Mixed-State Topological Phase: Quantized Topological Order Parameter and Lieb-Schultz-Mattis Theorem

Mixed-State T opological Phase: Quantized T opological Order Parameter and Lieb-Schultz-Mattis Theorem Linhao Li 1 and Y uan Y ao 2 , ∗ 1 Department of Physics, the P ennsylvania State University , University P ark, P ennsylvania 16802, USA 2 School of Physics and Astronomy , Shanghai Jiao T ong University , Shanghai 200240, China (Dated: Thursday 26 th March, 2026) W e in vestigate the e xtension of pure-state symmetry protected topological phases to mixed-state re gime with a strong U(1) and a weak Z 2 symmetries in one-dimensional spin systems by the concept of quantum chan- nels. W e propose a corresponding topological phase order parameter for short-range entangled mixed states by sho wing that it is quantized and its distinct values can be realized by concrete spin systems with disorders, sharply signaling phase transitions among them. W e also gi ve a model-independent way to generate two distinct phases by v arious types of translation and reflection transformations. These results on the short-range entangled mixed states further enable us to generalize the con ventional Lieb-Schultz-Mattis theorem to mixed states, e ven without the concept of spectral gaps and lattice Hamiltonians. Introduction. — Distinguishing and characterizing various quantum phases is a central subject in condensed matter and statistical physics. Symmetry has played an essential role in the phase identifications, such as the con ventional Landau-Ginzburg-W ilson spontaneously symmetry breaking framew ork [1]. Significant progress has been made beyond the Landau paradigm, e.g., the symmetry-protected topological (SPT) phases [2 – 7] where the ground state can be adiabatically deformed into product states by a finite-depth local unitary cir- cuit (FDLUC) — so-called short-range entangled (SRE) states [8], while such a deformation is forbidden in the presence of certain symmetry . Another notable example is the Lieb–Schultz–Mattis (LSM) theorem [9] and its generalizations [10 – 23], which exclude a SPT ground state when the system satisfies nontrivial symmetry data. The abo ve inv estigations of quantum phases focus on closed systems and characterize gapped phases by their ground states. Howe ver , disorders and decoher - ence are inevitable in real experiments, which leaves the systems as mixed states [24–32]. In this context, the familiar SRE structure is generalized to density ma- trices ρ . Correspondingly , the concept of symmetry is also extended; some symmetry G , called “weak sym- metry”, may not be respected by each single component of ρ , but it can be av eragely preserved by the entire en- semble g ρ = ρg, ∀ g ∈ G . In contrast, the so-called “strong symmetry” K is respected by every component of ρ . Great ef forts have been made for the mixed-state topological phases [33–42], e.g., average SPT (ASPT) phases [43 – 51], and LSM-type arguments [52 – 55]. Nev ertheless, it is challenging to diagnose and sharply characterize mixed-state SRE (mSRE) phases in the ab- sence of local order parameters. In analog to pure- state phase classifications, certain nonlocal string-order parameters may acquire nonzero expectation values in ASPT phases [47]. Ho wever , these signals can become arbitrarily small, prev enting a precise sharp distinction among phases. Furthermore, these string-order param- eters are strongly model-dependent and there is no uni- versal way to construct a suitable string operator based on ρ . Therefore, a model-independent topological or- der parameter (i) applicable to topological phases of, if exists, mixed states and (ii) sharply characterizing dis- tinct phases, e.g., quantized discretely v alued, remains an open question. Finally , a rigorous and systematic ex- tension of the LSM theorem, which forbids mSRE states by certain symmetry constraints, is still lacking, partially due to the absence of a commonly accepted definition of “gap” for mixed states. In this letter, we propose the nonlocal order param- eter