arXiv:2206.11662v1 [cond-mat.stat-mech] 23 Jun 2022

Wetting critical behavior within the Lindblad dissipative dynamics

Claudia Artiaco,1, ∗ Andrea Nava,2, 3 and Michele Fabrizio4

1Department of Physics, KTH Royal Institute of Technology, Stockholm 106 91, Sweden2Dipartimento di Fisica, Universita della Calabria, Arcavacata di Rende I-87036, Cosenza, Italy

3INFN - Gruppo collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy4Scuola Internazionale Superiore di Studi Avanza (SISSA), via Bonomea 265, 34136, Trieste, Italy

We investigate the critical behavior, both in space and time, of the wetting interface within thecoexistence region around the first-order phase transition of a fully-connected quantum Ising modelin a slab geometry. For that, we employ the Lindblad master equation in which temperature isinherited by the coupling to a dissipative bath rather than being a functional parameter as in theconventional Cahn’s free energy. Lindblad’s approach gives not only access to the dissipative dy-namics and steady-state configuration of the wetting interface throughout the whole phase diagrambut also shows that the wetting critical behavior can be successfully exploited to characterize thephase diagram as an alternative to the direct evaluation of the free energies of the competing phases.

I. INTRODUCTION

Wetting is generally defined as the ability of liquidsto maintain contact with solid surfaces. More specifi-cally, the study of wetting concerns the understanding ofthe relationship between bulk phase transitions and sur-faces [1–9]. Clearly, this problem is extremely vast andrich. Wetting phenomena have been investigated in a va-riety of systems ranging from classical ones, such as inliquid-vapor phase transitions or binary liquid mixturesof linear alkanes and methanol, to polymeric mixtures,superfluid 4He on thin cesium substrates, liquid 3He onsuperfluid 4He, dilute ultra-cold gases undergoing Bose-Einstein condensation, and many others [10–17].

In this study, we focus on the wetting layer that canform in the coexistence region accompanying a first-orderphase transition [18]. Indeed, even though the wettingphenomenon has been extensively studied in the last fortyyears [19, 20], there are still open issues about quantumfirst-order phase transitions. Several attempts to disclosethe wetting phenomenon in the quantum realm have re-lied on the quantum-classical mapping, i.e., on the ideathat the properties of d-dimensional quantum systemsat zero temperature and across a phase transition cor-respond to those of classical systems in higher dimen-sions [21]. Adopting a simple fully connected quantumspin Ising model, the authors of Ref. [18] observe thatthe critical properties of wetting in the quantum caseindeed correspond to the classical ones in higher dimen-sions, specifically d+ 1 in that simple mean-field model.However, the singular behavior of quantum fluctuationsis different from that of classical fluctuations at finitetemperature.

In this article, we perform a direct study of the dy-namics and the equilibrium configuration of the wettinglayer at phase coexistence in an open quantum system.We consider a slab geometry constituted by L layers,which is a discrete version of the model of Ref. [18]; each

∗ [email protected]

layer is modeled by a quantum Ising model with N fully-connected sites. In the thermodynamic limit, N → ∞,the mean-field approximation becomes exact, and theequilibrium state of the system can be found by solvinga set of self-consistency equations. The same single-layermodel already possesses a non-trivial phase diagram that,depending on the form of the spin-spin interactions, candisplay first- or second-order quantum and thermal phasetransitions. Thanks to its simplicity, such a model hasbeen widely employed in the past [22–24], for instance,to study the relaxation dynamics towards equilibrium inpresence of dissipation [25, 26].

Specifically, we define the single-layer Hamiltoniansuch that it may undergo a first-order phase transition,and set its parameters so that the layer is in the coexis-tence region. Each layer is in turn coupled to its nearestneighbors as well as to a dissipative bath. We fix theboundary conditions of the whole slab imposing that thefirst and last layers are, respectively, in the ordered anddisordered phases. In that way, we can easily study in-homogeneous phenomena, including wetting.

The presence of the dissipating baths coupled witheach layer makes the system an open one. In open quan-tum systems, one is usually interested in describing thedynamics of a system (S) coupled to an external environ-ment (E). In particular, one would like to derive a masterequation for S, integrating out the degrees of freedom ofE. In order to accomplish this task, we resort to the Lind-blad master equation (LE) formalism. The LE is amongthe most popular master equations and has been em-ployed in various and different contexts [27–33]. Startingfrom the microscopic Hamiltonian of the full system S+E,the LE can be derived within the weak S-E coupling ap-proximation. That entails three further approximations:the Born, the Markov, and the rotating-wave approxi-mation (see Ref. [27] for more details). Nonetheless, itcan be shown that the LE is the most general time-localgenerator that preserves hermiticity, complete positivity,and unity of the trace of the system’s density matrix, i.e.,it is a so-called Krauss operator (see Ref. [34]). Here, weassume a Lindblad dynamics of each layer with jumpoperators defined through the instantaneous mean-field

