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14Strong-randomness phenomena in quantum Ashkin-Teller models

Hatem Barghathi,1 Fawaz Hrahsheh,1, 2 Jose A. Hoyos,3 Rajesh Narayanan,4 and Thomas Vojta1

1Department of Physics, Missouri University of Science and Technology, Rolla, MO 65409, USA2Department of Physics, Jordan University of Science and Technology, Irbid 22110, Jordan

3Instituto de Fsica de Sao Carlos, Universidade de Sao Paulo,

C.P. 369, Sao Carlos, Sao Paulo 13560-970, Brazil4Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India

(Dated: April 10, 2014)

The N-color quantum Ashkin-Teller spin chain is a prototypical model for the study of strong-randomness phenomena at first-order and continuous quantum phase transitions. In this paper, wefirst review the existing strong-disorder renormalization group approaches to the random quantumAshkin-Teller chain in the weak-coupling as well as the strong-coupling regimes. We then introducea novel general variable transformation that unifies the treatment of the strong-coupling regime.This allows us to determine the phase diagram for all color numbers N , and the critical behaviorfor all N 6= 4. In the case of two colors, N = 2, a partially ordered product phase separates theparamagnetic and ferromagnetic phases in the strong-coupling regime. This phase is absent for allN > 2, i.e., there is a direct phase boundary between the paramagnetic and ferromagnetic phases.In agreement with the quantum version of the Aizenman-Wehr theorem, all phase transitions arecontinuous, even if their clean counterparts are of first order. We also discuss the various criticaland multicritical points. They are all of infinite-randomness type, but depending on the couplingstrength, they belong to different universality classes.

PACS numbers: 75.10.Nr, 75.40.-s, 05.70.Jk

I. INTRODUCTION

Simple models of statistical thermodynamics haveplayed a central role in our understanding of phase tran-sitions and critical phenomena. For example, Onsagerssolution of the two-dimensional Ising model [1] paved theway for the use of statistical mechanics methods in thephysics of thermal (classical) phase transitions. More re-cently, the transverse-field Ising chain has played a simi-lar role for quantum phase transitions [2].

The investigation of systems with more complex phasediagrams requires richer models. For example, the quan-tum Ashkin-Teller spin chain [35] and its N -color gen-eralization [68] feature partially ordered intermediatephases, various first-order and continuous quantum phasetransitions, as well as lines of critical points with continu-ously varying critical exponents. Recently, the quantumAshkin-Teller model has reattracted considerable atten-tion because it can serve as a prototypical model for thestudy of various strong-randomness effects predicted tooccur at quantum phase transitions in disordered systems[9, 10].

In the case of N = 2 colors, the correlation lengthexponent of the clean quantum Ashkin-Teller modelvaries continuously with the strength of the coupling be-tween the colors. The disorder can therefore be tunedfrom being perturbatively irrelevant (if the Harris cri-terion [11] d > 2 is fulfilled) to relevant (if the Harriscriterion is violated). For more than two colors, the cleansystem features a first-order quantum phase transition.It is thus a prime example for exploring the effects of ran-domness on first-order quantum phase transitions and fortesting the predictions of the (quantum) Aizenman-Wehr

theorem [12, 13].

In this paper, we first review the physics of the randomquantum Ashkin-Teller chain in both the weak-couplingand the strong-coupling regimes, as obtained by vari-ous implementations of the strong-disorder renormaliza-tion group. We then introduce a variable transformationscheme that permits a unified treatment of the strong-coupling regime for all color numbers N . The paper isorganized as follows: The Hamiltonian of the N -colorquantum Ashkin-Teller chain is introduced in Sec. II. Sec-tion III is devoted to disorder phenomena in the weak-coupling regime. To address the strong-coupling regimein Sec. IV, we first review the existing results and thenintroduce a general variable transformation. We also dis-cuss the resulting phase diagrams and phase transitions.We conclude in Sec. V.

