Origin of Topological Order in a Cooper Pair Insulator
OOrigin of Topological Order in a Cooper Pair Insulator
Siddhartha Patra ∗ and Siddhartha Lal † Department of Physical Sciences, Indian Institute of Science Education and Research-Kolkata, W.B. 741246, India (Dated: February 26, 2021)While a topologically ordered counterpart of the s-wave superconductor has been proposed inthe literature on phenomenological grounds, its microscopic origin remains unknown. In meetingthis goal, we employ the recently developed unitary renormalisation group (URG) method on ageneralised model of electrons in two spatial dimensions with attractive interactions. We show thatthe effective Hamiltonian obtained at the stable low-energy fixed point of the RG flow correspondsto a gapped, insulating state of quantum matter we call the Cooper pair insulator (CPI). Detailedanalyses show that the CPI ground state manifold displays several signatures of topological order,including a four-fold degeneracy when placed on the torus. Spectral flow based arguments revealthe emergent gauge-theoretic structure of the effective theory: the CPI effective Hamiltonian canbe written entirely in terms of non-local Wilson loops. Further, it contains a topological θ -termwhose coefficient is quantised in keeping with the requirement of invariance of the ground stateunder large gauge transformations. This θ -term is known to be equivalent to the Chern-Simonsterm in two spatial dimensions. Passage from the CPI to the metal by tuning the θ coefficientreveals a plateaux structure of the CPI ground state in terms of the steadily decreasing numberof condensed Cooper pairs. Investigations reveal that the long-ranged many-particle entanglementcontent of the CPI ground state is driven by inter-helicity two-particle scattering processes. Theplateau with θ = 0 possesses the largest bipartite entanglement entropy (EE), scaling logarithmicallywith subsystem size ( L ) and falling rapidly upon tuning θ towards the metal. The bipartite EE forinter-plateau transitions shows universal signatures in its variations with L and θ . While the EEsignatures for plateaux and transitions can be distinguished at low temperatures, such distinctionsare smeared out as temperature is raised. We also study the passage from the CPI to the s-wave BCSsuperconducting ground state under RG, and find that the RG flow promotes fluctuations in thenumber of condensed Cooper pairs and lowers those in the conjugate global phase of the ground statewavefunction. Consequently, we find that the distinct signatures of long-ranged entanglement in theCPI are replaced by the well-known short-ranged entanglement of the BCS state. Analysing theeffects of Josephson effects reveals that while a Cooper pair tunnel coupling between two CPI systemsdoes not lead to phase stiffening, breaking the global U (1) phase rotation symmetry in one of theminduces a phase coherence in the other. However, the generation of a Josephson current requires theseparate breaking of the U (1) symmetry in the two systems. Finally, we study the renormalisation ofthe entanglement in k -space for both the CPI and BCS ground states. The topologically ordered CPIstate is shown to possess an emergent hierarchy of scales of entanglement, and that this hierarchycollapses in the BCS state. Our work offers clear evidence for the microscopic origins of topologicalorder in this prototypical system, and lays the foundation for similar investigations in other systemsof correlated electrons. CONTENTS
I. Introduction 1II. Effective theory of the Cooper Pair Insulator(CPI) 4III. Topological Features of CPI 6A. Topological nature of the effective theory 6B. Topological degeneracy at Φ = 0. 8C. Spectral flow, plateau ground states andtopological quantum numbers 9IV. Entanglement Features of CPI 10A. Entanglement Spectrum 10B. Entanglement Entropy of the plateau groundstates and transitions 12V. Passage to the BCS ground state 14A. Properties of the ground state 14 B. The effect of a Josephson coupling 17VI. Entanglement Renormalisation 17VII. Conclusions and Discussions 20Acknowledgments 21A. Hamiltonian RG 22B. URG with symmetry breaking field 23References 23
I. INTRODUCTION
Superconductivity is undoubtedly one of the best stud-ied example of an emergent collective phenomenon in asystem of interacting electrons. While Cooper’s theory a r X i v : . [ c ond - m a t . s t r- e l ] F e b demonstrates that the presence of an attractive pair-ing interaction can lead to the formation and conden-sation of two-electron bound states (Cooper pair), thecelebrated BCS theory well describes the superconduct-ing nature of this condensate of Cooper pairs in termsof a ground state wavefunction with a fluctuation in thenumber of Cooper pairs. Importantly, the BCS theoryprovides microscopic insight into various superconduct-ing properties like the Meissner-Ochsenfeld effect, phasestiffness and the supercurrent, the transition tempera-ture etc. The phenomenological Ginzburg Landau the-ory captures well the criticality of the superconductingtransition in terms of a second order transition, involvingthe spontaneous breaking of the global U (1) symmetry ofthe electronic Hamiltonian; this is known to be equivalentto the abelian Higgs field theory. The many-particle en-tanglement properties of the BCS wavefuction have alsobeen established more recently .In the presence of a magnetic field and/or disorder ,a superconducting thin film has been observed to un-dergo a transition to an insulating state of matter (seeRef. for a review). Several experimental and theo-retical and works offer evidence shows that this insu-lator has Cooper pairs present in it. Recent experi-ments reveal the existence of an intervening metal-lic phase, called the “Bose metal” phase , lying betweenthe superconducting and insulating phases. Despite con-siderable effort, the precise nature of such a Bose metalremains unclear. Notable among various theoretical ef-forts on the Bose metal is the development of a gaugetheory of Josephson junction arrays (JJA) , whichshow the Bose metal can be described by a topologicalChern-Simons field theory emerging out of the nonlocalinteraction between the quasiparticle and vortex exci-tations of the superconducting system. Diamantini etal. show that such a topological insulating phase pos-sesses, at finite temperatures, a longitudinal conductancemediated by time-reversal symmetry preserved counter-propagating edge modes. Hansson et al. further estab-lished that the abelian Higgs model equivalent of the s-wave superconductor possesses other important featuresof topological order, i.e., a nontrivial ground state degen-eracy revealed on a multiply connected spatial manifold(such as a torus) and charge fractionalization. The de-generate ground states are labelled by topological quan-tum numbers corresponding to the eigenvalues of Wil-son and ’t-Hooft loops . More recently, the authorsof Ref. have shown that the gapped topological bulkfor s-wave pairing does not possess gapless edge states.Indeed, such a state displays a vanishing Hall conduc-tance. Further, they also study the nature of the topo-logical order in other spin-singlet superconductors . TheCooper pair insulating phase has also been studied inlattice bosonic superconductors using vortex-boson dual-ity , and its topological order investigated in the twodimensional variant .Insight into such a topologically ordered insulating phase(or Bose metal) from a microscopic approach, however, remains to be developed. Indeed, this is the primarygoal of this work. Thus, we are going to present mi-croscopic Hamiltonian for this novel phase of quantummatter, demonstrating it to be an insulator with topo-logical ordered gapped ground state manifold. Impor-tantly, we will find that such a state is emergent purelyfrom quantum fluctuations arising from inter-particle in-teractions. Further, such quantum fluctuations preservetranslational and time-reversal invariance, and are notnecessarily driven by either disorder or the coupling toan external magnetic field.In meeting this goal, we begin with a generalized Hamil-tonian for a Fermi liquid with a short-ranged repul-sive density-density interaction, as well as attractivepairing term. Suitably rewritten in terms of Ander-son pseudospins , this corresponds to a reduced BCSHamiltonian with additional repulsive interactions fa-miliar from Fermi liquid theory. Then, using the uni-tary renormalization group (URG) technique recently de-veloped by some of us , we resolve in a step-wisemanner the quantum fluctuations arising from the non-commutativity between the kinetic energy of the elec-trons and the inter-particle interactions. This involvesdecoupling one electronic Fock state in the momentumspace from all the other states it was connected to, suchthat the occupation number of the decoupled state is ren-dered as an integral of motion. The decoupling proceedsin a hierarchical fashion in terms of the kinetic energiesof the electrons, from high (near the Brillouin zone edge,UV) to low (near the Fermi surface, IR), and an effectiveHamiltonian is generated at every step. For the sake ofclarity, we have encapsulated the major aspects of theURG method in Appendix A. As shown in Sec. II, theRG flow stops at an IR fixed point, yielding a low energyeffective theory of a fixed number of condensed Cooperpairs but without any breaking of the macroscopic U (1)phase (i.e., with vanishing phase stiffness).We find that the emergent fixed point effective theory(eq.(11)) involves a non-local renormalised interactionbetween all pseudospins within the window that is emer-gent in k -space around the erstwhile Fermi surface, anddescribed by a collective zero-mode degree of freedomcomprised of these pseudospins. We call this symmetryunbroken phase of condensed Cooper pairs as the CooperPair Insulator (CPI). The RG procedure involves a novelenergy scale for quantum fluctuations ( ω ). In keepingwith this, the RG phase diagram obtained in Fig.1 clearlydisplays a quantum phase transition separating a CPIphase (at low ω ) and a gapless Fermi liquid metal (athigher ω ) for any repulsive interaction. The CPI Hamil-tonian corresponds to a collective quantum rotor modelcoupled to an effective Aharanov-Bohm (AB) flux (Φ).We demonstrate that there exist different ground statesof this CPI Hamiltonian related to one another by spec-tral flow upon tuning Φ, and displays the emergent quan-tisation of Φ under the RG flow. Importantly, we gainan idea of the accuracy of our method by numericallybenchmarking the energy per particle of the ground state(eq.(15)) of the CPI Hamiltonian obtained in the ther-modynamic limit with that obtained from exact diago-nalisation calculations (Fig.2).We have studied the topological features of the CPIHamiltonian in Sec.III. We establish first the gauge theo-retic nature of the emergent effective IR theory obtainedfrom the RG by demonstrating that the collective Hamil-tonian for the CPI can be written completely in terms ofa nonlocal Wilson loop operator. This is not surprising,given the zero-dimensional nature of the effective Hamil-tonian, where all degrees of freedom (within the emer-gent IR window in momentum space) are interacting withone another. The CPI Hamiltonian contains an topo-logical θ -term for a non-zero effective AB flux . Asshown by Yao and Lee , this zero-dimensional θ − termis in correspondence to a U (1) Chern-Simons topologicalterm in two spatial dimensions. This establishes the ef-fective theory for the CPI as the microscopic origin ofthe phenomenological U (1) Chern-Simons gauge theo-ries obtained by various people earlier . We thenreveal the nontrivial topological degeneracy and chargefractionalization signatures of the topologically orderedCPI ground state manifold through a flux-insertion spec-tral flow argument . These spectral flow argu-ments reveal a plateaux-like quantisation of the numberof Cooper pairs ( N , a topological quantum number) upontuning Φ through integer values (Fig.4), with the passagebetween the plateaux signifying topological transitions(at half-integer values of Φ). The collapse of the energyspectrum of the CPI phase with increasing Φ (Fig.5)yields another view of the transition between the CPIand metallic phases. As shown in Fig.6, the CPI stateis found to possess large helicity cross-correlations (Υ,a signature of inter-helicity two-particle scattering pro-cesses). Finally, in Fig.7, we track the passage into theCPI ground state by starting from a finite (but small)temperatures and lowering towards T = 0.In Sec.IV, we present a detailed analysis of the entan-glement features of the CPI ground state. The entangle-ment spectrum (ES) computed for the lowest CPI groundstate (Φ = 0) is found to be doubly degenerate for allpartitions of the system (Fig.8), reflecting the additionalparticle-hole symmetric nature of this CPI ground state.This degeneracy is lifted at the first topological transi-tion (Φ = 1 /
