Intertwining Markov Processes via Matrix Product Operators

In tert wining Mark o v Pro cesses via Matrix Pro duct Op erators Rouv en F rassek, 1 , ∗ Jan de Gier, 2 , † Jimin Li, 3 , ‡ and F rank V erstraete 3, 4 , § 1 University of Mo dena and R e ggio Emilia, Dep artment of Physics, Informatics and Mathematics, Via G. Campi 213/b, 41125 Mo dena and INFN, Sezione di Bolo gna, Via Irnerio 46, 40126 Bolo gna, Italy 2 Scho ol of Mathematics and Statistics, The University of Melb ourne, VIC 3010, Austr alia 3 Dep artment of Applie d Mathematics and The or etic al Physics, University of Cambridge, Wilb erfor c e R oad, Cambridge CB3 0W A, Unite d Kingdom 4 Dep artment of Physics and Astr onomy, Ghent University, Krijgslaan 281, S9, B-9000 Ghent, Belgium Dualit y transformations rev eal unexpected equiv alences betw een seemingly distinct models. W e in tro duce an out-of-equilibrium generalisation of matrix product op erators to implement duality transformations in one-dimensional boundary-driven Mark ov processes on lattices. In con trast to lo cal dualities asso ciated with generalised symmetries, here the duality op erator intert wines tw o Mark ov processes via generalised exc hange relations and realises the out-of-equilibrium dualit y glob- ally . W e construct these op erators exactly for the symmetric simple exclusion pro cess with distinct out-of-equilibrium boundaries. In this case, out-of-equilibrium boundaries are dual to equilibrium b oundaries satisfying Liggett’s condition, implying that the Gibbs–Boltzmann measure captures out- of-equilibrium physics when lev eraging the duality op erator. W e illustrate this principle through ph ysical applications. In tro duction. —Dualities hav e long b een used to de- fine equiv alences betw een models and hav e found wide applications in the theory of strongly correlated man y- b ody systems in equilibrium, ranging from the iden tifi- cation of critical p oin ts in lattice mo dels of statistical mec hanics to top ological phases of interacting quantum spin c hains [1–7]. Similarly , v arious dualities for classi- cal out-of-equilibrium many-bo dy systems, describ ed by con tinuous-time Marko v pro cesses, ha ve b een appreci- ated for decades—for example, the self-dualities of ex- clusion pro cesses [8–11] and rev erse duality [12]; see also the recent systematic approac hes using Lie algebras [13], Hec ke algebras and Macdonald polynomials [14], and solv able vertex mo dels [15]. The study of suc h equiv a- lence relations not only provides mathematical elegance but also leads to substantial improv ements in computa- tional efficiency [2]. In this Letter, we explore dualit y transformations of Mark ov pro cesses through the lens of tensor netw orks. T ensor netw orks provide a p ow erful framework for un- derstanding non-trivial correlations in strongly in ter- acting man y-b ody systems [16]. F or instance, using to ols from quantum information theory , it has b een sho wn that the ground states of large classes of one- dimensional lo cal Hamiltonians, H = P i h i,i +1 , are not structureless w av efunctions in the underlying Hilb ert space. Rather, they reside in a corner of the ex- p onen tially large Hilb ert space and are efficiently de- scrib ed b y matrix pro duct states (MPS) of the form Notes. —The authors of this paper were ordered alphab etically . ∗ rfrassek@unimore.it † jdgier@unimelb.edu.au ‡ Corresponding Author: jl939ph ysics@gmail.com § fv285@cam.ac.uk P { τ n } T r ( A τ 1 1 · · · A τ N N ) | τ 1 . . . τ N ⟩ , where A is a rank- three tensor. More sp ecifically , this class enjoys the frustration-free prop ert y h i,i +1 ( A i A i +1 ) = 0, which en- sures that the true many-bo dy ground state is annihi- lated b y every local term of the Hamiltonian. Exploit- ing the strong parallel b et w een quantum mechanics and Mark ov pro cesses [17], prop erties of MPS in quan tum systems often generalise directly to their stochastic coun- terparts, where the closest analogy for the frustration- free condition is detailed balance. Detailed balance is t ypically violated in one-dimensional b oundary-driv en lo- cal Marko v pro cesses, giving rise to richer phenomena than in their equilibrium counterparts [8]; the steady states are not described by the Gibbs–Boltzmann mea- sure and supp ort non-v anishing currents. The seminal work of Derrida, Ev ans, Hakim, and P asquier (DEHP) [18–21] iden tified a class of suc h Mark ov pro cesses in whic h detailed balance is strongly brok en, yet the exact steady states can still be con- structed using a matrix pro duct Ansatz (MP A). These steady states liv e in another corner of Hilb ert space com- pared to the ground states of frustration-free mo dels. This construction w as ac hieved b y imp osing the bulk cancellation mechanism h i,i +1 ( A i A i +1 ) = A i ¯ A i +1 − ¯ A i A i +1 , together with additional appropriate b oundary algebra, where ¯ A is another rank-three tensor. Each lo- cal term of the Marko v generator thereb y induces an al- gebraic relation when acting on the steady state, and summing ov er sites pro duces an exact cancellation. Similarly , matrix product descriptions of op erators ha ve b een widely employ ed in strongly correlated sys- tems, e.g., in the study of symmetries and dualit y trans- formations that intert wine the Hamiltonians of tw o dual mo dels. Duality op erators are ubiquitous, esp ecially in systems with