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Hamiltonian. This allows us to investigate arbitrarilylarge systems, and to explore the wetting phenomenonwithout resorting to Cahn’s free energy functionals inwhich temperature enters just as a functional parame-ter [1], while in our case temperature is inherited by thebath. We show that, within our model and LE scheme,the wetting phenomenon spontaneously emerges duringthe quantum dissipative dynamics. That allows us touncover in detail static and dynamic properties of thewetting interface as a function of the Hamiltonian pa-rameters and the bath temperature.

The paper is organized as follows. In Section II wepresent the model Hamiltonian of a single-layer fully con-nected quantum Ising model and review its dissipativedynamics yielded by the Lindblad equation. In Sec-tion III we extend the formalism discussed in the pre-vious Section to a multi-layer setup in which multiplecopies of the single-layer system are connected one afterthe other to form a slab of length L. Each bulk layer iscoupled to a dissipating bath while the states of the firstand last layers are kept fixed. Section IV is devoted tothe discussion of the relaxation and equilibrium proper-ties of the multi-layer system. In particular, we analyzethe behavior of the layer-resolved equilibrium energy, or-der parameter, and relaxation time. Finally, in SectionV we summarize our results and discuss possible furtherperspectives of our work.

II. SINGLE-LAYER SYSTEM

In this Section, we briefly mention the properties ofthe single layer when it is decoupled from all the others.In particular, we discuss its phase diagram and show howto construct the Lindblad jump operators to describe itsrelaxation dynamics to the closest stationary state.

A. The quantum spin model for the single-layersystem

We model each layer as a quantum Ising model on anN -site fully connected graph, described by the generalHamiltonian [22–24]

H = −hx∑i

σxi −Nm∑n=2

Jn

(1

N

∑i

σzi

)n(1)

where m ≥ 2 is an integer number, σαi , with α = x, y, z,are the Pauli matrices on site i = 1, . . . , N , hx is thetransverse magnetic field, and Jn are the n-spin exchangeconstants. In the following, we concentrate on the caseJ2 6= 0, J4 6= 0, and Jn 6=2,4 = 0, for which the model (1)undergoes a first-order phase transition [18, 35]: increas-ing either the temperature T or the transverse field hx,the system goes from an ordered, ferromagnetic phase(F) to a disordered, paramagnetic one (P).

FIG. 1. Phase diagram of the single-layer model. In theregion labeled as F, there is only a ferromagnetic free-energyminimum, and the Z2 symmetry is broken. Conversely, inregion P there is only a paramagnetic minimum. FP and PFdenote the coexistence regions: in FP, the absolute minimumis ferromagnetic; in PF, it is paramagnetic. The F-P phasetransition occurs along the dashed line separating FP fromPF, and it takes place when the free energies cross. The solidlines between F and FP, and between P and PF, are spinodallines where an additional metastable free energy minimumappears beside the stable one.

Thus, our single-layer system Hamiltonian reads

H = −hx∑i

σxi −J2N

(∑i

σzi

)2

− J4N3

(∑i

σzi

)4

. (2)

The phase diagram of the model (2) has been alreadystudied in the past: it is illustrated in Fig. 1.

Thanks to the full connectivity of Hamiltonian (2),mean-field approximation becomes exact in the thermo-dynamic limit N → ∞, as ensured by the vanishing ofthe covariance

〈σαi σβj 〉 − 〈σ

αi 〉〈σ

βj 〉 →

1

N−−−−→N→∞

0 . (3)

It follows that the equilibrium Boltzamnn distribution isgiven by

ρ −−−−→N→∞

∏i

ρi =∏i

e−βHi

Tr(e−βHi

) (4)

where ρi is a positive definite 4×4 matrix with unit trace,and the single site Hamiltonian Hi reads

Hi = −hx σxi − hz(m)σzi , (5)

with

hz(m) = 2J2mz + 4J4m3z. (6)

m = (mx,my,mz) indicates the Bloch vector, with com-ponents

mα :=1

N

∑i

〈σαi 〉. (7)

3

Notice that the Bloch vector verifies |m| ≤ 1, wherethe equality is fulfilled only by pure states, and thatthe single-site density matrix can be written as ρi =12 (1 +m ·σi). Thanks to the exact validity of the mean-field approximation, we can consider a single spin at atime and drop the index i.