2, Fig.8). Interestingly, all other CPI groundstates (corresponding to non-zero positive integer valuesof Φ) are observed to show the degeneracy of the ESfor only the equipartitioned system (Fig.9); the degener-acy is again lifted at the transitions. We find that thebipartite entanglement entropy (EE) has a logarithmicdependence on the subsystem length L (Fig.10). Fur-ther, Fig.11 shows that the Φ = 0 plateau possesses thelargest EE, and that this is rapidly lowered to zero asΦ is tuned through various plateaux towards the gap-less metal. The topological transitions between the CPIground states display universal signatures in the varia-tions of their EE with system size L and flux Φ: EEshows a non-monotonic variation with L (Fig.12), with a peak value at L ≡ L ∗ that is universal as Φ is tuned(Fig.13). Further, for L < L ∗ , the EE versus L data fallsonto a universal curve (Fig.12). Finally, L ∗ approachesthe equipartition value at Φ is tuned close to the transi-tion between the CPI state and the metal (Fig.13). Vari-ations of the EE for the CPI ground states and the tran-sitions with temperature also show interesting signatures(Fig.14). The EE versus temperature curves for all val-ues of Φ corresponding to a given ground state finallymerge at T →
0. Further, the EE curves for all transi-tions are clearly distinct from those associated with theCPI ground states. Interestingly, for large temperatures,the dominance of thermal fluctuations in smearing outthe distinction between the CPI ground states and thetransitions can be seen in the fact that all
EE curveshave a linear variation against temperature and with auniversal slope.Having studied the topologically ordered CPI phase indetail, in in Sec.V, we turn to its connection with thesymmetry broken BCS ground state. We note that thelow energy effective theory for the CPI phase correspondsto an isotropic Lipkin-Meshkov-Glick (LMG) model .Recent studies have also revealed the existence ofsuch an effective Hamiltonian within the global U (1)symmetry broken BCS phase, arising from degrees offreedom that are singular under the Bogoliubov-Valatintransformation and corresponding to a “thin spectrum”or Anderson tower of states . Upon taking the systemsize to the thermodynamic limit, the collapse of this thinspectrum is believed to engender the spontaneous break-ing of the U (1) symmetry. On the other hand, it is be-lieved that in a mesoscopic superconducting grain, thepresence of a large charging term helps in making thethin spectrum robust . Thus, in Appendix B, we inves-tigate the passage from the CPI phase to the BCS phaseunder RG upon adding a global U (1) symmetry breakingterm in the CPI Hamiltonian.In keeping with this, we show in Fig.15 of Sec.V that the U (1) symmetry breaking field promotes fluctuations inthe number of Cooper pairs, while lowering the fluctua-tions in the conjugate global phase. Further, in Fig.16,we show that symmetry breaking effectively destroys thehelicity cross-correlation (Υ) among the Cooper-pairs un-der RG. The BCS wavefunction is a product state inmomentum space, with vanishing inter- k entanglementbetween Cooper pairs and maximum inter-spin entangle-ment due to the singlet configuration of each pair . Wefind that an increasing symmetry breaking field lowersthe inter- k entanglement between Cooper pairs presentin the CPI to zero, while raising the inter-spin entangle-ment to its BCS value (Fig.17). Further, the equal-sizepartitioned entanglement entropy shows a monotonic de-crease of entanglement for the CPI ground states (andtheir intervening transitions) with an increasing symme-try breaking field (Fig.18), while the metallic phase showsa non-monotonic variation.Another way by which to distinguish the CPI phase fromthe BCS superconductor lies in investigating the effects ofa Josephson coupling. Thus, in a second part of Sec.V, weconsider the case of two CPI systems whose bulk is cou-pled via Josephson coupling (i.e., we are ignoring all ef-fects from gapless edge states), and one of whom is placedin an U (1) symmetry breaking field. In Fig.19, we findthat while a Josephson coupling between two pure CPIsystems (i.e., with the symmetry breaking of the secondCPI system switched off) does not lead to any phase stiff-ness in another another, breaking the symmetry in one ofthem does give a proximity induced phase coherence inthe other. Indeed, an increasing symmetry breaking fieldtogether with a large Josephson coupling turn a CPI intoa phase stiff ground state. The generation of a Josephsoncurrent through the phase-locking, however, requires theseparate symmetry breaking of the two individual sub-systems. This is demonstrated in Fig.20, through theobservation of a periodic variation of the ground stateenergy E ( φ ) with the phase difference φ upon the intro-duction of separate non-zero symmetry breaking fields inboth CPI systems. In addition to the recent transportmeasurements of Ref. on the Bose metal, these find-ings serve as predictions for the experimental search ofthe CPI state of quantum matter.Next, in Sec.VI, we have carried out an entanglement RGusing the technique developed by us in Refs. . Thisstrategy can be described briefly as follows. Our URGanalysis has helped obtain the IR fixed point Hamilto-nians for the CPI and BCS phases, their ground statewavefunctions as well as the unitary transformations ofthe RG flow that led to them. for corresponding CPIand BCS wavefunctions. The quantum circuits corre-sponding to these unitary transformations is shown forthe CPI and BCS cases in Figs.21 and 22 respectively.Now, applying these unitary operations in reverse on agiven IR wave function, we obtain a family of wave func-tions under RG leading towards that pertaining to theUV theory. We compute the entanglement entropy ofvarious sizes of partitions from this set of wavefunctions,revealing the evolution of the entanglement entropy withRG. As shown in Fig.23, the EE for the constituent sub-blocks of the emergent CPI ground state are clearly dis-tinguished from that for all other partitions: their EEvaries little under RG from UV to IR, while that of allothers is lowered under RG from UV to IR. Further, thisanalysis reveals a remarkable hierarchy of scales of en-tanglement possessed by the CPI ground state. Thishierarchy of scales of entanglement gradually collapsesupon tuning a symmetry breaking field (Fig.24), until itis no longer present in the BCS ground state (Fig.25).Further, we can distinguish scaling towards the BCS andCPI ground states under RG flow. Finally, we concludein Sec.VII with a discussion of some future directions. II. EFFECTIVE THEORY OF THE COOPERPAIR INSULATOR (CPI)
We begin by deriving an effective Hamiltonian for an in-sulating state of matter comprised of a fixed number ofCooper pairs (referred to as the Cooper pair insulator,or CPI, in the introduction). For this, we will carry outa renormalisation group (RG) calculation on a systemof electrons in two dimensions with a generalised pairingHamiltonian, H = (cid:80) q H q pair , where H q pair = (cid:88) k (cid:15) k n k − (cid:88) k (cid:54) = k (cid:48) ,σ | W qkk (cid:48) | c † k − q,σ c †− k, − σ c − k (cid:48) , − σ c k (cid:48) − q,σ + U (cid:88) k (cid:54) = k (cid:48) ( n k − / n k (cid:48) − / , (1)where with q denotes the pair-momenta, (cid:15) k the kinetic en-ergy for electrons about a circular Fermi surface, −| W qkk (cid:48) | is the attractive pairing interaction, U ( >
0) a repulsivedensity-density interaction and n k = (cid:80) σ c † k,σ c k,σ . Notethat the case of a constant | W qkk (cid:48) | ∀ ( k, k (cid:48) , q ) correspondsto the attractive Hubbard model . We proceed by us-ing Anderson’s pseudospin contruction in the subspace n k − q, ↑ = n − k, ↓ : (cid:126)S k = 12 φ k .(cid:126)τ .φ † k , (2)where (cid:126)τ = ( τ z , τ x , τ y ) are the Pauli matrices and φ k =( c k ↑ , c †− k ↓ ). The pseudospins obey the standard commu-tation relation for spin-1/2: [ S ik , S jk ] = i(cid:15) jkl S jk . Then,we write H q pair as H q pair = − (cid:88) k ˜ (cid:15) k,q ( S zk,q −
12 ) − (cid:88) k (cid:54) = k (cid:48) | W qkk (cid:48) | S − k,q S + k (cid:48) ,q + h.c.)+ U (cid:88) k (cid:54) = k (cid:48) S zk,q S zk (cid:48) ,q , (3)where ˜ (cid:15) k,q = (cid:15) − k + (cid:15) k + q is the kinetic energy for a pair ofelectrons. In order to ensure the extensivity of the model, | W qkk (cid:48) | = | V qkk (cid:48) | /N (where N corresponds to the totalnumber of pseudospins, and hence 2 N the total numberof electrons). The special case of H q =0pair with | W q =0 kk (cid:48) | and U = 0 is called the Richardson pairing model (see Ref. and references therein).Following the strategy developed in Refs. , we nowcarry out a renormalisation group analysis on H q pair ; seeappendix A for details. The RG equations obtained for˜ (cid:15) and | W qkk (cid:48) | are∆˜ (cid:15) ( j ) k (cid:48) ,q ∆ log Λ j Λ = 14 | W ( j ) k Λ k (cid:48) | (cid:18) ω − ˜ (cid:15) ( j ) k Λ ,q − U (cid:19) , (4)∆ | W q, ( j ) k (cid:48) k (cid:48)(cid:48) | ∆ log Λ j Λ = − | W q, ( j ) k Λ k (cid:48) || W q, ( j ) k Λ k (cid:48)(cid:48) | (cid:18) ω − ˜ (cid:15) ( j ) k Λ ,q − U (cid:19) , (5)where the index ( j ) represents the RG step number, | W q, ( j ) k (cid:48) k (cid:48)(cid:48) | = | V q, ( j ) k (cid:48) k (cid:48)(cid:48) | /N ( j ) (for N ( j ) being the number ofremnant pseudospins at the j th RG step) and k Λ the mo-mentum at a k -space window (Λ) lying on a radial andaround the circular Fermi surface. The symbol ω repre-sents an energy scale for the quantum fluctuations thatlead to UV-IR mixing. Further, we note that the RGstep index j starts from the number of pseudospin ( N )lying within the bare window Λ and proceeds to smallervalues. At every step of the RG, a pseudospin with mo-mentum k Λ lying on a given direction radial to the Fermisurface is disentangled from the rest ( ∀ k < k Λ ); the U (1)symmetry of the circular Fermi surface ensures that theRG is carried out simultaneously for all pseudospins withmomentum k Λ . Note that the repulsive coupling U doesnot flow under RG, as two-particle quantum fluctuationsdo not lead to the renormalisation of this term. Instead,it appears as a Hartree-shift in the pseudospin Greensfunction G ps = [ ω − ˜ (cid:15) ( j ) k Λ ,q − U ] − present in the RG equa-tions given above .It can be seen that the normalization for ˜ (cid:15) ( j ) k (cid:48) ,q is RG rele-vant for ω > (˜ (cid:15) k Λ ,q / U/ | W q, ( j ) k (cid:48) k (cid:48)(cid:48) | is RGrelevant for ω < (˜ (cid:15) ( j ) k Λ ,q / U/ (cid:15) − k = (cid:15) k fora circular Fermi surface, it is easily seen that ˜ (cid:15) ( j ) k,q ≥ (cid:15) ( j ) k .Therefore, given the denominator ω − ˜ (cid:15) ( j ) k Λ ,q / − U/ q -sector forthe lowest quantum fluctuation energyscale ( ω = 0) cor-responds to the case of q = 0 (i.e., Cooper pairs withzero centre of mass momentum). Thus, we will hence-forth study only the case of ˜ (cid:15) ( j ) k,q =0 ≡ ˜ (cid:15) ( j ) k = 2 (cid:15) ( j ) k and H q =0pair ≡ H .We define mode decompositions of the dispersion (cid:15) ( j ) k andthe pairing coupling | V ( j ) kk (cid:48) | as follows¯ (cid:15) ( j ) l = 1 √ N ( j ) (cid:88) (cid:126)k e ikl (cid:15) ( j ) k , ¯ V ( j ) ll (cid:48) = 1 N ( j ) (cid:88) k,k (cid:48) e i ( kl + k (cid:48) l (cid:48) ) | V ( j ) kk (cid:48) | , (6)such that the RG equations for ¯ (cid:15) ( j ) l =0 and ¯ V ( j ) l =0 ,l (cid:48) =0 areobserved to dominate under the RG flow over all othermodes for a thermodynamically large system :Re (cid:18) ∆¯ (cid:15) ( j ) l =0 ∆¯ (cid:15) ( j ) l (cid:54) =0 (cid:19) > , Re (cid:18) ∆ ¯ V ( j ) l =0 ,l (cid:48) =0 ∆ ¯ V ( j ) l (cid:54) =0 ,l (cid:48) (cid:54) =0 (cid:19) > . (7)The zero mode ¯ (cid:15) ( j )0 is related to the center of mass ki-netic energy ¯ (cid:15) ( j ) : ¯ (cid:15) ( j )0 = ( (cid:80) k (cid:15) ( j ) k ) / √ N ( j ) = √ N ( j ) ¯ (cid:15) ( j ) .Similarly, the zero mode ¯ V ( j )00 is connected to it’s centerof mass value : ¯ V ( j ) = ( (cid:80) kk (cid:48) V ( j ) kk (cid:48) ) / ( N ( j ) ) = V ( j )00 /N ( j ) .Thus, the RG relations of these zero modes is equivalentto the study of the center of mass degrees of freedoms∆¯ (cid:15) ( j ) = 14 | ¯ W ( j ) | (cid:18) ω − ¯ (cid:15) ( j ) − U (cid:19) = − ∆ | ¯ W ( j ) | , (8) where ¯ W ( j ) = ¯ V ( j ) /N ( j ) . In this way, we observe belowthe emergence of the well known reduced BCS model atthe stable fixed point of the RG eq.(8).The relation between the two RG equations (eq.(8)) leadsto a RG invariant: ¯ (cid:15) ( j ) + | ¯ W | ( j ) = C , C ∈ R . Fromthis invariant, it can now be seen that when the kineticenergy ¯ (cid:15) is RG relevant, the attractive coupling | ¯ W | is RGirrelevant and vice versa. We can now write the effectiveHamiltonian obtained at the stable fixed point of the RGflow as H coll = − (cid:15) ∗ N ∗ (cid:88) k ( S zk −
