Entropy- and Flow-Induced Superfluid States

Johan Carlström1 and Egor Babaev1,21Department of Theoretical Physics, The Royal Institute of Technology, Stockholm SE-10691, Sweden

2Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA(Received 13 March 2014; published 29 July 2014)

Normally the role of phase fluctuations in superfluids and superconductors is to drive a phase transitionto the normal state. This happens due to proliferation of topologically nontrivial phase fluctuations in theform of vortices. Here we discuss a class of systems where, by contrast, nontopological phase fluctuationscan produce superfluidity. Here we understand superfluidity as a phenomenon that does not necessarilyarises from a broken U(1) symmetry, but can be associated with a certain class of (approximate or exact)degeneracies of the system’s energy landscape giving raise to a U(1)-like phase.

DOI: 10.1103/PhysRevLett.113.055301 PACS numbers: 67.85.Fg

The phase transition from the superfluid to the normalstate is driven by phase fluctuations. Remarkably, in thecontext of superfluids it was first conjectured by Onsager[1] that the superfluid-to-normal phase transition is drivenby proliferation of topological defects in the form ofvortex loops. Due to the phase winding around a vortex,the presence of macroscopically large proliferated vortexloops disorders the phase φðrÞ of the complex orderparameter fields ψðrÞ ¼ jψðrÞjeiφðrÞ, leading to restorationof the U(1) symmetry so that hjψðrÞjeiφðrÞi ¼ 0. Thissituation also takes place in U(1) type-II superconductors,as was established in Ref. [2]. In that case proliferation ofvortex loops restores the local U(1) symmetry via theinverted-XY transition (see detailed discussion in, e.g.,Refs. [2–5]).In two dimensions, spin-wave-like phase fluctuations

play a relatively more important role. At any finite temper-ature they lead to algebraic decay of correlations and thus torestoration of U(1) symmetry [6]. However, it requiresproliferation of topological defects (vortex-antivortex pairs)to make correlations short range via the Berezinskii-Kosterlits-Thouless transition [7–9].In multicomponent systems, phase fluctuations in the

from of composite vortices generally result in the formationof paired states [10–12]. However these fluctuations still actto destroy superfluidity, albeit in some nontrivial channels.Indeed, more broadly we generally associate thermalfluctuations with symmetry restoration, and destructionof superfluidity or superconductivity. In other physicalsystems indeed exceptions exist. One notable example isthe Pomeranchuk and related effects [13,14]: that 3He canbe liquid at zero temperature but solidifies upon heating.The transition between different lattice types in crystals(see, e.g., Refs. [15,16]) is an example of a conventionalphase transition between two different broken symmetries.The question that we raise in this work is: are there

situations where fluctuations induce superfluidity or super-conductivity rather than destroy it, i.e., can a situation arise

where a system possesses a channel that is not superfluid orsuperconducting in the ground state, but fluctuations causethe system to exhibit superfluidity in this channel at, forexample, elevated temperature? We argue that indeed(quasi)superfluid states can be generated for entropicreasons due to phase fluctuations.Often the superfluid phase transition is associated with a

spontaneous breakdown of U(1) symmetry, but superflu-idity can also arise without symmetry breaking, like in theaforementioned two-dimensional case at finite temperature,or the one-dimensional case at zero temperature. Super-fluidity can in principle exist in macroscopic systems thatexplicitly break U(1) symmetry, although only weakly, thusallowing “pseudo” Goldstone bosons. Here we consider afurther possible generalization where a superfluid mode canbe associated with a U(1)-like degeneracy (which indeedcan be lifted by fluctuations), or nearly degeneracy notcorresponding to a U(1) symmetry of the effective poten-tial. Phase fluctuations and vortices can originate from thisdegeneracy.In the scenario that we discuss here, a superfluid system

when heated goes into an entropically induced state wherean additional superfluid channel is opened. We alsoconsider entropically induced states that are not associatedwith additional superfluid channels, but possess propertiesdifferent from those of the ground state.Here we discuss certain multicomponent models because

of the substantial progress in creation and investigation ofthese systems recently.In the cold atoms field, interest is driven by the extensive

possibilities to realize multicomponent superfluid states.These systems allow tuning of various forms of intercom-ponent interactions with high precision. In the context ofsuperconductivity, the interest was driven recently bydiscoveries of superconductors with nontrivial multibandorder parameters.We start the discussion with the multicomponent