II. N-COLOR QUANTUM ASHKIN-TELLER

CHAIN

The one-dimensional N -color quantum Ashkin-Tellermodel [68] consists of N identical transverse-field Isingchains of length L (labeled by the color index =1 . . .N) that are coupled via their energy densities. It isgiven by the Hamiltonian

H =N=1

Li=1

(JiS

z,iS

z,i+1 + hiS

x,i

)(1)

2Sx,i and Sz,i are Pauli matrices that describe the spin of

color at lattice site i. The strength of the inter-colorcoupling can be characterized by the ratios h,i = gi/hiand J,i = Ki/Ji. In addition to its fundamental interest,the Ashkin-Teller model has been applied to absorbedatoms on surfaces [14], organic magnets, current loopsin high-temperature superconductors [15, 16] as well asthe elastic response of DNA molecules [17]. The quan-tum Ashkin-Teller chain (1) is invariant under the dual-

ity transformation Sz,iSz,i+1 Sx,i, Sx,i Sz,iSz,i+1,

Ji hi, and J,i h,i, where Sx,i and S

z,i are the

dual Pauli matrices [18]. This self-duality symmetry willprove very useful in fixing the positions of various phaseboundaries of the model.In the clean problem, the interaction energies and fields

are uniform in space, Ji J , Ki K, hi h, gi g,and so are the coupling strengths h,i h and J,i J .In the present paper, we will be interested in the effects ofquenched disorder. We therefore take the interactions Jiand transverse fields hi as independent random variableswith probability distributions P0(J) andR0(h). Ji and hican be restricted to positive values, as possible negativesigns can be transformed away by a local transformationof the spin variables. Moreover, we focus on the case ofnonnegative couplings, J,i, h,i 0. In most of the paperwe also assume that the coupling strengths in the bareHamiltonian (1) are spatially uniform, J,i = h,i = I .Effects of random coupling strengths will be consideredin the concluding section.

III. WEAK COUPLING REGIME

For weak coupling and weak disorder, one can mapthe Ashkin-Teller model onto a continuum field theoryand study it via a perturbative renormalization group[1921]. This renormalization group displays runaway-flow towards large disorder indicating a breakdown of theperturbative approach. Consequently, nonperturbativemethods are required even for weak coupling.Carlon et al. [22] therefore investigated the weak-

coupling regime |I | < 1 of the two-color random quan-tum Ashkin-Teller chain using a generalization of Fishersstrong-disorder renormalization group [23, 24] of the ran-dom transverse-field Ising chain. Analogously, Goswamiet al. [21] considered the N -color version for 0 I c(N), the coupling strengths increase underthe renormalization group steps of Sec. III. If they get suf-ficiently large, the energy spectrum of the local Hamilto-nian changes, and the method breaks down. To overcomethis problem, two recent papers have implemented ver-sions of the strong-disorder renormalization group thatwork in the strong-coupling limit [25, 26].For large , the inter-color couplings in the second line

of the Hamiltonian (1) dominate over the Ising termsin the first line. The low-energy spectrum of the localHamiltonian therefore consists of a ground-state sectorand a pseudo ground-state sector, depending on whetheror not a state satisfies the Ising terms [25]. For differentnumbers of colors N , this leads to different consequences.For N > 4, the local binary degrees of freedom that

distinguish the two sectors become asymptotically freein the low-energy limit. By incorporating them into thestrong-disorder renormalization group approach, the au-thors of Ref. [25] found that the direct continuous quan-tum transition between the ferromagnetic and param-agnetic phases on the self-duality line Jtyp = htyp per-sists in the strong-coupling regime I > c(N). In agree-ment with the quantum Aizenman-Wehr theorem [13],the first order transition of the clean model is thus re-placed by a continuous one. However, the critical behav-ior in the strong-coupling regime differs from the randomtransverse-field Ising universality class that governs theweak-coupling case. The critical point is still of infinite-randomness type, but the additional degrees of freedomlead to even stronger thermodynamic singularities. Themethod of Ref. [25] relies on the ground-state and pseudoground-state sectors decoupling at low energies and thusholds for N > 4 colors only.We now turn to N = 2. The strong-coupling regime of

the two-color random quantum Ashkin-Teller model wasrecently attacked [26] by the variable transformation

zi = Sz1,iS

z2,i ,

zi = S

z1,i (9)

which introduces the product of the two colors as an in-dependent variable. The corresponding transformationsfor the Pauli matrices Sx1,i and S

x2,i read

xi = Sx2,i ,

xi = S

x1,iS

x2,i . (10)

r=

J

0ln e0

FM(Baxter)

PMr e/2c = ln( )

r 2/ec = ln( )

(1)

(3)

(2)

strong-randomnessproduct phase

MCP

FIG. 1. Schematic ground state phase diagram of thetwo-color random quantum Ashkin-Teller chain. For I 1, they are separated by a partially ordered prod-uct phase characterized by strong randomness and renormal-ization group flow towards infinite coupling. The two regimesare separated by a multicritical point (MCP) at = 1. (afterRef. [26]).

