Phase structure and phase transitions in a three dimensional SU(2) superconductor

1.1 MB·English

by  Egil V. Herland

Phase structure and phase transitions in a three dimensional SU(2) superconductor
Egil V. Herland,1 Troels A. Bojesen,1 Egor Babaev,2,3 and Asle Sudbø1
1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
2Physics Department, University of Massachusetts, Amherst, Massachusetts 01003, USA
3Department of Theoretical Physics, The Royal Institute of Technology, 10691 Stockholm, Sweden
We study the three dimensional SU(2)-symmetric noncompact CP1 model, with two charged
matter fields coupled minimally to a noncompact Abelian gauge-field. The phase diagram and the
nature of the phase transitions in this model have attracted much interest after it was proposed to
describe an unusual continuous transition associated with deconfinement of spinons. Previously, it
hasbeendemonstratedforvarioustwo-componentgaugetheoriesthatweaklyfirst-ordertransitions
3
1 may appear as continuous ones of a new universality class in simulations of relatively large, but
0 finite systems. We have performed Monte-Carlo calculations on substantially larger systems sizes
2 than those in previous works. We find that in some area of the phase diagram where at finite
sizes one gets signatures consistent with a single first-order transition, in fact there is a sequence
n
of two phase transitions with an O(3) paired phase sandwiched in between. We report (i) a new
a
estimate for the location of a bicritical point and (ii) the first resolution of bimodal distributions
J
in energy histograms at relatively low coupling strengths. We perform a flowgram analysis of the
4
directtransitionlinewithrescalingofthelinearsystemsizeinordertoobtainadatacollapse. The
1
datacollapsesuptocouplingconstantswherewefindbimodaldistributionsinenergyhistograms.

]
h PACSnumbers: 67.85.De,67.85.Fg,67.90.+z,74.20.De,74.25.Uv
c
e
m I. INTRODUCTION sionsofpairedphasesinvariousU(1) U(1)systems, see
×
Refs. 11, 21–23.) Furthermore, resorting to mean-field
-
t theory arguments, it has been pointed out that at least
a Recently, the CP1 model consisting of two matter
t inthevicinityofapairedstate(intheparameterspaceof
s fields coupled to an Abelian gauge field has been of
the model), the direct phase transition from a symmet-
. great interest in condensed matter physics. One of the
mat sources of interest is the proposed concept of deconfined rsihcoustldatebetofirasts-toartdeerw.1i0thSubbrosekqeunenUt(1M)×onUt(e1-C) asrylmomcaeltcruy-,
quantum criticality (DQC). It has been intensively de-
lations have reported a weak first-order phase transition
- bated as a possible novel paradigm for quantum phase
d fortheeasy-planeNCCP1 model.10,12 Theso-calledflow-
transitions.1–18 Such quantum criticality has been sug-
n grammethodhasalsobeenintroducedinRef.10,specif-
o gested to describe phase transitions that would not fit
ically to characterize weak first-order phase transitions.
c into the Landau-Ginzburg-Wilson (LGW) paradigm of a
Using this method, the direct phase transition from a
[ continuous (second-order) phase transition.1,2 In partic-
symmetric state to a state with broken U(1) U(1) sym-
1 ular, the continuous quantum phase transition from an metry,hasbeenclaimedtobefirst-orderfora×nynon-zero
v antiferromagneticN´eelstateintoaparamagneticvalence-
value of the coupling constant. The phase transitions in
2 bond solid (VBS) state,19,20 does not agree with the
theeasy-planelimitoftheNCCP1modelwerealsoexten-
4 LGW-description, according to which two phases with
sively studied in variety of other regimes in the context
1 different broken symmetries generically are separated by
3 of two-component superconductors with independently
a first-order phase transition. Recently, evidence for the
. conserved condensates.11,21–26
1 DQCscenariohasbeenclaimedinstudiesoftheso-called
0 J-Qmodel,3 whichisaHeisenbergmodelwithadditional For the SU(2)-symmetric case, Monte Carlo compu-
3 higher-order spin interaction terms. Namely, it was sug- tations have been performed in Ref. 9. Here, a direct
1
: gested that high-precision Quantum Monte Carlo sim- second-orderphasetransitionwassuggested,butthesys-
v ulations of this model support a continuous N´eel - VBS tem sizes that were considered were quite small. In a
Xi phasetransitioninaccordancewiththeDQCscenario.3–8 subsequentpaper,13 anextensivestudyofthemodelwas
It has been proposed that the critical field theory of a performed. In particular, for the direct transition line,
r
a continuous N´eel - VBS phase transition is the so-called a second-order phase transition was claimed. At higher
noncompact CP1 model (NCCP1), with a SU(2) sym- couplings to the gauge field, it was suggested to turn
metric field coupled to a noncompact U(1) gauge field in intoafirst-ordertransitionviaatricriticalpoint. Onthe
threedimensions(3D).1,2,9 Initialeffortsonstudyingthis other hand, in Ref. 14 (see also Ref. 27), it was argued
effective model were focused on the special case where thatthedirecttransitionlineisfirst-order. Theflowgram
the SU(2) symmetry was broken down to a U(1) U(1) methodemployedinRef.14showednoevidenceforatri-
×
symmetry, i.e., the easy-plane limit. For this case, a critical point along the direct transition line. Rather, in
continuous phase transition was claimed.9 However, in this work the large-scale behavior at small couplings to
Ref.10,theexistenceofapairedphaseintheU(1) U(1) thegaugefieldwasfoundtobethesameasforhighercou-
×
easy-plane action was pointed out. (For earlier discus- plings, where indications of a first-order transition were
2
seenbyresolvingabimodaldistributionintheenergyhis- where the components of σ are the Pauli matrices, the
tograms. NotethatatthesystemsizesstudiedinRef.14, NCCP1 model (1) can be rewritten as24
no bimodal distributions were resolved at small coupling
1 1
constants. Nonetheless,inRef.14, itwasconcludedthat H = [∂ n(x)]2+ [C(x)]2
8 µ 2
even for weaker couplings, bimodal distributions indica-
tive of a first-order phase transition would emerge for 1
+ (cid:15) ∂ C (x)
large enough system sizes. This conclusion was based on 2e2 µνλ ν λ
(cid:40)
thesimilarityofscalingofvariousquantitiesforlargeand (cid:104)
2
small couplings, as evidenced by the flowgrams. 1
n(x) ∂ n(x) ∂ n(x) , (5)
The weakness of the observed first-order phase transi- − 4 · ν × λ (cid:41)
tion, combined with the necessity of assessing the order (cid:105)
of the phase transitions also in the limit of vanishingly where sum over repeated indices is assumed. The model
small coupling strength, renders this problem computa- represents an O(3) nonlinear σ model coupled to a mas-
tionallyextremelydemanding. Inthiswork,wetherefore sive vector field C(x). The latter represents a charged
examine the phase diagram of the NCCP1 model at sub- mode, and its mass is the inverse magnetic field pene-
stantially larger systems sizes than what has been done tration length. At least for sufficiently large values of
in previous works.13,14 Performing the computations on electric charge coupling, the model can undergo a Higgs
largersystemsallowsustoperformaverydetailedinves- transition (where gauge field becomes massless) without
tigation of the range of parameters where a paired phase restoring simultaneously any broken global symmetries.
