Absorbing state phase transition with competing quantum and classical fluctuations

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by  M. Marcuzzi

Absorbing state phase transition with competing quantum and classical fluctuations
Matteo Marcuzzi,1 Michael Buchhold,2 Sebastian Diehl,2 and Igor Lesanovsky1
1School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
2Institut fu¨r Theoretische Physik, Universita¨t zu K¨oln, D-50937 Cologne, Germany
(Dated: January 28, 2016)
Stochastic processes with absorbing states feature remarkable examples of non-equilibrium uni-
versal phenomena. While a broad understanding has been progressively established in the classical
regime,relativelylittleisknownaboutthebehaviorofthesenon-equilibriumsystemsinthepresence
of quantum fluctuations. Here we theoretically address such a scenario in an open quantum spin
6 modelwhichinitsclassicallimitundergoesadirectedpercolationphasetransition. Bymappingthe
1 problem to a non-equilibrium field theory, we show that the introduction of quantum fluctuations
0 stemming from coherent, rather than statistical, spin-flips alters the nature of the transition such
2
thatitbecomesfirst-order. Intheintermediateregime,whereclassicalandquantumdynamicscom-

n pete on equal terms, we highlight the presence of a bicritical point with universal features different
a from the directed percolation class in low dimension. We finally propose how this physics could be
J explored within gases of interacting atoms excited to Rydberg states.

7
2
I. INTRODUCTION Directed percolation (DP) represents an instance of a

] classical, but intrinsically non-equilibrium system (for a
h
c Non-equilibrium phenomena are ubiquitous in nature, review,see[11]). Despiteitsrobustness,itsexperimental
e ranging from the microscopic scales of chemical reac- observation has so far been elusive [41], with a single re-
m tions to the macroscopic ones of disease-spreading. Re- markableexception[42,43]. Itwassuggestedrecentlyto
- markably, analogously to the equilibrium case, non- realize and explore DP dynamics in cold gases of atoms
at equilibriumensemblescanshowtheemergenceofuniver- excited to high-lying Rydberg states [44]. In that case,
t sal behavior, signaling the irrelevance of the microscopic the non-equilibrium nature of dynamics persists macro-
s
. detailsofthedynamicsformacroscopicobservables. This scopically, but the impact of quantum dynamics fades
at occurswhensuchout-of-equilibriumsystemsstarttoact out completely under coarse graining.
m collectively [1–4]. On a fundamental level, a distinction In this work, we harness the opportunities that result
- arises depending on the presence or absence of detailed from the fact that such Rydberg gases indeed represent
d balance [5–8], between systems which evolve towards a driven open quantum systems to go beyond the realm of
n stationaryequilibriumstate(e.g.,quenchedsystemscou- classical physics, and establish a novel absorbing state
o pled to thermal baths [9]) or that preserve their non- phase transition characterized by the interplay of classi-
c
[ equilibrium character even in the long-time limit, repre- cal and quantum terms on equal footing. This transition
senting flux equilibrium states. The universal dynamical doesnotfallintotheDPuniversalityclass,anditsorigin
1 featuresofpurelyclassical systemshavebeenextensively can be unambiguously traced back to the presence of co-
v studied and classified both for unbroken [10] and broken herent dynamics. More precisely, the latter introduces a
5
[11–14] detailed balance, i.e., genuine non-equilibrium strong coupling first-order non-equilibrium phase transi-
0
3 systems. Recently,experimentsinvariousplatformshave tionwithoutcounterpartinthepurelyclassicalDPprob-
7 startedtosystematicallyprobedrivenopenquantum sys- lem. Remarkably,thisdiscontinuousphasetransitionter-
0 tems. The spectrum includes light-driven semiconduc- minates in a novel bicritical point which even asymptot-
1. tor heterostructures [15], arrays of driven microcavities ically at large distances and in dimensions d < 2, does
0 [16,17],coldatomsinopticallattices[18],cavities[19,20] not feature the symmetries underlying DP, or any equi-
6 and microtraps [21–23]. Several among these instances librium problem.
1 employ excitation of the atoms to high-lying Rydberg
:
v orbitals[24–26]inordertoachievestronginteratomicin-
i teractions and to study cooperative effects [27–29]. In II. MODEL
X
all these systems, the driving/dissipation introduces co-
r herencelossandexplicitlyviolatestheequilibriumcondi- Inthefollowingwereproduceaquantumvariantofthe
a
tionsatthemicroscopiclevel[7,30]. Itisthusachallenge contactprocess (foranintroductionwerefertoRef.[11]).