T r ( ρU ) , where U ≡ exp  2 π i L P L j =1 j S z j  , to dis- tinguish and characterize mSRE phases in spin- S chains preserving strong U (1) and weak Z x 2 symmetry . The so- called twisting operator U was introduced in the proof of LSM theorem for pure states [9]. Recently , this or- der parameter was shown to be sharply quantized in the context of [ U (1) z ⋊ Z x 2 ] - SPT classification [56]. W e rig- orously prove that it can be generalized to mixed-state topological phases, which is not obvious as it appears since the previous pure-state theorem strongly relies on the lattice Hamiltonian and the gap concept. Our pow- erful theorem also enables us to extend LSM theorem to mixed states with weak (magnetic) translation symmetry or weak (magnetic) site-centered reflection symmetry . Preparations and the definitions —W e consider a spin- S chain with length L under periodic boundary condition (PBC) and focus on [ U (1) z ⋊ Z x 2 ] spin-rotation symmetry where U (1) z and Z x 2 are generated by S z tot and π -rotation R π x . S z tot = L X j =1 S z n ; R π x : R π x S z n ( R π x ) † = − S z n , (1) 2 where R π x can be interpreted as a π -rotation around x - axis. In the language of Hamiltonians, it was proven that the unique gapped ground state | G.S. ⟩ of a [ U (1) z ⋊ Z x 2 ] - symmetric SPT Hamiltonian satisfies ⟨ G.S. | U | G.S. ⟩ = ± 1 + O (1 /L ) → ± 1 corresponding to two distinct [ U (1) z ⋊ Z x 2 ] -SPT phases [56]. Since there is no Hamiltonian or gap concept for mixed states, we first, still in the pure-state re gime, ex- tend the above kno wn result so that its real parent Hamil- tonian is not referred to, by the powerful matrix product state (MPS) representations [57 – 59]. The SRE pure state, e.g., | G.S. ⟩ , can be represented as an injectiv e MPS with a bond dimension D bounded as D ≤ (2 S + 1) rd when L is suf ficiently lar ge [3, 60]. r is the range of the quantum gates and d is the circuit depth to deform it to a product state. Lemma 1 : For an y injectiv e MPS | Ψ ⟩ ⟨ s 1 , · · · , s L | Ψ ⟩ = T r [ A [1] s 1 · · · A [ L ] s L ] , (2) which is G -symmetric under ∀ g ∈ G , X s U s ′ s ( g ) A [ n ] s = exp[ iα n ( g )] V n − 1 ( g ) A [ n ] s ′ V † n ( g ) , (3) where U s ′ s ( g ) ≡ ⟨ s ′ | ˆ U ( g ) | s ⟩ is the unitary ph ysical transformation ˆ U ( g ) on the physical degrees of freedom | s ⟩ , exp( iα n ) is the g -eigen value per site, and V n is, up to a gauge transformation, the induced effecti ve transfor- mation on the virtual degrees of freedom, we can con- struct a G -symmetric parent local Hamiltonian H which is g apped with the unique ground state | Ψ ⟩ , as a ( k + 1) - neighboring interaction type (where the finite k is the injectivity length [61] ): H = P n h n,n +1 , ··· ,n + k . Pr oof: W e explicitly construct the Hamiltonian term h n,n +1 , ··· ,n + k = 1 − P n,n +1 , ··· ,n + k , where P n,n +1 , ··· ,n + k the orthogonal projection operator of the Hilbert space spanned by states of the form of: X s n , ··· s n + k T r ( A [ n ] s n · · · A [ n + k ] s n + k O ) | s n , · · · s n + k ⟩ , (4) where O exhausts all D × D matrices, and h n, ··· ,n + k is identity when acted on the Hilbert space of all the other sites (not n, · · · , n + k sites). By the cyclic property of the trace, the symmetry transformation (3) is effecti vely O 7→ exp   i n + k X j = n α j ( g )   V † n + k ( g ) OV n − 1 ( g ) , (5) which is bijectiv e on the above Hilbert space spanned by Eq. (4), thereby leaving P n,n +1 , ··· ,n + k and h n, ··· ,n + k G -in variant. It was proved in Ref. [58] that H is indeed gapped with | Ψ ⟩ its unique ground state. □ The abov e explicit construction giv es the following Hamiltonian-free Theorem: Theorem 2: For any [ U (1) z ⋊ Z x 2 ] -symmetric SRE pure state | Ψ ⟩ , we hav e ⟨ Ψ | U | Ψ ⟩ = ± 1 + O (1 /L ) . (6) Generalization to mixed states — W e are ready to e x- tend the phase characterization from the pure state to the mixed state by density matrices and quantum channels. W e focus on ρ that preserves strong U (1) z symmetry but only weak Z x 2 