Multiple universalities in order-disorder magnetic phase transitions

0.9 MB·

by  H. D. Scammell

Multiple universalities in order-disorder magnetic phase transitions
H. D. Scammell and O. P. Sushkov
School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia
(Dated: January 17, 2017)
Phase transitions in isotropic quantum antiferromagnets are associated with the condensation
of bosonic triplet excitations. In three dimensional quantum antiferromagnets, such as TlCuCl ,
3
condensation can be either pressure or magnetic field induced. The corresponding magnetic order
obeys universal scaling with thermal critical exponent φ. Employing a relativistic quantum field
theory, the present work predicts the emergence of multiple (three) universalities under combined
pressure and field tuning. Changes of universality are signalled by changes of the critical exponent
φ. Explicitly, we predict the existence of two new exponents φ = 1 and 1/2 as well as recovering
7 the known exponent φ=3/2. We also predict logarithmic corrections to the power law scaling.
1
0 PACSnumbers: 64.70.Tg,74.20.De
2

n
Pressureandmagneticfieldinducedcondensatephases B>0"
a B" T>0" T" B=0"
J inquantummagneticsystemshavebecomeinstrumental
6 to our understanding of universal, critical-phenomena. T=0" δBBEC" δTN"
1 A great effort (experimental, numerical and theoretical)
Disordered" BEC" Disordered" AFM"
hasbeendevotedtouncoveringandcategorisingtheuni-

] versal features of critical magnetic condensate phases.
l
e Thepresentworkconsidersthreedimensional(3D)quan- (a) p" p" (b) p" p"
- c c
r tum antiferromagnets (QAF), where the combined in-
st terplay between pressure, magnetic field and tempera- FIG. 1: Critical field and temperature power law shifts.
t. ture (p,B,T) remains theoretically unexplored, yet of- (a) Shift of critical field-pressure line with temperature
a fers an exiting arena for theorists and experimentalists δBBEC ∼Tφ. Solidbluecurveisatzerotemperature,dashed
m blue at non-zero temperature. (b) Shift of critical (N´eel)
aliketouncovernewuniversalbehaviour. InFigure1we
- temperature-pressure line with field δTN ∼ B1/φ. Solid red
present the generic phase diagrams of dimerised QAFs
d curve is at zero field, dashed red at non-zero field.
n suchasTlCuCl3,KCuCl3,andCsFeCl3. Panel(a)shows
o the magnon Bose condensation (BEC) line in the field-
c pressurediagram, andpanel(b)showstheantiferromag- φ=1/2
[
netic (AFM) transition line in the temperature-pressure φ=1

1 diagram. It is also instructive to look at Figure 2 which φ=3/2 TN(p)"
v
shows the 3D (p,B,T) phase diagram. The point of
1
primary interest is the critical field-critical temperature
8 T(B)"
1 power law, c
4 AFM"
0 a:δBBEC ∼Tφ, b:δTN ∼B1/φ, (1)
1. BEC" T" p"
TheshiftoftheBECtransitionlineatsmalltemperature
0
7 is shown schematically in Fig. 1a; while the shift of the B" Disordered" p"
1 AFM/N´eel transition line at small field is in Fig. 1b. Bc" c
: It is widely believed that at p < p , φ = 3/2 is the
v c
i universal BEC exponent, which can be obtained from FIG. 2: Multiple universalities in the (p,B,T) phase dia-
X the scaling arguments on the dilute Bose gas [1, 2] or gram. Blue curves correspond to the BEC transition lines;
r explicitly for magnon BEC [3, 4]. For a review see [5]. here p < pc and the critical exponent is φ = 3/2. Red
a curves correspond to the N´eel transition lines; here p > p
Ontheotherhand,experiment(onTlCuCl andKCuCl c
3 3 and the critical exponent is φ = 1/2. The dashed, black
[6–9]) and numerics [10] show 1.5 ≤ φ (cid:46) 2.3, depending
curve shows the critical pressure transition line, with critical
crucially on which temperature range is used for fitting exponent φ=1.