Inserting these transformations into the N = 2 versionof the Hamiltonian (1) gives

H =i

(Kizi

zi+1 + hi

xi )

i

(Jizi zi+1 + gi

xi )

i

(Jizi

zi+1

zi zi+1 + hi

xi

xi ) . (11)

An intuitive physical picture of the strong-couplingregime 1 close to self duality, htyp Jtyp, emergesdirectly from this Hamiltonian. The product variable is dominated by the four-spin interactions Ki whilethe behavior of the variable i which traces the originalspins is dominated by the two-spin transverse fields gi.All other terms vanish in the limit , i.e., the pairproduct variable and the spin variable asymptotically de-couple. The system is therefore in a partially orderedphase in which the pair product variable z developslong-range order while the spins remain disordered. Adetailed strong-disorder renormalization group study [26]confirms this picture and also yields the complete phasediagram (see Fig. 1) as well as the critical behaviors ofthe various quantum phase transitions. For example, thetransitions between the product phase and the param-agnetic and ferromagnetic phases (transitions 2 and 3 inFig. 1) are both of infinite-randomness type and in therandom transverse-field Ising universality class.

The strong-coupling behavior of the random quantumAshkin-Teller chains with N = 3 and 4 colors could notbe worked out with the above methods.

4B. Variable transformation for N = 3

In this and the following subsections, we present amethod that allows us to study the strong-couplingregime of the random quantum Ashkin-Teller model forany number N of colors. It is based on a generalizationof the variable transformation (9), (10) of the two-colorproblem. We start by discussing N = 3 colors whichis particularly interesting because it is not covered bythe existing work [25, 26]. Furthermore, it is the low-est number of colors for which the clean system featuresa first-order transition. After N = 3, we consider gen-eral odd and even color numbers N which require slightlydifferent implementations.In the three-color case, the transformation is defined

by introducing two pair variables and one product of allthree original colors,

zi = Sz1,i S

z3,i,

zi = S

z2,i S

z3,i,

zi = S

z1,i S

z2,i S

z3,i .(12)

The corresponding transformation of the Pauli matricesSx,i is given by

Sx1,i = xi

xi , S

x2,i =

xi

xi , S

x3,i =

xi

xi

xi . (13)

Inserting these transformations into the Hamiltonian (1)yields

H =i

gi (xi +

xi +

xi

xi ) (14)

i

Ki(zi

zi+1 +

zi

zi+1 +

zi

zi+1

zi

zi+1

)i

hi (xi +

xi +

xi

xi )

xi

i

Ji(zi

zi+1 +

zi

zi+1 +

zi

zi+1

zi

zi+1

)zi

zi+1.

We see that the triple product i does not show up inthe terms containing gi and Ki. In the strong-couplinglimit, 1, gi and Ki are much larger than hi andJi. The behavior of the pair variables i and i is thusgoverned by the first two lines of (14) only and becomesindependent of the triple products i. The i themselvesare slaved to the behavior of the i and i via the largebrackets in the third and fourth line of (14).The qualitative features of the strong-coupling regime

follow directly from these observations. The first twolines of (14) form a two-color random quantum Ashkin-Teller model for the variables i and i. As all termsin the brackets have the same prefactor, this two-colorAshkin-Teller model is right at its multicritical couplingstrength c (as demonstrated in Ref. [26] and shown inFig. 1). The i and i thus undergo a direct phase transi-tion between a paramagnetic phase for gtyp > Ktyp and aferromagnetic phase for gtyp < Ktyp. In agreement withthe quantum Aizenman-Wehr theorem, the transition iscontinuous; it is in the infinite-randomness universalityclass of the random transverse-field Ising model. More-over, in contrast to the N = 2 case, there is no additionalpartially ordered phase.

What about the triple product variables i? For largedisorder, the brackets in the third and fourth line of (14)can be treated as classical variables. If the i and i orderferromagnetically, xi +

xi +

xi

xi vanishes (for all sites

surviving the strong-disorder renormalization group atlow energies) while in the paramagnetic phase, zi

zi+1 +

zi zi+1+

zi

zi+1

zi

zi+1 vanishes. Thus, each i becomes a

classical variable that is slaved to the behavior of i andi. This means, the i align ferromagnetically if the iand i are ferromagnetic while they form a spin-polarizedparamagnet if i and i are in the paramagnetic phase.All these qualitative results are confirmed by a strong-

disorder renormalization group calculation which we nowdevelop for the case of general odd N .