to identify to what extent the non-equilibrium and the Its defining property is the following: in a lattice of “ac-
quantum nature of the dynamics impact on the macro- tive” and “inactive” sites, the former can spontaneously
scopic phase diagram and phase transition properties. decay to inactive, whereas activation can only occur in
Oftentimes, upon coarse graining such systems lose their the proximity of already active sites. Thus, the fully-
quantum character and equilibrium conditions are effec- inactive state is absorbing, i.e., once reached it cannot
tively restored [31–36]. But there are instances where be left. Specifically, we consider a lattice of quantum
non-equilibrium [37, 38] and quantum [39, 40] aspects two-level systems with spacing r. On every site k we
persist even at asymptotically large wavelength. definethebasis|a (cid:105)(active)and|i (cid:105)(inactive), theden-
k k
2
√
sity of active sites n = |a (cid:105)(cid:104)a | and the ladder oper- κn σ+ (an active site can activate a neighboring one
k k k j k √
ators σk+ = |ak(cid:105)(cid:104)ik| and σk− = |ik(cid:105)(cid:104)ak|. Under the ac- |ajik(cid:105) → |ajak(cid:105)) and coagulation Lc,j,k = κnjσk− (the
inverse process |a a (cid:105) → |a i (cid:105)). The operator Π in
j k j k k
H represents the simplest choice which effectively repro-
duces the requirement of an active site nearby to flip a
spin; this makes H the “minimal quantum equivalent”
of the noisy branching/coagulation above. Similar “con-
strained” Hamiltonians have been studied in the past
with a focus on many-body localization [47, 48].
III. EQUATIONS OF MOTION AND DENSITY
PATH INTEGRAL
We infer here the properties of the phase diagram by
exploiting an effective path integral description for the
density variable n alone. We start by deriving the
k
Heisenberg-Langevinequationsofmotion(EOM)[49]for
the single-site operators n , σx and σy = −iσ+ +iσ−.
k k k k k
Forconvenienceweintroducethecoordinationnumberz
(number of nearest neighbors per lattice site), the short-
hand Σx/y =σx/y(cid:80) σx, rescale time by t→τ =γt
Figure1. (Coloronline)(a)Fundamentalprocesses. Wecon- k k jnnk j
and the rates accordingly, i.e. χ=κ/γ and ω =Ω/γ:
sider a lattice whose sites admit two states: active (red) and
inactive (green). Active sites decay at a rate γ and become
inactive. Proliferationofactivesitesispossiblethroughclas- n˙k =−nk+[ωσky−χ(2nk−1)]Πk+ξˆkn, (2)
sical(rateκ)andquantum(strengthΩ)branching. (b)Phase
σ˙x =ωΣy − zχ+1σx−χσxΠ +ξˆx, (3)
diagram constructed from the effective action (5) in saddle- k k 2 k k k k
pointapproximation(colorcodecorrespondstodensityofac-
σ˙y =ωΣx− zχ+1σy−[ω(4n −2)+χσy]Π +ξˆy. (4)
tivesites). Intheclassicallimit(Ω=0)thesystemexhibitsa k k 2 k k k k k
continuous (2nd order) directed percolation phase transition
between an absorbing state and one with finite density. This The quantum noise terms ξˆα consider the fluctuations of
k
transition extends into the quantum regime (thick red line) the bath and depend on the structure of the jump op-
until the critical point α is reached. In the quantum limit erators. They show vanishing averages but non-trivial,
(κ = 0) a first-order transition is found which also extends Markovian correlations, which for the present setup are
into the classical regime (dashed yellow line) up to point α. (in rescaled units) (cid:104)ξˆxξˆx(cid:105) = (cid:104)ξˆyξˆy (cid:105) = δ , (cid:104)ξˆnξˆn(cid:105) =
In the neighborhood of this line, a narrow region of coexis- k k(cid:48) k k(cid:48) k,k(cid:48) k k(cid:48)
δ n , (cid:104)ξˆxξˆy (cid:105) = −iδ , (cid:104)ξˆnξˆx(cid:105) = −δ σ+ and
tence of two attractive stationary solutions is present, which k,k(cid:48) k k k(cid:48) k,k(cid:48) k k(cid:48) k,k(cid:48) k
is not resolved here. The high values of the density reached (cid:104)ξˆknξˆky(cid:48)(cid:105)=iδk,k(cid:48)σk+ uptoleadingorderinthedensity(see
intheactivephasestemfromneglectinghigherordersinnin Appendix A).
the action, which would otherwise prevent it from exceeding In the following, we work in the continuum limit
1/2. (k,t) → ((cid:126)x,t) ≡ X and derive an effective path integral
for the density field n via a Martin-Siggia-Rose (MSR)
X
tion of Markovian noise sources, the state ρ of the sys- construction [3, 50–52], presented in Appendix B. Cru-
tem evolves according to the Lindblad equation [45, 46] cially, the σx,y-fields are gapped, and thus can be inte-
ρ˙ =−i[H,ρ]+(cid:80)D[L ]ρ [see sketch in Fig. 1], where gratedoutperturbatively. Theresultinglongwavelength
a,k
a,k fieldtheorydependsonthedensityvariablenalone, and
is obtained by additionally performing a derivative ex-
(cid:88) (cid:88)
H =Ω Π σx with Π = n (1) pansion of the action. It reads
k k k j
k jnnk (cid:90) (cid:104) (cid:105)
S = n˜ (∂ −D∇2+∆)n +u n2 +u n3
is the quantum Hamiltonian, σx = σ+ + σ−, and n X t X 3 X 4 X
k k k X
“nnk” denotes nearest neighbors (nn) of site k; −(cid:90) (cid:2)1n˜2 n +µ n˜2 n2 (cid:3)≡S(1)+S(2), (5)
D[X]ρ=XρX†−(X†Xρ+ρX†X)/2 is the dissipator 2 X X 4 X X n n
X
and L are the so-called jump operators, having the
a,k
index a labeling the process type, and k the lattice site. where D = r2χ represents a diffusion constant
These jump operators are chosen to define a modified (lattice spacing r) and ∆=1−zχ− 8z2ω2 , u =
(zχ+1)3 3
contact process [11], which is known to feature a DP (cid:16) (cid:17)
transition, and include decay Ld,k =√γσk− (|ak(cid:105)→|ik(cid:105)) 2z χ− z2χzω+21 , u4 = 8zzχ2+ω12 and µ4 = (z2χz2+ω12)2 + (1z2χ8z+41ω)46
and — for every neighbor j of k — branching L = are the microscopic coupling constants. The response
b,j,k
3
field n˜ encodes the linear response properties of n under
small perturbations.
