Driven-dissipative systems can exhibit non-equilibrium phenomena that are absent in their equilibrium counterparts. However, phase transitions present in these systems generically exhibit an effectively classical equilibrium behavior in spite of their quantum non-equilibrium origin. In this paper, we show that multicritical points in driven-dissipative systems can give rise to genuinely non-equilibrium behavior. We investigate a non-equilibrium driven-dissipative model of interacting bosons that exhibits two distinct phase transitions: one from a high- to a low-density phase---reminiscent of a liquid-gas transition---and another to an antiferromagnetic phase. Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) Z2 symmetry. They, however, coalesce at a multicritical point giving rise to a non-equilibrium model of coupled Ising-like order parameters described by a Z2×Z2 symmetry. Using a dynamical renormalization-group approach, we show that a pair of non-equilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system. We elucidate various exotic features of these NEFPs. In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance. This further results in complex-valued critical exponents, spiraling phase boundaries, and a complex Liouvillian gap even close to the phase transition. As direct evidence of the non-equilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes "hotter" and "hotter" at longer and longer wavelengths. Finally, we argue that this non-equilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.

1 aYoung, Jeremy, T.1 aGorshkov, Alexey, V.1 aFoss-Feig, Michael1 aMaghrebi, Mohammad, F. uhttps://arxiv.org/abs/1903.0256901692nas a2200169 4500008004100000245006000041210006000101260001500161300001100176490000800187520120500195100001801400700002101418700002701439700001901466856003701485 2017 eng d00aCorrelated Photon Dynamics in Dissipative Rydberg Media0 aCorrelated Photon Dynamics in Dissipative Rydberg Media c2017/07/26 a0436020 v1193 aRydberg blockade physics in optically dense atomic media under the conditions of electromagnetically induced transparency (EIT) leads to strong dissipative interactions between single photons. We introduce a new approach to analyzing this challenging many-body problem in the limit of large optical depth per blockade radius. In our approach, we separate the single-polariton EIT physics from Rydberg-Rydberg interactions in a serialized manner while using a hard-sphere model for the latter, thus capturing the dualistic particle-wave nature of light as it manifests itself in dissipative Rydberg-EIT media. Using this approach, we analyze the saturation behavior of the transmission through one-dimensional Rydberg-EIT media in the regime of non-perturbative dissipative interactions relevant to current experiments. Our model is in good agreement with experimental data. We also analyze a scheme for generating regular trains of single photons from continuous-wave input and derive its scaling behavior in the presence of imperfect single-photon EIT.

1 aZeuthen, Emil1 aGullans, Michael1 aMaghrebi, Mohammad, F.1 aGorshkov, A, V uhttps://arxiv.org/abs/1608.0606802145nas a2200205 4500008004100000245005800041210005700099260001500156300001100171490000700182520152100189100002301710700002101733700002201754700002101776700002501797700002101822700002701843856006901870 2017 eng d00aEmergent equilibrium in many-body optical bistability0 aEmergent equilibrium in manybody optical bistability c2017/04/17 a0438260 v953 aMany-body systems constructed of quantum-optical building blocks can now be realized in experimental platforms ranging from exciton-polariton fluids to ultracold gases of Rydberg atoms, establishing a fascinating interface between traditional many-body physics and the driven-dissipative, non-equilibrium setting of cavity-QED. At this interface, the standard techniques and intuitions of both fields are called into question, obscuring issues as fundamental as the role of fluctuations, dimensionality, and symmetry on the nature of collective behavior and phase transitions. Here, we study the driven-dissipative Bose-Hubbard model, a minimal description of numerous atomic, optical, and solid-state systems in which particle loss is countered by coherent driving. Despite being a lattice version of optical bistability---a foundational and patently non-equilibrium model of cavity-QED---the steady state possesses an emergent equilibrium description in terms of a classical Ising model. We establish this picture by identifying a limit in which the quantum dynamics is asymptotically equivalent to non-equilibrium Langevin equations, which support a phase transition described by model A of the Hohenberg-Halperin classification. Numerical simulations of the Langevin equations corroborate this picture, producing results consistent with the behavior of a finite-temperature Ising model.