C. Variable transformation and strong-disorder

renormalization group for general odd N

For general odd N > 2, we define N 1 pair variablesand one product of all colors

z,i = Sz,i S

zN,i ( = 1 . . .N 1), zi =

N=1

Sz,i .

(15)The corresponding transformation of the Pauli matricesSx,i is given by

Sx,i = x,i

xi ( = 1 . . .N 1), SxN,i =

N1=1

x,i xi .

(16)In terms of these variables, the Hamiltonian (1) reads

H =i

gi

N1

5in the high-energy subspace. The Schroedinger equationcan then be written in matrix form

(H11 H12H21 H22

)(12

)= E

(12

)(18)

with Hij = PiHPj . Here, P1 and P2 project onthe low-energy and high-energy subspaces, respectively.Eliminating 2 from these two coupled equations givesH111 +H12(E H22)1H211 = E1. Thus, the effec-tive Hamiltonian in the low-energy Hilbert space is

Heff = H11 +H12(E H22)1H21 . (19)

The second term can now be expanded in inverse powersof the large local energy scale gi or Ki.

In the strong-coupling regime, 1, the strong-disorder renormalization group is controlled by the firsttwo lines of (17). It does not depend on the N -productsi which are slaved to the i and i via the large bracketsin the third and forth lines of (17).

If the largest local energy scale is the Ashkin-Tellerfield gi, site i does not contribute to the order parameterand is integrated out via a site decimation. The recur-sions resulting from (19) take the same form as in theweak-coupling regime, i.e., the effective interactions andcoupling strength are given by eqs. (2) to (4) [28].

What about the product variable i? The bracket inthe third line of the Hamiltonian (17) takes the value Nwhile the bracket in the fourth line vanishes. However,because hi gi, the value of xi is not fixed by the renor-malization group step. Thus i xi becomes a classicalIsing degree of freedom with energy Nhii that is inde-pendent of the terms in the renormalized Hamiltonian.This means, it is left behind in the renormalizationgroup step. Consequently, xi plays the role of the addi-tional internal degree of freedom first identified in Ref.[25].

The bond decimation step performed if the largest lo-cal energy is the four-spin interaction Ki can be derivedanalogously. The recursion relations are again identicalto the weak-coupling regime, i.e., the resulting effectivefield and coupling are given by eqs. (5) to (7). In thisstep, the bracket in the fourth line of the Hamiltonian(17) takes the value N while the bracket in the third linevanishes. Thus, the renormalization group step leavesbehind the classical Ising degree of freedom i zi zi+1with energy NJii. In the bond decimation step, theadditional internal degree of freedom of Ref. [25] is thus

given by i zi zi+1.All of these renormalization group recursions agree

with those of Ref. [25] where the renormalization groupwas implemented in the original variables for N > 4 col-ors.

D. Variable transformation and strong-disorder

renormalization group for general even N

For general even N 4, the variable transformation isslightly more complicated than in the odd N case. Wedefine N2 pair variables, a product of N1 colors anda product of all N colors,

z,i = Sz,i S

zN1,i ( = 1 . . .N 2),

zi =

N1=1

Sz,i , zi =

N=1

Sz,i. (20)

The Pauli matrices Sx,i then transform via

Sx,i = x,i

xi

xi ( = 1 . . .N 2),

SxN1,i =N2=1

x,i xi

xi , S

xN,i =

xi . (21)

After applying these transformations to the Hamiltonian(1), we obtain

H = i=1

gi

N2

6To substantiate these qualitative arguments, we haveimplemented the strong-disorder renormalization groupfor the Hamiltonian (22), using the projection methodas in the last subsection. In the case of a site decima-tion, i.e., if the largest local energy is the Ashkin-Tellerfield gi, we again obtain the recursion relations (2) to(4). The variable xi is not fixed by the renormalization

group. Thus i xi represents the extra classical Isingdegree of freedom that is left behind in the renormaliza-tion group step. Its energy is Nhii. If the largest localenergy is the four-spin interactionKi, we perform a bonddecimation. The resulting recursions relations agree withthe weak-coupling recursions (5) to (7). In this case,the product zi

zi+1 is not fixed by the decimation step.

