JHEP04(2021)148 Springer May 6, 2020 : April 15, 2021 March 9, 2021 : : February 20, 2021 : , Received Accepted Published -term. We demonstrate Revised Θ -term of the projected Hilbert space Published for SISSA by Θ https://doi.org/10.1007/JHEP04(2021)148 [email protected] , . 3 2005.02332 The Authors. Field Theories in Lower Dimensions, Holography and condensed matter c

We perform a unitary renormalization group (URG) study of the 1D fermionic , [email protected] [email protected] Department of Physical Sciences, IndianMohanpur Institute Campus, of West Science Bengal Education and —E-mail: Research 741246, — India Kolkata, Open Access Article funded by SCOAP higher order cumulants of the full countingKeywords: statistics. physics (AdS/CMT), Renormalization Group, Topological States of Matter ArXiv ePrint: entropy bound also distinguishes the gappedvery and different gapless holographic phases, spacetimes implying the acrosssurface generation the of as critical a point. quantum Finally,experiment impurity we is treat coupled devised the to in Fermi theimpurity, order and high to propose energy study ways electronic entanglement by states. entropy which generated A to by thought- measure isolating it the by studying the quantum noise and the nature ofweak-coupling RG RG flow analysis. aroundentanglement, the This as coincides critical the entanglement with point entropy aand RG is flow phase gapped identical shows transition distinct phases to in features dependingthe that for the on the Ryu-Takyanagi obtained many-particle critical entropy the from bound valuenetwork, the of for concretizing the the the relation topological many-body to eigenstates the holographic comprising duality the principle. EHM The scaling of the (EHM) tensor network description.of the A degrees topological of freedomtowards at either the the Fermi gapless surfaceThis Tomonaga-Luttinger are results liquid shown or to in govern gappedRG a the quantum phase nature nonperturbative liquid of diagram, version phases. RG of revealing flow a the Berezenskii-Kosterlitz-Thouless line (BKT) of intermediate coupling stable fixed points, while Abstract: Hubbard model. Thebody formalism eigenstates generates arranged a holographically familyURG across is of the realized effective tensor as network Hamiltonians from a and UV quantum many- to circuit, IR. leading The to the entanglement holographic mapping Anirban Mukherjee, Siddhartha Patra and Siddhartha Lal Fermionic criticality is shapedtopology: by Fermi a surface case studyliquid of the Tomonaga-Luttinger JHEP04(2021)148 ] are 16 ] arising from 18 28 , 20 17 16 ]. Careful studies involving 2 , 1 6 33 ]. Indeed, such boundary condition 19 11 38 45 – 1 – 5 ]. Transitions from the TLL to various gapped phases are 43 14 , 13 ], and are notably different from the tenfold classification of gapped 1 12 – 3 ]. Topological aspects of the zero mode of these vertex operators [ 15 , 1 BKT binding-unbinding transitions ofbe vortices studied that from describe the instabilitiestheory spin of [ and the charge TLL vertexunderstood can operators from of their connection theboundary to associated non-local condition sine-Gordon gauge changes field transformations on [ the Hilbert space [ One dimensional gapped quantumcal liquids properties can [ benon-interacting systems classified [ via theirdescribed by emergent the Berezinski-Kosterlitz-Thouless topologi- universality class.has Further, been the universality shown to extendThirring to model, a quantum variety of sine-Gordon models model in etc. 1D, e.g., Extensive the studies Schwinger have model, shown massive that the Electronic correlations in one dimensionliquids are generically that observed either to lead remain toLuttinger critical exotic liquid quantum (and (TLL)) belong ora to possess host the of universality a class analytical gappedtum of and liquids spectrum the numerical including [ methods Tomonaga the Luther reveal Emery a (LE) variety liquid of and low-energy the gapped Mott quan- insulating liquid (MI). 1 Introduction 9 Topological order and its10 observables Discussions and outlook A RG equation for the BCS instability 6 URG for Fermi surface instabilities:7 a topological Holographic viewpoint entanglement scaling towards the8 Fermi surface Observing the instability of the Fermi surface 3 Symmetries and topology of4 the Fermi surface Topological constraints on condensation 5 Structure of the Fermi surface pseudospin Hilbert space Contents 1 Introduction 2 Summary of main results JHEP04(2021)148 ]. Importantly, ] representation ] tensor network 51 25 – 48 – ] for gapless states, 49 , 23 , 27 , 38 6 , 26 4 ] Fermi surfaces. Even as we 52 ] via disentanglement of high-energy 47 – 41 ]. These insights inspire the following ques- 19 – 2 – ] or Dirac point-like [ ], and is consistent with other calculations for the EE 40 ]. There, we took careful account of the singular nature 32 – 37 , 28 36 ]. The boundary of the tensor network is in the UV, while the 38 ], e.g., the matrix product state (MPS) [ 22 – ]. Given the extensive developments in the understanding the physics of 20 ], as this method generates the RG flow of both Hamiltonians and their 35 40 – – 33 36 In this manner, the URG can be realized as a quantum circuit [ At the same time, the nature of many-body entanglement analyzed via tensor net- of the 2D Hubbardof model the [ Fermi surface,is as to it study contained the vantight-binding Hove chain, simplest singularities in possible at the Fermiregular half-filling. hope surface, connected of (non-singular) i.e., Here, gleaning [ we the insights aim two-point on Fermi higher surface dimensional of systems with the as the RG can beplayed implemented by on the a 1D topological momentum-space propertiesthe lattice, of RG we aim electronic flow to states towards clarifyFermi near a the surface the role variety topology Fermi of surface and gapped infor electronic or guiding the correlations gapless nested has fixed Fermi been points. surface recently of demonstrated The the by interplay tight-binding us between model on the square lattice in a study organization of effective Hamiltonians andtrum, entanglement features entanglement entropy (e.g., etc.) entanglement spec- fromgate layers, UV and to reveals IR the isrepresentation entanglement holographic carried of mapping out URG (EHM) via [ [ acollection collection of of layers comprising unitary the bulk of the network are directed towards the IR. as our goal, wesystems turn [ to the recentlyeigenbasis developed along the unitary holographic networkelectronic RG RG states direction (URG) (UV) [ for from fermionic theirstep low-energy of (IR) the counterparts. URG, one More UV specifically, degree at of every freedom is disentangled from the rest. The holographic area law [ interacting fermions in 1D, aenergy Hamiltonians unified of view quantum is liquidsmany-particle needed and Hilbert of the space. the associated emergence This entanglementelectronic of will content the correlations, surely of effective provide their low low- deeper spatial insights dimensions into and the many-body interplay of entanglement. With this for gapped statesprovides and another multiscale important entanglement pathway for RGentanglement classifying entropy (MERA) low (EE) of dimensional [ critical quantum 1D liquids.using quantum liquids The field has also theory been methods studiedarising extensively from [ a finite Fermi surface in higher spatial dimensions that show a violation of the studying this Fermi surface scatteringthe problem, phase and diagram if obtained yes,affirmative from answer how a would closely RG indicate does study that this theFermi of topological surface resemble the features can divergent of track quantum degrees the fluctuations? of low-energy freedom An physics at arising the fromwork UV-IR methods mixing. [ renormalization group (RG) phase diagramtions. [ Instabilities of afluctuations gapless that begin state with such scatterings as eventsof at the the its TLL Fermi neighbourhood. surface, are and outcomes Can leadthe to of we fermionic the relate divergent Hilbert gapping quantum space such at scattering the processes Fermi surface? to topological Can we properties build of a skeletal phase diagram by changes are associated with Berry phases that characterize the various phases in the BKT JHEP04(2021)148 , ]. ], ], ω 70 36 48 – ] and 68 54 ]. Unlike 53 ] and EHM [ 65 , 64 ] for the entanglement ] in our vertex RG equa- 63 67 , generated upon isolating a ) 62 R ( S ]. Various real space renormalisation diagrams [ 0 39 ], with every sublayer composed of a two- ] against various numerical methods [ ]. An important property of the URG tensor 38 36 61 . Upon reaching gapped phases at stable fixed – 3 – ]. The resummation process generically leads to – ] discussed earlier. By contrast, every unitary ω 39 55 48 ]. Further, the RG procedure reveals a multitude of 40 , 39 , 37 , 36 ) for quantum fluctuations that arise out of the non-commutivity between ω ], and shown the origin of BCS, ZS, ZS ], there exists an exact construction of the unitary disentangler within the URG ] is that it satisfies the Ryu-Takayanagi bound [ 66 ): this is a statement of the observation that 53 38 located at the boundary of the tensor network (the UV) from the bulk (the IR) S ] for a review), and we review briefly here those among them that are connected R 21 The URG procedure discussed earlier generates a family of RG scale dependent effec- A variety of tensor network RG methods have been dicussed in the literature (see bard model at, and away from, half-filling [ points as a functionpoints, of the fluctuation effective Hamiltonian scale arepairs written in of terms electronic of states, composite anddom. operators describe formed Importantly, the out we of condensation have also ofmarking demonstrated these with high quantitative emergent accuracy checks degrees the for of ground the free- state URG energy by and bench- double occupancy of the 2D Hub- obtain closed form analytic expressionsdenominators [ in the RG equationsrenormalized that self- are and dependent correlation on energies. thewe In quantum have a fluctuation also study scale established of an the 2Dformalism equivalence Hubbard of [ model the in URG ref. at [ tions. weak coupling As to a the direct functional consequence RG of the nonperturbative RG equations, we obtain stable fixed tive Hamiltonians, from which weand can higher extract order the vertices RG [ energy flow of scales various ( two-, four-,various six-point terms of the Hamiltonian.tices are Importantly, nonperturbative the in RG nature, equations as of they involve the the renormalized resummation of ver- loops to all orders to region is bounded by theis number also of shared links by otherproviding connecting a entanglement it RG connection to methods between the entanglement such scaling rest as and MERA of holographic [ the duality. system. This feature framework in terms of theapproaches similar electronic to Hamiltonian URG, in [ theof sense that Hamiltonian, they have lead appeared to the innetwork iterative refs. block [ diagonalization [ entropy ( qubit gate that disentangles aall given UV others, degree such of that freedomfeature (with its is, one however, quantum electronic similar state) number tothe from becomes the variational integral deep determination MERA of of (dMERA)dMERA motion. construction the [ of parameters The ref. of finite [ the depth unitary gates in MERA [ to the URG formalism.layers of quantum MERA circuits: isdirect (i) tensor product the unitary of network gate two-local comprisedthe layer unitary of has disentangled gates unit qubits a followed depth from by, collection (ii) and theis layer of is bulk distinct of of composed pairs isometries with the of of that tensor a gate regards removes network. layer to The of the latter the feature EHM of URG [ MERA has a finite depth [ the case of translationrequires invariant four-fermionic the interactions. linearisation of Further,any the our emergent electronic analysis phenomenon dispersion neither such near as the the Fermi separation surface, of nor chargeref. relies and on [ spin excitations. apply the URG to the 1D fermionic Hubbard Hamiltonian, our conclusions are relevant to JHEP04(2021)148 by showing that, 4 , we review the sym- 3 ] for the ML and MFL ground states of ] was recently achieved in the continuum 63 51 , 62 ]. Is it possible to reconcile the partitioning , – 4 – 71 38 ], we have already demonstrated this scheme for , we study the URG flow of the many-particle en- 7 38 , degrees of freedom formed by the pairing of fermions. 36 2 lead to a change in the center of mass Hilbert space by / F by showing that the pseudospin backscattering vertices con- k = 1 ± 5 S , and demonstrate how the topological features of the Fermi surface Hilbert 6 We now present an outline of the rest of the work. In section This leads us to ask the following questions in the present work. First, can the URG necting the Fermi points satisfying a constructive interferencement condition the URG for formalism scattering forin the processes. case section of We correlated then electronsspace on imple- guide the 1D the tight-binding RG chain flows. In section Fermi surface of the 1Din tight the binding presence of chain. ateractions, We Fermi a follow surface topological this instability property in arising ofimposes section from the a the Fermi constraint surface presence on Hilbert of the space inter-particleof condensation (the in- composite of first pseudospin four-fermion Chern class) vertices,This involving is the formation followed in section entanglement scaling of massless Dirac fermionsfield [ theory for gapless phases, wegapped aim as instead to well construct as quantum gapless circuit models phases for arising both from ametries Hamiltonian and of topological lattice-based properties electrons. of the Hilbert space for the degrees of freedom at the and holographic duality for thesimilar unitary fashion, quantum can circuit/tensor we networkfor of construct the the the entanglement URG. RG entanglement In flowin holographic a of this mapping correlated case tensor electrons as in network well?in 1D, terms Finally, and can of verify we two-local the provide unitary entropy a disentangling bound quantum maps? circuit While description a of quantum the circuit RG model flow for the tensor network representation of thephases URG, and in study the the entanglement BKTdistinct RG RG for entanglement the phase entropy various diagram. scalingTakayanagi features entanglement Will entropy from bound the [ such critical athe and 2D study? Hubbard gapped model, We fixed concretizing verified the points connection the reveal between Ryu- entanglement renormalization perform a reverse URGby on re-entangling by the theat UV low-energy eigenstates higher degrees of energy of the scales.the feedom, fixed 2D In generating point Mott refs. Hamiltonian thereby liquidMarginal [ many-body (ML) Fermi eigenstates ground liquid state (MFL) of metallic the state. 2D Hubbard In model, this as way, well we as aim its to parent obtain the EHM diagram has separatrices that separatethe relevant from separatrices irrelevant flows, meet while atof the a flows the along critical coupling point spaceFermi diagram [ surface obtained with from the analysisCan partitioning we of of diagonalise topological the the objects effective BKTURG living Hamiltonians flow RG to obtained at phase obtain from the diagram the the low-energy via stable spectrum fixed the and points the separatrices? associated of eigenstates? the If yes, we can procedure show the emergencetopology of of degrees nonlocal of constraints freedomto residing associated at the with (and condensation near the the) ofversion Hilbert Fermi composite of surface, space and degrees the how of sine-Gordon thiswill leads model, freedom? likely we yield As expect a our that non-perturbative starting an version point affirmative of is answer the to a BKT this lattice phase question diagram. The BKT phase JHEP04(2021)148 , showing 8 , and presenting 10 , we unveil signatures of topological order in 9 – 5 – , we then construct a skeletal counterpart of the BKT 4 ]. In section 72 , we conceptualize a quantum transport based thought experiment for , we demonstrate that the scaling of the entanglement entropy for the , we demonstrate the relation between Luttinger’s sum, the shift of the , we identify the subspaces of the many-body Hilbert space in which the , we perform a URG study of the primary instabilities and obtain a non- , we study a simplified problem of the dynamics of many-particle electronic 8 7 3 4 6 5 points are very different. the measurement of the fulla counting putative statistics Fermi of surface themeasurement instability. electrons of in the In the entanglement this background entropy way, of related we to provide the a instabilities. prescription for the indicate that all couplings diverge atHamiltonians strong coupling. and low Importantly, we energy obtain effective eigenstates for the new stable fixed points. effective theories obtained at the gapless TLL fixed points and the gapped fixed perturbative version of thefixed BKT points phase are found diagram.additionally to the While be existence identical all of to featuresThis those a of in obtained line the contrast originally of critical to by stable BKT, the fixed we perturbative points find nature at of intermediate the coupling. original RG calculation, which states on the Fermispaces surface of identified an in interacting section phase system. diagram From on the various theHilbert special basis space. sub- of topological arguments applied to the Fermi surface primary putative instabilities of theMott Tomonaga Luttinger instabilities, liquid can (TLL), arise. i.e., Withininstabilities BCS these through and subspaces, the we condensation study of the particular electronic pseudospin pairing degrees of freedom. center of mass momentuminteracting electrons and in a one firstthe spatial class topological dimension. Chern aspect Together,system invariant of these of for quantities the electrons. a represent two system point of Fermi non- surface for such a one dimensional In section In section In section In section In section In section 6. 