symmetry: U (1) z : exp( iθ S z tot ) ρ = exp( iα θ ) ρ, θ ∈ [0 , 2 π ) and Z x 2 : R π x ρ = ρR π x , where α θ = 0 or π due to R π x symmetry . Actually α θ = α θ =0 = 0 by the continuity of the U (1) z group and then S z tot ρ = 0 . (7) In this work, we will use Γ to denote the full symmetry group of ρ , which includes both the strong and the weak symmetries. U (1) z will be always taken as a strong sym- metry , and Z x 2 as a weak symmetry . A natural generalization of FDLUC for the pure states is the finite depth local channel (FDLC) which describe a generic locality-preserving ev olution of mixed states. A general FDLC transformation on any gi ven initial mixed state ρ 0 can be constructed in the following steps [Hast- ings, 2011]: • (a) Introduce additional qubits as the en viron- ment ( E ) on each site i of the system ( S ) to define an enlarged Hilbert space H S i ⊗ H E i per site. The en vironment is initialized in some product state | 0 E ⟩ . • (b) Apply a FDLUC D S ∪ E to the total system S ∪ E . • (c) T race out the en vironment. Thus a FDLC denoted as N can be written as N [ ρ 0 ] = T r E { D S ∪ E [ ρ 0 ⊗ | 0 E ⟩⟨ 0 E | ] D † S ∪ E } . (8) Moreov er , this FDLC is called locally weak Z x 2 - symmetric [3, 55], if (i) | 0 E ⟩ is a Z x 2 -symmetric prod- uct state, R E | 0 ⟩ E ∝ | 0 ⟩ E , where R E is the Z x 2 genera- tor on the en vironment, and (ii) the layer decomposition D S ∪ E = Q n layer j =1 D ( j ) S ∪ E satisfies [ D ( j ) S ∪ E , R π x ⊗ R E ] = 0 for each layer j . Similarly , e.g., for a locally str ong U(1) z -symmetric FDLC, R E in (i) is replaced by the en vironment U(1) z operator U θ E with θ ∈ [0 , 2 π ) , and R π x ⊗ R E in (ii) by exp( iθ S z tot ) ⊗ 1 E with 1 E the envi- ronment identity operator . FDLC is also symmetry pre- serving; N [ ρ 0 ] is Γ -symmetric if ρ 0 is Γ -symmetric and N is locally Γ -symmetric. 3 As FDLC generalizes the concept of FDLUC, we have the following e xtension of SRE for mixed states: Definition: ρ is a Γ ′ -mSRE state , if ρ = N [ ⊗ j ρ j ] , where N is a locally Γ ′ -symmetric FDLC and ⊗ j ρ j is a product of Γ ′ -symmetric mixed states ρ j each of which can include se veral lattice sites around j coordinates forming a j -“cell” and the cells in the product ⊗ j ρ j hav e no ov erlap with each other . Generally , the symmetry Γ of a Γ ′ -mSRE ρ can be strictly larger than Γ ′ ; Γ ′ reflects ho w the ρ is prepared while Γ is the symmetry of the produced state ρ . No w , we are ready to present our main result: Theorem 3: If ρ respects strong U (1) z -symmetry and weak Z x 2 -symmetry , then T r ( ρU ) = ± 1 + O (1 /L ) , (9) as long as ρ is (weak Z x 2 )-mSRE state. Remark: ρ needs to be a (weak Z 2 )-mSRE state rather than a general mSRE state, in contrast to its pure- state version as Theorem 4 . The extension to mixed states requires this nontrivial enriched symmetry struc- ture. Pr oof: F ollo wing Eqs. (27-29) of Ref.[55] which prov es that any (weak Z x 2 )-symmetric product mixed state can be purified by introduction of ancilla (A): ⊗ j ρ j = T r A ( | ψ S ∪ A ⟩⟨ ψ S ∪ A | ) (10) where an ancilla local Hilbert space H j A is a copy of the original spin Hilbert space H j S at each cell j . The state | ψ S ∪ A ⟩ is a product state satisfying [ R π x ⊗ ( R π x ) ∗ ] | ψ S ∪ A ( L ) ⟩ = | ψ S ∪ A ( L ) ⟩ , (11) where the complex conjugate ( R π x ) ∗ is act- ing on the ancilla. Recalling ρ = N [ ⊗ ρ j ] = T r E { D S ∪ E [( ⊗ j ρ j ) ⊗ | 0 E ⟩⟨ 0 E | ] D † S ∪ E } and Eq. (10), we obtain the following purification ρ = T r E ,A ( | Φ ⟩⟨ Φ | ) , (12) | Φ ⟩ ≡ ( D S ∪ E ⊗ I A ) | ψ S ∪ A ⟩ ⊗ | 0 ⟩ E , (13) where I A is the identity on the ancilla. Obviously , the pure state | Φ ⟩ defined in the total Hilbert space H S ⊗ H A ⊗ H E is in v ariant under the Z x 2 symmetry with representation: R x π ⊗ ( R π x ) ∗ ⊗ R E | Φ ⟩ = | Φ ⟩ . (14) It also satisfies the following extended U (1) z symmetry [62]: exp( iθ S z ) ⊗ I E ∪ A | Φ ⟩ = | Φ ⟩ , (15) where the U (1) z is defined to be tri vial on the artifi- cial ancilla and en vironment, which is