1 aFoss-Feig, Michael1 aNiroula, Pradeep1 aYoung, Jeremy, T.1 aHafezi, Mohammad1 aGorshkov, Alexey, V.1 aWilson, Ryan, M.1 aMaghrebi, Mohammad, F. uhttps://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.04382601202nas a2200169 4500008004100000245005700041210005600098260001500154300001100169490000700180520068400187100002600871700002700897700001900924700002000943856006900963 2017 eng d00aMulticritical behavior in dissipative {I}sing models0 aMulticritical behavior in dissipative I sing models c2017/04/26 a0421330 v953 aWe analyze theoretically the many-body dynamics of a dissipative Ising model in a transverse field using a variational approach. We find that the steady state phase diagram is substantially modified compared to its equilibrium counterpart, including the appearance of a multicritical point belonging to a different universality class. Building on our variational analysis, we establish a field-theoretical treatment corresponding to a dissipative variant of a Ginzburg-Landau theory, which allows us to compute the upper critical dimension of the system. Finally, we present a possible experimental realization of the dissipative Ising model using ultracold Rydberg gases.

1 aOverbeck, Vincent, R.1 aMaghrebi, Mohammad, F.1 aGorshkov, A, V1 aWeimer, Hendrik uhttps://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.04213301534nas a2200169 4500008004100000245006200041210005800103260001500161490000800176520102300184100002301207700002201230700002301252700002501275700002701300856003701327 2017 eng d00aA solvable family of driven-dissipative many-body systems0 asolvable family of drivendissipative manybody systems c2017/11/100 v1193 aExactly solvable models have played an important role in establishing the sophisticated modern understanding of equilibrium many-body physics. And conversely, the relative scarcity of solutions for non-equilibrium models greatly limits our understanding of systems away from thermal equilibrium. We study a family of nonequilibrium models, some of which can be viewed as dissipative analogues of the transverse-field Ising model, in that an effectively classical Hamiltonian is frustrated by dissipative processes that drive the system toward states that do not commute with the Hamiltonian. Surprisingly, a broad and experimentally relevant subset of these models can be solved efficiently in any number of spatial dimensions. We leverage these solutions to prove a no-go theorem on steady-state phase transitions in a many-body model that can be realized naturally with Rydberg atoms or trapped ions, and to compute the effects of decoherence on a canonical trapped-ion-based quantum computation architecture.

1 aFoss-Feig, Michael1 aYoung, Jeremy, T.1 aAlbert, Victor, V.1 aGorshkov, Alexey, V.1 aMaghrebi, Mohammad, F. uhttps://arxiv.org/abs/1703.0462600514nas a2200157 4500008004100000245006700041210006600108260001500174300001100189490000700200100002700207700001900234700002300253700002500276856005500301 2016 eng d00aCausality and quantum criticality in long-range lattice models0 aCausality and quantum criticality in longrange lattice models c2016/03/17 a1251280 v931 aMaghrebi, Mohammad, F.1 aGong, Zhe-Xuan1 aFoss-Feig, Michael1 aGorshkov, Alexey, V. uhttp://link.aps.org/doi/10.1103/PhysRevB.93.12512801582nas a2200169 4500008004100000245006700041210006600108260001500174300001100189490000700200520107500207100002701282700001901309700002301328700002501351856003601376 2016 eng d00aCausality and quantum criticality with long-range interactions0 aCausality and quantum criticality with longrange interactions c2016/03/17 a1251280 v923 a Quantum lattice systems with long-range interactions often exhibit drastically different behavior than their short-range counterparts. In particular, because they do not satisfy the conditions for the Lieb-Robinson theorem, they need not have an emergent relativistic structure in the form of a light cone. Adopting a field-theoretic approach, we study the one-dimensional transverse-field Ising model and a fermionic model with long-range interactions, explore their critical and near-critical behavior, and characterize their response to local perturbations. We deduce the dynamic critical exponent, up to the two-loop order within the renormalization group theory, which we then use to characterize the emergent causal behavior. We show that beyond a critical value of the power-law exponent of long-range interactions, the dynamics effectively becomes relativistic. Various other critical exponents describing correlations in the ground state, as well as deviations from a linear causal cone, are deduced for a wide range of the power-law exponent. 