5. 4. 3. 2. 1. experiment to track its(erstwhile) Fermi features surface. of We end from bysome the summarising future our properties directions. results of in section the degrees of freedom2 at the Summary of main results to gapless and gappedtribution quantum liquid between the phases. fundamental We and studythe emergent manifest dynamical degrees unitarity spectral of of weight the freedom redis- from RG in formalism the section by scattering tracking matrix thesome flow [ of of the the Friedel’s phase gapped shift quantum liquids attained from the URG flow, and design a thought tanglement and unveil the role played by the Fermi surface in distinguishing between flows JHEP04(2021)148 ] . , F ↓ = k k 73 † . At E = 0 (3.4) (3.3) (3.1) (3.2) P R . The i z , where ∗ † kσ = ˆ N )) ↑ , im P c t k iω E H z, k . ( h = 0 = + , )) , L ret aσ re z leave the Hamil- iη ω kσ G kσ 2 ˆ N − − = Z E z kσ (0 ∈ ) = ( 0 = P ˆ G ,T z, k ↓ = kσ (¯ † k , (ln symmetry mentioned above. c E ˆ 0 N ) is given by T adv aσ 2 T r = F > . Z G , † t ) − , k × T z H kσ F ( ↑ )) P 2 c † k k . = . The parity operation (P) Z . For all non-zero momenta, the poles iη † kσ − ) † T c c k , Im F T 1 ) k t R/L poles of the non-interacting single-particle − k ) associated with number of states at the (0 + cos ) , where the Fermi energy and velocity are + 1 0 = t . At zero temperature and with a chemical ) k ˆ ak 2 N G ( a F cos( = 0 † kσ E ,TH − k F c t – 6 – v E (ln iθ − kσ = ± X H e ), the poles appear at the points = z t k T r F 2 kσ ( = k F − † E ) = v − re,R θ P ) = ( , = ( t (accounting for spin degeneracy). ω and where ) = † F ≈ t 0 k U z, k F ˆ H spin-states due to the F G ( PH k = 4 and ) † kσ E c ↓ ) ret aσ 1 ) (ln , sin F − G θ N ↑ around the Fermi points ( t k )) ( phase-rotation symmetry of the Hamiltonian under the unitary k T r 2 ] U( − ω F − 3.3 : k v for commutes with the number operator cos ( ~ = ) t t F / ]. This arises from the dω∂ U(1) 2 v 0 F H U(1) , we study the origin of topological order in the low-energy subspace, − Z Λ 75 = , v , L, R , 9 ∈ ( = F = F 74 ) k v and time reversal transformation (T) 1 θ ~ re,L ak 2 N / , a description of the low-energy fermionic excitations about the sharp Fermi ω E cos U( 0 Z µ , t : Λ 2 ∈ F − k − and present various measurablestopology. that Our signal major changes predictions in heremodel include: Hilbert is (a) space two-fold the degenerate geometry ground at state and half-filling,of of (b) the mass the 1d entanglement Hilbert entropy Hubbard space in the hasthat center a interpolate topological between piece, the and ground (c) states. there are fractional excitations :[ In section − ,P = 0+ The retarded single-particle propagator (Green’s function) for states in the window = 0 7. kσ Λ F im − † Fermi energy [ Green’s function (eq. ( For the 1D Fermi surface, retarded Green’s function has poles inω the complex frequency plane at appear in pairs For states at the Fermithis energy point, ( we recall a topological invariant ( W and the advanced propagator by its complex conjugate: admitting a global operation potential surface is obtainedand by linearisation ofE the dispersion around the two Fermi points, tonian invariant Accordingly, the dispersion hasThe the Hamiltonian following symmetries: the 1D tight binding Hamiltonian in momentum-space for spinful electrons is given by [ and leads to the dispersion spectrumc 3 Symmetries and topologyWe recapitulate of the the symmetries Fermi andsurface surface topological of properties non-interacting of the electronsbe Fermi in volume critical one and in Fermi spatial dealing with dimension the in case this of section, interacting electrons as in these later will sections. To begin with, JHEP04(2021)148 . ) Z L/R 2 z, k (3.8) (3.6) (3.7) (3.5) I ( × . ret σa s 2 G as follows I symmetry . )  ↓ = 1 = ) , (SU(2)) = (2) ↑ ). This unites 4 3 ) ( 2 C I at a given state π k ω, k ( Z . The homotopy ⊗ ( 3 ) Z 1 SU ∈ dk k S ) is also associated σ,L − σa can then be seen as ∈ of the single-particle − F G C dn v ) ω groups for the spin (s) . K k, ~ L/R ∂ , the Fermi energy sub- σ,ak / 1 dk ) 0 n k, z − = = ( = 0 F 0 i ) v F 0 Λ ) = 0 ˜ 2Λ ~ k unit matrix, Φ( v a ω, k Λ ) is given by T ~ ( , SU(2) + z 4 σ,ak − ˜ k F z, k σa k F n ( = 1 ˆ × m k G − 4 ) 2 − ret σa 4 windows, the Fermi point singular- I , such that Z n ˜ † k ] G i Z h ω, k m πi 1 z ( ˜ k 2 1 ∂ + 0 . is the S=1/2 representative of the SU(2) E is the − σa ) = ˆ σ ∞ is associated with a non-trivial homotopy 4 G (2) = (1)) = I k ] is defined as an integral over the momentum i, C 1 ˜ k z, k by traversing a non-contractible closed path ∂ S , σ,L ( ) ( N ) 78 1 – 7 – kσ SU , + 0 ret σa , ν π ,H E k,z represents the compact space , ˜ ω, k × k 1 77 , where ( Φ( P − = i H n −∞ )) arising out of a pole of the single-particle Green’s · [ e σa dz G (2) symmetry, leading to an enhanced | ˜ k = 1 = ˜ ˆ k ) G σ X E 3.4 Z  2 | ) , L/R + t = k, z n k πi ] 1 | ( ( ]. Further, U(1) t ↓ 2 aI SU k G + H 76 | dk − σ,R dω T r = = × 2 ), along with compactifying the boundaries of the momentum- , c a near the left or right Fermi point ( dn I ↑ U k (2) ) = possesses a further particle-hole symmetry ( σ,ak 3.7 k s − dk dk n , c F 0 k, z SU I v ↓ F 0 Λ ( = 0 ~ v k Λ characterizing the winding around the sharp Fermi surface. for a sharp Fermi surface within a window ~ G representation of symmetry group SU(2) is associated with a topological + πi 1 , c − F and frequency-space =0 2 Z unit matrices are Casimir invariants of the ↑ 2 K k F k k ] / k ) in eq. ( 2 c = F H Z I Λ v ) = = 0 ~ 3.6 = 1 1 = [ aσ πi 1 , S T ν 2 S ( ˜ k F Λ around the singularity. The number of electrons ( v 1 topological invariant (eq. ( − ~ m = π 1 [ 1 space, i.e., the Bloch sphere. The rotations on the Bloch sphere is generated by the S A pole in the complex frequency plane of the single-particle Green’s function The We can represent the Hamiltonian of non-interacting electrons as a sum over a set of N belongs to the homotopy group of 1 σ,R ≡ ν Using eq. ( space ities lead to first class Chern invariants on the frequency-momentum torus vector at kσ The orientation of the two Fermi points [ group at a given momentum with a residue. ThisGreen’s residue function is pickedP up by the phase field group of the compactThe space topological SU(2) invariants (with of asurface this constraint space as are a Chern sharpthe invariants. boundary The in existence energy-momentum of spacefunction the at at Fermi zero frequency [ points is seen to be CP set of unitary operations group and the constraint and the two and chirality (L/R) sectorsHamiltonian respectively. For thewith poles the at number conservation for states at the Fermi surface. Thus, the complete symmetry group for the two Fermi sub-Hamiltonians, each of which governs a group of four states where JHEP04(2021)148 ] = ) 82 ˆ T , (3.9) E (3.16) (3.12) (3.13) (3.14) (3.15) (3.10) (3.11) 81 , 18 , for the spin ]. Note that 2 σ,R 81 . J , , − ) , causing a shift in σR sectors respectively 80 ν . The Atiyah-Singer , s σ,L − fermions with opposite 17 J L/R = 0) , ↓ j i σL = = ( , and ν ↓ 5 z i a σ tot c (SU(2)) ( ψ and 3 z = tot ,S π ↑ iπφ , ,S ) − ∈  = 0 e . The non-commutativity between ) , z , tot ↓ R S = 0 e † around in a local patch (P) given by jσ cm ˆ − ψ U = 1 N ˆ ˆ ) = T n i ) P ↑ † R | cm S U NS N ψ z jσ † tot c/s P | c/s )( π/N ˆ fermions, leading to a gain in momentum 1 = O ( = − ) in the presence of an electric field ( P ,S . The action of the Wilson loop operators πν = 2 ˆ c eE , J T ↓ iφ F 2 j ) degrees of freedom (d.o.f) with positions L is related to the anomaly of the axial charge S i π/N ˆ = ( c k , and can be related to the net axial current O ( c/sα ψ e C 5 ↓ , and we have kept a factor of = ˆ = 0 c/s e T e − − ˆ † L iφ , O J ˆ ˆ – 8 – , jσ † = O N  ↓ and 2 ψ = 2 , = , ψ ∆ cm ( F ψ α i T e U F 0 ) ↑ /N ˆ † k = c P T ↑ v † jσ | Λ ˆ dx z are the net flux in the tot σR = O ~ U z tot ψ ν = ˆ † ˆ c/s s k, ω T , where = 0) S Z − ,S i = ˆ ( c i ˆ Z O = − ) j C  /N  ˆ s O 2 ( ˆ aσ T defines Wilson loop operators on a two-tori [ 5 = gives σ ˆ ↑ , we obtain the twisted state respectively by a twist operation [ = 0 d s × O z tot jσ ) due to spin excitations of the Fermi sea σL dt G J } s i , α ψ ˆ ν ˆ S n 1 e k O ↓ ) ) and charge ( 1 ˆ Z -component of the total spin angular momentum and c/s cm 2 cm ∂ − s z tot ± α X ) z iπφ = P = P O e |  , α changes boundary conditions for ,S log 5 = + s Ψ( leads to twisted boundary conditions (TBC) in real space. Thus, F | 0 s ν (0 ] c/s k, ω ↑ (here 1 v dt ˆ Λ ( ˆ σ O ) = dQ ~ Z S = 0 ∈ 1 X jσ N, is the − aσ  ˆ σL,R n N × = 2 and j ν = 5 cm σ z c/s j . By taking a state / 1 5 πφX = J eG e 2 P ˆ c S S | 2 X N J =1 ˆ j i j , N j N T 2 ( = = i ) ∆ ≡ ,X / ↑ 1 P 5 P ↓ e σ 2 ψ = ) J σ,a X N , α = c J ∆ N → ˆ is the global translation operator. On the other hand, the charge twist operator (0 Z − = exp [ = ] ] is then obtained as s 7→ T (2 z is the charge of the electron and ↑ tot σ ∈ c ˆ / ) O 77 S cm ν log e U ] X φ ) across the gapless Fermi points ( c,s on the state iP 5 For fermions on a 1D lattice, the axial current is generated by momentum imparted 79 = 1 = [ causes an equal amount of twist for J and the twist operators ≡ { s σ 5 c c/s φ, X ∆ ˆ ˆ where emanating from P ( where Z for the center of mass of the Fermi sea where the total no. ofT electrons is given by where exp [ O where the spin twist operator phases center of mass momentum ( to the center ofX mass spin ( X degeneracy. The twist operatorthe is center defined of in mass terms of the following unitary operations on where Q index [ ( It can be seen that JHEP04(2021)148 . , ) ) = π 2 are M due = α , α 2 − γ (3.20) (3.18) (3.19) (3.17) 1 M c/s T α S J and = ( = ( ) such that 2 S α p/q N/ that form a Möbius = i e ) , such that α ] (where i ,N ) . Ψ( | α 80 . ]. In the presence of both q p , = 0 π Ψ( 87 18 | cm . In the space ) of the twisted tight-binding – 1 ) M ∇ . eP α φ = 2 | i ( ) , 85 , α − + Ψ α π )) is proportional to the center of | c/s γ 74 2 ) c/s (i.e., † E c/s cm Ψ( e φ J ˆ h ˆ ( 2 P O 3.17 / = × c/s cm = = E P ∇ 1 c/s c/s = 1 cm · † 1 α α (eq. ( = c/s P ˆ α ˆ ν n O space ˆ i γ c/s O ∆ ∆ – 9 – e (represented by the red vector) gathers a phase J cm Ψ α e α | represent the eigenstates ˆ 2 i P = d ) 1 2 = † c/s α π α T ˆ ] on the torus can now be computed in terms of the 2 ˆ O O c/s 0 cm t ) corresponding to the change in the persistent current c/s c/s Z Ψ( ˆ P | 2 1 H 84 E , π c/s ∆ 2 2 2 1 and dα 0 e c/s d 83 Z ˆ 2 O ≡ i /dα α 1 along a Möbius strip. In the above relation, 2 . 0 ] can now be computed 2 + c/s c/s = ) induced in the ring under the adiabatic twisting of boundary Φ J T E π 88 2 γ γ c/s , 2 c/s d cm ˆ X P 17 = , the ray field ∆ = )[ 1 e cm 2 † c/s P α represents the zero mode of the charge/spin current generated on 1 ˆ T ∆ ) α ) = ]. For this, we note that the eigenvalue ( , α π c/s ∆ 0 2 ˆ X Φ c/s 88 , , 2 2 . The red vectors on the torus is the Luttinger’s sum for charge excitations [ α c/s to ˆ 17 ,J T ν 1 [ . Additionally, the first Chern class α 0 α π Spectral flow arguments further reveal the significance of the momentum shift ( = cm c/s J = 1 P ˆ x. J where the persistent current momentum generated in the ring is given by The energy curvature ( ( conditions ( mass “momentum shift” obtained upon adiabaticfrom twisting of the flux over one closed circuit ∆ Hamiltonian is given by is the Luttinger’s sumthe for total spin spin excitations quantum ofp/q number the and Fermi sea magnetisation [ time-reversal of and the particle-hole Fermi symmetries, seaγ respectively), while As shown in figure as it traverses the torus where we define the( persistent currentto vector TBC. The firstcurl Chern of class the [ nonlocal current vector in the Figure 1 strip for the centre of mass Hilbert space, where the angle between initial and final vector is JHEP04(2021)148 , ], ∈ is 2 J 2 0 ) in , 0) Φ 1 , C (3.21) / (0 1 c/s J ) and spin − ↓ . The above = Φ ] 0) π φ , of the 1D tight- , k 2 0 ↑ , F k (Φ [0 k ± ∈ c/s 1 J c/s = ) X . 5 c/s 1 F c/s J J k ∆ )) is related to a topological in- − ∆ )). While the energy curvature is , ≡ 3.19 3.17 F c/s k states upon tuning through a complete ( ) = 2 R ν (eq. ( . , the relation can also be interpreted as the 0 = 2 . γ − ) 2 ]. We will discuss the effect of such instabilities L R C ν ) of the conformal field theory for a gapless 1D ν 2 2 – 10 – , c Φ = Φ ] for a system possessing Lorentz invariance [ 1 − = ( L . This is the well-known phenomenon of anomalous 88 ν 5 , around the two Fermi energy points ( c/s J 1 1 2 17 R )) for the states J ∆ ν ], seen either from the perspective of the center of mass = ∆ 3.8 c 90 , and 1 ], F c/s L 89 k ν 92 , 91 represents the Fermi momentum of the charge ( , 35 is the AB flux), and is connected to the Atiyah-Singer index computed ), together with the ejection of one electron from the right Fermi point F c/s C k c/s ), leads to a net transfer of Φ C ≡ − ) belonging to the first Chern class ) excitations of the Fermi sea. . Chern numbers ≡ 1 ↓ γ − k (where − = 1 The same thought experiment can also be carried out with the persistent current Further, it can be shown that a net persistent current = , 1] ↑ L R , ν ν k axial-symmetry breaking [ or that of theapparent anomalous Fermi breaking surface. of the When symmetryis is taken directly restored. together, related Finally, the these to Atiyah-Singerelectronic the two index system anomalies central [ charge cancel ( and the an anomaly in the bulk( of Fermi sea: the injection( of one electron from thecircuit. left At Fermi point the sameaction time, as of shown independent in monopole figure leading sources to at an the anomalous L/R current Fermi points (the Chern nos. In the above, ( i.e., by now takingrelation the depicts center that of the mass net on current a accumulated by complete the circuit center of of mass is associated with [0 from the Chern numbers (eq. ( can lead to a RGsociated relevant change vertex, in resulting the energy in curvatureassociated the [ with creation topology of a changing many-body terms in gap a and an later as- section. accumulated by taking the center of mass on a one-flux quantum circuit of In this way, wevariant find ( that the energyproportional curvature to (eq. the Drude ( weightthis [ is generally not trueImportantly, for a a lattice Fermi based surface non-relativistic instability system of arising interacting from electrons. Umklapp back-scattering processes Figure 2 binding electronic chain take part inboundary the conditions spectral by flow a of full the center flux of quantum mass spectrum upon changing JHEP04(2021)148 π ) = = 2 ) to = σ (4.1) (4.2) U γ /S 1 )sgn( symmetric k C U(1) = sgn( . This scattering η ⊗ † jσ ψ . ikj L/R π e , j ↓ i = 2 ), the matching of helicities n ) that are either occupied or P ↑ SU(2) i γ 3 N 2 n = 2 2 ⊗ projected Hilbert spaces to form d.o.fs formed by pairing i s √ co X / π ] . N 2 L/R U σ 93 γ/ − + = 1 Hilbert space for the emergent degrees SU(2) k . The Hubbard term has a spin backscat- − 1 kσ † kσ n SU(2) c ± iσ 2 = c = / ψ † kσ = ˆ η c that, in the presence of a Fermi surface insta- † iσ by showing that the pseudospin backscattering lead to a change in the center of mass Hilbert ) ψ kσ = 1 – 11 – 4 ). It is important to note the topological origin 5 k 1 n F S = k SU(2) )) defining the center of mass topology of the metal : ˆ ± , where iσ CP cos( 1 n ↓ → C 1 3.17 k,σ k X and chirality c ensures that each paired-fermion state is equivalent to a t s ↑ 2 L/R 1 1 k − (eq. ( C − c π = ↓ SU(2) q SU(2) = H − 1 γ ⊗ k † − s c ↑ q + ]) upon adding the Hubbard term (with on-site repulsion strength 1 SU(2) † k 2 c : the Chern class This constraint can be classified in terms of matching of helicity ( While the results presented in this section are for the case of a two-point Fermi surface, 1 ) across the Fermi surface. Starting with the C 1 of freedom whose condensationrespected characterizes via the the formation instability.unoccupied. of paired-fermion Time The states reversal ( constraint symmetrytwo-level is system (i.e., isomorphicof to ± many-body fermionic Fock spaceleads of to the a locking metala of (see spin new section helicity process operates in therespecting low-energy parity subspace and with time reversal zero symmetries, momentumconstraint as and seen on from opposite the the spin many-body application of pairs fermionic the Fock following space [ where the local electronic densitytering is vertex We will now look intoFermi the surface topological (e.g., features either thatFermi the arise surface out spin [ of backscattering variousthe or instabilities tight-binding the of Hamiltonian the Umklapp scattering across the fermions. We will follow thisvertices up in connecting section the Fermispace points by satisfying the constructive interference condition 4 Topological constraints on condensation introduced in this sectioninteractions, symmetry will and be topology shownsurface. in to Specifically, shaping we the be will many-body see usefulbility instabilities in arising in of section from understanding the the Fermi theimposes presence a interplay topological of of constraint inter-particle on interactions, thethe condensation the formation of of first four-fermion vertices. composite Chern This pseudospin class involves the formalism adopted by ussystems is of equally noninteracting applicable electrons. totees Fermi the surfaces Further, existence of the of higher notionthe a dimensional of presence Fermi adiabatic of surface electronic continuity inproperties guaran- correlations. Landau’s of formulation the It of Fermi is, liquid the thus, in Fermi possible an liquid analogous to even manner. explore in In various what topological follows, many of the ideas JHEP04(2021)148 ] i ↑ 2 ∈ 0 = κ / | Λ ,η η , 0 , the ( (4.4) (4.3) 0 is the ω κ . The 3 = 1 Λ A H 1) − S , for 1 = [ − , . In this way, = 1 0 i Λ η (Λ , σ W ˆ s , or pseudo-spin Λ k 2 1) / − . , = 1 i ≡ | ) classes, 3 : as shown in figure 1 η κ 1 κ , | S . c C 2 2 κ within the oval) is composed of / c 1 co U † κ c N † κ 1) = (Λ is the antisymmetrizer and c = − 3 A π ,κ leads to formation of SU(2) is the number of electrons. A subspace 2 states in the window 2 = in shaping the projective Hilbert space e ,κ X 0 1 η γ/ 1 N Λ ( πS κ C κ N , where U = 2 V 8 that extract four-fermion vertices involving | = 2 N and for 4 1 = ) – 12 – } ⊗ γ ] for right and left Fermi points respectively). The 1 − , ... H − 1 1 ( σ ). The momentum wave-vectors in the window are h.c. ) A π, π − − C F particle antisymmetrized Hilbert space of a 1D system = k j,l + k − ( = 2> 1> [ − κ η N − n c N 4 , ∈ T r and ) 4 is the normal distance from the two point Fermi surface and l F + ˆ (a) (b) F H k via the relation Λ can be grouped into two helicity ( and condenses pairs of fermions forming , k † κ kσ = 1 i 1 i c 1 is constituted of ˆ ( 0 n κ ˆ s C | | 1) | C κ l N k † − 4 c , T r 1 H , where e s =1 − l N . Hence, the transform via imposition of the constraint ⊆ F , pseudospin vertices. ↓ Q 0 0 +Λˆ 2 { Λ (Λ Λ ˆ s / N N F 8 4 := or k maps a subset of four-fermion scattering vertices involving zero-momentum F ⊗ = 1 1 N = 1) H F C S , . The constraint Λ A B 1 . (a) The four-fermionic vertex (designated by the N k , 4 = lattice sites is defined as . (b) The pseudo-spins scatter via spin-flip (1>) and non-spin-flip vertices (2>). 