It is easy to show that, when the system is at equi-librium with a bath at temperature T , its state is deter-mined by a set of self-consistency equations:

m = tanhβh(m)(

cos θ(m) , 0 , sin θ(m))

(8)

where

tan θ(m) =hz(m)

hx, (9)

h(m) =√h2x + hz(m)2 . (10)

B. Lindblad master equation for the single layer

In the following, we describe the dissipative dynam-ics of open quantum systems in terms of LE formalism.Under the Born approximation the system-environmentdensity matrix is factorized at any time, i.e.,

ρS+E(t) ' ρS(t)⊗ ρE(t) , ∀ t. (11)

Integrating out the environment degree of freedom underthe Markov approximation one arrives at the followinggeneral form of the LE:

ρS(t) = −i[HS , ρS(t)

]+∑λ

[γλ

(2LλρS(t)L†λ − {L

†λLλ, ρS(t)}

)+ γλ

(2L†λρS(t)Lλ − {LλL†λ, ρS(t)}

)], (12)

where HS indicates the system Hamiltonian. The LE de-scribes a dynamical map that is linear, completely posi-tive, and trace preserving.

Neglecting pure dephasing processes, the dissipativedynamics of the system can be described by non-Hermitian jump operators that produce transitions be-tween two eigenstates of the system Hamiltonian HS :

Lλ(m,n) = |m〉〈n|, En < Em. (13)

It is easy to verify that the Boltzmann distribution isa stationary solution of Eq. (12) if γλ/γλ = e−βελ withελ = Em − En > 0. In the following, we set

γλ = Γf(− βελ/2

)(14)

with f(x) the Fermi-Dirac distribution function, and Γthe overall bath-system coupling strength.

Ref. [25] extensively discusses many possible ways inwhich the Lindblad jump operators can be defined to

capture the physics of the simple model (2). Here, weconsider one choice that, we will argue, allows correctlyreproducing the dynamics of the wetting interface in themulti-layer system, see Sec. IV, and recovering the semi-classical results of Ref. [18].

Starting from a factorized density matrix, the full con-nectivity of the model (2) ensures that it remains factor-ized at any time:

ρS(t) −−−−→N→∞

∏i

ρi(t) (15)

where ρi(t) describes the time evolution of the spin i,coupled to a bath at temperature T and in the presenceof a time-dependent magnetic field given by Eqs. (5)-(6):

h(t) := h(m(t)) (16)

:=(hx , 0 , 2J2mz(t) + 4J4(mz(t))

3)

(17)

with

m(t) =1

N

∑i

Tr(ρS(t)σi

), (18)

which is self-consistently determined by the system’s timeevolution. Notice that the mean-field nature of the modelis at the origin of the self-consistency of the dissipa-tive dynamics. Hence, we can formally define a time-dependent system Hamiltonian as

Ht := −h(t) · σ := −∣∣h(t)

∣∣v3(t) · σ, (19)

which is just a two-level system Hamiltonian with time-dependent magnetic field. From Eq. (13), we can writethe instantaneous Lindblad jump operators

L(t) = |1〉〈0| =(v1(t)− iv2(t)

)· σ/2

:= v−(t) · σ/2, (20)

and its Hermitian conjugate

L†(t) = |0〉〈1| := v+(t) · σ/2, (21)

where

v+(t) ∧ v−(t) = 2v3(t), (22)

v+(t) =(v−(t)

)∗. (23)

The energy difference between the eigenvalues of the in-stantaneous Hamiltonian is simply ε(t) = 2

∣∣h(t)∣∣. Thus,

γ(t) and γ(t) depend on time.In turn, this yields a LE with time-dependent param-

eters, where the expectation value of the spin operator isgiven by

m(t) = Tr(ρS(t)σ

)= −2h(t) ∧m(t)

− γ(t)

2

[4(v3(t) + m(t)

)− v−(t)

(v+(t) ·m(t)

)− v+

(v−(t) ·m(t)

)]+γ(t)

2

[4(v3(t)−m(t)

)+ v−(t)

(v+(t) ·m(t)

)+ v+(t)

(v−(t) ·m(t)

)]. (24)

4

In Ref. [25], it has been shown that the Lindblad jumpoperators (20)–(21) are not able to capture the longtime dynamics of the single-layer system, which remainstrapped at all times in the closest stationary state, even ifmetastable. To describe the full dynamics, both at shortand long times, and the relaxation to the true equilib-rium state, one needs to write the master equation as asum of competing terms, one for each phase (either sta-ble or metastable). In this way, both supercooling andMpemba effect emerge during the dissipative dynamics.Fortunately, we will not need such a complicated masterequation to describe the relaxation and equilibrium dy-namics of the wetting layer, as will become clear in thenext Sections.

