Thermodynamics of Statistical Anyons
TThermodynamics of Statistical Anyons
Nathan M. Myers ∗ and Sebastian Deffner
1, 2, † Department of Physics, University of Maryland Baltimore County, Baltimore, MD 21250, USA Instituto de F´ısica ‘Gleb Wataghin’, Universidade Estadual de Campinas, 13083-859, Campinas, S˜ao Paulo, Brazil (Dated: February 23, 2021)In low-dimensional systems, indistinguishable particles can display statistics that interpolate be-tween bosons and fermions. Signatures of these “anyons” have been detected in two-dimensionalquasiparticle excitations of the fractional quantum Hall effect, however experimental access to thesequasiparticles remains limited. As an alternative to these “topological anyons,” we propose “statisti-cal anyons” realized through a statistical mixture of particles with bosonic and fermionic symmetry.We show that the framework of statistical anyons is equivalent to the generalized exclusion statistics(GES) pioneered by Haldane, significantly broadening the range of systems to which GES apply. Wedevelop the full thermodynamic characterizations of these statistical anyons, including both equilib-rium and nonequilibrium behavior. To develop a complete picture, we compare the performance ofquantum heat engines with working mediums of statistical anyons and traditional topological anyons,demonstrating the effects of the anyonic phase in both local equilibrium and fully nonequilibriumregimes. In addition, methods of optimizing engine performance through shortcuts to adiabaticityare investigated, using both linear response and fast forward techniques.
I. INTRODUCTION
A unique aspect of quantum particles is that they maybe truly identical - that is to say there exists no methodor feature by which we would be able to distinguish onefrom another. The linear nature of quantum mechan-ics accounts for this in the form of a superposition statefor the system of all possible permutations of individ-ual particle states [1]. Such a composition is not unique,with two possible solutions distinguished by a phase of ± exchange forces . A simpleevaluation of the separation distance between two identi-cal particles (see for example [1]) shows that bosons willtend to bunch together, while fermions will tend to befound farther apart. In fact, for fermions the probabilityof two particles being found in the exact same state isidentically zero, which is the origin of the familiar Pauliexclusion principle.The field of quantum statistics was significantly ex-panded when it was discovered by Leinaas and Myrheimthat the differences in topology in one- and two-dimensional systems allows for the existence of a con-tinuum of fractional statistics solutions, represented by ageneral phase factor of e iπν [2],Ψ( r , r ) = e iπν Ψ( r , r ) . (1)For ν = 2 n , where n = 0 , , , ... , the bosonic case is re-covered, and for ν = 2 n + 1 we obtain the fermionic case. ∗ [email protected] † deff[email protected] Shortly thereafter Wilczek proposed a realization of these“any”-ons in the form of a two-dimensional quasiparticlemade up of a charged particle orbiting a magnetic fluxtube [3]. The name anyons signifies that interchangeof particles (accomplished by successive half-rotations ofeach quasiparticle around the other) can produce an ar-bitrary phase between that of fermions and bosons as aconsequence of the Aharonov-Bohm effect [3].The study of anyons was brought into the forefrontwhen it was proven by Arovas, Schrieffer, and Wilczekthat quasiparticles entering the fractional quantum Halleffect posses not only fractional charge, but also obeyfractional statistics [4]. Interest in anyons received an-other boost when it was found that non-abelian anyons[5] could be used as key components in the developmentof a fault-tolerant quantum computer. For non-abeliananyons, exchange (“braiding”) does not just introducea complex phase, but acts as a unitary transformationon the state. In this manner, combinations of braidscan act as quantum logic gates [6]. As small local per-turbations do not change the braiding properties of theanyons, this method of quantum computation is veryrobust against noise [6]. Experimental access to non-abelian anyons remains a challenge, however indirect ev-idence of non-abelian anyonic states has been found [7–9].See Ref. [10] for a more detailed review on topologicalquantum computation and the role of anyons. While non-abelian anyons are necessary for implementing topologi-cal quantum computation, understanding the propertiesof abelian anyons can provide important insight into thegeneral behavior of anyons that may be useful in control-ling their non-abelian counterparts.Discussion of anyons in one-dimension requires someadditional subtlety, as exchange by rotation is no longervalid. In this context, Haldane introduced a dimension-independent approach to anyonic statistics, generalizedexclusion statistics (GES), based on a generalization ofthe Pauli exclusion principle [11]. In Haldane’s approach, a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b particle nature is quantified by a statistical interactionthat determines the degree to which two identical par-ticles can occupy the same state [11]. Notably, this ap-proach classifies particle type solely on degree of repulsionand as such identifies “hardcore-bosons” and fermionsas equivalent, despite differences in low-energy behavior[11]. While closely linked to Wilczek’s formulation ofanyons, Haldane’s GES anyons are distinctly different,as their behavior is based on generalizing the exclusionprinciple, rather than the exchange behavior [12].Haldane’s work was expanded upon by Wu, who de-rived the statistical distribution for an ideal anyon gas[13]. This approach continued to receive attention whenit was shown by Yang and Yang that a 1D system of in-teracting bosons could be treated as non-interacting 1Danyonic system obeying Haldane’s generalized exclusionstatistics [14, 15]. Similar mappings to interacting any-onic systems have been demonstrated for more complexpotentials [16].While the theoretical study of anyons is well-established, it is only in recent years that experimentaltechniques have advanced to the point that detection ofanyons is feasible. For the case of fractional quantum Hallstate anyons, a popular detection method is based on im-plementing quasiparticle interferometers whose interfer-ence effects depend directly on the anyonic phase [17–20].Numerous experimental attempts have followed this ap-proach [21–26] but distinguishing signatures of the any-onic phase from interference effects arising from other fac-tors, such as Coulomb blockading and Aharonov-Bohminterference, has proven elusive until this year [27].The difficulty in accessing fractional quantum Hallstate anyons has led to alternative methods of experi-mentally studying their properties. A promising routethat we will focus on in this work is motivated by thecelebrated Hong-Ou-Mandel (HOM) effect first observedin photonic interferometry [28].In section II we establish an implementation of anyonsfrom a statistical mixture of boson and fermion pairs,which we refer to as “statistical anyons.” We show thatstatistical anyons, despite different construction, behaveequivalently to Haldane’s generalized exclusion principleanyons. In section III we examine the equilibrium ther-modynamic properties of statistical and abelian topologi-cal anyons, such as those proposed by Leinaas, Myrheim,and Wilczek, including entropy, heat capacity, and freeenergy. In section IV we compare the performance ofan endoreversible quantum Otto engine with a work-ing medium of statistical anyons to one with a workingmedium of topological anyons. In section V, we extendour analysis of the statistical anyon engine to the fullynonequilibrium regime. In section VI we explore the roleof anyonic statistics in the implementation of shortcuts toadiabaticity , which can be used to enhance thermal ma-chine performance by increasing power without loss ofefficiency. In section VII we conclude with a perspectiveon future research directions. II. “STATISTICAL” ANYONS
In the typical HOM effect a pair of entangled photonsare incident symmetrically on a two-port 50/50 beam-splitter. Assuming all other degrees of freedom of thephotons are identical (such as frequency and polariza-tion), the initial bosonic spatial state is described by asymmetric superposition of each input, (cid:12)(cid:12) ψ Bi (cid:11) = 1 √ (cid:16) | a (cid:105) | b (cid:105) + | b (cid:105) | a (cid:105) (cid:17) . (2)The operation of the beamsplitter evolves state | a (cid:105) → √ ( | c (cid:105) + i | d (cid:105) ) and state | b (cid:105) → √ ( i | c (cid:105) + | d (cid:105) ) withthe imaginary component denoting the phase shift of π picked up upon reflection. Carrying out this evolutionon each input state while keeping track of the particleindices we find that the only states to survive are thosein which both photons exit the same beamsplitter port, (cid:12)(cid:12) ψ Bf (cid:11) = i √ (cid:16) | c (cid:105) | c (cid:105) + | d (cid:105) | d (cid:105) (cid:17) . (3)Physically, this is a manifestation of the effective attrac-tion between bosons (typically this is referred to as “bo-son bunching”) [28].The HOM effect can be extended to cases in which theother degrees of freedom of the photons are not identical[29]. Consider a case in which the photons are preparedin a Bell state basis in polarization. The four possibleBell pairs are, | Φ A (cid:105) = 1 √ (cid:16) | (cid:105) + | (cid:105) (cid:17) , | Φ B (cid:105) = 1 √ (cid:16) | (cid:105) − | (cid:105) (cid:17) , | Φ C (cid:105) = 1 √ (cid:16) | (cid:105) + | (cid:105) (cid:17) , | Φ D (cid:105) = 1 √ (cid:16) | (cid:105) − | (cid:105) (cid:17) , (4)where here | (cid:105) and | (cid:105) represent orthogonal polarizationstates. We can use this additional degree of freedom toencode fermionic, and ultimately anyonic, statistics intothe behavior of the photons [30–33]. In this manner thephotons can be thought of as a “quantum substrate” onwhich we will construct our desired statistics.We note that | Φ A (cid:105) , | Φ B (cid:105) , and | Φ C (cid:105) are symmetric un-der exchange, while | Φ D (cid:105) is antisymmetric. As photonsare bosons, their overall wave function must still be sym-metric. This can occur in one of two ways: (i) a symmet-ric spatial state paired with one of the three symmetricpolarization states, (cid:12)(cid:12) Ψ Bi (cid:11) = 1 √ (cid:16) | a (cid:105) | b (cid:105) + | b (cid:105) | a (cid:105) (cid:17) ⊗ | Φ j (cid:105) , (5)where j = { A, B, C } , or (ii) an antisymmetric spatialstate paired with the antisymmetric polarization state, (cid:12)(cid:12) Ψ Fi (cid:11) = 1 √ (cid:16) | a (cid:105) | b (cid:105) − | b (cid:105) | a (cid:105) (cid:17) ⊗ | Φ D (cid:105) , (6) ab cd (i) ab cd (ii) ab cd (iii) ab cd (iv) (a) ab cdab cd (b) FIG. 1. (a) All possible photon transmission and reflectioncombinations. For symmetric (bosonic) states possibilities (i)and (iv) interfere deconstructively while for antisymmetric(fermionic) states they interfere constructively. (b) Represen-tation of statistical anyons for N = 5. Top figure: N particlepairs are incident, with blue particles representing pairs withsymmetric entanglement and red pairs with antisymmetricentanglement. Bottom figure: The equivalent statistical any-onic representation for each particle pair, with particle sizerepresenting the probability density of finding a particle atthat output port. Assuming it is non-polarizing, the beamsplitter acts onlyon