0 ∈ F We now present a precise mathematical formulation of how the symmetries and topol- Λ N † N 2 denotes the orientation ( 8 ˆ The single-electron states 1) = (Λ ogy of the Fermiof surface the guides emergent the condensate. constraint Inwe order first to obtain carry a out compactwe the notation define transformation partial for of trace the the operators one-electronfrom Hilbert states: the space, Hamiltonian given by s states within constraint pairs onto of single-particle projective Hilbert spacethe in basis which the many-bodyF states are representedaround in the two Fermi points( As a consequence ofHilbert the spaces from constraint thestate original has a many-body partner fermionic Fock space. Due to TRS, every Figure 3 c, c SU(2) is connected to constraint JHEP04(2021)148 . , . is 2 π ) ] in , we π/ = = 0 σ (4.7) (4.8) (4.5) (4.6) 2 is the TRS- 94 . We ± γ ) ) . p 0 π/ ) 0 = σ is ⇒ ± )sgn( = 0 leads to a (Λ − F s 0 . As shown σ, σ p , k 6= 2 ) 0 κ N j / ˆ − s γ/π F n − k σ, , , = j 0 . The non-trivial and = 1 , with magnitude − = sgn(ˆ θ . ( S ,η κ S η 1 is the width around 2 Λ = 0 κ h.c. , 0 → c A 2 ) = ( ). = (Λ , 0 1 Λ ) 3 + 3 ] κ ↓ ,η 0 1 S c ˆ . We also note that the κ s j 0 , σ . Note that if 93 κ c − ). The resulting Anderson Λ † κ 0 , 2 3 ˜ c c ˆ s κ Λ 3 k † κ c = 0 , σ σ 4.2 , | ↓ih↓ | 1 c Λ ˆ 1 n ) arises an emergent symmetry † κ ,η 1 k 2 ,η σ ,η and, j c ). Here, + − ( 0 ± Λ k Λ † Λ + 1 X ↑ κ,κ ) 4.2 ˜ c 0 2 ˆ A † κ s A 4.6 , ˆ s 0 c S 2 Λ U V , σ k Λ 1 0 as η,η = TRS pairs, where ˆ κ n s k δ = X 0 , 2 algebra | ↑ih↑ | 0 ,η + Λ / . On the other hand, for j , U V 1 Λ = Λ = 0 , h.c. correspond to the momentum wave-vectors δ nπ + 1)(ˆ S = p I = 1 s S + , = (Λ 0 i ijk ( SU(2) ] = 0 ˆ  s with a monopole of charge = 2 momentum subspace can be written purely in S κ j S 1 c h.c. net-momentum electronic pairs 2 n 1 0 – 13 – n κ ˆ s κ · = 1 T ] = + c , 0 | ). This allows identification of the first Chern class j for the Λ = 0 ) (where κ i )) ,η = 0 Λ k ,η ) 0 σ c ↓ p and with a parameter space 0 0 I Λ S k j Λ H + p κ κ − j 0 . Here, 2 S − s c c | † κ 2 / 0 iθ ˆ Λˆ s, ,A c † κ )) , k ( + 1) 1) ) follow the c ,η ↑ − 0 H k , κ i Λ = 1 0 S are defined as earlier. We now define Nambu spinors [ − 0 exp[ ( † κ | A = ( 0 [ 0 1 S c T r , Λ S κ ( i and orientations κ † κ =1 , 0 ,η N Y SU(2) ↓ j n within the electronic states, we obtain an effective model containing off- c , κ ) s = (Λ † Λ k = ( 0 i 1 1 d ,η ∈ κ − = + ˆ κ 0 0 κ 2 κ Λ T r | j 1 κ , κ ,η n † κ ˆ , L † Λ , such that 0 T r ,η c κ ↑ Λ n c 3 spheres explore the topology of the enclosed null vectors at the origin , κ ( Λ k ˆ s ) n . κ κ 3 3 , A 1 0 ,η ) is extracted as follows | A X Λ j κ | S ,η ↓ 0 , k ]( † Λ T r ˆ s, σ . The three spin orientations are given as + 1)(ˆ ↑ z Λ c , 93 A S of the resulting projected Hamiltonian eq. ( = 0 ( and ,η = ( )) on the center of mass torus z Λ a, the S 0 2 n ˆ s ,η Λ = (Λ representation A j 2 4 N = † Λ Λ 1 2 κ ˜ 3.17 c k 2 ⊗ | , Within the projective Hilbert space enforced via eq. ( The instability in the electronic Hilbert space due to the backscattering processes re- A subset of the four-fermion vertices containing the zero pair-momentum 1 CP ,η ˆ s Λ 1 (eq ( Λ A where by figure of the angular momentum sphere: SU(2) both the Fermi points within whichnumber the of putative electronic condensation states takes within place, the and window.prescription in We will a demonstrate this later via section. the unitary Further, RG this space of SU(2) transformations is spanned by and the Casimir two-level system ( γ sults in the condensation ofpseudo-spins electronic pairs, [ as seen from eq. ( The wavevectors the the helicity. From the spinors,| we define Anderson pseudospinsHartree [ processses terms of Summing over alldiagonal the scattering terms involving where can easily see that the sum of momentum wave-vectors belonging to k the Umklapp process is allowed,have implying only invariant pair ( three normal distances JHEP04(2021)148 . ) σ . with − (4.9) , (4.11) (4.10) B p 1 Λ is given = − N 0 , Λ , , is governed (Λ ˆ LB )). The twist W = 2 0 Λ ˆ s,σ , N j cm 2 and 3.11 Λ P ⊗ ˆ n ) ) that wraps around a j 1 3 , σ (eq. ( S b, the geometry of the jsσ X CP c +1 . The zero mode of the ) 4 ˆ O Λ ,η , p, = ( Λ 1 N 4 − A / tot Λ A (Λ + · H P , = (1 +1 (b) 1 = − A Λ cm η centred around a monopole of charge ) span the space A Λ ˆ under unitary operation: X A group generated by the center of mass · ω 2 U N 1 S A 2 . Therefore, a vanishing center of mass i} 0 Λ + z Λ = πS, N + A A 2 is seen, therefore, from the homotopy group . As shown in figure † = 0 ,η Λ ⊗ ˆ L N Λ . This symmetry in the projective Hilbert space via the twist operator z Λ + 2 . 4 U ω N ω – 14 – Λ N 0 0 cm σ A 2 2 F A S Λ Λ , can be generated by applying twisted boundary cm − ≡ {⊗| Λ ⊗ U(1) ˆ L eP ˆ p, = ( X W 2 to every electronic pair, resulting in the pseudospins B + ω 6= 0 ˆ X k L p =0 → − F p Λ=0 , the Λ , and the number of states within window 0 SU(2) n cm v traces a sphere j implies the presence of parity, and leads to a vanishing Λ H cm P F j = ˆ = ,η i X Λ δk : + Λ Ψ , ensures the stability of the composite objects. A shift of | =0 k,σ (a) A p F   = n z k j = : H . j = 0 S ω 1 i j = , θ S I k C ) Ψ j , . (b) A representation of the geometry of the Hilbert space of the angular 2 ) in the Fock space z | being constituted of electronic states k z ω / j S † c 2 X , n U A Λ2 i ˆ 0 cm O + 1) , P and   ,η , and is a reflection of the existence of angular momentum spheres of radius S Λ 1 ( Z S A S = (0 c is the total pseudospin vector given by coinciding with the null vector ˆ p ∈ j O = exp . (a) The unitary operations η n z z = = A U . Further, | U γ/π Λ ,η ,η The dynamics of the modified projective Hilbert space, For Λ N Λ = (SU(2)) = p, A 3 S | where Hamiltonian is given by This momentum gain creates aa collective modified Cooper-pair constraint persistent current by the projected Hamiltonian kinetic energy, the center of massconditions momentum, for the states in theoperator window imparts a netA momentum where by total current for composite objects: reflects in the invariance of the basis translation operators is topology of the symmetryπ group ( angular momentum vector 2 Figure 4 monopole of charge momentum vectors JHEP04(2021)148 . 1 π 2 − . We / γ/ , such (4.13) (4.14) (4.15) (4.12) F π . ) at the E . π = 2 = = +1 /  F =  ˆ s in the center 3 2 k 3 2 2 2 γ = 1 + )) is given by T + i n S 2 n /   1 . has eigenvalues and , representation of the i 2 F / + 1) k 2 + 1) 2 / h.c. , and where n ± † n 0; 1 ) | (2 + )) and the lowest positive H ¯ (2 ω i = 1 c Λ − 2 − − R / † ω S 1 c A is given by , , ( Λ 0 2 + 1) R σ + L + 1) Λ / ) ] p , and which can be classified in terms A A N . Subsequently, we will study their ¯ p ω ( ) 2 h · 2 ( c between the highest negative energy 95 p σ 1; 1 p ⊗  C | L Λ  † ω − ⊥ 0 c N A gaps the spin excitations around J ( Λ Λ or ˆ s, 4 Λ U S , = 0 U N N E 1 U SU(2) N 2 + Λ 2 = C 2 ∆ and does not cause the coupling of helicities. F − Λ ˆ s = = E , – 15 – z R 1 = Λ 2 − A = ( F A Λ k A ¯ ω 2 z L . The Umklapp instability Hamiltonian is governing n,m,p n,m,p , a pseudo-spin singlet ( ± + A F ) 0 = projective Hilbert space of spins with survives after taking account of divergent fluctuations ] methods. Λ Λ p . Λ and || ak . The robust nature of this charge gap will be confirmed 2 +1 N 2 N J 2 ,E ) ,E H , Λ E 2 A i i 1 (8 N 2 2 ∆ L/R . In turn, this leads to pairs of fermions from the same side / 2 ˆ / = s, σ / | ). The Fermi momentum satisfies the condition π SU(2) , 1 U F | U/ k . This shows that = + 1 + 1 3 , a pseudo-spin triplet ( − 2 Λ ) = ± n / γ n − is the total pseudospin vector that acts as the generator of global = (Λ Λ N 0 = p 2 ˆ Λ s E = N = ω , = Λ 1 H ∆ R (8 U/ = +1 − / A A | 1 ˆ s A = Λ E U around the Fermi energy = | 0 = P , where L E E ¯ ω = A ] and bosonization [ +1 = n − ; ∆ A ˆ p s 0 1 ; = 97 , E E A group for pseudospin operators p = ω 96 n Λ = = The projection mechanism leads to the condensation of spinors representing pairs of A similar analysis of the Umklapp (charge) scattering instability leads to the constraint : A | A E 2 | In the next section,Fermi we surface will arising see from theeffect the on formation the constraints of center topological of objects mass Hilbert (with space due to the ensuing instability. The gap around the Fermiand energy triplet for states: charge excitationsvia exists RG between in pseudospin a singlet laterof section. this Hamiltonian For the sake of generality, we will study the anisotropic version where rotations in the projectiveeigen vectors Hilbert given space. by The Hamiltonian represents the two chiralities fermions with total momentum the dynamics in the projective Hilbert space that the particle-hole symmetry enforcesof the first mass Chern Hilbert class for space, of the torus the Fermi sea formingThis a projective HilbertSU(2) space is again associated with a forward scattering events inRG gap [ opening of the 1D Fermi surface hasC been confirmed by of the chirality ( ∆ will show subsequently that in a renormalizationscattering group physics (RG) is formalism. givenHence, by the latter Unlike does backscattering not lead processes, to instabilities forward of the Fermi surface. The RG irrelevance of The gap state ( energy state ( and possesses eigenstates and eigenvalues given by [ JHEP04(2021)148 ) ) ] = on := 2 c/s − π and ˆ / 2 O s form F b z π (5.3) (5.4) (5.1) (5.2) , Z A 2 = → C = + : (0 + . In the γ ) 1 ) = η 1 F a z S and containing 2 F a/b A U(1) ( 1 ∈ ) operating on i} ≡ C F ∈ cm F b H is invariant under ] can be studied via ↑ P x is composed of the (SU(2) i} = 2 1 3 : 1 ↑ F F a 1 π due to the transfer of F i} F F 1 F H ↓ k iθD ˆ S N π | ↓ is the Chern number of F ∓ , k 0 i , the pseudospin flip term ↓ ∓ S 1 + 2 etc. 0 F F b SU(2) F ↑ k , , ↓ ) 0 F cm = 2 | = exp [ x k + F b F b F a P 0 1 D | 2 = A F A Λ h.c. . . Here, the indices = U i · space. = b C + F a N + γ i 2 1 & A F a ≡ {| ↑ b F s/c 2 2 c/s U/ A . The subspace + 2 1 a F s,c CP b ˆ | ↓ Z = ( SU(2) O Λ F , + + | ↓ i cm a U N † F 2 z ⊗ ≡ ∈ , , where ↑ Hilbert space manifold, and will be discussed 2 P i 1 F ) A a D ] ↓ F k is equivalent to the Lieb-Schultz-Mattis crite- z Λ F cm → ∓ C k – 16 – π P 1 N h.c. SU(2) ∓ ↓ πD 2 1 cm F 2 + SU(2) ↑ = 2 i k → P = 0) = = 2 F U/ ). The backscattering leads to helicity ( 1 b k | − F b F F − γ 1 ˆ | A 2 ] N = , defined on the compact space δk correspond to the pseudospin projective Hilbert space of = 2 = ( i = + F a 19 ) 1 [ cm on the subspace = exp [ SU(2) i ,σ F A F s,c P ; ( F ) . The projective Hilbert space 1 H s,c H b F × 2 ; c/s R/L ak , and the spinor U a ˆ a F O ˆ (2) n F h.c. b A ,σ := ( + ⊗{| ↑ ± σ SU c = † F SU(2) ≡ {| ↑ − F b a × . This leads us to conclude that, in basis A A ]. The relation ) P F 8 π + F a (2) 2 81 = a , A γ/ F 19 2( SU , and is associated with a topological ˆ / N 1 1 = ( F allows the identification ⊂ ], and allows either a gapless unique ground state or a doubly-degenerate gapped and the index 1 projective Hilbert space at the Fermi surface (FS), supporting topological objects D F ) 18 Tunnelling between the two degenerate levels of the subspace As seen from the center of mass projective Hilbert space, a doubled twist operator = 0 − / is given by z F the effective Hamiltonian where a Unitary transformation (corresponding to a vertex operatorthe in double-twist the operator equivalent sine-Gordon theory) is equivalent to 2 electrons [ rion [ state of matter.between The paths gapped on a state non-simply connected offurther matter in section is associated with constructive interference SU(2) states in causes the total momentumto ( shift by a reciprocal lattice vector flip piece electrons ( symmetry breaking: A the effective monopole charge associated with the homotopy group (+ spin and charge instabilityF sectors respectively. The sub-Hamiltonian ( and possesses a resonant backscattering at the FS in terms of the action of the pseudospin- belonging to projective Hilbert space In this section, we willSU(2) see how the instabilitylike associated Dirac with strings constraints andthe magnetic center monopoles. of mass Furthermore, HilbertThe the space states first will at Chern be the seen class FS to are characterize given the by topological objects at FS. 5 Structure of the Fermi surface pseudospin Hilbert space JHEP04(2021)148 , ) ν − dx D (5.6) (5.7) (5.8) (5.5) for a ] (see Dirac iφ ∧ e ) 2 , µ i / 102 − 0 ]) ψ dx ν π Φ iφ 2 µν D − , | e F ψ + µ ) is then given by = ) arising from the D = (0 b D i h γ − )( ( i E , φ 2 2 Dψ / Im = π/ + (b) | ∧ | = = E = (1 arises from a monopole of θ ). Parametrizing the Bloch . The Dirac string [ b , θ Dψ → µν . π D h γ winds around the Dirac-string iγ F γ . π e γ 2 = 1 2 = ] = 0 ˆ U φ r = F ( , punctures the XZ plane from the · Ω = R 0 , † 100 d l ( Ω = 2 , π ˆ b d γ Φ T 2 2 π iγ π 2 e = 2 † c/s dφ = , the WZNW topological term. (Right) Topo- ˆ , acts as a half-flux quantum 2 O γ/ = 2 γ/ S F ˆ Z F T † 2 d traced by , a closed path is traced such that the eigen- ˆ T | = 2 2 c/s † 2 1 i Λ – 17 – ) = ˆ ˆ S φ O T Φ = 2 N 2 2 dφ ˆ dA, dA 2 D ˆ θ<π/ T ≡ Z π 1 (a) θdφ = I ˆ 2 b U/ T π 2 i | (U(1)) = γ is the first Chern class of the gapless FS. This leads iγ h 2 2 ± 1 e π = is a gauge-invariant derivative on the projective Hilbert -term for a monopole with a Dirac string. γ ds . = = ], we define the generators 1 ), and is revealed by using Gauss’ law = Θ i + sin b ∈ b R F γ γ † 2 bS 2 → gives the non-commutativity between the unitary operators 101 = exp dψ ˆ γ T | /γ 2 dθ 1 † ) 1 b , the − ˆ ( ψ T ˆ S 1 T γ 2 ih 2 F r = ], where ˆ θ, φ 2 T ψ ( 1 S traces a great circle on the Bloch sphere described by Fubini-Study and ˆ of radius bN T 102 γ 1 i [ = 2 θ i − | -term, . The topological phase accrued by the closed circuit ( i S ˆ D φ ψ Θ CP | ∂ 2) ) γ/π Dψ ] | = /i π/ i ∈ = ( i ) 99 i , Dψ h . (Left) Topological object in subspace R S = (1 ( 98 (right)) carries an effective flux 2 Dψ | ≡ h ψ φ | 5 2 D = exp ds 1 ˆ The associated string for the Möbius stripsphere projective made Hilbert by space the encircling subspace the great circle of the Bloch This shows thatcharge the nesting instabilityto associated the equivalence: with figure North/South pole ( where, following Schwinger [ and sphere by the angles T space. The greatassociated with circle the monopole ofenergy charge degeneracy point.can The integrated be two-form, writtenhemisphere in of terms the of Bloch sphere the (with Berry solid curvature angle [ state metric [ where Figure 5 logical object in subspace parameter space JHEP04(2021)148 . : b ) ) 2 2 ] is is 2 2 = γ F → )). F || F − F b 2 b and H J low- b γ (5.9) F A (5.10) (5.11) SU(2) − lies at = 1 5.10 F ↑↓−↓↑ F 2 + F a ⊥ H S A possesses =0 ,E 2 J r SU(2) | 1 ] ↓↑ ] (eq. ( = vectors form F = ( ⊗ + , the Hamil- σ for which we i} ⊂ · r b ⊥ a S ↑↓ 103 ˆ n 1 + F ↓ J r ) and the sym- ¯ a F . Now, A ,E ± , ) ) and the Hamil- ), the four energy + b = 2 and representation. − F ↓ ⊥ ⊥ I | ↓ 2 a ¯ = 0 = [ SU(2) 5.4 Hamiltonian can be , 2 A )) ↓ F . The 1 2 i ,J / ,J || 2 || σ b ) anisotropic C || F || F J J ↑ F ,E , + F = J J b a H . A compact form for ¯ ) H ( = 1 ↑ 5.2 A 2 2 a F S ↑ F S ρ rearrange themselves. Thus, h.c. E [ = , or 2) + ) 1 / ≡ {| ↑ 1 ) plane, n . The S ))(1+sgn( ⊥ b F − F b φ min ⊥ ) , the low-energy subspace is acts in sub-basis 2 r | . The monopole charge J ∂ 0 A F f ,J = x Z is given in eq. ( ± symmetry of × || + F a 4 D . Further, precisely at the crossing J 1 sgn( ∈ − | b / n SU(2) r < A ⊥ F 2 θ || ,E ( − + ) γ J i , where ∂ a 1 H J ⊗ Λ , the low-energy subspace is ( 3 )( ⊥ ± 0 · + − SU(2) a 4(1 N f S and J = Eψ 4 2 z z F 0 = > γ/ ∈ xn A D = SU(2) + ] 2 associated with a Dirac string is given by || – 18 – || < i 4 )+ d ˆ n (i.e., once again a Bloch sphere with a monopole J J SU(2) . The density matrix ) ψ , where || || z F b · + sgn( 2 1 ⊥ F Z J − J Wess-Zumino-Novikov-Witten term [ ↑↓±↓↑ 1 A F f or . The family of unitary operations that keep = 0 | S H E S term ,J (SU(2) H = ( D 0 z F a , the Hamiltonian vanishes ( CP = 3 || r b , iθ ⊥ 1 π − A b ↓ J ⊕ = 2 F 4) a J γ . For ))sgn( as Λ 1 / Θ ∈ || 2 r . By varying the coupling values ( H r > ± F 0 + 1 || N F J 2 2 ih↓ 2 J H 6= 0 b C F = ↓ π/ r C a = = ) = 0 = ) = exp [ || 2 is represented by subspace b = such that ˆ F n ↓ J F | ↓ a 2 wraps the FS metric singularity of the projective Hilbert space H ↓ θ, , θ 2(1+sgn( H F θdφ ( ) + I 3 E } 2 2 θ | 2 i γ/ S ,F S ψ = b )( || { (left), and arises from an equivalence of the projective Hilbert space homeomorphic to ↑ cos b =0 . Along two special lines on the ( ⊥ ↑ J a b + sin F 2 5 a J − + ↑ 2 S δk a ih↑ = ). The topological term associated with the homotopy group of this (a) is a skeletal phase diagram that depicts the topological objects residing . This E 2 b has the enhanced symmetry U (1 dθ ( 2 S ↑ ( Z 6 π 2 / 2 a 2 F F r ⊗ = sgn( 2 H H | ↑ b = 1 f S + Ω = 2 a S Figure We have seen earlier that By making the two-spin interaction of the Hamiltonian eq. ( = = , π 2 2 Ω 2 F in the projectivecouplings Hilbert characterized space by ofassociated the with Fermi a surface monopoleOn degrees with the strength of other given freedom by hand, Chern for for number anisotropic either is determined by whethercan the be low written energy down subspace in is terms in of ( where have computed the topological γ eigenvalues the existence of topological objectsenergy subspace like ( magnetic monopoles or Dirac strings in the lowest where the Chern numbers shown in figure to a two-level system formed by pairing of two pseudospins of a unit sphere of charge projective Hilbert space is the tonian I invariant is and ds the origin of the Bloch sphere, reaffirming the U(1) tonian point of themetry two lines, group isblock emergently decomposed restored as to we extend our argumentsers to the a symmetry two-dimensional coupling associated space: with its projective Hilbert space from