[5, 11]. We understand recent data on 3D QAF CsFeCl
3
[12], taken along the thick blue-red solid lines in Fig. 2,
as a hint for a significant and unexpected evolution of pends on the fitting range; even if a priori the range
the index φ along the line. seems to be very narrow. We will show that answers to
The primary goal of the present work is to derive the both questions are related to the quantum critical point
evolution of the critical index φ across the phase dia- (p,B,T) = (p ,0,0). Ultimately, the quantum critical
c
gram. Another goal is to explain why the index de- point (QCP) governs the evolution of the critical index
2
φ across the phase diagram. This is illustrated in Fig. 2.
0.8
Previous theoretical approaches were concentrated at D+ 1.5 D+Ê
theBECtransition,p<p . TheyemployedadiluteBose 0.6 Ê
c V D V D Ê
gas model [4, 5] and/or bond-operator technique [13]. me D0 me1.0 Ê D0
In the end, these techniques rely on the Hartree-Fock- s0.4 @ s ‡‡ @ Ï Ï Ï Ï
D D
Popov approximation, yet it is known that the Hartree- 0.2 D- 0.5 ‡ ‡ D-
Fock-Popov approximation breaks down in the vicinity ‡ ‡
of a critical point [14]. In the present work we employ a 0.00.0 0.2 0.4 0.6 0.8 1.0 1.2 0.00 2 4 6 8
quantumfieldtheoryapproachwhichnaturallydescribes (a) p kbar (b) B T
quantum critical points.
FIG. 3: Excitation gaps ∆ : (Left) pressure driven at fixed
Thequantumphasetransition(QPT)betweenordered @ D σ @ D
field B = 0.2 meV and T = 0. (Right) field driven at p = 0
and disordered phases is described by the effective field kbar and T = 1.5 K. Solid lines are theoretical results de-
theory with the following Lagrangian [15, 16], rived in this paper. Markers indicate experimental data for
TlCuCl [17, 18].
3
1 1 1 1
L= (∂ ϕ(cid:126)−ϕ(cid:126)×B(cid:126))2− (∇(cid:126)ϕ(cid:126))2− m2ϕ(cid:126) 2− α ϕ(cid:126) 4.
2 t 2 2 0 4 0
(2) Beyond mean-field: Everywhere in the text m2 =
0
γ2(p −p) and α represent the zero temperature mass
0 c 0
The vector field ϕ(cid:126) describes staggered magnetisation, B tuning parameter and coupling constant without quan-
is an external applied field, and for now we set gµB =1. tum fluctuation corrections. Taking into account quan-
We now briefly outline the mean-field phase transitions tum and thermal fluctuation corrections due to interac-
captured by this Lagrangian. Consider first B = 0, the tion term 1α ϕ(cid:126) 4, we will denote the renormalized pa-
4 0
pressureinducedQPTresultsfromtuningthemassterm, rameters m2 → m2 and α → α . The explicit form
0 Λ,σ 0 Λ
m20, for which we take the linear expansion m20(p) = form2 =m2 (p,T,B)dependsonthelocationwithin
γ2(pc−p), where γ2 >0 is a coefficient and p is the ap- the phΛa,σse diaΛg,rσam, and polarisation σ. Full details are
pliedpressure. Varyingthepressureleadstotwodistinct presentedinSupplementaryMaterialCandD,whileex-
phases; (i) for p < pc we have m20 > 0, and the classi- pressions are presented below. The strength of the cou-
cal expectation value of the field is zero ϕ2c = 0. This plingαΛ determinesthestrengthofallinteractionsinthe
describes the magnetically disordered phase, the system theory, and is dependent on the energy scale Λ. Generi-
has a global O(3) rotational symmetry, and the excita- cally, the one-loop renormalized coupling takes the form
tionsaregappedandtriplydegenerate. (ii)Forpressures [19, 21]
p>p we have m2 <0, and the field obtains a non-zero
mclaasgsnicceatilceaxllpyecotradtei0roend,vaalnuteifϕer2cro=m|amαg020n|e.tTichpishdasees.crVibaersytinhge αΛ = 1+11α0/(8απ02)ln(Λ0/Λ). (3)