consistent with R π x S z n ( R π x ) † = − S z n , where S z E ,n = S z A,n ≡ 0 . By Eq. (13), Φ is an SRE pure state, and Theor em 2 gives ⟨ Φ | U S ∪ A ∪ E | Φ ⟩ = ± 1 + O (1 /L ) , (16) where U S ∪ A ∪ E ≡ exp " 2 π i L L X n =1 n ( S z n + S z A,n + S z E ,n ) # = U ⊗ I E ∪ A . (17) Due to Eqs. (12,17), we obtain T r ( ρU ) = Tr S ∪ A ∪ E ( | Φ ⟩⟨ Φ | U S ∪ A ∪ E ]) = ⟨ Φ | U S ∪ A ∪ E | Φ ⟩ = ± 1 + O (1 /L ) , (18) which completes the proof of Theorem 3 . □ Mixed-state topological phase and topological order parameter — The classification of topological phases can be extended to the mix ed-state regime as follo ws. Definition: (Mixed-state topological phase) T wo den- sity Γ -symmetric matrices ρ 1 , ρ 2 are said to belong to the same Γ -symmetric phase, called Γ -symmetric phase, if they are two-way connected by a pair of locally Γ - symmetric FDLCs [33] The discretely v alued ⟨ U ⟩ = T r ( ρU ) as L → ∞ im- plies that it can be used to distinguish at least two distinct mixed-state topological phases. It can be seen by the fol- lowing continuity consideration. Let us assume there is a FDLC connecting ρ in and ρ out satisfying the condition of Theorem 3 . In the aforementioned general construction step (b) of FDLC, the FDLUC therein can be simulated with a continuous unitary ev olution by a local Hamil- tonian for a finite time t ∈ [0 , 1] [3]. Therefore, if this unitary ev olution is from t = 0 only until t = s followed by the step (c) of tracing the en vironment out, we hav e a continuous s -family [63] of FDLCs and thereby a s - parameterized density matrices ρ s with ρ s =0 = ρ in and ρ s =1 = N [ ρ in ] = ρ out . The simulating unitary ev olution until any s also preserves [ U (1) z ⋊ Z x 2 ] symmetry since the simulating local Hamiltonian, e.g., i log( D ( j ) S ∪ E ) for each layer j , preserves [ U (1) z ⋊ Z x 2 ] . Thus, ρ s satisfies the condition of Theorem 3 for any s ∈ [0 , 1] , which enables us to calculate and conclude that I s ≡ lim L →∞ T r ( ρ s U ) ∈ {± 1 } , (19) which must be a constant function about s since it is discretely ( ± 1) -v alued and s -continuous. Particularly , I s =1 = I 0 , which shows that, con versely , different val- ues of I ≡ lim L →∞ ⟨ U ⟩ , as a quantized topological or - der parameter , sharply signal distinct mixed-state matter of phases [64]. 4 Let us confirm our rigorous statement by illustrating examples. Both I = ± 1 can be indeed realized on lat- tices by the follo wing exactly solvable model. The clean Hamiltonian is a spin-1/2 chain H 0 = − P N n =1  S 2 n ·  S 2 n +1 , and the disorder is ∆ H ( B ) = N X n =1 2 h n ( − S z 2 n +1 + S z 2 n +2 ) , (20) where h n are independent random variable ( B ≥ 0) : P ( h n = B ) = 50% , P ( h n = − B ) = 50% . (21) By a transferred matrix technique [65], we find two phases separated by B c = 1 : [up to O (1 /L ) ] ⟨ U ⟩ = ( − 1 + π 2 8 L (2 + ∆ 2 B ) , 0 ≤ B < B c ; 1 − π 2 8 L (∆ 2 B + 8∆ B + 9) , B > B c , (22) where ∆ B ≡ 4 B 1+ √ 1+4 B 2 + 4 B 2 1+ √ 1+4 B 2 . T w o states at B = 0 and B → ∞ are both (weak Z x 2 )-mSRE [65], and we find two mixed-state realizations predicted by Theorem 3 . Although Eq. (22) is analytically true, a di- rect numerical simulation in FIG. 1 illustrates the sharp topological phase transition at B = B c . B FIG. 1: Numerical calculation of ⟨ U ⟩ with v arying L and B identifies two topological phases: (i) I = − 1 when 0 ≤ B < 1 , and (ii) I = +1 when B > 1 . Actually , there is a model-free understanding of exis- tence of tw o phases by various translation symmetries as shown belo w . Lieb-Schultz-Mattis (LSM) theorem for the mixed state — W e observe that both two distinct mixed-state phases abov e fail to preserve weak translation symme- try . Actually , the translation symmetry can be shown to “connect” distinct phases: Theorem 4: If a half-integral spin chain is in a (weak Z 2 )-mSRE state ρ respects strong U (1) z -symmetry and weak Z x 2 -symmetry , then I [ ρ ] = −I [ T ρT − 1 ] = −I [ ˜ T ρ ˜ T − 1 ] = −I [ R ρR − 1 ] = −I [ ˜ Rρ ˜ R − 1 ] . (23) Here T is the lattice translation