is sandwiched in between the fully disordered and fully In that case, the remaining broken global symmetry is
orderedstate. Thismeansthatthesetwophasesaresep- O(3) which is described by the order parameter n(x).
arated, not by a direct transition, but by two separate If one introduces an easy-plane anisotropy for the vec-
transitions. At small system sizes, these two separate tor field n(x), this would break the symmetry of the
phasetransitionsinfactgivesignatureswhichwouldlead model to U(1) U(1), and the separation of variables
×
one to conclude that the system features one single first- yieldsaneutralandachargedmode,thephysicsofwhich
order phase transition. The existence of a paired phase has been extensively studied.10,11,21–26 However, there is
sandwiched in between the fully ordered and disordered onesubstantialdifferenceinthecaseofSU(2)symmetry.
statesemergesonlywhenoneconsiderslargeenoughsys- Thechargedandneutralsectorsarecoupledthroughthe
tems. Our study thus allows us to provide improved es- last term in Eq. (5). Another difference compared to
timates for the location of a bicritical point where the the U(1) U(1) case is that in two dimensions, stable
×
direct transition line splits into two. singly quantized vortex lines do not exist in a type-II
SU(2) model (the same applies to vortex lines in three
dimensions).28 On the other hand, a type-I SU(2) model
has energetically stable counterparts of ordinary singly
II. MODEL
quantized type-I vortices. Since composite vortices are
topological excitations which lead to the occurrence of
The continuum NCCP1 model is written as paired states in U(1) U(1) systems, this aspect makes
×
the phase diagram of SU(2) theory an especially inter-
esting problem to study.
Z = Ψ Ψ† A e−βH, (1) In the Monte Carlo simulations, we employ a lattice
D D D realization of this model on a cubic lattice with size L3
(cid:90)
1 and with lattice constant a = 1. The fields ψ (x) are
H = d3x [∇ ieA(x)]Ψ(x)2+[∇ A(x)]2 , c
2 | − | × then defined on the vertices r ixˆ+jyˆ+kˆzi,j,k
(cid:90) (cid:110) (cid:111)(2) 1,...,L of the lattice, ψc(x∈) { ψc,r. For t|he firs∈t
{term in (}2}), we rescale the gauge fi→eld by e−1 and invoke
where β is the inverse temperature and Ψ†(x) = the gauge invariant lattice difference,
(ψ∗(x),ψ∗(x))aretwocomplexfieldsthatarecoupledto ∂
an1oncom2pactgaugefieldA(x)withchargee. Thefields ∂x −ieAµ(x) ψc(x)→ψc,r+µˆe−iAµ,r −ψc,r, (6)
ψ (x), c 1,2 , obey the CP1 constraint, Ψ(x) =1. (cid:20) µ (cid:21)
c
∈{ } | |
The model can be mapped onto a nonlinear O(3) σ where µ x,y,z and r + µˆ denotes the nearest-
∈ { }
model coupled to massive vector fields.24 By introducing neighborlatticepointtovertexrintheµ-direction. The
the fields, gaugefieldAµ,r livesonthe(r,r+µˆ)linksofthelattice.
For the Maxwell term we get
i
C(x)= 2 [ψc(x)∇ψc∗(x)−ψc∗(x)∇ψc(x)]−eA(x), [∇×A(x)]µ →e−1 (cid:15)µνλ∆νAλ,r, (7)
(cid:88)c (3) (cid:88)ν,λ
where ∆ is the forward finite difference operator,
n(x)=Ψ†(x)σΨ(x), (4) ν
∆ A A A , and (cid:15) is the Levi-Civita
ν λ,r λ,r+νˆ λ,r µνλ
≡ −
3
symbol. Inaddition,byinvokingtheCP1 constraintand the flow of replicas in parameter space, essentially by
discarding constant factors in the partition function Z, shifting coupling values towards the bottlenecks.36 How-
we obtain the following lattice realization of the NCCP1 ever, in cases with no severe bottleneck, the optimal set
model: ofcouplingswasfoundbydemandingthattheacceptance
ratesforswappingneighboringreplicaswereequalforall
Z = A 1 u 2π θ 2π θ e−βH, (8) couplings.37 Irrespective of how the set of couplings was
D D D 1 D 2 found,itwasalwaysascertainedthatreplicaswereableto
(cid:90) (cid:90)0 (cid:90)0 (cid:90)0
traverse parameter space sufficiently many times during
H = √ur√ur+µˆcos(∆µθ1,r Aµ,r) production runs. The measurements were postprocessed
(cid:34)− − by multiple histogram reweighting.38 Random numbers
r,µ
(cid:88) weregeneratedbytheMersenne-Twisteralgorithm.39Er-
√1 u 1 u cos(∆ θ A )
− − r − r+µˆ µ 2,r− µ,r rors were determined by the jackknife method.40
(cid:112) 2 Asmentionedintheintroduction,theNCCP1modelis
1
+ (cid:15) ∆ A , a difficult model on which to perform Monte Carlo com-
2e2  µνλ ν λ,r (cid:35) putations. In Ref. 14, the NCCP1 model was mapped
ν,λ
(cid:88)
  to a so-called J-current model, which allows simulations
where u = ψ 2 = 1 ψ 2 and where ψ is the based on the worm algorithm.41,42 (For the sake of com-
r 1,r 2,r c,r
amplitudean|dθ | isthe−ph|ase|ofthecomplex| field|sψ . pleteness, and since to our knowledge the details of the
c,r c,r
mappinghavenotbeenpublished,wepresentthederiva-
tionofthismappinginAppendixA.)Anapproachbased
ontheJ-currentmodelwasattemptedaswell. However,
III. DETAILS OF THE MONTE CARLO
duetothepresenceoflong-rangeinteractionsinthisfor-
SIMULATIONS
mulation, it was difficult to work with the lattice sizes
above L 40. Hence, the computations were performed
TheMonteCarlosimulationsareperformedonacubic on the m∼odel in the original NCCP1 formulation, using
latticewithperiodicboundaryconditionsinalldirections
thePTalgorithmwithwhichitiseasytogrid-parallelize
and with size L3 where L 8,...,96 . Up to 4.0 107
∈ { } · the lattice.