m20 from positive to negative spontaneously breaks the Specifically for the problem at hand, the coupling runs
O(3) symmetry of the system. with scale Λ = max{m ,B,T}. Accordingly, there is
Λ,σ
Next consider non-zero B at fixed p < pc: For B < just a single point on the phase diagram at which all
Bc =m0 the system has O(2) symmetry, and the degen- energy scales vanish Λ → 0; the quantum critical point
eracy of the triplet modes is lifted by Zeeman splitting. (p ,0,0), see Fig. 2. At this point the coupling runs to
c
The field induced QPT results from tuning B > m0. zero αΛ → 0 (asymptotic freedom). The running of the
The condensate field is ϕ2 = B2−m20. To determine the coupling constant will play an essential role in resolving
c α0
order-disorder(BECorAFM)transitionlineonecanap- our main goals/questions: Why the index φ depends on
proach the transition starting from either the ordered or the location within the phase diagram, and; why the ex-
disordered phase. In this work we start from the lat- pected index φ=3/2 in the BEC regime depends on the
ter;allresultsarederivedstartingfromdisorderedphase. fitting range.
There are three magnetic excitations with ladder polar- In the disordered phase the Euler-Lagrange equation
isation σ = −,0,+. The polarisation is the projection with (2) results in the following dispersion
of angular momentum on the direction of magnetic field.
(cid:113)
In Figure 3 we summarise the results for the evolution ωσ = k2+m2 +σB. (4)
k Λ,σ
of the three mode gaps through the field and pressure
quantum phase transitions, separately. Explicit parame- where m is the renormalised mass. Note that the σB
Λ,σ
ters correspond to those found in Ref. [19] for TlCuCl . term is not renormalised. This is a consequence of a
3
Here we disregard the small easy-plane anisotropy seen Ward identity (Larmor theorem). While the stationary
in TlCuCl , which has been shown to have negligible in- states (4) have a fixed ladder polarisation, technically it
3
fluence on the critical properties [19], see also comment is more convenient to calculate fluctuation corrections in
[20]. theCartesianbasisϕ(cid:126) =(ϕ ,ϕ ,ϕ ). LetusdenotebyV
x y z
3
thepartoftheLagrangian(2)independentofderivatives. pressure, p = p ; (III) below the critical pressure, when
c
Then, using a Wick decoupling of the interaction term T = T . At zero magnetic field, the critical temper-
c BEC
1α ϕ(cid:126) 4, in the single loop approximation we find ature in case (I), Eq. (8), is identical to the equation for
4 0
the N´eel temperature derived in Ref. [19].
∂2V
=m2−B2+3α (cid:104)ϕ2(cid:105)+α (cid:104)ϕ2(cid:105)+α (cid:104)ϕ2(cid:105) Consider case (I); p > pc. In this case according to
∂ϕ2 0 0 x 0 y 0 z Eq. (1b)theN´eeltemperaturevariesinaweakmagnetic
x
∂∂2ϕV2y =m20−B2+α0(cid:104)ϕ2x(cid:105)+3α0(cid:104)ϕ2y(cid:105)+α0(cid:104)ϕ2z(cid:105) fideislpde.rTsiooncsalωcuk+la=teωΣk−T a=t ωBk0→=0kw.eHteankceethΣeTcr=itic51aα2lΛlTin2e,
where T = T +δT ; T is the N´eel temperature in
∂2V N0 N N0
=m2+α (cid:104)ϕ2(cid:105)+α (cid:104)ϕ2(cid:105)+3α (cid:104)ϕ2(cid:105) (5) zero magnetic field. Hence using Eq.(8) we find
∂ϕ2 0 0 x 0 y 0 z
z
6 B2
where (cid:104)ϕ2(cid:105) is the loop integral over the Green’s function (I): δT = at B (cid:28)T . (9)
x N 5α T N0
of field ϕ . An explicit calculation shows (cid:104)ϕ2(cid:105) = (cid:104)ϕ2(cid:105), Λ N0
x x y
hence from equations (5), we have rather trivially satis-
So the critical index in Eq. (1b) is φ=1/2.