At this point we emphasize two key properties of the
action (5): First, the absence of a density indepen-
dent Markovian noise level ∼ Tn˜2 (which is necessar-
X
ily present in any classical system in thermal equilib-
rium). This is characteristic of DP dynamics, which fea-
ture the absence of density fluctuations in the absorbing
state n = 0 and consequently a multiplicative kernel
X
∼ n . An additive noise introduced by the dissipative
X √
terms L = γσ− only occurs in the eliminated spin
d
variables σx,y. Second, the presence of a non-zero coher-
ent coupling ω (cid:54)= 0 – i.e. the intrinsic quantum effect
– leads to the appearance of non-zero couplings u and
4
µ as well as a negative contribution to u . This new
4 3
“quantum” scale ω breaks a fundamental symmetry of Figure 2. Effective potential and phase transitions. (a) Be-
the DP class (specified below) and strongly modifies the havior of the effective potential Γ(n) (arbitrary units) across
phase diagram compared to the purely dissipative model the second order phase transition. Dots mark the minima of
[see Fig. 1]. Γ(n). The transition occurs when ∆ in Eq. (6) changes sign.
(b)Stationarystatedensityintheclassicallimit(ω=0)asa
functionofχ(chainof200sites,averageover103 realizations
per point), obtained via Monte Carlo simulations starting
IV. EFFECTIVE POTENTIAL AND
from a completely active configuration and stopped at time
MEAN-FIELD PHASE DIAGRAM γt=104. Thedatashowthecharacteristicbehaviorofasec-
ondorderphasetransitionaroundχ ≈6.2. (c)Effectivepo-
c
The discussion of the various phases and phase transi- tentialΓ(n)(dashedlines)andcorresponding“optimal-path”
tionsofthesystemisconsiderablysimplifiedbyrealizing potentialW(n)(solidlines),seeEq.(7),acrossthefirst-order
thatthedeterministiccontributiontotheactionS(1) can transition. Atthetransitionpoint,W(n1 =0)=W(n2)=0.
(cid:104) (cid:105) n (d)Steady-statehistogramofthedensityinthequantumlimit
be written as (cid:82)Xn˜X ∂tnX −D∇2nX + δΓδ(nnXX) , where χ=0 (12 spins) obtained via a quantum-jump Monte Carlo
(QJMC) method, indicating a first-order transition (ω ≈2)
c
∆ u u asωincreases. Twostablestationarysolutions,onewithzero
Γ(n)= n2+ 3n3+ 4n4 (6)
2 3 4 andonewithfinitedensity,emerge. Theinsetdisplaysasec-
tion of the histogram taken at ω=8.
is a local effective potential. In the absence of fluctu-
ations Γ characterizes the mean-field phases, which are
(cid:112)
determined by the properties around its minima. correlation length ξ = 1/ |∆| < ∞. The form of the
The corresponding phase diagram is shown in effective potential Γ(n) suggests a first-order transition
Fig.1(b). Theactivephaseisidentifiedby∆<0,u ≥0 line in this regime featuring the coexistence of the zero
4
and u > 0, which leads to a single minimum of the ef- and finite-density solutions. This case, however, requires
3
fective potential at finite density. On the other hand, additional care due to the specific form of the noise, as
when both ∆ and u are positive, there is a local mini- detailed further below.
4
mum of Γ at n=0. For negative and sufficiently strong The α point in Fig. 1(b) located at ∆ = u = 0 rep-
√ 3
cubiccouplingu <−2 u ∆,thereexistsasecondlocal resents a bicritical point at which both the line (∆ >
3 4 √
minimum at finite density n > 0. In this regime, the 0,u =−2 ∆u )andthelineofsecondordertransitions
3 4
mean field evolution features two attractive fixed points (∆=0,u >0)terminate. Atthispoint,thequarticpo-
3
and the thermodynamic phase is determined within the tential term u provides the leading non-linearity.
4
optimal path approximation in phase space [53].