12 ) − ¯ V ∗ N ∗ (cid:88) k (cid:54) = k (cid:48) ( S + k S − k (cid:48) + h.c.)+ U (cid:88) k (cid:54) = k (cid:48) S zk,q S zk (cid:48) ,q , (9)where ¯ (cid:15) ∗ , ¯ V ∗ and N ∗ are the fixed point values of ¯ (cid:15) ( j ) ,¯ V ( j ) and N ( j ) respectively reached at the endpoint of theRG flow (and ¯ V ∗ /N ∗ = C − ¯ (cid:15) ∗ ). Finally, by defining thecomposite pseudospin (cid:126)S = (cid:80) k (cid:126)S k , we can rewrite theHamiltonian H coll (upto additive constants) as H coll = − (cid:15) ∗ N ∗ S z − ¯ V ∗ N ∗ ( S + S − + S − S + ) + U S z = − (cid:15) ∗ N ∗ S z − ¯ V ∗ N ∗ ( S − S z ) + U S z . (10)While the first term arises from the electronic kinetic en-ergy, the second the potential energy saved by the forma-tion of pairs (i.e., the condensation energy) and the thirdrepresents the repulsive charging energy cost of the elec-trons that form the Cooper pairs. Note that the Hamil-tonian eq.(9) has the global U(1) symmetry of the gen-eralised pairing Hamiltonian eq.(1). This is expected, asRG transformations are symmetry preserving.We present the RG phase diagram below in Fig.1 bya numerical solution of the RG equations for the elec-tronic dispersion along a radial to the circular Fermisurface being (cid:15) k = 2 t cos( k ), a bare window near theFermi energy v F Λ = 0 . t , a constant bare attractivecoupling | V qkk (cid:48) | = 4 t/N and the total number of pseu-dospins N = 51. The phase diagram is presented in theplane of the effective quantum fluctuation energy scale ω and the repulsive coupling U (and both are in units of thekinetic energy bandwidth 4 t ). It clearly shows that theCooper pair insulator (CPI) is stabilised at lower valuesof ω for all U , and that a metallic phase (lying at highervalues of ω ) is obtained through a quantum phase tran-sition into a gapless Fermi liquid metallic phase.For the sake of simplicity, we will henceforth focus on thecase of U = 0 H coll = − (cid:15) ∗ N ∗ S z − ¯ V ∗ N ∗ ( S − S z ) , = ¯ V ∗ N ∗ ( S z − Φ) − ¯ V ∗ N ∗ S , (11)where Φ ≡ ¯ (cid:15) ∗ / ¯ V ∗ and we have ignored a constant( ∝ (¯ (cid:15) ∗ /N ∗ ) ). Further, for N ∗ pseudospins, 0 ≤ S ( ∈ FIG. 1. Renormalisation group phase diagram for the effec-tive pairing Hamiltonian given in eq.(3). The y-axis repre-sents the energyscale ( ω ) for quantum fluctuations that areresolved under the RG flow, while the x-axis represents the re-pulsive interaction ( U ) in the parent metal (whose electronsfeel the additional attractive pairing). The phase diagramclearly shows the existence of an emergent Cooper pair insu-lator (CPI) phase at low ω (i.e., energyscales proximate tothe Fermi surface of the parent metal) for all U ≥ Z ) ≤ N ∗ / − S ≤ S z ( ∈ Z ) ≤ S and the number ofCooper pairs is given by N c = S − S z . It is easily seenthat both S and S z commute with H coll . Further, H coll arises a global collective angular momentum degree offreedom, S z , and possesses the form of a quantum parti-cle whose dynamics is confined to a circle and coupled toan (effective) dimensionless Aharanov-Bohm (AB) fluxΦ. As well will discuss in a later section, this pointsto a topological property possessed by the ground statemanifold of its Hilbert space.We note here, however, that the emergent Hilbert spacecorresponding to H coll possesses the property of spectralflow . First, observe that minimisation of the energy isachieved under RG flow for the case of Φ = ¯ (cid:15) ∗ / ¯ V ∗ → S = N ∗ / S z = 0,corresponding to the largest number of Cooper pairs( N c = N ∗ / S z , that can be reached under RG for valuesof the effective AB flux Φ (cid:29) C and the quantumfluctuation scale ω . As shown in Appendix A, the finalfixed point value of ¯ (cid:15) ∗ is given by ¯ (cid:15) ∗ = 2 ω − U . Usingthis together with the relation for the RG invariant is C = ¯ (cid:15) ∗ + | ¯ W | ∗ , we find the effective flux at the IR fixed point isΦ = ¯ (cid:15) ∗ | ¯ V ∗ | = 2 ω − U N ∗ | ¯ W ∗ | = 2 ω − U N ∗ ( C − ω + U ) ≡ n ∈ Z . (12)For the case of U = 0, this leads to C = 2 ω ( 1 + nN ∗ nN ∗ ) → ω for n (cid:29) . (13)Finally, we note that the ground state wavefunction ofthe U (1)-symmetric CPI state with S z = 0 (i.e., at strongcoupling) is given by | ψ g (cid:105) = N (cid:18) (cid:88) k c †− k ↓ c † k ↑ (cid:19) S | vac (cid:105) (14)where | vac (cid:105) is the state that contains no Cooper pairs,and N is a normalisation factor. By acting with H coll on | ψ g (cid:105) , we obtain the ground state energy density as E g N ∗ = − ¯ V ∗ N ∗ S = − ¯ V ∗ N ∗ N ∗ N ∗ − ¯ V ∗ N ∗ ) (cid:39) − ¯ V ∗ N ∗ >> . (15)In order to gauge the accuracy of the effective Hamil-tonian (eq.(11)) and ground state wavefunction (eq.(14))obtained from the RG procedure, we compare the groundenergy density value obtained in the thermodynamiclimit (eq.(15)) from a finite-size scaling analysis with thatobtained from a finite-size scaling for exact diagonaliza-tion (ED) studies of small systems of the bare Hamilto-nian (eq.(1)) for U = 0 , | V qkk (cid:48) | = 2 (in units of a hop-ping parameter t ), | W qkk (cid:48) | = 2 /N . For a U (1)-symmetricFermi surface, it suffices to compare the energy densityvalue obtained along any one diameter of the sphericalFermi volume. As is shown in Fig.2, we find excellentagreement between the results obtained in the thermody-namic limit from the two approaches: E g /N ∗ (cid:39) − . t from the RG, as against E g /N ∗ (cid:39) − . t obtained fromED. This indicates the efficiency of the RG method inpreserving the spectral content during the flow towardsthe stable IR fixed point, and offers confidence in theanalyses of subsequent sections that offer insight into theproperties of the CPI phase. III. TOPOLOGICAL FEATURES OF CPI
We will, in this section, study the topological propertiesof the many-body system described by the stable fixedpoint effective Hamiltonian H coll (eq.(11)) obtained fromthe RG. A. Topological nature of the effective theory
We begin by showing that the effective Hamiltonian forthe plateau state in the strong coupling limit (i.e., with
FIG. 2. Finite-size scaling of the ground state en-ergy density E g /N ∗ under RG for system sizes N =100 , , , , , , , , , N ∗ is the number of pseudospinswithin the emergent subspace at the stable fixed point of theRG. The red dash-dot line shows a linear fit to the finite-sizescaling data. The blue dash-dot line shows the thermody-namic limit value for E g /N obtained from a finite-size scalingexact diagonalisation study of small systems ranging between5 −
15 pseudospins. Inset: Zoom of RG finite-size scaling datafor N = 3200 , , , , k -space window (dark grey) of size 2Λaround the Fermi surface (FS, yellow circle) formed under RGfor the CPI phase. ˆ s represents a given direction in k -spacenormal to the FS. (Right) Construction of the respective twist(( ˆ O ˆ s , ˆ O Λ ) and translation operators (( ˆ T ˆ s , ˆ T ˆ s ⊥ ) defined on thetorus created by imposing periodic boundary conditions onemergent window in k -space (i.e., on all directions ˆ s ). ¯ (cid:15) ∗ = 0) system) is purely topological. This will be doneby rewriting H coll (¯ (cid:15) ∗ = 0) = − ¯ V ∗ N ∗ ( S x + S y ) in termsof emergent Wilson loop operators defined on a toruscreated by imposing periodic boundary conditions in theΛ-direction (i.e., the window in k -space that defines theCPI condensate, see Fig.3 below).We define the k -space translation ( T ˆ s , brown curved linein Fig.3) and twist operators ( ˆ O i ˆ s , blue dashed line inFig.3) T ˆ s : S i Λ , ˆ s → S i Λ+ δ Λ , ˆ s , ˆ O i ˆ s = exp (cid:20) πN ∗ i N ∗ − (cid:88) n =0 nS in Λ , ˆ s (cid:21) , (16)where the twist operator ˆ O i ˆ s spans all values of Λ in the ˆ s direction, and imparts a gradual twist to the pseudospins S in Λ , ˆ s such that the total twist imparted across the ˆ s di-rection is 2 π . Further, we can also define a compositetwist operator that spans the entire torus shown in Fig.3˜ O i = N ∗ − (cid:89) n =0 ˆ O R n ˆ s = exp (cid:20) πiN ∗ N ∗ − (cid:88) n,m =0 mS imδ Λ ,R n ˆ s (cid:21) . (17)Then, we compute the following (nonlocal) Wilson loopoperator W i ( i = ( x, y )) defined in terms of T ˆ s and ˜ O i W i = T ˆ s ˜ O i T † ˆ s ˜ O i † = exp (cid:20) πiN ∗ N ∗ − (cid:88) m,n =0 S im Λ ,R n ˆ s (cid:21) × exp (cid:20) πi N ∗ − (cid:88) n =0 S i Λ=0 ,R n ˆ s (cid:21) , (18)where the first term on the right hand side imparts thetwist to the centre of mass of the torus of pseudospins.The second denotes the trivial phase twist accumulatedat a virtual boundary defined on the torus by the curveΛ = 0, exp (cid:20) πi N − (cid:88) n =0 S i Λ=0 ,R n ˆ s (cid:21) = 1. Thus, we obtain thecomposite pseudospin ( S i , i = ( x, y )) in terms of W i as S i = (cid:88) m,n S im Λ ,R n ˆ s = N ∗ π Im (cid:2) ln( W i ) (cid:3) , (19)such that the U (1)-symmetric effective Hamiltonian ob-tained from the RG can be written purely in terms of theemergent W i as H coll = − ¯ (cid:15) ∗ π Im [ln( W z )] − N ∗ ¯ V ∗ π (cid:88) i = x,y (cid:18) Im (cid:2) ln( W i ) (cid:3) (cid:19) . (20)This shows us that the U (1) symmetry is encoded in theinvariance of the Wilson loops W i large gauge transfor-mations , and that the nonlocal nature of their dy-namics is encoded in the dependence of H coll on W i . Inthis way, we can clearly see the emergence of an effectivegauge theory from the microscopic Hamiltonian eq.(1).We note that H coll can also be written in terms of anotherset of emergent Wilson loop operators W i obtained froma different pair of translation and twist operators definedon the torus (see Fig.3) T ˆ s ⊥ : S i Λ , ˆ s → S i Λ ,R ˆ s , ˆ O i Λ = exp (cid:20) πN ∗ i N ∗ − (cid:88) n =0 nS i Λ ,R n ˆ s (cid:21) , (21)such that we can redefine as earlier the following compos-ite twist ( ˜ O i ) and pseudospin ( S i ) and the Wilson loop W i as˜ O i = N ∗ − (cid:89) m =0 ˆ O mδ Λ = exp (cid:20) πN ∗ i N ∗ − (cid:88) n,m =0 mS imδ Λ ,R n ˆ s (cid:21) , (22) S i = N ∗ π Im (cid:2) ln( e iπ W i ) (cid:3) , W i = T ˆ s ⊥ ˜ O i T † ˆ s ⊥ ˜ O i † . (23)We can once again write the effective Hamiltonian H coll as H coll = − ¯ (cid:15) ∗ π Im [ln( W z )] − N ∗ ¯ V ∗ π (cid:88) i = x,y (cid:18) Im (cid:2) ln( W i ) (cid:3)(cid:19) , (24)where ¯ (cid:15) ∗ = ¯ (cid:15) ∗ − N ∗ V ∗ / H coll ) obtainedfrom the RG can be written completely in terms ofglobal collective gauge degrees of freedom (i.e., the Wil-son loops W i and W i ) is not surprising. Indeed, followingRefs. on the effective theory for the CPI phase beinga Chern-Simons gauge field theory, we expect that theeffective Hamiltonian for the CPI cannot be written interms of local degrees of freedom. Thus, for finite andnon-zero ¯ (cid:15) ∗ , the association of a U (1)-symmetric Chern-Simons gauge field theory with the effective quantum ro-tor Hamiltonian H coll (eq.(11)) in 0-spatial dimensionscan be argued for as follows. The action correspondingto H coll contains a 0-dimensional topological θ -term θ = i ¯ (cid:15) ∗ N ∗ ¯ V ∗ (cid:90) β dτ ∂φ∂τ , (25)written in terms of a global phase φ conjugate to S z , suchthat − i (cid:126) ∂/∂φ ≡ S z = − i (cid:126) [ S x , S y ] = iN ∗ π (cid:126) [ln W x , ln W y ] , (26)and with a Berry phase given by γ = 2 π ¯ (cid:15) ∗ ¯ V ∗ = 2 π Φ .It was shown by Yao and Lee that such a θ -term in0-spatial dimensions is in precise correspondence witha U (1) Chern-Simons topological term in 2-spatial di-mensions. Hansson et al. show that the Chern-Simonsterm encodes a topological coupling of the vorticity (orwinding number part) of the global phase field φ to afield associated with the quasiparticle excitations. In thisway, they show that the system, in the presence of adynamical gauge field, possesses gauge invariance underlarge gauge transformations. Further, they argue thatthe time-reversal invariance of the original problem ne-cessitates that the K -matrix of the equivalent 2-flavourmixed Chern-Simons theory is K = 2 σ x (see also Ref. ).Then, the topological ground state degeneracy on a torusof genus g is given by | Det ( K ) | g = 4 g . This clari-fies that the topologically ordered condensate of vorticesobserved in Ref. , arising from the coupling the globalphase of the superconducting ground state to dynamicalelectromagnetic gauge fields, corresponds to the Cooperpair number fixed insulating state of matter (the CPIphase) found at the stable fixed point of the RG flow.Next, we will demonstrate the 4-fold degeneracy for thespecial case of ¯ (cid:15) ∗ = 0 = Φ. B. Topological degeneracy at
Φ = 0 . In order to unveil a ground state degeneracy at Φ = 0,we follow the adiabatic flux insertion treatment of Os-hikawa . For this, we define the following momentumtranslation ( ˆ T ˆ s ⊥ ) (see Fig.3) and twist ( O ˆ s ⊥ ph ) operatorsˆ T ˆ s ⊥ = e i ˆΘ ˆ s ⊥ , O ˆ s ⊥ ph = exp (cid:26) i πN N − (cid:88) p =0 kS z ( R p k ˆ s ) (cid:27) , (27) S z ( R p k ˆ s ) = (cid:88) Λ ( Rpk ˆ s ) S z Λ ( Rpk ˆ s ) , (28)where ˆΘ ˆ s ⊥ denotes the center of mass angular positionalong ˆ s ⊥ . With this, we find T ˆ s ⊥ O ˆ s ⊥ ph T † ˆ s ⊥ = O ˆ s ⊥ ph exp (cid:20) i πN (cid:18) N S z − N − (cid:88) k ˆ s =0 S zk ˆ s (cid:19)(cid:21) , = O ˆ s ⊥ ph exp (cid:20) iπ (2 n + 1) (cid:21) = O ˆ s ⊥ ph e iπ , (29)where we have set (cid:80) N − k ˆ s =0 S zk ˆ s = 0 and S z = (2 n + 1) (and 2 n + 1 being the number of pseudospin states inthe direction ˆ s ) in the second line in order to obtain thethird. Thus, {T ˆ s ⊥ , O ˆ s ⊥ ph } = 0 , [ H ( ∗ ) , T ˆ s ⊥ ] = 0 = [ H ( ∗ ) , O ˆ s ⊥ ph ] . (30)These relations imply the existence of two degeneratestates labelled by the center of mass angular positionalong ˆ s ⊥ (Θ ˆ s ⊥ ) | Θ ˆ s ⊥ = 0 (cid:105) , | Θ ˆ s ⊥ = π (cid:105) , (31)with transitions from one to the other taking place viathe twist operator O ph O ˆ s ⊥ ph | Θ ˆ s ⊥ = 0 (cid:105) = | Θ ˆ s ⊥ = π (cid:105) (32)We now unveil another two-fold degeneracy of the groundstate manifold. By first defining pseudospin degrees offreedom along a given direction of momentum space (ˆ s )that are resolved in terms of the eigenvalue of the helicityoperator ( η = ± S ˆ s,z = (cid:88) k δ η k , − S ˆ s,zk,η k , ˜ S ˆ s,z = (cid:88) k δ η k , +1 S ˆ s,zk,η k , (33)we define a helicity twist operator O ˆ sH O ˆ sH = e i π (0 . ˜ S ˆ s,z +1 . ˜ S ˆ s,z ) . (34)Then, we define the helicity inversion operator T H T ˆ sH ≡ e i ˆ H ˆ se : S ˆ s,zk,η k → S ˆ s,zk, − η k , (35)where ˆ H ˆ se corresponds to the generator of helicity inver-sion of the center of mass along ˆ s . These helicity twistand translation operators follow the algebra T ˆ sH O ˆ sH T ˆ s, † H = O ˆ sH × exp (cid:20) i π (cid:18) S ˆ s,z − [ ˜ S ˆ s,z + ˜ S ˆ s,z ] (cid:19)(cid:21) = O ˆ sH × exp (cid:18) i π ˜ S ˆ s,z (cid:19) , = O ˆ sH × e i π (2 n +1) = O ˆ sH e iπ , (36)where we have set [ ˜ S ˆ s,z + ˜ S ˆ s,z ] = 0 and ˜ S ˆ s,z = (2 n + 1) in the second line in order to obtain the third. Thus, {T ˆ sH , O ˆ sH } = 0 , [ H ( ∗ ) , T ˆ sH ] = 0 = [ H ( ∗ ) , O ˆ sH ] . (37)Again, these relation imply the existence of two degener-ate states labelled by the eigenvalue of the generator ofhelicity inversion ( ˆ H ˆ se ) of the center of mass along ˆ s | H ˆ se = 0 (cid:105) , | H ˆ se = π (cid:105) , (38)with transitions from one to the other taking place viathe twist operator O ph T ˆ sH | H ˆ se = 0 (cid:105) = | H ˆ se = π (cid:105) . (39)Importantly, we find that[ T ˆ sH , T ˆ s ⊥ ] =0= [ O ˆ sH , O ˆ s ⊥ ph ] , {T ˆ sH , O ˆ sH } =0= {T ˆ s ⊥ , O ˆ s ⊥ ph } . (40)Thus, these four operators together label the four-folddegenerate ground state manifold. As noted above, thismatches the result for the phenomenological BF Chern-Simons gauge field theory formulation of Hansson et al. Finally, the topological order is protected by the spectralgap ∆ top = E S z =1 − E S z =0 = V ∗ N ∗ (41)separating the degenerate ground state manifold fromthe lowest lying excited state of the effective Hamilto-nian H coll (eq.(11)). Further, these ground states arealso separated from the single-particle excitations bya many-body gap (∆ MB ) that arises from the helicitybackscattering term, ( ¯ V ∗ /N ∗ ) (cid:80) k ( S + k S −− k + h .c. ) , con-tained within the effective Hamiltonian H coll (eq.(11)) ∆ MB = (cid:104) ψ | ¯ V ∗ N ∗ (cid:88) k ( S + k S −− k + h.c.) | ψ (cid:105) . (42) C. Spectral flow, plateau ground states andtopological quantum numbers