generalised symmetries. Matrix pro duct op erators (MPO) provide a systematic framew ork for 2 h L L h L Z Z L A L Z Z L L B L A R B R L L L L V V V W W W I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k 5 P R I N p r o j e c t C UP - E 53D 23002220006. N o te s . —T h e au t h or s of t h i s p ap e r w e r e or d e r e d al - p h ab e t i c al l y . A L A R B L B R h → jl 9 3 9 @ c a m . a c . u k [1] H . A . K ra m e rs a n d G . H . W a n n i e r, S t a t i s t i c s o f t h e t w o - d i m e n s i o n a l fe rro m a g n e t . P a rt I , P h y s i c a l R e v i e w 60 ,2 5 2 (1941). [2] L . L o o t e n s , C . De l c a m p , a n d F . V e rs t ra e t e , E n t a n g l e - m e n t a n d t h e d e ns i t y m a t ri x re n o rm a l i z a t i o n g ro u p i n t h e g e n e ra l i z e d L a n d a u p a ra d i g m , N a t . P h y s . 21 ,1 6 5 7 (2025). [3] D. V . E l s e , S . D. B a rt l e t t , a n d A . C . Do h e rt y , H i d d e n s y m m e t ry - b re a k i n g p i c t u re o f s y m m e t ry - p ro t e c t e d t o p o - l o g i c a l o rd e r, P h y s . R e v . B 88 ,0 8 5 1 1 4 ( 2 0 1 3 ) . [4] L . L o o t e n s , C . De l c a m p , D. W i l l i a m s o n , a n d F . V e r- s t ra e t e , L o w - d e p t h u n i t a ry q u a n t u m c i rc u i t s fo r d u a l i t i e s in on e -d imen sion al qu an tu m lattice mo d els, Ph ys. Rev. Lett. 134 ,1 3 0 4 0 3 ( 2 0 2 5 ) . [5] L . L i , M . O s h i k a w a , a n d Y . 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W a n n i e r, S t a t i s t i c s o f t h e t w o - d i m e n s i o n a l fe rro m a g n e t . P a rt I , P h y s i c a l R e v i e w 60 ,2 5 2 (1941). [2] L . L o o t e n s , C . De l c a m p , a n d F . V e rs t ra e t e , E n t a n g l e - m e n t a n d t h e d e ns i t y m a t ri x re n o rm a l i z a t i o n g ro u p i n t h e g e n e ra l i z e d L a n d a u p a ra d i g m , N a t . P h y s . 21 ,1 6 5 7 (2025). [3] D. V . E l s e , S . D. B a rt l e t t , a n d A . C . Do h e rt y , H i d d e n s y m m e t ry - b re a k i n g p i c t u re o f s y m m e t ry - p ro t e c t e d t o p o - l o g i c a l o rd e r, P h y s . R e v . B 88 ,0 8 5 1 1 4 ( 2 0 1 3 ) . [4] L . L o o t e n s , C . De l c a m p , D. W i l l i a m s o n , a n d F . V e r- s t ra e t e , L o w - d e p t h u n i t a ry q u a n t u m c i rc u i t s fo r d u a l i t i e s in on e -d imen sion al qu an tu m lattice mo d els, Ph ys. Rev. Lett. 134 ,1 3 0 4 0 3 ( 2 0 2 5 ) . [5] L . L i , M . O s h i k a w a , a n d Y . Z h e ng , N o n i n v e rt i b l e d u a l i t y t ra n s fo rm a t i o n b e t w e e n s y m m e t ry - p ro t e c t e d t o p o l o g i c a l a n d s p o n t a n e o u s s y m m e t ry b re a k i n g p h a s e s , P h y s . R e v . B 108 ,2 1 4 4 2 9( 2 0 2 3 ) . [6] T. M . Lig g ett, Int er act ing Part icle Syst ems ( S p ri n g e r, 1985). [7] G . M . S c h ¨ u t z , E x a c t l y s o l v a b l e m o d e l s fo r m a n y - b o d y s y s t e m s fa r fro m e q u i l i b ri u m , i n Phas e T r an s i ti ons and Crit ic al Phenomena , V o l . 1 9 ( E l s e v i e r, 2 0 0 1 ) . [8] C . G i a rd i n ` a , J . K u rc h a n , F . R e d i g , a n d K . V a fa y i , Du - a l i t y a n d h i d d e n s y m m e t ri e s i n i n t e ra c t i n g p a rt i c l e s y s - tems, J . Stat. Ph ys. 135 ,2 5( 2 0 0 9 ) . [9] C . G i a rd i n ` a a n d F . 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S a sa mo t o , a n d M . W ad a t i, A symmet - ri c s i m p l e e x c l u s i o n p ro c e s s w i t h o p e n b o u n d a ri e s a n d Ask ey–Wilson p olynomials, J . P h ys. A 37 ,4 9 8 5 ( 2 0 0 4 ) . [14] R . A . B l y t h e a n d M . R . E v a n s , N o n e q u i l i b ri u m s t e a d y s t a t e s o f m a t ri x - p ro d u c t fo rm : A s o l v e r’ s g u i d e , J . P h y s . A : M a t h . T h e o r. 40 ,R 3 3 3( 2 0 0 7 ) . [15] K . T e m m e a n d F . V e rs t ra e t e , S t o c h a s t i c m a t ri x p ro d u c t sta tes, Ph ys. R ev . Lett. 104 ,2 1 0 5 0 2( 2 0 1 0 ) . [16] L . L o o t e n s , C . De l c a m p , G . O rt i z , a n d F . V e rs t ra e t e , Du - a l i t i e s i n o n e - d i m e n s i o n a l q u a n t u m l a t t i c e m o d e l s : Sy m - 5 P R I N p r o j e c t C UP - E 53D 23002220006. N o te s . —T h e au t h or s of t h i s p ap e r w e r e or d e r e d al - p h ab e t i c al l y . A L A R B L B R h → jl 9 3 9 @ c a m . a c . u k [1] H . A . K ra m e rs a n d G . H . W a n n i e r, S t a t i s t i c s o f t h e t w o - d i m e n s i o n a l fe rro m a g n e t . P a rt I , P h y s i c a l R e v i e w 60 ,2 5 2 (1941). [2] L . L o o t e n s , C . De l c a m p , a n d F . V e rs t ra e t e , E n t a n g l e - m e n t a n d t h e d e ns i t y m a t ri x re n o rm a l i z a t i o n g ro u p i n t h e g e n e ra l i z e d L a n d a u p a ra d i g m , N a t . P h y s . 21 ,1 6 5 7 (2025). [3] D. V . E l s e , S . D. B a rt l e t t , a n d A . C . Do h e rt y , H i d d e n s y m m e t ry - b re a k i n g p i c t u re o f s y m m e t ry - p ro t e c t e d t o p o - l o g i c a l o rd e r, P h y s . R e v . B 88 ,0 8 5 1 1 4 ( 2 0 1 3 ) . [4] L . L o o t e n s , C . De l c a m p , D. W i l l i a m s o n , a n d F . V e r- s t ra e t e , L o w - d e p t h u n i t a ry q u a n t u m c i rc u i t s fo r d u a l i t i e s in on e -d imen sion al qu an tu m lattice mo d els, Ph ys. Rev. Lett. 134 ,1 3 0 4 0 3 ( 2 0 2 5 ) . [5] L . 