sweeps over the lattice were performed for the largest
systems, while up to 1.0 107 sweeps were used for initial
·
equilibration and initialization of the coupling distribu-
IV. OBSERVABLES AND FINITE-SIZE
tion (see below). Monte-Carlo time-series were routinely
SCALING
inspected for equilibration. To test for ergodicity, typi-
cally4independentlargesimulationswereperformedfor
Perhaps the most familiar quantity that is used to ex-
the largest system sizes. Histograms based on raw data
plore phase transitions, is the specific heat C . The spe-
and reweighted data were also compared for consistency. v
cific heat is given by the second moment of the action,
Formostofthesimulations, theparalleltempering(PT)
algorithm was employed.29–31 To be specific, we fix the β2
couplingeandperformthecomputationsonanumberof Cv = L3 (H −(cid:104)H(cid:105))2 , (9)
replicas (typically from 8 to 32 depending on the system (cid:68) (cid:69)
size L and the range of β values) in parallel at differ- where brackets ... denote statistical averages. In most
(cid:104) (cid:105)
ent values of β. A Monte Carlo sweep consists of sys- cases, Cv exhibits a well-defined peak at the phase tran-
tematically traversing all lattice points with local trial sition. For a continuous phase transition the correlation
movesofallsixfieldvariablesbytheMetropolis-Hastings length diverges with critical exponent ν as ξ t−ν,
∼ | |
algorithm.32,33 For ur, the proposed new values are cho- with t=(β−βc)/β being the deviation from the critical
sen with uniform probability within the interval [0,1], coupling βc. The critical exponent α is defined by the
aunndifofromr θpcr,or,batbhielitpyrowpiotsheidn tnheewinvtaelruveasla[0re,2cπho.seFnorwtihthe sliimngitueldarsypsatretmofofCsviz,egLiv3e,nthbeyfiCnivte∼-siz|te|−scαa.linTgh(eFnS,Si)noaf
noncompactgaugefield,theproposednewvalu(cid:105)esarecho- the specific heat is given by
senwithinsomelimitedincrement(typically[ π/4,π/4])
from the old values.34 There is no gauge fixi−ng involved Cv ∼C0+C1Lα/ν, (10)
inthesimulations. Inadditiontotheselocaltrialmoves,
where C and C are non-universal coefficients. For
0 1
the Monte Carlo sweep also includes a PT trial move of
a first-order transition, with two coexisting phases and
swapping replicas at neighboring β values.
no diverging correlation length, there is indeed no crit-
All replicas were initially thermalized from an ordered
ical behavior. Still, first-order transitions exhibit well-
ordisorderedstartconfiguration. Then,initialrunswere
behaved FSS with “effective” exponents, α = 1 and
performedinordertoproduceanoptimaldistributionof ν = 1/3.43,44 Hence, the peak of the specific heat scales
couplings for the simulation. In some cases, the set of
as
couplings was found by measuring first-passage-times.35
In this approach, the optimal set of couplings maximizes C L3, (11)
v
∼
4
for a first-order transition. Distinguishing between con- when β < β , scales as χ L2−η at β = β . Hence,
c c
∼
tinuous and first-order transitions is an important issue wemaydeterminetheanomalousscalingdimensionη by
inthepresentwork. Forthatpurpose,FSSofthespecific FSS of χ measurements obtained at β .
c
heat peak will play an important role. The exponent ν can, alternatively, be determined by
We also investigate the third moment of the action calculatingthelogarithmicderivativeofthesecondpower
given by45,46 of the magnetization,51
β3 ∂ m2H
M3 = L3 (H −(cid:104)H(cid:105))3 . (12) ∂β ln m2 = m2 −(cid:104)H(cid:105). (21)
(cid:68) (cid:69) (cid:10)(cid:104) (cid:105)(cid:11)
(cid:10) (cid:11)
Inthevicinityofthecriticalpoint,thisquantitytypically
The FSS of this quantity is ∂ ln m2 L1/ν. Since
features a minimum point and a maximum point [see for ∂β ∼
the logarithmic derivative exhibits a peak that is associ-
instance the inset in panel (b) of Fig. 2]. The difference (cid:10) (cid:11)
ated with the critical point, it is possible to extract ν by
in the M value of these two extrema scales as
3 measuring the logarithmic derivative at the pseudocriti-
(∆M3)height L(1+α)/ν, (13) cal point, without an accurate determination of βc.