Three different types of phase transitions from the ac-
tive to the inactive state can be thus identified, their V. FLUCTUATIONS AT THE CONTINUOUS
nature depending on the specific choice of parameters TRANSITION
and the dimensionality. When the gap ∆ vanishes with
both u3,u4 > 0 the system undergoes a second order The competition between quantum and classical dy-
phase transition [see Fig. 2(a)], corresponding to a di- namics strongly affects the nature of the active-to-
(cid:112)
verging correlation length ξ =1/ |∆|→∞. Numerical inactive transition. In the absence of the coherent cou-
evidence for this transition is presented in panel (b) of pling, u ,µ = 0, the action (5) is equivalent to the so-
4 4
Fig. 2, which displays the stationary density of active called Reggeon field theory for classical DP [54]. It fea-
sites obtained for ω = 0 in a chain of 200 sites. For tures — upon rescaling the fields — the characteristic
√
∆ > 0 and u ≤ −2 u ∆, the transition from the ac- rapidity inversion symmetry,whichleavesthesystemin-
3 4
tive to the inactive phase takes place instead at finite variant under the transformation n ↔ −n˜ and t → −t
4
[3, 7, 53]. For u > 0, this symmetry is broken by the W(n ) = 0, which identifies the non-equilibrium first-
4 2
microscopic action. The implications depend on the di- order line [dashed line in Fig. 1(b)]. Due to the non-
mensiond: Ford>2,u isRGirrelevantandcanbedis- equilibrium nature of the fluctuations, this does not co-
4
carded in the infrared-dominated dynamics close to the incide with the naive prediction Γ(n ) = 0, as shown in
2
secondordertransition. Consequently,ind>2,rapidity- Fig. 2(c). In Fig. 2(d) we report the full-counting statis-
inversionisrestoredandthelineofcontinuoustransitions tics of the density n obtained via QJMC techniques [55]
displaysuniversalscalingbehaviorcorrespondingtoclas- for a chain of 12 spins. Despite the presence of strong
sical DP. finite-size effects, a bimodal structure is still highlighted
Attheα point[whitedotinFig.1(b)], u =0andthe forlargevaluesofω. Thisimpliesthattrajectoriesbunch
3
leading-ordercouplingbecomesu . Ford>2,thesecond together around two possible values, the absorbing one
4
order transition at this point is governed by mean-field and a finite-density one and is a signature of the afore-
scaling behavior, since u is RG-irrelevant and cannot mentioned coexistence.
4
introduce infrared divergent corrections to the vanishing
couplings u ,∆. On the other hand, for d < 2, u be-
3 4
comes RG relevant and generates a non-trivial RG flow VII. REALIZATION WITH RYDBERG ATOMS
of ∆ and u on the entire second order transition line.
3
This leads to a violation of rapidity-inversion which per-
Instances of this crucial competition between classical
sistsatlongwavelength,andthusdrivesthesystemaway
and quantum processes can be implemented with cold
fromtheDPcriticalpointtoanewnon-equilibriumuni-
atoms excited to Rydberg states [18, 56–60]. They are
versality class, without specific symmetries. In d < 2,
represented with two internal states, the ground state
therefore, only the isolated point χ = 1/z, ω = 0 lies in
|GS(cid:105)≡|i(cid:105)(inactivesite)andtheexcitedone|Ryd(cid:105)≡|a(cid:105)
theDPclass,whilethepresenceofquantumfluctuations
(activesite). Rydberggasesfeaturestrongvan-der-Waals
imprints a new universal scaling behavior on the entire
interactions in state |a(cid:105) [24–26], which rapidly decay as
line,includingtheαpoint. Formarginalu ind=2,the
4 r−6 with the interparticle distance r. For the sake of
scaling of the fluctuation corrections to u determines
4 simplicity, we approximate it here as a nearest-neighbor
whether this coupling becomes relevant, making the sce-
interaction of strength V in a one-dimensional configu-
nario equivalent to d < 2, or irrelevant, which has to be nn
ration.
determined by an RG analysis of the problem.
Quantum branching/coagulation is realized via coher-
entdrivingbyalaserfieldofRabifrequencyΩanddetun-
ing ∆ with respect to the atomic transition frequency;
L
VI. NON-EQUILIBRIUM DISCONTINUOUS fixing∆ =−V enablesan“anti-blockade”[58,61,62]
L nn
TRANSITION
mechanism which favors the excitation of a Rydberg
√ atom next to an already excited one, e.g. |iai(cid:105) → |iaa(cid:105).
For (∆ > 0,u3 < −2 ∆u4) the effective potential Differently from the idealized model above, the con-
Γ displays two distinct minima, n1 = 0 and n2 = straint requires here a single excitation nearby, and pro-
|u3|+( u23 − ∆)1/2,suggestingafirst-orderphasetransi- cesses such as |aia(cid:105) → |aaa(cid:105) are highly suppressed. The
2u4 4u24 u4 Hamiltonian is therefore approximately given by H =
tion. The actual transition line, however, lies where the ryd
Ω(cid:80) Π(cid:48)σx where Π(cid:48) =n +n −2n n .