Therefore, the left-behind Ising degree of freedom in thisdecimation step is i zi zi+1 with energy NJii.The above strong-disorder renormalization group

works for all even color numbers N > 4. For N = 4,an extra complication arises because the left-behind in-ternal degrees of freedom i do not decouple from therest of the Hamiltonian. For example, when decimatingsite i (because gi is the largest local energy), the

z termin the the third line of (22) mixes the two states of theleft-behind xi degree of freedom in second order pertur-bation theory. An analogous problem arises in a bonddecimation step. Thus, for N = 4 colors, the internal idegrees of freedom need to be kept, and the renormal-ization group breaks down. In contrast, for N > 4, thecoupling between the internal i degrees of freedom andthe rest of the Hamiltonian only appears in higher orderof perturbation theory and is thus renormalization-groupirrelevant.

E. Renormalization group flow, phase diagram,

and observables

For color numbers N = 3 and all N > 4, the strong-disorder renormalization group implementations of thelast two subsections all lead to the recursion relations (2)to (7). The behavior of these recursions has been studiedin detail in Ref. [25]. In the following, we therefore sum-marize the resulting renormalization group flow, phasediagram, and key observables.According to (4) and (7), the coupling strengths flow

to infinity if their initial value I > c(N). Moreover,the competition between interactions Ki and fields giis governed by the recursion relations (3) and (6) whichsimplify to

K =Ki1Ki

2(N 2)gi , g =gigi+1

2(N 2)Ki (23)

in the large- limit. They take the same form asFishers recursions of the random transverse-field Isingmodel [24]. (The extra constant prefactor 2(N 2)is renormalization-group irrelevant). The renormaliza-tion group therefore leads to a direct continuous phasetransition between the ferromagnetic and spin-polarized

FIG. 2. Schematic of the renormalization-group flow diagramon the self-duality line of the random quantum Ashkin-Tellermodel with N = 3 or N > 4 colors in the disordercouplingstrength parameter space. For < c(N) (left arrows), thecritical flow approaches the usual Ising infinite-randomnesscritical point of Ref. 24. For > c (right arrows), we find adistinct infinite-randomness critical point with even strongerthermodynamic singularities (after Ref. [25]).

paramagnetic phases on the self-duality line gtyp = Ktyp(or, equivalently, htyp = Jtyp) . The renormalizationgroup flow on this line is sketched in Fig. 2. In theweak-coupling regime, I < c(N), the flow is towardsthe random-transverse field Ising quantum critical pointlocated at infinite disorder and = 0, as explained inSec. III. In the strong-coupling regime, I > c(N), theN -color random quantum Ashkin-Teller model (N = 3and N > 4) features a distinct infinite-randomness crit-ical fixed point at infinite disorder and infinite couplingstrength. It is accompanied by two lines of fixed pointsfor r = ln(gtyp/Ktyp) > 0 (r < 0) that represent theparamagnetic (ferromagnetic) quantum Griffiths phases.The behavior of thermodynamic observables in the

strong-coupling regime at criticality and in the Grif-fiths phases can be worked out by incorporatingthe left-behind internal degrees of freedom in therenormalization-group calculation. This divides therenormalization group flow into two stages and leads totwo distinct contributions to the observables [25]. Forexample, the temperature dependence of the entropy atcriticality takes the form

S = C1

[ln

(IT

)] 1

ln 2 + C2

[ln

(IT

)] 1

N ln 2,

(24)

where = 1/2 is the tunneling exponent, = 12 (1+5),

C1 and C2 are nonuniversal constants, and I is the bareenergy cutoff. The second term is the usual contributionof clusters surviving under the strong-disorder renormal-ization group to energy scale = T . The first termrepresents all internal degrees of freedom left behinduntil the renormalization group reaches this scale. As > 1, the low-T entropy becomes dominated by the ex-tra degrees of freedom S Sextra [ln(I/T )]1/().

7Analogously, in the Griffiths phases, the contribution ofthe internal degrees of freedom gives

Sextra |r|(T/I)1/(z+Az) ln 2, (25)

which dominates over the regular chain contribution pro-portional to T 1/zN ln 2. Here, = 2 is the correlationlength critical exponent, and z = 1/(2|r|) is the non-universal Griffiths dynamical exponent. Other observ-ables can be calculated along the same lines [25].The weak and strong coupling regimes are separated

by a multicritical point located at r = 0 and I = c(N).At this point, the renormalization group flow has twounstable directions, r = ln(gtyp/Ktyp) and I c(N).The flow in r direction can be understood be insertingc(N) into the recursion relations (2) and (5) yielding

J =Ji1Ji

(1 + (N 1)c)hi , h =hihi+1

(1 + (N 1)c)Ji . (26)