∼ Similar to Ref. 13, we search for the critical point of
and the difference in the coupling values scales as the Higgs transition by measuring the dual stiffness
(∆M ) L−1/ν, (14) 2
3 width ∼ ρµµ (q)= r,ν,λ(cid:15)µνλ∆νAλ,reiqr , (22)
for a continuous phase transition. For a first-order tran- dual (cid:42)(cid:12)(cid:12)(cid:80) (2π)2L3 (cid:12)(cid:12) (cid:43)
sition, the FSS is (cid:12) (cid:12)
which is the Fourier space correlator of the magnetic
(∆M ) L6, (15)
3 height field. This order parameter for the Higgs transition is
∼
dual in the sense that it is finite in the high-temperature
and
phase and zero in the low-temperature phase. Like in
(∆M ) L−3. (16) Ref. 13, this quantity is measured at the smallest avail-
3 width
∼ able wavevector q = 0. We chose to measure ρzz at
(cid:54) dual
As was mentioned above, one may construct a three q = (2π/L,0,0). At the critical point, the quantity
min
component gauge neutral field n ,24 given by Lρµµ (q ) is universal, such that the finite-size cross-
r dual min
ings of Lρµµ (q ) can be used to estimate the critical
nr =Ψ∗rσΨr, (17) pointof thdeuaHliggmsintransition. Inaddition, measuringthe
coupling derivative of Lρµµ (q ) can be used to esti-
wherethecomponentsofσ arethePaulimatrices. Since dual min
mate the correlation length exponent ν as
it is a unit O(3) vector, we can introduce a “magnetiza-
tion”, ∂
Lρµµ (q ) L1/ν, (23)
∂β dual min ∼
M= n . (18)
r
at the critical point.
r
(cid:88)
The order parameter m , where m = M/L3, signals
(cid:104) (cid:105)
the onset of order in the O(3) gauge neutral vector field V. NUMERICAL RESULTS
n , and the critical point of this transition can be accu-
r
rately determined by a proper analysis of the finite-size
A. Outline of the phase diagram
crossings of the associated Binder cumulant,47–49
5 3 M4 The phase diagram of the NCCP1 model is presented
U4 = 2 − 2(cid:104)M2(cid:105)2. (19) in Fig. 1. For small values of β, there is a normal phase
(cid:104) (cid:105) thatcanberecognizedbyadisorderedgaugeneutralvec-
The finite-size crossings of the Binder cumulant are tor field nr and a massless gauge field. Hence, m = 0
known to converge rapidly towards the critical coupling and ρµµ (q) = 0 in this phase. For large val(cid:104)ues(cid:105)of e
dual (cid:54)
β . Hence, β can be accurately determined by a simple and higher values of β, there is a transition into a phase
c c
extrapolation of the finite-size crossings to the thermo- that we label the O(3) phase. Here, the vector field
dynamic limit or by invoking scaling forms that account nr is ordered (the O(3) symmetry is spontaneously bro-
for finite-size corrections.49,50 ken), m =0, whereas the gauge field remains massless,
A number of quantities related to magnetization may ρµµ (q(cid:104))(cid:105)=(cid:54) 0. In the case of U(1) U(1) symmetric su-
dual (cid:54) ×
be used to extract critical exponents from the Monte perconductors, this phase is sometimes denoted a metal-
Carlo simulations. The magnetic susceptibility, given by lic superfluid or a paired phase, with long-range order
in the gauge neutral linear combination of the phases
χ=L3β m2 , (20) (in the U(1) U(1) case), but not in the individual
×
(cid:10) (cid:11)
5
3.5 1. O(3) line
3 O(3)phase
In Refs. 14, 54, and 55, the existence of an inter-
SU(2)phase
2.5 mediate paired phase, separating a fully ordered state
from a fully disordered one, was shown in the SU(2)-
2 symmetric theory. The nonlinear σ model mapping pre-
β sented above, suggests that the transition line between
1.5 thenormalphaseandtheO(3)phaseshouldbeacontin-
uous transition in the O(3) universality class, at least in
1
thelimitfarfromthebicriticalpoint. Wehaveconsidered
Normalphase
thisforthecasee=6.0,andtheFSSresultsaregivenin
0.5
Fig. 2. A log-log plot of the FSS of the peak height in
∂ ln m2 is given in panel (a), and the measured peak
0 ∂β
0 1 2 3 4 5 6 7 heights fall on a straight line for L 20. The best fit to
e the fo(cid:10)rm (cid:11)∂ ln m2 L1/ν yields ν≥=0.715 0.004. In
∂β ∼ ±
Figure1. (Coloronline)PhasediagramoftheNCCP1 model. panel (b), we a(cid:10)lso m(cid:11) easure (∆M3)height, and this quan-
SU(2) phase: Fully ordered phase where the O(3) symme- tity exhibits negligible finite-size corrections to scaling
try is spontaneously broken, (cid:104)m(cid:105) (cid:54)= 0, and the gauge field is at least for L 10. The best fit according to Eq. (13)
massive, ρµµ (q) = 0. O(3) phase: O(3) symmetry is spon- yieldsα= 0.1≥17 0.011,wherethevalueofν obtained
dual − ±
taneously broken, (cid:104)m(cid:105) =(cid:54) 0, but the gauge field is massless, above was used. In this case, it was found that ν was
ρµµ (q) (cid:54)= 0. Normal phase: O(3) symmetry is restored, most precisely determined by measuring the peak height
dual
(cid:104)m(cid:105)=0,andthegaugefieldismassless,ρµµ (q)(cid:54)=0. Thedi- in ∂ ln m2 rather than measuring (∆M ) . The
recttransitionlinefromtheSU(2)phasetdouatlhenormalphase ∂β 3 width
maximum peak in M is not very sharp [see the inset of
is denoted by +-markers and a solid red line. The Higgs (cid:10) (cid:11) 3
panel (b)]. Thus, the error bars in (∆M ) are large.