the spatial portion of the wavefunction. For case (ii),as shown in Eq. (6), the output spatial state is, (cid:12)(cid:12) ψ Ff (cid:11) = 1 √ (cid:16) | c (cid:105) | d (cid:105) − | d (cid:105) | c (cid:105) (cid:17) , (7)guaranteeing that both photons exit opposite ports ofthe beamsplitter. This is a result of the effective repul-sion between fermions, and can be seen as a manifesta-tion of the Pauli exclusion principle. Despite the factthat photons are fundamentally bosons, since the beam-splitter only accesses the antisymmetric portion of theoverall state, they behave exactly as fermions. Figure1a summarizes the possible outcomes for the two-photoninput states. Notably, both bunching and anti-bunchingbehavior vanishes if the photons become distinguishable(say by increasing the flight time for just one port of thebeamsplitter) [29].This effect can be extended to anyonic statistics by in-troducing a phase that tunes between the symmetric andantisymmetric Bell states. Phase control can be achievedusing a rotated polarization beamsplitter [33], adjustingpath length [30, 31], or introducing phase plates [32],making this method of simulating anyonic behavior veryexperimentally accessible. Here we propose introducingthis anyonic phase through a statistical mixture of sym-metrically and antisymmetrically entangled photon pairs.Given N photon pairs incident on the beam splitter, theaverage behavior of each photon pair can then be repre- sented by an anyonic wave function in which the anyonicphase gives the probability that any two incident parti-cles will exit the same port, p B , (cid:12)(cid:12) ψ Ai (cid:11) = 1 √ (cid:16) | a (cid:105) | b (cid:105) + e iπν ( p B ) | b (cid:105) | a (cid:105) (cid:17) . (8)This behavior is shown pictorial in Fig. 1b.We refer to this implementation as “statistical anyons”in contrast to “topological anyons,” that appear inWilczek’s charge and flux tube realization [3], or in quasi-particle excitations in the fractional quantum Hall regime[4]. While both formulations pick up an anyonic phaseunder particle exchange and display behavior interpolat-ing between that of fermions and bosons, in the presentstatistical anyon framework this is a result of averagingover the behavior of a large number of particles, whilefor topological anyons it is a property of the quasiparti-cles themselves. Note that when we refer to topologicalanyons, we are considering specifically abelian particles,whose exchange introduces a phase a e iπν [34].From a topological standpoint, there is another im-portant distinction between statistical and topologicalanyons. It is well established that the configuration spaceof topological anyon statistics is a representation of thebraid group, rather than the permutation group, as isthe case for bosonic and fermionic statistics [2, 34]. Thisdifference arises as in two dimensions, two repeated ex-changes is a topologically distinct operation from doingnothing, while in three dimensions these operations aretopologically equivalent [2, 34]. This leads to the phys-ical consequence that the direction of the exchange issignificant for topological anyons, as two exchanges inthe same direction will result in a wavefunction differentfrom the original by a phase of e iπν . Since statisticalanyons are constructed from an average over a mixture ofbosons and fermions, whose permutation group statisticsensures that two repeated exchanges will always returnthe original wavefunction, the anyonic phase of statis-tical anyons is independent of exchange direction. As aresult, the space of statistical anyon phases is of a dimen-sion smaller than the space of topological anyon phases.We can represent the two-dimensional space of possibletopological anyon phases as a circle, with bosonic andfermionic phases given by diametrically opposite points.The space of possible statistical anyon phases is then rep-resented by the diameter line of the circle. In this geo-metric picture, the statistical anyon phase correspondingto the topological anyon phase of e iπν would be the pro-jection of that radial vector onto the diameter line. Thesedifferences between topological and statistical anyons areillustrated graphically in Fig. 2.We see from this comparison that, as a representationof the braid group, topological anyons show increasedcomplexity in comparison to statistical anyons. Thisleads to the motivating questions of this work – How doesthe thermodynamics of topological and statistical anyonscompare? Can the simpler, more experimentally accessi-ble statistical anyons replicate any of the unique proper- +1−1 𝑒 !" 𝑒 !" ! ) (a) 𝑒 !" ! ) 𝜓 ’ 𝜓 ( 𝜓 ( 𝜓 ’ 𝜓 ( 𝜓 ’ 𝑒 !" 𝜓 ’ 𝜓 ( 𝜓 ( 𝜓 ’ 𝜓 ( 𝜓 ’ 𝑒 !" 𝜓 ’ 𝜓 ( 𝜓 ( 𝜓 ’ 𝑒 ’!" 𝜓 ( 𝜓 ’ (b) FIG. 2. (a) Comparison between the exchange phases of topological and statistical anyons. For topological anyons thedirection of the exchange (clockwise or counterclockwise) determines the sign of the anyonic phase. As such, the space ofpossible topological anyon exchange phases can be represented by a circle, with bosonic and fermionic symmetry lying onopposite sides. For statistical anyons, the anyonic phase is an average over N B bosonic phase factors and N F fermionic phasefactors. As such, the space of possible statistical anyon phases is confined to a line between − e iπν (left diagram), while two exchanges inopposite directions leaves the wavefunction unchanged (middle diagram). For statistical anyons the direction of the exchangeis irrelevant, with two exchanges always resulting in an unchanged wavefunction (right diagram). ties of topological anyons? Do statistical anyons them-selves display intricate thermodynamic behavior that canbe exploited? A. Statistical Anyon Wavefunction
To better understand the relationship between statisti-cal and topological anyons, let us consider extending thenotion of statistical anyons beyond the realm of quantumoptics. For a single pair of bosonic particles the spatialwave function is given by,Ψ B ( x ) = 1 (cid:112) δ n ,n ) [ ψ n ( x ) ψ n ( y ) + ψ n ( y ) ψ n ( x )] , (9)where ψ n ( x ) is the normalized single-particle eigen-state corresponding to quantum number n and x =( x, y ). Note that the normalization coefficient for bosonschanges when both particles occupy the same state, as inthis case the un-normalized wavefunction becomes iden-tical to the wavefunction of two distinguishable particleswith an additional factor of two. Similarly, the wavefunction of a fermionic particle pair is,Ψ F ( x ) = 1 √ ψ n ( x ) ψ n ( y ) − ψ n ( y ) ψ n ( x )] . (10)Here we require no delta function in the normalization, astwo fermions will never occupy the same state. The totalwave function for N independent particle pairs, of which N B of the pairs are symmetric under exchange and N F = N − N B of the pairs are antisymmetric under exchangeis then,Ψ( x , x , ..., x N ) = N B (cid:89) j =1 Ψ B ( x j ) N (cid:89) k = N B +1 Ψ F ( x k ) . (11)This wave function can be equivalently represented in theframework of statistical anyons as,Ψ( x , x , ..., x N ) = N (cid:89) j =1 Ψ A ( x j ) , (12)where,Ψ A ( x j ) = 1 (cid:112) δ n ,n ) [ ψ n ( x j ) ψ n ( y j )+ e iπν j ψ n ( y j ) ψ n ( x j )] . (13)We pause here to note a significant difference betweenstatistical and topological anyons. As we see in Eq. (13),the statistical anyon wave function can always be con-structed from a superposition of the single particle wavefunctions. This is not generally true for the wave func-tion of topological anyons, outside of the bosonic andfermionic limits of the anyonic phase [34–36].The product in Eq. (12) must give N B symmetric wavefunctions and N F antisymmetric wave functions. Thiscondition is fulfilled if the anyonic phase in Eq. (13) isgiven by ν j = Θ ( j − N B − · ) is the Heav-iside step function, using the convention Θ(0) = 1. Inthis framework, the anyonic factor in Eq. (13) can bethought of as giving the “average” phase picked up underexchange for any one of the N pairs of particles. For large N we can express the number of bosonic and fermionicparticle pairs as N B = N p B and N F = N p F = N (1 − p B )where p B ( p F ) is the probability of a particle pair havingbosonic (fermionic) symmetry.In the limit of a single two-particle system, the anyonicwave function becomes,Ψ A ( x ) = 1 (cid:112) δ n ,n ) (cid:104) ψ n ( x ) ψ n ( y )+ e iπ ( p B +1) ψ n ( y ) ψ n ( x ) (cid:105) , (14)where p B = { , } , as in reality each individual pair ofparticles must be either bosons or fermions. B. Statistical Anyons in Second Quantization
The statistical anyon framework can also be extendedto second quantization, from which we can determinethe appropriate commutation relations. For topologicalanyons, the creation and annihilation operators can bedetermined from the bosonic or fermionic operators viathe Jordan-Wigner transformation, and obey the follow-ing commutation relations [37], a ( x C ) a ( y C ) − e − iπν a ( y C ) a ( x C ) = 0 ,a ( x C ) a † ( y C ) − e iπν a † ( y C ) a ( x C ) = 0 , (15) a † ( x C ) a ( y C ) − e iπν a ( y C ) a † ( x C ) = 0 .a † ( x C ) a † ( y C ) − e − iπν a † ( y C ) a † ( x C ) =0 , Note that the commutation relations are dependent onthe curve C , which indicates the direction of the exchangerotation. As the topological anyon operators are a repre-sentation of the braid group, the phase picked up dependson whether or not the exchange was performed clockwiseor counterclockwise (as illustrated in Fig. 2). We seethat ν = 0 restores the boson commutation relations and ν = 1 restores the fermion anticommutation relations.Notably, for particles at the same position the anyonicphase cancels out and the canonical commutation rela-tion reduces to [36, 37], a ( x C ) a † ( x C ) ± a † ( x C ) a ( x C ) = 1 , (16)with the (+) occurring if the anyonic operators are con-structed from transformed fermionic operators, and the( − ) if they are constructed from transformed bosonic op-erators. In general, topological anyons act as “hard-core”particles for all ν other than ν = 0, obeying an exclusionprinciple [36]. In this sense, the anyonic phase parametercan be thought of as quantifying the degree of repulsionbetween two identical particles [36].To determine the commutation relations for statisti-cal anyon creation and annihilation operators, we follow a similar approach to the construction of the statisti-cal anyon wave function in the first quantization. Letus consider N Fock states. Starting each in the vacuumstate, we can construct a single occupancy state from theapplication of the corresponding creation operator. Weapply the bosonic creation operator to N B states and thefermionic creation operator to N F = N − N B states, | (cid:105) | (cid:105) ... | (cid:105) N = N B (cid:89) j =1 b † j | (cid:105) j N (cid:89) k = N B +1 f † k | (cid:105) k . (17)We can equivalently represent this state in the statisticalanyon picture using a statistical anyon creation operator, | (cid:105) | (cid:105) ... | (cid:105) N = N (cid:89) j =1 s † j | (cid:105) j . (18)The operator s † j must reduce to the bosonic creation op-erator for j ≤ N B and to the fermionic creation operatorfor j > N B . We can similarly define a statistical anyonannihilation operator that must reduce to the bosonicand fermionic annihilation operators under the same con-ditions. With these restrictions, we can construct thestatistical anyon commutation relations as follows, s † j s j − e iπ Θ( j − N B − s j s † j = 1 ,s j s † j − e iπ Θ( j − N B − s † j s j = 1 , (19) s † j s † j − e iπ Θ( j − N B − s † j s † j = 0 ,s j s j − e iπ Θ( j − N B − s j s j = 0 . In the limit of a single particle the canonical commutationreduces to, s j s † j + e iπ ( p B +1) s † j s j = 1 , (20)where p B = { , } , as in reality the particle is eithera boson or fermion. This echoes Eq. (16), where theform of the commutation relation depends on whetherthe anyonic operators are constructed from transformedfermionic or bosonic operators. C. Statistical anyons and Generalized ExclusionStatistics
