Dissipative free fermions in disguise

Dissipativ e free fermions in disguise Kohei F uk ai, 1 Hironobu Y oshida, 1, 2 and Hosho Katsura 1, 3, 4 1 Dep artment of Physics, Gr aduate Scho ol of Scienc e, The University of T okyo, 7-3-1, Hongo, Bunkyo-ku, T okyo, 113-0033, Jap an ∗ 2 None quilibrium Quantum Statistic al Me chanics RIKEN Hakubi R ese ar ch T e am, RIKEN Pione ering R ese ar ch Institute (PRI), Wako, Saitama 351-0198, Jap an 3 Institute for Physics of Intel ligenc e, The University of T okyo, 7-3-1 Hongo, T okyo 113-0033, Jap an 4 T r ans-Scale Quantum Science Institute, The University of T okyo, 7-3-1, Hongo, T okyo 113-0033, Jap an Recen tly , a class of spin c hains known as “free fermions in disguise” (FFD) has been disco vered, whic h p ossess hidden free-fermion spectra ev en though they are not solv able via the standard Jordan- Wigner transformation. In this work, we extend this FFD framework to op en quantum systems go verned by the Gorini-Kossak owski-Sudarshan-Lindblad (GKSL) equation. W e establish a general class of exactly solv able open quan tum systems within the FFD framew ork: if the Liouvillian frus- tration graph is cla w-free and has a simplicial clique, the Liouvillian p ossesses a hidden free-fermion sp ectrum. In particular, the (even-hole, claw)-fr e e condition automatically guarantees this, enabling exact computation of the Liouvillian gap and an infinite-temp erature auto correlation function. Our results provide the first realization of the FFD mechanism in op en quantum systems. Intr o duction .— Exact solutions play a pivotal role in understanding the dynamics of quan tum many-bo dy sys- tems. While the generalized Gibbs ensemble and gen- eralized hydrodynamics describ e the equilibration and transp ort in integrable mo dels [ 1 – 4 ], constructing mo d- els where the full spectrum and dynamics can be accessed exactly remains a central challenge. Recen tly , F endley constructed a spin c hain that is quartic in Jordan-Wigner fermions yet p ossesses a hidden free-fermion sp ectrum [ 5 ], now known as “free fermions in disguise” (FFD) [ 6 – 20 ]. Unlik e conv en tional free- fermion-solv able spin mo dels [ 21 – 23 ], FFD mo dels can- not be diagonalized by the Jordan-Wigner transforma- tion [ 8 ]. F ree-fermion solv ability is no w understo o d in graph-theoretic terms [ 8 , 9 , 24 , 25 ]: the (even-hole, claw)- fr e e (ECF) condition on the frustration graph guarantees FFD solv ability [ 8 ], later extended to claw-free graphs with a simplicial clique [ 9 ]. While these developmen ts hav e deep ened our un- derstanding of solv able closed systems, realistic quan- tum systems are inevitably coupled to an environmen t, leading to dissipativ e dynamics. When the system- en vironment coupling is Mark ovian, the time ev olu- tion of the density matrix is go verned b y the Gorini- Kossak owski-Sudarshan-Lindblad (GKSL) equation [ 26 – 28 ]. Although dissipation generally breaks in tegrability [ 29 , 30 ], finding exactly solv able Lindbladians is of great imp ortance, as they can serv e as b enc hmarks for appro xi- mate metho ds [ 31 ] and offer insights into non-equilibrium steady states (NESS) and relaxation dynamics [ 32 , 33 ]. Sev eral strategies hav e b een dev elop ed to find exact solutions for the GKSL equation [ 34 , 35 ]. A prominen t class consists of