As mentioned in the previous subsection, by tuning theratio Φ ≡ ¯ (cid:15) ∗ / ¯ V ∗ , we can access ground states with differ-ent number of Cooper pair bound states (i.e., the eigen-value of the operator S z ). We will now study the passagebetween these ground states, and also show the journey FIG. 4. Variation of (cid:104) S z (cid:105) (red curve, and y-axis on left) and (cid:104) ∆ S z (cid:105) (blue curve, and y-axis on right) with effective ABflux Φ at small temperature T = 0 .
01 (in units of k B ). (cid:104) S z (cid:105) and (cid:104) ∆ S z (cid:105) are computed using the effective CPI Hamiltonianeq.(11). See text for discussion. towards a metallic (gapless) ground state (i.e., with avanishing number of Cooper pair bound states).We recall that for Φ = 0, the ground state is given by | ψ g (cid:105) = | S = N ∗ / , S z = 0 (cid:105) , i.e., a state with N ∗ Cooperpairs. The action of S + on | ψ g (cid:105) is S + (cid:12)(cid:12)(cid:12)(cid:12) N ∗ , (cid:29) = (cid:112) ( S − S z )( S + S z + 1) (cid:12)(cid:12)(cid:12)(cid:12) N ∗ , (cid:29) = (cid:114) N ∗ N ∗ (cid:12)(cid:12)(cid:12)(cid:12) N ∗ , (cid:29) , (43)i.e., lowers the Cooper pair number by 1. Energetically,this is equivalent to a value of the parameter Φ in H coll within the range 0 . < Φ < .
5. In the same way, m < Φ < m + 1 ( m ∈ Z ) leads to a ground state | N ∗ / , m (cid:105) , such that for Φ > ( N ∗ − /
2, we attain aground state with a vanishing number of Cooper pairs.This amounts to reducing the spectral gap of the CPIphase in a step-like manner, until a gapless spectrum(the “metal”) is attained. Thus, tuning the parameterΦ amounts to a process of spectral flow between variousground states. Each gapped ground state (correspondingto different values of S z ) possesses topological features(as discussed in the previous subsection).At zero temperature, in the presence of a pairing-inducedgap ∆, these gapped ground states will show plateaux ina variation of (cid:104) S z (cid:105) with Φ = ¯ (cid:15) ∗ / ¯ V ∗ (see Fig.4 for a small k B T = 0 . m + 1 /
2, the ground state be-comes degenerate via level-crossings, i.e., a linear com-bination of | S = N ∗ / , S z = m (cid:105) and | S = N ∗ / , S z = m + 1 (cid:105) | ψ m PT (cid:105) = c | N ∗ , m (cid:105) + c | N ∗ , m + 1 (cid:105) , (44)where | c | + | c | = 1. As shown in Fig.4, these corre-spond to transitions between plateaux in (cid:104) S z (cid:105) and leadto large fluctuations ( (cid:104) ∆ S z (cid:105) ) in S z . It can be shown0 FIG. 5. Variation of the energy spectrum ( E , y-axis) of theeffective CPI Hamiltonian (eq.(11)) with the effective AB fluxΦ (x-axis) for a system of 200 electrons. The colour scale rep-resents the number of Cooper pairs ( N C ). The bright yellowborder represents the transition between the metal (white re-gion) and CPI phases upon tuning Φ.FIG. 6. Variation of the normalized helicity cross correlation(Υ) with the ground state eigenvalue of S z ( M ) for a systemof 200 electrons. M is increased by increasing the flux Φ.The large values of Υ for M → → M → that the largest (cid:104) ∆ S z (cid:105) is obtained for c = 1 / √ c .As noted above, the final level-crossing is attained atΦ = ( N ∗ − /
2. In Fig.5, we show a variation of theenergy for excited states obtained from H coll (computedwith respect to the ground state energy) with the pa-rameter Φ, and where the colour scale denotes the num-ber ( N c ) of Cooper pairs in a given state. The plotclearly shows the collapse of the excitation spectrum ofthe gapped plateaux as Φ is tuned towards passage fromthe final plateau into the gapless metal (white space inFig.5).This is reinforced by a study of the helicity cross corre-lation (Υ), i.e., inter-helicity two-particle scattering, de-fined as Υ = (cid:104) S + S − (cid:105) + (cid:104) S − S + (cid:105) − (cid:104) S + (cid:105)(cid:104) S − (cid:105) , (45)where the expectation value is taken with respect to theground state. For | ψ g (cid:105) = | S, S z = M (cid:105) , it can be shownthat Υ = 2( S + S − M ). A plot of Υ versus M in FIG. 7. Variation of (cid:104) S z (cid:105) and (cid:105) ∆ S z (cid:105) with lowering temper-ature (from T = 1000 to T = 0 (in units of k B ), and in-dicated through arrows) and flux Φ = 0 towards the threeCPI ground states S z = 0 , ± S = 1system. (cid:104) S z (cid:105) and (cid:105) ∆ S z (cid:105) are computed using the CPI Hamil-tonian (eq.(11)). See text for discussion. Fig.6 shows that the strength of inter-helicity scatteringgradually reduces as M increases, i.e., the parameter Φis tuned towards the gapless metal.Upon increasing the temperature, the plateaux aresteadily degraded and the fluctuations (cid:104) ∆ S z (cid:105) at the tran-sitions increase in strength (Fig.4). In Fig.7, we show aplot of (cid:104) ∆ S z (cid:105) against (cid:104) S z (cid:105) obtained from H coll for differ-ent values of the parameter Φ and temperature T forthe case of S = 1 , S z = 0 , ±
1. The blue curves arefor − ≤ Φ < − ≤ (cid:104) S z (cid:105) <
0) and the greencurves are for 0 < Φ ≤ < (cid:104) S z (cid:105) ≤ T generically leads to aplateau ground state ( (cid:104) S z (cid:105) = 0 , ± , (cid:104) ∆ S z (cid:105) = 0). On theother hand, there also exist special cases when lowering T leads to (unstable) ground states located precisely at theplateau transitions ( (cid:104) S z (cid:105) = 1 / ±(cid:104) ∆ S z (cid:105) ). While thefigure shows the numerical computation for H coll with S = 1, we have observed that a similar plot for a muchlarger value of S also shows the same “dome”-like struc-ture of the curves. IV. ENTANGLEMENT FEATURES OF CPIA. Entanglement Spectrum
We begin our discussion of the entanglement features ofthe CPI phase with an investigation of the entanglementspectrum for the plateau ground states, whose wavefunc-tion is given by | ψ P (cid:105) = | S, S z (cid:105) for Φ = m , m ∈ Z .We will also present the entanglement spectrum at theplateau transitions where the ground state is degenerate, ψ PT = √ ( | S, S z (cid:105) + | S, S z + 1 (cid:105) ) for Φ = m/ , m ∈ Z .We first Schmidt decompose the state | ψ P (cid:105) (with n =1 FIG. 8. Entanglement spectrum (ES, ξ i = − log λ i ) of asubsystem of size L in the ground state wavefunction at Φ = 0(eq.(14), upper plot) and at Φ = 1 / N ∗ = 488 Cooper pairs and as a function of thesubsystem size L . The index i labels the ES eigenvalues. Theupper inset shows the double degeneracy for all levels, whilethe lower inset shows that the degeneracy is lifted at Φ = 1 / S z + N ∗ / ↑ -pseudospins) into subsystems of length L and N ∗ − L (with l and n − l ↑ -pseudospins respectively) | ψ P (cid:105) = l max (cid:88) l = l min λ nl | L, l (cid:105) ⊗ | N ∗ − L, n − l (cid:105) , (46)where λ nl are the Schmidt coefficients. The number l ranges within l min and l max given by l max = n for n ≤ L and (47)= L for n > L , (48) l min = 0 for n ≤ ( N ∗ − L ) and (49)= n − ( N ∗ − L ) for n ≤ ( N ∗ − L ) . (50)From the pure state density matrix ρ P = | ψ P (cid:105)(cid:104) ψ P | , wecan then obtain a reduced density matrix ρ P,L ( n ) for asubsystem of L pseudospins ρ P,L ( n ) = l max (cid:88) l = l min ( λ nl ) | L, l (cid:105)(cid:104)
L, l | . (51) FIG. 9. Entanglement spectrum (ES, ξ i = − log λ i ) of asubsystem of size L in the ground state wavefunction at Φ =240 (eq.(14), upper plot) and at Φ = 240 . N ∗ = 488 Cooper pairs and as a functionof the subsystem size L . The index i labels the ES eigenvalues.The inset in the lower figure shows the lifting of degeneraciesobserved in the upper plot. The Schmidt coefficients ( λ nl ) are determined by the com-binatorial factor that specifies the number of ways onecan choose l ↑ spins from n ↑ spins:( λ nl ) = C Ll C N ∗ − Ln − l C N ∗ n = L !( N ∗ − L )! n !( N ∗ − n )! l !( L − l )!( n − l )!( N ∗ − L − n + l )! N ∗ ! . (52)At the plateau transitions, we start with a pure statedensity matrix obtained from the linear superpositionstate | ψ PT (cid:105) , ρ PT = | ψ PT (cid:105)(cid:104) ψ PT | , and proceed identicallyas above to obtain the reduced density matrix ρ PT,L ( n ).The entanglement spectrum (ES) is obtained from theSchmidt eigenvalues, ξ ni = − log ( λ ni ), for ρ P,L ( n ) and ρ PT,L ( n ) with given values of N ∗ and L .For a fixed N ∗ = 488 pseudospins, we plot the ES forvarious values of the reduced partition size L for the caseof the plateau at strong coupling Φ = 0 and the firstplateau transition at Φ = 1 / L at Φ = 0 is revealed by the small splitting revealedat Φ = 1 /
2. This double degeneracy for the entire spec-trum reflects the additional particle-hole symmetric na-2
FIG. 10. The red curve shows the variation of the entangle-ment entropy (EE) with ln ( L ) ( L is the subsystem size) forthe CPI ground state at Φ = 0 for a system of N ∗ = 488Cooper pairs. The blue curve is a linear fit to the form1 / ( L ) + 1. See text for discussion. ture of the CPI ground state at Φ = 0, and corroboratedby the degeneracy lifting precisely at the transition point(Φ = 1 / m, m ∈ Z , m > only a bi-partitioning of thesystem L = N ∗ /
2. For instance, in Fig.9, we present theES at a weak coupling plateau Φ = 240 and plateau tran-sition Φ = 240 . N ∗ = 488 pseudospins.Here too, the plots clearly show the double degeneracy ofthe plateau and the degeneracy lifting at the transition.The restricted degeneracy of the ES for all bi-partitionedCPI plateaux ground states with Φ > B. Entanglement Entropy of the plateau groundstates and transitions
We now compute the bipartite entanglement entropyin momentum space for various plateau ground states ob-tained by tuning the parameter φ . As before, we take asystem where N ∗ and L are the total number of pseu-dospins and number of pseudospins within the reducedsubsystem, while n and l are the number of ↑ -pseudospinswithin N ∗ and L respectively. The bipartite entangle-ment entropy ( S EE ) can simply from the Schmidt coeffi-cients ( λ nl ) via the following formula S EE ( n, L ) = − l max (cid:88) l = l min d l | λ nl ( L ) | log | λ nl ( L ) | , (53)where d l is the degeneracy factor for the l th state ofthe entanglement spectrum. We have observed earlier inFig.8(a) that for the case of the strong coupling groundstate at Φ = 0, d l = 2 ∀ l due to the two topologically FIG. 11. Plot of bipartite entanglement entropy for variousCPI ground states corresponding to different various flux Φof a system of N ∗ = 500 Cooper pairs. Inset shows the rapidfall of the bipartite EE upon approaching the CPI to Metal(Φ = 500) transition. distinct sectors X = ±
1. The appearance of the constant d l = 2 thus signifies the influence of the topological na-ture of the ground state manifold on S EE . In Fig.10, wesee that S EE varies linearly with log ( L ) for N ∗ = 10000for a large range of L , departing from the linear variationonly very near the equipartitioning value of L = N ∗ / /
2) as Φ is varied through the first 2000 plateaux.However, it is not clear whether this indicates a universal-ity of plateaux ground states observed at strong couplingwith those at intermediate coupling. Unlike the obser-vation of logarithmic scaling of S EE with subsystem sizein 1+1D quantum critical systems (see Ref. and refer-ences therein), the log-scaling observed by us in Fig.10 isindicative of the physics of a gapped ground state of theeffectively zero-dimensional Hamiltonian H coll (eq.(11))obtained from the RG .The value of the intercept (1, in units of log (2)) is theentanglement entropy of a subsystem size of L = 1 andcorresponds to a maximally mixed pseudospin. The in-tercept is, however, observed to decrease steadily beyondthe first 200 plateaux as Φ is varied, indicating that a sin-gle pseudospin’s entanglement with the rest of the systemwithin a plateau ground state is lowered as Φ is tuned to-wards weak coupling. Further, in Fig.11, we present thevariation of the bipartite S EE with the parameter Φ fora system with N ∗ = 1000 = 2 L . The plot clearly showsthat the Φ = 0 plateau possesses the largest entangle-ment content, and that this is rapidly lowered to zero asΦ is tuned through various plateaux towards the gaplessmetal .In contrast to Fig.10 above, the entanglement entropyat the first transition (Φ = 0 .