Ginzburg-Landau functional, although our results do not

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rely on the existence of the Ginzburg-Landau expansion inthe sense that the effective potential does not have to berepresented in this particular form of series in powers of ψ i.The free energy density is given by

F ¼X

i

j∇ψ ij22

þX

i;j

ηijψ iψ�j þ

X

i;j;k;l

νijklψ iψ jψ�kψ

�l ; ð1Þ

where summation is conducted over N components(complex fields ψ i). These components can originate fromCooper pairing in different bands (see, e.g., Refs. [17–22])or Josephson-coupled superconducting layers, or corre-spond to different components of Bose-Einstein-condensedcold atoms. The intercomponent interaction terms in Eq. (1)make up the first- and second-order Josephson couplings.Such couplings can be realized and controlled inJosephson-coupled multilayers and Josephson-junctionarrays. Intercomponent couplings of this kind can alsobe realized in cold atoms. This problem becomes especiallyrich in the four-component case, with features such as avariety of accidental degeneracies [20] appearing even atthe level of only bilinear couplings.Under certain conditions, there appear local minima of

the free energy [Eq. (1)]. This feature is also present in theLondon limit. Consider a three-component lattice Londonmodel containing only phases:

H ¼ −X

i;hj;kicosðφi

j − φikÞ þ

X

k

Vðφ12k ;φ13

k Þ; ð2Þ

where φijk ¼ φj

k − φik. The first term describes the gradient

energy and contains a summation over superconductingcomponents i, and nearest neighbors hj; ki. The secondterm contains a summation over all lattice points k, anddescribes the potential energy. We will consider threeexamples of different potentials, which are constructedto represent minimalistic models of the situations men-tioned above, with local minima in the free energy land-scape. The potentials are chosen so that they lock phasedifferences, explicitly breaking the symmetry down toU(1). The local minima correspond to a different phaselocking pattern with slightly higher energy.The potentials are displayed and defined in Fig. 1. Two

of them, V1 and V2, are specifically constructed asarchetypical representations of two kinds of energy land-scapes with strong features that facilitate numerical study,while V3 is built from interaction terms of the form ψ�

iψ jand ψ�

iψ�jψkψ l, which constitute the first- and second-order

Josephson harmonics. This should give rise to two states,one that is centered around the minimum, which we denote(↓) which can also have a degeneracy, and another statecentered around the local minimum, which we denote (↑).Another feature of these potentials is that they exhibit acomparatively steep slope around the global minimum,while the local minimum is surrounded by a flatter slope in

the energy landscape. Consequently, the energy cost ofphase-difference excitations is smaller in the state (↑) thanin (↓).We are interested in the role of phase fluctuations in

these systems. The fact that these are energetically cheaperin the state (↑) implies a lower free energy at a certaintemperature, resulting in a transition to a an entropicallystabilized state. The normal role of phase fluctuations is torestore the U(1) symmetry, but in models 1 and 3 thesituation is reversed as the state (↑) actually possesses anadditional superfluid mode.To investigate the entropically stabilized superfluid

state we have conducted Monte Carlo simulations basedon the metropolis algorithm with two types of update. Thefirst is a local update concerning only a local spin. Thesecond is a global update where one of the phases isselected and an update is proposed so that φi → φi þ Δπon all lattice points simultaneously. See also the remarkin Ref. [23].During the simulation several quantities were monitored.

The coherence of the individual phases was measured as

oγðTÞ ¼�����

1

L3

X

j

eiφγ;j

�����; ð3Þ

where the index 1 ≤ γ ≤ 3 denotes the different phases. Toidentify the two states we also introduce

S12 ¼ hcosðφ2 − φ1Þi: ð4Þ

In the state (↓) we should have S12 ∼ −1 while in the state(↑) we expect S12 ∼ 1. To determine the properties of thesector φ1;3, we introduce the following two functions.When the system is in a state where φ13 has an appreciablemass (which turns out to be the case in model 2 but in factalso in model 3) we calculate

S13 ¼ hcosðφ3 − φ1Þi; ð5Þ

which has a preferential value that reflects degeneracylifting. When the mode becomes superfluid (which occursin model 1), this is best illustrated by the correlator

C13 ¼1

N

Zdr½hXðrÞXð0Þi − hXðrÞihXð0Þi�;

X ¼ fcosφ13; sinφ13g; N ¼ LD: ð6Þ

Here superfluid modes are recognized as having nonzerocorrelations even in large systems. Consequently, in a largesystem C13 takes a finite value in a superfluid while itapproaches zero for a massive mode.The results of the simulations are summarized in Fig. 1.