A notable difference between statistical and topologicalanyons is the lack of an exclusion principle, except in thefermionic limit. Instead, statistical anyons admit “par-tially occupied” states that arise from averaging over theoccupancy of all N systems. In this sense, the statisticalanyon framework is more akin to Haldane’s generalizedexclusion statistics [11]. GES is constructed by extend-ing the Pauli exclusion principle through the definitionof a parameterized differential relation that quantifies thechange in the dimension of the Hilbert space of a discrete-state system upon a change in the particle number [11],∆ d GES = − g ∆ N. (21)For bosons, with infinite possible state occupancy, thedimension is independent of the particle number, thus g = 0. For fermions, subject to the full exclusion prin-ciple, the dimension scales directly with each additionalparticle, thus g = 1. GES statistics anyons have beenshown to manifest in confined, interacting gasses, such asthe Calogero-Sutherland model gas, consisting of bosonsor fermions in a harmonic potential with an inversesquare law interaction [38–41]. Notably, this implementa-tion of GES anyons can be directly mapped to topologicalanyons confined to the lowest Landau level [13, 42, 43].Other physical systems shown to host GES anyons in-clude Lieb–Liniger and hard core Tonks-Girardeau gasseswith δ -function potentials [16, 44–54] as well as Hubbardchains [55, 56], whose anyonic behavior can be imitatedin ultracold gasses [57].In the statistical anyon framework, the change in theHilbert space dimension will be given by the sum of thedimension change if the particles are bosons and the di-mension change if they are fermions, weighted by therespective probabilities,∆ d SA = p B ∆ d B + p F ∆ d F (22)Noting ∆ d B = 0 and ∆ d F = − ∆ N , this simplifies to,∆ d SA = − p F ∆ N. (23)Comparing Eq. (23) to Eq. (21) we see that the param-eter g in the GES framework is identical to the fermionprobability in the statistical anyon framework. Thisdemonstrates that the statistical anyon framework is fullyequivalent to GES anyons.For bookkeeping ease, we conclude this section by com-paring the features of topological, statistical, and GESanyons in Table I. III. EQUILIBRIUM THERMODYNAMICS OFANYONSA. 1D Statistical Anyons
The first step in understanding the thermodynamicsof anyonic systems is to examine how the thermody-namic quantities such as internal energy, entropy, andheat capacity depend on the anyonic phase. To de-termine this, we need the proper partition function forour system. Motivated by the manifestation of GES intrapped, interacting gasses, let us consider a gas of twostatistical anyons in a 1D harmonic potential. However,in this case we will consider the particles to be non-interacting . Trapped boson-fermion mixtures have seenprevious study both theoretically [58–60] and experimen-tally [61–63], but this work has focused primarily on ef-fects on the ground state configurations that arise frominteractions between the boson and fermion species. Inthe statistical anyon framework we instead consider the average collective behavior that arises in the ideal gaslimit of such a mixture.The Hamiltonian for our system of harmonically con-fined non-interacting statistical anyons reads, H = p + p m + 12 mω ( x + x ) . (24)In the statistical anyon framework the partition functionfor a pair of anyons evolving under this Hamiltonian is, Z SA = ( Z B ) p B ( Z F ) − p B , (25)where Z B is the partition function for two bosons and Z F is the partition function for two fermions. See Ap-pendix A for a full derivation of the partition function.Note that Eq. (25) is identical to that of a GES Calogero-Sutherland model gas [38]. This is further demonstrationof the equivalence of statistical and GES anyons, how-ever, we see that for statistical anyons the anyonic be-havior arises purely out of the average properties, ratherthan from an interaction term in the Hamiltonian.Using Eq. (25) we can determine the internal energy,free energy, entropy, and heat capacity as follows, E = − ∂∂β ln( Z SA ) , F = − β ln( Z SA ) , (26) S = k B β ∂F∂β , C = − k B β ∂E∂β , where β is the inverse temperature and k B is Boltzmann’sconstant. Plugging Eq. (25) into Eq. (26) we find, E = 12 (cid:126) ω [3 coth( β (cid:126) ω ) + csch( β (cid:126) ω ) − p B + 1] F = 1 β ln (cid:20)
18 csch ( β (cid:126) ω −
14 csch( β (cid:126) ω ) (cid:21) − p B (cid:126) ωS = 12 k B β (cid:126) ω [3 coth( β (cid:126) ω ) + csch( β (cid:126) ω ) + 1] (27)+ k B ln (cid:20)
18 csch ( β (cid:126) ω −
14 csch( β (cid:126) ω ) (cid:21) C = 12 k B β (cid:126) ω csch ( β (cid:126) ω ) [cosh( β (cid:126) ω ) + 3]Plots of each as a function of temperature are shownin Fig. 3. We see that both the internal energy andfree energy are shifted by a constant proportional to p B .Physically this offset arises from the generalized exclusionprinciple, as outlined in Section II. The constant shiftsthe lowest energy state of the system from the bosoniclimit, with both particles in the ground state of the os-cillator, to the fermionic limit, with one particle in theground state and the other in the first excited state.We find that the dependence on p B cancels out ex-actly in the entropy and heat capacity, leaving them in-dependent of the anyonic phase. This is expected, as inthe thermal equilibrium state both fermions and bosonshave an equivalent, countably infinite, number of avail-able states. Note that the behavior of the heat capacity is TABLE I. Comparison of the features and behavior of topological, statistical, and GES anyons.
Dim. Rep. Group Exclusion Principle Origin of Anyonic BehaviorTopological
2D Braid (abelian) Hard core for all ν (cid:54) = 0 Complex phase introduced by exchange Statistical
Arbitrary Permutation Generalized exclusion Statistical average of bosonic and fermionic behavior
GES
Arbitrary Permutation Generalized exclusion Generalization of state occupancy arising frominterparticle interactions / βℏω / ℏω (a) / βℏω - - - - - / ℏω (b) / βℏω / k B (c) / βℏω / k B (d) FIG. 3. Equilibrium (a) internal energy, (b) free energy, (c) entropy, and (d) heat capacity for two statistical anyons in onedimension with anyonic phase corresponding to p B = 1 (blue, dashed), p B = 1 / p B = 1 (green, dot-dashed). consistent with that of a one-dimensional, two-oscillatorEinstein solid [64]. However, since topological anyons donot exist in one dimension, in order to properly com-pare their thermodynamic behavior to that of statisticalanyons, we must extend the above analysis to two dimen-sions. B. 2D Statistical Anyons
We can repeat the thermodynamic analysis for two sta-tistical anyons in a two-dimensional harmonic potential.To avoid clutter, we give the full expressions for the in-ternal energy, free energy, entropy, and heat capacity inAppendix B. We plot each as a function of temperaturein Fig. 4. In contrast to the one-dimensional case, we seethat now the entropy and heat capacity do depend on p B .The origin of this difference is clear if we think of the en-tropy for bosons and fermions in the zero-temperaturelimit. For bosons, only one configuration is available –both particles in the ground state in both dimensions.However, the fermion ground state is degenerate. ThePauli exclusion principle requires that one particle mustbe in the ground state and the other in the first excitedstate, but the excited state can be in either dimension.This results in a non-zero entropy at T = 0. The en-tropy of generic statistical anyons interpolates smoothlybetween the boson and fermion limits as p B changes fromone to zero. C. 2D Topological Anyons
The partition function for two topological anyons in atwo-dimensional harmonic potential has been previouslyderived (see for example [34, 35]). It is given by, Z TA = e − β (cid:126) ω (2+ ν ) + e − β (cid:126) ω (4 − ν ) (1 − e − β (cid:126) ω ) ( e − β (cid:126) ω ) , (28)where ν is the anyonic phase. The first thing we noteis that, unlike the statistical anyon/GES anyon partitionfunction, the topological anyon partition function can-not be expressed as a function of the boson and fermionpartition functions. This arises from the increased com-plexity of the topological anyon wave function. In the twoanyon problem, the the anyonic phase can be consideredas a shift in the value of the relative motion angular mo-mentum [36]. However, accounting for the fact that thephase depends on the direction of rotation (a considera-tion unique to the braid group-based topological anyons)leads to a multi-valued wave function [35]. Accountingfor both branches of the wave function results in the twoseparate anyonic phase-dependent terms seen in the nu-merator of Eq. (28).Following the same process as for the statisticalanyons, we determine the equilibrium internal energy,free energy, entropy, and heat capacity for the two anyonsystem, with the full expressions given in Appendix B.Plots of each as a function of temperature are given inFig. 5. Here we see qualitatively similar behavior tothe two-dimensional statistical anyons, with some no-table differences. We see that the entropy of topologicalanyons converges to zero in the zero-temperature limit,indicating the existence of a unique ground state config-uration for all values of ν except ν = 1, corresponding / βℏω / ℏω (a) / βℏω - / ℏω (b) / βℏω / k B (c) / βℏω / k B (d) FIG. 4. Equilibrium (a) internal energy, (b) free energy, (c) entropy, and (d) heat capacity for two statistical anyons in twodimensions with anyonic phase corresponding to p B = 1 (blue, dashed), p B = 1 / p B = 1 (green, dot-dashed). / βℏω / ℏω (a) / βℏω - / ℏω (b) / βℏω / k B (c) / βℏω / k B (d) FIG. 5. Equilibrium (a) internal energy, (b) free energy, (c) entropy, and (d) heat capacity for two topological anyons withanyonic phase corresponding to ν = 0 (blue, dashed), ν = 1 / ν = 1 (green, dot-dashed). to pure fermions. Another significant discrepancy is seenin the behavior of the heat capacity. For intermediatevalues of the anyonic phase we see that, at low tempera-tures, the topological anyon heat capacity is higher thanthat of both the bosonic and fermionic values.Comparing Figs. 4 and 5 we see that, in each plot, forthe statistical anyons the line corresponding to p B = 1 / ν = 1 / ν [35, 36]. In the ground state, the energy eigenvaluescorresponding to the two branches of the wave functiononly coincide in the fermionic limit, making ν = 1 theonly degenerate ground state and giving rise to the ob-served zero-temperature limit of the entropy. However,as temperature increases, the hard-core nature of topo-logical anyons biases their thermodynamic behavior moretowards the fermionic limit. In contrast, the statisticalanyon energy spectrum is a much simpler weighted aver-age over the bosonic and fermionic spectrums, leading tothermodynamic behavior that is evenly spaced betweenthe bosonic and fermionic limits for p B = 1 /
2. This hasnotable ramifications for thermodynamic applications ofanyons, as statistical anyons will retain their “intermedi-ate” behavior at higher temperature regimes.