JHEP04(2021)148

+ – 19 –

+ + (b) (a) . (a) corresponds to topological objects lying between the separatrices of the coupling phase diagram obtained by studying the effective problem at the Fermi surface. (b) ) ⊥ ,J || J ( corresponds to theorigin). topological Together, objects these living twotext diagrams on for reveal detailed the the discussions. separatrices, shape of and the at BKT the RG phase critical diagram. point See (the main Figure 6 JHEP04(2021)148 . . r ∈ i} 4 b π and γ = 1 ). 2 term given block b (5.12) (5.13) {| S γ F γ/ 1 ) Θ 1 = F = ⊕ 0 associated = 2 2 of section π S F 2 π,J> (with = ∓ ∪ 2 = 3 ,C (SU(2) 1 = 2 = 1 γ 1 3 b F ⊗ γ π C . As depicted in F C acting on the Fermi 2 2 B F (with Chern number ∈ γ . i 2 2 = F F b )). Two special half-lines and where C γ . In the following section, = 0 )) and b , 1 5.8 z F F was observed to be a monopole 5.12 , i} and ,A b | seen from a ) C -term b γ lead to the condensation of four- SU(2) ) ) b | γ (eq. ( b 2 Θ , the subspace , 2 γ + F i 2 × ih b (eq. ( a − C b γ γ a 0 rotations of spin representation ( F b γ | i = 0 ↓ e J 4 and ⊥ 3 F a π,J< SU(2) (SU(2) J 1 ± SU(2) 3 | ↓ − SU(2) = C π , b π 2(1 + = i γ , and reflects the qualitative accuracy of the / ± b ∈ (a) and (b). B F b || – 20 – = γ ↑ b J 6 1 (b), the topological objects describing this sub- γ = 1 F I F a 6 F C 4 J , the first Chern class A projective Hilbert space associated with the putative | {| ↑ 5 . In this section, we will show how FS topology shapes = 0 lead to the low-energy subspace ) and a Dirac string associated with F = = F Λ ) 2 i 0 N H b 2 F γ = 0 ⊗ | of the gapless FS became the Chern number belonging to the r π,J< 0 . In this way, RG flows will be dictated by the discontinuous ± Λ -term coefficient is given by )) = N b (SU(2) 8 SU(2) ⊥ Casimir invariant under Θ γ 3 F J π 3 B where × 0 ∈ is an outcome of an emergent 3 sgn( ). The co-existence of two topological objects in the low-energy subspace , the low-energy subspaces , 0 ), possessing a Dirac string given by the . Further, in section ) is a r J < < Z = 0 = 1 π , we showed that the constraints γ/π || = ± ≡ 2 4 (U(1)) , r sgn( || = S ,J 1 b 0 , J γ ) π || I ± = 0 ∈ J = 2 J > = r 2 (SU(2)) /γ F ⊥ 3 b (URG) method by decoupling electronicthe states emergence starting of from IR fixed the points UV.Furthermore, governed In we by this the show topological way, we that constraints show URG the is essential features dictated ofskeletal by the phase a diagrams RG topological shown phase in term diagram figures obtained via homotopy group of the SU(2)π condensate projective Hilbert space,of i.e., charge ( surface projective Hilbert space the instabilities via aelectronic renormalization model group for (RG) strong formalism. correlations, For and this, perform we the start with unitary an renormalization group In section fermion vertices into BCS and Mott instabilitieswith respectively. the This Fock way, space the first Chern class shown above, such that(sgn( the relevance or irrelevance ofchanges in a boundary flow conditions is accounted decided for by by the the quantities topological6 object at the FS URG ( for Fermi surface instabilities: a topological viewpoint switch. Precisely atpossesses the both critical the point earlier, Chern displaying numbers the enhancedwe symmetry will of treatprocedure. the instabilities We of will the see Fermi there surface that via the a RG renormalization flows group are (RG) characterized by the quantity decomposition of the Hamiltonian where For the skeletal phasespace diagram is of a figure C monopole associated withγ projective Hilbertfor space J This special subspace contains states belonging to both a Dirac string whose JHEP04(2021)148 , − N ] is 1) (6.7) (6.4) (6.5) (6.6) (6.1) (6.2) (6.3) = 1 − 40 j – ( } ) H 36 j [ † represents ( . . The . ) = j . Note that is the bare ) j 1 2 ) , η ) ( is given by j ) j j , ( c j ( = 0 U ) )) ≥ ) ( s, an effective − H j j ) i H  ( η H ( j ) j ) i { ( 6.5 j j H ∆ n ( H = ω c ( ) is iteratively block- ∆ Ψ j j ) | for = ( , leading to the flow ) ) N j H T r j j ( ( H ( j † . τ ( c j H H i U j 1) , ∆ n , | T r − )ˆ ) ] = 0 † j (using eq. ( j j j i ... c ( ( , ˆ ˆ ) n n j . ) j Φ Ψ , 1) D ) ( n | j h ) D ] = 0 , j ( )ˆ − j X ( i j j ( H ) = H = N ˆ n ˆ j n H ( ( H ( 1 , i ) − i [ j ) j U ) D ( j j Ψ − 1) j ( | ( , ˆ ω − T r H ) . The unitary operation 1) j ) H ( Ψ ( i 1 . These constitute the natural quantum [ i | N − j ) ) − ( ) j ], the Hamiltonian j Ψ X , Ψ ) ( ) ( | N | U ) j j i ( T r D ( ( j ( 40 ω † H ( ˆ n Ψ – ˆ ω E H | − U , + ) ) 36 accounts for the residual quantum fluctuations j i j = − = , in increasing order of bare one-particle energy – 21 – = ( ) } ( ] 1) ) ) j ) i † j ˆ j i ( ω j − j H ) ( ( † j ( ) j , D ( , η Ψ ( ,N Ψ j ˆ ω ω j | | ( j ( c ) H Ψ [1 c | , η j U ) | , the Hamiltonian renormalization ) ) ( ) ) ) ) ) j j j ∈ ) = ( j U j ( ( = j ( ( † ) ( Φ j j H H η η h = H 1) ( † j † j ˆ − c i ω and eigenvalues c − = j ( represents the anti-commutator and ( ( 1) i ) i j ) ) − H ( ) in the above expression satisfy the algebra T r j j T r j η ( ( ( {  BA ) ) j j Ψ Ψ ω j j | | † τ τ ( ( ) + . The operator η In the URG scheme [ ) ) Φ( H (1 + j | j = = D ( , ( . Electronic states in the UV are disentangled first, eventually scaling 2 ) ) 1 . In terms of of n j -th electronic state that is disentangled and H j AB N ) √ ( ( 2ˆ j  j i η D ( ) − H = j − represents the diagonal and the off-diagonal components of = ) ( ≤ H j ∆ ) ) } ( Ψ j j | X − ( ( ω ... (ˆ ) U | H is the j ) ( ] = 1 A, B j ≤ ) ) is essentially a rewriting of the Hamiltonian RG flow equation: ( ω j j { † ( 2 Φ  and = ˆ h are eigenstates 6.5 , η ) ) ) ) ≤ j j j j D ( ( ( ( 1 η ˆ energyscales arising fromHamiltonian the RG off-diagonal flow blocks.many-body is energy obtained, For spectrum. describingobtained each the from the of renormalization The vanishing the of of condition the a for matrix element sub-part reaching of a the stable RG fixed point is where the disentangled degree ofω freedom. Associated with the quantum fluctuation operator H eq. ( H [ due to renormalized off-diagonal blocks, and is defined as where the initial (bare)given eigenstate by is The operators This guarantees the preservation of the many-body eigenspectrum numbers, as theyelectronic commute states with are labelled the as Hamiltonian, towards the IR. Concomitantly, thiseigenstates involves the entanglement renormalization within the diagonalized by a successionequation of unitary maps where Hamiltonian. The occupancies of the disentangled electronic states become good quantum URG algorithm. JHEP04(2021)148 ) ) j j )) † ( ( We U at a H ) 6.4 (6.9) (6.8) j (6.10) ) ∆ ( ↓ H , . , ) j ↑ j . The RG ) ( (eq. ( ( Λ p 0 ( U ) 4 j < ˆ s,σ ( = j, V Λ U 1) , 2 − − ) ), we find that the  j ) is number diagonal: η )) ) ( > . . . > 0 j p ) − κ for details) H 6.7 ( j 1 p,η  p, ( ) ( ( − j 1 ˆ ω 1 j ( A + κ , where κ , Λ c | c j K ) ) ) κ ( ˆ s,σ j  ( > η, j, ( p,η, − Φ j 1 2 ( U ). Following this identification, 0 1 p, Λ ih 1 ( κ ) ) and eq. ( − 0 1 . It is important to note that in A c ± j κ 1 2 ) ( ). The redefinition of momentum = ω c ) 6.6 Φ V ,σ p,η | = ( 1 , 0 p,η 0 j ± ) † κ ( ) = 0 κ 0 j c = p † κ > . . . > τ ) κ ˆ s . In obtaining the above equations, we ( c  ) ) ) j Q p,η N = σ ( ( + p,η Λ † κ j − ( j V = κ )). As discussed earlier, an important feature , two electronic states with spins † κ κ τ j ˆ s, ) ∆  V c j is the helicity. We have included the spin- ( 6.5 ( – 22 – − , unveils a natural labelling scheme for states in ) and forward scattering processes for the vari- + Kc 1 2 . The rotated Hamiltonian U  η ) i p, 4 K p ( ) correspond to stable fixed points in theory space. is ) (see also appendix ) , σ 4 + j ,p,η ˆ s ( 0 ω < j j Fermi points are simultaneously disentangled. The net j X 6.9 6.6 V ) Λ κ,κ p k R | ( − ) ’s (se eq. ( + j ) = (Λ ( ω 0 j κ in section κ and ˆ V n 0 j  s ) κ pair momentum vertices (see appendix κ L p  + ( . Finally, by comparing eq. ( ) ] for simple Hamiltonians. j κ j + Λˆ κ ( X 6= 0 and  ˆ s ( p K 104 momentum pairs (with coupling F ) ] = 0 = 1 2 ) k p j 1 D ( − ˆ s, σ = and H , ω ,H j s | Λˆ ) j k = 0 ( from both the ) = Φ p p j = (Λ ) is that the denominator in the RG equations originates from the denominator in is the pair momentum and ( ih Λ ) at only the one-particle level. Also, we have not accounted for the feedback of the ) j j j ( p κ 6.9 ( ω K Φ | We investigate the RG equation in the regime Next, from the Hamiltonian RG flow equation, we extract the vertex RG flow equations ) ∆ j ( ω A similar form of thewas obtained RG in equation ref. (i.e., [ with a pole structure related to RG fixed points) scales renormalized vertices into the of eq. ( the Hamiltonian flow equation eq. ( we note that the poles of RG eq. ( where have chosen the intermediate configuration arriving at the above RG equations, we have accounted for the quantum fluctuation energy disentangles the electronic state thus obtained is off-diagonal with respect to electronic statesfor at all distances the terms of normal distancestransformations from then Fermi points disentangle electronicscaling states gradually farthest towards from it.distance the At Fermi RG points,unitary step while transformation at the step where backscattering process (with coupling ous opposite spin, wavevectors the BCS instability by starting from the model [ poles of the Hamiltonianvanish. RG flow equation are fixedURG points study at of which the elementsadapt BCS of the and URG procedure Mott to instabilities a of 1D 1D model of correlated strongly electrons. correlated electrons. We first address This implies that the projected subspace generated by JHEP04(2021)148 . , ) = = − − )), V 1 0 and 0 1 ω | κ i 6.4 K (6.13) (6.14) (6.11) (6.12) κ → , | sgn( ) − /ω (eq. ( ˆ s, σ Λ ) − j F ) = ( , 1 v U ~ V . , | ) ), the signatures of  ˆ . Further, in those = (Λ s ) sgn( ˆ s 0 − (0) 1 ) p, p, j ( 6.10 κ ( ( < 1 1 . and κ , κ , ) V c ) c 4 4 p ) ) 4 ∆ , the ratio / ( / | 1 ˆ s, ) / ˆ K s, 1 0 1 2 1 j 1 V p, 2 − V ( 1 V ( 1 < V K 0 1 p, → | ( K − κ − K sgn( ) 0 1 − c , and defined the couplings 1 κ p ) 1 Λ V,V − 0 1 ˆ ( s c Λ ) ) Λ p, j ˆ s = ( are given by ( 0 . The pair of electronic states p, | ) ) = † κ 0 ( V i 0 j Λ 1 log c σ Λ 0 1 † κ ) 1 Λ)) + Λ ∆ ˆ d s Λ)) + c κ K | − is the system dimension), the relation is | ) ) along with forward scattering processes F p, momentum pairs constitute the dominant ˆ 0 s dV ω, ( , log v ˆ s, ω, momentum vertices) RG flow equations in | ( p, L † κ ( ~ K ( d . We have additionally reasoned that c . This ensures that both RG equations are . In the regime of eq. ( sgn( 0 p, † κ ) 1 and (0) c − , = 0 )) exists for ωG 0 p ) V ωG Λ) = 0 j i 4 – 23 – p ω ( + , as 1 Λ+ − F K | → / 6.9 + − 1 p v  κ 1 K − V | V j 1 (where ~ > (1 V 1 (1 . By constructing the unitary maps 1 ∆ −

) 1 | ,p,η − ) ) 0 − K K V (2Λ  σ X and ω ) = ( 1 < /L ( 2 p F /ω κ,κ − ˆ s | = = / v K Λ ) (1 = ~ ˆ s, + + p, by the differential p F 0 0 O ( ( j Λ Λ 0 0 κ v − (Λ) Λ Λ ) j 0 1 Λ 1 ≈ j Λ ~ κ ˆ Λ n 1 Λ Λ ( ∝ V p, κ  ) dV  dK − log log 0 j K . In obtaining the continuum RG equations, we have replaced = V dK F log κ d d and Λ  v ∆ κ 1 d Λ + X F ) ~ V ∆ log v + 1 − ~ ; as − = j | ⇒ | − ˆ s, σ κ 2 ω ω = , V and  = (

) | Λ) = (1 ( 0 H 1 V Λ) κ > ω, ( F | ) v Λ) = ) = (Λ p ~ and upon scaling towards the Fermi surface ˆ s are related to those of ωG and ω, − 0 1 + ) p, ( ( V ω j G ( κ / ˆ s, σ . The resulting electronic states Similarly, in order to study the Mott instability, we study the Hamiltonian that includes ω < scatter onto the opposite side of the Fermi surface, such that net momentum transfer (2Λ π and − ) is given by i 2 , F (Λ) 0 1 F v κ where | is (Λ (with coupling Umklapp scattering processes (with coupling K K For such that eventually dominated by the second term, i.e., where the discrete difference naturally satisfied for largethat system of sizes. the With dominantthe these 2-particle continuum arguments, vertex limit we ( restrict our study to Here, the net kinetic energyE for the pair of~ electronic states near the Fermi surface (about points of the abovecases, RG the equations two-particle (eq. vertices scattering ( RG flows as it corresponds to the vicinity of Fermi energy. Note that the only set of non-trivial fixed JHEP04(2021)148 ) ). + 2 0 (a) j √ κ 6 / ].  (6.18) (6.15) (6.16) (6.17) → ( )) and ⊥ . How- 1 2 , where J 2 || || 0 . 108 J and BCS J ) – = 6.17 p 0 ( − ) . 4 p > ⊥ j 106 2 ( ⊥ | 0 J F J v V ~ (0) = 2 ) − j and + ( r )) ) 0 p 0 j 2 ω ( κ V / )  j ]] ∆ ( | 0 J + , ]. We also note that the − j K < 0 κ ( 1 2 , | 4  V ) = ) 37 ( 0 1 || , p 4 || 1 2 || J ( − J K ) J 36 ( j 1 − ⊥ ( − 0 J 1 ω = V 0 ∆ = | Λ Λ 0 ) = 1 0 ) as , p Λ Λ | ( ⊥ dV ) log 6.6 (a) and (b) were obtained purely from the j ln ( dJ ) are equivalent to the non-perturbative RG d (0) 0 6 ) d j , V ( 0 , )) can be written in a compact form as 4 ∆ 6.14 The continuum RG equations for the Mott 0 / || – 24 – 1 K 4 0 for the Mott and BCS cases respectively. These 1 J , term in the denominator of both RG equations V 2 ⊥ ) ∆ V . Remarkably, the essential structure of this RG 1 0 6.12 1 | 1 p J − . The RG equations for the Mott (eq. ( ( K 7 − ) − K 4 1 j < ) ( 1 2 0 /ω | ) or ) = V S p p = 0 || 1 ( ) and eq. ( ( (Λ) 0 ) J ) 0 − Λ j 0 K Λ j || ] (upon reparameterization ( Λ ) ( ]. This diagrammatic contributions in URG have also been 0 Λ V 0 6.8 0 1 j 0 − = ln κ dJ V 67 K  105 ) = d (1 dK log ⊥ p ∆ 0 ( + 1 J d ) V j j ( κ 0 ,  ⇒ | ( and K 1 2 /ω p 1 F V − dominate the RG flows as seen below v (Λ) 0 )) instabilities include all higher order contributions of the ZS ω π ~ ], and are precisely identical to them at weak coupling (i.e., for leads to non-trivial fixed points. In this case, the pairs with net momentum or , and investigate the above RG equations in the regime K 0 + 0 = p 1 ]. The critical and intermediate coupling stable fixed point features of the RG 6.12 ˆ V F s ) = = 110 , > , v 40 p )) and BCS instabilities (eq. ( Λ 0 ( 1 ) – ~ = ) − p j K k ( || ( 109 36 ω < ω 0 ) 6.17 J j ≈ − + ( K ˆ s ) , ∆ 0 j Λ κ pling [ phase diagram are depictedphase in diagram is figure capturedand by (b). the We skeletal recall phase thatconsideration diagrams the of presented diagrams the earlier in topological in figure features figure of the Fermi surface Hamiltonian. equations have thetions same [ form asThey the also possess Berezinskii-Kosterlitz-Thouless the (BKT) sameever, RG RG the invariant equa- labelling presence each ofrepresents RG trajectory, a the new non-perturbative featureseen obtained to from be the responsible URG for formalism, the and will RG be flows reaching stable fixed points at intermediate cou- with equations obtained in ref.for [ the XXZ spin model belonging to theRG Gaussian model and universality class(eq. Fermi [ ( surface topology. where BCS (eq. ( diagrams respectively. Thisonly is the in leading contrast diagrams weakdemonstrated [ coupling for 1-loop the RG, 2dcontinuum which RG Hubbard equations accounts eq. model for ( by some of us [ In the continuum limit, the RG equations attain the form  V,V k We again make the choice of electronic states in the vicinity of the Fermi surface, we obtain the coupling RG equations from eq. ( JHEP04(2021)148 7→ ) ⊥ (6.19) (6.20) ,J || J )). Indeed, . ( )]) , is associated b ) 5.12 z Λ J A ( − )). We recall that a (eq. ( sgn a γ 0 z Λ A 3.17 = . ,J> b b 2( ) γ γ r − , ) iπ/ | trajectories). The red line drawn B (eq. ( r 4 J | sgn( π | to 2 r / = 0 ) sgn( 0 = | b r γ 2 r − | = exp [ | γ ,J< − J b 2 2 γ ⊥ | − | γ ( / triggers the back-scattering instability, i J 1 helicity-inversion symmetry, 1 2 e B b ⊥ coupling, i.e., ˆ U 2 q γ J ) − Z J || ≡ q 1 J ) 2 ( || ˆ / K J ) to write the two RG equations in a combined ( 4 b – 25 – γ sgn r ) sgn − ) has a γ ⊥ ( i J − b ( e 1 γ 6.20 = sgn | ), this change in = J term | ln Λ d )) with the sign of d Θ ⊥ 6.20 ln Λ dJ d 5.11 ) in terms of the topological properties of the Fermi surface, i.e., the J and the ≡ γ ⊥ ), it is possible to simplify this RG equation for the case of the WZNW lines J ± -term (eq. ( 5.11 Θ = , given by the unitary operation: ) . Nonperturbative BKT RG phase diagram for the LE/Mott instabilities of the TLL. The || ⊥ ,J J − Further, the RG equation eq. ( We can now use the RG invariant , = 0 || J r with a change in theas Fermi surface seen subspace from from RGleading flow to eq. an ( irrelevant coupling turning dangerously relevant. ( where particle-hole/time reversal symmetrychange leads in to Using eq. ( ( first Chern class driven Fermi surface topology changing (Lifshitz) transitions of the Fermi surface. fashion Figure 7 WZNW transitions are depicted by theacross separatrices trajectories (i.e., the shows thephases transition (red from circles). the TLL The fixed dashed points line (blue (including circles) the to origin) the corresponds LE/Mott to a line of interaction JHEP04(2021)148 . - ) ) )). Θ for π 0) and and 2 R , 1 > (6.21) (6.22) (6.23) J 3.10 F c/s || , the C X : [0 ) ,J and 0 6= 0 (as shown in π/γ L cm r ) J (2 P i e , π r < ( [0 between clockwise , there is no Dirac ) of the associated = or ∈ c π 0) WZNW separatrices leads to the coupling 0 < cm fixed point theory. The )) RG flows, we obtain phase factor in the RG 2 c/s 0 = 2 || ). Remarkably, it is also P b = 0 ˆ 1 O 5 $ r r > ,J can now be