fied the O(2) Ward identity: ∂2V/∂ϕ2 −∂2V/∂ϕ2 = 0.
x y In Ref. [19] the set of parameters describing TlCuCl
Further details are presented in Supplementary Material 3
was determined
A and B.
Quantum corrections corresponding to (5) come from
p =1.01 kbar, γ =0.68 meV/kbar1/2,
the scale Λ < q < Λ . Hence they must be accounted c
0 α
viasinglelooprenormalizationgroup(RG).Thethermal 0 =0.23, Λ =1 meV. (10)
8π 0
partof(5)comesfromq ∼T,henceherethesimplesingle
loop approximation is sufficient. All in all, calculations
When fitting experimental data in Ref. [19] the ther-
presented in Supplementary Material D give
mal line-broadening had been accounted via ω = k →
(cid:112)
ω = k2+ξ2T2, ξ = 0.15. Therefore, if we use the
∂2V
=m2 (T)−B2 set of parameters (10) to determine the value of the run-
∂ϕ2 Λ,±
i ning coupling constant α , Eq. (3), the coefficient in
Λ
∂∂2ϕV2 =m2Λ,0(T) (6) F(9i)g.h4awsetoillbuestcroartreeEctqe.d(9a)cbcyorddainshgleyd;y5eα6lΛlow→li1n.e14o5rαi6gΛin.aItn-
z
ing from T = 2.8K. The couling constant is α /8π =
N0 Λ
where ϕ ={ϕ ,ϕ }, and the renormalised masses are
i x y α /8π = 0.107. For comparison, the solid blue line
TN0
originatingfrom2.8KrepresentsexactsolutionofEq.(8)
(cid:20) (cid:21)5
m2 =m2 αΛ 11+Σ with coupling constant running along the line.
Λ,± 0 α T
0 Considercase(II);tuningexactlytothequantumcrit-
m2 =m2(cid:20)αΛ(cid:21)151+α (cid:88)1/ω0{n(ω+)+n(ω−)+3n(ω0)} ical point, p = pc, TN0 = 0. Again, to calculate ΣT
Λ,0 0 α Λ k k k k at B → 0 we have to take the critical line dispersions
(cid:88)0 k ωk+ =ωk− =ωk0 =k and hence again ΣT = 51α2ΛT2. Sub-
Σ ≡α 1/ω0{2n(ω+)+2n(ω−)+n(ω0)}. (7) stitution into (8) gives
T Λ k k k k
k
(cid:114)
5α
Heren(ωk)=1/(eωTk −1),andweintroducethefunction (II): Bc = 12ΛT at Bc (cid:28)T. (11)
Σ for brevity. Obviously, expansions of Eq.’s (7) in
T
powersofBcontainonlyevenpowers. Interestinglythese TheconditionB (cid:28)T issatisfiedatsufficientlylowtem-
c
expansions are different for mΛ,± and mΛ,0. Therefore peraturessincethecouplingconstantsdecayslogarithmi-
therelationωk+−ωk0 =ωk0−ωk−,whichisexactatT =0, cally,αΛ ∝1/ln(cid:0)ΛT0(cid:1). Henceinthiscase(II),thecritical
does not hold at non-zero T. At non-zero T the relation index of Eq. (1) is φ = 1, and we find that, in addition
is valid only up to the linear in B approximation. to the exponent, there is nontrivial logarithmic scaling.