finite-density minimum becomes statistically preferred. k k k k k−1 k+1 k−1 k+1
To generate the incoherent branching/coagulation the
Inequilibrium,thiswouldbethepointatwhichthemin-
atomsarecoupled(withcouplingg)toasecondequally-
ima of Γ are at the same height. However, the present
detuned light field with strong phase noise (dephasing
non-equilibrium noise shows more pronounced fluctua-
rateλ(cid:29)g)[63];foracorrelationlengthshorterthanthe
tions at larger densities and thus favors the absorbing
interatomicdistance,thebathismodeledasindependent
minimum n with respect to n . To estimate the steady
1 2
state distribution function P(n), we apply the optimal bosonic modes bk, b†k acting on each lattice site. The
path approximation to the action [3, 53]; this involves effective equation of motion for the atoms is obtained
treating the coefficient Ξ(n) = 1n + µ n2 of n˜2 as a by performing second order perturbation theory in the
2 4
kind of mean-field and density-dependent temperature. small parameter g/λ [44, 60, 64]. The resulting master
It yields (see Appendix D) equation for the reduced atomic density matrix ρ is
(cid:90) n 4g2 (cid:88)(cid:16) (cid:17)
P(n)= Z1e−V W(n), with W(n)= dm∂ΞΓ(/m∂m) ,(7) ρ˙ = λ (cid:104)b†kbk(cid:105)D[Π(cid:48)kσk+]+(cid:104)b†kbk+1(cid:105)D[Π(cid:48)kσk−] ρ.
0 k
with volume V and normalization Z. Both potentials For sufficiently high ((cid:104)b†b (cid:105) (cid:29) 1) and homogeneous
W(n) and Γ(n) vanish in n and share the finite-density k k
1 ((cid:104)b†b (cid:105) ≈ (cid:104)b† b (cid:105)) intensity, one can identify κ =
mP(inni)m→umδ(nn2−.nlI)n,wthheerethle=rm1o,2dydneapmenicdinligmoitnwVhi→cho∞ne, (4gk2(cid:104)kb†kbk(cid:105))/λ,mlemading√to the branching/co√agulation
is the global minimum of W, accounting for the physi- jumpoperators: Lryd = κΠ(cid:48) σ+andLryd = κΠ(cid:48) σ−.
b,k k k c,k k k
cal constraint n ≥ 0. The transition takes place when The final process is radiative decay of an atom from its
5
Rydbergstatetothegroundstate, modeledbythejump Master equation
√
operator Lryd = γσ− [26].
d,k k
Althoughthemicroscopicformulationofthedynamics O˙ =i[H,O]+(cid:88)D(cid:48)[L ]O+ξˆO, (8)
a,k
is slightly different from the previously-discussed model,
a,k
theresultingphasestructureissimilar,astheEOMsonly
differ from Eqs. (2-4) by RG irrelevant higher order den- where D(cid:48)[X]O = X†OX −(X†XO+OX†X)/2 and ξˆO
sity terms. In particular, they leave the universal prop-
is the quantum noise term for the operator O. The
erties near the continuous transition points unchanged.
noise-less equation of motion is only formally correct on
the level of single-operator expectation values, while the
noise contributes by preserving the (anti-)commutation
relations of the operators during the evolution. [49].
VIII. OUTLOOK
For a linear coupling of the system to the bath, the
noise is typically Gaussian, with zero mean, but non-
Wehaveinvestigatedtheeffectsofquantumdynamical
vanishing time- and space-local correlations. This pre-
processes on a prototypical absorbing-state phase tran-
scriptionleadstoEqs.(2)-(4);belowweprovidethemain
sition. We highlighted the emergence of a richer struc-
conceptual steps.
tureinthephasediagram, whichincludesbothadiscon-
There exist several canonical (and equivalent) strate-
tinuous and a continuous non-equilibrium transition. In
gies to determine the properties of the noise operators
low dimension d < 2 the presence of a quantum coher-
ξˆn,x,y, as for instance outlined in Ref. [49]. Here, we
ent process leads to a breaking of the only fundamental
follow a path relying on the unitary Heisenberg equa-
symmetry of DP in a way that persists at long wave-
tions of motion for system operators in the presence of
lengths,andthusleadstoaphasetransitionofadifferent
a bath. As a simplifying assumption, we imagine the
nature. In equilibrium, the interplay between classical
spatial correlations of this bath to be shorter than the
(thermal) and quantum fluctuations typically leads to a
typical interparticle distance in the system. This allows
dimensional crossover [2, 65]. The present work shows
us to describe every spin as coupled to its own bath. We
that out of equilibrium the picture is not as straight-
can therefore focus on a single spin as a representative
forward and opens the path for further investigations in
andwemodelthespontaneousemissiondynamicsviathe
this field, including the quantitative characterization of
simple Hamiltonian
the new universality class.
(cid:88) (cid:88)
H(cid:101) =Hs-b+Hb = λq(σ+bq+b†qσ−)+ ωqb†qbq, (9)
q q
ACKNOWLEDGMENTS
where the b s represent a set of bosonic bath operators.
l
We further assume that this bosonic reservoir is kept at
M.M. and I.L. wish to express their gratitude for the
zero temperature and that the number of modes is suf-
insightful discussions with J.P. Garrahan and for access
ficiently large to allow a continuum description with a
totheUniversityofNottinghamHighPerformanceCom- (cid:80)
density of states D(ω) = δ(ω − ω ). Taking the
putingFacility. I.L.acknowledgesthattheresearchlead- q q
von Neumann equation for the global (spin plus bath)
ing to these results has received funding from the Eu-
density matrix and eliminating the bath degrees of free-
ropean Research Council under the European Union’s √
dom leads then to the jump operator L = γσ− with
Seventh Framework Programme (FP/2007-2013) / ERC d
γ =2π[λ(0)]2D(0)beingproportionaltothebathdensity
GrantAgreementn. 335266(ESCQUMA).Furtherfund-
of states D(0) and the couplings λ(0) evaluated at zero
ing was received through the H2020-FETPROACT-2014
frequency (see e.g. Chapter 8 of [49]). The Heisenberg
grant No. 640378 (RYSQ) and from EPSRC Grant no.