D. Thermodynamic Equivalence
In this section we have seen that statistical and topo-logical anyons display different thermodynamic behavior.This brings up the question: Is it possible to mimic thericher behavior of topological anyons using the mathe-matically and experimentally simpler framework of sta-tistical anyons?To determine the relation between the topologicalanyon parameter ν , and the statistical anyon parame-ter p B we can set their partition functions equal to eachother and solve for p B . This yields, p B = ln [cosh(( ν − β (cid:126) ω )]ln [cosh( β (cid:126) ω )] . (29)We see that in order to capture the more complicatedthermodynamic behavior p B becomes dependent on thetemperature and frequency parameters. If we take thehigh temperature limit of Eq. (29) we find a simpler,parameter-independent relation, p B = ( ν − . (30)We note that the approximate relation becomes exact for ν = 0 ,
1, as the behavior of the statistical and topologicalanyons must converge to the same bosonic and fermioniclimits. Notably, previous comparisons between Haldane’sGES parameter and the topological anyon phase us-ing the second Virial coefficient have also determined aquadratic polynomial relation [12].Plotting the internal energy, free energy, entropyand heat capacity for topological anyons and statisticalanyons using Eq. (30) in Fig. 6 we see that the statis-tical and topological anyon behavior rapidly converges.We note that the parameter-independent approximationdoes not capture the low-temperature behavior of thetopological anyon heat capacity. To fully imitate thisbehavior, we must use Eq. (29).The ability to mimic the thermodynamic properties oftopological anyons using statistical anyons has importantramifications from an experimental standpoint. The dif-ficulty of detecting and manipulating topological anyonsin two-dimensional materials makes probing their ther-modynamics exceedingly challenging. Statistical anyonsprovide an straightforward alternative, both as a modelto test thermodynamic control of topological anyons oras a replacement in applications that would rely on theirthermodynamics properties.
IV. ENDOREVERSIBLE ANYONIC ENGINE
Having established the equilibrium thermodynamic be-havior of both statistical and topological anyons, we nowcontinue our exploration of their behavior in the contextof heat engines. Studying the thermodynamic propertiesof a system using the framework of cyclic heat engineshas a rich tradition as old as thermodynamics itself [65].In equilibrium thermodynamics the optimal efficiency ofany heat engine cycle is bounded by the Carnot efficiency,regardless of the properties of the working medium [66].However, this efficiency is obtained in the limit of in-finitely slow, quasistatic strokes, resulting in zero poweroutput. A figure of merit of more practical use, the efficiency at maximum power (EMP), was introducedby Curzon and Ahlborn using the framework of endore-versible thermodynamics [67–69]. Curzon and Ahlbornfound the EMP of a endoreversible Carnot engine to be, η CA = 1 − (cid:114) T c T h (31)where T c ( T h ) is the cold (hot) reservoir temperature [67].In endoreversible thermodynamics the system is as-sumed to be in a state of local equilibrium at all times,but with dynamics that occur quickly enough that globalequilibrium with the environment is not achieved. Thisresults in a process that is locally reversible, but globallyirreversible [69]. It has been shown that the performanceof a quantum Otto engine is dependent on the strokeprotocol [70–83] and the nature of the working medium[84–98], with the EMP in particular being determinedby the form of the fundamental relation of the workingmedium [99]. In Ref. [91] it was shown that the EMPof an endoreversible quantum Otto engine with a singleparticle working medium can exceed the Curzon-Ahlbornefficiency. It is of interest then to examine the role ofquantum statistics in the operation of such an engine.In the following analysis we closely follow the methodestablished in Ref. [91]. Let us consider a workingmedium of two anyons in a harmonic potential, evolvingunder the Hamiltonian given in Eq. (24) with a time- dependent frequency. The Otto cycle consists of fourstrokes summarized graphically in Fig. 7:(1) Isentropic compression
During this stroke the working medium remains in a stateof constant entropy, exchanging no heat with the environ-ment. Using the first law ∆ E = Q + W we can identifythe change in internal energy completely with work, W comp = E ( T B , ω ) − E ( T A , ω ) . (32)(2) Isochoric Heating
During this stroke the externally-controlled work param-eter (the trap frequency in the case of the harmonic en-gine) is held constant, resulting in zero work. By thefirst law we can then identify the change in internal en-ergy completely with heat, Q h = E ( T C , ω ) − E ( T B , ω ) . (33)Recalling the conditions of endoreversibility we note thatthe working medium does not fully thermalize with thehot reservoir during this stroke, giving us the condition T B ≤ T C ≤ T h [67]. The change in temperature dur-ing the stroke depends on the properties of the workingmedium can be determined using Fourier’s law [66], dTdt = − α h ( T ( t ) − T h ) , (34)where α h is a constant determined by the heat capacityand thermal conductivity of the working medium.(3) Isentropic expansion
In exactly the same manner as the compression stroke,we can identify the change in internal energy during theexpansion with work, W exp = E ( T D , ω ) − E ( T C , ω ) . (35)(4) Isochoric Cooling
As in the heating stroke, we identify the change in inter-nal energy during this stroke with heat, Q c = E ( T A , ω ) − E ( T D , ω ) . (36)The temperature change can again be determined fromFourier’s law, dTdt = − α c ( T ( t ) − T c ) , (37)where T D > T A ≥ T c .The efficiency of the engine is given by the ratio of thetotal work and the heat exchanged with the hot reservoir, η = − W comp + W exp Q h , (38)and the power output by the ratio of the total work tothe cycle duration, P = − W comp + W exp γ ( τ h + τ c ) . (39)Note that only the durations of the heating and coolingstrokes are accounted for explicitly, with γ serving asa multiplicative factor that implicitly incorporates theduration of the isentropic strokes [91].0 / βℏω / ℏω (a) / βℏω - / ℏω (b) / βℏω / k B (c) / βℏω / k B (d) FIG. 6. Comparison of (a) internal energy, (b) free energy, (c) entropy, and (d) heat capacity of two topological anyons withphase parameter ν , against two statistical anyons using phase parameter p B = ( ν − . The lines correspond to ν = 0 , p B = 1(blue, dashed), ν = 1 , p B = 0 (green, dot-dashed), ν = 1 / p B = 1 / A BCD T h T c FIG. 7. Energy-frequency diagram of a quantum Otto cyclefor a harmonic trapping potential with a working medium oftwo anyons.
A. 1D Statistical Anyons
Combining the internal energy and entropy from Eq.(27) with Eqs. (32), (33), and (35) and plugging it allinto Eq. (38) yields a complicated expression that canbe considerably simplified. First we note that from theisentropic strokes we have the conditions, S ( T A , ω ) = S ( T B , ω ) and S ( T C , ω ) = S ( T D , ω ) . (40)Using Eq. (27) it is straightforward to verify that theconditions in Eq. (40) are satisfied by, T A ω = T B ω and T C ω = T D ω . (41)Furthermore, Eq. (34) and Eq. (37) can be fully solvedto yield, T C − T h = ( T B − T h ) e − α h τ h ,T A − T c = ( T D − T c ) e − α c τ c , (42) where τ h ( τ c ) is the duration of the heating (cooling)stroke. Combining Eq. (38) with Eq. (41) and Eq. (42)yields a much simplified form for the efficiency, η = 1 − κ, (43)where κ ≡ ω /ω is the compression ratio . We note thatthis efficiency is identical to classical, reversible Otto ef-ficiency, as well as the single particle quantum Otto effi-ciency found in Ref. [91] and is completely independentof the quantum statistics of the working medium.To find the EMP we next need to compute the power,given by Eq. (44). Eliminating free parameters using thesame simplification process as we did for the efficiency wearrive at the much more cumbersome expression, P = (1 − κ ) ω (cid:126) γ ( τ h + τ c ) [3 coth(Γ) − − csch (Λ)] , (44)where,Γ = κω (cid:126) ( e α c τ c + α h τ h − k B [ κT h ( e α h τ h − e α c τ c + T c ( e α c τ c − , Λ = κω (cid:126) ( e α c τ c + α h τ h − k B [ κT h ( e α h τ h −
1) + T c ( e α c τ c − e α h τ h ] . (45)The first thing that we can note about this expres-sion is that the endoreversible power output does notdepend on the statistics of the working medium for one-dimensional statistical anyons. This is consistent withthe results of Ref. [96], which showed differences inthe performance of a one-dimensional harmonic quantumOtto engine arising from the bosonic or fermionic natureof the working medium are a feature of nonequilibriumperformance. In the case of endoreversible operation, theonly effect of the generalized exclusion principle is to shiftthe value of the work expended during the compressionstroke by (cid:126) p B ( ω − ω ) and the work extracted duringthe expansion stroke by (cid:126) p B ( ω − ω ). Thus when W comp and W exp are summed to determine the total work, thesecontributions exactly cancel each other out.We observe that the power vanishes for the case ofΓ = Λ. Examining Eq. (45) we see that these expressionsbecome equivalent in the limit κ → T c /T h . This matches1 T c / T h FIG. 8. EMP as a function of the ratio of bath temperaturesfor two distinguishable quantum particles (dot-dashed, blue)and two indistinguishable statistical anyons (dashed, green)in one dimension. The Curzon-Ahlborn efficiency (bottomsolid, red) and the Carnot efficiency (top solid, black) aregiven in comparison. Operation is in the quantum regimecorresponding to (cid:126) ω /k B T c = 10. Parameters are α c = α h = γ = 1, and τ c = τ h = 0 . with our physical intuition, as this limit corresponds toquasistatic operation at Carnot efficiency. This providesus with the parameter range Γ < Λ where the poweroutput is positive and the cycle operates as an engine.This is typically referred to as the positive work condition .To find the EMP we maximize Eq. (44) numericallywith respect to the compression ratio. The EMP asa function of the ratio of bath temperatures for one-dimensional statistical anyons is shown in Fig. 8, alongwith the EMP of distinguishable particles, the Curzon-Ahlborn efficiency, and the Carnot efficiency for com-parison. Notably we see that, while the EMP does notdepend on the anyonic phase, it does depend on whetheror not the particles of the working medium are distin-guishable . In this case, distinguishable refers to particlesthat remain sufficiently spatially separated such that theoverlap of their wave functions is negligible, negating anybehavior that would arise from the exchange forces. Ex-perimentally, we can consider an engine consisting of twodistinguishable particles as equivalent to the joint outputof two separate single particle engines situated across thelab from each other. We see that, at low temperature ra-tios, the EMP of the indistinguishable, anyonic workingmedium outperforms that of two distinguishable quan-tum particles. We also note that for both indistinguish-able and distinguishable quantum working mediums theEMP is greater than the Curzon-Ahlborn efficiency, con-firming the results found in [91].