In a magnetic field, the condition of condensation fol- In Fig.4 we illustrate the asymptotic (11) by dashed yel-
lows from Eq. (4), mΛ,±−Bc =0. Using (7) this equa- low line originating from B =T =0. The solid blue line
tion can be rewritten as originatingfromthesamepointrepresentsexactsolution
of Eq.(8).
(cid:20) (cid:21)5
ΣT =Bc2−m20 ααΛ0 11 . (8) InFtihniasllcyasweeoncloynstihdeerωt−hediBspEeCrsiocnasbera(nIIcIh),ispcr<iticpacl.,
k
Therearethreedistinctcases: (I)Abovethecriticalpres- ωk− ≈ 2k∆20, where ∆0 = B0 is the gap at B = 0.
sure, when Tc = TN i.e. critical temperature equals The other two branc√hes are gapped. Calculation of ΣT
the AFM/N´eel temperature; (II) exactly at the critical gives ΣT = αΛζ(√3/2) ∆0T3/2, where ζ is Riemann’s ζ-
π 2π
4
8 7 7 ÊÊÊ ‡
BT46ÊÊÊÊÊÊÊÊÊÊÊÊ@DÊÊÊÊÊÊ ÊÊÊÊÊÊ ÊÊÊÊ ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ BT3456 p<@Dpc BT3456 ÊÊÊ@DÊpÊ<ÊÊpÊÏʇcʇÊÊÏÊÊÊÏÊʇʇÚÏʇ‡‡ÏÚÏÊÙÚÚÚÊÙÚÏÊÙÚÊÙÚχχ‡ ÊÊÊ
ÚÚÚ Ï ‡
2 12 p= pc p>pc 12 p~pc ÊÙÚÚÚÊÙÊÚÙÏÚÏÏχ‡Ï‡‡‡‡ ÊÊÊÊpÊ>pc
00 1 2 3 4 5 00 2 4 6 8 00 1 2 ÏÏ3‡‡‡ 4 ÊÊÊ 5
T K (a) T K (b) T K
FIG. 4: Critical field vs temperature: Dashed yellow curves FIG.5: Multipleuniversalities: Variouscurvesshowthecrit-
showsolutionstoscalingequatio@nsD (9),(11)and(12). Dashed ical field Bc(T)@atD various pressures ranging p@<D pc, p = pc
maroon shows solution of (8) that accounts for thermal mix- to p>pc. (a) Solutions to (8) with parameters for TlCuCl3.
ing of non-critical modes, but does not account for running (b) Data for quantum antiferromagnet CsFeCl3 [12].
coupling; coupling is at fixed value α →α =0.169×8π.
Λ ∆0
Solid blue lines are the solution to (8) with full account of
non-critical modes and logarithmic running coupling. Blue
Ontheotherhand,theBECregime(III)hasbeenconsid-
points are experimental data from [9, 22, 23].
eredinanumberofpublicationsusingtheHartree-Fock-
Popov approximation for hard core bosons, from which
function. Hence, using Eq. (8) we find simple T3/2 dependence is predicted. Our conclusion is
that such an approximation is only valid at vanishingly
δB ζ(3/2)(cid:18) T (cid:19)3/2 small temperatures and the region of validity shrinks to
(III): c =α at δB (cid:28)∆ .