1 aMaghrebi, Mohammad, F.1 aGong, Zhe-Xuan1 aFoss-Feig, Michael1 aGorshkov, Alexey, V. uhttp://arxiv.org/abs/1508.0090601255nas a2200157 4500008004100000245006900041210006900110260001500179300001100194490000700205520078100212100002700993700002201020700001901042856003601061 2016 eng d00aFlight of a heavy particle nonlinearly coupled to a quantum bath0 aFlight of a heavy particle nonlinearly coupled to a quantum bath c2016/01/28 a0143090 v933 a Fluctuation and dissipation are by-products of coupling to the `environment.' The Caldeira-Leggett model, a successful paradigm of quantum Brownian motion, views the environment as a collection of harmonic oscillators linearly coupled to the system. However, symmetry considerations may forbid a linear coupling, e.g. for a neutral particle in quantum electrodynamics. We argue that nonlinear couplings can lead to a fundamentally different behavior. Specifically, we consider a heavy particle quadratically coupled to quantum fluctuations of the bath. In one dimension the particle undergoes anomalous diffusion, unfolding as a power-law distribution in space, reminiscent of L\'{e}vy flights. We suggest condensed matter analogs where similar effects may arise. 1 aMaghrebi, Mohammad, F.1 aKrüger, Matthias1 aKardar, Mehran uhttp://arxiv.org/abs/1508.0058201808nas a2200205 4500008004100000245007600041210006900117260001500186300001100201490000700212520120100219100001901420700002701439700001301466700002301479700002101502700002401523700001901547856003601566 2016 eng d00aKaleidoscope of quantum phases in a long-range interacting spin-1 chain0 aKaleidoscope of quantum phases in a longrange interacting spin1 c2016/05/11 a2051150 v933 aMotivated by recent trapped-ion quantum simulation experiments, we carry out a comprehensive study of the phase diagram of a spin-1 chain with XXZ-type interactions that decay as 1/rα, using a combination of finite and infinite-size DMRG calculations, spin-wave analysis, and field theory. In the absence of long-range interactions, varying the spin-coupling anisotropy leads to four distinct phases: a ferromagnetic Ising phase, a disordered XY phase, a topological Haldane phase, and an antiferromagnetic Ising phase. If long-range interactions are antiferromagnetic and thus frustrated, we find primarily a quantitative change of the phase boundaries. On the other hand, ferromagnetic (non-frustrated) long-range interactions qualitatively impact the entire phase diagram. Importantly, for α≲3, long-range interactions destroy the Haldane phase, break the conformal symmetry of the XY phase, give rise to a new phase that spontaneously breaks a U(1) continuous symmetry, and introduce an exotic tricritical point with no direct parallel in short-range interacting spin chains. We show that the main signatures of all five phases found could be observed experimentally in the near future. 1 aGong, Zhe-Xuan1 aMaghrebi, Mohammad, F.1 aHu, Anzi1 aFoss-Feig, Michael1 aRicherme, Philip1 aMonroe, Christopher1 aGorshkov, A, V uhttp://arxiv.org/abs/1510.0210801599nas a2200145 4500008004100000245006500041210006400106260001500170300001100185490000700196520116200203100002701365700002501392856003601417 2016 eng d00aNonequilibrium many-body steady states via Keldysh formalism0 aNonequilibrium manybody steady states via Keldysh formalism c2016/01/27 a0143070 v933 a Many-body systems with both coherent dynamics and dissipation constitute a rich class of models which are nevertheless much less explored than their dissipationless counterparts. The advent of numerous experimental platforms that simulate such dynamics poses an immediate challenge to systematically understand and classify these models. In particular, nontrivial many-body states emerge as steady states under non-equilibrium dynamics. While these states and their phase transitions have been studied extensively with mean field theory, the validity of the mean field approximation has not been systematically investigated. In this paper, we employ a field-theoretic approach based on the Keldysh formalism to study nonequilibrium phases and phase transitions in a variety of models. In all cases, a complete description via the Keldysh formalism indicates a partial or complete failure of the mean field analysis. Furthermore, we find that an effective temperature emerges as a result of dissipation, and the universal behavior including the dynamics near the steady state is generically described by a thermodynamic universality class. 