quadratic open systems, which can be diagonalized using third quantization tec hniques [ 36 – 39 ]. In the context of transp ort, b oundary-driv en spin chains ha ve b een extensively studied, often allowing for the ex- plicit construction of the NESS [ 32 , 40 – 46 ]. More re- cen tly , solv abilit y has b een established for superop era- tors exhibiting a triangular structure [ 47 , 48 ] and for some Y ang-Baxter integrable models [ 49 – 53 ]. Despite this progress, the FFD framework has not yet b een ex- tended to open quan tum systems. It remains an op en question whether dissipative couplings can b e designed suc h that the Liouvillian sup erop erator exhibits a hid- den free-fermion sp ectrum. In this pap er, w e answer this question affirmatively . W e sho w that if the Hamiltonian and jump op erators are designed so that the Liouvillian frustration graph is cla w-free and has a simplicial clique, the Liouvillian is solv able by hidden free fermions. In particular, we show that any ECF frustration graph yields an exactly solv able dissipativ e spin chain. Crucially , this solv ability holds for arbitrary coupling constants without fine-tuning, and a non-lo cal mapping b ey ond the standard Jordan-Wigner transformation is utilized. W e can explicitly derive the Liouvillian sp ectrum, the finite-size scaling of the Liou- villian gap, and an infinite-temp erature auto correlation function, demonstrating that the FFD mec hanism can b e successfully extended to the realm of op en quantum systems. The GKSL master e quation .— W e consider a spin-1 / 2 system whose Marko vian dynamics is describ ed b y the Gorini-Kossak owski-Sudarshan-Lindblad (GKSL) mas- ter equation [ 26 , 28 ]. The time evolution of the density matrix ρ ( t ) is given by ˙ ρ ≡ L ρ = − i[ H , ρ ] + X a  ℓ a ρℓ † a − 1 2 { ℓ † a ℓ a , ρ }  . (1) Here, L is the Liouvillian sup erop erator, H is the Hamil- tonian for the unitary dynamics, and ℓ a are the jump op erators describing dissipative pro cesses. T o analyze the sp ectrum of the Liouvillian, it is con- v enient to employ the vectorization formalism. W e map the densit y matrix ρ to a state vector | ρ ⟩⟩ in the dou- 2 bled Hilbert space H ⊗ H , treating the Liouvillian as a non-Hermitian Hamiltonian acting on the ket and bra spaces following the argumen t in Refs. [ 38 , 50 , 52 , 54 , 55 ]. Sp ecifically , op erators acting from the left (righ t) on ρ are mapp ed to op erators acting on the ket (bra) space. The Liouvillian L in the doubled Hilb ert space reads H ≡ i L = H (1) − H (2) ∗ + i X a  ℓ (1) a ℓ (2) ∗ a − 1 2 ℓ (1) † a ℓ (1) a − 1 2 ℓ (2) T a ℓ (2) ∗ a  . (2) Here, T denotes transp ose, ∗ complex conjugation, and A (1) = A ⊗ I (resp. A (2) = I ⊗ A ) acts on the ket (resp. bra) space. In this pap er, we construct a class of Liou- villians L exactly solv able via the FFD framework. F or example, this general construction also cov ers the F endley mo del [ 5 ] with dissipation on a chain of M sites. Its Hamiltonian is H = P M j =1 b j h j , where the b j are ar- bitrary coupling constants and h j = σ z j − 2 σ z j − 1 σ x j ( j = 3 , . . . , M ) , (3) while h 1 = σ x 1 and h 2 = σ z 1 σ x 2 [ 56 ]. Here σ x j and σ z j denote the standard Pauli op erators acting on site j . As explained below, the corresp onding frustration graph has v ertex set V ( G ) = { 1 , . . . , M } and is the zigzag ladder graph sho wn in Fig. 2 (b). One exactly solv able c hoice of dissipation is the single right-boundary jump op erator ℓ = √ γ σ z M . W e first review the general FFD framework, within which this mo del will b e identified as a particular case. F r e e fermions in disguise .