5, see Fig.12) computedusing eq.(44) for c = 1 / √ c for various systemsizes N ∗ displays a non-monotonic variation of S EE withthe subsystem size L . Remarkably, S EE displays a com-3 FIG. 12. Variation of the entanglement entropy ( S EE ) withsubsystem size L at the first plateau transition (Φ = 0 .
5) forvarious system sizes in the range 100 ≤ N ∗ ≤ mon peak at L ∗ = 7 for system sizes ranging between100 ≤ N ∗ ≤ L ≤ L ∗ . This suggests that theentanglement content at the first plateau transition isdominated by small subsystem size. While the lastdata point in Fig.12 corresponds to the equipartition S EE ( L = N ∗ /
2) for N ∗ = 100, we have checked that S EE ( L = N ∗ /
2) falls logarithmically with N ∗ . As shownin Fig.13, we have also observed that the maximum valueof S EE observed in Fig.12 remains unchanged (as Φ istuned across various plateau transitions for a system ofsize N ∗ = 1000) until almost the very last few transi-tions, where it falls rapidly to zero. On the other hand, L ∗ increases gradually with Φ, climbing rapidly at thelast few transitions. This clearly demonstrates that thepeak in S EE is a universal feature of the plateau transi-tions. We have also observed that the S EE for a singlepseudospin (i.e., L = 1) falls to zero gradually from itsvalue at the first transition as Φ is varied.Finally, we present the computation of the entanglemententropy of the plateau ground states and transitions atfinite temperature. The thermal density matrix for theplateau ground state | N ∗ = 2 S, n = S z + N ∗ (cid:105) can bewritten as ρ ( β ) = (cid:88) n e − βE n Z | N ∗ , n (cid:105)(cid:104) N ∗ , n | , (54)where β is the inverse temperature and Z the partitionfunction. Equipartitioning the system precisely as de-scribed earlier in eqs.(51)-(52) (with L = N ∗ / ρ L ( β ) = (cid:88) l,n e − βE n Z ( λ nl ) | L, l (cid:105)(cid:104)
L, l | . (55)The reduced density matrix is easily seen to be diagonal[ ρ L ( β )] l,l (cid:48) = δ l,l (cid:48) (cid:88) n ( λ nl ) e − βE n Z , (56)
FIG. 13. Plot of the maximum entanglement entropy ( S maxEE ,red curve, left y-axis) and the corresponding subsystem size( L ∗ , blue curve, right y-axis) at the transitions between vari-ous CPI ground states as Φ is varied for a system of N C = 500Cooper pairs. Inset: Rapid variation of S maxEE and L ∗ with Φupon approaching the CPI to Metal (Φ = 500) transition. such that the entanglement entropy at a non-zero tem-perature is obtained as S EE ( L, β ) = − (cid:88) l [ ρ L ( β )] l,l log ([ ρ L ( β )] l,l ) . (57)Precisely the same formalism can also be carried out withthe state equal admixture state at the plateau transition( | ψ (cid:105) P T ). FIG. 14. Variation of the equipartition entanglement entropy( S EE ( β )) with temperature ( β − ) for a subsystem size L = 4Cooper pairs. Various coloured curves correspond to the CPIground states and transition ground states at different valuesof the flux Φ. See text for discussion. In Fig.14, we present a numerical evaluation of S EE ( L, β )for a subsytem of L = 4 pseudospins with varying k B T = β − for the first four plateaus (coloured curvescentered about Φ = 0 , , . , . , . . N ∗ = 8) forthe sake of visual clarity, we have checked that all featuresof the plot are qualitatively unchanged for larger N ∗ . Re-markably, the plot shows that the S EE corresponding tothe transitions clearly separates all curves arising fromneighbouring plateaus for temperatures k B T << V ∗ .The S EE curves for all Φ corresponding to a plateau col-lapse to a universal value at T = 0 characteristic of thatplateau. Thermal fluctuations are observed to affect thecurves of both the plateaus and the transitions in thesame manner. For instance, the position of the diver-gence of the curves for a given plateau as T is increasedsuggests the robustness of that plateau to thermal transi-tions. Clearly, the Φ = 0 plateau (strong coupling) is themost robust, the Φ = 1 slightly less and so on, ending atthe last plateau (at S EE ( T = 0) = 0) beyond which liesthe gapless metal. Similarly, the curves for transitionsassociated with higher plateaus depart from their initialflat behaviour at lower temperatures in comparison tothat for lower plateaus. The domination of thermal fluc-tuations as T is raised is also clearly observed: various S EE curves corresponding to a particular T = 0 plateaushow a linear increase with T asymptotically, and with aslope common to that of the S EE curve for the transitionthat leads to the next plateau (e.g., the slopes of the S EE curves for Φ = 0 , .
25 and 0 . T etc.). V. PASSAGE TO THE BCS GROUND STATE
In order to chart the passage from the number-fixed CPIground state to the conjugate phase-fixed BCS groundstate, we will carry out the RG analysis upon adding aglobal U (1)-symmetry breaking term ( − B (cid:80) k S xk , B >
0) to the collective Hamiltonian H coll (eq.(9)) with therepulsive density-density interaction U = 0. The B fieldrepresents a Josephson coupling to an external phase-fixed BCS superconductor. We will show that at large B , the RG flow leads to a BCS-like ground state. Thiswill also be reinforced by studying the variation of sev-eral quantities with the symmetry-breaking field B , e.g.,inter- k entanglement, helicity-partitioned entanglement,helicity cross-correlation, pair number fluctuation etc.Thus, we begin with the Hamiltonian H SB = − (cid:15)N (cid:88) k S zk − ¯ V N (cid:88) kk (cid:48) ( S + k S − k (cid:48) + h.c.) − | B | (cid:88) k S xk . (58)The RG equations for ¯ (cid:15) and ¯ V are those given earlier ineq.(8) (with ¯ W = ¯ V /N ), while the RG for the symmetry-breaking field B is found to be∆ | B ( j ) | ∆ log Λ j Λ = − | ¯ W ( j ) || B ( j ) | (cid:18) ω − (cid:15) ( j ) − U (cid:19) . (59) In the regime ω < (cid:15) ( j ) + U , both ¯ W and B are found tobe RG relevant. From the RG eqs.(8) and (59), we find∆( (cid:15) )∆ | ¯ W | = − , ∆( (cid:15) )∆ B = − | ¯ W | B ⇒ ∆ | ¯ W | ∆ B = | ¯ W | B , (60)indicating that while both | ¯ W | and B grow to strongcoupling under the RG flow, the ratio | ¯ W | /B remainsinvariant. This shows that while the original BCS mean-field Hamiltonian is achieved only in the limit of the RGinvariant | ¯ W | /B →
0, a U (1)-symmetry broken BCS-likeeffective Hamiltonian is emergent from the RG flow atstrong coupling H SB = H coll − B ∗ S x , (61)where H coll is given in eq.(10). Further, in appendix B,we show that the familiar form of an exponentially smallspectral gap is obtained from the RG flow to strong cou-pling in B . In considering the effective Hamiltonian ob-tained from the RG, we will henceforth drop the ∗ sym-bol from all couplings. Clearly, as [ S z , H SB ] (cid:54) = 0, thetotal Cooper pair number operator (proportional to S z )is no longer a conserved quantity. Further, the topolog-ical order parameter Z encountered earlier is no longera good order parameter, as [ Z, H SB ] (cid:54) = 0: the effectiveHamiltonians H coll ≡ H SB ( B = 0) and H SB ( B (cid:54) = 0)are topologically inequivalent. We will demonstrate be-low that, as B is tuned to larger values, the Cooper pairnumber fluctuations increase rapidly while the fluctua-tions in the conjugate U (1) global phase is lowered. Thisindicates that a BCS-like ground state is attained underthe RG flow of B to strong coupling.In commonality with the BCS ground state, the groundstate wavefunction for H SB is given by a linear superpo-sition of states with different S z (eq.(14)) | ψ ( B ) (cid:105) = S (cid:88) S z = − S α B ( S z ) | S, S z (cid:105) = S (cid:88) Sz = − S α B ( S z ) (cid:18) (cid:88) k c † k ↑ c †− k ↓ (cid:19) S − S z | vac (cid:105) , (62)where the (normalised) coefficients α B ( S z ) are functionsof the symmetry-breaking field B . We will also showbelow that several properties of | ψ ( B ) (cid:105) closely resemblethose of | ψ BCS (cid:105) as B is tuned to large values. For in-stance, we will show that at large B , | ψ ( B ) (cid:105) leads tovanishing inter- k entanglement. We recall that a vanish-ing inter- k entanglement entropy is a special property ofthe BCS ground state, arising from the fact that different k -momenta electron-pair states are decoupled from oneanother. A. Properties of the ground state
In order to obtain various properties of the ground stateof the effective Hamiltonian H SB , we carry out exact5diagonalization computations for system sizes of N ∗ =50 Cooper pairs. We compute various quantities relatedto the ground state, e.g., fluctuation in the Cooper pairnumber and the conjugate global phase, helicity crosscorrelations, various measures of entanglement etc.In Fig.15, we present the variation of the fluctuations inthe Cooper pair number ( (cid:104) ∆ N (cid:105) ≡ (cid:104) ∆ S z (cid:105) , blue curve)and conjugate global phase ( (cid:104) ∆ φ (cid:105) , red curve). The plotclearly shows the rapid decline in (cid:104) ∆ φ (cid:105) as B is increased,together with an equally rapid growth in (cid:104) ∆ N (cid:105) . Wehave also observed that a plot of the number fluctua-tions (cid:104) ∆ S z (cid:105) versus (cid:104) S z (cid:105) for different values of Φ and B is strikingly similar to Fig.7. Further, we shall see belowthat the entanglement content of the fluctuations inducedby a non-zero B are very different from that induced bythermal fluctuations.The red curve in Fig.16 shows the variation of the he-licity cross correlations (Υ, eq.(45)) with B . The plotdisplays the rapid decline of Υ that is characteristic ofthe CPI ground state towards zero as B is tuned to largevalues. This is expected, as Υ vanishes for the BCSground state. The blue curve in Fig.16 shows the vari-ation of the helicity-partitioned entangled entropy (Ξ)with B . Ξ is derived from the reduced density ma-trix obtained by tracing out one of the two helicities( η ± = sgn ( k ) sgn ( σ ) = ± | ψ g ( B ) (cid:105) = S (cid:88) m = − S C m ( B ) | S, S z = m (cid:105) , | S, m (cid:105) = (cid:88) m η + ,m η − D mm η + ,m η − | S/ , m η + ; S/ , m η − (cid:105) , (63)where D mn ’s are Clebsch-Gordon coefficients given by D mm η + ,m η − = δ m,m η + + m η − × (cid:115) ( S !) ( S + m )!( S − m )!(2 S )!( S − m η + )!( S + m η + )!( S − m η − )!( S + m η − )! . (64)In the ground state, the total spin ( S ) is maximised, en-suring that the total spin within each helicity sector isalso maximised (and taken to be S/ η + ), we obtain a reduced density ma-trix ρ η − = T r η + | ψ g ( B ) (cid:105)(cid:104) ψ g ( B ) | . The set of eigenval-ues ( { λ i } ) btained by diagonalising ρ η − then gives thehelicity-partitioned entanglement entropy Ξ (i.e., a mea-sure of the entanglement between the opposite helicities)Ξ( B ) = − (cid:88) i λ i ( B ) log ( λ i ( B )) . (65)The variation of this entanglement entropy with B isshown via the blue curve in Fig.16, displaying a rapiddecline towards zero in the entanglement between thehelicities η + and η − as B is tuned to large values. Thisis consistent with the fact that the BCS ground statedoes not possess helicity entanglement; this arises simplyfrom the fact that the BCS ground state wavefunction FIG. 15. Variation of fluctuation in number of Cooper pairs( (cid:104) ∆ N (cid:105) , blue curve, left y-axis) and global phase ( (cid:104) ∆ φ (cid:105) , redcurve, right y-axis) with the global U (1) symmetry breakingfield B for a system of N ∗ = 50 Cooper pairs.FIG. 16. Variation of helicity cross correlations (Υ, red curve,left y-axis) and helicity partitioned entanglement entropy (Ξ,blue curve, right y-axis) in number of Cooper pairs ( (cid:104) ∆ N (cid:105) ,blue curve, left y-axis) with the global U (1) symmetry break-ing field B for a system of N ∗ = 50 Cooper pairs. is a direct product state of pairs of electronic momenta( k, − k ).On the other hand, as Cooper pairs in the s-wave BCSstate are spin singlets, there is a non-zero entanglementbetween the two spins of a Cooper pair . Thus, in orderto distinguish the CPI and BCS ground states further,we compute the entanglement entropies S kEE and S SEE by partitioning the ground state | ψ g ( B ) (cid:105) for an analyti-cally tractable system of two Cooper pairs (( k ↑ , − k , ↓ )and ( k ↑ , − k ↓ )) in the momentum variable ( k , k )and the spin variable ( ↑ , ↓ ) respectively for the case of¯ (cid:15) = 0 (strong coupling limit) in the Hamiltonian H coll (eq.