equations of motion for the operators are therefore
EP/J009776/1. M.B.andS.D.acknowledgefundingby
theGermanResearchFoundation(DFG)throughtheIn- (cid:88)
stitutional Strategy of the University of Cologne within σ˙+ =i[H(cid:101),σ+]=−i λqb†qσz, (10)
the German Excellence Initiative (ZUK 81). q
(cid:88)
n˙ =i λ (b†σ−−σ+b ), (11)
q q q
q
Appendices b˙† =iλ σ++iω b†. (12)
q q q q
Formally solving Eq. (12) yields
A. HEISENBERG-LANGEVIN EQUATIONS OF
MOTION (cid:90) t
b†(t)=b†(0)eiωqt+iλ dt(cid:48)σ+(t(cid:48))eiωq(t−t(cid:48)). (13)
q q q
0
In order to derive the Heisenberg-Langevin equations
of motion of an observable O we employ the conjugate InsertingthissolutionintoEqs.(10),(11)andperforming
6
the Born-Markov approximation leads to B. MARTIN-SIGGIA-ROSE CONSTRUCTION
γ (cid:88)
σ˙+ =−2σ++i λqb†q(0)σzeiωqt, (14) Inthissection,weprovidethederivationoftheMartin-
q Siggia-Rose(MSR)pathintegralforthepresentquantum
(cid:124) (cid:123)(cid:122) (cid:125)
contact process, which results in the effective long wave-
ξ˜+(t)
length action for the density, Eq. (5). As a first step, we
(cid:88)
n˙ =−γn+i λq(b†q(0)σ−eiωqt−σ+bq(0)e−iωqt). take the continuum limit of the equations of motion for
q n , σx and σy, such that
(cid:124) (cid:123)(cid:122) (cid:125) k k k
ξ˜n(t) (cid:88) n →(r2∇2+z)n , (23)
(15) j x
j nnx
Defining ξ˜−(t) = (ξ˜+(t))† and taking the bath to be in wherez isthecoordinationnumber,r isthelatticespac-
the vacuum state (corresponding to spontaneous emis- ing, ∇ is the the common d-dimensional gradient and
sion), we find therefore the noise properties in the Born x = rk the position. We then re-interpret the opera-
Markov approximation to be tors as stochastic fields subject to the continuum noise
(cid:104)ξ˜+(t)ξ˜+(t(cid:48))(cid:105)=(cid:104)ξ˜+(t)ξ˜−(t(cid:48))(cid:105)=(cid:104)ξ˜+(t)(cid:105)=(cid:104)ξ˜n(t)(cid:105)=0, sources ξXx, ξXy , ξXn — where X =(t,x) is shorthand for
the spatio-temporal argument — which have vanishing
(cid:104)ξ˜−(t(cid:48))ξ˜+(t)(cid:105)=γδ(t−t(cid:48)). (16) (cid:68) (cid:69)
mean and correlations ξi ξj =γδ(X−Y)Mij, where
X Y
By noticing that ξ˜n ≡ −ξ˜+σ− −σ+ξ˜− one gets all the
remaining non-vanishing correlations  1 0 −σx 
2
(cid:104)ξ˜n(t)ξ˜n(t(cid:48))(cid:105)=γnδ(t−t(cid:48)), (17) M = 0 1 −σ2y . (24)
−σx −σy n
(cid:104)ξ˜n(t)ξ˜+(t(cid:48))(cid:105)=−γσ+δ(t−t(cid:48)), (18) 2 2
(cid:104)ξ˜n(t)ξ˜+(t(cid:48))(cid:105)=−γσ−δ(t−t(cid:48)). (19) The equations of motion can thus be expressed as
Rotating into the (x,y,n) basis and introducing ξ˜x = n˙X =Fn(nX,σXx,σXy )+ξXn, (25)
ξ˜++ξ˜− and ξ˜y = −iξ˜++iξ˜− (and analogously σx and σ˙x =F (n ,σx,σy )+ξx, (26)
(cid:68) (cid:69) X σx X X X X
σy) one finds ξ˜i(t)ξ˜j(t(cid:48)) =γδ(t−t(cid:48))Mij with σ˙y =F (n ,σx,σy )+ξy , (27)
X σy X X X X
 1 −i −σx−iσy  where
2
M =−σxi+iσy iσx+1iσy −iσnx2+σy . (20) Fn =−nX +[ωσXy −χ(2nX −1)](r2∇2+z)nX, (28)
2 2
F =−zχ+1σx −χσx(r2∇2+z)n +
The dependence of the ξn noise on the density keeps σx 2 X X X
track of the fact that the absorbing configuration n = 0 +ωσy (r2∇2+z)σx, (29)
X X
represents a fluctuationless state in the entire parameter
regime,whichforbidsadensityindependentcontribution Fσy =−zχ2+1σXy −[ω(4nX −2)+χσXy ](r2∇2+z)nX+
to ξn in Eq. (17). The Markovian noise level introduced +ωσx(r2∇2+z)σx. (30)
bythedecaytermsL =γσ−onlyappearsasanadditive X X
d
noise in the σx,y variables. AsshowninRefs.[3,53],theMSRconstructiondefines
Including classical coagulation and branching pro- a path integral in the variables σy ,n for the equations
X X
cesses yields additional noise terms. However, due to of motion (25)-(27). The MSR partition function repre-
(cid:80)
the presence of the term n these contributions will sents the sum over all allowed field configurations, i.e.
j j
always be higher-order in the density and are therefore
(cid:90) (cid:90)
subleading with respect to the ones derived above in the Z = D[n ,σx,σy ] D[ξn,ξx,ξy ]P(ξn,ξx,ξy )×
absorbingphase. Extendingthesystemfromasinglespin X X X X X X X X X
toalatticeofindividualspins,anequivalentcomputation ×J[n ,σx,σy ]δ(n˙ −F (n ,σx,σy ))×
X X X X n X X X
shows
×δ(σ˙x −F (n ,σx,σy )) δ(σ˙y −F (n ,σx,σy )),
(cid:104)ξˆn(t)ξˆn(t(cid:48))(cid:105)=n γδ δ(t−t(cid:48))+O(n2), (21) X σx X X X X σy X X X
k k(cid:48) k k,k(cid:48) (31)
(cid:104)ξˆxξˆx(cid:105)=γδ δ(t−t(cid:48))+O(n)=(cid:104)ξˆyξˆy (cid:105). (22)
k k(cid:48) k,k(cid:48) k k(cid:48) where the integral (cid:82) D[ξ(cid:126)]P(ξ(cid:126)) averages over all noise
To leading order in the density, this yields the same
configurations described by the Gaussian noise distri-
noise terms reported above. Since the coherent branch- (cid:110) (cid:16) (cid:17)(cid:124) (cid:111)
bution P(ξ(cid:126)) = exp −1(cid:82) ξ(cid:126) M−1ξ(cid:126) . The factor
ingandcoagulationdoesnotproduceanadditionalnoise, 2 X X X
thisconcludesthederivationoftheHeisenberg-Langevin J[n ,σy ] is a Jacobian which, for our present purposes,
X X
equations. can be conveniently set to 1 after choosing a proper,
7
retarded regularization [3, 53]. Introducing three sets where, up to leading order in a (spatial) derivative ex-
of imaginary response fields n˜ ,σ˜x,σ˜y , exploiting the pansion, the action reads
(cid:82)X X X
Fourier transform δ(f(n)) = Dn˜exp(−n˜f(n)) and in-
tegrating over the noise variables ξn/x/y, Z can be cast
into a path-integral form
(cid:90)
Z = D[n˜ ,n ,σ˜xσx,σ˜y σy ] e−S. (32)
X X X X X X
(cid:90) (cid:20) 1 (cid:21)
S = n˜ ∂ −D∇2−(zχ−1)− n˜ n +2zχn˜ n2
X t 2 X X X X
X
(cid:90) (cid:20) 1 (cid:21) 1
+ σ˜y ∂ + zχ+1 +zχn + n˜ σy − (σ˜y )2+σ˜y (zω(4n2 −2n ))−σy (zωn˜ n ) (33)
X t 2 X 2 X X 2 X X X X X X X
X
(cid:90) (cid:20) 1 (cid:21) 1
+ σ˜x ∂ + zχ+1 +zχn +zωσy + n˜ σx − (σ˜x)2−zωσ˜y (σx)2.
X t 2 X X 2 X X 2 X X X
X
Since zχ+1 ≥1/2 throughout the physical parameter re- classicaldirectedpercolation. Itfeaturesthecharacteris-
2
gion χ≥0, both the σx and the σy fields remain gapped tic rapidity inversion symmetry under n ↔−n˜, t →−t.
and one can therefore neglect the subleading derivative For ω > 0, the relevance of the u coupling has to be
4
andfluctuatingtermswithinsquarebrackets. Thisyields considered, which is determined by the scaling behavior
an action which is separately quadratic in (σx,σ˜x) and of the fields n,n˜. In the absence of a thermal fluctuation
X X
(σy ,σ˜y ). These modes can be actually integrated out dissipation relation, both fields n,n˜ typically have the
X X
exactly and, up to RG-irrelevant terms, one obtains ac- same scaling dimension [3, 53, 67]. This leads to an up-
tion(5). Thecouplingscorrespondtothefollowingcom- percriticaldimensionofd =4forthecubiccouplingu
c 3
binations of miscroscopic parameters: and an upper critical dimension of d =2 for the quartic
c
coupling u . In dimensions d > 2, u renormalizes to
4 4
∆=1−zχ− 8z2ω2 , (34a) zerointheRGflowandtherapidityinversionsymmetry
(zχ+1)3
(cid:16) (cid:17) is restored in the infrared regime. Hence, the effective
u3 =2z χ− z2χzω+21 , (34b) low frequency theory, and therefore the long time dy-
namics, is again described by the directed percolation
u = 8z2ω2, (34c)
4 zχ+1 class. On the other hand, for d<2, u4 is relevant in the
2z2ω2 128z4ω4 renormalization group sense and the absence of rapidity
µ = + . (34d)
4 (zχ+1)2 (zχ+1)6 inversion introduces a different non-equilibrium dynam-
ics at the phase transition, which is not captured by the
Ourprocedurediffersconceptuallyfromtheapproachad- DPuniversalityclass. Ind=2,thequarticcouplingsare
vocatedin[66]. Wefounditnecessarytoaccuratelycap- marginalandwhethertheybecomerelevantorirrelevant
ture the short distance physics of the problem. intheRGflowhastobedeterminedbyarenormalization
group analysis of the problem.