— W e briefly review the FFD framew ork [ 5 , 8 , 9 ]. W e first introduce a Hamiltonian rep- resen ted by a graph-Clifford algebra defined on a frustra- tion graph G : H G = X j ∈ V ( G ) b j h j , (4) where V ( G ) denotes the vertex set of G , b j are arbitrary coupling constants, and h j are generators asso ciated with eac h vertex j satisfying ( h j ) 2 = 1 , h j h ℓ = ( − 1) A j ℓ h ℓ h j , (5) where A is the adjacency matrix of G with A j ℓ ∈ { 0 , 1 } for j  = ℓ and A j j = 0. Th us, generators associated with adjacent vertices anticomm ute, while those asso ci- ated with non-adjacen t v ertices comm ute. An y graph- Clifford algebra ( 5 ) naturally admits a P auli-string rep- resen tation on | V ( G ) | spin-1 / 2 sites; an explicit construc- tion is given in End Matter [ 19 ]. The solv ability of H G is go verned by the graph- theoretic properties of G . A subgraph of G is induc e d if, for every pair of its vertices, the tw o vertices are adja- cen t in the subgraph exactly when they are adjacent in G . A graph is claw-fr e e if it has no claw as an induced (a) Claw K 1 , 3 C 4 C 6 · · · (b) Even holes FIG. 1. F orbidden induced subgraphs for the ECF condition; a subgraph is induc e d if it contains all edges of the original graph betw een its v ertices. (a) The cla w K 1 , 3 : a cen ter v ertex connected to three mutually non-adjacent leav es. (b) Ev en holes C 4 , C 6 , . . . : induced cycles of even length. 1 2 3 4 5 (a) Path graph 1 3 5 7 2 4 6 8 (b) Zigzag ladder FIG. 2. Examples of ECF frustration graphs. (a) Path graph corresp onding to the Ising/XY c hain, in whic h vertices j and k are adjacent when | j − k | = 1. (b) Zigzag ladder corresponding to the F endley mo del ( 3 ) for M = 8, in which vertices j and k are adjacent when | j − k | ≤ 2. The shaded vertices pro vide examples of indep enden t sets, with | S | = 2 in (a) and | S | = 3 in (b). subgraph (Fig. 1 (a)), and even-hole-fr e e if it has no even hole as an induced subgraph (Fig. 1 (b)). A graph sat- isfying b oth conditions is called even-hole-fr e e, claw-fr e e (ECF), whose examples are shown in Fig. 2 . The path graph (a) corresp onds to the Ising/XY c hain, solv able by the standard Jordan–Wigner transformation [ 24 , 25 , 57 ]. The zigzag ladder graph (b), whic h corresp onds to the F endley mo del ( 3 ), represen ts a new class of exactly solv- able models beyond the Jordan–Wigner paradigm [ 5 , 11 ]. When G is ECF, the Hamiltonian ( 4 ) is exactly diag- onalized [ 5 , 8 ]: H G = α G X k =1 ε k [Ψ k , Ψ − k ] , (6) where α G is the independence n um b er of G explained b elo w, Ψ ± k are hidden fermion op erators satisfying { Ψ k , Ψ k ′ } = δ k + k ′ , 0 , and ε k is the single-particle en- ergy . The eigenenergies are given b y 2 P k ε k s k + E 0 with s k ∈ { 0 , 1 } , where E 0 ≡ − P α G k =1 ε k is the ground-state energy . The single-particle energies ε k are determined b y the indep endenc e p olynomial of G . An indep endent set S ⊆ V ( G ) is a subset of m utually non-adjacen t v ertices, so that the corresp onding