(10)). For this system of two coupled pseudospins, S = 1 and the ground state wavefunction is | ψ g ( B ) (cid:105) = (cid:88) α = − C α ( B ) | S = 1 , S z = α (cid:105) , (66)where the coefficients C α are a function of the field B FIG. 17. Variation of spin partitioned entanglement entropy( S SEE , blue curve, left y-axis) and momentum partitioned en-tanglement entropy ( S kEE , red curve, right y-axis) with theglobal U (1) symmetry breaking field B for a prototypical sys-tem of N ∗ = 2 Cooper pairs. and the coupling V /N given by { C α } = N (cid:18) , − α + (cid:112) α + 8 β β , (cid:19) β>> −→ (cid:18) , − √ , (cid:19) , (67)where N is the normalisation factor, α = VN and β = B √ .By writing the states | S, S z (cid:105) in the basis of | n k ↑ n − k ↓ (cid:105) ⊗| n k (cid:48) ↑ n − k (cid:48) ↓ (cid:105) , i.e., the states {| ↑ ↓ (cid:105) k ⊗ | ↑ ↓ (cid:105) k (cid:48) , | ↑ ↓ (cid:105) k ⊗| ↑ ↓ (cid:105) k (cid:48) , | ↑ ↓ (cid:105) k ⊗ | ↑ ↓ (cid:105) k (cid:48) , | ↑ ↓ (cid:105) k ⊗ | ↑ ↓ (cid:105) k (cid:48) } , the den-sity matrix ρ ( B ) = | Ψ g ( B ) (cid:105)(cid:104) Ψ g ( B ) | is found to be ρ ( B ) = | C | C C √ C C √ C C − C C √ | C | | C | C C − √ C C √ | C | | C | C C − √ C C − C C − √ C C − √ | C − | . (68)The momentum-partitioned reduced density matrix isthen obtained by tracing out the k (cid:48) pseudo-spin fromthe density matrix (68). The reduced density matrix ρ k is written in the basis {| ↑ ↓ (cid:105) k , | ↑ ↓ (cid:105) k } ρ k = T r k (cid:48) ρ ( B ) (69)= | C | + C C C √ + C C − C C √ + C C − √ | C − | + C The inter-k entanglement is S kEE calculated from the den-sity matrix ρ k . As shown via the red curve in Fig.17, S kEE reduces monotonically from its largest value at B = 0 as B is increased, displaying the destruction of the inter- k entanglement of the CPI ground state in the passagetowards the BCS ground state.Similarly, for the spin-partitioned entanglement en-tropy, we trace out a given spin sector, say ↓ .Then, the reduced density matrix in the basis {| k k (cid:48) (cid:105) ↑ , | k k (cid:48) (cid:105) ↑ , | k k (cid:48) (cid:105) ↑ , | k k (cid:48) (cid:105) ↑ } is ρ ↑ = T r ↓ ρ ( B ) (70)= | C | | C | | C |
00 0 0 | C − | The spin-partitioned entanglement entropy S kEE is ob-tained from ρ ↑ . As shown via the blue curve in Fig.17, S ↑ EE increases steadily from its smallest value at B = 0as B is increased, and saturates as B >>
1. This showsthe growth of the inter-spin entanglement of the BCSground state in the passage from the CPI ground state.The limiting values of S ↑ EE at B = 0 and B >> B = 0,we get C = 1 , C − = C = 0, giving S ↑ EE for theCPI ground state as log B become ( , , , ). This shows that ρ ↑ ( B >>
1) becomes maximally mixed in nature, leadingto S ↑ EE ( B >>
1) saturating to the value seen in Fig.17 S = − × log
14 = 2 log . (71)This is a clear signature of the maximal entanglement ofthe Cooper pair singlets in the BCS ground state.Another diagnostic of the difference between the groundstates at B = 0 and B >> k -momentum electron (cid:104) n k (cid:105) = T r (ˆ n k ρ ↑ ) = | C | | C − | . (72)While at B = 0, (cid:104) n k (cid:105) ≡ (cid:104) n k (cid:105) ( α, β ), (cid:104) n k (cid:105) → / B >>
1. Given that (cid:104) n k (cid:105) follows the Fermi-Diracdistribution for the BCS ground state , the result of (cid:104) n k (cid:105) = 1 / B >> H SB (eq.(61)) in the presence of alarge U (1) symmetry breaking coupling B describes theBCS superconductor near the Fermi surface.We now present the entanglement entropy ( S EE ) com-puted from partitioning the ground state | ψ g (cid:105) (with spin S ) of Hamiltonian H SB into two equal subsystems A andB such that S = S A + S B , S A = S/ S B using thestrategy adopted in eqs.(46)-(53). In Fig.18, we present avariation of S EE with B computed for a system of 8 pseu-dospins, and for ground states of H SB ( B = 0) ≡ H coll atvarious values of Φ = (cid:15)/V . The plot shows the monotonicdecrease for S EE with B for all ground states with a non-zero number of Cooper pairs (0 ≤ Φ < S EE computed for the gapless ground state at Φ = 4 shows anon-monotonic variation with B . The latter case corre-sponds to the entanglement related to superconductingphase fluctuations in a mean-field BCS Hamiltonian, i.e.,a Hamiltonian H SB in which the BS x term induces pair-ing in the gapless spectrum of H coll . While the BCSground state corresponding to a vanishingly small S EE FIG. 18. Variation of the equipartition entanglement entropyfor different CPI ground states and transition ground states(corresponding to different integer and half-integer values of0 ≤ Φ ≤
4) with the global U (1) symmetry breaking field B for a system of N ∗ = 8 Cooper pairs. Note that the non-monotonic behaviour of the purple curve arise from the factthat this corresponds to the ground state precisely at thetransition from the CPI to the parent metal. is obtained for all these curves in the limit of large B ,the approach of the mean-field ground state is clearlydifferent from those with pre-existing Cooper pair boundstates: the peak in the curve for φ = 4 likely arises dueto the creation of Cooper pairs in a gapless system. B. The effect of a Josephson coupling
We end with a brief presentation of the effects of aJosephson coupling between the bulk of two CPI systemsA and B (i.e., we are ignoring all effects from gapless edgestates), each of which is modelled by H SB (eq.(61)) H µ = − (cid:15) µ V µ (cid:88) k ∈ µ S zk − V µ N µ (cid:88) k (cid:54) = k (cid:48) ∈ µ (cid:18) S + k S − k (cid:48) + h.c. (cid:19) − B µ (cid:88) k ∈ µ S xk , µ =A,B H AB = T (cid:88) k ∈ A,k (cid:48) ∈ B (cid:18) e iφ S + k S − k (cid:48) + e − iφ S − k S + k (cid:48) (cid:19) , (73)where H AB is the Josephson coupling between the sys-tems A and B , with the phase φ dependent on the ex-ternally applied voltage difference between the two sys-tems . We have simulated the equations in eq.(73) fortwo systems comprised of 4 pseudospins each.First, we set the field B A = 0, such that system A isin a U(1) symmetric CPI phase and couple it with thesystem B ( H B ) for several values of the field B B andthe Josephson coupling T . The values of the parameters FIG. 19. Plot of the induced phase stiffness (cid:104) S xA (cid:105) in CPI sys-tem A due to a Josephson coupling (with strength T ) to CPIsystem B (in presence of a global U (1) symmetry breakingfield ( B B )). Various curves correspond to different values of B B . (cid:15) A = (cid:15) B =, V A /N A = V A /N B = 1 and φ = π . In Fig.19,we study the phase coherence being generated in the sys-tem A by computing (cid:104) S xA (cid:105) in the ground state of the totalsystem ( H = H A + H B + H AB ). The blue line in Fig.19clearly shows that a Josephson coupling between two sys-tems that are individually in CPI phases ( B A = 0 = B B )cannot lead to phase coherence being induced in eithersystem A or B. On the other hand, for non-zero valuesof B B , the other curves in Fig.19 shows that as systemB already possesses some degree of phase coherence, anincreasing non-zero phase coherence is induced in systemA via the Josephson coupling with increasing B B . Note,however, that while this demonstrates the breaking of theU(1) symmetry of the system A via the Josephson cou-pling to the symmetry-broken system B, the phases ofthe two systems are locked to one another with zero rel-ative phase difference . This is demonstrated in a plotof the total ground state energy E ( φ ) as a function ofthe phase φ : the blue line in Fig.20 clearly shows that aJosephson current ( J ∝ ∂E ( φ ) /∂φ ) cannot be generatedin the coupled system for B A = 0. On the other hand, inthe presence of a non-zero symmetry breaking field B A , E ( φ ) shows a cosinusoidal variation with φ in Fig.19.This shows that when the symmetry is separately bro-ken in the two systems, a Josephson coupling certainlyinduces a Josephson current. The results of this subsec-tion serve as predictions for the experimental search ofsystems in the CPI ground state. VI. ENTANGLEMENT RENORMALISATION
Having explored the entanglement features of the topo-logically ordered CPI and symmetry broken BCS groundstates at some length in previous sections, we now presentan analysis the T = 0 RG evolution of the many-particle8 FIG. 20. Plot of the total ground state energy E ( φ ) for aCPI system (placed in a gradually increasing U (1) symmetrybreaking field B A ) coupled to a BCS superconductor throughJosephson tunneling as a function of their phase difference φ .The various curves correspond to different values of B A . entanglement content of these ground states. For this,we follow the strategy for entanglement renormalisationthat was developed in Refs. . For the sake of com-pleteness, we outline briefly the strategy below.As we have seen earlier, the URG proceeds by disen-tangling electronic states sequentially from the UV to-wards the IR by the application of many-particle uni-tary transformations ( U , see Appendix A for further de-tails). At the IR stable fixed point, we have identifiedthe ground state wavefunction. Now, by reversing theRG flow through the sequential applications of the appro-priate U † s, we generate a family of ground state wave-functions ranging towards the UV. This allows for thecomputation of several entanglement features from eachmember of the family of wavefunctions, thereby gener-ating the RG flow of these entanglement features. Asdiscussed in detail in Refs. , the unitary operators U of the URG method can be implemented as a quan-tum circuit, i.e., in terms of a combination of universal2-qubit gates (e.g., Hadamard, C-NOT and phase-shiftgates). Below, in Figs.21 and 22, we show the quantumcircuit realisations that implement the reverse URG flowalong one radial direction in k -space for the CPI and BCSwavefunctions respectively.As shown in Fig.21, the nodes 9 and 19 refer to thefermion states residing just outside and just inside theFermi surface respectively. The distance from the Fermisurface increases with passage between states 9 to 0 (alloutside the Fermi surface), and with passage betweenstates 19 to 10 (all inside the Fermi surface). As indi-cated by the quantum circuit diagrams, the reverse RGflow starts from the emergent CPI phase (described by FIG. 21. Quantum circuit representation of (left panel) theground state of the fixed point CPI Hamiltonian and (rightpanel) the ground state after the first step of the reverse uni-tary RG step. Both are for a system of N ∗ = 2 Cooper pairs.FIG. 22. Quantum circuit representation of (left panel) theground state of the fixed point BCS Hamiltonian and (rightpanel) the ground state after the first step of the reverse uni-tary RG step. Both are for a system of N ∗ = 2 Cooper pairs. the effective Hamiltonian in eq.(11)) obtained at the sta-ble fixed point and with a window of electronic statesgiven by N ∗ ( N ∗ = 2 in the figures). The reverse RGflow proceeds by the re-entangling of two electronic stateslying outside the window at each step of the RG. We nowpresent the results of the RG evolution for the entangle-ment entropy of a block in k -space along a given radialdirection for CPI and the symmetry broken phases. InFig.23, we present the RG variation of the entanglemententropy computed for a block (lying outside the Fermisurface) of varying size ranging from one to ten fermionicstates. The two plots are for different sizes of the win-dow ( N ∗ ) for the emergent CPI phase: the upper plotis for N ∗ = 2 (i.e., comprised of states 9 and 19 only),while the lower is for N ∗ = 8 (i.e., comprised of state 6-9and 16-19). Further, the reverse RG process increasesstepwise from step 0 (in the IR) towards the UV.The upper panel of Fig.23 shows that block entropyfor all block sizes terminates at a universal value of S = 0 .