seen as follows. − → 6.21 2 0 ) holds here as well. . R = Λ ν SU(2) r = 0 r For RG invariant , . r < iγ ⊗ − ( ⊥ − e − r r a L 2 ⊥ 2 ,J ⊥ . This tracks the passage through a − ν 6= 0 = 0 regime indicate stable fixed points at − J ), these conserved currents gave rise J } 2 0 ⊥ (see section 2 (for either ⊥ q q = 0 J J 3.8 1 4 1 4 SU(2) < , r γ = 0 F r q q | 2 − ⊥ − , r 1 / = , and the J 1 ) ) − 1 b b ) ¯ b n ) b γ γ r γ J γ ( = 0 | and − – 26 – − − b i$ γ > 0 ( γ e γ γ i ]. | ( e ¯ n > )) and anisotropic (eq. ( = 19 J [ = | ⊥ | (a): ) = ( ) J r | 6.20 ⊥ 6 ( (irrelevant RG flows) and ]. Instead, for the cases J ⊥ | log Λ $ J h.c. d 0 | log Λ . In the presence of an electric field, this gapless spectrum d ) RG flows (above and below the d b + d leads to constructive interference 110 γ ν . This change shows up as a distinction between irrelevant > . . . > , < for the − b | )) changes from γ 0) 1 = 7 A − 109 b + a 0 respectively, to bind via the spin/charge backscattering term and → < γ n 5.11 A log Λ a Λ ( || ν : 0 J c/s ⊥ | /d ,J θ | symmetric theories ending at a J X ) 0 > J b | ≡ | + d in eq. ( 0 π/γ a ) n , from (2 Λ f i r < is given by γ J − h.c. ) − $ e = {| r SU(2) ( + b = ) and relevant ( $ γ = 0 ( = b 2 c/s in the pseudo-spin composite operator basis γ b The red dots in figure The enhanced symmetry of the critical point ˆ O = 0 γ 2 ( † b 2 c/s F ⊥ γ ˆ This is equivalent to thefor unbinding the of BKT real transition space [ string vortex-anti vortex in pairs, the as subspace isequation, (see well and figure known will lead to destructiveanticlockwise interference senses between in paths the traversed center in of clockwise mass and projective Hilbert space O J intermediate coupling, and lead totriplet (charge/spin) condensates which vortex-antivortex are pseudospin odd singlet and and even under (helicity/chirality) exchange respectively. Changing & anticlockwise pathsThis in causes the vortices center and of anti-vortices mass in projective momentum-space, Hilbert space the RG equation where respectively) in the equation By combining the isotropic (eq. ( Topological structure of theterm RG ( phase diagram. to ( a consequence of thethe independent gapless conservation laws Fermi for surface.to the We Chern chiral recall currents invariants thatwill in eq. again ( display theSimilarly, axial the anomaly relation seen observedconformal earlier field earlier theory the between and non-interacting the the metal central Atiyah-Singer indices (eq. charge ( ( ( For sequence space of enhanced symmetry of theC critical point is associated with two Chern invariants JHEP04(2021)148 ) ) We = 0 6.23 6.12 b (6.24) (6.25) γ ) governs 6.20 . In eq. ( γ , corresponding 7 ]. The dashed line in can now be associated $ 111 is constructed from a pair , ,a . 74 ]. Finally, note that the uni- Λ ) 19 A as )) in the RG eq. ( h.c. ), we have shown the equivalence 7 + 5.11 3.17 − b , where A ). We note that a similar conclusion was ) operations of the center of mass position and the first Chern class ,a + a (eq. ( ˆ corresponding to the critical and gapped T Λ A . This change in b cm ( ) ∗ cm π γ Energy A P r . In the next section, we present the math- ∗ P ⊥ ∗ ( H 2 corresponds to non-critical phases and ∆ J Λ 7 ∆ $ , are collective pseudospins constructed by sum- < π – 27 – + z b Λ b and z b = A P A ∗ z a 0 b A z a γ A = H || A -term with , a ∗ ∗ and 1 || Θ J J A a ) and translation ( A = = ˆ O ∗ ∗ 0 π )). Recall that below eq. ( H H 6.23 respectively. 2 (eq. ( ) γ/ r are the magnitudes of the couplings for the Ising and pseudospin scattering ( ± ∗ ⊥ $ J ). Here, pseudospins . Destructive interference in the scattering processes involving the center of mass Hilbert and ). The irrelevant RG flows then lead to a line of blue dots in figure corresponds to a line of Lifshitz transitions of the Fermi surface labelled by the topo- ∗ || 7 8 6.17 ,J || , , with phases ∗ 1 J terms at the fixed points,and and ( can be reconstructed fromming over their individual definitions pseudospins below eqs. ( construct the effective Hamiltonians fixed points (blue dots and red dots respectively in figure ematical forms of the effective Hamiltonians, as wellEffective as Hamiltonians their obtained eigenstates at andjust the eigenvalues. saw stable that fixed the points appearancethe of of nature the a of RG stable flow. fixedcorresponds points, to i.e., the critical phases. From the coupling RG flows for these two cases, we can ated with the change inwith a the topological vanishing of angle the “momentumreached shift” for ( the 1d Hubbardtary and sine-Gordon RG models procedure in generates ref.and effective [ Hamiltonian gapped at fixed the points gapless (red fixed dots) points in (blue figure dots) to gapless Tomonaga-Luttinger liquid (TLL)figure metallic system [ logical angle of the shift of the centerabove, of we mass have momentum shown that the RG relevance of an Umklapp instability is directly associ- space (left) leads to atrajectories gapless involving TLL the metal twist (right).X ( The interference shows up between two opposing figure Figure 8 JHEP04(2021)148 ] ; ∗ for N | 113 ∗ , (6.27) (6.26) Λ = 2 48 F centered from the v m ] ]. Further, ~ ∗ ) are tabu- R ]. The URG Λ − , ω 38 114 ∗ | 6.14 , Λ 4 , − − 64 , i [ , eq. ( i ∗ . = 2 48 2 N m ∗ | H r J = ), the eigenstates at the = ω −| π b z b and = 4 = = ,A ML ) and ∗ ∗ ∗ || || b , m, A 2 ∗ γ J 1 N N 6.8 J 2 = . In subsequent sections, we will 2 = z a | a m ∗ eq. ( || , Λ 1 ∗ ,A 1 m, A F < m < J | H v and with r 0 = ~ = 0 z a −| z b − ) = = 0 A ,A = ω m )), we obtain the corresponding eigenspectrum LE | r ( ∗ + || 4 – 28 – , E N ∗ and its complement. Earlier works based on real 1 z a , where − of an interacting quantum system is a measure of 6.25 J = R ) = b ∗ R || m, A + 1)] ( J ,A (eq. ( ∗ S = ∗ π N b N ( ) H b ∗ A γ = N ] revealed distinct scaling features of EE for gapped as against , the eigenfunctions are determined completely from for the Mott liquid phase. As the LE fixed point is reached in + a 2 0 a for the ML fixed point. Similarly for the critical phases governed ω A 0 π − | 112 A < | ∗ = = 4 -term ( = 0 = J 0 ∗ Θ + 1) i π J . Magnitude of couplings at critical and non-critical fixed points. m i Ψ m | Ψ ( )), the many-body eigenfunctions are given by | . Along the WZNW lines ( m 1 [ ∗ 6.24 Table 1 J is the number of pseudospins within each emergent window ∗ ) = (eq. ( ] shows that the many-body states generated by the entanglement renormalisation N m ∗ 0 38 ( The values of the fixed point couplings at the Luther-Emery (LE) and Mott liquid (ML) H E represents the nonlocal unitary disentanglementunitary transformations operations, providing as thereby a a entanglement product holographicor of mapping (EHM) tensor two-local [ network representationentanglement of RG along URG. the The holographic scaling URGref. direction of method [ the generates EHM [ Hamiltonian and rest of the system.between It degrees quantifies the of information freedom lostspace in with entanglement region RG regards [ to quantumgapless correlations phases. Using theRG URG flow formalism, towards some various of IR usthe fixed entanglement have points scaling recently of features studied the of entanglement 2D the Hubbard normal model, and distinguishing Mott thereby insulating states [ 7 Holographic entanglement scalingThe towards Entanglement the entropy Fermi (EE) surface many-particle quantum entanglement that is generated upon isolating a region stable fixed point are given by and the corresponding eigenspectrumshow is that the electronic statesof at the the LE Fermi and surface ML are phases. witness to the low energy features of electronic states in theabove low-energy fixed window at point the Hamiltonians, fixed we point. have Note only that accounted in for obtainingphases only the arising the from dominant the RG RGlated flows. flow in of table- Hamiltonians as the LE liquid, and the attractive regime on the other hand, by where about the Fermi points. From JHEP04(2021)148 ) 9 are that, (7.1) 6.24 1 ) − j , ( j U Λ . , A leads to the ) i ) ∗ † ∗ j ( j ) and eq. ( ( U and H Ψ 6.25 +1 , +1) j comprise the entangled ∗ +1) Λ j ∗ ( j † ( U A −−−−−→ (eq. ( 19) U represent psuedospins with , ←−−−−−| ) ∗ i (9 j 19 ( H +1) ...... ∗ j ,..., ( 2) Ψ 10 − N ( U −−−−→ +2) is associated with the highest energy. The ∗ j 1) † ( 9) , while − U , from within the bulk of the network. ], i.e., the entanglement entropy generated ←−−−−−| N (0 ( 63 H ... , = +1 – 29 – ) ) η 62 N N † ( ( U U −−−→ ←−−− ) i ) , forms a monotonically decreasing sequence such that the N ( N minimal curve ( ... H , Ψ | )) of the effective Hamiltonian 11) , (2 from its complement is bounded from above by the number of 6.27 : : , R 10) IR , UV (1 URG −−−→ ) and eq. ( ). On the other hand, along the reverse RG flow, two pseudospins re-entangle UV 6.26 1 . EHM representation of URG flow towards the LE ground state. The black nodes reverse URG −−−−−−−−→ represent pseudospins with helicity . Together, they comprise the holographic boundary in the UV. Pseudospins are labelled pair is located at the Fermi energy. The yellow block represents the unitary map 1 9 (eq. ( IR − As noted above, upon scaling from UV towards the IR fixed point, the URG generates ) ∗ 19) j = , ( ,..., URG, two pseudospin degreesdisentangled, generating of a freedom Hamiltonian(see of flow table- opposite towards, say, helicities theat each Luther-Emery step, fixed enabling point thedisplays reconstruction the of EHM the eigenstates construction at for high the energy scales. entanglement RG Figure flow from the UV towards the We now discuss some important features of this scheme. Recall that at each step of the effective Hamiltonians at each step.Ψ Starting with therespectively), ground and state performing at reverse thereconstruction URG IR of steps fixed the point, using states the in unitary the maps UV energy scale: respects the Ryu-Takyanagi EEupon bound isolating [ region links between them. Inholographic this EE bound section, possess we distinct will features in demonstrate the that TLL the phase and RG Luther flow Emery phases. of EE and the energy of the(9 pairs at every step, disentangles thebulk highest of energy the pair EHM). of Theground pseudospins dotted state (represented circle emergent by in around red the theis nodes IR. pair being in The of isolated grey the black by box nodes the represents blue the dotted region (nodes 0 to 9) in the UV that Figure 9 0 η in descending order of energy, such that the pair JHEP04(2021)148 ). )). η (7.2) ) can i 6.26 2 ). Note , it has 1 decouple Ψ | † 10 − 3 . Here, the of opposite 17 i , , generating U i 7 i s | U = ... 19 | , i 18 , i 3 , 7 =11 i 18 i Ψ Ψ U | | † 1 18 19 | i ⊗ , with the disentangled U i into two-qubit disentan- 7 , † 2 s i i ]. For the present EHM and with helicity | U 2 π U U i † i 3 17 38 Ψ ) are located farthest from 2 | =0 | 8 i U η 19 Ψ | , | i of the LE phase (eq. ( )), we have only accounted for i ⊗ , decouples the pseudospins/qubits Ψ π i | 2 (orange block in figure 17 π and has depth (d=number of orange † 6.8 , 1 and Ψ | 7 U carry a decreasing sequence of net i i Ψ U | j U † | 2 19 = | 18 U ... i , , ) . The complete ground state at the IR i 7 , ∗ i = +1 i ), and j )) map the ground state from IR to UV U i 1 2 18 ( | Ψ i η | 19 ]. Such a decomposition will allow us to i , , Ψ 12 † | 1 7.1 = i | | is located at the Fermi points. The dotted 7 9 48 , U , 1 | U i i 8 , − , 2 i 18 (connecting states | 19 8 – 30 – | 17 | from the pseudospins ,A 3 , eq. ( to , U 1 2 ), ( i U 9 9 , i 9 with opposite helicities ( | 11 = 17 i 11 | U = +1 with helicity 10 , +1 | i i and η = , 1 9 | i ,A | 3 8 0 | . An orange block operates on the two pseudospins demarcated by U to = 0 , we show some aspects of the equivalent quantum circuit in 0 = 5 z 9 d configurations of the individual pseudospins. The reverse unitary ,A being the fixed point): = 1 | ⇓i with helicity = 1 i A | 7 ∗ | represents the entangled groundstate and j (yellows blocks in figure = 9 † i π U (red nodes) from the pseudospins disentangles the pair of pseudospins | ⇑i . Decomposition of nonlocal unitary disentanglement map Ψ i | . The second last yellow block 7 i,j | U 10 An essential feature of the EHM network is that it can be represented entirely as and i are the i 17 where that in the parent Hamiltonianthe for backscattering the diagrams that RG couple analysis pairs (eq.In of ( the electronic second states last with opposite RGthe helicity step, ( pseudospin the collection of unitary disentanglers network shown in figure figure be decomposed as follows (with RG step thereby the EHM tensor network. a product of two-qubitinterpret the disentangling URG gates as a [ quantum circuit renormalization group [ Comprised of the pseudospins the form fixed point can be obtainedpseudospin states by labelled performing a tensors product of operations such that pair of pseudospins the Fermi points and naturallysuccession associated of with pseudospin the pairs highest ( electronic electronic energy, pair-energy. such Thus, that a oval the in pair figure | blocks acting sequentially) the lines crossing its edges. LE ground state in the IR. Pseudospin states are labelled in descending order of energy, Figure 10 glers represented as orange blocks (within the right figure). JHEP04(2021)148 , . ) 9 64 , i, j )) to from (7.4) (7.3) ( of the j and so 48 d 7.2 18) and , i (7 , , that disentangle ) ]. The grey block | momentum-space 8 , the circuit depth generated at each 17) , ) possesses an upper j , j ( is the reduced den- 17 63 . The depth ) are constrained by , 10 (7 )). This allows us to ... R U Ψ . On the other hand, ) ] l 9 , 62 , i ( ih 7.2 i by doing reverse URG: at UV, while the dotted ) 1 j 3 17 j 6.27 = +1 ρ − i ( U R ) Ψ ⇑ | η ∗ Ψ , and obtain therefrom the ) that are entangled [ | j at the Fermi point increases ( ) ( 17 +1 . This shows that the entan- , l 0) 7 9 Ψ R in eq. ( (eq. ( )) determined by the renormal- − . | ( U i 10 n )) 1 2 a,b ( 6.4 18 i , ,..., ( with the maximum one-pseudospin 7 ,..., U j i − | ⇓ 8 9 R = log 2 ) = U 1 , S ( , − R 1 T r S (9 (eq. ( i ( , as shown in figure ) involve the vanishing of mutual infor- ⇓ z − R with helicity 3 j 3 j ∈ n i i Ψ +1 U U max ,A | 10 9 ) = | l 1 2 ,... ] for the pair of qubits ( 17 × | ⇑ , j [ 7) 7 show the entanglement entropy RG flow for ) 1 = , U . For and R − 8 – 31 – , ρ 115 represents the region j ( , , 2 i 11 , such that j z +1 )) U is generated from j 8 9 operations carried out sequentially (in eq. ( l | √ (9 n 48 A ( [ i | , j (see figure ) ≤ = ) j a,b ρ ( 3 8) = block sizes, suggesting perfect disentanglement of the ) i U , Ψ from U π i i, j R | ( 0 ( (9 1 Ψ , ) log j in figure all | S l Ψ − | S ( . We also provide a reverse URG formulation for the TLL 9 j − i (9) ) at RG step = ρ ) ∗ ( to j j ) η ( ( the EE for the lowest block size i 0 ( T r S Ψ j | − 11 is the joint entropy associated with the isolating pair of qubits S ∗ . Many-body states j ) ) ) + j i ( ( ) = is the entanglement entropy associated with isolating qubit i, j H l is the size of the momentum-space block and S ( ( ...U ) j of helicity j 1 S . This is followed by the next set of unitary maps ( S i − 1 10] † ) = j , j ,S 17 − U | ) : † [1 j i ( i = U ( ∈ S η ) I l given above, generating the URG flow for the block entanglement entropy of the l = ( ]. . The form of the individual qubit disentanglers j i Finally, we turn to display the holographic feature of the EHM: when computed from Upon implementing the reverse URG, we isolate at each RG step ) S 38 10 ]. To compute the upper bound, we multiply j ( = 5 Ψ entanglement entropy This leads to the Ryu-Takyanagi formularanging for from entanglement entropy pseudospin [ pseudospins. In this way, we observe distinct scaling features forthe the bulk TLL of and the LE EHMbound phases. tensor related network, to the the EE number114 associated of degrees with of region freedom in along the RG flow,glement terminating between at Fermi a points finalthe at value maximally the of entangled IR state fixedfor pointis the enchanced, TLL leading phase, toto the formation momentum-space zero of block along EE the is RG observed flow to decrease for monotonically different momentum-space block sizesforward in RG the flow, LE the and successivetion TLL disentanglement of phases of block pseudospins respectively. EE: leads the Alongsharing to disentanglement the between of a IR UV gradual and degrees reduc- UVthe of degrees left freedom of reduces panel freedom. the of entanglement figure It is important to note, however, that in sity matrix. For| this, the state phase starting from theconstruct state a EHM quantum circuit realizationand of the URG for theTLL. TLL The similarly left to figures and right panels of figure blocks of increasing lengths EE where ized Hamiltonian sublayer of the circuit composing mation on [ the system and pseudospin orange block equals the numbercomplete of the disentanglement operation d the analytical form of the complete unitary map helicity JHEP04(2021)148 − ) R from ( j )) with R n 7.4 ) = R ( ) for (a) the LE 1 − 10 j is the entanglement n to , (9) 9 1 S . Left panel represents the 0) ,..., ) and the gapped LE phase (red is the maximum single-pseudospin (9 7 ranging from (1) l max leading to the EE upper bound for the S (1) units, max S – 32 – 9 renormalization for varying sizes of momentum-space (i.e.g, block size ) l ( 0) S reduces. In the present case of figure ) ,..., 8 R , ( j (9 n ,..., 7) , 8 , for both the LE and TLL phases. The blue curve shows the holographic (9 } ). First, we note that the , 9 , we confirm the holographic entanglement entropy relation (eq. ( 7 8) , 12 . Entanglement Entropy (9 . Blue and orange curves show the RG scaling of the holographic entanglement entropy the size of the minimal surface in the bulk of the EHM (varies with the RG step). See ,..., , 1 n , (9) 0 { from the bulk of the EHM. Importantly, the distinct holographic entanglement entropy = . In figure R scaling features imply that itcritical is TLL a phase witness to (blue the dotsdots entanglement in phase the of transition phase figure between diagram the figure LE and TLL phases are distinct: while it arises entirely from