transition line between the SU(2) phase and the O(3) phase 3 width
In order to determine η, the FSS of the magnetic sus-
isdenotedby∗-markersandadottedblueline. Thetransition
linebetweentheO(3)phaseandthenormalphaseisdenoted ceptibility χ is given in panel (c). Here, χ is measured
by×-markersandadashedgreenline. Linesareguidetothe at the critical coupling βc = 2.7894 0.0003, which was
±
eyes. determined by fitting the Binder crossings of L and L/2
to a function that accounts for power-law finite-size cor-
rections. The best fit of χ(L) was determined for sizes
L 12,...,64 to yield η = 0.024 0.014. All the ex-
∈ { } ±
ponents listed above correspond well with the exponents
of the O(3) universality class.56,57
ones.10,11,21–23,26,52 From the O(3) phase, by reducing
the value of e, one enters an ordered phase that we label
2. Superconducting transition.
the SU(2) phase. Going into this phase, the gauge field
dynamically acquires a Higgs mass and the system be-
comes a two-component NCCP1 superconductor. Note Computations have also been performed along the
that the Higgs transition is related to a local symmetry, transition line between the O(3) phase and the SU(2)
and indeed, is not associated with spontaneous symme- phase. In analogy with the paired phase of the
trybreaking.53 Thisaspectshouldbekeptinmindwhere U(1) U(1)model10,11,21–23,25,26 (i.e., themetallicsuper-
×
we for brevity refer to the fully ordered state as “broken fluid), the transition to the O(3) sector should be associ-
SU(2)” or “fully broken state” to distinguish it from a ated with the proliferation of single-quanta vortices. In
paired state. The SU(2) phase is recognized by measur- the U(1) U(1) model, such vortices have similar phase
ing m =0 and ρµµ (q)=0. windings×in both complex fields and are topologically
(cid:104) (cid:105)(cid:54) dual
well-defined objects. In the SU(2) case, such vortices
can have either similar phase windings in both compo-
It is generally expected that at small values of e, the nents, or a phase winding only in one component if the
SU(2) phase may also be entered directly from the nor- othercomponentexistsonlyinthevortexcoreofthefor-
mal phase, i.e., without going through the intermediate mer. Such objects are non-topological, and are unstable
paired phase. The nature of the phase transition along in type-II SU(2) superconductors.28 This suggests that
this direct transition line in this and related multicom- the system should be a type-I SU(2) superconductor in
ponent models has been intensively debated due to its ordertofeatureaphasetransitionintoapairedphase. In
relevance to deconfined quantum criticality. We will re- analogy with single-component type-I superconductors,
turn to the direct transition line in Secs. VB and VC. onewouldthenexpectafirst-orderphasetransition.58,59
First, we present results for the two separate transition A different viewpoint is based on mean-field arguments,
lines. which suggest that the transition line could be a first-
6
102
60
i
2
m 40
∂2mln∂βhiβ=βpc(cid:17)(cid:16)102 ∂ln∂βh200 2.β8 2.85 zzLρ(q)]mindualβ=βc(cid:12)10(cid:12)1(cid:12) β2.736
[
101 (a) ∂∂β(cid:12)(cid:12)(cid:12) 2.734
0 0.05
L 1
−
100
103 0 10 50
3
M
L
)height −102 2.7 2.9 Fcriigtuicrael3p.oinLtoβg-,loagsaplfoutnoctfio∂n∂βoLfρszdyzusatle(qmmsiniz)emLe.aTsuhreecdhaartgethies
M3 β e=5.0. Measucrements are performed for 15 different system
∆
( sizes L ∈ {8,...,64}. The derivative was found by calcu-
lating the differences of ρzz (q ). The solid straight line
dual min
102 is the best fit obtained with a fit function on the form aLb
where a and b are two free parameters. The inset shows the
(b)
Lρzz (q )crossingsforsystemsLandL/2asafunctionof
dual min
L−1. Thesecrossingswereusedtoestimatethecriticalpoint,
β =2.7347±0.0005. Errorsindeterminingβ aretakeninto
c c
account by also considering the sensitivity of ν with respect
104 χ104 to β when estimating the uncertainty in the exponent.
0
=βc 2.β8 2.85 ordertransitionlineinthevicinityofabicriticalpoint.10
β
χ Other objects which can disorder the Higgs sector, are
103 Hopfions24,60. In this work we have made no serious at-
tempts at resolving such topological defects.
To check the universality class of this line, FSS results
(c) of∂/∂β[Lρzz (q )],obtainedatthecriticalpointwith
dual min
e = 5.0, are given in Fig. 3. First, the critical coupling
10 50
was determined to be β =2.7347 0.0005, by consider-
c
L ingthecrossingsofLρzz (q )(s±eetheinsetofFig.3).
dual min
Then, the correlation length exponent was estimated to
Figure 2. FSS results for the transition between the normal
be ν = 0.664 0.039. This value is consistent with an
phase and the O(3) phase when e = 6.0. 13 system sizes
L ∈ {8,...,64} are used. In all panels, the solid straight inverted 3Dxy±transition line.61 We have not been able
line is the best fit obtained for a fitting function on the form to resolve a first-order phase transition at this line.
aLb with two free parameters a and b. Panel (a): Log-
log plot of the maximum in the logarithmic derivative of
the second power of the magnetization (∂/∂βln(cid:104)m2(cid:105))
β=βpc B. Estimate for a bicritical point
[see Eq. (21)] as a function of L. The best fit is obtained
for sizes L ∈ {20,...,64}. The inset shows the measure
of (∂/∂βln(cid:104)m2(cid:105)) in the case when L = 40. Panel In Ref. 14, the flowgram method has been suggested
β=βpc
(b): Log-log plot of the third moment height difference as a useful tool to assess whether or not there is a tri-
(∆M ) as a function of L. The best fit is obtained critical point at weak couplings to the gauge field. This
3 height
for sizes L ∈ {10,...,64}. The inset shows the measure method relies on resolving a first-order phase transition
(∆M3)height in the case when L = 14. Panel (c): Log-log at stronger couplings, just below the bicritical point at
plot of the magnetic susceptibility measured at the critical
which the paired phase opens up between the normal
coupling χ as a function of L. The best fit is obtained
β=βc phase and the SU(2) phase. It is thus important to be
forsizesL∈{12,...,64}. Theinsetshowsχ forthecase
β=βc abletodeterminethebicriticalpointaccurately. Forthis
whenL=40,andthearrowheadsindicatethatχismeasured
purpose, we will focus on the region slightly above the
at the same fixed coupling β for all sizes.
c
bicritical point and establish when two separate phase
transitions are clearly resolved. In this way, we can de-
termine an upper bound on the bicritical point.