B. 2D Statistical Anyons
It is straightforward to extend the previous analysisto two dimensions. As noted in Section IV, in two di- mensions the entropy is no longer independent of theanyonic phase, in the form of p B . Similarly, in the two-dimensional endoreversible engine the p B dependence inthe work done on expansion and expended during com-pression no longer cancel each other out, resulting in ananyonic phase dependent power output, P = (1 − κ ) ω (cid:126) γ ( τ h + τ c ) (cid:104) − / − coth(Λ /
2) + p B tanh(Λ) − p B tanh(Γ) (cid:105) , (46)where Λ and Γ are the same as given in Eq. (45). Thisdependence manifests in the EMP at small values of thebath temperature ratio, shown in Fig. 9. We see that,as in one-dimension, all indistinguishable working medi-ums show greater EMP than a working medium of twodistinguishable quantum particles. We see further thatthe anyonic phase that gives maximum performance de-pends on the bath temperature ratio. For temperatureratios between around 0.1 and 0.25 we see that bosonicsymmetry ( p B = 1) gives the greatest enhancement toEMP over distinguishable particles and fermionic sym-metry ( p B = 0) the least. However, around T c /T h = 0 . p B interpo-late smoothly between these limits. Interestingly, thistransition indicates the existence of a critical point atwhich the EMP becomes equivalent for all values of p B .The origin of this behavior can be traced to the factthat, unlike in the one-dimensional case, the energy shiftarising from the generalized exclusion principle is tem-perature dependent. In one dimension, while the magni-tude of each energy eigenvalue depends on whether theparticles are bosons or fermions, the degeneracy doesnot. In two dimensions, both the magnitude and de-generacy of each energy state differ between bosons andfermions, leading to the temperature-dependent shift inthe internal energy. Due to this temperature dependence,in general the contributions to the work from p B on thecompression and expansion strokes no longer cancel out.The critical point then corresponds to the unique ratio ofbath temperatures such that these contributions becomeexactly equal, leading to p B -independent performance. C. 2D Topological Anyons
Using the partition function in Eq. (28) we can carryout the endoreversible analysis for a harmonic quan-tum Otto engine with a working medium of topologi-cal anyons. This results in an expression for the powerthat is very similar to the two-dimensional statisticalanyon power, but with an additional factor dependenton the anyonic phase within the phase-dependent hyper-bolic trigonometric terms,2 T c / T h T c / T h T c / T h FIG. 9. EMP as a function of the ratio of bath temperatures for two statistical anyons with p B = 1 (long dashed, brown), p B = 1 / p B = 0 (dotted, cyan), and two distinguishable quantum particles (dot-dashed, blue) in twodimensions. The Curzon-Ahlborn efficiency (solid, red) is given in comparison. The bottom inset highlights the range of bathtemperature ratios at which bosonic working mediums display the greatest EMP and the top inset highlights the critical pointand transition to the region where fermionic working mediums begin to outperform bosonic ones. Operation is in the quantumregime corresponding to (cid:126) ω /k B T c = 10. Parameters are α c = α h = γ = 1, and τ c = τ h = 0 . P = (1 − κ ) ω (cid:126) γ ( τ h + τ c ) (cid:110) − / − coth(Λ / − ν ) tanh (cid:0) (1 − ν )Λ (cid:1) − (1 − ν ) tanh (cid:0) (1 − ν )Γ (cid:1)(cid:111) . (47) ν FIG. 10. Power as a function of the anyonic phase fortwo topological anyons at T c /T h = 0 . T c /T h = 0 . T c /T h = 0 . α c = α h = γ = τ c = τ h = 1, and κ = 0 . (cid:126) = k B = 1. This results in richer behavior, with the power nolonger always being maximized by either the bosonic orfermionic limit. Fig. 10 shows the power as a functionof the anyonic phase for several different bath tempera-ture ratios. We see that as the bath temperature ratioincreases, the anyonic phase that maximizes the poweroutput shifts from more fermionic to more bosonic. Again maximizing with respect to the frequency, wefind an EMP with complex dependence on the anyonicphase, shown in Fig. 11. Consistent with the pure power,we see that the anyonic EMP is no longer bounded by thebosonic and fermionic limits of the anyonic phase. Thephase that provides the maximum EMP is highly tem-perature dependent, and we see intermediate values thatprovide both better and worse EMP than either bosonsor fermions. Furthermore, the anyonic EMP can evenfall below that of the distinguishable particles. One suchexample for the case of ν = 0 . ν -dependent hyper-bolic tangent terms in Eq. (47). However, the morecomplicated energy spectrum arising from the hard-corerestriction and multivalued wave function results in anadditional ν -dependence in the argument of the trigono-metric functions. This highly non-linear dependence isresponsible for the large variations in performance forintermediate values of ν seen in Figs. 11 and 12.From the standpoint of pure performance, the exis-tence of temperature regimes where intermediate valuesof ν exceed the bosonic EMP demonstrates an advantagefor topological anyons over statistical anyons as a working3medium. However, the complex dependence of the poweron ν also has the consequence that small variations in theanyonic phase can lead to vastly different performance, asseen for the case of ν = 0 .
8. From a quantum metrologystandpoint, the fact that performance is linked directlyto the anyonic phase for both statistical and topologicalanyons indicates that thermal machines may be a usefultool for detecting signatures of anyonic statistics.
V. ANYONIC ENGINE: BEYONDENDOREVERSIBILITY
Under the conditions of endoreversiblity we have seenthat the engine performance depends on the anyonicphase for both statistical and topological anyons in twodimensions, but not for statistical anyons in one dimen-sion. We now move beyond the assumption of local equi-librium, and consider a finite-time quantum Otto cyclewith fully nonequilibrium isentropic strokes. To focuson the effects on engine performance arising solely fromthe quantum statistics, we will make the standard as-sumption that the thermalization time is short enoughand the isochoric strokes long enough that the workingmedium is in a state of thermal equilibrium with the hot(cold) bath at point C (A) in the cycle, removing theneed to explicitly model the interaction with the heatbaths [71, 73, 74, 76, 96, 97, 100, 101].For the full nonequilibrium treatment we restrict ouranalysis to the case of one-dimensional statistical anyons,as of the three working mediums we have explored sofar this was the only one that has not yet shown effectsarising from the nature of the quantum statistics. Inorder to calculate the efficiency, power, and EMP of theengine, we must determine the internal energies at pointsA, B, C, and D in the cycle. These can be found fromthe density operator in the typical fashion, (cid:104) H (cid:105) = tr { ρH } . (48)For N independent particle pairs, the total density op-erator is given by the product of the individual densityoperators for each pair, ρ N = N B (cid:89) j =1 ρ ( j )B N (cid:89) k = N B +1 ρ ( k )F , (49)where ρ B and ρ F are the density operators of particlepairs with bosonic and fermionic symmetry, respectively.Combining Eqs. (48) and (49) we have, (cid:104) H N (cid:105) = N B (cid:88) j =1 tr (cid:110) ρ ( j )B H j (cid:111) + N (cid:88) k = N B +1 tr (cid:110) ρ ( k )F H k (cid:111) (50)= N B (cid:104) H B (cid:105) + ( N − N B ) (cid:104) H F (cid:105) . Using N B = N p B we arrive at the expression for theinternal energy of a single pair of statistical anyons, (cid:104) H SA (cid:105) = p B (cid:104) H B (cid:105) + (1 − p B ) (cid:104) H F (cid:105) (51) In Ref. [96] both the thermal state and time-evolveddensity operators are derived for two bosons and fermionsin a harmonic potential (for completeness, the full expres-sions are provided in Appendix C). The correspondinginternal energies are, (cid:104) H (cid:105) A = (cid:126) ω β (cid:126) ω ) + csch( β (cid:126) ω ) ∓ , (cid:104) H (cid:105) B = (cid:126) ω Q ∗ (3coth( β (cid:126) ω ) + csch( β (cid:126) ω ) ∓ , (cid:104) H (cid:105) C = (cid:126) ω β (cid:126) ω ) + csch( β (cid:126) ω ) ∓ , (52) (cid:104) H (cid:105) D = (cid:126) ω Q ∗ (3coth( β (cid:126) ω ) + csch( β (cid:126) ω ) ∓ . Here the ( − ) corresponds to bosons and the (+) tofermions. Q ∗ and Q ∗ are protocol-dependent dimen-sionless parameters that measure the degree of adiabatic-ity of the isentropic strokes [102]. Using Eqs. (51) and(52) we can determine the engine efficiency and poweroutput. The full expressions are cumbersome and givenin Appendix D. We note that the engine behavior we findhere is equivalent to that found in Ref. [88] for an Ottoengine with a working medium of a Calogero–Sutherlandgas. However, our underlying construction is very dif-ferent, with the anyonic nature of the working mediumarising from a simple statistical average over bosons andfermions rather than an additional inter-particle interac-tion term.To continue our analysis we must pick a specific proto-col for the compression and expansion strokes. For sim-plicity we choose the “sudden switch” protocol, whichcorresponds to an instantaneous quench from the initialto final frequency. For the sudden switch the adiabaticityparameters are, Q ∗ = Q ∗ = 1 + κ κ , (53)where κ = ω /ω .Following the same method of performance analysisthat we used in the endoreversible case, we numericallymaximize the power with respect to the frequency ratioin order to determine the EMP. The EMP as a functionof the bath temperature ratios is shown in Fig. 13. Weimmediately see that in the case of nonequilibrium op-eration the performance is no longer independent of theanyonic phase. We note fermionic symmetry gives theworst EMP and bosonic symmetry the best, with inter-mediate values of p B falling between. In the parameterregimes explored we see no transitions between anyonicphases that provides the optimal EMP, unlike the two-dimensional endoreversible engines.In general we see that the EMP of the nonequilibriumengine is significantly worse than in the endoreversiblecase, no longer outperforming the Curzon-Ahlborn effi-ciency. This is unsurprising, as the sudden switch pro-tocol we have employed is far from adiabatic, resulting4 T c / T h T c / T h T c / T h FIG. 11. EMP as a function of the ratio of bath temperatures for two topological anyons with ν = 0 (long dashed, brown), ν = 1 / ν = 0 .
25 (dot-dash-dashed, orange), ν = 1 (dotted, cyan), and two distinguishable quantumparticles (dot-dashed, blue) in two dimensions. The Curzon-Ahlborn efficiency (bottom solid, red) and Carnot efficiency (topsolid, black) are given in comparison. The bottom inset highlights a parameter region where an intermediate anyonic phase( ν = 0 .