the degree of freedom at the blue line is the minimaldeep surface, within or, the EHM. equivalently, the Evidently, the numberbulk minimal of of surface the links shrinks EHM cut as such that to we1 proceed isolate deeper into the R entropy upper bound, while the orange curve shows the entropy computed for the region curves for the LEentropy phase, and generated right by panel isolatingEE for block and the TLL of phase. size discussion The in legend main text. Figure 12 bound and the entropy scaling of the momentum-space block Figure 11 blocks and (b) the TLL phases. See discussion in main text. JHEP04(2021)148 = )), s δ (8.2) (8.3) (8.4) (8.1) 3.17 (i.e., of ) for the ∗ 9 n L (eq. ( π 2 , γ/ ) to the scattering ) for the LE phase, δ 1 0 F H -electron subspace 1 1 2 F has the following matrix F ). The vertex is described 0 ˆ 1 H G to step , F 3.3 ˆ 9 S − ], and its discrete counterpart π , within the window 1 = 2 1 1 ] = F i 117 F )) 0 , 0 , ˆ G F b ˆ πS G ↑ π   2 ( 116 2 t F a ln iδ 0 − ( ] ∂ω e 1 ], this relation holds even in the presence πν , We recall that a topological constraint F s 18 0 iδ = 1 e i ± | ↓ 119 = 2 1 ∂T r   – 33 – ) to the IR (Fermi surface pseudospin F F b are the Luttinger sum for the charge and spin 0 ˆ S ↓ = iπ dω 2 , 1 / F a π , γ F )) ∞ ˆ S | ↑ 1 −∞ F = 2 Z . We recall that the change in center of mass momentum = 1 ˆ S )). The scattering matrix ], relating the scattering phase-shift ( 2) [ γ ( 4) = S F √ ln F 5.4 / ( 118 N ) along the RG trajectory (step 1 πJ/ N ]), this will need further investigation and will be presented (1 9 F and − ( 65 1 T r 2 − / ), in the presence of p-h and TRS symmetries) respectively. As seen = 2 δ = 1 2 = tan 3.17 )), and satisfies the criterion [ π t . Due to Kohn’s theorem [ 2 , δ 5.4 π γ/ 2 is given in (eq. ( ) 4) 1 is the non-interacting Greens’ function given in eq ( 1 F γ/ = F ˆ 0 S )) ensured that the FS backscattering vertex acts on the (eq. ( πJ/ ˆ ν H = G 1 (3 4.3 F 1 cm − H P tan under twist via∆ a full flux-quantumof is electronic related interactions. tototal the Thus, spectral in Luttinger weight the sum associated presence with of the a FS subspace putative instability of the FS, the representation in the basis where the pseudospin singlet/triplet scattering phase shifts are given respectively by matrix ( where where excitations (eq. ( earlier, backscattering leads togiven by the Friedel’s formation sum of rule composite [ objects with spectral weight where by 8 Observing the instability ofDynamical the Fermi spectral surface (eq. ( weight transfer. AdS-CFT conjecture for continuumin field lattice theories [ fieldelsewhere. theories Though [ we haveshown not for shown the the LE analysis phase here, are precisely also similar obtained results for to the those gapped ML phase described earlier. it shifts gradually from UVTLL as (pseudospin the RG proceeds.different: Second, while the shape it of scalesclearly the linearly non-linear upper with for bound the the for TLL logarithmic the phase.of two RG holographically phases While step generated is the size entanglement quite latter spacetime for is expected the reminiscent for of LE a the phase, gapless rapid phase it expansion (i.e., is the Fermi surface (pseudospin JHEP04(2021)148 , , − γ 1 = 2 ] = 4 (8.8) (8.5) (8.6) (8.7) − = , and ) simul- F ∗ F b E 121 2 γ J E ( N I 1 along the )[ F s s 0 δ δ , ˆ G and We have seen )) . 1 1 ) I F 1 ) F in a ring geometry ) = ( S ˆ S 2 ( ˆ . For the E ) ( ] that tracks spectral π ln D 1 π ( . F (ln( . Further, for ˆ (3 122 G γ [ T r 1 and T r 2 − 1 1 to ¯ N )) = 0 − , but with oppositely directed 1 D 1 γ F F ))) + ± ˆ − . N G ( iη | = = ln J − 4) = tan η | ( ], ln Λ s / d δ π d ∗ (0 2 T r 1 2 2 F 121 / πJ J − , ) ˆ G 2 b d (3 ( ln Λ 4 γ F π and detectors 1 9 − d 120 N ln 2 − , the setup has two open identical 1D systems γ ( changes from ( I i = – 34 – e b 1 + 13 T r γ )) + )) 1 and = − ˆ F = tan G and the Friedel’s scattering phase shift ( ) 1 ( ˆ I ∗ s s S ))) F ln δ ln Λ (( ( N dδ iη ∂ω 1 d ln F ( (0 + T r . The injectors and detectors are momentum resolved, such ∂T r 1 0 is then given by F Φ ˆ respectively. The injectors are switched off immediately after the iπ G d dω J / ln Λ ( F d = ln k ∞ ( −∞ + ⊥ = Φ Z J T r λ = = = and 1 || for the WZNW flows as ¯ N π F J γ 2 k − = 0 . The RG equation for the pseudo-spin singlet scattering phase-shift J 1 − In order to track the BCS instability in the two 1D systems, injectors ) 1 F taneously inject an electron each withmomenta a given helicity injection, and the detectorsbackscattering are in switched each on of simultaneously.involve the The two-particle arms, injected interfering and electrons pathways reach between suffer the the two two detectors arms through that trajectories together that enclose the that are tunnel coupled toenclosing injectors a flux that they inject andwith extract electrons well in defined a helicity/chirality.1-particle resonant superpositions manner Further, across at the the the injection two Fermi arms and energy of extraction the events ring. involve attributes that track thenow propose two- a particle “two-path” thought scattering experiment forelectronic induced a system. instabilities. ring-like geometry This of gedanken In the aimsbackscattering interacting this to 1D dynamics measure spirit, of the FS we various electrons momentsa (full of putative counting the instability. statistics spin/charge As (FCS)) shown in in the figure presence of Measuring full countingin statistics earlier for sections Fermi that surface the electronic electrons. degrees of freedom at the FS possess topological The dynamical spectralwhere weight transfer the stops scattering eventuallyparticle phase subspace at shift at the the IRweight Fermi redistribution surface, fixed in we point terms define of a quantity This relation shows that the Friedel’s phasepoint shift changes non-analytically across the critical unitarity dictates that the increasingand dynamical composite spectral degrees weight of transfer freedom between within fermions the LEB be obtained from WZNW line given by a generalized Luttinger’s sum rule [ where the fermionic single-particleH Greens’ function is given by the composite objects, together with that of any remnant fermionic degrees of freedom) is JHEP04(2021)148 = as ) is 1 i (8.9) ) F 5 (8.14) (8.10) (8.11) (8.12) (8.13) ω ( ). The acquires ψ | 1 5 ω F ∂ | , acquiring a . ) ) , ω of the injected π )] . ( ↓ 2  ) i , ψ F t i t h k δ δ [0 ˆ F b φ ˆ ) in the basis n . For instabilities in F b i ↓ 2 ↑ ∈ − cos sin 8.3 F a = r ↑ F a − − θ F , and k s s | ↑ A t || ↓ − δ δ n 1 t iθ (ˆ | t e θ iω s sin cos e γ iγλ =  π + − 2 ω ˆ i φ . )) π are the reflection and transmission t (see discussion in section ω 2 δ F b , t ↓ = exp [ = iδ r −   e , this initial state scatters to λ s s F a γ/π ω = arctan = λ δ ˆ ¯ Z − t t r r t , || ↑ = s   r A | cos( iδ ( · S 1 2 e in superposition between the two arms and are r 2 − – 35 – ω iθ 1 = , = e d I (1 1 t γ F , = I , θ coefficients can be represented as follows 1)] t ˆ S i = iδ t  = − e s t t F b s ↓ δ γ δ ↓ are deactivated). This thought experiment is, therefore, γ + F and k s , F a 1 n r iδ † D e | ↑ λ + ˆ c/s + sin + cos 1 , computed from the berry potential ˆ ¯ ↑ s s F O 2 = δ δ ˆ F s S I k † γ c/s r of the composite pseudospin instability operator (see section n ˆ ¯ = , t T sin (ˆ cos is acquired as follows. The initial state r i i}  ) λ c/s iθ s iγλ ˆ ¯ ω e F b O γ (while ( s ↑ π ψ γ 2 2 | c/s F a ˆ ¯ T D r are the pseudospin singlet/triplet phase shifts defined earlier for the Fermi | ↓ = = exp [ t = = arctan , . This two-particle interference thus provides information related to the correla- i λ c r r , δ λ ˆ ¯ λ c/s θ s Z F b ˆ ¯ δ Z ↓ We define flux-resolved momentum space Wilson loop operators that encode the The two-electron scattering matrix at the Fermi surface eq. ( F a amount of effective flux observed by charge/spin degrees of freedom scattering berry-phase follows, the two arms and in the absence of the AB flux This scattered state then traverses a closed loop over a geometric phase associated with thegeometric monopole phase of charge electrons is an equal amplitude superposition of the two arms where the two-electron wavefunction upon scattering in projective Hilbert space where surface electrons, and coefficients respectively. The variant of this gedanken would involve injection of electrons and{| extraction ↑ of holes. given by tions accrued from the backscatteringtwo processes electrons in are the instead two injectedextracted arms. from at For the Motta instability, two-particle fermionicinterplay Hanbury-Brown-Twiss of interparticle setup correlations whose and Fermi purpose surface is topology. An to Andreev expose scattering the AB flux JHEP04(2021)148 . i + and i F b ↓ 2) (8.15) (8.16) (8.17) (8.18) F a F a λ/ | ↑ ( | ↑ , F c/s 2) ) translation ψ c   | ˆ ¯ T λ/ 1) ( = 1 − F i S ↓ ) . Superposing these k i λ n ( = )). Further, they track (at the brink of LE/Mott created by the scattering F b via Y junctions. The ring + ˆ ↓ 2 ) i F c/s . 2 λ ↑ ] Ψ ( F a 2) k 3.17 | D ), backscattering processes feed two electrons ( F s n and , cm F (ˆ 2 | ↑ λ/ 1 ) and charge ( Ψ ˆ 1 k ( 2 I k X s eq ( | 2) D ) ˆ ¯ ± T , e F F γ c/s . 2 k λ/ | 2 ψ and | λ/ < c/s − X | λ/ 1 ( ˆ ¯ † k c/s Z N/N | I 1 1 ˆ ¯ ( Z i F i F 1 . See main text for discussions.   S S F 2 2) S = 2 λ/ i λ/ † c/s and detectors λ/ c/s = exp[ ˆ ¯ − Z 2) 2 ˆ ¯ = exp ( Z s I 1 c , ˆ ¯ F λ/ – 36 – ˆ ¯ O 1 T S I − , 2) = ( ] 2) = , λ/ F c/s and cm λ/   ( ˆ ψ − ) X | 1 ( 2) ↓ i ) twist operators defined as 1 F , we have used the spin ( k c F λ/ S ˆ n ˆ ¯ ( O S λ c/s 1 ˆ ¯ − 1 F Z ↑ S = exp[ k . This results in a superposition state c n 2 (ˆ ˆ ¯ . The Y junctions of injectors O k λ | or F 1 k | < X | ) (and which led to the first Chern class k ). Along path-1, a flux is accrued by FS electrons via the Wilson loop ) and charge ( | s i ˆ ¯ O   , leading to the modified S-matrix 3.14 13 2 λ/ . Thought experiment for probing the entanglement entropy related to two-particle ˆ ¯ Z = exp s ¯ ˆ T ) to either system i F b Along path-2, the modified S-matrix is and the scattered state istwo given by scattered wavefunctions gives the total wavefunction as paths (see fig operator This leads to a scattered two-particle wavefunction: These Wilson loopshown operators in eq. are ( theboundary momentum-space condition duals changes of atIn the the the Fermi real-space scattering surface variants problem forcan the between be the spin/charge visualized degrees two on Fermi of the points freedom. ring ( in terms of interfering clockwise (1) and a anticlockwise (2) and the spin ( process described by matrices In the above expressionsoperators for given by Figure 13 scattering through transport measurements. Two openinstability) 1D are systems coupled to electrongeometry injectors encloses a flux ↓ JHEP04(2021)148 i is ) λ ( )) λ (8.25) (8.23) (8.24) (8.19) (8.20) (8.21) (8.22) ( F c/s Ψ F | c/s defined as ) ) λ M i ( ( ) 4 λ ( , F c/s Im . Ψ c/s i h ˆ  I ) observables, the h , 2 λ i (0) ) )  i λ , , we can generate all F c/s ( λ t 2) , F ψ c/s δ | ψ λ/  γλ , 2 F a + 2) − F c/s ( λ s/c ˆ  s / I . Similarly, π s A , can be determined from the ˆ ¯ h γ Z 2 2 ↓ δ λ . , containing the FCS for the F c/s | cos ) )) 2) F i ψ 2 − = 1 k ) λ λ | − | γλ . ( ( (0) t λ  † ∓ λ/ | 1 S 2) ( c ) † ( − F F ↑  c/s F c/s c/s ( cos F c/s 4 F F λ/ c/s 1 ψ + 2 ) ln tan =0 ˆ k 2 ln M M ψ I h t F | − λ ( h 2 † − t δ ˆ | ( | i S  | c ) − = , r s | γλ . π − λ Re + ) F + c/s i  γ ( 2 = s λ + 2 i ψ 2 λ F δ 2) ( ) h sin 2 F a + − | λ G + F b − 2 ) and current ( ( r λ/ | A F c/s 1 i Re | )) = 2  t ( m ) cos( |  m )). Alternatively, the quantity 2 λ 2) – 37 – F c/s 2 λ )Ψ ( ,A π F c/s ( ˆ F c/s d γ − λ Q ↓ 2 dλ λ/ ψ ( h F ψ ( c/s F | 8.19 s π † 1 )! k + γ  2 m ), the even cumulants yield the entanglement entropy )) = 2 F c/s F F c/s 2) M † ± = 2 + 2 = 2 + 2 2 m ˆ ( 2 ˆ λ λ c S ) = π Q Ψ ( γ α ln ↑ i λ/ h 2 λ (2  ) e F S ( Re 8.11 s π − F 0 † k † . By varying the counting field λ c/s γ ( ln ) = 2 ( c ) F c/s F c/s = λ M X λ m> − − ψ ( F ( c/s ( F c/s = i ) = M  ψ λ Ψ = = = h F ( | e c/s S F c/s Im ) † F c/s + F a ˆ Q λ M c/s F A Ψ h FS = ( ) = ˆ Q S i ) = λ F c/s ) ( λ λ ( Ψ ( F h c/s F c/s ) = ln F ˆ c/s I M is given by λ ˆ I ( h ) , and leads to the following two-particle interference pattern for spin/charge ] together with eq. ( F i λ ( G 2) † 123 F c/s λ/ Ψ − between the helicity/chirality sectors (see section ( F c/s FS ψ function: cumulants of the spinformula or [ charge distributions atS the FS. Further, using the Klich-Levitov Thus, in termstwo-particle FCS of generating function the can be charge written ( as A cumulant generating function can now be constructed from the moment generating evaluated from the expectation value of the composite current operator where and, as shown earlier, is a measure of the expectation value of the charge operator two-particle back-scattering at the FS The real part of the momenttwo-electron generating interference function, pattern (eq. ( currents carried by the pseudospin composite degrees of freedom (see section We can now define a moment generating function, | JHEP04(2021)148 , ]. π ) in to )). π 1 3 → (9.2) (9.1) 123 , and is the − (8.26) √ h.c. 5.11 γ ( i} : 0 / + ,σ = , π − b c/s +Λ A b (eq. ( = 4 F X γ k + a = 0 π J n ) reappears in A . ) cm at the RG fixed → Λ)ˆ top π P ~ 0 S ? ⊥ is the ) given by − {| J ( + 2 1 ). We will see in the = Z F = 4 − is the gap in the center from k π ∗ WZNW lines from ~ = 2 B J ( tan )) s/c and the high-energy triplet r d , ,σ ( ( − ∗ E , for the (spin/charge) gapped Λ )  (where 1 ∆ π  < π = 0 SU(2) sgn F | 2 = c/s ) is a function of the pseudospin b Λ b (3 Z | πj γ N )). 1 X 2 A B P − dyn π , γ 5.3 2 + S = 2)(1+ i ) to = i a  γ γ/ , k depends purely on the first Chern class A i cm = tan j = cos P → ∞ = ( = b s/c c/s N γ ) and basis , φ E X = ln 2 c/s – 38 –  jumps across the 0 (unitary limit). Interestingly, for cm X π is purely topological. In this way, a measurement 2 φ | ) γ t 2 H + ∆ r δ i ± | ), see also eq. ( π ( γ in the section below. ikj 2 − ln e 2 tan cm M = 0 − ln 2 P s/c 6.25  j , sgn X E − 2 = 0 = = c/s ∆ ) ln N depends on the renormalized coupling = 1 X < π/ φ | cm √ top ) in the final line. The first piece of this entanglement entropy h ) are given by || . This reflects on the additional entanglement content of the = S H J 1 FS s/c π cm = 2 cos( S F s E i 8.11 δ H π ∆ k γ/ γ | 2 as  q . = 0 = = . The cosine potential arises from the backscattering term ) and will be quantified in the next section. This is in clear contrast with ] to = 1 i 0 ∗ ∗ → dyn 1 (in the thermodynamic limit b B ), leading to a value at the IR stable fixed point ( S Λ , π M , such that 0 , 1 A 2 2 FS ∗ S for the metal (described by / S Λ + = 0 is the total mass of the system, and π/ ' 2 λ > − cm =0 a [ b P cm = γ N A M ln P t B Z | = δ The precence of the cosine potential changes the gauge symmetry of the state manifold /d | , together with a jump in the Berry phase − and accounts for dimension of projective Hilbert space 0 cm J | s insulator (described by the gap obtained from 1-loopfixed RG point). (and We where will the quantify coupling diverges at thefrom strong coupling symmetry of thebasis cosine potential). The translation- and gauge-invariant wavefunction window the fixed point Hamiltonian (eq.of mass ( spectrum between thestates low-energy singlet point (see table H where center of mass momentum of the electronic states within the emergent momentum space 9 Topological order andWe its begin observables by clarifyinginstability. how Recall that the for center+1 of mass HilbertThese space changes topology are is associated shaped with a by modification the of the center of mass Hamiltonian from Note that δ of the FCS should enable that of the associated many-particle entanglement entropy [ projective Hilbert space many-body wavefunction arising from thepiece dynamical spectral renormalizes weight incrementally transfer. asd This second the instability hits the Fermi surface ( is purely topological, i.e., γ next section that thisthe topological center of piece mass of Hilbertsinglet the space. and entanglement The entropy triplet second ( piece scattering ( phase shifts, and is dependent on the geometry of the where we have used eq ( JHEP04(2021)148 : r = cm ∗ n H (9.4) (9.5) (9.3) cm ). In L -term, P Θ ih is formed 6.25 ,a = 0 Λ cm A P | = π , P , # )) with Chern coefficient # )). The Hamiltonian 1 4 ) is directly related to the 0 || 1 , J − . 9.8 5.10 ) arises from the destructive 4 ] = 2 0 / || is the ratio of the renormalized 0 || − 1 1 d M J 0 − 1 4 2 for the BKT transition in terms ) " /J Λ 2 / (eq. ( ]) " 1 ∗ / ⊥ 2 ∗ J π [ 2 − cm coh exp are gapped. This gap is the analogue − Λ B ξ P exp leads to purely topological dynamics 2 ∗ ) is equivalent to the gap between the 4 (16 r n 2 MB exp[ / We have seen already that the e 1 t L c E + l /S 0 ! 2 = 0 ∆ 0 || ) = 4 = 4 )) in the IR. This should be contrasted with ∗ 2Λ || J π r = 1 J P

√ || – 39 – / t q ∗ 6.18 1 cm ( n 1-loop e MB for the MI and LE phases. The spin/charge gap at J at the FS, and can be attributed to the fact that the c 4 H l = 4 E / 1 π i 1 . ) between the pseudospin singlet and triplet states ) possess the same couplings. Following Ambegaokar ∗ 0 ∆ P (eq. ( ∗ ∗ ! ⊥ Λ Λ Λ ) = 0 ], J || = 0 t 4 ∗ ⊥ J b 4.15 coupling under RG, eventually ending at an asymptotically 2 is the bandwidth and , J /N