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S a sa mo t o , a n d M . W ad a t i, A symmet - ri c s i m p l e e x c l u s i o n p ro c e s s w i t h o p e n b o u n d a ri e s a n d Ask ey–Wilson p olynomials, J . P h ys. A 37 ,4 9 8 5 ( 2 0 0 4 ) . [14] R . A . B l y t h e a n d M . R . E v a n s , N o n e q u i l i b ri u m s t e a d y s t a t e s o f m a t ri x - p ro d u c t fo rm : A s o l v e r’ s g u i d e , J . P h y s . A : M a t h . T h e o r. 40 ,R 3 3 3( 2 0 0 7 ) . [15] K . T e m m e a n d F . V e rs t ra e t e , S t o c h a s t i c m a t ri x p ro d u c t sta tes, Ph ys. R ev . Lett. 104 ,2 1 0 5 0 2( 2 0 1 0 ) . [16] L . L o o t e n s , C . De l c a m p , G . O rt i z , a n d F . V e rs t ra e t e , Du - a l i t i e s i n o n e - d i m e n s i o n a l q u a n t u m l a t t i c e m o d e l s : Sy m - 5 P R I N p r o j e c t C UP - E 53D 23002220006. N o te s . —T h e au t h or s of t h i s p ap e r w e r e or d e r e d al - p h ab e t i c al l y . A L A R B L B R h → jl 9 3 9 @ c a m . a c . u k [1] H . A . K ra m e rs a n d G . H . W a n n i e r, S t a t i s t i c s o f t h e t w o - d i m e n s i o n a l fe rro m a g n e t . P a rt I , P h y s i c a l R e v i e w 60 ,2 5 2 (1941). [2] L . L o o t e n s , C . De l c a m p , a n d F . V e rs t ra e t e , E n t a n g l e - m e n t a n d t h e d e ns i t y m a t ri x re n o rm a l i z a t i o n g ro u p i n t h e g e n e ra l i z e d L a n d a u p a ra d i g m , N a t . P h y s . 21 ,1 6 5 7 (2025). [3] D. V . E l s e , S . D. B a rt l e t t , a n d A . C . Do h e rt y , H i d d e n s y m m e t ry - b re a k i n g p i c t u re o f s y m m e t ry - p ro t e c t e d t o p o - l o g i c a l o rd e r, P h y s . R e v . B 88 ,0 8 5 1 1 4 ( 2 0 1 3 ) . [4] L . L o o t e n s , C . De l c a m p , D. W i l l i a m s o n , a n d F . V e r- s t ra e t e , L o w - d e p t h u n i t a ry q u a n t u m c i rc u i t s fo r d u a l i t i e s in on e -d imen sion al qu an tu m lattice mo d els, Ph ys. Rev. Lett. 134 ,1 3 0 4 0 3 ( 2 0 2 5 ) . [5] L . L i , M . O s h i k a w a , a n d Y . Z h e ng , N o n i n v e rt i b l e d u a l i t y t ra n s fo rm a t i o n b e t w e e n s y m m e t ry - p ro t e c t e d t o p o l o g i c a l a n d s p o n t a n e o u s s y m m e t ry b re a k i n g p h a s e s , P h y s . R e v . B 108 ,2 1 4 4 2 9( 2 0 2 3 ) . [6] T. M . Lig g ett, Int er act ing Part icle Syst ems ( S p ri n g e r, 1985). [7] G . M . S c h ¨ u t z , E x a c t l y s o l v a b l e m o d e l s fo r m a n y - b o d y s y s t e m s fa r fro m e q u i l i b ri u m , i n Phas e T r an s i ti ons and Crit ic al Phenomena , V o l . 1 9 ( E l s e v i e r, 2 0 0 1 ) . [8] C . G i a rd i n ` a , J . K u rc h a n , F . R e d i g , a n d K . V a fa y i , Du - a l i t y a n d h i d d e n s y m m e t ri e s i n i n t e ra c t i n g p a rt i c l e s y s - tems, J . Stat. Ph ys. 135 ,2 5( 2 0 0 9 ) . [9] C . G i a rd i n ` a a n d F . R e d i g , Du a l i t y f o r M a r k o v P r o c e s s e s : A L i e A lgebr a i c A p p r o a ch ( S p ri n g e r N a t u re , 2 0 2 6 ) . [10] J . I . C i ra c , D. P ´ e re z - G a rc ´ ı a , N . S c h u c h , a n d F . V e r- s t ra e t e , M a t ri x p ro d u c t s t a t e s a n d p ro je c t e d e n t a n g l e d p a i r s t a t e s : C o n c e p t s , s y m m e t ri e s , t h e o re m s , R e v . M o d . Ph ys. 93 ,0 4 5 0 0 3 ( 2 0 2 1 ) . [11] F . C . A l c a ra z , M . Dro z , M . H e n k e l , a n d V . R i t t e n - b e rg , R e a c t i o n - d i ! u s i o n p ro c e s s e s , c ri t i c a l d y n a m i c s , a n d qu a n tu m c h a in s, An n . Ph ys. 230 ,2 5 0 ( 1 9 9 4 ) . [12] B . De rri d a , M . R . E v a n s , V . H a k i m , a n d V . P a s q u i e r, E x a c t s o l u t i o n o f a 1 D a s y m m e t ri c e x c l us i o n m o d e l u s i n g a m a t ri x fo rm u l a t i o n , J . P h y s . A 26 ,1 4 9 3 ( 1 9 9 3 ) . [13] M . U c h iy a ma , T. S a sa mo t o , a n d M . W ad a t i, A symmet - ri c s i m p l e e x c l u s i o n p ro c e s s w i t h o p e n b o u n d a ri e s a n d Ask ey–Wilson p olynomials, J . P h ys. A 37 ,4 9 8 5 ( 2 0 0 4 ) . [14] R . A . B l y t h e a n d M . R . E v a n s , N o n e q u i l i b ri u m s t e a d y s t a t e s o f m a t ri x - p ro d u c t fo rm : A s o l v e r’ s g u i d e , J . P h y s . A : M a t h . T h e o r. 40 ,R 3 3 3( 2 0 0 7 ) . [15] K . T e m m e a n d F . V e rs t ra e t e , S t o c h a s t i c m a t ri x p ro d u c t sta tes, Ph ys. R ev . Lett. 104 ,2 1 0 5 0 2( 2 0 1 0 ) . [16] L . L o o t e n s , C . De l c a m p , G . O rt i z , a n d F . V e rs t ra e t e , Du - a l i t i e s i n o n e - d i m e n s i o n a l q u a n t u m l a t t i c e m o d e l s : Sy m - 5 T h e c ol l e c t i v e m ot i on of f e r m i on i c o c c u p at i on s an d an t i - p h as e d om ai n w al l s i s c ap t u r e d b y t h e f ol l o w i n g T D L- c ou p l e d l o c al op e r at or s H M ,A = J 1 ! i " b M i , 1 + b M ,A i , 1 # + J 2 ! i " b M i , 2 + b M ,A i , 2 # , ( 20) w h e r e t h e l o c al op e r at or s ar e d e fi n e d b y b M ,A i , 1 := m 1 m 1 m 1 1 m 1 m 1 m 1 1 + m m 1 1 1 m 1 m m 1 1 1 m 1 an d b M ,A i , 2 := 1 m m 1 1 1 1 1 m m 1 m m 1 . Not e t h at i n t h e J 1 t e r m , t h e n on - t r i v i al T D L c h an ge s i t s p os i t i on , w h i l e t h e p os i t i on i s fi x e d i n t h e J 2 te rm . T he f e r m i on i c Ham i l t on i an i n E q . ( 18) an d t h e e n t i r e p h y s i - c al Hi l b e r t s p ac e ar e r