7
1. Signatures of an intermediate paired phase at e=4.2. (a) L=16
60 L=24
L=32
50 L=40
In order to discern two separate, but close-lying phase L=48
L=56
transitions, we need to establish signatures that can be 40
taken as evidence for splitting of a transition line. To 30
this end, results are presented for the case when e=4.2. Cv
20 60
We find unambiguous evidence for two separate phase
transitions. Remarkably, at smaller system sizes we find 10 max
characteristics of the phase transition consistent with a 0 Cv,
20
first order transition, and it was interpreted as such in -10
Ref. 14. (e = 4.2 corresponds to g ≈ 1.88 in the units 6 104 10 L 50
of Ref. 14. This Reference gave the estimate for the po- · ((bb))105
ssaietedi,oiffnpeeroreffnotrthmceionnbgciclcuroistmiiocpnau.ltpaTtohiionentrseaoatnsoglna≈rigse2tr.h0say.)tstfieAnmistsewl-eseaizdsesheatflo-l 4·104 M)3height104
fects will disguise the existence of separate transitions ∆
(
and make them appear as one. 2 104 103
· 10 L 50
In Fig. 4, results are presented for four different ob-
servables obtained at 12 different system sizes, L M3
∈ 0
8,...,56 ,inacouplingrangecoveringbothphasetran-
h
{sitions. I}n panel (a), results for the specific heat are )widt10−2
given. When system sizes are small, it is only possible 2 104 M3
to resolve one peak in the specific heat. However, when − · ∆
(
L=40, it is possible to resolve a bump to the left of the 10−3
10 L 50
peak. The bump, which corresponds to the O(3) order- 4 104
− ·
(c)
ing phase transition, becomes more pronounced when L 0.9
2.35
increases. Thisbehaviorsuggeststhattherearetwotran-
0.85
sitionsinsteadofone. Moreover,intheinsetofpanel(a)
westudythescalingofthepeakonalog-logscale. When U4 0.8 βcross2.348
L is small, there is a rather steep and slightly increasing 0.75 2.346
slope. However, at higher values of L there is a definite 0.7
change in the slope towards smaller values, correspond- 0 0.05 0.1
0.65
ing to a sudden slowing down in the growth of the peak. (L1L2)−1/2
0.6
Thisbehaviorshouldclearlybeassociatedwithresolving
(d)
separate transitions with increasing L.
In panel (b) of Fig. 4, results for the third moment of 0.4
) 2.356
tirtheseipsaoocntndiloiynngpaotrsoesipabrleessientnogtleerdesp.ohlWavseheeantrcahsnyassritatecimotner.siiszteHiscoawforeervmesmr,caoalrlt-, zzρ(qmindual 0.35 βcross2.354
L 40, a secondary form is developing to the left of L
0.3 2.352
≥
the original form, resolving the O(3) ordering transition. 0 0.05 0.1
When studying the scaling of the quantities (∆M3)height 0.25 (L1L2)−1/2
and(∆M3)width intheinsetsofthepanel,itisclearthat 2.34 2.345 2.35 2.355
theybothexhibitslopechangesassociatedwithresolving β
both transitions.62
The Binder cumulant is given in panel (c) of Fig. 4, Figure4. (Coloronline)MonteCarloresultsforfourdifferent
and its crossings are given in the inset of the panel. quantitiesand12differentsystemsizesobtainedforacoupling
rangecoveringtwoseparate,butclose-lyingphasetransitions.
By considering the crossings with largest L, we find
The gauge field coupling e = 4.2. For clarity, the panels
that the critical point of the O(3) ordering transition is
only show results for L ∈ {16,24,32,40,48,56}, but insets
β = 2.347 0.001, a value that corresponds well with
c ± include all 12 sizes, L ∈ {8,...,56}. Panel (a) shows results
the leftmost transition point in panel (a) and (b). Note
forthespecificheatC ,andtheinsetshowsthescalingofthe
v
that there is a non-monotonic behavior in the coupling peak C in a log-log scale. Panel (b) shows the results
v,max
values of the Binder crossings. Hence, by studying small for the third moment of the action M , and the insets show
3
systems only, one might be misled to overestimate the the scaling of (∆M ) and (∆M ) in a log-log scale.
3 height 3 width
critical point of the phase transition. Panel(c)showstheBindercumulantU ,andtheinsetshows
4
In panel (d) of Fig. 4, we show results for the quantity thecouplingβcrosswheretheBindercurvescrossasafunction
Lρzz (q ), and the corresponding crossings are given of(L1L2)−1/2whereL1andL2arethetwoactualsizes. Panel
dual min (d) shows the quantity Lρzz (q ) and the inset shows the
in the inset. We estimate the critical point of the Higgs dual min
coupling where the curves cross. Lines are guide to the eyes.
8
e=3.0 e=3.0
e=3.2 e=3.2
e=3.4 106 e=3.4
e=3.6 e=3.6
100 e=3.8 e=3.8
e=4.0 e=4.0
e=4.2 e=4.2
Cv,max∝L3 105 (∆M3)height∝L6
Cv,max M)3height
∆ 104
(
103
10
10 100
L
Figure 5. (Color online) Log-log plot of the value of the spe-
cificheatpeakC asafunctionofsystemsizeLforseven
v,max
different values of e ∈ {3.0,...,4.2}. The dotted line corre-
sponds to the slope expected for a first-order transition, ac-
cording to Eq. (11). For e≥3.8, the scaling of C shows 10−2
v,max
anegativecurvature, insteadofcurvinguptowardsthefirst- dth
wi
order characteristic scaling line. From this, our upper bound )
M3
on the position of the bicritical point in the phase diagram
∆
would be e=3.8. Lines are guide to the eyes. (
e=3.0
e=3.2
10−3 e=3.4
transition to be β = 2.353 0.001 by a crude extrap- e=3.6
c ± e=3.8
olation to the thermodynamic limit. Hence, the critical e=4.0
e=4.2
pointoftheHiggstransitionissignificantlydifferentfrom (∆M3)width∝L−3
the critical point of the O(3) ordering transition. 10 100
The results in Fig. 4 show that it is of particular im- L
portancetosimulatelargesystemsinregionswherethere
Figure 6. (Color online) Log-log plot of the FSS of the
might be multiple phase transitions in multicomponent
height (upper panel) and the width (lower panel) of the
gauge theories. Discarding data points for L > 20, the
third moment of the action, for seven different values of
crossings in panel (c) and (d) appear to converge to the
e ∈ {3.0,...,4.2}. The dotted lines correspond to the slope
same coupling. In panel (a) and (b), we would only re-
expected for a first-order transition, according to Eqs. (15)
solveasinglephasetransitionwithratherstrongthermal
and (16). Lines are guide to the eyes.
signatures.
