+∞

er(ln K−I(a)) θ(0 < a ≤ amax )

(54)

This is the first indication that the ordered spin-glass cluster remains very sparse near the critical point, as confirmed by the analysis of the renormalized transverse field in the next section. V.

STATISTICAL PROPERTIES OF THE RENORMALIZED TRANSVERSE FIELD

In this section, we focus on the RG flow of Eq. 28 for the renormalized transverse field ln

r

hR 0 r−1 h0R

≃−

K K+1 K 1 X X X ln 1 + Pi21 ,..,ir−1 ,ir .. 2 i =1 i =1 i =1 1

2

(55)

r

which can be evaluated in terms of the large deviation analysis of Eq. 48 concerning the K r products P (r) Rr Z 1 amax h0 r(ln K−I(a)) −2ar ≃ − ln dae ln 1 + e r−1 r→+∞ 2 amin h0R

(56)

9 A.

Paramagnetic phase for amin > 0

In the paramagnetic phase amin > 0, Eq. 56 becomes Rr Z Z 1 amax 1 amax h0 r(ln K−I(a)−2a) ≃ − dae = − daer(I(amin )−I(a)−2a) ln r−1 r→+∞ 2 amin 2 amin h0R The integral is dominated by the lower boundary amin of the integral, and one obtains the exponential decay Rr h0 ∝ −e−2amin r ln r−1 r→+∞ h0R

(57)

(58)

r

By integration, one obtains that hR 0 remains finite as r → +∞ Rr Z r ′ h0 1 − e−2amin r ln ∝ − dr′ e−2amin r ∝ − r→+∞ r→+∞ h0 amin 1 The typical asymptotic value hR 0

∞

(59)

diverges with the following essential singularity near the transition amin → acriti min = 0 R∞ 1 h0 ∝ − (60) ln amin →0 h0 amin B.

Spin-Glass phase for amin < 0

In the spin-glass phase amin < 0, it is convenient to evaluate separately the contributions of the two regions a < 0 and a > 0 in the integral of Eq. 56. The contribution of the region a > 0 is dominated by the lower boundary a = 0 of the integral Z amax Z amax r(ln K−I(a)) −2ar daer(ln K−I(a)−2a) ≃ er(ln K−I(0)) = er(I(amin )−I(0)) (61) ≃ dae ln 1 + e r→+∞

0

0

while the region a < 0

Z

0

daer(ln K−I(a)) ln 1 + e−2ar

amin

≃

r→+∞

Z

0

daer(ln K−I(a)) (−2ar)

(62)

amin

is dominated by the upper boundary a = 0. In summary, the RG flow of renormalized transverse field of Eq. 56 is dominated by the exponentially big term of coefficient (I(amin ) − I(0)) > 0 Rr Z r ′ h0 er(I(amin )−I(0)) ln dr′ er (I(amin )−I(0)) ∝ − ∝ − (63) r→+∞ r→+∞ h0 (I(amin ) − I(0)) 1 This behavior confirms the indication of Eqs 53 and 54 concerning the renormalized couplings : near the critical point, the ordered spin-glass cluster remains very sparse. The number NSG of spins involved in this ordered spin-glass cluster grows exponentially with the distance r NSG ∝ er(ln K−I(0))

(64)

but is only sub-extensive with respect to the total number of spins N = K r NSG ∝ er(ln K−I(0)) = N θ

(65)

in the whole region of the phase diagram where the continuously varying exponent θ =1−

I(0) I(0) =1− ln K I(amin

(66)

remains in the interval θcriti = 0 < θ < 1 = θext

(67)

10 At criticality, the vanishing exponent θcriti = 0 corresponds to the logarithmic growth with respect to N criti NSG ∝ ln N = r ln K

(68)

meaning that only a finite number of the branches sustain the spin-glass order. The location where the spin-glassordered cluster becomes extensive θext = 1 corresponds to the vanishing of the large deviation rate function I ext (0) = 0, i.e. to the vanishing of the typical value aext = 0 (Eq. 34) : it is then clear that an extensive number K r of branches 0 become ordered. The finite region of the phase diagram corresponding to Eq. 67 where the ordered spin-glass cluster remains subextensive is somewhat formally reminiscent of the delocalized non-ergodic phase existing in the Anderson Localization model defined on the Cayley tree [65–67], i.e. in exactly the same geometry as in the present paper, and for the same technical reasons based on large deviations on the branches of the Cayley tree [65]. Let us mention however that the existence of delocalized non-ergodic phase remains very controversial for the Anderson Localization model on Random Regular Graphs [68–73] or for Many-Body-Localization models [74–78], where an analogy with the Anderson Localization transition in an Hilbert space of ’infinite dimensionality’ has been put forward [65, 79–82], while the properties of the delocalized non-ergodic phase can be explicitly computed in some random matrix models [83–86]. C.

Finite-size scaling in the critical region

The above results for the renormalized transverse field as a function of the radial distance r can be summarized by the following finite-size scaling form in the critical region Rr 1 h0 ln (69) ∝ −rψ G r ν (J − Jc ) r→+∞ h0 with the exponent ψ=1

(70)

ν=1

(71)

and the correlation length exponent

as in many other phase transitions on the Cayley tree. The scaling function G(x) is constant at the origin G(0) = cst, behaves as G(x)

∝

x→−∞

−

1 x

(72)

to reproduce the behavior of Eq. 60 in the paramagnetic phase J < Jc , and as G(x)

ex − 1 x→+∞ x ∝

(73)

to reproduce the behavior of Eq. 63 in the spin-glass phase J > Jc . VI.

CONCLUSION

We have introduced a simple Real-Space-Renormalization procedure in order to construct the whole set of eigenstates for the quantum Ising model with random couplings and random transverse fields on the Cayley tree of branching ratio K. The analysis of the renormalization rules via large deviations was described to obtain the critical properties of the phase transition between the paramagnetic and the spin-glass Many-Body-Localized phases. In particular, we have found that the renormalized transverse field of the center site involves the activated exponent ψ = 1 and the correlation length exponent ν = 1. The spin-glass-ordered cluster containing NSG spins was found to be extremely sparse with respect to the total number N ∝ K r of spins : its size grows only logarithmically at the critical point criti NSG ∝ ln N , meaning that only a finite number O(1) of the branches are long-ranged-ordered, while the other

11 branches display exponentially decaying correlations. In addition, the size NSG spin-glass-ordered cluster is subextensive NSG ∝ N θ in the finite region of the spin-glass phase where the continuously varying exponent θ remains in the interval 0 < θ < 1.

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