693 = ln 2, corresponding to the entanglement forthe N ∗ = 2 pseudospins that form the emergent CPI win-dow in the IR. Further, the plots demonstrate that theblock entanglement entropy of block size 1 (i.e., for thestate 9, one of the two states that form the CPI groundstate in the IR) increases slowly with the RG flow from9 FIG. 23. Plot for the RG variation of the entanglement en-tropy S ( n ) for various block sizes 1 ≤ n ≤
10 (in differentcolours) for a CPI system with N ∗ = 2 (upper panel) and N ∗ = 8 (lower panel) Cooper pairs. See text for discussion. UV to IR. On the other hand, the block entropy of allother block sizes (greater than one) decreases with thestepwise decoupling of electronic states. Further, the en-tanglement entropy of the blocks varies non-linearly withthe RG steps. Additionally, in the block entropy plots for N ∗ = 8 (lower panel of Fig.23), we see that the entangle-ment entropy for all block sizes less than 4 (i.e., the size ofthe four states 9-6 that are part of the CPI ground statein the IR) are affected very little by the RG flow. This isa remarkable display of the fact that the entanglement ofthe electronic states proximate to the Fermi surface (andthat eventually form a part of the emergent window) isquite robust under RG evolution, and distinguishes themfrom those that are decoupled along the flow. Further,the CPI ground state possesses a hierarchy of scales ofentanglement defined by the various block sizes.Next, we present the entanglement RG results for a sys-tem in the presence of a bare symmetry breaking field FIG. 24. Plot for the RG variation of the entanglement en-tropy S ( n ) for various block sizes 1 ≤ n ≤
10 (in differentcolours) for a CPI system with N ∗ = 2 (upper panel) and N ∗ = 8 (lower panel) Cooper pairs in the presence of a weakglobal U (1) symmetry breaking field B = 5 × − (in unitsof the attractive pairing coupling V ). Note that the curvesfor n = 8 , ( B ). In Fig.24, we see that the presence of a very weakbare U (1) symmetry breaking field B ∼ × − (in unitsof V ), the entanglement RG flows from UV to IR are verysimilar to those shown in Fig.23 above for the CPI (i.e.for the case of B = 0), with only one difference: the finalvalue of the block entropies in the IR here is reduced withrespect to those obtained for the CPI. This indicates agradual collapse of the hierarchy of scales of entangle-ment of the CPI upon tuning a symmetry breaking field.Finally, in Fig.25, we present that the entanglement RGflows for a system with N ∗ = 4 for the case of a slightlylarger (but still weak) bare U (1) symmetry breaking field B ∼ × − (in units of V ). Here, we find that the en-tanglement curves for various block sizes is very differentto those obtained for the CPI (see Fig.23). For instance,the block entropy for both block sizes one and two (i.e.,0corresponding to the two possible subblocks of the emer-gent BCS ground state in the IR) have zero entanglemententropy throughout the RG. There is, thus, no longer anyway to distinguish between the constituent blocks of theBCS ground state under the RG. Further, the entangle-ment entropy varies linearly with the RG steps for blocksizes ≥ FIG. 25. Plot for the RG variation of the entanglement en-tropy S ( n ) for various block sizes 1 ≤ n ≤
10 (in differentcolours) for a CPI system with N ∗ = 2 (upper panel) and N ∗ = 8 (lower panel) Cooper pairs in the presence of a strongglobal U (1) symmetry breaking field B = 25 × − (in unitsof the attractive pairing coupling V ). Note that the curvesfor n = 8 , VII. CONCLUSIONS AND DISCUSSIONS
A body of theoretical work has proposed , on phe-nomenological grounds, the existence of a topologicallyordered counterpart of the superconductor. This novelstate of quantum matter, which we call the Cooper pairinsulator (CPI) is expected to be a condensate of a fixednumber of Cooper pairs, but without any phase stiffness.Instead, the CPI would correspond to a gapped systemin the bulk and with gapless states at the boundaries.In keeping with this proposal, recent experimental stud-ies of the superconductor to insulator transition (SIT) inthin films suggests the existence of such a CPI lying pre-cisely at the transition . While the phenomenologicalgauge field theories proposed for the CPI offer some in-sight into its properties, a microscopic approach remainsabsent. Thus, a major finding of our work is the deriva-tion of an effective microscopic Hamiltonian for the CPIthat is emergent from a unitary renormalization group(URG) analysis.For this, we have worked on a generalized model of ametallic system (i.e., with a repulsive density-density in-teraction U ) as well as an attractive pairing interaction.Our URG study of this model offers a phase diagram in terms of a quantum fluctuation scale ( ω ) and U , clearlydisplaying the existence of a CPI phase of quantum mat-ter at small ω (i.e., corresponding to energyscales forexcitations proximate to the Fermi surface) and for all U . The low energy fixed point effective Hamiltonian ob-tained for the CPI phase is then studied in detail. Asmentioned earlier, the symmetry-preserved CPI phaseis found to possess a fixed number of Cooper-pairs butwithout any global phase coherence among them. Subse-quently, we have carried out a detailed analysis of vari-ous topological and many-particle entanglement featuresof this state of quantum matter, establishing therebythe emergence of topological order in the CPI. We havealso contrasted the properties of the CPI ground statewith its (BCS s-wave) superconducting counterpart, andbelieve that some of our results provide experimentallytestable predictions. Importantly, we have also bench-marked numerically the ground state energy density ofthe CPI (in the thermodynamic limit) obtained from afinite-size scaling analysis for the RG against a similarfinite size scaling analysis of exact diagonalisation calcu-lations. We now end with a discussion of the broadersignificance of our findings with regards to the subject oftopological order.Topological order is proposed to describe the ordering ofinteracting many-particle quantum system beyond theGinzburg-Landau-Wilson (GLW) paradigm (see Ref. for a recent review). The GLW paradigm describes or-der arising the spontaneous breaking of symmetries, mea-sured in terms of real-space local order parameters andassociated with a phase transition whose universality iscaptured by a set of scaling exponents. On the otherhand, a topologically ordered ground state does not arisefrom breaking any symmetries and thus lacks a local or-der parameter. Instead, such ground states are invariantunder large gauge transformations, can be representedpurely in terms of non-local gauge operators (e.g., Wil-son loops etc.), and their quantum dynamics can be cap-tured by a topological gauge field theory. When placedon a multiply connected manifold (e.g., a torus), a topo-logically ordered system displays a non trivial degener-acy of the ground state manifold (protected by a non-zero energy gap), as well as the existence of fractionallycharged topological excitations that interpolate betweenthe ground states. While the bulk of such a system isan incompressible insulating state of matter (due to thespectral gap), it can possess gapless current-carrying de-grees of freedom at its boundaries. It has also been shownthat the ground states of a topologically ordered sys-tem can possess signatures of non-trivial many-particleentanglement, e.g., an entanglement entropy (due to areal-space bipartitioning) proportional to the degeneracycount of the ground state manifold (called the quantumdimension). While all of these properties are widely be-lieved to be the features and diagnostics of a topologi-cally ordered system, an overarching theoretical frame-work for this subject remains an outstanding challenge.As the pairing instability of the Fermi surface represents1a paradigmatic phenomenon for a system of interactingelectrons, our insights into the CPI represents an oppor-tunity towards learning the inner workings of emergenttopological order in such systems, as well as how it is dif-ferent from the order captured by ground states belong-ing to the GLW paradigm (e.g., the BCS ground state).The body of results presented for the CPI phase clearlysatisfy the diagnostics described above. We have es-tablished analytically the topological degeneracy of theground state manifold using flux insertion arguments,and shown that the zero mode collective effective Hamil-tonian for the CPI can be written in terms of Wilsonloop operators. This then paves the way for connect-ing the topological θ term in the effective theory for theCPI with the 2+1 dimensional topological Chern-Simonsgauge field theory proposed for such systems . Wehave shown the origin of the spectral gap that protectsthe ground state manifold, and shown the spectral flowproperty of such ground states with a variation in the θ parameter: ground states form plateaux in θ labelled by atopological quantum number and with topological quan-tum phase transitions separating them. Indeed, much ofthe phenomenology observed by us is common with theproperties of topologically ordered fractional quantumHall ground states (see Ref. and references therein). Itwill be interesting to test these conclusions for systemsof interacting electrons in the presence of disorder orincommensuration .Our investigations of the entanglement features showclear universal signatures that distinguish the topolog-ically ordered CPI ground states (plateaux) from thosefound at the transitions between plateaux. The passageto the metallic state upon tuning the effective Aharanov-Bohm flux of the fixed point Hamiltonian is charted atzero as well as finite temperatures, yielding clear signa-tures once again in the entanglement for the CPI groundstates. By carrying out the RG analysis in the presence ofa global U (1) symmetry breaking term, a detailed com-parison between the CPI and BCS ground states is alsooffered. This allows us to demonstrate the clear distinc-tions between these two kinds of ground states in terms ofmany-particle entanglement and many-body correlations:unlike the BCS state, the CPI ground state is found topossess various measures of entanglement. Further, weshow that, as CPI ground states lack phase stiffness, theycannot show the Josephson effect (i.e., upon coupling twosuch CPI systems through Cooper pair tunneling).All of this leads us to conjecture that our results on theCPI offer a broad framework for understanding topolog-ical order. Specifically, we believe that various quan-tum liquid systems displaying the hallmark signaturesof topological order described above are likely to be de-scribed by effective zero mode collective Hamiltoniansdescribed in terms of Wilson loop like non-local degreesof freedom. Using similar flux insertion arguments, itshould be possible to show that the ground state man-ifolds of such Hamiltonians display topological degener- acy on the torus etc. Indeed, similar conclusions havebeen reached by some of us for the Mott liquid groundstates of the 2D Hubbard model discovered recently inRefs. , and the spin liquid ground states of quantumspins coupled through antiferromagnetic exchange on ge-ometrically frustrated lattices . It should be possi-ble, therefore, to chart out in a similar fashion the mi-croscopic origins of various kinds of topologically orderedquantum liquids. This will go a long way in establishinga detailed understanding of the universality of such phe-nomena.We end with a brief discussion on where to search forsuch CPI ground states. As we have seen here, theCPI state reached from a generic non-nested Fermi sur-face is strongly susceptible towards the effects of spon-taneous symmetry breaking and the emergence of theBCS s-wave superconducting ground state. As men-tioned earlier, some hints of the CPI have been foundto lie at the superconductor to insulator transition inrecent experiments on thin films. Based on our recentstudy of the 2D Hubbard model , and its relevanceto the physics of the high-temperature superconductinghole doped cuprate Mott insulators, we believe that theCPI ground states may well be observed in those materi-als too. Specifically, in Refs. , we observed the existenceat T = 0 of a pseudogapped CPI state of quantum mat-ter lying above the d-wave superconducting “dome” ob-tained upon optimally doping the Mott insulating groundstate of the 1 / / k -space that could be described interms of a state of matter containing condensed Cooperpair bound states but without any global phase coher-ence. The large superconducting phase fluctuations ob-served in this pseudogapped phase are a signature ofthe CPI, and are reminiscent of the findings from Nernsteffect measurements on the pseudogap phase of the dopedcuprates . We believe, therefore, that the cuprates areexcellent candidate systems in which to search for theexistence of the CPI phase. Following the suggestion ofRef. , pressurised solid H S may be another interestingcandidate system in which to search for the CPI.