1 aMaghrebi, Mohammad, F.1 aGorshkov, Alexey, V. uhttp://arxiv.org/abs/1507.0193901535nas a2200193 4500008004100000245005200041210005100093260001500144300001100159490000700170520099900177100001901176700002701195700001301222700002201235700002301257700002501280856003601305 2016 eng d00aTopological phases with long-range interactions0 aTopological phases with longrange interactions c2016/01/08 a0411020 v933 a Topological phases of matter are primarily studied in quantum many-body systems with short-range interactions. Whether various topological phases can survive in the presence of long-range interactions, however, is largely unknown. Here we show that a paradigmatic example of a symmetry-protected topological phase, the Haldane phase of an antiferromagnetic spin-1 chain, surprisingly remains intact in the presence of arbitrarily slowly decaying power-law interactions. The influence of long-range interactions on the topological order is largely quantitative, and we expect similar results for more general systems. Our conclusions are based on large-scale matrix-product-state simulations and two complementary effective-field-theory calculations. The striking agreement between the numerical and analytical results rules out finite-size effects. The topological phase considered here should be experimentally observable in a recently developed trapped-ion quantum simulator. 1 aGong, Zhe-Xuan1 aMaghrebi, Mohammad, F.1 aHu, Anzi1 aWall, Michael, L.1 aFoss-Feig, Michael1 aGorshkov, Alexey, V. uhttp://arxiv.org/abs/1505.0314601570nas a2200133 4500008004100000245010700041210006900148260001500217520110300232100002701335700001901362700001901381856003601400 2015 eng d00aContinuous symmetry breaking and a new universality class in 1D long-range interacting quantum systems0 aContinuous symmetry breaking and a new universality class in 1D c2015/10/053 aContinuous symmetry breaking (CSB) in low-dimensional systems, forbidden by the Mermin-Wagner theorem for short-range interactions, may take place in the presence of slowly decaying long-range interactions. Nevertheless, there is no stringent bound on how slowly interactions should decay to give rise to CSB in 1D quantum systems at zero temperature. Here, we study a long-range interacting spin chain with U(1) symmetry and power-law interactions V(r)∼1/rα, directly relevant to ion-trap experiments. Using bosonization and renormalization group theory, we find CSB for α smaller than a critical exponent αc(≤3) depending on the microscopic parameters of the model. Furthermore, the transition from the gapless XY phase to the gapless CSB phase is mediated by the breaking of conformal symmetry due to long-range interactions, and is described by a new universality class akin to the Berezinskii-Kosterlitz-Thouless transition. Our analytical findings are in good agreement with a numerical calculation. Signatures of the CSB phase should be accessible in existing trapped-ion experiments.1 aMaghrebi, Mohammad, F.1 aGong, Zhe-Xuan1 aGorshkov, A, V uhttp://arxiv.org/abs/1510.0132501507nas a2200229 4500008004100000245005700041210005700098260001500155300001100170490000800181520087400189100002701063700002101090700001601111700001301127700001501140700002001155700001801175700002101193700002501214856003801239 2015 eng d00aCoulomb bound states of strongly interacting photons0 aCoulomb bound states of strongly interacting photons c2015/09/16 a1236010 v1153 a We show that two photons coupled to Rydberg states via electromagnetically induced transparency can interact via an effective Coulomb potential. This interaction gives rise to a continuum of two-body bound states. Within the continuum, metastable bound states are distinguished in analogy with quasi-bound states tunneling through a potential barrier. We find multiple branches of metastable bound states whose energy spectrum is governed by the Coulomb potential, thus obtaining a photonic analogue of the hydrogen atom. Under certain conditions, the wavefunction resembles that of a diatomic molecule in which the two polaritons are separated by a finite "bond length." These states propagate with a negative group velocity in the medium, allowing for a simple preparation and detection scheme, before they slowly decay to pairs of bound Rydberg atoms. 