25) displays the greatest EMP and the top inset highlights a region of low temperature ratios that exhibits multiplecrossings. Operation is in the quantum regime corresponding to (cid:126) ω /k B T c = 10. Parameters are α c = α h = γ = 1, and τ c = τ h = 0 . T c / T h FIG. 12. EMP as a function of the ratio of bath tem-peratures for two topological anyons with ν = 0 . (cid:126) ω /k B T c = 10. Parameters are α c = α h = γ = 1, and τ c = τ h = 0 . in significantly lower power due to the loss of energy tononadiabatic excitations. We also compare the EMPas we transition from the deep quantum regime char-acterized by (cid:126) ω /k B T c = 10 to a more classical regimecharacterized by (cid:126) ω /k B T c = 1. We see that in thedeep quantum regime the indistinguishable particles giveworse performance than distinguishable quantum parti- cles. As we transition toward the more classical regimethe gaps in EMP between different values of the anyonicphase shrink, but we also see that the indistinguishableparticles begin to outperform the distinguishable ones.Fig. 13b examines the engine performance in the sameparameter regime studied in Ref. [96], where we see thesame bosonic advantage emerge. VI. OPTIMIZATION OF ANYONIC ENGINES:SHORTCUTS TO ADIABATICITY
As we saw in the previous section, nonadiabatic drivingsignificantly hinders engine performance through the lossof energy to nonadiabatic excitations - an effect typicallyreferred to as “quantum friction.” To achieve completelyfrictionless strokes, the cycle driving must be fully adia-batic, requiring infinite time and leading to zero poweroutput. This trade-off can be circumvented throughthe use of “shortcuts to adiabaticity” (STA). STA re-fer to a set of techniques that can produce the same fi-nal state of a system in finite time that it would haveachieved under adiabatic driving [103]. There are nu-merous established techniques to achieve this, includingcounterdiabatic driving [104–107], dynamical invariants[108], inversion of scaling laws [109], the fast forward ap-proach [110–114], optimal protocols [115–118], and time-rescaling [119, 120]. For a recent review on the topic ofshortcuts see Ref. [121]. In this section we will examineimplementing optimal protocol and fast forward STA forharmonic anyonic systems in order to determine if the5 T c / T h (a) T c / T h T c / T h (b) FIG. 13. Nonequilibrium EMP as a function of the ratio of bath temperatures for two statistical anyons in one dimension with p B = 1 (long dashed, brown), p B = 1 / p B = 0 .
25 (dot-dash-dashed, orange), p B = 0 (dotted, cyan line),and two distinguishable quantum particles (dot-dashed, blue). The Curzon-Ahlborn efficiency (lower solid, red) and Carnotefficiency (top solid, black) are given in comparison. Plot (a) shows operation in the regime corresponding to (cid:126) ω /k B T c = 10and (b) in the regime corresponding to (cid:126) ω /k B T c = 1. We have set τ = 1. anyonic phase plays a role in shortcut design. A. Optimal Protocol Shortcut
We will first examine the optimal protocol shortcutusing the phenomenological framework of linear responsetheory [116, 122]. This shortcut is based on separatingthe total nonequilibrium work into two contributions, thequasistatic work and the excess work, (cid:104) W (cid:105) = (cid:104) W qs (cid:105) + (cid:104) W ex (cid:105) . (54)Here (cid:104) W qs (cid:105) corresponds to the work carried out were theprocess to be fully quasistatic, and (cid:104) W ex (cid:105) the work lostdue to nonequilibrium excitations. It has been shownthat there exist optimal protocols for which (cid:104) W ex (cid:105) van-ishes, leading to quasistatic performance in finite time.This is a true STA, as for protocols where excess workvanishes no nonadiabatic transitions between eigenstatesoccur [116].Let us consider a process in which we begin with aquantum system in thermal equilibrium with a reser-voir at inverse temperature β . The system is then de-coupled from the reservoir and driven by Hamiltonian H ( t ) ≡ H ( λ t ) where λ t is a time-dependent externalcontrol parameter, λ t ≡ λ + δλg ( t ). This process corre-sponds exactly to the isentropic strokes of the harmonicOtto engine, with ω t as the external control parameter.If the external driving can be considered a weak pertur-bation, we can derive an expression for the excess workentirely from the equilibrium thermodynamic propertiesof the system using the tools of linear response theory. In this framework the excess work is given by [116], (cid:104) W ex (cid:105) = − ( δλ ) (cid:90) t f t dt ∂∂t g ( t ) (cid:90) t − t ds R ( s ) ∂∂s g ( t − s ) , (55)where R ( t ) is the relaxation function. Note that typicallythe relaxation function is denoted by Ψ( t ), as in Refs.[116, 122]. Here we use R ( t ) to avoid confusion with thewave function. The relaxation function is determined bythe quantum response function, φ ( t ) = − ∂∂t R ( t ) , (56)which is in turn found from the equilibrium state, φ ( t ) = 1 i (cid:126) tr { ρ [ A , A t ] } . (57)Here A t is the generalized force, A = ∂∂λ H ( λ ) . (58)Let us first determine the excess work for two bosonsand two fermions in one dimension. With these resultswe will be able to construct the excess work for one-dimensional statistical anyons. From the Hamiltonian inEq. (24) we determine the time-dependent generalizedforce, A t = 12 m (cid:2) x ( t ) + x ( t ) (cid:3) . (59)With Heisenberg’s equation of motion and some elemen-tary commutator algebra we determine x ( t ), and fromthere the commutator [ A , A t ]. Plugging this commuta-tor into Eq. (57) we can then take the trace. To simplify6the necessary integrals we first convert the thermal den-sity operator to its Wigner distribution representation.The full expression for the Wigner distribution for bosonsand fermions is given in Appendix C. In the Wigner dis-tribution representation x and p are converted from op-erators to simple commuting variables. The resulting re-sponse functions are, φ ( t ) = (cid:126) ω [3 coth( β (cid:126) ω ) + csch( β (cid:126) ω ) ∓
1] sin(2 tω )(60)where the ( − ) corresponds to bosons and the (+) tofermions. From the response functions we can determinethe relaxation functions with a trivial integral, R ( t ) = (cid:126) ω [3 coth( β (cid:126) ω ) + csch( β (cid:126) ω ) ∓
1] cos(2 tω ) . (61)We now have all the pieces we need to calculate theexcess work from Eq. (55). For simplicity we take t = 0and t f = τ . We pick a simple linear protocol g ( t ) = t/τ ,as it has been shown that there exist zeros of the excesswork for a single particle in harmonic potential [116]. Wefind the excess work for two bosons or fermions to be, (cid:104) W ex (cid:105) = (cid:126) ( δω ) ω τ sin ( ω τ ) × (cid:104) β (cid:126) ω ) + csch( β (cid:126) ω ) ∓ (cid:105) . (62)In Eqs. (49) – (51) we showed that, as the N parti-cle density operator is simply the product of the densityoperators of each individual particle pair, the statisticalanyon internal energy is simply given by the weightedsum of the boson and fermion internal energies. The ex-act same mathematical process can be applied here in thecalculation of the statistical anyon Wigner distribution,response function, relaxation function, and excess work.Thus we have, (cid:104) W SAex (cid:105) = p B (cid:104) W Bex (cid:105) + (1 − p B ) (cid:104) W Fex (cid:105) . (63)The statistical anyon excess work is plotted as a func-tion of τ in Fig. 14. We see first that excess work varieswith the anyonic phase, with fermions having the great-est excess work and bosons the least. We note that itszeros, however, are independent of the statistics. This isclear when examining Eq. (62). The statistics only comeinto play in the form of a ∓ τ = nπ/ω , where n ∈ Z .In Ref. [116] it was also shown that there exists a fam-ily of degenerate shortcut protocols that produce zerosof the excess work for a single particle in a parametricharmonic potential. This family take the form of, g ( t ) = tτ + α sin( κπt/τ ) (64) τ < W ex > FIG. 14. Excess work under linear driving as a function ofthe driving time for two statistical anyons with p B = 1 (longdashed, brown), p B = 0 (dotted, cyan), p B = 1 / p B = 3 / p B = 1 / ω = β = (cid:126) = 1,and δω = 0 . τ < W ex > FIG. 15. Excess work under shortcut protocol driving asa function of the driving time for two statistical anyons with p B = 1 (long dashed, brown), p B = 0 (dotted, cyan), p B = 1 / p B = 3 / p B = 1 / ω = β = (cid:126) = 1, α = 1, κ = 2 and δω = 0 . where κ ∈ Z and α ∈ R . The excess work from this proto-col is shown for one dimensional statistical anyons in Fig.15. We arrive at the same conclusion as in the case ofthe linear protocol. While the excess work itself dependson the statistics, the optimal shortcut protocol does not.Examining Eq. (55) we can extend this conclusion to anyarbitrary protocol for the parametric harmonic potential.Since the factor in the expression for the excess work bywhich the bosons and fermions differ is independent oftime, it will be identical for any g ( t ).7 B. Fast Forward Shortcut
Having shown that the optimal protocol for achievinga shortcut to adiabaticity for one-dimensional statisti-cal anyons in a harmonic Otto engine is independent ofthe quantum statistics, we next turn to a different short-cut implementation, the fast-forward method. Unlike theoptimal protocol, in which the shortcut is determinedentirely from the form of the thermal state and systemHamiltonian, the fast forward shortcut is constructed us-ing the instantaneous eigenstates. Since these eigenstatesdepend explicitly on the anyonic phase, we expect thisdependence to carry through to the shortcut. To applythis method we introduce an auxiliary potential to theSchr¨odinger equation, i (cid:126) ∂∂t Ψ( x , t ) = − (cid:126) m ∇ Ψ( x , t ) + ( V + V aux )Ψ( x , t ) , (65)where the form of V aux ensures the final state of thesystem after a finite time evolution is identical to thatachieved after an adiabatic evolution of the unperturbedsystem. Let us consider the following ansatz for our time-dependent wave function,Ψ( x , t ) = ψ ( x , R t ) e if ( x ,t ) e − i (cid:126) (cid:82) t ds (cid:15) ( R s ) , (66)where ψ ( x , R t ) is the instantaneous eigenstate of the un-perturbed Schr¨odinger equation with eigenenergy (cid:15) ( R t ), (cid:15) ( R t ) ψ ( x , R t ) = − (cid:126) m ∇ ψ ( x , R t ) + V ψ ( x , R t ) . (67)Our goal is now to find the phase, f ( x , t ), and auxiliarypotential V aux such that the above ansatz and equationare true. Imposing the condition that f ( x ,
0) = f ( x , τ )(where τ is the duration of the driving) ensures that thefinal state is identical to that of the adiabatically drivenunperturbed equation, and our shortcut is achieved.To simplify the following analysis we express ψ ( x , R t )in polar representation, ψ ( x , R t ) = µ ( x , R t ) e iγ ( x ,R t ) . (68)Combining Eqs. (68), (66), and (65), simplifying, andseparating the real and imaginary components we find, V aux = − (cid:126) ∂γ∂t − (cid:126) ∂f∂t − (cid:126) m ∇ f ∇ γ − (cid:126) m ( ∇ f ) , (69)and, 2 m ∂µ∂t + 2 (cid:126) ∇ f ∇ µ + (cid:126) µ ∇ f = 0 , (70)where we have written µ ( x , R t ) = µ , γ ( x , R t ) = γ , and f ( x , t ) = f to simplify notation.The shortcut method has a major caveat in that theauxiliary potential often becomes singular at the nodes ofthe instantaneous eigenstates, limiting its applicability tothe ground state [123, 124]. However, for the Lewis-Leachfamily of Hamiltonians, of which the harmonic oscillatoris a member [125], it has been shown that the fast forwardpotential is independent of the energy level [112, 123].