In Fig. 6, we show the FSS of (∆M ) and
3 height
2. Monte Carlo results for e∈{3.0,...,4.6} (∆M ) . Observe that the same signatures of split-
3 width
tingappearsfore 3.8,4.0 asfoundfore=4.2above,
∈{ }
We first turn our attention to the region with e < 4.2 namelythattheslopeof(∆M3)heightchangestoasmaller
tolookforthesignaturesthatwehaveestablishedabove. value and the slope of (∆M3)width changes to a higher
Fig.5showstheFSSofthepeakintheheatcapacityfor value. This is again inconsistent with the scaling of a
e 3.0,...,4.2 . The results show that there is a def- single first-order transition. For a first-order transition
in∈ite{change in t}he slope of the scaling of C , also the slopes should converge towards the scaling for first-
v,max
for e=4.0 and 3.8. Note that this signature of splitting ordertransitions,giveninEqs.(15)and(16)(seeRef.12
appears at higher L when e is reduced, corresponding to for an example).
thecouplingdifferencebetweenthetwotransitionsbeing To determine the positions of the O(3) ordering tran-
smaller. TheslopeofthedottedlineinFig.5istheslope sition and the Higgs transition, the finite size crossings
of a first-order transition [see Eq. (11)]. For all values of of U and Lρzz (q ) are given in Fig. 7 for eight dif-
4 dual min
einFig.5, wefindthatforsmallandintermediateLthe ferent values of e 3.2,...,4.6 . For e 4.0,...,4.6 ,
slope is steep and increasing, and one might be tempted the U crossings ∈an{d the Lρzz }(q ) c∈ro{ssings clearl}y
4 dual min
toconcludethattheyallarefirst-ordertransitions. How- extrapolates to different couplings as expected for two
ever,thechangetowardsasmallerslope,thatwefindfor separate transitions. Also note the corresponding non-
largeLande 3.8,4.0,4.2 ,isindeedinconsistentwith monotonic behavior for the Binder crossings. When the
∈{ }
a single first-order phase transition. couplingdifferencebetweenthetwophasetransitionsde-
9
Lρzz (q )
dual mUin4 2.188 0.03
2.54
2.186
β2.52 β βc 0.02
2.184 ∆
2.5
2.182 0.01
e=4.6 e=3.8
2.48 2.18
0
2.104 3.4 3.6 3.8 4 4.2 4.4 4.6
2.44 e
β β2.102
e =3.8
2.43 ∗
e =3.6
2.1 ∗
e =3.4
∗
e =3.2
2.42 e=4.4 2.098 e=3.6 0.01 e∗∗ =3.0
βc e∗ =2.8
2.022 ∆ e∗ =2.6
2.355
2.02
β 2.35 β
2.018
0.001
2.345 2.016 0.1 1
e e
e=4.2 e=3.4 ∗
2.34 2.014 −
2.272
1.942 Figure 8. (Color online) Plot of the difference in the critical
2.27 coupling between the Higgs transition and the O(3) ordering
1.94
transition, ∆β . ∆β is determined by calculating the differ-
2.268 c c
β β1.938 ence between the Lρzz (q ) crossing and the U crossing,
dual min 4
2.266 1.936 andaveragingoverfourofthesedifferenceswithlargestvalue
of(L L )1/2 (i.e.,thefourleftmostdatapointsfromthepan-
2.264 1.934 1 2
els in Fig. 7). We only include results for e ≥ 4.0 where the
e=4.0 e=3.2
2.262 1.932 non-monotonic behavior of the Binder crossings can clearly
0 0.05 0.1 0 0.05 0.1
be resolved. Upper panel: ∆β as a function of e. Lower
(L1L2)−1/2 (L1L2)−1/2 panel: Log-log plot of ∆β as acfunction of e−e∗ where e∗
c
isgiveninthekey. Positivecurvaturesuggeststhate∗ >e ,
Figure7. (Coloronline)Plotsofthefinitesizecrossingsofthe bc
negative curvature suggests that e∗ <e and a straight line
BindercumulantU [Eq.(19)],andthequantityLρzz (q ) bc
4 dual min suggests that e∗ ≈e . Lines are guide to the eyes.
for eight different values of e ∈ {3.2,...,4.6}. The x-values bc
are given by (L L )−1/2 where L and L are the two sizes
1 2 1 2
that form the crossing.
and a negative curvature suggests that e∗ < e . Since
bc
there is a clear positive curvature both for e = 3.8 and
creases,largersystemsareneededtoresolvethisfeature. e = 3.6, this suggests that ebc < 3.6. Note that the
For e = 3.8, we observe that the leftmost U crossing results given in the lower panel of Fig. 8 essentially is
4
(L1 = 80,L2 = 96) deviates, consistent with the non- an extrapolation of the difference ∆βc (which also is an
monotonic behavior for the larger e values. For the sizes extrapolation) in the upper panel to find the point ebc
available,thecrossingsseemtoconvergetothesamecou- where ∆βc = 0. As it will be clear below, even at the
pling value for e 3.2,...,3.6 . largest system sizes accessible for us, we could not prove
The results in∈F{igs. 5, 6 and} 7, show that there are that there is a single first-order transition at e = 3.6.
twoseparatetransitions whene 3.8. Wethusestimate Therefore, simulations of even larger systems are needed
that the bicritical point must be≥below e=3.8. Clearly, to determine more accurately the existence and the po-
the system sizes we are able to reach are too small to sition of ebc.