III. MULTI-LAYER SYSTEM

In this Section, we introduce the multi-layer modelthat we are going to investigate in the next Sections.We discuss how we introduce the inhomogeneities at theboundaries, and how we couple each layer in the bulk toa bath in order to study the relaxation and equilibriumdynamics of the wetting interface.

A. The quantum spin model for the multi-layersystem

Let us now consider a multi-layer system composed ofL layers, where each layer is modeled by the Hamiltonianin Eq. (2), and it is coupled to its nearest neighbor layersvia quadratic and a quartic terms:

HT =

L∑`=1

H` −L−1∑`=1

{J2N

(∑i∈`

σzi

)( ∑i∈`+1

σzi

)

+J4N3

(∑i∈`

σzi

)2( ∑i∈`+1

σzi

)2}, (25)

where H` is the Hamiltonian (2) for layer `.

In the thermodynamic limit, the mean-field single siteHamiltonian for layer `, dropping the site index, reads

H∗ ` = −hx σx` −(

2J2mz` + 4J4m

z`3)σz`

− J2(mz`−1 +mz

`+1

)σz`

− 2J4

(mz`−1

2 +mz`+1

2)mz` σ

z` . (26)

In the following, we will set Jn = Jn/2; hence, when themulti-layer system is in the homogeneous case, i.e., allthe layers are in the same state, Eq. (26) reduces to thesingle-layer Hamiltonian of Eq. (5).

FIG. 2. Graphical representation of the model. We considera multi-layer system composed of L layers. The first and thelast layers are fixed: they are set to the ferromagnetic (F) andparamagnetic (P) state, respectively. All the other layers arecoupled to a heat bath at temperature T , as schematized bythe springs.

B. Lindblad master equation for the multi-layersystem

We wish to study the dynamics of the wetting layerwhen the single layer is within the coexistence phase il-lustrated in Fig. 1. Within the coexistence phase, thesingle layer presents both stable and metastable phases.In the FP region, the ferromagnetic minima are stable(i.e., have a lower free-energy), and the paramagnetic ismetastable; viceversa in the PF region. To model thepresence of the wetting layer, we fix the first and the lastlayers of the multi-layer system in the ferromagnetic (F)and paramagnetic (P) phase, respectively. All the otherlayers are coupled to a heat bath and are free to evolvein time. The system is depicted in Fig. 2.

Following the same line of reasoning of Sec. II for thecase of a single layer, we can write the dissipative dy-namics of the full system considering a time-dependentmagnetic field for each layer ` which is self-consistentlydetermined by the system dynamics. From Eq. (26), thetime-dependent magnetic field, accounting both for theintra- and inter-layer interactions is

h`(t) :=∣∣∣h`(m`(t),m`−1(t),m`+1(t)

)∣∣∣v3`

(m`(t),m`−1(t),m`+1(t)

)(27)

:=

[hx , 0 , J2m

z` (t) + 2J4m

z` (t)

3

+ J2/2(mz`−1(t) +mz

`+1(t))

+ J4(mz`−1(t)2 + (mz

`+1(t)2)mz` (t)

]. (28)

From Eq. (27), we can define the single-layer time-dependent jump operators, similarly to Eqs. (20)–(21),the only difference being that both the magnitude and

5

the direction of the time-dependent magnetic field h`(t)vary between layers. In fact, for each layer `, we obtaina LE for the expectation value of the magnetization sim-ilar to Eq. (24), in which h(t),v3(t),v+(t),v−(t) dependon m`(t),m`−1(t),m`+1(t). Notice that the presence ofthe coupling between layers gives a set of coupled first-order non-linear differential equations that determine thewetting dynamics. Our implementation of the numericalsolver for such a set of differential equations is providedat Ref. [36].

IV. RESULTS

In this Section, we discuss the results of the relaxationdynamics and equilibrium configuration of the multi-layer quantum spin model described in Sec. III, whereinhomogeneities are introduced across the boundaries ofthe system in the presence of a first-order phase tran-sition. Having set Jn = Jn/2 in the original Hamilto-nian (25), the equilibrium phase diagram of the homoge-neous multi-layer model reduces to the one of the single-layer case in Fig. 1 with the presence of a coexistenceregion where both the ferromagnetic, F, and paramag-netic, P, phases are minima of the free energy. In thefollowing, we always set T and hx within the coexistenceregion. In order to study interface phenomena, we con-sider a finite length L, multi-layer system, as depicted inFig. 2, with the first and last layers, i.e., the boundaries,fixed in the F and P state respectively. At equilibrium,we expect that when T and hx are in the FP (PF) phase,the bulk of the system lies in the F (P) phase while asmall but finite region, i.e. the wetting region, formsnear the last (first) layer of the slab. In the following,we discuss the energy cost and thickness of the wettingregion as a function of the temperature and the mag-netic field. Moreover, we discuss the dependence of therelaxation time on the bath coupling strength.