These recursions are again of Fishers random transverse-field Ising type (as the prefactor (1 + (N 1)c) isrenormalization-roup irrelevant). Thus, the renormaliza-tion group flow at the multicritical point agrees with thatof the weak-coupling regime. Note, however, that the Ntransverse-field Ising chains making up the Ashkin-Tellermodel do not decouple at the multicritical point. Thus,the fixed-point Hamiltonians of the weak-coupling fixedpoint and the multicritical point do not agree.The flow in the direction can be worked out by

expanding the recursions (4) and (7) about the fixedpoint value c(N) by introducing J,i = J,i c andh,i = h,i c. This leads to the recursions

J = J,i+J,i+1+Y hi , h = h,i+h,i+1+Y J,i (27)

with Y = c/[(1+ (N 1)c)(1+ (N 2)c)]. Recursionsof this type have been studied in detail by Fisher in thecontext of antiferromagnetic Heisenberg chains [29] andthe random transverse-field Ising chain [24]. Using theseresults, we therefore find that scales as

typ() Y I , Y = 12(1 +

5 + 4Y ) (28)

with the renormalization group energy scale =ln(I/). The crossover from the multicritical scalingto either the weak-coupling or the strong-coupling fixedpoint occurs when |typ| reaches a constant x of orderunity. It thus occurs at an energy scale x = |x/I |1/Y .

V. CONCLUSIONS

To summarize, we have investigated the ground statephase diagram and quantum phase transitions of theN -color random quantum Ashkin-Teller chain which isone of the prototypical models for the study of vari-ous strong-disorder effects at quantum phase transitions.

After reviewing existing strong-disorder renormalizationgroup approaches, we have introduced a general variabletransformation that allows us to treat the strong-couplingregime for N > 2 in a unified fashion.

For all color numbersN > 2, we find a direct transitionbetween the ferromagnetic and paramagnetic phases forall (bare) coupling strengths I 0. Thus, an equivalentof the partially ordered product phase in the two-colormodel does not exist for three or more colors. In agree-ment with the quantum version of the Aizenman-Wehrtheorem [13], this transition is continuous even if the cor-responding transition in the clean problem is of first or-der. Moreover, the transition is of infinite-randomnesstype, as predicted by the classification of rare regions ef-fects put forward in Refs. [9, 30] and recently refined inRef. [31]. Its critical behavior depends on the couplingstrength. In the weak-coupling regime < c(N), thecritical point is in the random transverse-field Ising uni-versality class because the N Ising chains that make upthe Ashkin-Teller model decouple in the low-energy limit.In the strong-coupling regime, > c(N), we find a dis-tinct infinite-randomness critical point that features evenstronger thermodynamic singularities stemming from theleft-behind internal degrees of freedom.

The novel variable transformation also allowedus to study the multicritical point separating theweak-coupling and strong-coupling regimes. Itsrenormalization-group flow has two unstable directions.The flow for r = ln(gtyp/Ktyp) 6= 0 and I c(N) = 0is identical to the flow in the weak-coupling regime im-plying identical critical exponents. The flow at r = 0in the direction is controlled by different recursions for = c(N) which we have solved for general N .So far, we have focused on systems whose (bare) cou-

pling strengths are uniform J,i = h,i = I . What aboutrandom coupling strengths? If all J,i and h,i are smallerthan the multicritical value c(N), the renormalized de-crease under the renormalization group just as in the caseof uniform bare . If, on the other hand, all J,i and h,iare above c(N), the renormalized values increase underrenormalization as in the case of uniform bare . There-fore, our qualitative results do not change; in particular,the bulk phases are stable against weak randomness in. The same holds for the transitions between the ferro-magnetic and paramagnetic phases sufficiently far awayfrom the multicritical point. Note that this also explainswhy the randomness in produced in the course of thestrong-disorder renormalization group is irrelevant if theinitial (bare) are uniform: All renormalized values areon the same side of the multicritical point and thus floweither to zero or to infinity.

In contrast, the uniform- multicritical point itself isunstable against randomness in . The properties of theresulting random- multicritical point can be studied nu-merically in analogy to the two-color case [26]. This re-mains a task for the future.

8ACKNOWLEDGEMENTS

We are grateful for the support from NSF under GrantNos. DMR-1205803 and PHYS-1066293, from Simons

Foundation, from FAPESP under Grant No. 2013/09850-7, and from CNPq under Grant Nos. 590093/2011-8 and305261/2012-6. J.H. and T.V. acknowledge the hospital-ity of the Aspen Center for Physics.

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