conclusively determine if there are separate transitions Our estimates for the bicritical point differ from the
for e < 3.8. However, in order to estimate the bicriti- results in Refs. 13 and 14 which studied substantially
cal point e , in Fig. 8 we show results for the coupling smallersystems. Ourupper bounde <3.8corresponds
bc bc
difference between the two phase transitions ∆β as a to >0.151 in Ref. 13. This means that a part of the
c bc
K
function of the coupling e. To estimate when ∆β 0, line that was interpreted as a direct first-order transi-
c
in the lower panel, we show ∆β as a function of e→e∗ tion in that work, in fact are two separate transitions.
c
on a log-log scale where e∗ is some trial value as lab−eled Moreover, the upper bound e < 3.8 corresponds to
bc
inthekeyofthefigure. Ife∗ e ,astraightlineshould g <1.65 in Ref. 14 where the bicritical point was esti-
bc bc
beexpected. Apositivecurva≈turesuggeststhate∗ >e mated to g 2.0.
bc
≈
10
60 e=4.0 LL==3224 60 e=3.6 LL==3224 Also, as mentioned above, the instability of composite
L=48 L=48 vorticesintype-IISU(2)theorysuggeststhatthesystem
3/L)40 LL==8604 3/L)40 LLL===986604 shouldbeatype-Isuperconductorintheproximityofthe
H H paired phase (since the paired phase results from prolif-
P(20 P(20 erationofcompositevortices),withapossibilityofafirst
order transition via Halperin-Lubensky-Ma mechanism.
0 0
-1.44 -1.42 -1.4 -1.38 -1.34 -1.32 -1.3 -1.28 We did not consider large enough system sizes to resolve
H/L3 H/L3
this issue.
60 60
e=3.8 LL==3224 e=3.4 LL==3224 Combining the results in Figs. 5, 6 and 9, it appears
L=48 L=48
3/L)40 LLL===986604 3/L)40 LL==8604 tphoainttf,orthceorueplairnegsstsrloignhgtlythaebrmovaeltshigeneasttuimreasteind bteicrrmitsicoafl
H H
P(20 P(20 broad energy distributions and rapidly increasing peaks
in the specific heat and the third moment of the action.
0 0 However, when system sizes are larger, we can explicitly
-1.38 -1.36 -1.34 -1.28 -1.26 -1.24
H/L3 H/L3 seesignaturesofsplittingfore 3.8. Wecannotexclude
≥
the possibility that this may also be the case for some of
Figure9. (Coloronline)Histogramsoftheprobabilitydistri- the couplings with e < 3.8. Indeed, the crude extrapo-
butionoftheenergypersiteH/L3,fore∈{3.4,3.6,3.8,4.0}.
lation in Fig. 8 suggests that e = 3.6 also is above the
Ineverycasetheflattest(ormostbimodal)energyhistograms
bicritical point. If so, we should expect to see signatures
were found by reweighting in the vicinity of the pseudocrit-
ofsplittingforsystemsizeslargerthanthoseavailablein
ical coupling corresponding to the peak of the specific heat,
this work. On the other hand, the strong thermal signa-
C . The areas under the curves are normalized to unity.
v,max
tures we find for e < 3.8 can also be consistent with a
weaksinglefirst-ordertransition. Inthatcase,weshould
expecttoseethatproperfirst-orderscalingisobeyedfor
3. Signatures of a weak first-order transition
larger system sizes.
Summarizing this part, we find that the strongest
Although we are led to a different conclusion concern-
signatures for a single first-order phase transition were
ing the phase diagram than Refs. 13 and 14 for e 3.8,
≥ found at e=3.4 and e=3.6. Previous works on smaller
we find some of the same thermal signatures. As men-
systems did not resolve bimodal structure at these cou-
tioned above (see Figs. 5 and 6), when systems are too
plings. For e < 3.4, we did not find any bimodal struc-
small to resolve two phase transitions, the Monte Carlo
ture in the energy histograms at the system sizes which
results show that the scaling of C and (∆M )
v,max 3 height we can reach.
are almost as one would expect for a single first-order
transition. Moreover, when investigating the energy dis-
tributionsfore 3.8,4.0 inFig.9,wefindthatthehis-
tograms are bro∈ad{. Also,}in contrast to previous works, C. The flowgram method
we have resolved bimodal structures for e 3.4,3.6 .
∈ { }
This could be interpreted as evidence of a first order Toanalyzesituationswhereitisdifficulttoresolveand
phase transition. At the same time we note that they analyze bimodal structures in histograms such as those
only appear at the largest system sizes. Thus, it is dif- considered above, the authors of Ref. 10 proposed the
ficult to determine if the correct scaling for first-order flowgram method. By rescaling the linear system size
transition is obeyed.63,64 The histograms that appear at L C(g)L, where g = e2/(4β) and where C(g) is a
→
the largest system sizes, have not yet started to evolve monotonous scaling function of the parameter g, it may
into distributions resembling delta functions. In particu- be possible to collapse curves for various physical quan-
lar, for the system sizes which we can access the dips in tities computed at the phase transition, for different sys-
between the peaks are still increasing with system size, temsizesandcouplingconstants,ontoasinglecurve.27 If
ratherthandecreasing. Thelatterisrequiredfordrawing such uniform scaling is found for all coupling constants,
a firm conclusion that there is a direct first-order phase one may conclude that a phase transition has the same
transition at e = 3.4 and e = 3.6. Although rare, there characteristics for all these coupling constants. For in-
are examples in the literature where bimodal energy dis- stance, if a first order phase transition were to be found
tributions are found in cases with no first-order phase for large coupling constants, and the scaled plots fall on
transition.65–68 a single line for all other coupling constants, one may
For e = 3.8, we do not resolve any bimodality, but conclude that the transition is first order for all these
the histograms are wide. The width of the histograms coupling constants. To draw such a conclusion, it is very
decreases and the flat top structure disappears when L important that a broad enough window of systems sizes
increases. This is not consistent with a single first-order L is considered, such that there is adequate overlap of
transition. Notethatifthispointislocatedslightlyabove datapoints for all coupling constants, when the data are
the bicritical point, then according to a mean-field ar- plotted in terms of C(g)L.
gument, the Higgs transition should be first-order.10,13 In Fig. 10, we show results of a flowgram analysis