All three models exhibit a transition to an entropicallystabilized state that is driven by nontopological phasefluctuations. These fluctuations are energetically cheaper

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in the state (↑), something that is reflected in the fact thatthe phase coherence (oi) at a given coupling strengthis smaller in this state [rows (A)–(C)]. The transition isassociated with a change of the parameter S12, which is∼ − 1 in the state (↓) and ∼þ 1 in (↑) [row (D)].The simplest situation takes place in model 2. It has two

minima: a deep and narrow well (↓) and a slightly moreshallow well with a flatter slope (↑) with no additional

degeneracy. Thus, in this case the phase fluctuations candrive a transition between two states with different phaselockings and different normal modes, giving raise tosubstantially different coherence lengths.In model 1, the ground state (↓) breaks U(1) symmetry,

while in the state (↑), the U(1)-like degeneracy is broken inaddition. This is reflected in the appearance of a nonzerocorrelator (C13) and thus entropically induced superfluidity.

(a)

(b)

(c)

(d)

(e)

FIG. 1 (color online). Top row: potential 1 exhibits a global minimum at φ12 ¼ φ13 ¼ π, and a local minimum corresponding toφ12 ¼ 0. The potential is given by V1ðφ12;φ13Þ ¼ tanhð3–5 cosφ12Þ þ 2.2 tanhð8 cosφ12 þ 8 cosφ13 þ 16Þ. Potential 2 possessesminima in the form of two wells: one deep but narrow situated in φ12 ¼ φ13 ¼ π, and one slightly more shallow, though wider, located inφ12 ¼ φ13 ¼ 0. The potential is given by V2ðφ12;φ13Þ ¼ 2.05 tanhð8 cosφ12 þ 8 cosφ13 þ 16Þ þ 2 tanhð2 − cosφ12 − cosφ13Þ.Potential 3 possesses a global minimum in φ12 ¼ φ13 ¼ π, and a local minimum corresponding to φ12 ¼ 0. The potential is givenby V3ðφ12;φ13Þ ¼ − cos 2φ12 − 0.18 cosφ12 þ 0.2ð1 − cosφ12Þ cosφ13. Rows (A)–(E): summary of the simulation results. The threecolumns correspond to potentials 1, 2, and 3. The first three rows (A)–(C) give the averages o1, o2, o3 [Eq. (3)] for system sizes8 ≤ L ≤ 40. These are zero if a system is in the normal state. Row (D) gives S12 [Eq. (4)] while row (E) shows the correlatorC13 [Eq. (6)]for model 1 and S13 [Eq. (5)] for models 2 and 3. In rows (D) and (E) data are only displayed for the system size L ¼ 40. In all plots thereare two curves corresponding to the two different states (↓, ↑). These two states coexist in part of the parameter range, particularly inmodels 1 and 2, due to hysteresis.

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Model 3 is similar to model 1 in the sense that it exhibitsa ground state where the energy is invariant under U(1)transformations as well as a local minimum correspondingto φ12 ¼ 0. In the local minimum the energy is independentof φ13, implying that it has an additional degeneracy.However, in this state the energy is only independent ofφ13 when φ12 ¼ 0 exactly, i.e., precisely at the bottom ofthe “valley.” At any nonzero temperature, thermal fluctua-tions render this mode massive (although with a compa-ratively very small mass). This is reflected in the fact thatS13 [shown in the third column of row (E)] is nonzero evenfor the state (↑). At finite length scale, this state, however,shares properties with a Uð1Þ × Uð1Þ superfluid.Importantly, in these types of systems, the transitions

between the states can occur not only because of thermalfluctuations, but also as a result of an applied phase twist.This can be demonstrated as follows. To simplify calcu-lations, consider a system similar to model 1, but with aninfinitely narrow well. Let the energy be 0 in the well, V inthe valley, and U everywhere else so that 0 < V < U.Suppose φ1;3 is twisted by N2π along a system of size L. Inthe state (↓) the twist results in kinks of width lw ¼π

ffiffiffiffiffiffiffiffiffi2=U

pand energy Ew ¼ 2π

ffiffiffiffiffiffiffi2U

p. In the state (↑) the

twist instead results in an evenly distributed phase gradient.The energy difference between the two solutions can thenbe written