e al i s e d b y t ak i n g D = C = Ve c Z 2 an d M = sV ec . F u r t h e r m or e , w e n ot e t h at E q . ( 18) p r e - s e r v e s t h e t ot al n u m b e r of an t i - p h as e d om ai n w al l s , w h i c h m e an s t h at t h e t e n s or p r o d u c t Hi l b e r t s p ac e d e c om p os e s i n t o s e c t or s l ab e l l e d b y t h e t ot al n u m b e r of T D Ls k . D u e t o t h e u n c on v e n t i on al T D L c ou p l i n g, t h e f e r m i on i c Ham i l t on i an on t h e f u l l p h y s i c al Hi l b e r t s p ac e i n E q . ( 18) i s n ot gau ge - i n v ar i an t , an d t h e t op ol ogi c al s e c t or s ar e i l l - d e fi n e d . W h i l e t h e c h ar ge s e c t or i s d e - t e r m i n e d b y t h e gl ob al Z 2 s y m m e t r y , t h e T D L i n t e r ac t s wi t h t he J 1 t e r m i n a w a y t h at go e s b e y on d m e r e l y t w i s t - i n g t h e b ou n d ar y c on d i t i on s . T h u s , c h an gi n g t h e m o d u l e c at e gor y f r om sV ec to Ve c f or H M ,A d o e s n ot i m p l e m e n t a p r op e r d u al i t y t r an s f or m at i on . Ho w e v e r , t h e f e r m i on i c Ham i l t on i an c on t ai n s an e x - t e n s i v e n u m b e r of gau ge - i n v ar i an t s u b s p ac e s f or e ac h k , w h i c h c an b e c on s t r u c t e d b y G † A H sV e c ,A G A .W i t h i n t h i s gau ge - i n v ar i an t s u b s p ac e w e r e c o v e r t h e M P O i n - t e r t w i n e r i n E q . ( 16) . I n t h e Z 2 e v e n c h ar ge s e c t or s , e ac h gau ge - i n v ar i an t s u b s p ac e w i t h an e v e n ( o d d ) k i s m ap p e d to the Z 2 e v e n ( o d d ) c h ar ge s e c t or of t h e XXZ m o d e l i n E q . ( 19) u n d e r p e r i o d i c b ou n d ar y c on d i t i on s . T h i s i s b e - c au s e t h e gau gi n g m ap f u s e s T D Ls , w h i c h d i v i d e s t h e t ot al n u m b e r of T D Ls i n t o e v e n or o d d . F u r t h e r m or e , al l t h e o d d c h ar ge s e c t or s ar e t r i v i al i s e d b y t h e gau gi n g m ap an d t h e r e f or e d o n ot ad m i t d u al m o d e l s . V. S UMMAR Y I n t h i s w or k , w e h a v e e x p l or e d a p h y s i c al ap p l i c at i on of ge n e r al i s e d s y m m e t r y t o q u an t u m m an y - b o d y s y s t e m s on l at t i c e s . W e c on s t r u c t an u n c on v e n t i on al ap p l i c a- t i on of T D Ls t h r ou gh a n o v e l i n t e r a c t i on b e t w e e n T D Ls an d s y m m e t r i c t e n s or s , ad d i n g t o t h e k n o w n e x am p l e s of T D Ls . W e al s o u s e M P O an d c at e gor i c al m e t h o d s t o b u i l d t h e c or r e s p on d i n g Ham i l t on i an , w h i c h i s n ot gau ge - i n v ar i an t . W h i l e t h e t op ol ogi c al s e c t or s ar e n ot w e l l d e fi n e d i n t h i s c as e , s u c h Ham i l t on i an s p os s e s s gau ge - i n v ar i an t s u b s p ac e s , an d d u al m o d e l s c an b e f ou n d u s i n g t h e c at e gor i c al r e c i p e . W e d e m on s t r at e t h e s e ab s t r ac t i d e as u s i n g a c on c r e t e f e r m i on i c m o d e l , w h i c h i s e q u i v al e n t t o t h e t - J z mo del i n t h e N ´ e e l s e c t or . W e s h o w t h at t h e f e r m i on i c m o d e l i s d u al t o an e x t e n s i v e n u m b e r of c op i e s of t h e t w i s t e d XXZ m o d e l w i t h i n t h e gau ge - i n v ar i an t s u b s p ac e s . T h e s e h i d - d e n s t r u c t u r e s ar e d i ! c u l t t o d i s c o v e r u s i n g t r ad i t i on al met ho ds. W e h op e t o r e p or t s i m i l ar c on s t r u c t i on s of p h y s i c al Hi l b e r t s p ac e s f or ot h e r c on s t r ai n e d Hi l b e r t s p ac e s , i n t h e c on t e x t of u n d e r s t an d i n g q u an t u m t h e r m al i s at i on an d s l o w d y n am i c s . An ot h e r i n t e r e s t i n g d i r e c t i on i s t o g e n e r - al i s e ou r c on s t r u c t i on t o h i gh e r d i m e n s i on s an d i n v e s t i - gat e h o w s i m i l ar T D Ls m an i f e s t i n t h e d u al gau ge t h e or y . AC K N OW L E D G M E N T S J L t h an k s B r am V an c r ae y n e s t - D e C u i p e r f or p r o v i d i n g op e n ac c e s s t o t h e T i k Z fi l e s . J L an d F V ac k n o w l e d ge s u p p or t b y t h e UK R I G r an t No. E P /Z 003342/1. J L, F V, an d LL w ou l d l i k e t o t h an k t h e I s aac Ne w t on I n s t i - t u t e f or M at h e m at i c al S c i e n c e s , C am b r i d ge , f or s u p p or t an d h os p i t al i t y d u r i n g t h e p r ogr am m e ”Q u an t u m F i e l d T h e or y w i t h B ou n d ar i e s , I m p u r i t i e s , an d D e f e c t s ” w h e r e w or k on t h i s p ap e r w as u n d e r t ak e n . T h i s w or k w as s u p - p or t e d b y E P S R C gr an t n o E P /R 014604/1. Ap p en d ix A: Min im ia l Ca t egor y T h eor y h ˜ h 5 T h e c ol l e c t i v e m ot i on of f e r m i on i c o c c u p at i on s an d an t i - p h as e d om ai n w al l s i s c ap t u r e d b y t h e f ol l o w i n g T D L- c ou p l e d l o c al op e r at or s H M ,A = J 1 ! i " b M i , 1 + b M ,A i , 1 # + J 2 ! i " b M i , 2 + b M ,A i , 2 # , ( 20) w h e r e t h e l o c al op e r at or s ar e d e fi n e d b y b M ,A i , 1 := m 1 m 1 m 1 1 m 1 m 1 m 1 1 + m m 1 1 1 m 1 m m 1 1 1 m 1 an d b M ,A i , 2 := 1 m m 1 1 1 1 1 m m 1 m m 1 . Not e t h at i n t h e J 1 t e r m , t h e n on - t r i v i al T D L c h an ge s i t s p os i t i on , w h i l e t h e p os