1 aMaghrebi, Mohammad, F.1 aGullans, Michael1 aBienias, P.1 aChoi, S.1 aMartin, I.1 aFirstenberg, O.1 aLukin, M., D.1 aBüchler, H., P.1 aGorshkov, Alexey, V. uhttp://arxiv.org/abs/1505.03859v101233nas a2200145 4500008004100000245009700041210006900138260001500207300001100222490000800233520076600241100002701007700001601034856003701050 2015 eng d00aEntanglement entropy of dispersive media from thermodynamic entropy in one higher dimension0 aEntanglement entropy of dispersive media from thermodynamic entr c2015/04/16 a1516020 v1143 a A dispersive medium becomes entangled with zero-point fluctuations in the vacuum. We consider an arbitrary array of material bodies weakly interacting with a quantum field and compute the quantum mutual information between them. It is shown that the mutual information in D dimensions can be mapped to classical thermodynamic entropy in D+1 dimensions. As a specific example, we compute the mutual information both analytically and numerically for a range of separation distances between two bodies in D=2 dimensions and find a logarithmic correction to the area law at short separations. A key advantage of our method is that it allows the strong subadditivity property---notoriously difficult to prove for quantum systems---to be easily verified. 1 aMaghrebi, Mohammad, F.1 aReid, Homer uhttp://arxiv.org/abs/1412.5613v201270nas a2200193 4500008004100000245005700041210005700098260001500155300001100170490000700181520071900188100002700907700002000934700002100954700001700975700002200992700002501014856003701039 2015 eng d00aFractional Quantum Hall States of Rydberg Polaritons0 aFractional Quantum Hall States of Rydberg Polaritons c2015/03/31 a0338380 v913 a We propose a scheme for realizing fractional quantum Hall states of light. In our scheme, photons of two polarizations are coupled to different atomic Rydberg states to form two flavors of Rydberg polaritons that behave as an effective spin. An array of optical cavity modes overlapping with the atomic cloud enables the realization of an effective spin-1/2 lattice. We show that the dipolar interaction between such polaritons, inherited from the Rydberg states, can be exploited to create a flat, topological band for a single spin-flip excitation. At half filling, this gives rise to a photonic (or polaritonic) fractional Chern insulator -- a lattice-based, fractional quantum Hall state of light. 1 aMaghrebi, Mohammad, F.1 aYao, Norman, Y.1 aHafezi, Mohammad1 aPohl, Thomas1 aFirstenberg, Ofer1 aGorshkov, Alexey, V. uhttp://arxiv.org/abs/1411.6624v101362nas a2200181 4500008004100000245005800041210005800099260001500157300001100172490000800183520083900191100002701030700002101057700002201078700002501100700001701125856003801142 2015 eng d00aParafermionic zero modes in ultracold bosonic systems0 aParafermionic zero modes in ultracold bosonic systems c2015/08/06 a0653010 v1153 a Exotic topologically protected zero modes with parafermionic statistics (also called fractionalized Majorana modes) have been proposed to emerge in devices fabricated from a fractional quantum Hall system and a superconductor. The fractionalized statistics of these modes takes them an important step beyond the simplest non-Abelian anyons, Majorana fermions. Building on recent advances towards the realization of fractional quantum Hall states of bosonic ultracold atoms, we propose a realization of parafermions in a system consisting of Bose-Einstein-condensate trenches within a bosonic fractional quantum Hall state. We show that parafermionic zero modes emerge at the endpoints of the trenches and give rise to a topologically protected degeneracy. We also discuss methods for preparing and detecting these modes. 1 aMaghrebi, Mohammad, F.1 aGaneshan, Sriram1 aClarke, David, J.1 aGorshkov, Alexey, V.1 aSau, Jay, D. uhttp://arxiv.org/abs/1504.04012v201828nas a2200145 4500008004100000245015000041210006900191260001400260490000700274520129600281100002701577700002201604700001901626856003701645 2014 eng d00aNonequilibrium quantum fluctuations of a dispersive medium: Spontaneous emission, photon statistics, entropy generation, and stochastic motion 0 aNonequilibrium quantum fluctuations of a dispersive medium Spont c2014/7/160 v903 a We study the implications of quantum fluctuations of a dispersive medium, under steady rotation, either in or out of thermal equilibrium with its environment. A rotating object exhibits a quantum instability by dissipating its mechanical motion via spontaneous emission of photons, as well as internal heat generation. Universal relations are derived for the radiated energy and angular momentum as trace formulas involving the object's scattering matrix. We also compute the quantum noise by deriving the full statistics of the radiated photons out of thermal and/or dynamic equilibrium. The (entanglement) entropy generation is quantified, and the total entropy is shown to be always increasing. Furthermore, we derive a Fokker-Planck equation governing the stochastic angular motion resulting from the fluctuating back-reaction frictional torque. As a result, we find a quantum limit on the uncertainty of the object's angular velocity