generators all commute. F or example, the shaded vertices in Fig. 2 form indep en- den t sets: S = { 2 , 4 } for the path graph in (a) and S = { 1 , 4 , 8 } for the zigzag ladder in (b), the frustra- 3 tion graph of the F endley mo del ( 3 ). The indep endenc e numb er α G is the maximum size of an indep endent set. The indep endence p olynomial is defined as P G ( x ) ≡ X S ∈S G ( − x ) | S | Y j ∈ S b 2 j , (7) where S G denotes the collection of all indep enden t sets. The single-particle energies are giv en b y ε k = 1 /u k , where u 1 , . . . , u α G are the ro ots of P G ( u 2 ) = 0. F or claw- free graphs, if all b j are real, so that the vertex weigh ts b 2 j are non-negativ e, then P G ( x ) has only p ositiv e real ro ots [ 58 ]. Therefore the single-particle energies are p os- itiv e. The free-fermion operators in Eq. ( 6 ) are built from an e dge op er ator χ and a transfer matrix T G ( u ) [ 5 , 8 , 9 ]: Ψ ± k = 1 N k T G ( ∓ u k ) χT G ( ± u k ) , (8) where u k is again a ro ot of P G ( u 2 ) = 0. The normaliza- tion constant N k is given in End Matter [see Eq. ( 26 )]. The edge op erator χ and the transfer matrix T G ( u ) are defined b elo w. The edge op erator χ is asso ciated with a clique K s ⊆ V ( G ), i.e., a subset of m utually adjacen t vertices. It satisfies χ 2 = 1 and { h j , χ } = 0 for j ∈ K s , [ h j , χ ] = 0 for j / ∈ K s . (9) In the frustration graph, this corresp onds to adding a new vertex χ adjacent to ev ery vertex of K s and to no others, giving an extende d gr aph G χ . The clique K s is called simplicial if G χ is also claw-free [ 9 , 59 ]. Equiv alently , denoting the closed neighborho od of j b y Γ[ j ] ≡ { j } ∪ { ℓ ∈ V ( G ) | A j ℓ = 1 } , K s is simplicial if and only if Γ[ j ] \ K s is a clique for all j ∈ K s . Figure 3 sho ws (a) a simplicial clique and (b) a non-simplicial clique. The edge op erator χ can be represen ted as a P auli string [ 19 ]; see Eq. ( 25 ) in End Matter. In fact, the ev en-hole-free condition can b e relaxed: an y cla w-free frustration graph with a simplicial clique yields hidden free-fermion solv abilit y [ 9 ], and the ECF is a sp ecial case [ 5 , 8 ]. The transfer matrix T G ( u ) is defined as a generating function of the conserved charges Q ( n ) G asso ciated with indep enden t sets of size n : Q ( n ) G ≡ X S ∈S G | S | = n Y j ∈ S b j h j , T G ( u ) ≡ α G X n =0 ( − u ) n Q ( n ) G , (10) with Q (0) G ≡ 1 . It satisfies T G ( u ) T G ( − u ) = P G ( u 2 ). F or claw-free graphs, these c harges commute, so [ T G ( u ) , T G ( v )] = 0 [ 5 , 8 , 9 ]. In this work, we extend the FFD framework to dissi- pativ e systems. As we show below, χ also plays a central role in the dissipative setting. A B χ C D (a) Simplicial A B χ C D (b) Not simplicial FIG. 3. Simplicial vs. non-simplicial cliques. The claw- free graph G consists of four vertices A, B , C , D , and G χ is obtained b y adding the vertex χ (green) connected to the clique K s = { A, B } (orange). (a) G χ is also claw-free, so K s is simplicial. (b) Without the edge C - D , Γ[ A ] \ K s = { C, D } is not a clique, so K s is not simplicial; indeed, adding χ creates a claw centered at A whose leav es are drawn in red. FFD solvable Liouvil lian .