i t i on i s fi x e d i n t h e J 2 te rm . T he f e r m i on i c Ham i l t on i an i n E q . ( 18) an d t h e e n t i r e p h y s i - c al Hi l b e r t s p ac e ar e r e al i s e d b y t ak i n g D = C = Ve c Z 2 an d M = sV ec . F u r t h e r m or e , w e n ot e t h at E q . ( 18) p r e - s e r v e s t h e t ot al n u m b e r of an t i - p h as e d om ai n w al l s , w h i c h m e an s t h at t h e t e n s or p r o d u c t Hi l b e r t s p ac e d e c om p os e s i n t o s e c t or s l ab e l l e d b y t h e t ot al n u m b e r of T D Ls k . D u e t o t h e u n c on v e n t i on al T D L c ou p l i n g, t h e f e r m i on i c Ham i l t on i an on t h e f u l l p h y s i c al Hi l b e r t s p ac e i n E q . ( 18) i s n ot gau ge - i n v ar i an t , an d t h e t op ol ogi c al s e c t or s ar e i l l - d e fi n e d . W h i l e t h e c h ar ge s e c t or i s d e - t e r m i n e d b y t h e gl ob al Z 2 s y m m e t r y , t h e T D L i n t e r ac t s wi t h t he J 1 t e r m i n a w a y t h at go e s b e y on d m e r e l y t w i s t - i n g t h e b ou n d ar y c on d i t i on s . T h u s , c h an gi n g t h e m o d u l e c at e gor y f r om sV ec to Ve c f or H M ,A d o e s n ot i m p l e m e n t a p r op e r d u al i t y t r an s f or m at i on . Ho w e v e r , t h e f e r m i on i c Ham i l t on i an c on t ai n s an e x - t e n s i v e n u m b e r of gau ge - i n v ar i an t s u b s p ac e s f or e ac h k , w h i c h c an b e c on s t r u c t e d b y G † A H sV e c ,A G A .W i t h i n t h i s gau ge - i n v ar i an t s u b s p ac e w e r e c o v e r t h e M P O i n - t e r t w i n e r i n E q . ( 16) . I n t h e Z 2 e v e n c h ar ge s e c t or s , e ac h gau ge - i n v ar i an t s u b s p ac e w i t h an e v e n ( o d d ) k i s m ap p e d to the Z 2 e v e n ( o d d ) c h ar ge s e c t or of t h e XXZ m o d e l i n E q . ( 19) u n d e r p e r i o d i c b ou n d ar y c on d i t i on s . T h i s i s b e - c au s e t h e gau gi n g m ap f u s e s T D Ls , w h i c h d i v i d e s t h e t ot al n u m b e r of T D Ls i n t o e v e n or o d d . F u r t h e r m or e , al l t h e o d d c h ar ge s e c t or s ar e t r i v i al i s e d b y t h e gau gi n g m ap an d t h e r e f or e d o n ot ad m i t d u al m o d e l s . V. S UMMAR Y I n t h i s w or k , w e h a v e e x p l or e d a p h y s i c al ap p l i c at i on of ge n e r al i s e d s y m m e t r y t o q u an t u m m an y - b o d y s y s t e m s on l at t i c e s . W e c on s t r u c t an u n c on v e n t i on al ap p l i c a- t i on of T D Ls t h r ou gh a n o v e l i n t e r a c t i on b e t w e e n T D Ls an d s y m m e t r i c t e n s or s , ad d i n g t o t h e k n o w n e x am p l e s of T D Ls . W e al s o u s e M P O an d c at e gor i c al m e t h o d s t o b u i l d t h e c or r e s p on d i n g Ham i l t on i an , w h i c h i s n ot gau ge - i n v ar i an t . W h i l e t h e t op ol ogi c al s e c t or s ar e n ot w e l l d e fi n e d i n t h i s c as e , s u c h Ham i l t on i an s p os s e s s gau ge - i n v ar i an t s u b s p ac e s , an d d u al m o d e l s c an b e f ou n d u s i n g t h e c at e gor i c al r e c i p e . W e d e m on s t r at e t h e s e ab s t r ac t i d e as u s i n g a c on c r e t e f e r m i on i c m o d e l , w h i c h i s e q u i v al e n t t o t h e t - J z mo del i n t h e N ´ e e l s e c t or . W e s h o w t h at t h e f e r m i on i c m o d e l i s d u al t o an e x t e n s i v e n u m b e r of c op i e s of t h e t w i s t e d XXZ m o d e l w i t h i n t h e gau ge - i n v ar i an t s u b s p ac e s . T h e s e h i d - d e n s t r u c t u r e s ar e d i ! c u l t t o d i s c o v e r u s i n g t r ad i t i on al met ho ds. W e h op e t o r e p or t s i m i l ar c on s t r u c t i on s of p h y s i c al Hi l b e r t s p ac e s f or ot h e r c on s t r ai n e d Hi l b e r t s p ac e s , i n t h e c on t e x t of u n d e r s t an d i n g q u an t u m t h e r m al i s at i on an d s l o w d y n am i c s . An ot h e r i n t e r e s t i n g d i r e c t i on i s t o g e n e r - al i s e ou r c on s t r u c t i on t o h i gh e r d i m e n s i on s an d i n v e s t i - gat e h o w s i m i l ar T D Ls m an i f e s t i n t h e d u al gau ge t h e or y . AC K N OW L E D G M E N T S J L t h an k s B r am V an c r ae y n e s t - D e C u i p e r f or p r o v i d i n g op e n ac c e s s t o t h e T i k Z fi l e s . J L an d F V ac k n o w l e d ge s u p p or t b y t h e UK R I G r an t No. E P /Z 003342/1. J L, F V, an d LL w ou l d l i k e t o t h an k t h e I s aac Ne w t on I n s t i - t u t e f or M at h e m at i c al S c i e n c e s , C am b r i d ge , f or s u p p or t an d h os p i t al i t y d u r i n g t h e p r ogr am m e ”Q u an t u m F i e l d T h e or y w i t h B ou n d ar i e s , I m p u r i t i e s , an d D e f e c t s ” w h e r e w or k on t h i s p ap e r w as u n d e r t ak e n . T h i s w or k w as s u p - p or t e d b y E P S R C gr an t n o E P /R 014604/1. Ap p en d ix A: Min im ia l Ca t egor y T h eor y h ˜ h FIG. 1. Diagrammatic illustration of the out-of-equilibrium MPO intert winer, see Eqs. (5) and (6) for the details. dualities of quantum spin chains with generalised sym- metries [22, 23], encompassing man y translationally in- v ariant Marko v pro cesses [24, 25]; within this frame- w ork, dualities are implemented lo c al ly —eac h local term of the Hamiltonian h i,i +1 is mapp ed to its dual counter- part. As a canonical example, the