in steady rotation. Finally, we show in some detail that a rotating object drags nearby objects, making them spin parallel to its axis of rotation. A scalar toy model is introduced in the first part to simplify the technicalities and ease the conceptual complexities; a detailed discussion of quantum electrodynamics is presented in the second part. 1 aMaghrebi, Mohammad, F.1 aJaffe, Robert, L.1 aKardar, Mehran uhttp://arxiv.org/abs/1401.0701v101278nas a2200205 4500008004100000245009000041210006900131260001400200490000700214520065300221100001600874700001300890700002000903700002700923700002100950700001800971700002500989700002101014856003701035 2014 eng d00aScattering resonances and bound states for strongly interacting Rydberg polaritons 0 aScattering resonances and bound states for strongly interacting c2014/11/30 v903 a We provide a theoretical framework describing slow-light polaritons interacting via atomic Rydberg states. We use a diagrammatic method to analytically derive the scattering properties of two polaritons. We identify parameter regimes where polariton-polariton interactions are repulsive. Furthermore, in the regime of attractive interactions, we identify multiple two-polariton bound states, calculate their dispersion, and study the resulting scattering resonances. Finally, the two-particle scattering properties allow us to derive the effective low-energy many-body Hamiltonian. This theoretical platform is applicable to ongoing experiments. 1 aBienias, P.1 aChoi, S.1 aFirstenberg, O.1 aMaghrebi, Mohammad, F.1 aGullans, Michael1 aLukin, M., D.1 aGorshkov, Alexey, V.1 aBüchler, H., P. uhttp://arxiv.org/abs/1402.7333v101052nas a2200145 4500008004100000245005700041210005600098260001500154490000700169520062400176100002700800700002300827700001900850856003700869 2013 eng d00aQuantum Cherenkov Radiation and Non-contact Friction0 aQuantum Cherenkov Radiation and Noncontact Friction c2013/10/210 v883 a We present a number of arguments to demonstrate that a quantum analog of Cherenkov effect occurs when two dispersive objects are in relative motion. Specifically we show that two semi-infinite plates experience friction beyond a threshold velocity which, in their center-of-mass frame, is the phase speed of light within their medium. The loss in mechanical energy is radiated away through the plates before getting fully absorbed in the form of heat. By deriving various correlation functions inside and outside the two plates, we explicitly compute the radiation, and discuss its dependence on the reference frame. 1 aMaghrebi, Mohammad, F.1 aGolestanian, Ramin1 aKardar, Mehran uhttp://arxiv.org/abs/1304.4909v201136nas a2200145 4500008004100000245005800041210005600099260001300155490000700168520070900175100002700884700002300911700001900934856003700953 2013 eng d00aA Scattering Approach to the Dynamical Casimir Effect0 aScattering Approach to the Dynamical Casimir Effect c2013/1/70 v873 a We develop a unified scattering approach to dynamical Casimir problems which can be applied to both accelerating boundaries, as well as dispersive objects in relative motion. A general (trace) formula is derived for the radiation from accelerating boundaries. Applications are provided for objects with different shapes in various dimensions, and undergoing rotational or linear motion. Within this framework, photon generation is discussed in the context of a modulated optical mirror. For dispersive objects, we find general results solely in terms of the scattering matrix. Specifically, we discuss the vacuum friction on a rotating object, and the friction on an atom moving parallel to a surface. 1 aMaghrebi, Mohammad, F.1 aGolestanian, Ramin1 aKardar, Mehran uhttp://arxiv.org/abs/1210.1842v201667nas a2200145 4500008004100000245006400041210006200105260001400167490000700181520123200188100002701420700001801447700001901465856003701484 2012 eng d00aPolymer-mediated entropic forces between scale-free objects0 aPolymermediated entropic forces between scalefree objects c2012/12/30 v863 a The number of configurations of a polymer is reduced in the presence of a barrier or an obstacle. The resulting loss of entropy adds a repulsive component to other forces generated by interaction potentials. When the obstructions are scale invariant shapes (such as cones, wedges, lines or planes) the only relevant length scales are the polymer size R_0 and characteristic separations, severely constraining the functional form of entropic forces. Specifically, we consider a polymer (single strand or star) attached to the tip of a cone, at a separation h from a surface (or another cone). At close proximity, such that h<