A. Energy

Let us start by looking at the relaxation dynamics inthe T ' 0.07 (i.e., T ≈ 0) limit, setting Jn = 1, inwhich case the coexistence region extends from hx ' 2to hx ' 2.83 while the critical magnetic field that sepa-rates the FP and PF phases is hc ' 2.63113. At t < 0,the system is prepared in its equilibrium minimum, i.e.,all the layers are in the F (P) phase for hx lower (higher)than hc. The state of each layer is specified by the Blochvector mF (mP) self-consistently determined by Eqs. (8)–(10). At t = 0 we fix the first and last layers into the Fand P phase, respectively; at the same time, each layerin the bulk is connected with its own bath (see Fig. 2).Suddenly, the Bloch vector of each layer, m`, starts toevolve in time while each layer exchanges energy withits neighbors and with the bath in order to reach thenew equilibrium configuration that minimizes the energy

FIG. 3. Layer-resolved time evolution of the energy for asystem of L = 50 layers for Γ = 0.2, in the FP phase (hx =2.55, top panel), and in the PF phase (hx = 2.65, bottompanel). In both cases, the first layer is constrained into theF phase, the last layer into the P phase. The dot-dashed lineis the energy of the homogeneous F phase (εF), the dashedline is the energy of the homogeneous P phase (εP). Onlythe layers closer to the corresponding metastable boundaryare explicitly shown as the energy of the layers in the bulkoverlaps with the energy of the stable minima, i.e. min[εF, εP].The label ` of each layer is shown near each curve. Coloronline.

of the multi-layer system, compatibly with the inhomo-geneity introduced via the boundaries.

In Fig. 3 we plot the time evolution of the single-layerenergy, defined as

ε` = −hxmx` −

J22mz`2 − J4

2mz`4

− J24

(mz`−1 +mz

`+1

)mz`

− J44

((mz

`−1)2 − (mz`+1)2

)mz`2, (29)

obtained numerically by integrating the nonlinear LEgiven by Eqs. (24) and (27). We consider a bath cou-pling strength Γ = 0.2 and magnetic field along x be-low, hx = 2.55 (top panel), and above, hx = 2.65 (bot-

6

tom panel), the critical value hc. From now on, we re-fer with εF (εP) to the single-layer energy of a homoge-neous system with magnetization vector mF (mP), i.e.,εF ≡ ε`(m` = mF ∀`) (εP ≡ ε`(m` = mP ∀`)).

Notice that the single-layer energy has an intra-layercontribution plus an inter-layer term. At t = 0, whenthe inhomogeneities at the boundaries are created, all thelayers deep in the bulk of the system have the same en-ergy as in the stable homogeneous configuration. On thecontrary, a large energy contribution emerges from theinter-layer term at the metastable boundary, due to thediscontinuity in the magnetization between the boundarylayer in the metastable phase (` = L for the FP case and` = 1 for the PF one) and its neighboring layer (see thebrown curve in both panels of Fig. 3). In the early stageof dynamics, in order to reduce the total energy, the orderparameters of the layers around the metastable bound-ary start to rearrange assuming intermediate values be-tween mF and mP in order to make the discontinuitysmoother. Doing so, while the intra-layer energy contri-bution increases, the inter-layer energy of the boundarydrastically decreases driving the system, at large t, intoa new inhomogeneous equilibrium configuration with theformation of a wetting interface.

At equilibrium, moving along the wetting interface,from the metastable boundary towards the bulk, thesingle-layer energy ε` is a non-monotonous function ofthe distance from the boundary. Plotting the equilib-rium values of ε` as a function of the magnetization mz

`(see Fig. 3), we observe that the energy increases andthen decreases until it reaches the stable value min[εF, εP]inside the bulk as if it were virtually climbing up the(pseudo)potential barrier that separates the metastableand stable phases at the boundaries. This interpreta-tion becomes clearer if we refer to the energy that eachlayer would have in the homogeneous case at fixed mz

` ,which we plot in Fig. 4 (blue dots) together with theequilibrium energies (red dots) of the layers forming thewetting interface, same data of Fig. 3. We note that theblue dots strictly follow the energy landscape, climbingup the potential barrier that separates the metastableminima from the stable one, while the red dots describea new energy landscape of the inhomogeneous system.