1. Statistical Anyons
As in the optimal protocol analysis let us first considerthe fast forward shortcut for the case of two bosons andtwo fermions, from which we will be able to construct thestatistical anyon shortcut. It is straightforward to extendEqs. (69) and (70) to a two particle system, V aux = − (cid:126) ∂γ∂t − (cid:126) ∂f∂t − (cid:126) m ( ∇ f ∇ γ + ∇ f ∇ γ ) − (cid:126) m (cid:2) ( ∇ f ) + ( ∇ f ) (cid:3) , (71)2 m ∂µ∂t + 2 (cid:126) ( ∇ f ∇ µ + ∇ f ∇ µ ) + (cid:126) µ ( ∇ f + ∇ f ) = 0 . (72)As we seek shortcuts to optimize performance of ourharmonic Otto engine let us again consider a one-dimensional harmonic potential. In this case we have R ( t ) = ω ( t ). We can express µ ( x , x , ω t ) for both bosonsand fermions in terms of the single particle harmonic os-cillator eigenstates in the typical fashion, µ ( x , x , ω t ) = 1 (cid:112) δ n ,n ) (cid:104) µ n ( x ) µ n ( x ) (73) ± µ n ( x ) µ n ( x ) (cid:105) , where, µ n ( x ) = 1 √ n n ! (cid:18) mωπ (cid:126) (cid:19) / e − mωx (cid:126) H n (cid:18)(cid:114) mω (cid:126) x (cid:19) . (74)For a single particle in a harmonic potential Eq. (70)is solved by [110, 112], f ( x, ω t ) = − m ˙ ω t πω t x . (75)With this in mind, we take our ansatz for the two particlesolution to be, f ( x , x , ω t ) = − m ˙ ω t πω t (cid:0) x + x (cid:1) . (76)We can see that for any protocol with ˙ ω = ˙ ω τ the short-cut condition will be fulfilled, as f ( x ,
0) = f ( x , τ ). Asimple example of a protocol that fulfills this condition isthe linear protocol, ω ( t ) = ω + αt , where α is a constant.With Eqs. (76) and (73) we can directly verify thatour ansatz is correct by plugging in to Eq. (72). Since γ ( x , x , ω t ) = 0 for both bosons and fermions, we canimmediately see that not only the phase, but also theauxiliary potential will be identical for both bosons andfermions. Using Eq. (71) we determine the explicit formof the auxiliary potential to be, V aux = − m ˙ ω t ω (cid:0) x + x (cid:1) . (77)8Following the same method we can extend this analysisto two dimensions. For this case we take the ansatz, f ( x , x , y , y ω t ) = − m ˙ ω t πω t (cid:0) x + x + y + y (cid:1) , (78)which we verify is a solution to Eq. (72). As in theone-dimensional case, γ ( x , x , y , y , ω t ) = 0 so the aux-iliary potential will again be identical for both bosonsand fermions.Since the statistical anyon state is constructed froma statistical average over the boson and fermion states,we know that if there is no difference in the shortcut forbosons and fermions, there will be none for statisticalanyons. We conclude that in both one and two dimen-sions the implementation of a fast forward STA is inde-pendent of the quantum statistics. This result is counterto our original hypothesis that the difference in phaseof the instantaneous eigenstates should carry through tothe design of the auxiliary potential. From the form ofEqs. (76) and (78) we can see that, physically, the twoparticle shortcut for the harmonic oscillator potential isaccomplished by driving each particle individually.
2. Topological Anyons
Our previous analysis has shown that the behavior oftopological anyons can have a richer dependence on theanyonic phase. Furthermore, topological anyon eigen-states are significantly more complex, and unlike bosonicor fermionic states can not generally be separated intoa superposition of the single particle states [36]. As thefast-forward shortcut depends on the stationary eigen-state of the system it is of interest to explore whether ornot a shortcut for topological anyons will show depen-dence on the anyonic phase.Let us again consider the situation of two anyons in aharmonic potential. The time-independent Schr¨odingerequation for this problem can be solved by separatingthe Hamiltonian in Eq. (24) into the center of mass andrelative components [34–36], H = (cid:126) ω (cid:18) − ∂ ∂Z∂Z ∗ − ∂ ∂z∂z ∗ + | Z | + | z | (cid:19) (79)where we have used the complex coordinates z j = (cid:112) mw/ (cid:126) ( x j + iy j ) with j ∈ ,
2. Here Z is the centerof mass coordinate, given by Z = ( z + z ) / z therelative coordinate, given by z = z − z . In two dimen-sions the anyonic exchange symmetry is satisfied by twoseparate families of eigenstates. The harmonic oscillatorground states for each are [35], µ ( I ) g = A ν ωz ν e −| Z | − | z | ,µ ( II ) g = A ν ω ( z ∗ ) − ν e −| Z | − | z | , (80)where A ν is an anyonic phase-dependent normalizationfactor. We can construct any desired excited state from these ground states by applying the appropriate raisingoperators [35].In order to determine the fast forward shortcut for thetopological anyon system it will be convenient to convertEqs. (71) and (72) into the center of mass and relativecoordinate systems, V aux = − ω ∂γ∂t − ω ∂f∂t − ∂f∂Z ∂γ∂Z ∗ − ∂γ∂Z ∂f∂Z ∗ − ∂f∂Z ∂f∂Z ∗ − ∂f∂z ∂γ∂z ∗ − ∂γ∂z ∂f∂z ∗ − ∂f∂z ∂f∂z ∗ , (81) ∂µ∂t + ω (cid:104) ∂f∂Z ∂µ∂Z ∗ + ∂µ∂Z ∂f∂Z ∗ + µ ∂ f∂Z∂Z ∗ + 4 ∂f∂z ∂µ∂z ∗ + 4 ∂µ∂z ∂f∂z ∗ + 4 β ∂f∂z∂z ∗ (cid:105) = 0 . (82)Motivated by our results for the statistical anyons, wechoose the same ansatz for f (converted into the relativeand center of mass coordinate system), f ( Z, z, ω t ) = − ˙ ω t ω (cid:18) | z | + 2 | Z | (cid:19) . (83)Plugging Eq. (83) and Eq. (80) into Eq. (82) we seethat our ansatz is indeed a solution to the differentialequation. As before, we have γ ( Z, z, ω t ) = 0 in the centerof mass and relative coordinate system. Using Eq. (81)we can construct the auxiliary potential, V aux = ˙ ω t ω (cid:18) | z | + 2 | Z | (cid:19) . (84)We see that, as in the case of the statistical anyons, thefast forward STA for topological anyons is independentof the anyonic phase. This indicates that, counter to ouroriginal intuition, the harmonic oscillator fast forwardshortcut is truly independent of any exchange behavior,not just the bosonic and fermionic limits.While this result was derived for the ground state weknow that, since the harmonic oscillator Hamiltonian is amember of the Lewis-Leach family [125], the above short-cut will also hold for any excited state. This providesa physical motivation for why the shortcut is indepen-dent of the anyonic nature of the particles. Both themodified hard-core nature of topological anyons and thegeneralized exclusion statistics of statistical anyons affecthow the particles will be distributed among the availablestates. Since the same harmonic oscillator shortcut holdsfor all states, it makes sense that it will be independentof the anyonic phase. VII. CONCLUDING REMARKSA. Summary
In this work we have presented a dimension-independent formulation of anyons, dubbed “statistical9anyons,” in which anyonic properties arise from averag-ing over the behavior of a system consisting of a statisti-cal mixture of particles with antisymmetric and symmet-ric exchange properties. Motivated by the HOM effect,we outlined a quantum optics implementation of statis-tical anyons. We showed that the statistical anyons arephysically equivalent to Haldane’s generalized exclusionstatistics anyons, broadening the applicability of GES toany system of indistinguishable particles for which sucha mixture can be constructed.We determined the thermodynamic properties of sta-tistical anyons in one and two dimensions, and comparedthem to the thermodynamic properties of topologicalanyons. We found that in two dimensions the internalenergy, free energy, entropy, and heat capacity all dis-play a dependence on the anyonic phase, but it is not asrich a dependence as the topological anyons. However,we determined that, with a parameter-dependent choiceof the anyonic phase, statistical anyons can exactly imi-tate the thermodynamic behavior of topological anyons.With the thermodynamic properties established, weconsidered a harmonic quantum Otto engine with a work-ing medium of statistical anyons. We found that in twodimensions endoreversible engine performance dependson the anyonic phase for both statistical anyons and topo-logical anyons. We found that for the nonequilibriumregime, even in one-dimension, the engine EMP for sta-tistical anyons depended on the anyonic phase.Lastly, we examined the role of the anyonic phase intwo STA, the optimal protocol shortcut and the fast for-ward shortcut. We found both shortcut methods are in-dependent of the anyonic nature of the particles. In thecase of the fast forward method, this independence can beconsidered to arise from the fact that the shortcut doesnot depend on the energy level for a harmonic potential.
B. Impacts and Future Directions
Previous work has been primarily focused on determin-ing the thermodynamic properties of anyons through thecalculation of the partition function or Virial coefficientsof an anyonic gas [12, 34–36, 39, 126–131], anyonic phasetransitions [132], and the distribution of the anyon gas[13, 133, 134]. In this work we have taken a differentapproach by examining the thermodynamic properties ofanyons in the context of heat engines. This method ismotivated by the history of thermodynamics, as a fielddeveloped around the optimization of thermal machines,and is an approach commonly used in modern quantumthermodynamics [65]. To the authors’ knowledge, thisapproach has only been applied in two works, both basedaround Calogero–Sutherland GES anyons [74, 88].While functionally equivalent to GES, statisticalanyons present a new paradigm both theoretically andexperimentally that extends GES to a range of newsettings, including Bose-Fermi mixtures [58–63], opti-mechanical systems [135], and surface plasmonics [136]. While they lack the non-abelian properties necessary forimplementing topological quantum computation, they doprovide an experimentally tractable method of examin-ing the thermodynamic behavior of abelian topologicalanyons. This opens up the door to searching for generalthermodynamic signatures that may provide alternativemethods of detecting and controlling both abelain andnon-ableian anyons.Using statistical anyons we have connected topicsacross various subfields of physics, including the Hong-Ou-Mandel effect from quantum optics, heat enginesfrom quantum thermodynamics, GES from quantum sta-tistical mechanics, fractional exchange statistics fromtopological states of matter, and shortcuts to adiabatic-ity from quantum control. The establishment of the sta-tistical anyon framework opens up a multitude of newpossible research directions and questions for the studyof anyons. There remains much to be explored about thethermodynamics of statistical anyons, including Gibbsmixing for statistical anyons [137] and the behavior of au-tonomous quantum engines with statistical anyon work-ing mediums. We have shown in this work that the per-formance of a cyclic quantum engine is sensitive to theanyonic phase, and we would expect the same to be truefor autonomous engines. Implementing the latter in anoptical or plasmonic setting has metrological implicationsas an alternative device for detecting signatures of any-onic behavior.It would be of interest to compare statistical anyons toother methods of generating anyonic behavior not cov-ered in this work, such as N00N states subject to Blochoscillations [138], quasi-holes in Bose-Einstein conden-sates [139], or particles possessing ambiguous statistics[140]. Statistical anyons may also have the potential toprovide simpler implementations or experimental ana-logues for other applications of anyons, including Hal-dane insulators [141, 142] or anyon beams [143]. Anyonicstatistics has also been applied to more exotic systems,such as black hole gasses [140, 144], for which statisticalanyons may provide a tractable theoretical tool. Finally,a particularly interesting possibility for statistical anyonsis the introduction of interparticle interactions. We haveshown that non-interacting statistical anyons are capableof recreating GES, which historically has been limitedto interacting systems. By combining these paradigms,we add significant complexity to statistical behavior ofthe particles. It is intriguing to wonder if this additionalcomplexity may be leveraged to expand the state space ofpossible anyonic phases, effectively recreating the braid-ing statistics of topological anyons. We leave this multi-tude of questions open for exploration in future works.