Kramers–W annier du- alit y relates the tw o transv erse-field Ising Hamiltonians P i ( − J σ z i σ z i +1 − g σ x i ) and P i ( − J σ x i − g σ z i σ z i +1 ), where σ k denote spin-1 / 2 Pauli matrices, g and J are parameters. The corresp onding duality op erator admits a MPO rep- resen tation, ⊗ i (H i CZ i,i +1 ) (up to a half-site translation), where H is the Hadamard gate and CZ the controlled-Z gate. Each lo cal term σ z i σ z i +1 ( σ x i ) is mapped to lo cal term of the dual mo del σ x i ( σ z i σ z i +1 ). In this Letter, we lift the MP A to a higher rank and establish a nov el MPO framework for dualit y operators of Marko v pro cesses that do not satisfy detailed balance and need not p ossess generalised symmetries. Dualities are implemented glob al ly , such that the conv en tional lo- cal duality relations are violated [22, 23], but dualities of the whole system emerge through a cancellation mec ha- nism of MPO. Our construction bridges sp ectral proper- ties and physical observ ables. It is particularly p ow erful when one of the dual pro cesses is in equilibrium, as cal- culations p erformed using the Gibbs–Boltzmann measure yield direct information ab out the out-of-equilibrium sys- tem [26–28]. Out-of-equilibrium MPO intert winers. —W e consider a one-dimensional lattice consisting of N sites, where each site τ i can host a finite num b er d of states. The Hilbert space is the N -fold tensor-pro duct v ector space H of total dimension d N . Let H T b e the generator of a Marko v pro cess and H b e the sto c hastic Hamiltonian. The time evolution of the man y-b ody probabilit y distribution is gov erned b y the Master equation d | P ( τ 1 . . . τ N ) ⟩ /dt = H | P ( τ 1 . . . τ N ) ⟩ . Giv en t wo processes defined by H 1 = A L + N − 1 X i =1 h i,i +1 + A R (1) and H 2 = B L + N − 1 X i =1 ˜ h i,i +1 + B R , (2) where the bulk terms h i,i +1 and ˜ h i,i +1 acting only on adjacen t sites. In addition, there are sto c hastic bound- ary terms acting on the first and last sites, denoted by the subscripts L and R . W e say that H 1 and H 2 are dual pro cesses if there exists the following in tertwining relation H 1 G = G H 2 . (3) The dualit y op erator G relates tw o exp onen tially large op erator spaces. W e construct an efficient represen tation of the dualit y op erator in the standard MPO form, G = d X τ 1 ,...,τ N =1 τ ′ 1 ,...,τ ′ N =1 ⟨ W | L τ 1 τ ′ 1 1 · · · L τ N τ ′ N N | V ⟩| τ 1 . . . τ N ⟩⟨ τ ′ 1 . . . τ ′ N | . (4) Here, L is a rank-four tensor, and W ( V ) denotes the left (righ t) b oundary vector. F urthermore, we imp ose that the MPO in tertwiner satisfies the following algebraic relations in the bulk, whic h w e refer to as the out-of- equilibrium generalised exc hange relation (or generalised pulling-through relation using tensor-netw ork terminol- ogy), h i,i +1 L i L i +1 − L i L i +1 ˜ h i,i +1 = L i Z i +1 − Z i L i +1 , (5) where Z is another rank-four tensor. In addition, tw o b oundary algebraic relations are required, ⟨ W | ( A L L − Z ) = ⟨ W | L B L , ( A R L + Z ) | V ⟩ = L | V ⟩ B R , (6) see Fig. 1 for a diagrammatic representation. This nov el set of generalised exchange relations consists of the bulk part as w ell as non-trivial boundary parts. Unlike in con ven tional local dualit y , the bulk term h i,i +1 is not mapp ed to the dual bulk term ˜ h i,i +1 when pulled through the tensor L i L i +1 ; instead, it generates a lo cal divergence of tensors. The b oundary relations are chosen such that the divergence is cancelled and the boundaries of the tw o pro cesses A L,R and B L,R are interc hanged. W e note that the bulk relation tak es the form of the Sutherland equation [29] for h i,i +1 = ˜ h i,i +1 , whic h arises from the Y ang-Baxter equation, cf. [30]. In analogy , the b oundary relations can be seen as a consequence of the b oundary Y ang-Baxter equation or equiv alently as the lifting of the Ghoshal-Zamolo dc hiko v relations to a higher rank [31, 32], see also [33, 34]. W e no w show that these generalised exc hange relations indeed imply the dualit y transformation Eq. (3). W e first consider pulling all bulk terms through the MPO, ( P N − 1 i =1 h i,i +1 ) G . Using the bulk relation, this produces 3 h L L h L Z Z L A L Z Z L L B L A R B R L L L L V V V W W W I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k I n t e r t w i ni n g no n-e q ui l i bi r um a nd e q ui l i br i um SSE P v i a M a t r i x P r o duct O p e r a t o r s J an d e G i e r , 1 R ou v e n F r as s e k , 2 J i m i n Li , 3, → an d F r an k V e r s t r ae t e 3, 4 1 De p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , T h e U n i v e r s i t y o f M e l b o u r n e , V I C 3 0 1 0 , A u s t r a l i a 2 U n i v e r s i t y o f M o d e n a a n d R e g g i o E m i l i a , De p a r t m e n t o f P h y s i c s , Informat ics and Mat hemat ics, Via G . Campi 213/b, 41125 Mo dena, It aly 3 De p a r t m e n t o f A p p l i e d M a t h e m a t i c s a n d T h e o r e t i c a l P h y s i c s , U ni ver s i ty o f C a mb ri d ge, Wi lb e rf o r c e R o a d , C a mbr i d ge C B3 0 W A , U ni te d Ki ngd o m 4 De p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , G h e n t U n i v e r s i t y , K r i j g s l a a n 2 8 1 , S 9 , B - 9 0 0 0 G h e n t , B e l g i u m INTR O DUCTIO N ω ε ϑ ϖ 1 AC K N OW L E D G E M E N T S → jl 9 3 9 @ c a m . a c . u k FIG. 2. Illustration of the SSEP . Arro ws (grey circle) indicate