( t A 1 . We will shortly see how this establishes topological order. ∗ ⊥ ⊥ = 4 ) = 4 from the triplet of excited states. That this degeneracy is 4 J J ∗ + = coh vanishes, where the projection operator n a ξ + 1 . Thus, ], where the coupling diverges under RG flow towards strong s/c | MB π = 2 A s/c (Λ | π C E ) as, B E || E 1 . The fixed point theory resides in the low-energy window J ∆ ∆ , )). The gap in the center of mas spectrum is determined from the = s/c = ∆ ) and eq. ( WZNW term (of the form shown in eq. ( and d E 9.1 represents the two degenerate singlet states in center of mass spectrum, i , with the many-body gap ( D 5.9 = 0 ] ∆ cm . The two-fold ground state degeneracy ( i ∗ → ∞ 2 P π | n b = 1 , π ih Λ ∗ 0+1 = b n π F (Λ) Λ γ v A || = 0 ~ = J A + ], we can now define the coherence length , is determined from the table is the bare coupling, is the vortex core size and we have used the relation between the RG invariant a ∗ for cm + ∗ 0 n || || c P cm A l 124 a | J , leads to an increasing J | Λ ∗ P γ | F n = 1 Along-with the many body gap for single particle excitation discussed above, we find v Λ = + ~ cm A b | | − et al. [ of the RG invariant where and the final fixed point coupling where the intermediate-coupling fixed pointstates ( Hamiltonians eq. ( cussion below eq. ( spin/charge gap ( ( this manner, all single particleof excitations within that obtained fromliquid the and 1D one-loop Mott BKT insulator RG [ for the 1Da superconducting substructure associated Luther-Emery with the spectrum of the center of mass Hamiltonian (see dis- where and bare energy window cutoffs.for Within every the pair many body of gap, electronic a states, pseudospin and compose the effective Hamiltonian eq. ( safe stable fixed point the one-loop RG resultcoupling: [ [ governed by a C Chern coefficient as Two spectral gaps andγ topological order. protected by the gap not lifted due tointerference tunneling in the (via manifold the of kinematic quantumprojected states term onto within the basis 0 where JHEP04(2021)148 > > (9.6) (9.7) (9.8) |1 ) and |0 , c/s i ˆ O demarcating = 0 Wilson loop ) around the b 2 symmetry and + s/c Z MB a 2 E E objects encircling Z ∆ ∆ ( 2 γ, A e/ = (b) cm by the (spin/charge) gap P | i , , = i ) operations of the center of mass i = 1 ˆ = 0 T , b ] traversed in clockwise sense) and = 0 γ + spin/charge gap = 0 ] a b cm , b + A i ,P + | , π cos a a | c/s 2 arises from charge = 0 X ,A Ψ [ to be π, A = [0 iγ b γ || Energy e 3 + e 1 = a = 0 c/s )(left). The interference shows up between two Ψ . The zero mode of the square-root of the ver- | c/s X 5 ˆ c/s cm O 9.2 – 40 – ) within the center of mass Hamiltonian. See main γ, A ˆ ) P X ) and translation ( + 2 T | | 2 2 X corresponds to one of the possible conclusions of the ˆ | = O = 2 = c/s 2 π ˆ ˆ i O T Ψ 2 cm | in eq. ( cos(2 = along with the standard many-body gap P Ψ c/s | | π + i γ ˆ 1 s/c O B 2 | | = , E 1 for i i . This leads to destructive interference in a multiply-connected and ∆ Ψ respectively. This leads to a | and (right). The green curve within the left figure corresponds to the π i s/c i = 0 = 0 γ ∗ = 0 E n | b b ± = 0 ) involving path-1 ( L + + ] discussed in section ∆ ) a a b (a) traversed in anti-clockwise sense ) between the two degenerate states π ] is a nonlocal gauge transformation with an associated + 18 i , γ ] 2 a ,A ,A , 19 . The Aharonov-Bohm phase [ , π [0 2 ,A ] = between = 0 = 0 Ψ 2 c/s ≡ ) operators was shown in section F cm iγ ˆ = [0 = 0 ˆ O E 1 T e cm cm ,P S P P , with phases . Constructive interference phenomenon for scattering processes on the center of mass c/s | | c/s = c/s X 1 X X 2 | c/s X given above. We recall that the commutation relation between the twist ( Ψ [ ˆ O = The modified center of mass projective Hilbert space has a emergent s/c i 1 E Ψ where the Dirac stringdegeneracy in with the a gap FSLSM-type projective criterion Hilbert [ space.tex operator Indeed, the finding of topological geometry ( path-2 ( | ∆ translation ( where associated doubly-degenerate ground states separated from the lowest-lying excited states with the low energyinteraction window induced potential ( text for discussions. Figure 14 Hilbert space describedopposing by trajectories basis involving the twistposition ( Fermi energy JHEP04(2021)148 , , , ) 0 r π → 19 || . In (9.9) J = 2 (9.11) (9.12) (9.13) (9.10) ] have ). sgn( ( r > 2) ]. The )) that b changes χ 5 γ − ) 128 − ∆ r 3.11 π ( 126 spin/charge , (4 We now iden- = 0) 125 π/ [ (eq. ( || ] . Importantly, we , the gapless TLL ∗ J 0 ⊥ ( J c/s ) = 127 χ . 2 ˆ → = 0 O )[ ) show that for = b for the classical 2D XY S =0 ξ , as the change in /S γ λ ) χ | χ , there exists a universal π ) ) 6.18 ∆ 2 λ = 1 ( ) γ/ || 5.9 || G ( J J → ∞ n . ( n 2 . χ d i dλ . Nelson and Kosterlitz [ 2 : 0 − γ ∞ → c/s . ))sgn( − || : ˆ of the center of mass position operator →∞ r π X lim J || ξ L n , → − = 1 = 0) n n C πJ 0 ) ) b i − h i γ || π γ π = J : − 2 th derivative of the gauge-invariant cumulant ( 2 c/s (2 ( ∆ b n T χ ˆ r > γ (1 + sgn( X s – 41 – B h = = ρ π 1 k = γ 2 , the superfluid stiffness . On the other hand, for 1 4 n C 2 = 0 ~ C C χ 2 m = 14 , ∆ ξ ∆ in the Hamiltonian eq. ( i b r > = π = || γ 2 λ s/c χ J 2 ˆ and C O = h 2 is related to the localization length ( 1 changes sign, C 2 /S || C , topological degeneracy, topological excitations and J are defined as the π ) = ln ) B λ n λ discussed earlier and are indicative of fermionic criticality, while the ( ( C , as 2 G G 0 / = 4 = 1 -term associated with the Dirac string in the projective Hilbert space [ ). This Wilson loop operator reveals the presence of a vortex condensed 2 = 1 Θ ]. Our asymptotically safe RG equations eq. ( r > can be tracked by the cumulants S /γ 3.14 2 128 +1 [ π , the change in the localization length is 4 0 2 to 1 → < π/ − || . The cumulants The second cumulant J ]. Now, for : 0 ) = c/s || For depicts the transition from a metal to a (spin/charge) gapped state. The localization length asymptotically free BKT RG equations aresee obtained for that the the limit crossover of RG flowas is well also as associated a with jump a in jump in the the first Berry cumulant phase jump in the superfluid∞ stiffness across the transitionJ given by this light, the asymptoticfixed point safety at of our RG results arise from the intermediate coupling They then employ thefor asymptotically the free BKT nature transition of the to weak show coupling that perturbative for RG is related to the 127 shown that for crossover RGmodel flows is related to the coupling generating function The first cumulant, tify observables associated withprobe the changes in center Hilbert of spacemodification mass geometry in twist and center topology of operator mass acrossfrom Hilbert the space RG as phase the diagram. signX The of the RG invariant sgn topological state of mattermodified Hilbert at space the intermediategap coupling are fixed depicted pictorially point metal in corresponds figure to the second conclusion ofObservables the for LSM-type changes in criterion (see Hilbert section space geometry and topology. given by eq. ( JHEP04(2021)148 , 2 is is 2 . , 2 0 and , we × B } 2] λ (9.20) (9.17) (9.18) (9.19) (9.14) (9.15) (9.16) ⊗ H 1 λλ cm via the π/ 1 ⊗ H g 2 3 )). Now, H 1 S , r < . As both ⊂ B λλ The center 0 π H g ∈ π ˆ n 3.17 ] B 2 < into || ,X 129 and J i 0 (eq. ( π ˆ n 2 = representation basis ,...,π,..., 2 . SU(2) 2 T 2 X i / 2 1 . ≡ . , π/ | | X , ) . 2 1 0 1 i cm = 1 1 i| = [ π X ω . In order to compute the is given by λλ ) ( ) 1 X = H g λ 2 0 0 1 π ,S ih | cm 2 s/c √ X ω 2 1 H ), the combined Hilbert space γ , Ψ( 0 σ X 2 λ dλ 1 {| / π − ∂ | | π 2 are both topological in origin. The Re can then be represented in terms of ) 2 ,S π SU(2) = [0 =  0 0 2 λ π Z 2 π 2] = 1 γ B dω 2 B r c/s Ψ( π 1 π/ = 0) : 2 ∞ − X 3 ,S 0 2 ω i ± : 1 Z = ( 2 H  i − |h ξ X ) s/c cm 0 ,..., + λ ) = , σ 1 SU(2) S 0 . Within this formalism, the basis | ⇒ – 42 – . The center of mass coordinate in ) are functions of the quantum metric ω . In the light of the arguments presented above, } π X 1 ) ( n ) Ψ( 1 ˆ n 1 X . The basis | λ λλ . By performing an average over the gauge cm γ 0 γλ M S ∂ g 0 is spanned by a s/c 1 π | 2 is cut at diametrically opposite points ,..., γ 2 i in the (spin/charge) gapped phase ih σ 2 ) B S 2 = γ SU(2) e 1 = 1 λ ˆ ) and Im O dλ 2 ∈ r X cm π π/ λ = ⊗ 2 Im 0 ξ S 1 ( X Ψ( { | n b 1 π = 0 λ π G π γ 4 i ∈ i γ ∂ + ˆ → = n ,X 2 ) h ω i 0 λ = , π + ˆ ˆ n for = cm SU(2) = 1 =0 = lim 2 1 , the reduced density matrix for 1 Ψ( 1 ) + ln(1 + 2 | X = 0 X cm dξ λλ H π λλ 0 H 1 g ρ 2 g | n H or ,S , cm i γ/ P = ˆ π to the fluctuations encoded in the quantum metric [ | are labelled by eigenvalues of the number operators cm B =0 is the first Chern class on the center of mass torus 2 ) 1 i ˆ c/s O S π ω X ,π ˆ of the center of mass Hilbert space. Further, they will all show universal ( X 0 0 ⊕ = 1 ) = ln( γ {| represent two-level systems ( | s/c i ∈ )) embedded on a circle ( λ , γ ) σ cm ( i 2 = 1 λ is isomorphic to ,π G 9.2 1 H S 0 2 Im Ψ( B 0 | | = 0 : 1 → and (eq. ( ⊗ H cm ω H S 1 1 has an interpretation as a geometric distance on the center of mass Hilbert space. To see π Recall that by partial tracing over H represented as is subject to the constraint entangled states where where H B entanglement entropy, the circle This results in a surgery of the center of mass Hilbert space Berry phase jumps across the transition. Entanglement entropy and noise in theof center mass of Hilbert mass Hilbert space space. Thus, the drastic jumps incumulant both generating function given by we can conclude that all cumulants ( We can now relate thelim imaginary part of thefluctuation-dissipation (spin/charge) and conductivity Kramers-Kronig at relations zero frequency this, we define the quantum metric of the center of mass Hilbert spacewhere as obtain ξ JHEP04(2021)148 ] 123 (9.21) (9.22) (9.23) ) ] and other ], as well as within the log 40 9.19 37 , 2 36 . π observed earlier. We = 2 T . c/s tracks the number fluctu-  X , where the λ π γ ˆ π 2 O = − γ 1 . ,  i ] m π 2 ] for ˆ log n ) on the torus C  )! 130 m , is thus dependent on the center of mass iλγ π = 2 [ γ 2 ) m 2 1 α d (2 H − ]. Given the simplicity of the two-point Fermi 0 ρ exp [ ) (i.e. fluctuation associated with the center of 1 h – 43 – ln  = ln 2 X 40 m> 9.9 , 1 − H = 39 top ρ ) = ln are the Bernoulli numbers. The cumulant generating π S ( γ 2 λ top ( m Tr S 2 in eq. ( G − log B m π 2 = γ 2 C S − , and | = m 2 S B | ], Fermi surfaces in higher dimensions. In this sense, fermionic critical- m 2 )) can therefore be written in terms of basis states of eq. ( ) 52 π 9.9 = (2 m 2 α models of correlated lattice electronssurface [ at hand, weuniversal, expect i.e., that they these shed insights lightDirac also into point-like on [ the the nature instabilitiesity of of appears fermionic regular to criticality connected be are [ shaped by the global (topological) features of the Fermi surface. It will skeletal phase diagram thatdiagram. shares the This essential skeletal features phaseand characteristic which diagram of are highlights the separated regions BKT byto possessing phase transitions the different projected involving symmetries, changes Hilbert in spacedriven topological at Fermi the quantities surface Fermi related surface. topology-changingcritical In (Lifshitz) point, this transition way, similar we of observe to an the the interaction 1D transitions metal observed across in the the 2D Hubbard [ fixed points. While the natureis of preserved, critical RG the flows nonperturbative comprisingnon-Abelian the nature constraints BKT of leading RG to phase the diagram We a URG also family formalism perform of shows a gapped the quantumface, intermediate mechanical emergence coupling and analysis fixed of compute of points. topological scattering processes terms at induced the by Fermi sur- interactions. In doing so, we obtain a 10 Discussions and outlook In summary, we have appliedproblem of the interacting unitary electrons renormalization inlogical 1D, group unveiling features (URG) thereby the technique in role to guiding played the by the emergent topo- RG flow towards critical (TLL) and stable (LE and MI) This shows that changing boundaryations conditions associated via with operator the state of the center of mass position where function (eq. ( refers to the two-fold topologicalrecall degeneracy that the ( same topologicalthe entanglement scattering entropy dynamics was at derived thebetween in the Fermi an surface. even earlier cumulants section As by mass seen position earlier, vector) there and exists the a entanglement relation entropy [ Hilbert space topology and is given by Clearly, the entanglement entropy is The entanglement entropy, JHEP04(2021)148 ]. We also demonstrate 133 ]. 132 – 44 – ] or incommensuration [ 131 The MI and LE phases emerge respectively from the charge gapping and spin gapping The many-particle entanglement renormalization comprising the EHM tensor network Vidhyadhiraja, P. Majumdar, S.A. Jalal M. and and M. S. Patra P.S. for thank L. several the thanks discussions CSIR, the and DST, Govt.which feedback. a Govt. of part of of India this India for work for was fundingcomments. carried funding through out. through research We a also fellowships. Ramanujan thank an Fellowship during anonymous referee for insightful Acknowledgments The authors thank R. K. Singh, A. Dasgupta, A. Bhattacharya, A. Taraphder, N. S. tering between the Fermi surfaceresults and obtained the from excitations.many-particle our An entanglement experimental gedanken and verification the would of Fermi pave the surface. the way towards systematic studies of the degrees of freedomto at other the (“bath”) two-point degrees Fermithought-experiment surface for of the as freedom topological a at andthis “quantum entanglement higher way, features impurity” we energies, of track coupled the the we entanglement Fermithe present entropy surface. rest generated a In of by scattering isolating the thestudying matrix system. Fermi the surface We quantum from also noise suggest and ways higher of order measuring cumulants this generated entanglement by entropy by two-electron scat- manifold) protected by a manyture body of gap, emergent charge phasesthe fractionalization is emergent etc. the pairs of An phenomena electronic important of statesysis and fea- dynamical of the spectral the fundamental scattering electrons. weight phase transfer Usingand shift, the quantify between we URG the compute anal- dynamical nonperturbatively spectral the weight net transferred Friedel from phase UV shift to IR. Finally, by treating tanglement RG flows from UV toto IR speculate for both that the these gapped are and again gapless phases. universal findings, It and is tempting willof be the explored Fermi surface in and future itsfeatures works. neighbourhood. of topological We find order: that ground these state phases degeneracy possess (for the the essential system placed on a periodic EHM network, we demonstrate the emergencepoint of in the two distinct dual spacetimes gravity theory acrossphases (i.e., the respectively), corresponding critical such to the that gaplessgraphic the TLL dual critical and spacetime gapped point generated LE/MI appears fromthe to important entanglement RG role act played flow as by [ the a degrees horizon of in freedom at the the holo- Fermi surface in guiding the en- in the presence of disorder [ show distinct features for the criticalsition and across gapped the phases. critical Thus,ment point the phase of topological transition. phase the By tran- BKT verifying the phase Ryu-Takayanagi holographic diagram entropy bound also for corresponds the to a entangle- be interesting to test these conclusions for one dimensional systems of interacting electrons JHEP04(2021)148 ] . # ) (A.4) (A.1) (A.2) (A.3) as an ) j X ( η ) ) − j η H ( p, − ˆ ( ω 0 j p, ) ( κ η 0 j + ∆ ) vertices are τ − κ ) τ V η ) p, ) ( η ... − η 0 j − κ p, − + ( τ p, j p, ) ( arXiv:0903.1069 ( 0 j κ η  j [ τ κ − κ 1 2 , . ) τ τ ) p, p ) ) ) ( η ( η η ) − j p ) η − κ − − ( b j ( ) ) τ − p, ˆ of the operator . p, p, n j ( η ) ( ( ( p, ) j V 0 j 0 j p ( − η κ ( 0 j κ κ ω V p, τ )   − − κ τ τ ) ( j 2 ) ) τ 0 j ( p 1 2 p, 1 η η ) ( κ ( )) η − 0 j ) V − − τ (2009) 155131 ] p to only the one-particle dispersion j − κ ) − . ( ( p, p, η τ ( ( ) ) − a p, ω j,p 0 j j j j ) and backscattering ( j ( 80 − K ˆ ( ( n 1 κ j κ κ G p, , Clarendon, Oxford, U.K. (2004).  ˆ κ τ τ [ ω K Bosonization and strongly correlated − (  ] K ) ) τ j ) ), and accounting only for two-particle ( ) − p p η κ 1) ( ( j,p p τ ) − ), we obtain the renormalisation of the ) ) − ( 2 η 6.6 j j j ) G p, ( ( ( [ j ( − , )) ( 1 6.8 ) 4 a,b p, p V V κ ( ( V Γ c j ) p,η ) ) − − – 45 – ( κ j p η ( 1 τ 1 1 Phys. Rev. B ( a,b κ X j − ) − − , K c κ j ] ] p, ) (  . ( + 4( 0 1 j,p j,p K p,η κ −   ( c 4 G G 0 1 1 2 1 2 ) [ [ ω κ c ), which permits any use, distribution and reproduction in p,η − − ) = ( 0 i i Tensor-entanglement-filtering renormalization approach and 1 † κ ) = ) = ˆ ˆ p,η n n c ( − j,p p p 0 )   ( ( † κ G ) ) ) ) c j j p,η ) ( ( ( 1) 1) † κ − − V K p,η c j j ( ( i ( i ∆ CC-BY 4.0 † κ ∆ + c Quantum physics in one dimension " This article is distributed under the terms of the Creative Commons + Σ + Σ i i ,p,η   1 ( ( ]. X i i , Cambridge University Press, Cambridge, U.K. (2004). κ,κ X X = ≈ = SPIRE ) ) j j IN ( ( systems symmetry protected topological order [ T. Giamarchi, Z.-C. Gu and X.-G. Wen, A.O. Gogolin, A.A. Nersesyan and A.M. Tsvelik, ˆ H ω The vertex RG flow equations for the forward ( ∆ [2] [3] [1] Attribution License ( any medium, provided the original author(s) and source areReferences credited. Open Access. and study thereby the RG equations as a functionthen of obtained as We interpret the variousinherent quantum quantum parameter fluctuation arising eigenvalues out of the renormalization of the interaction vertices, Importantly, we note that in obtaininglimited the the denominator study of the of above the RG quantum equation, fluctuation we have scale Here, Starting from thevertices Hamiltonian RG in flow the eq.Hamiltonian Hamiltonian as ( given in eq. ( A RG equation for the BCS instability JHEP04(2021)148 , , 42 . , ]. Phys. , (2017) ]. 95 (2019) 538 1 Phys. Rev. B SPIRE , IN ][ . arXiv:0910.1811 ]. (2012) 075125 [ Phys. Rev. B , 85 SPIRE (2011) 235128 interacting spinless Fermi gas IN Instanton Aharonov-Bohm (1997) 1142 [ (2011) 165139 . 84 D Gapless topological phases and (1964) A171 1 Nature Rev. Phys. . 