ACKNOWLEDGMENTS
S.D. acknowledge support from the U.S. National Sci-ence Foundation under Grant No. DMR-2010127.0
Appendix A: Statistical Anyon Harmonic Oscillator Partition Function
In this appendix we derive an expression for the partition function of two statistical anyons in a harmonic potential.We begin with the definition of the partition function for N independent pairs of particles in the position basis, Z N A = tr (cid:8) e − βH (cid:9) = N (cid:89) j =1 (cid:90) dx j (cid:90) dy j (cid:104) x j y j | e − βH j | x j y j (cid:105) , (A1)where, H j = p x j + p y j m + 12 mω ( x j + y j ) . (A2)Inserting the identity in the energy basis I = (cid:80) ∞ n ( j )1 ,n ( j )2 =0 (cid:12)(cid:12)(cid:12) n ( j )1 n ( j )2 (cid:69) (cid:68) n ( j )1 n ( j )2 (cid:12)(cid:12)(cid:12) twice leads to, Z N A = N (cid:89) j =1 (cid:90) dx j (cid:90) dy j ∞ (cid:88) n ( j )1 ,n ( j )2 =0 ∞ (cid:88) m ( j )1 ,m ( j )2 =0 Ψ ∗ A ( x j , y j )Ψ A ( x j , y j ) e − β (cid:126) ω ( m ( j )1 + m ( j )2 +1) (cid:68) n ( j )1 n ( j )2 | m ( j )1 m ( j )2 (cid:69) . (A3)Here Ψ A ( x j , y j ) is given by Eq. (13) where, ψ n ( x ) = 1 √ n n ! (cid:18) mωπ (cid:126) (cid:19) / e − mωx (cid:126) H n (cid:18)(cid:114) mω (cid:126) x (cid:19) . (A4)After evaluating the inner product, the integration over each x j and y j can be carried out with application of theHermite polynomial orthogonality to yield, Z N A = N (cid:89) j =1 ∞ (cid:88) n ( j )1 ,n ( j )2 =0 e − β (cid:126) ω ( n ( j )1 + n ( j )2 +1) (2 + e − iπ Θ( j − N B +1) δ n ( j )1 ,n ( j )2 + e iπ Θ( j − N B +1) δ n ( j )1 ,n ( j )2 ) . (A5)where δ n ,n is the Kronecker delta. In order to evaluate the step functions we separate the product into one from 1to N B and a second from N B + 1 to N , Z N A = N B (cid:89) j =1 ∞ (cid:88) n ( j )1 ,n ( j )2 =0 e − β (cid:126) ω ( n ( j )1 + n ( j )2 +1) (1 + δ n ( j )1 ,n ( j )2 ) N (cid:89) j = N B +1 ∞ (cid:88) n ( j )1 ,n ( j )2 =0 e − β (cid:126) ω ( n ( j )1 + n ( j )2 +1) (1 − δ n ( j )1 ,n ( j )2 ) . (A6)Here we can immediately recognize the individual partition functions for two bosons and two fermions in harmonicpotential, Z B = ∞ (cid:88) n ,n =0
12 (1 + δ n ,n ) e − β (cid:126) ω ( n + n +1) , Z F = ∞ (cid:88) n ,n =0
12 (1 − δ n ,n ) e − β (cid:126) ω ( n + n +1) . (A7)Thus we can re-write our anyonic partition function in the simple form, Z N A = N B (cid:89) j =1 Z ( j )B N F (cid:89) k =1 Z ( k )F = Z N B B Z N F F = ( Z p B B Z p F F ) N . (A8)Here we have used the fact that for large N we can express the number of bosonic and fermionic particle pairs as N B = N p B and N F = N p F = N (1 − p B ) where p B ( p F ) is the probability of a particle pair having bosonic (fermionic)symmetry.The boson and fermion partition functions can be written in closed-form as, Z B = 18 csch (cid:18) βω (cid:126) (cid:19) + 14 csch( βω (cid:126) ) , Z F = 18 csch (cid:18) βω (cid:126) (cid:19) −
14 csch( βω (cid:126) ) . (A9)With this, we arrive at a closed-form expression for the partition function of a single pair of statistical anyons, Z A = (cid:20)
18 csch (cid:18) βω (cid:126) (cid:19) + 14 csch( βω (cid:126) ) (cid:21) p B (cid:20)
18 csch (cid:18) βω (cid:126) (cid:19) −
14 csch( βω (cid:126) ) (cid:21) − p B . (A10)1 Appendix B: Equilibrium Thermodynamic Quantities
In this appendix we give the full expressions for the internal energies, free energies, entropies, and heat capacitiesof statistical anyons and topological anyons in one- and two-dimensional harmonic potentials.
1. 2D Statistical Anyons
Plugging the respective partition functions for two bosons and two fermions in a harmonic potential into Eq.(25), and substituting that into Eq. (26) we arrive at the following expressions for the thermodynamic properties oftwo-dimensional statistical anyons: E = (cid:126) ω (cid:20) β (cid:126) ω ) + tanh( 12 β (cid:126) ω ) − p B tanh( β (cid:126) ω ) (cid:21) (B1) F = − β ln (cid:26)
18 [cosh( β (cid:126) ω )] p B csch ( 12 β (cid:126) ω )csch ( β (cid:126) ω ) (cid:27) (B2) S = k B β (cid:126) ω (cid:20) β (cid:126) ω ) + tanh( 12 β (cid:126) ω ) − p B tanh( β (cid:126) ω ) (cid:21) + k B ln (cid:26)
18 [cosh( β (cid:126) ω )] p B csch ( 12 β (cid:126) ω )csch ( β (cid:126) ω ) (cid:27) (B3) C = 12 k B β (cid:126) ω (cid:20) ( 12 β (cid:126) ω ) − sech ( 12 β (cid:126) ω ) + 2 p B sech ( β (cid:126) ω ) (cid:21) (B4)
2. 2D Topological Anyons
Plugging Eq. (28) into Eq. (26) we determine the following expressions for the thermodynamic properties oftopological anyons: E = (cid:126) ω (cid:20) coth( 12 β (cid:126) ω ) + 2 coth( β (cid:126) ω ) + ( ν −
1) tanh( β (cid:126) ω (1 − ν )) (cid:21) (B5) F = 1 β ln (cid:20) β (cid:126) ω ( ν − ( 12 β (cid:126) ω ) sinh ( β (cid:126) ω ) (cid:21) (B6) S = k B β (cid:126) ω (cid:20) coth( 12 β (cid:126) ω ) + 2 coth( β (cid:126) ω ) + ( ν −
1) tanh( β (cid:126) ω (1 − ν )) (cid:21) + k B ln (cid:20)
18 cosh( β (cid:126) ω ( ν − ( 12 β (cid:126) ω )csch ( β (cid:126) ω ) (cid:21) (B7) C = 12 k B β (cid:126) ω (cid:20) csch ( 12 β (cid:126) ω ) + 4csch ( β (cid:126) ω ) + 2( ν − sech ( β (cid:126) ω ( ν − (cid:21) (B8) Appendix C: 1D Boson and Fermion Density Operators and Wigner Distributions
In this appendix we give the full expressions for the thermal state and time evolved density operators for a nonequi-librium harmonic quantum Otto engine with a working medium of 1D bosons and fermions. We also include thethermal state Wigner distribution applied in the determination of the optimal protocol shortcut. In each expressionthe top sign of each plus/minus denotes the boson expression and the the bottom sign the fermion expression.The thermal state density operator in position representation is given by, ρ ( x , x , y , y ) = 1 Z mω π (cid:126) sinh ( β (cid:126) ω ) (cid:20) e − mω (cid:126) { [( x + y ) +( x + y ) ] tanh ( β (cid:126) ω/ x − y ) +( x − y ) ] coth ( β (cid:126) ω/ } ± e − mω (cid:126) { [( x + y ) +( x + y ) ] tanh ( β (cid:126) ω/ x − y ) +( x − y ) ] coth ( β (cid:126) ω/ } (cid:21) . (C1)2Solving the time dependent Schr¨odinger equation by means of the appropriate evolution operator for the isentropicstrokes of the engine yields the time-evolved density operator, ρ t ( x , x , y , y ) = mω π (cid:126) ( Y t + X t ω ) (cid:0) e ∓ β (cid:126) ω − (cid:1) × (cid:26) e m (cid:126) ( Y t + X t ω [ i ( x + x − y − y )( Y t ˙ Y t + X t ˙ X t ω ) − ω ( x + x + y + y )coth( β (cid:126) ω )+2 ω ( x y + x y )csch( β (cid:126) ω ) ] ± e m (cid:126) ( Y t + X t ω [ i ( x + x − y − y )( Y t ˙ Y t + X t ˙ X t ω ) − ω ( x + x + y + y )coth( β (cid:126) ω )+2 ω ( x y + x y )csch( β (cid:126) ω ) ] (cid:27) . (C2)Here X t and Y t are solutions to the equation of motion of the classical time-dependent harmonic oscillator,¨ X t + ω ( t ) X t = 0 . (C3)The two particle Wigner distribution is found by carrying out the integral, W ( x , p , x , p ) = 14 π (cid:126) (cid:90) du (cid:90) du ρ (cid:16) x + u , x + u , x − u , x − u (cid:17) e − ip u (cid:126) e − ip u (cid:126) . (C4)For the thermal state the Wigner distribution is, W ( x , p , x , p ) = sech ( β (cid:126) ω/ π (cid:126) (csch ( β (cid:126) ω/ ± β (cid:126) ω )) × (cid:18) e − ( p p m x x ω β (cid:126) ω/ mω (cid:126) ± e − ( p p m x x ω β (cid:126) ω )+2( p p m ω x x β (cid:126) ω ) mω (cid:126) (cid:19) . (C5) Appendix D: Nonequilibrium Engine Characterizations: 1D Statistical Anyons
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