— The graph-theoretic solv- abilit y criterion extends to op en quantum systems. If the Hamiltonian and jump operators are designed so that the frustration graph e G of the non-Hermitian Hamiltonian H in Eq. ( 2 ) is claw-free and has a simplicial clique, H is solv able by hidden free fermions [ 9 ]. F or an ECF graph G , choose an edge op erator χ asso- ciated with a simplicial clique K s ⊆ V ( G ), which can b e c hosen Hermitian; see Eq. ( 25 ) in End Matter. T ake a single jump op erator ℓ = √ γ χ. (11) The resulting e G satisfies the solv ability criterion ab o v e. The corresp onding non-Hermitian Hamiltonian on the doubled space in Eq. ( 2 ) b ecomes H e G = X j ∈ V ( G ) b j  h (1) j − h (2) j  + i γ d − i γ , (12) where we defined d ≡ χ ⊗ χ . The op erators { h (1) j } and { h (2) j } form t w o independent copies of the original graph- Clifford algebra, and d satisfies d 2 = 1 and { d, h ( p ) j } = 0 ( j ∈ K s ) , [ d, h ( p ) j ] = 0 ( j / ∈ K s ) , (13) where p = 1 , 2 labels the t w o copies. The Liouvillian frustration graph e G consists of tw o disjoin t copies G (1) and G (2) of G , together with one extra v ertex d connected to all v ertices in K (1) s ∪ K (2) s . If G is ECF and K s is simplicial, then e G is again ECF: even holes cannot span b oth copies, and cla ws inv olving d are excluded by the simplicialit y of K s . F or p = 1 , 2, define e K ( p ) s ≡ { d } ∪ K ( p ) s . (14) These are simplicial cliques of e G . W e denote by e χ the edge op erator of e G asso ciated with e K (1) s . Therefore the FFD construction of Refs. [ 5 , 8 , 9 ] applies also to the non-Hermitian Hamiltonian H e G . 4 e χ FIG. 4. Liouvillian frustration graph e G M for the b oundary- driv en F endley mo del. The left (blue) and right (green) sub- graphs are the t wo copies G (1) and G (2) of the zigzag ladder in Fig. 2 (b). The central orange v ertex is d = χ ⊗ χ = σ z M ⊗ σ z M , sho wn here for the single-vertex b oundary clique K s = { M } . The extra blue v ertex represents the edge operator e χ , and the dotted contour marks the simplicial clique e K (1) s = { d } ∪ K (1) s . As a concrete example, for the F endley model with dissipation introduced ab o ve, taking K s = { M } and χ = σ z M yields the Liouvillian frustration graph e G M sho wn in Fig. 4 (ignore the dotted lines for the moment). The indep endence p olynomial of e G splits according to whether an indep enden t set contains d : P e G ( u 2 ) = P G ( u 2 ) 2 + γ 2 u 2 P G \ K s ( u 2 ) 2 = e P + G ( u ) e P − G ( u ) , (15) where the first term coun ts indep enden t sets excluding d , and the second those including d , and we define e P ± G ( u ) ≡ P G ( u 2 ) ± i γ uP G \ K s ( u 2 ) . (16) Because the added vertex d carries the imaginary weigh t i γ , the usual real-ro otedness result for weigh ted indep en- dence p olynomials of claw-free graphs does not apply here [ 60 , 61 ], so the corresp onding single-particle ener- gies can b e complex. W e next define the free-fermion modes of H e G using the FFD construction: e Ψ ± k = 1 e N k T e G ( ∓ e u k ) e χT e G ( ± e u k ) , (17) where e P + G ( e u k ) = 0 for k = 1 , . . . , α e G , e N k is a nor- malization factor whose explicit form is given in End Matter, Eq. ( 27 ), and α e G = max  2 α G , 1 + 2 α G \ K s  . Both e P ± G ( u ) are degree- α e G p olynomials in u , related by e P − G ( u ) = e P + G ( − u ) and e P ± G ( u ) = e P ± G ( − u ). W e exp ect Im e u k > 0 to hold for general ECF graphs, as