At the point α, u vanishes microscopically and the
3
C. ADDITIONAL DETAILS ON THE NATURE
rescaling leading to the action (35) is not defined. In di-
OF THE OBSERVED PHASE TRANSITIONS
mensions d > 2 this point features a second order phase
transitionintheabsenceoftherapidityinversionsymme-
Except for the α point, in the proximity of the transi-
try. Sincetheleadingordertermintheeffectivepotential
tions u (cid:54)= 0 and one can rescale the fields according to
3 √ Γ (Eq. (6)) is RG irrelevant in d>2, one expects mean-
n→Kn, n˜ →n˜/K with the choice K =1/ 2u . Thus,
3 field scaling behavior at this point. On the other hand,
one finds
in d < 2 the effective theory at the α-point corresponds
(cid:90) (cid:104) (cid:105) to the new universality class in the absence of rapidity
S = n˜X ∂t−D∇2+∆+κ3nX + 2uu43n2X nX inversion.
X
−(cid:90) n˜2 (cid:2)κ n +µ n2 (cid:3). (35) In any experiments with cold atoms, the presence of
X 3 X 4 X small fluctuation-inducing terms ∼ ∆ σx or ∆ σy is
X x y
hardly avoidable at the microscopic level. These will
(cid:112)
with κ =Ku =1/2K = u /2. In the absence of the generate fluctuations on top of the absorbing state and
3 3 3
u coupling, i.e. for ω = 0, this is the action describing lead to a temperature type term ∼ n˜2 T in the action
4 X
8
(35), with T ≈ ∆ ,∆ . The present discussion of the this, there exists the non-trivial solution
x y
non-equilibrium phase transitions is then valid on length
√ Γ(cid:48)(n )
scales l−1 ≥ T. n˜op = t . (37)
t Ξ(n )
t
Considering only configurations which correspond to the
optimal path the action becomes
(cid:90) ∞ (cid:90) n Γ(cid:48)(m)
D. DETAILS ON OPTIMAL PATH W(n)= n˜op∂ n dt= dm, (38)
APPROXIMATION 0 t t t n0 Ξ(m)
with the change of variable ∂ n dt → dm and where n
t t 0
In the present setting, the noise Ξ ≡ 1n +µ n2 is the initial condition — whose specific value is irrele-
increases monotonically with the denXsity. 2AXs a co4nseX- vant — and n the steady-state value of the density. The
quence, it favors the (fluctuationless) zero density solu- corresponding density distribution function is
tion over the finite-density one. In order to determine
1
the distribution function for the density variable in the P(n)= e−VW(n), (39)
Z
vicinity of the active-to-inactive transition, we apply the
optimalpathapproximation[3,53]tothepartitionfunc- withZ =(cid:82) e−VW(n),iftheintegralexists. Forasystem
n
tion. Note that the system remains gapped for ∆ > 0 in thermal equilibrium, Ξ(n)∝T is simply proportional
(wherethefirst-ordertransitionisexpectedtotakeplace) to the temperature and one recovers the naive expecta-
and therefore we can — as a first approximation — ne- tion P(n)∼exp(−Γ(n)/T). For the present noise terms
glect spatial fluctuations and approximate n ,n˜ by
X X (cid:104) (cid:105)
spatially homogeneous but temporally fluctuating fields W(n)= 1 ∆l+u (n− l )+u 4nµ4(nµ4−1)+2l , (40)
n ,n˜ . This yields the action µ4 3 2µ4 4 16µ34
t t
with l=log[1+2nµ ].
4
(cid:90) (cid:104) (cid:105) (cid:114)The minima of W(n) are n1 = 0 and n2 = −2uu34 +
S =V n˜t∂tnt+n˜tΓ(cid:48)(nt)−n˜2tΞ(nt) (36) u23 − ∆ and coincide with the ones of Γ(n). The two
t 4u24 u4
functionals, however, may differ significantly. In partic-
ular, the global minimum of W does not necessarily co-
with the shorthand Γ(cid:48)(n) = δΓ/δn. The optimal path incidewiththeglobalminimumofΓ, suchthatthepres-
for the configurations n ,n˜ corresponds to the configu- enceofanon-equilibriumnoisetermcanstronglymodify
t t
rations for which the non-fluctuating part of the action the phase boundary as a function of the noise strength.
vanishes, i.e. for which n˜ Γ(cid:48)(n )−n˜2Ξ(n )=0. This Since W(0) = 0 for all parameters, the first order tran-
t t t t
equation shows the two trivial solutions n˜ = 0 and n sition line separating the active from the inactive phase
t t
arbitrary as well as n =0 and n˜ arbitrary. Apart from for ∆>0 is determined by the equation W(n )=0.
t t 2
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