55 84 , Entanglement spectrum of a Symmetry protection of topological ]. (2010) 064439 133 arXiv:1306.2164 [ ]. Phys. Rev. B 81 Classification of topological quantum Symmetry-protected topological phases of , SPIRE Quantum information meets quantum IN (2016) 035005 Phys. Rev. B ][ , (2002) 153110 Phys. Rev. B Phys. Rev. B 88 Classifying quantum phases using matrix product Phys. Rev. , , (2014) 117 , arXiv:1905.06969 65 – 46 – , ]. Two soluble models of an antiferromagnetic chain ]. ]. Phys. Rev. B Complete classification of one-dimensional gapped 349 , models and a special group supercohomology theory Nonstandard symmetry classes in mesoscopic arXiv:1108.4996 σ SPIRE [ SPIRE ]. SPIRE IN IN IN [ ][ Lattice twist operators and vertex operators in sine-Gordon [ arXiv:1201.2648 Symmetry-protected topological orders for interacting fermions: Phys. Rev. B [ Rev. Mod. Phys. Topological phase in a one-dimensional interacting fermion system , SPIRE , IN Annals Phys. , ][ ]. ]. ]. ‘Luttinger liquid theory’ of one-dimensional quantum fluids. I. Properties (2011) 195107 (1961) 407 , New York, NY, U.S.A. (2019). (1981) 2585 84 16 14 . (2014) 115141 Theory of the insulating state A practical introduction to tensor networks: matrix product states andTensor projected networks for complex quantum systems arXiv:1610.05706 90 Springer [ , cond-mat/9602137 arXiv:0909.4059 arXiv:1010.3732 arXiv:1103.3323 theory in one dimension entangled pair states effect and macroscopic quantum coherence(1990) in 7614 charge-density-wave systems Annals Phys. matter with symmetries of the Luttinger modelJ. and Phys. their C extension to the general matter normal-superconducting hybrid structures [ one-dimensional interacting fermions with245108 spin-charge separation symmetry-enriched quantum criticality [ fermionic topological nonlinear Rev. B [ Phys. Rev. B phases in one-dimensional quantum spin systems topological phase in one dimension quantum phases in interacting[ spin systems states and projected entangled pair states M. Nakamura and J. Voit, R. Orús, R. Orús, W. Kohn, E.H. Lieb, T. Schultz and D. Mattis, F.D.M. Haldane, E.N. Bogachek, I.V. Krive, I.O. Kulik and A.S. Rozhavsky, A. Altland and M.R. Zirnbauer, C.-K. Chiu, J.C. Teo, A.P. Schnyder and S. Ryu, R. Verresen, R. Thorngren, N.G. Jones and F. Pollmann, B. Zeng, X. Chen, D.-L. Zhou and X.-G. Wen, A. Montorsi, F. Dolcini, R.C. Iotti and F. Rossi, Z.-C. Gu and X.-G. Wen, H. Guo and S.-Q. Shen, F. Pollmann, E. Berg, A.M. Turner and M. Oshikawa, X. Chen, Z.-C. Gu and X.-G. Wen, N. Schuch, D. Pérez-García and I. Cirac, F. Pollmann, A.M. Turner, E. Berg and M. Oshikawa, [9] [7] [8] [5] [6] [4] [19] [20] [21] [17] [18] [15] [16] [13] [14] [11] [12] [10] JHEP04(2021)148 ]. ] Phys. (2005) SPIRE , ]. IN 0708 [ 38 (2006) (2010) J. Stat. ]. , 96 ]. SPIRE ]. 105 IN ][ SPIRE ]. . SPIRE J. Phys. A IN , IN J. Stat. Mech. cond-mat/0505563 ][ , ][ [ (2020) 115163 SPIRE IN arXiv:1708.00006 960 ][ , (2020) 115170 Phys. Rev. Lett. , Phys. Rev. Lett. quant-ph/0304108 , (2020) 063008 ]. [ 960 ]. arXiv:1410.4211 [ 22 (2005) P07007 SPIRE SPIRE quant-ph/0211074 IN IN ]. ]. quant-ph/0504151 [ Finitely correlated states on quantum spin Entanglement in quantum critical [ Nucl. Phys. B (2004) 79 0507 [ ][ , ]. hep-th/9403108 [ 116 Nucl. Phys. B SPIRE SPIRE , – 47 – Geometric and renormalized entropy in conformal (2016) 104425 IN IN New J. Phys. Entanglement and alpha entropies for a massive [ [ Entanglement in xy spin chain ]. SPIRE , ]. (1992) 443 IN 93 ][ (2003) 227902 (1994) 443 (2006) 100503 SPIRE 144 Tensor networks in a nutshell J. Stat. Mech. SPIRE ]. ]. Entanglement entropy and quantum field theory IN 90 , hep-th/0405152 Quantum spin chain, Toeplitz determinants and the IN 96 ][ 424 J. Stat. Phys. [ ][ Holographic entanglement renormalisation of topological order in Holographic unitary renormalization group for correlated Holographic entanglement renormalisation of topological order in Holographic unitary renormalization group for correlated Scaling theory for Mott-Hubbard transitions. II. Quantum , ]. Fermionic multiscale entanglement renormalization ansatz . SPIRE SPIRE Entanglement entropy of fermions in any dimension and the Widom IN IN Phys. Rev. B , ][ ][ arXiv:2003.06118 arXiv:2003.06118 , , arXiv:0705.2024 An area law for one-dimensional quantum systems [ (2004) P06002 Phys. Rev. Lett. Nucl. Phys. B Violation of the entropic area law for fermions Entanglement entropy and the Fermi surface Phys. Rev. Lett. , (2009) 165129 , , ]. Effective field theory for one-dimensional valence-bond solid phases and their quant-ph/0503219 arXiv:0908.1724 Commun. Math. Phys. 80 [ [ 0406 , quant-ph/0409027 [ SPIRE arXiv:2004.06897 arXiv:2004.06900 IN electrons. I. A tensor[ network approach a quantum liquid [ criticality of the doped Mott insulator a quantum liquid conjecture 050502 electrons. II. Insights on fermionic criticality Fisher-Hartwig conjecture 010404 field theory 2975 Mech. Dirac field in two[ dimensions symmetry protection phenomena Rev. B chains (2007) P08024 A. Mukherjee and S. Lal, A. Mukherjee and S. Lal, A. Mukherjee and S. Lal, A. Mukherjee and S. Lal, B. Swingle, A. Mukherjee and S. Lal, B.Q. Jin and V.E. Korepin, M.M. Wolf, D. Gioev and I. Klich, C. Holzhey, F. Larsen and F. Wilczek, A.R. Its, B.Q. Jin and V.E. Korepin, P. Calabrese and J.L. Cardy, H. Casini, C.D. Fosco and M. Huerta, G. Vidal, J.I. Latorre, E. Rico and A. Kitaev, P. Corboz and G. Vidal, M. Fannes, B. Nachtergaele and R.F. Werner, M.B. Hastings, Y. Fuji, J. Biamonte and V. Bergholm, [39] [40] [37] [38] [35] [36] [32] [33] [34] [30] [31] [28] [29] [26] [27] [23] [24] [25] [22] JHEP04(2021)148 , 43 79 , . JHEP , (2019) (2016) 21 (2010) 567 49 832 arXiv:1309.6282 Phys. Rev. B ]. (2016) 104205 , Phys. Rev. Lett. , arXiv:1905.08821 , (2014) 076 , 93 01 J. Phys. A SPIRE New J. Phys. , IN , ][ (2007) 220405 ]. JHEP ]. Nucl. Phys. B 99 ]. , dimensions , Holography from conformal field . Phys. Rev. B SPIRE , SPIRE Quantum circuit approximations and IN 1 + 1 SPIRE IN ][ IN ]. ][ ][ Many-body localization from the perspective of On the holographic renormalization group arXiv:1407.6552 ]. ]. SPIRE [ Phys. Rev. Lett. . IN , (2017) 1600322 – 48 – Towards a derivation of holographic entanglement ][ ]. ]. Random antiferromagnetic chain SPIRE SPIRE Space-time and the holographic renormalization group 529 IN IN Correlated spin liquids in the quantum kagome ][ ][ hep-th/9903190 Holographic and Wilsonian renormalization groups Entanglement holographic mapping of many-body localized (2014) 280 [ Explicit local integrals of motion for the many-body localized SPIRE SPIRE arXiv:1102.0440 ]. arXiv:0907.0151 . [ IN IN ]. ]. 87 [ (2016) 010404 ][ ][ Algorithms for entanglement renormalization Robust entanglement renormalization on a noisy quantum SPIRE 116 SPIRE SPIRE hep-th/9912012 IN [ IN IN (1999) 3605 Annalen Phys. ][ (2011) 036 , ][ ][ (2009) 079 83 05 arXiv:0707.1454 arXiv:1010.1264 10 [ [ Flow towards diagonalization for many-body-localization models: adaptation of Eur. Phys. J. B . arXiv:1711.07500 , (2000) 003 Holographic description of quantum field theory Quantum renormalization group and holography Entanglement renormalization Exact holographic mapping and emergent space-time geometry Advances on tensor network theory: symmetries, fermions, entanglement, and , ]. ]. JHEP arXiv:1602.03064 arXiv:1811.05222 JHEP , 08 [ [ , Phys. Rev. Lett. , (2011) 031 SPIRE SPIRE IN cond-mat/0512165 arXiv:1305.3908 IN arXiv:0912.5223 305002 integrals of motion (1979) 1899 system by spectrum bifurcation renormalization group state the Toda matrix differential flow to random quantum spin chains computer (2009) 144108 entanglement renormalization for the[ Dirac field in antiferromagnet at finite field:023019 a renormalization group analysis [ holography 06 [ [ [ entropy Phys. Rev. Lett. JHEP theory L. Rademaker, M. Ortuño and A.M. Somoza, S.K. Ma, C. Dasgupta and C.K. Hu, L. Rademaker and M. Ortuño, C. Monthus, I.H. Kim and B. Swingle, G. Evenbly and G. Vidal, Y.-Z. You, X.-L. Qi and C. Xu, S. Pal, A. Mukherjee and S. Lal, G. Vidal, R. Orús, F. Witteveen, V. Scholz, B. Swingle and M. Walter, S.-S. Lee, X.-L. Qi, S.-S. Lee, H. Casini, M. Huerta and R.C. Myers, I. Heemskerk and J. Polchinski, J. de Boer, E.P. Verlinde and H.L. Verlinde, I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, V. Balasubramanian and P. Kraus, [58] [59] [56] [57] [53] [54] [55] [52] [49] [50] [51] [47] [48] [44] [45] [46] [42] [43] [41] JHEP04(2021)148 , . , 81 66 Rev. , (1988) , Bull. (2012) (2006) . , 67 84 08 Z. Phys. Functional (1985) 220 (2012) 065007 JETP , Cambridge, . 86 , JHEP 159 , Oxford, U.K. , cond-mat/0505529 , Rev. Mod. Phys. (2017) 125125 , . He film 95 3 (in German), Rev. Mod. Phys. Static and dynamical ]. , (2015) 041041 Phys. Rev. D Annals Phys. ]. 5 , . , SPIRE (1992) 10741 IN SPIRE ][ Phys. Rev. B 45 IN , [ Oxford University Press (1982) 538 , Cambridge University Press , Phys. Rev. X 114 , ]. Hybrid-space density matrix renormalization group ]. – 49 – (1960) 1153 Phys. Rev. B , hep-th/0603001 ]. SPIRE [ SPIRE IN ]. 119 . ]. IN Physica A ][ [ (1988) 123]. , The index of elliptic operators on compact manifolds SPIRE ]. Aspects of holographic entanglement entropy Holographic derivation of entanglement entropy from AdS/CFT ]. IN 94 SPIRE [ SPIRE IN IN ][ SPIRE ][ Phys. Rev. SPIRE (2002) 7472 IN (2006) 181602 , (1969) 422 IN [ ][ Solutions of the two-dimensional Hubbard model: benchmarks and results 96 69 117 Fermi surface and some simple equilibrium properties of a system of (1975) 773 The universe in a helium droplet The Schwinger model and its axial anomaly The renormalization group: critical phenomena and the Kondo problem Lectures on quantum mechanics Renormalization group approach to interacting fermions Numerical canonical transformation approach to quantum many-body problems Luttinger’s theorem and bosonization of the Fermi surface Entanglement renormalization and holography Constructing holographic spacetimes using entanglement renormalization An analog of the quantum Hall effect in a superfluid 47 cond-mat/9307009 . [ Bremsvermögen von Atomen mit mehreren Elektronen Decimation method of real-space renormalization for electron systems with ]. Zh. Eksp. Teor. Fiz. arXiv:1105.5289 hep-th/0605073 [ [ [ SPIRE IN arXiv:0905.1317 Am. Math. Soc. [ interacting fermions (2009). 1804 U.K. (2009). (1933) 363 properties of doped Hubbard clusters Mod. Phys. (1994) 129 from a wide range of numerical algorithms study of the doped two-dimensional Hubbard model [ renormalization group approach to correlated299 fermion systems Phys. Rev. Lett. arXiv:1209.3304 application to random systems J. Chem. Phys. 045 M.F. Atiyah and I.M. Singer, F. Haldane, N.S. Manton, G.E. Volovik, G. Volovik, S. Weinberg, F. Bloch, J.M. Luttinger, E. Dagotto, A. Moreo, F. Ortolani, D. Poilblanc and J.K.G. Riera, Wilson, J. LeBlanc et al., G. Ehlers, S.R. White and R.M. Noack, W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden and K. Schönhammer, R. Shankar, S. Ryu and T. Takayanagi, B. Swingle, B. Swingle, S.R. White, S. Ryu and T. Takayanagi, H. Aoki, [77] [78] [79] [75] [76] [72] [73] [74] [70] [71] [68] [69] [66] [67] [63] [64] [65] [61] [62] [60] JHEP04(2021)148 , , ]. 66 ] (1969) (1990) Phys. 60 , SPIRE Phys. Rev. 65 , IN J. Stat. [ ]. , (1969) 2426 SPIRE ]. hep-th/9210046 Rev. Mod. Phys. 177 IN (1991) 38 , ][ Nuovo Cim. A , 207 Phys. Rev. Lett. SPIRE . , IN . Theoretical Advanced Study , (1992) [ [ model . . Phys. Rev. σ , in , ]. (1984) 2453 in the Annals Phys. (1961) 345 . (2000) 1535 SPIRE , . 17 cond-mat/0002392 γγ (1997) 1984 (1997) 1110 IN [ 84 122 ][ → 78 79 Nonperturbative approach to Luttinger’s Magnetization plateaus in spin chains: “haldane 0 Insulator, metal, or superconductor: the criteria π ]. ]. . – 50 – (1986) 3844 J. Phys. A Spontaneous symmetry breaking and decoherence in , SPIRE SPIRE 33 (2000) 3370 (2008) 064523 Phys. Rev. Statistical distance and the geometry of quantum states IN IN Dynamical model of elementary particles based on an , [ Ground state energy of a many fermion system. II 84 ][ 77 ]. Geometry of quantum evolution Haldane gap and fractional oscillations in gated Josephson Entanglement entropy and quantum field theory Impurity effect, degeneracy, and topological invariant in the Phys. Rev. Lett. . hep-th/0405152 , (1998) 4484 Phys. Rev. Lett. Phys. Rev. Lett. Quantised adiabatic charge transport in the presence of substrate [ , ]. , SPIRE A PCAC puzzle: 81 IN [ Phys. Rev. B (1994) 3439 , SPIRE ): from black holes and strings to particles IN (1993) 7995 Phys. Rev. B 72 Random-phase approximation in the theory of superconductivity [ 92 , Phys. Rev. Lett. Topological approach to Luttinger’s theorem and the Fermi surface of a Commensurability, excitation gap, and topology in quantum many-particle ]. Effective field theory and the Fermi surface 47 , (2004) P06002 Renormalization group approach to interacting fermions cond-mat/9307009 ]. Axial vector vertex in spinor electrodynamics [ Edge waves in the quantum Hall effect (1960) 1417 ]. ]. SPIRE Phys. Rev. Lett. 0406 , IN SPIRE 118 (1958) 1900 [ IN SPIRE SPIRE [ IN IN [ 1697 Phys. Rev. Lett. superconductors (1994) 129 Institute (TASI 112 analogy with superconductivity. I [ Mech. Phys. Rev. B 47 Rev. systems on a periodic lattice Kondo lattice quantum Hall effect disorder and many-body interaction gap” for half-integer spins theorem in one dimension arrays J. Anandan and Y. Aharonov, S.L. Braunstein and C.M. Caves, R. Shankar, J. Polchinski, P.W. Anderson, Y. Nambu and G. Jona-Lasinio, J. van Wezel and J. van den Brink, S.L. Adler, M. Stone, P. Calabrese and J.L. Cardy, D.J. Scalapino, S.R. White and S. Zhang, J.S. Bell and R. Jackiw, M. Oshikawa, M. Oshikawa, R. Tao and F.D.M. Haldane, Q. Niu and D.J. Thouless, J.M. Luttinger and J.C. Ward, M. Yamanaka, M. Oshikawa and I. Affleck, E. Altman and A. Auerbach, M. Oshikawa, M. Yamanaka and I. Affleck, [98] [99] [96] [97] [93] [94] [95] [90] [91] [92] [88] [89] [86] [87] [83] [84] [85] [81] [82] [80] JHEP04(2021)148 32 Adv. , , (2011) Phys. Rev. , 145 Phys. Rev. B (1998) 253 , Nucl. Phys. B , Cambridge, 2 , ]. Proc. Roy. Soc. Lond. Prog. Low Temp. Phys. Sov. Phys. JETP Proc. Roy. Soc. Lond. A , , , , J. Stat. Phys. SPIRE , (2007) 220405 ]. ]. IN [ 99 ]. ]. SPIRE Cambridge University Press SPIRE , IN , Academic Press, New York, NY, IN ]. [ ][ ]. SPIRE ]. SPIRE IN Conformal invariance and the spectrum of IN Adv. Theor. Math. Phys. ]. (1975) 3908 [ , ]. Cambridge University Press SPIRE , 12 SPIRE SPIRE IN IN . [ IN SPIRE Phys. Rev. Lett. (1979) 318 [ . Two-dimensional physics ][ , IN SPIRE 5 Extracting entanglement geometry from quantum – 51 – ][ IN (1987) 771 (1971) 907] [ 121 ][ hep-th/9711200 58 [ 59 Correlation functions on the critical lines of the Baxter and (1950) 544 Chapter Universality, marginal operators, and limit cycles limit of superconformal field theories and supergravity Phys. Rev. B ]. 5 (1961) 1242 , (2017) 140502 ]. N Tensor network states and geometry ]. Calculation of critical exponents in two-dimensions from quantum 123 Exact holographic mapping in free fermion systems 119 ]. SPIRE ]. (1998) 231 Annals Phys. IN SPIRE cond-mat/0303297 , 2 [ ][ SPIRE IN arXiv:1503.08592 IN ][ The large [ SPIRE Destruction of long range order in one-dimensional and two-dimensional cond-mat/9712314 Remarks on Bloch’s method of sound waves applied to many-fermion [ Phys. Rev. Lett. Low energy effective Hamiltonian for the XXZ spin chain IN [ , Quantised singularities in the electromagnetic field, [ Quantum theory of angular momentum Lectures on quantum mechanics Phys. Rev. Quantal phase factors accompanying adiabatic changes . Field theories of condensed matter physics Zh. Eksp. Teor. Fiz. , Anti-de Sitter space and holography Cyclotron resonance and de Haas-van Alphen oscillations of an interacting Prog. Theor. Phys. Entanglement renormalization , (1984) 45 Phys. Rev. Lett. (2004) 094304 , arXiv:1106.1082 (1931) 60 (1998) 533 [ (2016) 035112 69 392 (1978) 371 hep-th/9802150 cond-mat/0512165 [ U.K. (2015). electron gas states Theor. Math. Phys. [ 93 891 7 problems the XXZ chain systems having a continuous(1971) symmetry 493 group. [ I. Classical systems 522 field theory in one-dimension Ashkin-Teller models Cambridge, U.K. (2013). B A U.S.A. (1965). 133 S. Weinberg, W. Kohn, K. Hyatt, J.R. Garrison and B. Bauer, J.M. Maldacena, E. Witten, C.H. Lee and X.-L. Qi, G. Evenbly and G. Vidal, J. Kosterlitz and D. Thouless, S.-I. Tomonaga, G. Vidal, F.C. Alcaraz, M.N. Barber and M.T. Batchelor, V.L. Berezinsky, A. Luther and I. Peschel, L.P. Kadanoff and A.C. Brown, E. Fradkin, S.D. Glazek and K.G. Wilson, S.L. Lukyanov, J. Schwinger, P.A.M. Dirac, M.V. Berry, [118] [119] [115] [116] [117] [113] [114] [110] [111] [112] [108] [109] [106] [107] [103] [104] [105] [101] [102] [100] JHEP04(2021)148 , , , , ] 61 102 Phys. , , . Phys. Rev. B ]. , Phys. Rev. Lett. , arXiv:1603.08509 [ SPIRE IN Dynamics of superfluid films [ ]. . arXiv:0908.2418 , . (2016) 044 SPIRE IN 09 ][ (1977) 1201 (2000) 1666 Superconductors are topologically ordered ]. 39 ]. 62 (1982) 362 ]. JHEP , SPIRE 48 Polarization and localization in insulators: – 52 – SPIRE IN ]. [ SPIRE IN [ IN Universal jump in the superfluid density of Friedel sum rule for a system of interacting electrons ][ Phase diagrams of lattice gauge theories with Higgs fields SPIRE Anderson localization and interactions in one-dimensional cond-mat/0404327 IN Phys. Rev. B [ [ . Phys. Rev. Lett. , Riemannian structure on manifolds of quantum states . , (1980) 289 (1988) 325 . Quantum noise as an entanglement meter Phys. Rev. Lett. 76 , 37 One-dimensional fermions with incommensuration (2004) 497 (1979) 3682 (1980) 1806 (1961) 1090 arXiv:0804.1377 Fermi-surfae sum rule and its consequences for periodic Kondo and 19 [ 21 313 Single-particle Green’s functions and interacting topological insulators . 121 (2011) 085426 Horizon as critical phenomenon Entanglement entropy in many-fermion system ]. Phys. Rev. B 83 , SPIRE IN [ metals (2000) 9001 two-dimensional superfluids Commun. Math. Phys. Phys. Rev. D Annals Phys. generating function approach (2009) 100502 Phys. Rev. B Phys. Rev. mixed-valence systems Rev. B T. Giamarchi and H.J. Schulz, D. Sen and S. Lal, S.-S. Lee, D.R. Nelson and J.M. Kosterlitz, J.P. Provost and G. Vallee, W. Ding, T.H. Hansson, V. Oganesyan and S.L. Sondhi, I. Souza, T. Wilkens and R.M. Martin, I. Klich and L. Levitov, V. Ambegaokar, B.I. Halperin, D.R. Nelson and E.D. Siggia, E.H. Fradkin and S.H. Shenker, R.M. Martin, V. Gurarie, J.S. Langer and V. Ambegaokar, [131] [132] [133] [128] [129] [130] [126] [127] [123] [124] [125] [121] [122] [120]