observed n umerically in the boundary-driven F endley model, so that e ε k ≡ 1 / e u k satisfies Im e ε k < 0. W e tak e e Ψ − k as the annihilation operator and e Ψ k as the creation op er- ator for the mo de with single-particle energy e ε k . Be- cause H e G is non-Hermitian, e Ψ k is not in general the Hermitian adjoin t of e Ψ − k , namely e Ψ † k  = e Ψ − k . How- ev er, the fermionic an ticommutation relations still hold: { e Ψ k , e Ψ k ′ } = δ k + k ′ , 0 [ 38 , 62 ]. Then, the Liouvillian ( 12 ) is diagonalized as H e G = α e G X k =1 e ε k h e Ψ k , e Ψ − k i − i γ = 2 α e G X k =1 e ε k e n k , (18) where e n k ≡ e Ψ k e Ψ − k is the o ccupation num b er of the mo de k . Comparing the co efficien t of u in Eq. ( 16 ) with that in e P + G ( u ) = Q k (1 − u/ e u k ) giv es P k e ε k = − i γ . Using [ e Ψ k , e Ψ − k ] = 2 e n k − 1, the comm utator term contributes the constan t − P k e ε k = i γ , which cancels − i γ in the middle expression of Eq. ( 18 ) and yields the right-hand side. The configuration with e n k = 0 for all k there- fore gives the zero Liouvillian eigenv alue. Eac h o ccupied mo de then contributes − 2i e ε k to the Liouvillian eigen- v alue, so λ = − 2i P k e ε k s k for s k ∈ { 0 , 1 } , satisfying Re λ ≤ 0. F or generic couplings [ 63 ], these 2 α e G Liouvil- lian eigenenergies are distinct. In addition to the α e G hidden fermion mo des e Ψ ± k , there exist zero mo des Ξ j satisfying Ξ 2 j = 1, [Ξ j , H e G ] = 0, Ξ l Ξ k = ± Ξ k Ξ l , and e Ψ k Ξ l = ± Ξ l e Ψ k [ 12 , 17 ]. These generate a 2 2 | V ( G ) |− α e G -fold degeneracy at eac h eigenen- ergy . Accordingly , the zero-eigen v alue eigenspace Ker L is 2 2 | V ( G ) |− α e G -dimensional. Physical steady states are the p ositiv e, trace-one op erators in this eigenspace. Disp ersion r elation .— Since e P + G ( u ) can be obtained ex- plicitly for an y ECF graph, so can the dispersion relation. Using λ = − 2i P k e ε k s k and Im e ε k ≤ 0, the Liouvillian gap is given by g ≡ − max λ  =0 Re λ = − 2 max 1 ≤ k ≤ α e G Im e ε k . (19) F or the boundary-driven F endley mo del [Eq. ( 3 ), Fig. 4 ] with uniform couplings b j = 1 and M = 3( L − 1), the Liouvillian gap is given by g = γ π 2 3 + γ 2 1 L 3 . (20) The standing-w av e analysis leading to Eq. ( 20 ) is omit- ted here. This L − 3 scaling is consisten t with that ob- serv ed in other integrable chains with b oundary dissi- pation [ 36 , 55 , 64 – 66 ]. Note that the isolated F endley mo del has an energy gap scaling as L − 3 / 2 [ 5 ]. F or in- homogeneous couplings, a phase with an exponentially small Liouvillian gap also app ears [ 55 ]. Infinite-temp er atur e auto c orr elation function .— As a represen tative case, w e consider the infinite-temp erature auto correlation function of the edge op erator χ , B ( t ) ≡ ⟨ χ ( t ) χ (0) ⟩ ∞ , (21) with χ ( t ) = e t L † [ χ ] and ⟨·⟩ ∞ ≡ tr( · ) / tr( 1 ), where L † is the adjoin t of the Liouvillian giv en in Eq. ( 28 ). Ex- tending the Krylov-space metho d of Refs. [ 12 , 67 ] to the b oundary-driv en Liouvillian (see End Matter for the deriv ation), we obtain B ( t ) = − α e G X k =1 P G \ K s ( e u 2 k ) e u k e P + ′ G ( e u k ) exp[ − 2( γ − i e ε k ) t ] , (22) where e P + ′ G ( u ) ≡ ∂ u e P + G ( u ). 