We emphasize that, with our choice of jump operators,albeit the single-layer would remain forever trapped inthe metastable phase [25], the same unphysical metasta-bility does not occur in the inhomogeneous slab. Indeed,we find that the profile of the wetting region is indepen-dent of the initial state and the bath-system couplingstrength Γ. For instance, if we initialize the bulk lay-ers in the metastable minimum, only the dissipation dy-namics and the total relaxation time are affected but notthe equilibrium configuration, which is always the sta-ble one. The same happens if we start with an inhomo-geneous configuration where P and F regions alternate.This guarantees that the approach we take yields physi-cally sound results.We also mention that, if we start in the FP phase and

FIG. 4. Black line: energy per layer, ε`, of the homogeneoussystem as a function of the order parameter for hx = 2.55(FP, left panel) and hx = 2.66 (PF, right panel). Blue dots:energy per layer in a homogeneous system at fixed magne-tization mz

` . Red dots: equilibrium energy of the layerscloser to the metastable boundary in the presence of the wet-ting interface (i.e., corresponding to the equilibrium values inFig. 3). The red line is a guide for the eye, representing the(pseudo)potential for the inhomogeneous system.

initialize the bulk with domain walls between F+, i.e.,mz` > 0, and F−, i.e., mz

` < 0, regions, interfacesemerge with mz

` switching from positive to negative pass-ing through layers in the disordered phase mz

P. Such aninterface has exactly the same shape as the wetting in-terface at the boundary between the P and F phase butis now doubled around the central P layer. Althoughthese composite F± − P − F∓ domain walls have an en-ergy cost, they can move inside the system and annihilatewith each other. This behavior is similar to the forma-tion and propagation of solitons in trans-polyacetylenechains [37]. Studying the dynamics of domain walls be-tween two equivalent stable minima in the coexistenceregion goes beyond the scope of this work and will bediscussed in a forthcoming publication. This effect doesnot emerge in the PF phase where the F± − F∓ domainwalls are destroyed by the bath in favor of the stableordered phase P.

B. Thickness

The wetting layer extent is expected to increase in sizeas the critical magnetic field, hc, is approached. Indeed,as h → hc, one has |εF − εP| → 0. It follows that, inorder to smooth out the discontinuity, the system prefersto unpin more and more layers at the boundary from thestable phase towards the metastable one. In fact, thenearer we are to the critical magnetic field, the lower theintra-layer energy contribution is, while the inter-layerone becomes dominant.

In the PF phase (h > hc), we define the amplitude of

7

FIG. 5. Layer-resolved equilibrium value of the order param-eter in the FP (top panel) and PF (bottom panel) phase forΓ = 0.2 and different values of the magnetic field along x,hx. Only the eight nearest layers to the metastable boundaryare shown. Top panel: hx = 2.55 (blue circles), 2.6 (yellowsquares), 2.628 (green diamonds), 2.6295 (red triangles), 2.631(purple inverted triangles). Bottom panel: hx = 2.632 (bluecircles), 2.635 (yellow squares), 2.64 (green diamonds), 2.7(red triangles), 2.8 (purple inverted triangles). The dashedlines correspond to the fit obtained through Eq.32.

the wetting interface due to the F phase as

AF =

L∑`=1

mz`

mzF

, (30)

while in the FP phase (h < hc) we define the wettingamplitude due the P phase as

AP =

L∑`=1

(1− mz

`

mzF

)= L−AF . (31)

By looking at the wetting surface shown in Fig. 5, weobserve that, even for |hc − hx| ≈ 10−4, only a smallfinite number of layers around the metastable boundaryis involved, so that the equilibrium configuration is notaffected by the system length L, i.e. AF/P � L. Clearly,in the caseAF/P ≈ L the system length must be increasedaccordingly in order to avoid finite size effects. The data

FIG. 6. Blue curve: wetting amplitude, A = min[AF,AP], ofthe wetting region as a function of the transverse magneticfield, hx, for T = 0.2 and Γ = 0.2. Red curve: logarithmic fitf(hx) := a−b ln |hx − hc| with a = 0.75, b = 0.38 for hx < hc,and a = 1, b = 0.43 for hx > hc.

in Fig. 5 can be fitted by a two parameter function of theform

f (α, β) =tanh (βL− α)− tanh (β`− α)

tanh (βL− α)− tanh (β − α), (32)

similar two the solitons bond-alternation domain wallsin polyacetylene [37], or the non-equilibrium station-ary state occupation number profile of an interactingfermionic chain [38]. This behavior is substantially differ-ent from the exponential decay expected for the second-order phase transitions [39].

In Fig. 6 we show the wetting amplitude A as a func-tion of the magnetic field hx for a system at finite tem-perature T = 0.2. As discussed, the wetting inter-faces increases approaching the critical magnetic fieldand diverges at hx = hc, where the phase transitionbetween the FP and PF phases takes place. In agree-ment with the continuum limit discussed in Ref.18, wefind that the wetting thickness diverges logarithmicallyas AF/P = aF/P − bF/P ln |hx − hc|, similarly as in theclassical case at the thermal phase transition [3].