ACKNOWLEDGMENTS
The authors thank A. Mukherjee, S. Pal, R. K. Singh,A. Dasgupta, A. Ghosh, S. Sinha, G. Baskaran, S. Mo-roz and A. Taraphder for several discussions and feed-back. S. P. thanks the CSIR, Govt. of India and IISERKolkata for funding through a research fellowship. S. L.thanks the DST, Govt. of India for funding through aRamanujan Fellowship during which a part of this workwas carried out.2
Appendix A: Hamiltonian RG
We first briefly recapitulate the unitary RG method developed in Refs. , and then derive the RG equations forthe generalised pairing Hamiltonian eq.(3). The RG method adopted uses a unitary transformation to decouple onesingle-particle Fock state | kσ (cid:105) from the rest of the states it is interacting with. Very generally, one can write themany-particle Hamiltonian as ˆ H = ˆ H D + ˆ H Xkσ + ˆ H ¯ Xkσ , where ˆ H D contains all single-particle and many-particle numberdiagonal (kinetic energy and interaction) terms. ˆ H Xkσ represents all the off-diagonal interaction terms connected to thesingle-particle state | kσ (cid:105) , while ˆ H ¯ Xkσ represents all off-diagonal interaction terms among all (say, 2 N − ) single-particlestates other than | kσ (cid:105) . Considering a many-particle eigenstate of the Hamiltonian | Ψ (cid:105) (a member of the full 2 N dimensional Hilbert space), we can writeˆ H | Ψ (cid:105) = ( ˆ H D + ˆ H Xkσ + ˆ H ¯ Xkσ ) | Ψ (cid:105) = ¯ E | Ψ (cid:105) , (A1)where ¯ E is the eigenvalue for | Ψ (cid:105) . One can rewrite the wavefunction | Ψ (cid:105) in a Schmidt decomposed form as follows | Ψ (cid:105) = a | Ψ (cid:105) ⊗ | kσ (cid:105) + a | Ψ (cid:105) ⊗ | kσ (cid:105) , (A2)where {| kσ (cid:105) , | kσ (cid:105)} live in a 2-dimensional single-particle Fock space and {| Ψ (cid:105) , | Ψ (cid:105)} lives in the remaining 2 N − dimensional Hilbert space. We then proceed to remove all quantum fluctuations connected between | k, σ (cid:105) with theother | k (cid:48) (cid:54) = k, σ (cid:105) states. For this, one can define transition operators ˆ η kσ and ˆ η † kσ as follows a | Ψ (cid:105) ⊗ | kσ (cid:105) = ˆ η † kσ a | Ψ (cid:105) ⊗ | kσ (cid:105) , a | Ψ (cid:105) ⊗ | kσ (cid:105) = ˆ η kσ a | Ψ (cid:105) ⊗ | kσ (cid:105) , (A3)where | kσ (cid:105) and | kσ (cid:105) represent the n kσ = 1 and n kσ = 0 states respectively, andˆ η kσ = 1ˆ ω − T r kσ ( ˆ H D (1 − ˆ n kσ ))(1 − ˆ n kσ ) T r kσ ( c † kσ ˆ H ) c kσ . (A4)Here, T r kσ ( ) represents a partial trace in the Fock space over the state | k, σ (cid:105) . These transition operators have afermionic nature ˆ η † kσ ˆ η kσ = ˆ n kσ = 1 − ˆ η kσ ˆ η † kσ , { ˆ η † kσ , ˆ η kσ } = 1 , [ˆ η † kσ , ˆ η kσ ] = 2ˆ n kσ − , ˆ η kσ = 0 . (A5)Using the transition operators ( η kσ , η † kσ ) and the eqs.(A3), one can see that | Ψ (cid:105) = a | Ψ (cid:105) ⊗ | kσ (cid:105) + a | Ψ (cid:105) ⊗ | kσ (cid:105) = a | Ψ (cid:105) ⊗ | kσ (cid:105) + ˆ η kσ a | Ψ (cid:105) ⊗ | kσ (cid:105) = a (1 + ˆ η kσ ) | Ψ (cid:105) ⊗ | kσ (cid:105) = a e ˆ η kσ | Ψ (cid:105) ⊗ | kσ (cid:105) . (A6)Thus, one can construct a unitary operator U kσ = 1 √ η † kσ − η kσ ) , (A7)that rotates the many-particle basis in such way that U kσ | Ψ (cid:105) = N | α (cid:105) , α = 0 or N normalization constant.This unitary rotaion removes all quantum fluctuations between the states | kσ (cid:105) and | kσ (cid:105) . Further, using the unitaryoperator, the Hamiltonian can be written in the rotated basis as U kσ ˆ HU † kσ = 12 T r kσ ( ˆ H ) + τ kσ T r kσ ( Hτ kσ ) + τ kσ { c † kσ T r kσ ( ˆ Hc kσ ) , ˆ η kσ } . (A8)It is important to note that while ˆ n kσ ˆ H (1 − ˆ n kσ ) (cid:54) = 0 (i.e., there existed non-trivial quantum fluctuations in theoccupation of single-particle Fock state given by n σ ) prior to the application of the unitary operator, subsequent toits application we find ˆ n kσ U kσ ˆ HU † kσ (1 − ˆ n kσ ) = 0 ⇒ [ˆ n kσ , U kσ ˆ HU † kσ ] = 0 . (A9)The degree of freedom n kσ is thus rendered an integral of motion (IOM) of the RG flow. The RG equations can thenbe obtained from the condition eq.(A9).Coming to the problem at hand, in the generalised pairing Hamiltonian eq.(3), we are working in the subspace givenby n k,σ = n − k,σ . Thus, at every step of the RG, we are disentangling two single-particle states | k, σ (cid:105) , | − k and − σ (cid:105) simultaneously. We now proceed by rewriting the Hamiltonian in terms of Anderson pseudospins H q pair = − (cid:88) k ˜ (cid:15) k,q ( S zk,q −
12 ) − (cid:88) k (cid:54) = k (cid:48) | W qkk (cid:48) | S − k,q S + k (cid:48) ,q + h.c.) + U (cid:88) k (cid:54) = k (cid:48) S zk,q S zk (cid:48) ,q , (A10)3such that the part of the Hamiltonian associated with the k N pseudo-spin is given by H qN = − ˜ (cid:15) k N ,q τ zk,q − (cid:88) k (cid:54) = k N | W qkk N | τ − k N ,q S + k,q + τ + k N ,q S − k,q ) + U . (A11)Applying the RG formalism to H qN , one obtains from the condition eq.(A9) the operator level RG equation for theHamiltonian in the low-energy sector for the quantum fluctuation scale ω as∆ H = (cid:18) (cid:88) k (cid:54) = k N | W qkk N | τ + k N S − k (cid:19) G k N (cid:18) (cid:88) k (cid:48) (cid:54) = k N | W qk (cid:48) k N | S + k (cid:48) τ − k N (cid:19) . (A12)From this, we derive the RG equations in the relevant channel ( τ zk,q = + ) for Cooper pair condensation as∆˜ (cid:15) ( j ) k (cid:48) ,q ∆ log Λ j Λ = 14 | W ( j ) k Λ k (cid:48) | (cid:18) ω − ˜ (cid:15) ( j ) k Λ ,q − U (cid:19) , ∆ | W ( j ) k (cid:48) k (cid:48)(cid:48) | ∆ log Λ j Λ = − | W ( j ) k Λ k (cid:48) || W ( j ) k Λ k (cid:48)(cid:48) | (cid:18) ω − ˜ (cid:15) ( j ) k Λ ,q − U (cid:19) . (A13) Appendix B: URG with symmetry breaking field
We begin by including a global U (1) symmetry breaking term ( − | B | (cid:88) k S xk ) to the pairing Hamiltonian eq.(A10)(but with the repulsion coupling U = 0). Naturally, the symmetry breaking term now appears in the Hamiltonianinvolving the node k N (eq.(A11)), as well as the operator RG equation (eq.(A12)). Subsequently, in the sector τ zk N = +1 /
2, we get the RG equations for ˜ (cid:15) k (cid:48) ,q and | W k (cid:48) k (cid:48)(cid:48) | precisely as in eqs.(A13) (but with U = 0). Further, weobtain a RG equation for the symmetry breaking field | B | ∆ | B ( j ) | ∆ log Λ j Λ = − | B ( j ) || W ( j ) k Λ k (cid:48)(cid:48) | (cid:18) ω − ˜ (cid:15) ( j ) k Λ ,q (cid:19) . (B1)We now compute the spectral gap of the symmetry broken BCS superconducting phase. For this, taking | W ( j ) k Λ k (cid:48)(cid:48) | ≡ W (0) (a constant independent of k Λ and k (cid:48)(cid:48) ), we note that the solution to the RG equation for B is given by B ( j ) = B (0) | W (0) | j − (cid:88) l =0 (cid:15) ( j ) − ˜ ω . (B2)The strong coupling RG fixed point of B → ∞ is reached when the denominator of the above relation for B vanishes: − | W (0) | = j − (cid:88) l (cid:15) ( l ) − ˜ ω ≈ (cid:90) E F + (cid:126) v F Λ ∗ E F + (cid:126) v F Λ (0) N ( E ) dE E − E F ) , (B3)where we have replaced the sum by an integral, N ( E F ) is the electronic density of states (DOS) at the Fermi energy( E F ), (cid:15) ( l ) by the continuous energy variable E and ˜ ω by 2 E F . Λ (0) and Λ ∗ correspond to the bare and final k -spacecutoffs of the RG flow. From here, we obtain the well-known relation for the (exponentially small) BCS gapΛ ∗ = Λ (0) exp (cid:18) − | W | N ( E F ) (cid:19) . (B4) ∗ [email protected] † [email protected] L. N. Cooper, Phys. Rev. , 1189 (1956). J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. , 1175 (1957). V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. ,1064 (1950). X. M. Puspus, K. H. Villegas, and F. N. C. Paraan, Phys.Rev. B , 155123 (2014). M. Di Tullio, N. Gigena, and R. Rossignoli, Phys. Rev. A , 062109 (2018). A. F. Hebard and M. A. Paalanen, Phys. Rev. Lett. ,927 (1990). A. Yazdani and A. Kapitulnik, Phys. Rev. Lett. , 3037(1995). T. I. Baturina, D. R. Islamov, J. Bentner, C. Strunk, andM. R. B. . A. Satta, Journal of Experimental and Theo-retical Physics Letters , 337 (2004). A. Doron, I. Tamir, S. Mitra, G. Zeltzer, M. Ovadia, andD. Shahar, Phys. Rev. Lett. , 057001 (2016). A. M. Goldman and N. Markovi´c, Physics Today (1998), doi: 10.1063/1.882069. V. F. Gantmakher and V. T. Dolgopolov, Physics-Uspekhi , 1 (2010). S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar,Rev. Mod. Phys. , 315 (1997). G. Sambandamurthy, L. W. Engel, A. Johansson, andD. Shahar, Phys. Rev. Lett. , 107005 (2004). H. Q. Nguyen, S. M. Hollen, M. D. Stewart, J. Shainline,A. Yin, J. M. Xu, and J. M. Valles, Phys. Rev. Lett. ,157001 (2009). M. Feigel’man, L. Ioffe, V. Kravtsov, and E. Cuevas, An-nals of Physics , 1390 (2010), july 2010 Special Issue. Nature , 876 (2007). M. A. Paalanen, A. F. Hebard, and R. R. Ruel, Phys. Rev.Lett. , 1604 (1992). B. Sac´ep´e, C. Chapelier, T. I. Baturina, V. M. Vinokur,M. R. Baklanov, and M. Sanquer, Phys. Rev. Lett. ,157006 (2008). Haviland, D. B., Y. Liu, and A. M. Goldman, Phys. Rev.Lett. , 2180 (1989). H. M. Jaeger, D. B. Haviland, B. G. Orr, and A. M.Goldman, Phys. Rev. B , 182 (1989). A. F. Hebard and M. A. Paalanen, Phys. Rev. Lett. ,927 (1990). van der Zant, H. S. J., Fritschy, F. C., Elion, W. J., L. J.Geerligs, and J. E. Mooij, Phys. Rev. Lett. , 2971(1992). N. Mason and A. Kapitulnik, Phys. Rev. Lett. , 5341(1999). N. Markovi´c, C. Christiansen, A. M. Mack, W. H. Huber,and A. M. Goldman, Phys. Rev. B , 4320 (1999). M. A. Steiner, N. P. Breznay, and A. Kapitulnik, Phys.Rev. B , 212501 (2008). A. T. Bollinger, G. Dubuis, J. Yoon, D. Pavuna, and J. M.. I. Boˇzovi´c, Nature , 458 (2011). S. Eley, S. Gopalakrishnan, and P. M. G. . N. Mason,Nature physics , 59 (2012). A. Allain and Z. H. . V. Bouchiat, Nature materials ,590 (2012). Z. Han, A. Allain, H. Arjmandi-Tash, K. Tikhonov,M. Feigel’man, and B. S. . V. Bouchiat, Nature physics , 380 (2014). F. Cou¨edo, O. Crauste, A. A. Drillien, V. Humbert,L. Berg´e, and C. A. M.-K. . L. Dumoulin, Scientific Re-ports , 35834 (2014). S. Park and J. S. . E. Kim, Scientific Reports , 42969(2017). N. P. Breznay and A. Kapitulnik, Science Advances (2017), 10.1126/sciadv.1700612. C. G. L. Bøttcher, F. Nichele, M. Kjaergaard, H. J. Suomi-nen, J. Shabani, and C. J. P. . C. M. Marcus, Naturephysics , 1138 (2018). D. Das and S. Doniach, Phys. Rev. B , 1261 (1999). M. Diamantini, P. Sodano, and C. Trugenberger, NuclearPhysics B , 641 (1996). M. Diamantini, P. Sodano, and C. Trugenberger, NuclearPhysics B , 641 (1996). M. Diamantini, A. Y. Mironov, S. Postolova, X. Liu,Z. Hao, D. Silevitch, Y. Kopelevich, P. Kim, C. Trugen-berger, and V. Vinokur, Physics Letters A , 126570(2020). T. Hansson, V. Oganesyan, and S. Sondhi, Annals ofPhysics , 497 (2004). E. Fradkin and S. H. Shenker, Phys. Rev. D , 3682(1979). E. Fradkin,
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