∆0 Λ (2π)32 ∆0 c 0 zero upon approaching the critical pressure QCP. This
is illustrated in Fig. 4 by lines originating from points
(12)
B = 4.73T and B = 2.36T at T = 0. Our exact theo-
0 0
AsexpectedthecriticalindexinEq. (1a)isφ=3/2. To retical solutions (blue solid lines) differ from the simple
understand the region of validity of Eq.(12) we compare T3/2 dependence (dashed yellow) due to two effects; in-
with TlCuCl data [9, 22, 23]. The value of the gap at fluence the non-critical excitations, and the running of
3
T = p = B = 0 is ∆ = m = 0.64meV [24]. The the coupling constant. Both effects are governed by the
0 Λ,±
BEC critical field for T = p = 0 is B = 4.73 T [25]. magnetic quantum critical point (p ,0,0) and cannot be
0 c
Hence, we obtain the g-factor, which is defined as B → accounted within a hard core boson model; whether it
gµ B, g = 2.35 [20]. In Fig. 4 the dashed yellow line be Hartree-Fock-Popov approximation or even an exact
B
originating from B = 4.73T shows B versus T at solution. Including these effects, the present analysis re-
0 BEC
p=0calculatedwithEq. (12). Thevalueofthecoupling solves the long standing problem of the BEC critical ex-
constant in this equation is obtained from Eq.’s (3) and ponent, which has been consistently reported at higher
(10), α /(8π) = α /(8π) = 0.169. Experimental data value; 3/2≤φ≤2.3 [5–11].
Λ ∆0
[9, 22, 23] are shown by circles. We see that Eq. (12) is
The existence of three critical exponents φ = 3/2, 1
valid only at T ≤1K.
and 1/2, and even logarithmic corrections to these expo-
There are two physical effects accounted in (8), but
nents,isareadilytestableresultandconstitutesourmost
neglect in (12). These are (i) the influence of the non-
important prediction for experiment. Figure 4 provides
critical (gapped) modes ω+,ω0; (ii) the logarithmic run-
k k predictions directly for TlCuCl3. In Figure 5a we plot
ning of α . To illustrate the importance of non-critical
Λ the predicted critical field in TlCuCl vs temperature at
3
modes, the dashed maroon line originating from 4.73 T
various pressures. For comparison in Fig. 5b we present
in Fig. 4 shows solution solution of Eq. (8) with ac-
asimilarexperimentalplotforquantumantiferromagnet
countofallthreemodes,butwithfixedcouplingconstant
CsFeCl published very recently [12]. Unfortunately we
3
α /(8π)=0.169. Finally,thesolidbluelineoriginating
∆0 cannot perform exact quantitative calculations (includ-
from 4.73 T shows solution of (8) with account of both
ing all pre-factors) for CsFeCl . Existing data for this
3
(i) and (ii). Agreement with experiment is remarkable.
compound are not sufficient to perform analysis similar
Westressthatthereisnofittinginthetheoreticalcurve.
to [19] for TlCuCl . However, the data [12] supports the
3
The set of parameters (10) was determined in Ref. [19]
proposed multiple critical exponent theory.
from data unrelated to magnetic field. To be consistent
In summary, employing a quantum field theoretic ap-
with this set when generating the solid blue and dashed
proach, our work predicts multiple critical exponents,
maroon curves in Fig. 4 we use the same line broaden-
(cid:113) and their corresponding logarithmic corrections, on the
ing as in [19], ωkσ → k2+m2Λ,σ+Γ2T +σB, ΓT = ξT, pressure, magnetic field and temperature - phase digram
ξ =0.15. for3Dquantumantiferromagnetsinvicinityofthequan-
Regimes(I)and(II)haveneverbeenconsideredbefore. tumcriticalpoint. ForTlCuCl wedemonstrateremark-
3
5
ableagreementwithexistingdata,andprovidequantita- [10] S.Wessel,M.Olshanii,andS.Haas,Phys.Rev.Lett.87,
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[12] N.KuritaandH.TanakaPhys.Rev.B94,104409(2016).
We thank Christian Ru¨egg for a very important, stim-
[13] Jesko Sirker, Alexander Weiße and Oleg P. Sushkov, J.
ulatingdiscussionandYaroslavKharkovfordrawingour
Phys. Soc. Jpn. Vol. 74 (2005) Suppl. pp. 129?134
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the Australian Research Council grant DP160103630. 304 (1998) 1.
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Univ. Press, 2011).
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