5 F or the uniform b oundary-driv en F endley mo del, the long-time asymptotics in the thermo dynamic limit is B ( t ) ∼ γ 2 ( γ 2 + 3) (1 + γ 2 ) 2 exp  − 2 γ 1 + γ 2 t  , t ≫ 1 . (23) The relaxation rate 2 γ 1+ γ 2 increases monotonically for γ < 1. How ever, for γ > 1, it decreases monotonically with γ , whic h is a typical signature of the contin uous Quan tum Zeno effect [ 68 , 69 ]. W e also note that, in the closed- system limit γ = 0, this exp onen tial decay is replaced by the algebraic tail B ( t ) ∼ t − 2 / 3 [ 12 ]. Conclusion .— By extending the FFD solv abilit y to op en quan tum systems, we established a new class of exactly solv able Lindbladians that are not reducible to quadratic forms in Jordan–Wigner fermions, distinct from the third-quan tization approach to quadratic open systems [ 36 , 37 ]. F or generic couplings, any ev en-hole- free, claw-free frustration graph yields an exactly solv- able Lindbladian once an edge op erator asso ciated with a simplicial clique is taken as the jump operator. The exact solution ( 18 ) enables the calculation of the Liou- villian gap and yields a closed-form expression for the infinite-temp erature auto correlation function ( 22 ). More generally , the Liouvillian remains free-fermion solv able whenever its frustration graph is claw-free and has a simplicial clique [ 9 ]. F or example, adding a sec- ond jump operator at the opp osite edge of the boundary- driv en F endley mo del creates ev en holes in the Liouvillian frustration graph, but the resulting graph remains claw- free and has a simplicial clique, so the Liouvillian is still free-fermion solv able [ 70 ]. The same criterion allows ex- actly solv able dissipativ e extensions of other FFD mo dels whose frustration graphs are claw-free and ha ve a simpli- cial clique, including the Kitaev honeycomb mo del [ 23 ]. K.F. was supp orted by JSPS KAKENHI Grant No. JP25K23354. H.K. w as supported b y JSPS KAK- ENHI Gran ts No. JP23K25783 and No. JP23K25790. K.F. and H.K. were supp orted by MEXT KAKENHI Gran t-in-Aid for T ransformative Research Areas A “Ex- treme Universe” (KAKENHI Grant No. JP21H05191). ∗ k ohei.fuk ai@ph ys.s.u-tokyo.ac.jp [1] M. Rigol, V. 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In the earlier FFD literature [ 5 , 8 , 9 ], coprimal- it y app ears to be assumed implicitly; when P G and P G \ K s share a ro ot x k = u 2 k , the normalization factor N k in Eq. ( 8 ) v anishes, but this degenerate case do es not app ear to be addressed. Such a coincidence can be engineered by fine tuning: when G is the path graph 1–2–3–4 with uni- form couplings and K s = { 2 , 3 } , P G ( x ) = 1 − 4 x +3 x 2 and P G \ K s ( x ) = (1 − x ) 2 share the ro ot x = 1. At a shared ro ot x ∗ = u 2 ∗ , e P ± G ( ± u ∗ ) = 0 and the single-particle en- ergies e ε = ± 1 /u ∗ are real, leading to nonzero Liouvillian eigen v alues on the imaginary axis. The resulting non- deca ying oscillatory sector, constrained by Lindbladian symmetries and in v ariant subspaces [ 73 – 76 ], can produce p ersisten t oscillations in the long-time dynamics [ 77 – 79 ]. END MA TTER Defining r epr esentation of the gr aph-Cliffor d algebr a .— Cho ose V ( G ) = { 1 , . . . , M } with M ≡ | V ( G ) | . Ref. [ 19 ] giv es a canonical op erator-state corresp ondence that re- alizes the abstract graph-Clifford algebra on a spin-1 / 2 c hain of length M . One of the t wo defining spin-chain represen tations is h j =   Y ℓ