symmetry and R is the site-centered reflection symmetry . ˜ T is a general magnetic translation and ˜ R is a general magnetic site- centered reflection : ˜ T = T ◦ Q ◦ Θ , ˜ R = R ◦ Q ◦ Θ , (24) where Θ is the con ventional time-rev ersal and Q is any unitary onsite operator that commutes with S j z . Pr oof: After applying the translation operation to ρ , we obtain T r ( T ρT − 1 U ) = Tr  ρU exp(2 π iS z 1 ) exp( − 2 π i L S z tot )  = ( − 1) 2 S T r ( ρU ) = − Tr ( ρU ) (25) where we have used Eq. (7) that S z tot ρ = 0 due to the weak Z x 2 symmetry . Next, without loss of generality , we consider a reflec- tion R centered at site L/ 2 [66]. This yields T r ( RρR − 1 U ) = Tr  ρU † exp(2 π iS z L )  = ( − 1) 2 S T r ( ρU † ) = − T r ( ρU ) , (26) where we hav e used that ρ respects weak Z x 2 symmetry to deduce T r ( ρU † ) = T r ( ρU ) . The translation transformation T and reflection R abov e can be replaced by ˜ T and ˜ R respectively with no substantial change to the proof. □ Theorem 4 directly gives a realization of two phases once we are given one state ρ in whichever phase. It also naturally implies the following mixed-state version of the Lieb–Schultz–Mattis (LSM) theorem: Corollary 5: (Mixed-state LSM theor em) A mixed state ρ of a spin chain respects strong U (1) z -symmetry , weak Z x 2 -symmetry and weak (magnetic) translation symmetry or weak (magnetic) site-centered reflection symmetry . Then it cannot be a weak Z x 2 -mSRE state, if the spin chain has half-integral spin per unit cell. □ Corollary 5 is thus an extension of the LSM theorem not only to mixed states but also to translation symmetry beyond the unitary case. The original LSM proof relies on the unitarity of translation to define a well-defined lat- tice momentum, which is unav ailable in the antiunitary case. Furthermore, compared with Ref. [55], which also discusses the LSM theorem for mixed states, our result does not require the state to be a (weak T )-mSRE state 5 and includes the case of magnetic translations and (mag- netic) reflections, so it is strictly stronger than the result in Ref. [55]. Illustrating examples of the mixed-state LSM theo- rem — Let us explicitly see a mixed state model as an application of the LSM theorem. W e consider the 1d chiral scalar triple-product spin model: H ch = X r ( − 1) i J  S i · (  S i +1 ×  S i +2 ) . (27) with ground state | G.S. ⟩ and the channel ∀ p ∈ [0 , 1] N = ⊗ i N z i , N z i [ ρ ] = (1 − p ) ρ + 4 pS z i ρS z i (28) The Hamiltonian and the channel preserve the strong- U (1) z and weak- Z x 2 symmetry since S z i in the channel commutes with U (1) z symmetry and anticommutes with R x π . Moreover , this model also preserves a magnetic translation symmetry: ( T ◦ Θ)  S r = −  S r +1 ( T ◦ Θ) . (29) Therefore, Corollary 5 implies that strong U (1) z - and weak Z 2 -symmetric N [ | G.S. ⟩⟨ G.S. | ] cannot be a weak Z 2 -mSRE state. Actually , there is a model-dependent proof [65], where we show the spin–spin correlation function decays as a power la w . Conclusions and discussions — W e propose a quan- tized topological phase order parameter to distinguish different mSRE states and such an order parameter is model-independent and can sharply detect v arious topo- logical phase transitions, for which we present an ex- actly solvable model. Our results enable us to extend the LSM theorem to mixed states without the concept of spectral gaps. The generalization to SU(N) spin systems is also straightforward and natural [56]. As another application of Corollary 5 for mSRE state, we can conclude that if an mSRE state ρ re- spects strong U (1) z -symmetry , weak Z x 2 -symmetry and weak (magnetic) translation symmetry or weak (mag- netic) site-centered reflection symmetry . then ρ cannot be a (weak Z x 2 )-mSRE. Thus, it must be in a nontrivial weak- Z x 2 symmetric topological phase if the spin chain has half-integral spin per unit cell, since there is no two- way locally (weak Z x 2 )-symmetric FDLCs connecting ρ and a Z x 2 -symmetric product mixed state. A lattice con- struction of such a nontri vial phase along this direction is left for future work. Ackno wledgments. — The authors thank Akira Fu- rusaki for useful discussions. The w ork of Y . 