the allow ed stochastic rules (particle). P N − 1 i =1 ⟨ W | L 1 · · · L i − 1 ( L i Z i +1 − Z i L i +1 ) L i +2 · · · L N | V ⟩ + G ( P N − 1 i =1 ˜ h i,i +1 ), which forms a telescoping sum of op- erators, leaving tw o residual b oundary con tributions. Next, w e pull through the remaining b oundary terms and use the b oundary relations to generate additional terms, whic h cancel precisely with the remaining terms A L G + A R G = ⟨ W | Z 1 L 2 · · · L N | V ⟩ − ⟨ W | L 1 · · · L N − 1 Z N | V ⟩ + G B L + G B R . W e thus establish the global duality via the out-of-equilibrium pulling-through mec hanism. Symmetric Simple Exclusion Pro cess. — In the follo wing, we demonstrate the out-of-equilibrium MPO in tertwiner for the symmetric simple exclusion pro cess (SSEP), a canonical mo del for studying non-equilibrium transp ort that is known to be related to the integrable XXX Heisenberg c hain [9, 30] The mo del describ es sto c hastic hopping of hard-core particles. Each site of the one-dimensional lattice is either empty ( τ i = 1) or o ccupied by a hard-core particle ( τ i = 2), corresp onding to d = 2. Each particle hops to either adjacent site with probabilit y dt , provided the target site is empt y . In ad- dition, the lattice is coupled to t w o particle reserv oirs. A particle is added to the first (last) site with probabil- it y αdt ( δ dt ) if the site is empty , and remo ved from the first (last) site with probability γ dt ( β dt ) if it is o ccupied. W e denote the four stochastic b oundary parameters b y { α i } i =1 ,..., 4 = { α , β , γ , δ } . Out-of-equilibrium b ehaviour arises for generic b oundaries satisfying αβ − γ δ  = 0. In the tensor-pro duct basis of τ i , the out-of-equilibrium sto c hastic Hamiltonian H NE is given b y h i,i +1 =    0 0 0 0 0 − 1 1 0 0 1 − 1 0 0 0 0 0    , (7) and the b oundary terms are A L =  − α γ α − γ  A R =  − δ β δ − β  . (8) W e first consider the MPO realisation of mapping from the out-of-equilibrium pro cess H 1 = H NE to an equilib- rium pro cess H 2 = H E via Eq. (3). H E will b e deter- mined explicitly b elo w. This map coincides with the one obtained in [28] using a p erturbativ e approach [17]. The dualit y G = G defined b y Eq. (4) and the generalised exc hange relations Eqs. (5) and (6) are realised by L = LY − 1 L =  − F E F D  Z =  0 − 1 0 1  (9) and h i,i +1 = ˜ h i,i +1 , where Y is a pro duct op erator de- fined b elo w, and F , E , and D satisfy the follo wing bulk algebra [ E , F ] = F [ D , F ] = − F [ D , E ] = D + E . (10) In addition, the b oundary algebra reads ⟨ W | ( αE − γ D ) = ⟨ W | and ( β D − δ E ) | V ⟩ = | V ⟩ . W e find the following bidiagonal bi-infinite-dimensional matrix representation of this algebra, the non-v anishing matrix elements are D n,n = − E n,n = n + 1 β + δ F n,n +1 = 1 r n D n +1 ,n = δ ( α + γ ) r n δ + β E n +1 ,n = β ( α + γ ) r n δ + β r n =  β + δ αβ − γ δ   1 α + γ + 1 β + δ + n  , (11) where n, m ∈ Z . Using the standard orthonormal basis {| n ⟩} n ∈ Z and ⟨ n | m ⟩ = δ nm , the b oundary vectors are giv en by | V ⟩ = | 0 ⟩ and | W ⟩ = P n ∈ Z | n ⟩ . W e emphasise that the sp ecial structure of this rep- resen tation implies that the MPO can b e realised ex- actly for finite systems with matrix dimensions (b ond dimensions) that are line ar in the system size and effec- tiv ely given by 2 N + 1. Matrix representations of the algebra are not unique; for instance, one may conjugate L and the b oundary v ectors b y an arbitrary similarity transformation. Another infinite matrix representation can b e constructed [35] by exploiting prop erties of La- guerre p olynomials, reflecting the connections b etw een MPS constructions and orthogonal p olynomials [19, 36]. A crucial difference from the bi-infinite matrix represen- tations in Eq. (11) is that n is restricted to non-negative in tegers in [35]. Finally , we define the pro duct op erator Y to complete the explicit MPO realisation of G . There are tw o equiv- alen t constructions Y R = − 1 α + γ β β + δ 1 α + γ δ β + δ ! ⊗ N Y L =  − 1 β + δ γ α + γ 1 β + δ α α + γ  ⊗ N . (12) F or Y = Y R , the dual H E is defined b y B R = A R and B L = r A R , where r = α + γ β + δ . F or the alternative choice Y L , the stochastic b oundaries of H E b ecome B L = A L and B R = 1 r A R . Notably , distinct H NE ma y b e dual to the same H E , leading to the closed MPO algebra dis- cussed b elo w. Next, w e establish the dualit y transformation from equilibrium to out-of-equilibrium. Consider another set of rates { α ′ i } i =1 ,..., 4 = { α ′ , β ′ , γ ′ , δ ′ } , and let H 1 = H ′ E , H 2 = H ′ NE and G = G ′ . The MPO intert winer Eqs. (4) 4 and generalised exchange relations Eqs. (5) and (6) are realised by L = Y ˜ L ˜ L =  D ′ − E ′ F ′ F ′  Z = −  1 1 0 0  . (13) The bulk algebra remains unchanged Eq. (10), but the b oundary relations turn into ⟨ W ′ | ( αE ′ − γ D ′ ) = −⟨ W ′ | and ( β D ′ − δ E ′ ) | V ′ ⟩ = −| V ′ ⟩ , where E ′ , D ′ , and F ′ are defined by Eq. (11) ev aluated at {− α ′ i } , i.e., ˜ L ( {− α ′ i } ). And Y ( { α ′ i } ) is given by Eq. (12). Note that G ′ has the same computational complexity as G . Moreov er, G ′ re- duces to the in verse of G in the sp ecial case of { α i = α ′ i } , and surprisingly the MPO structure is preserved under op erator in version. Lastly , we complete the MPO intert winer b e- t ween t wo distinct out-of-equilibrium b oundaries { α i } and { α ′ i } b y composing G and G ′ asso ciated with the equiv alent H E . Under the conditions α + γ = α ′ + γ ′ and β + δ = β ′ + δ ′ , we define ˜ G ( { α i } , { α ′ i } ) = G ( { α i } ) Y ( { α i } ) Y − 1 ( { α ′ i } ) G ′ ( { α ′ i } ), whic h obeys H NE ˜ G = ˜ GH ′ NE . W e remark that ˜ G exhibits an imp ortan t prop ert y from the theoretical p ersp ectiv e of tensor netw orks. The fusion of tw o MPOs generates a close d MPO algebr a ˜ G ( { α i } , { α ′ i } ) ˜ G ( { α ′ i } , { α ′′ i } ) = ˜ G ( { α i } , { α ′′ i } ) (14) with the appropriate con tinuous b oundary parameters. Remark ably , the b ond dimension of this MPO algebra scales with system size, in contrast to conv entional MPO algebras with a fixed b ond dimension [16]. Ph ysical Applications. —In the follo wing, w e dis- cuss the physical applications of duality op erators for the SSEP . W e b egin with the duality transformation of eigenstates and tak e Y = Y R WLOG. F rom the inter- t wining relation Eq. (3), one immediately obtains the mapping b et ween eigenstates, | P 1 ⟩ = G | P 2 ⟩ , for the en- tire spectrum. In particular, the most physically relev ant eigenstate is the steady state, defined by H | P ss ⟩ = 0, whic h enco des the long-time behaviour and all static observ ables. F or out-of-equilibrium b oundaries, the ex- act steady state of H 1 is a highly correlated probabil- it y distribution supp orting a non-v anishing current in- duced b y the b oundary driv e. Its MPS represen tation reads | P ss 1 ⟩ = P 2 τ 1 ,...,τ N =1 ⟨ W | A τ 1 1 · · · A τ N N | V ⟩| τ 1 . . . τ N ⟩ , where A 1 = E and A 2 = D . By con trast, for equilib- rium b oundaries, the steady state | P ss 2 ⟩ = 1 β + δ  β δ  ⊗ N is given b y a Bernoulli measure and therefore exhibits no spatial correlations. It is straightforw ard to v erify that | P ss 1 ⟩ = G | P ss 2 ⟩ . W e next consider the ev aluation of ph ysical observ- ables in the steady state. In the MPS framework, lo cal observ ables are extracted from the transfer ma- trix constructed from A , whic h requires sp ectral in- formation of an infinite-dimensional matrix in the out- of-equilibrium case. By con trast, for the equilibrium pro cess, the pro duct structure of the steady state al- lo ws direct contraction with observ ables. By further exploiting the MPO intert winer, one can access out-of- equilibrium prop erties via computations p erformed in the Gibbs–Boltzmann ensemble. As a concrete exam- ple, consider the multi-point density correlation function in the out-of-equilibrium steady state, ⟨ + | n i · · · n j | P ss 2 ⟩ , where n i = 1 2 ( 1 − σ z i ) and ⟨ + | denotes the uniform probabilit y vector such that ⟨ + | H = ⟨ + | . This quan- tit y can equiv alently b e computed b y a veraging ov er the uncorrelated Bernoulli measure, ⟨ + | ˜ n i ··· j | P ss 1 ⟩ , where ˜ n i ··· j = ⟨ W | L 1 · · · X i · · · X j · · · L N | V ⟩ and X =  0 0 F D  , thereb y recov ering the standard result [19, 37]. Discussion. —W e hav e established a matrix pro duct approac h to the duality op erator for b oundary-driven Mark ov pro cesses. By introducing a nov el set of gen- eralised exc hange relations, we lift the DEHP Ansatz to op erators and apply the telescopic cancellation mecha- nism to implement the duality transformations. Out- of-equilibrium dualities are realised globally , in contrast to local dualities associated with generalised symme- tries. The MPO intert winer establishes a sp ectral equiv- alence betw een distinct Mark ov pro cesses with stochastic b oundaries and provides a direct correspondence betw een their eigenstates and observ ables, including the surpris- ing mappings b etw een out-of-equilibrium and equilib- rium. A related dualit y was previously explored in the h ydro dynamic approximations of SSEP and describ ed as a “miracle” in Ref [27], and understo od using changes of (field) v ariables [38]; our approach provides an exact man y-b o dy lattice realisation of this dualit y using MPO. The intimate connection to quantum integrabilit y is ex- plored in a separate companion work [39], where an MPO is deriv ed from in tegrable defects [40]. W e hope to rep ort analogous constructions for the asymmetric simple exclu- sion pro cess and XXZ in future w ork [9, 41, 42]. Another direction is to in vestigate the application of our construc- tion to op en quantum systems [43]. Ac kno wledgements. —W e thank Cristian Giardin` a for helpful discussions. JL ackno wledges discussions on related topics with Paul F endley , Juan P . Garrahan, Rob ert Jack, Hosho Katsura, Katja Klobas, Linhao Li, Laurens Lo otens, and W ei T ang. RF thanks Zolt´ an Ba jnok and Istv´ an M. Sz´ ecs´ enyi for v ery inspiring dis- cussions on isosp ectral spin chains, defects and matrix pro duct op erators during an early stage of this pro ject. JL and FV ac knowledge supp ort by the UKRI Gran t No. EP/Z003342/1. JL and FV would lik e to thank the Isaac Newton Institute for Mathematical Sciences, Cam bridge, for support and hospitalit y during the pro- gramme “Quan tum Field Theory with Boundaries, Im- purities, and Defects” where work on this pap er was un- dertak en. This work was supported b y EPSR C gran t No. EP/R014604/1. RF was supported by INdAM (GNFM), the F AR UNIMORE pro ject CUP-E93C23002040005, and by the PRIN pro ject CUP-E53D23002220006. 5 [1] H. A. Kramers and G. H. 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