The results of Fig. 6 can be extended at any tempera-ture and any magnetic field within the coexistence regionof the homogeneous system, see Fig. 1. In particular, thediscussions above remain valid by reinterpreting our re-sults in terms of free energy rather than energy at zerotemperature [18]. In Fig. 7, we report the wetting am-plitude for all the (T, hx) values corresponding to the co-existence region of the single-layer phase diagram. Justby looking at the divergence of the wetting amplitude A,we are able to recover FP-PF critical line, as can be ob-served by comparing Fig. 7 with the single-layer phasediagram in Fig. 1.

This result is nontrivial. In order to access the crit-ical line that separates the two metastable phases in afirst-order phase transition, standard approaches require

8

FIG. 7. A = min[AF,AP] of the wetting layer as a functionof the magnetic field and temperature, for Γ = 0.2.

solving self-consistent equations of the form of Eqs. (8)–(10) and then comparing all the free energy minima ata given value of the parameters (such as temperatureand magnetic field). This requires computing, as soonas T 6= 0, the entropy of the system, which is often acumbersome task. In addition, standard LE approachesare only able to recover the F-FP and PF-P critical lines,while failing to recover the FP-PF transition line withinthe coexistence region [25]. On the contrary, in the multi-layer setup discussed in this paper, thanks to the inho-mogeneities introduced by the boundary conditions, itis possible to implement the standard self-consistent LEapproach to the full phase diagram. Within the LE ap-proach, the only required ingredients are the instanta-neous Hamiltonian eigenvectors that define the Lindbladjump operators, see Eq. (24), at the given bath tempera-ture, see Eq. (14). It follows that the full phase diagramat any finite T can be easily derived.

Before moving on to the next Section, let us com-pare the results obtained within the LE approach for themulti-layer system with the results obtained within thesemiclassical analysis on a continuum semi-infinite slab,see Eq. (35) in Ref. [18]. In Fig. 8, we plot, for differ-ent values of hx, mz

`/mzF as a function of the layer in-

dex `, obtained within the LE approach (dots), togetherwith the same quantity computed within the continuumlimit formula [18] (dashed lines). We observe a remark-able quantitative agreement between the results of thetwo approaches as long as hx is not too close to hc. Ashx → hc, the wetting interface of the discrete model isthicker than the continuum one (the leftmost in the toppanel, the rightmost in the bottom panel). Such dis-agreement simply derives from the fact that the interfacewidth is controlled in the continuum limit by the stiffnessterm, second-order expansion in the interlayer distanceof the coupling among layers, and that can well changequantitatively the results, but not the critical behavior.

FIG. 8. Layer-resolved equilibrium value of the order param-eter, normalized to the corresponding mz

F, in the FP (toppanel) and PF (bottom panel) phase, for Γ = 0.2 and differ-ent values of the transverse magnetic field, hx. The markersrepresent the discrete model, the dashed lines the continuumlimit within the semiclassical approximation (see main text).Only the eight nearest layers to the metastable boundaryare shown. Top panel: hx = 2.55 (blue circles), 2.6 (yel-low squares), 2.628 (green diamonds), 2.6295 (red triangles),2.631 (purple inverted triangles). Bottom panel: hx = 2.632(blue circles), 2.635 (yellow squares), 2.64 (green diamonds),2.7 (red triangles), 2.8 (purple inverted triangles).

C. Time

In this Section, we discuss the relaxation time, τ , de-fined as the time required to reach equilibrium, as afunction of the magnetic field, hx, and the bath cou-pling strength, Γ. In Fig. 9 we show τ as a functionof hx at T = 0.2 and three different values of the bathstrength. Far from hc, the relaxation time is indepen-dent of hx and reaches a steady value that, as intuitivelyexpected, decreases for increasing bath strength Γ, i.e.,a stronger bath dissipates faster. Furthermore, for eachvalue of Γ we observe a critical slowing down: the re-laxation time has a power-law divergence approachinghc, τ ∼ |hx − hc|−α, with the critical exponent that isfunction of the bath coupling strength, α (Γ). We findα(0.2) = 1.4, α(0.5) = 1.8, and α(0.9) = 2.1, respec-

9

FIG. 9. Relaxation time to reach the equilibrium configura-tion for a system initialized in the F phase, if h < hc, and Pphase, if h > hc, as a function of hx, for T = 0.2, and Γ = 0.1(blue/upper), 0.2 (yellow/middle) and 0.5 (green/lower). Thefirst layer is constrained into the F phase and the last layerinto the P phase. Inset: relaxation time for a second orderphase transition, where the critical magnetic field is hc = 2.Color online.

tively.