E↑ − E↓ ¼ Lð−ρEw þ 2π2ρ2 þ VÞ; ρ ¼ NL: ð7Þ

This expression is valid provided 0 ≤ ρ ≤ 1=lw (since thesize of the kinks is lw). If ρ ¼ 0, this is positive and (↓) isthe lowest energy state. Inserting ρ ¼ 1=lw we insteadobtain E↑ − E↓ ¼ V −U < 0, and the lowest energy stateis (↑). Thus, above some critical phase twist ρc, it isenergetically beneficial to go to the superfluid state (↑)where the system can take advantage of the (near) degen-eracy in the energy landscape. Below it, (↓) becomes thepreferred state.Finally, we address the question of mass generation

through thermal fluctuations.The fact that the induced superfluid state does not

correspond to an underlying symmetry of the Hamiltonianmeans that fluctuations can result in this mode becomingmassive.The extent to which this occurs very much depends on

the shape of the potential, as well as the temperature. Inmodel 3 this phenomena is directly visible even at relativelylow temperature in a finite system, where it renders S13nonzero in the state (↑). In contrast, model 1 is unaffectedby this process within the accuracy of the simulations.When the fluctuations render the mode massive the systembehaves as a superfluid on length scales smaller than theinverse mass of the mode. Figure 2 shows the correlatorC13 at higher temperature for model 1 and system sizes

4 ≤ L ≤ 28. At these sizes the φ13 sector remains super-fluid until vortex proliferation takes place at β ≈ 0.45.In conclusion, we demonstrated the existence of the class

of systems that possesses a channel that is not superfluid inthe ground state but upon heating becomes superfluid due toentropy associated with nontopological phase fluctuations.We also discussed that such systems can have upper andlower critical superfluid velocities. The superfluid responsein the considered example is associated with a (near)degeneracy, which does not originate from a symmetry,yet it allows us to define a superfluid phase variable andquasitopological defects (vortices). It would be especiallyinteresting to realize such systems in cold atoms experiments.

We thank A. Kuklov, B. Svistunov, N. Prokof’ev,M. Wallin, and J. Lidmar for discussions. This workwas supported by the Knut and Alice WallenbergFoundation through a Royal Swedish Academy ofSciences Fellowship, by the Swedish Research Council,and by the U.S. National Science Foundation underCAREER Grant No. DMR-0955902. The computationswere performed using resources provided by the SwedishNational Infrastructure for Computing (SNIC) at theNational Supercomputer Center in Linkoping, Sweden.

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1.0 1.5 2.0 2.5

0.2

0.4

0.6

0.8L=4

L=28

Cor

rela

tor

C13

L=4L=6L=8L=10L=12L=16L=20L=24

Inverse Temperature

FIG. 2 (color online). Correlator C13 for model 1 at highertemperature. System sizes range from 4 to 28. The point on thehorizontal axis β ≈ 0.45, where C13 approaches zero, correspondsto the transition where superfluidity is destroyed by vortexproliferation.

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[10] E. Babaev, arXiv:cond-mat/0201547.[11] E. Babaev, A. Sudbo, and N.W. Ashcroft, Nature (London)

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[22] A. Gurevich, Physica (Amsterdam) C456, 160 (2007).[23] The updates are accepted with probability e−βΔE. For each

model, 256 different coupling strengths (inverse temper-atures) were selected. For each coupling strength, twosimulations corresponding to the two possible states(↑, ↓) were prepared. The equilibration was conducted intwo stages. First, an initial 50 000 local updates were carriedout (i.e., each lattice point was subject to 50 000 proposedupdates). Second, another 50 000 updates were carried out.But this time, every three updates, a global update wasproposed. After that, the equilibration, data was collected asfollows. Local updates were proposed 200 times for everylattice point. For every three such updates, a global updatewas proposed. Then, simulation data was collected. Thisprocess was repeated 28 000–100 000 times depending onsystem size, resulting in the same number of data points.The simulations were conducted on a cubic lattice withperiodic boundary conditions and system sizes 8 ≤ L ≤ 40.Finally, points in the parameter space where the systemkeeps tunneling between the two states were discarded (thismainly occurred for smaller systems in model 3).

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