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Phys. 93 , 045003 (2021). [60] M. Fannes, B. Nachter gaele, and R. F . W erner, Commun. Math. Phys. 144 , 443 (1992). [61] Injectivity length k refers to the smallest integer such that the tensors A [ n ] s n · · · A [ n + k ] s n + k is injective when viewed as a linear transformation C D → C d k × C D , where D and d are the dimensions of virtual space and physical space, respectiv ely . [62] This can be easily verified by performing the Schmidt de- composition of | Φ ⟩ under the bipartition S ∪ AE . [63] It is generally not s -differentiable for multi-layer cases. [64] W e note that a refined definition of mixed-state phase equiv alence—based on one-way connectivity via locally rev ersible channel circuits—was recently introduced in Ref. [36], motiv ated by the existence of a counterexam- ple in two-dimensional classical statistical mechanics un- der the original two-way connectivity definition. It has been sho wn in [34] that two states being equiv alent un- der this refined definition implies their equi valence under two-way connectivity . Thus two states with distinct val- ues of I must also belong to distinct mixed-state matter of phases ev en under this refined definition. [65] See Supplemental Materials. [66] The length L must be ev en due to S z tot ρ = 0 . [67] P . Schmoll, A. Haller, M. Rizzi, and R. Or ´ us, Phys. Rev . B 99 , 205121 (2019). 7 Mixed topological phases and phase transitions W e consider a spin-1/2 chain with an ev en length L = 2 N under the periodic boundary condition (PBC). The clean Hamiltonian is the dimer model: H 0 = − N X n =1 (2 P 2 n, 2 n +1 − 1) , (30) where P 2 n, 2 n +1 is the singlet projection operator on the sites 2 n and 2 n + 1 , and H 0 has the unique ground state as dimer formed by (2 n, 2 n + 1) sites. W e add the follo wing disorder H ′ ( B ) = N X n =1 2 h n ( − S z 2 n +1 + S z 2 n +2 ) , (31) where h n are independent random variable with B ≥ 0 : P ( h n = B ) = 50% , P ( h n = − B ) = 50% . (32) W e define each “cell” j as 2 n + 1 and 2 n + 2 lattice points and then the concept of product state in Theorem 5 can be defined. At B = 0 , the ground state of the system can be prepared from a Z x 2 symmetric product state | →← · · · →←⟩ which is obviously a product of mixed (actually pure) states of each cell, by the following Z x 2 symmetric FDLUC Q n U 2 n, 2 n +1 where U 2 n, 2 n +1 = 1 √ 2 ( | →←⟩ − | ←→⟩ ) ⟨→← | + 1 √ 2 ( | →←⟩ + | ←→⟩ ) ⟨←→ | + | ←←⟩⟨←← | + | →→⟩⟨→→ | . (33) At B = ∞ , the density matrix is already a product mixed state under the current definition of cells. Let us calculate the av erage v alue of the twisting operator U = exp 2 π i L L X n =1 nS z n ! , (34) and a local staggering magnetism: M s ≡ − 4 N N X n =1 S z 2 n S z 2 n +1 . (35) Calculation of a general local operator Let us calculate the average v alue of a general product operator ⊗ N n =1 O n , in which O n includes the Hilbert space at the sites 2 n and 2 n + 1 . Obviously , the Hamiltonian, at any arbitrary realization H 0 + H ′ ( B ) , has a ground state as a product of states Ψ n which stays within the Hilbert space of the sites 2 n and 2 n + 1 . Focusing on this pair of sites, we define the “disorder” pseudospin space ( a ) p,q = h n − 1 + h n =  2 B 0 0 − 2 B  p,q , (36) for 4=2 × 2 possibilities corresponding to  h n − 1 = h n = B , h n − 1 = − h n = B − h n − 1 = h n = B , h n − 1 = h n = − B  (37) 8 and we denote the four-dimensional local Hilbert space as | ϕ ⟩ = | s = 0 , s z = 0 ⟩ ; | ψ m ⟩ = | s = 1 , s z = m ⟩ ( m ∈ { 0 , ± 1 } ) . (38) Since H 0 + H ′ ( B ) preserv es U (1) z , only | ϕ ⟩ and | ψ 0 ⟩ can be mix ed with each other . W e simply separate se veral special cases: • 0 ≤ B < 1 The four ground states corresponding to the abov e four possibilities can be put into a 2 × 2 matrix: (Ψ n ) p,q = 1 r 1 + a 2 p,q (1+ √ a 2 p,q +1) 2   | ϕ ⟩ + − a p,q (1 + q a 2 p,q + 1) 2 | ψ 0 ⟩   , ( p, q ∈ { 1 , 2 } ) . (39) • B > 1 (Ψ n ) 2 , 1 = | ψ + ⟩ , (Ψ n ) 1 , 2 = | ψ − ⟩ , (40) while (Ψ n ) 1 , 1 and (Ψ n ) 2 , 2 still has the same expression as in Eq. (39). Therefore, ⟨⊗ n O n ⟩ = 1 2 N T r N Y n =1 ⟨ Ψ n | O n | Ψ n ⟩ , (41) where the matrix ( ⟨ Ψ n | O n | Ψ n ⟩ ) p,q ≡ ⟨ (Ψ n ) p,q | O n | (Ψ n ) p,q ⟩ (42) A verage value of the twisting operator T aking O n = exp  2 π i L [2 nS z 2 n + (2 n + 1) S z 2 n +1 ]  , (43) then, due to the difference at the boundary , U = ( − 1) N Y n =1 O n . (44) we obtain that • 0 ≤ B < 1 ⟨ Ψ n | O n | Ψ n ⟩ = ⟨ Ψ n | exp  2 π i L S z 2 n +1  | Ψ n ⟩ =  cos π L + i ∆ B sin π L cos π L cos π L cos π L − i ∆ B sin π L  , (45) since S z 2 n + S z 2 n +1 = 0 , where ∆ B ≡ 4 B 1+ √ 1+4 B 2 + 4 B 2 1+ √ 1+4 B 2 , and we hav e also used the fact that S z 2 n +1 = − τ x 2 , (46) exp  2 π i L − τ x 2  = cos π L − iτ x sin π L , (47) 9 under the two-dimensional basis {| ϕ ⟩ , | ψ 0 ⟩} . Diagonalizing the n -independent ⟨ Ψ n | O n | Ψ n ⟩ gi ves the analytical result ⟨ U ⟩ = ( − 1) X η = ± 1  1 2  cos π L + η r cos 2 π L − ∆ 2 B sin 2 π L  L/ 2 = − 1 + π 2 8 L (2 + ∆ 2 B ) + O (1 /L 2 ) . (48) • B > 1 ⟨ Ψ n. | O n | Ψ n ⟩ = ⟨ Ψ n. | exp  2 π i L S z 2 n +1  | Ψ n ⟩ =  cos π L + i ∆ B sin π L exp  − 2 π i L  2 n + 1 2  exp  2 π i L  2 n + 1 2  cos π L − i ∆ B sin π L  . (49) Thus, after using the identity to eliminate the n -dependence within each multiplier T r N Y n =1  x + iy z exp( − inθ ) z exp( inθ ) x − iy  = T r (  exp( − i σ z 2 θ )  x + iy z z x − iy  N exp  i σ z 2 N θ  ) , (50) we obtain the analytical result ⟨ U ⟩ = X η = ± 1 " 1 2 cos π L cos 2 π L − ∆ B sin π L sin 2 π L + η s 1 −  cos π L sin 2 π L + ∆ B sin π L cos 2 π L  !# L/ 2 = 1 − π 2 8 L (∆ 2 B + 8∆ B + 9) + O (1 /L 2 ) , (51) where the “ ( − 1) ” in Eq. (44) is cancelled by exp( iN θ σ z / 2) in Eq. (50) because θ = 4 π /L = 2 π / N . Calculation of staggering magnetism Due to the weak translation symmetry , ⟨ M s ⟩ = ⟨− 4 S z 2 S z 3 ⟩ . (52) W e notice that under the s z = 0 basis {| ϕ ⟩ , | ψ 0 ⟩} , − 4 S z 2 S z 3 =  1 1 1 1  . • 0 ≤ B < 1 ⟨ M s ⟩ = 1 2 N T r  1 1 1 1  N = 1 . (53) • B > 1 ⟨ M s ⟩ = 1 2 N T r "  1 − 1 − 1 1   1 1 1 1  N − 1 # = 0 . (54) 10 The abo ve calculations show that the transition at B c = 1 is of first order . Thus, B = B c is a singular point leading to discontinuous jumps of various quantities. These two phases correspond to two average SPT (ASPT) phases protected by the U (1) z ⋊ Z x 2 symmetry [47], which generalizes the notion of SPT phases to mixed states. The classification is given by the group cohomology H 2 ( U (1) z , H 1 ( Z 2 , Z x 2 )) = Z 2 . In the case of the onsite U (1) z ⋊ Z x 2 symmetry , the corresponding definition of “cell” will define the concept of “tri vial” and “nontri vial” phases as follows. Each cell is required to form a U (1) z ⋊ Z x 2 representation, so it must consist of ev en number of spin-1/2’ s. There are two typical definitions of the cell: (a) each cell by (2 n ) and (2 n + 1) sites; (b) each cell by (2 n + 1) and (2 n + 2) sites. In the e xample abo ve, at B = 0 the system is in a pure state as a trivial U (1) z ⋊ Z x 2 ASPT phase under the definition (a) since it is a product state while nontrivial under the definition (b). By contrast, at B = ∞ the system reduces to a product mixed state under the definition (b) thereby belonging to the tri vial ASPT phase. Ho wever , under the definition (a) and its different topological order parameter ⟨ U ⟩ from B = 0 , it must belong to the nontrivial ASPT phase. Therefore, for our current lattice models without any interest in defining the artificial choice of the cell, only the distinction of topological phase is practically essential while which one is trivial or nontri vial is not substantial. Here we remark that the mixed state at B = ∞ indeed satisfies a quantum information definition of ASPT in Ref. [47] where an ASPT state is defined as a symmetrically in vertible gapped state. First, the “gapped” property of ρ ( B = ∞ ) = ⊗ n 1 2 ( | ↑↓⟩⟨↑↓ | + ( | ↓↑⟩⟨↓↑ | ) 2 n − 1 , 2 n follows from the f act that its conditional mutual information vanishes: I ( A, B , C ) = S AB + S B C − S B − S AB C = 0 (55) for any tripartition A, B , C of the chain and S M is the entropy of the reduced density matrix in the region M . Thus the state contains no long-range correlations in the quantum-information sense. Second, symmetric in vertibility requires the existence of an auxiliary mixed state ˜ ρ on an auxiliary Hilbert space such that one can define a diagonal onsite U (1) z ⋊ Z x 2 generated by S z tot ⊗ ˜ S z tot and R π π ⊗ ˜ R π π where the tilde operators satisfy the same group relations. The condition is that ρ ⊗ ˜ ρ is two way connected to a pure product through some locally symmetric finite-depth channels. For ρ ( B = ∞ ) , we can sho w this property by considering two copies: ρ ( B = ∞ ) ⊗ ρ ( B = ∞ ) and apply the one-layer unitary circuit: U 1 = Y n 1 +  σ 2 n ·  ˜ σ 2 n − 1 2 (56) which swaps the physical spin at site 2 n site and the auxiliary spin on 2 n − 1 site. This unitary transformation preserves the diagonal U (1) z ⋊ Z x 2 symmetry . After this transformation, the state becomes a product state state ρ ′ = ⊗ n ρ ′ n = ⊗ n 1 2  | ↑ ˜ ↓⟩⟨↑ ˜ ↓| + ( | ↓ ˜ ↑⟩⟨↓ ˜ ↑|  n for either choice (a) or (b) of the unit cell. On each site n , this mixed state is two way connected to the single state | ψ − ⟩ = 1 √ 2 ( | ↑ ˜ ↓⟩ − | ↓ ˜ ↑⟩ ) via symmetric finite-depth channels. Explicitly , N 12 [ | ψ − ⟩⟨ ψ − | ] = ρ ′ n , K 12 0 = I , K 12 1 = σ z , N 21 [ ρ ′ n ] = | ψ − ⟩⟨ ψ − | , K 21 0 = | ψ − ⟩⟨↑ ˜ ↓| , K 21 1 = | ψ − ⟩⟨↓ ˜ ↑| , K 21 2 = | ↑ ˜ ↑⟩⟨↑ ˜ ↑| , K 21 3 = | ↓ ˜ ↓⟩⟨↓ ˜ ↓| (57) where K are the corresponding Kraus operators. Thus ρ ′ i locally-symmetrically two-way connected to a product of singlet state a product of singlet states, which establishes the symmetric in vertibility of ρ ( B = ∞ ) . W e therefore include that ρ ( B = ∞ ) realizes an ASPT state in the sense of Ref. [47]. In conclusion, we ha ve the statement: If ρ belongs to a [ U (1) z ⋊ Z x 2 ] ASPT phase with strong U (1) z and weak Z x 2 symmetry , then T r ( ρU ) = ± 1 + O (1 /L ) and its thermodynamic limit identifies two distinct ASPT phases. Spin–spin correlation in chiral scalar triple-pr oduct model W e consider chiral scalar triple-product spin model H ch = X r ( − 1) i J  S i · (  S i +1 ×  S i +2 ) . (58) 11 with ground state | G.S. ⟩ and the channel N = ⊗ i N z i , N z i [ ρ ] = (1 − p ) ρ + 4 pS z i ρS z i (59) When p = 0 , i.e., N [ | G.S. ⟩⟨ G.S. | ] = | G.S. ⟩⟨ G.S. | , an iDMRG study [67] shows that it exhibits critical behavior which belongs to the same uni versality class as the spin-1/2 Heisenber g chain. In particular , the spin–spin correlation function decays as a power la w ⟨ G.S. | S z i S z j | G.S. ⟩ ∼ (log | i − j | ) 0 . 5 | i − j | (60) When the channel is included with nonzero p , the spin–spin correlation function remains in variant: T r ( N [ | G.S. ⟩⟨ G.S. | ] S z i S z j ) = T r ( N [ | G.S. ⟩⟨ G.S. | S z i S z j ]) = ⟨ G.S. | S z i S z j | G.S. ⟩ (61) where we use the fact that the channel commutes with S z i . Hence, the spin–spin correlation function always exhibits power -law decay , independent of p . For such a mixed state, we can pro ve that it cannot be weak Z x 2 -mSRE by contradiction without inv oking Corol- lary 7 as follows. Suppose that N [ | G.S. ⟩⟨ G.S. | ] = T r E { D S ∪ E [ ρ 0 ⊗ | 0 E ⟩⟨ 0 E | ] D † S ∪ E } (62) Then, if the distance between i and j is sufficiently large, we obtain T r S ( N [ | G.S. ⟩⟨ G.S. | ] S z i S z j ) = T r S E { D S ∪ E [ ρ 0 ⊗ | 0 E ⟩⟨ 0 E | ] D † S ∪ E S z i S z j } = T r S E { [ ρ 0 ⊗ | 0 E ⟩⟨ 0 E | ] ˜ S z i ˜ S z j } = T r S E { [ ρ 0 ⊗ | 0 E ⟩⟨ 0 E | ] ˜ S z i } T r S E { [ ρ 0 ⊗ | 0 E ⟩⟨ 0 E | ] ˜ S z j } =0 (63) where ˜ S z i is D † S ∪ E S z i D S ∪ E is a local operator that is odd under R x π ⊗ R E . Here we also use the fact that ρ 0 is a product of weak Z x 2 -symmetric mixed states. This result contradicts with the power -law decaying behavior in Eq.(61), which completes the proof.