As a consequence, relaxation time curves for differentΓ intersect with each other for some value of the mag-netic field hx. It follows that for each value of hx we candefine an optimal dissipation through the finite value ofΓ that maximizes the system-bath energy exchange rate.This dissipative optimal working point, represented bya minimum of τ(Γ) as shown in Fig. 10 for four differ-ent values of the magnetic field, is similar to the non-equilibrium optimal working point that emerges in thenon-equilibrium stationary state of systems coupled withtwo baths [38, 40, 41]. Indeed, when a system is cou-pled to two different baths that induce a particle/energyflow, one observes a change in the monotonicity of thenon-equilibrium stationary current as a function of theapplied bias, which represents the optimal performanceof the bath. The existence of an optimal working pointis a consequence of the presence, in dissipative dynamics,of two timescales: an intrinsic timescale induced by theHamiltonian of the system, and a dissipative timescale setby the bath strength Γ. As long as the former is shorterthan the latter, increasing Γ increases the performanceof the bath, i.e., reduces τ . When the two timescales arecomparable, the system and the bath are in resonanceand the energy transfer is maximum. Finally, when thebath becomes too strong, the natural evolution inducedby the Hamiltonian and the dissipative one induced bythe bath through the jump operators (that are themselvesa function of the Hamiltonian) are desynchronized, andthey slow each other down. Consequently, the relaxationtimes increase with Γ. Finally, we observe that movingtowards the critical line the optimal working point movesto lower values of the coupling Γ. It is also worth notingthat, albeit in our discussion we made use of values of

FIG. 10. Relaxation time to reach the equilibrium configura-tion for a system initialized in the F phase, if h < hc, and Pphase, if h > hc, as a function of Γ for T = 0.2 and differentvalues of the magnetic field in proximity of hc. The marksrepresent the optimal working points (minima of the curve)for hx = 2.60 (blue square), 2.62 (yellow rombus), 2.62 (greentriangle) and 2.66 (red cicle). The first layer is constrainedinto the F phase and the last layer into the P phase.

the coupling Γ of the order of 10−1, for better plot ren-dering, the same behavior is observed for lower values ofthe coupling, i.e., the ultra-weak regime, where the LEis, in general, more physically sound.

We verified the existence of an optimal working pointalso in a single-layer system with second-order phasetransition like the one described by Eq. (1) for Jn = 0,∀n 6= 2 (see also Ref. [25]) as shown in the inset of Fig. 9.In such a way, we support the idea that the presence ofthe optimal working point is not a consequence of the in-homogeneities induced by the fixed boundary conditionbut, rather, an intrinsic property of the self-consistentdissipative dynamics.

The results shown in Fig. 9 can be extended to thefull phase diagrams as done for the wetting amplitude Asuggesting that also the relaxation time can be used toextract, numerically or experimentally, the critical line inthe coexistence region.

V. CONCLUSIONS

We have discussed the main static and dynamic fea-tures of the wetting interface that emerges in the co-existence region of a first-order transition, both quan-tum and thermal, when at the surface the metastablephase is favored over the stable one present in the inte-rior of the bulk. For that, we have considered a proto-type mean-field model that displays a first-order phasetransition both at zero and finite temperatures, a fully-connected quantum Ising model with two and four spinexchange in a slab geometry. Instead of using a time-dependent Cahn’s free-energy functional, as often done,

10

we have simulated the dynamics through the Lindbladmaster equation, where the temperature is directly in-herited by the coupling to a dissipative bath. In thisway, we are able to study the wetting phenomenon atany temperature and Hamiltonian parameters. In par-ticular, we reproduce the known critical behavior of thewetting interface length as the first-order transition is ap-proached [3]; we also identify a critical behavior of therelaxation time, with bath-dependent exponents, whichreveals the emergence in the parameter space of a dissi-pative optimal working point where the relaxation timeis minimum.

Moreover, our analysis suggests a way to characterizethe phase diagram alternative to the direct comparisonbetween the free energies of the coexisting phases, whichexploits the critical behavior in space and time of thewetting interface upon approaching the phase transition.

The reliability of the approach in recovering physicallysound results, combined with its simplicity and versatil-ity, could make it a precious tool to investigate both equi-librium and non-equilibrium phase transitions in openquantum systems paving the way to search for novelphases or phase transitions arising in spin models [42, 43]or junctions of interacting fermionic systems [44–50].

ACKNOWLEDGEMENTS

This work received funding from the European Re-search Council (ERC) under the European Union’sHorizon 2020 research and innovation program, Grantagreements No. 101001902 (C. A.) and No. 692670(M. F.). A. N. acknowledges financial support fromItaly’s MIUR PRIN projects TOP-SPIN (Grant No.PRIN 20177SL7HC).

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