SciPost Phys. 8, 044 (2020)

Yang-Baxter integrable Lindblad equations

Aleksandra A. Ziolkowska and Fabian H.L. Essler

The Rudolf Peierls Centre for Theoretical Physics,Oxford University, Oxford OX1 3PU, UK

Abstract

We consider Lindblad equations for one dimensional fermionic models and quantumspin chains. By employing a (graded) super-operator formalism we identify a numberof Lindblad equations than can be mapped onto non-Hermitian interacting Yang-Baxterintegrable models. Employing Bethe Ansatz techniques we show that the late-time dy-namics of some of these models is diffusive.

Copyright A. A. Ziolkowska and F. H. L. Essler.This work is licensed under the Creative CommonsAttribution 4.0 International License.Published by the SciPost Foundation.

Received 30-12-2019Accepted 13-03-2020Published 19-03-2020

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doi:10.21468/SciPostPhys.8.3.044

Contents

1 Introduction 2

2 Lindblad equations for lattice models 32.1 A useful decomposition for N = n2

B + n2F , M = 2nBnF for integer nB, nF 4

2.2 Super-operator formalism for Lindblad equations 42.3 Fermionic jump operators 5

3 Lindblad equations as non-Hermitian two-leg ladders 63.1 Single-site jump operators 63.2 Single-bond jump operators 73.3 General form of the Liouvillian 7

4 Generalized Hubbard models 74.1 SU(2) Hubbard model 7

4.1.1 Associated Lindblad equation 84.2 Integrable structure of generalized Hubbard models and associated Lindblad

equations 84.3 USW model 9

4.3.1 Associated Lindblad equation 104.3.2 Differential equations for correlation functions 10

4.4 Maassarani models 114.4.1 3-state Maassarani model 114.4.2 4-state Maassarani model 124.4.3 Bethe Ansatz solution 134.4.4 String solutions and vanishing of the Liouvillian gap in the thermody-

namic limit 14

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4.5 GL(N , M) Maassarani models 154.5.1 3-state GL(1, 2) model 15

5 Other integrable two-leg ladder models 155.1 GL(N2) magnets 16

5.1.1 Representation as a 2-leg ladder 165.1.2 Associated Lindblad equation 165.1.3 Twisting the boundary conditions 175.1.4 Example: GL(4) spin ladder 18

5.2 GL(n2B + n2

F |2nBnF ) magnets 185.2.1 Associated Lindblad equation 19

5.3 Integrable spin ladder model of Refs [62,63] 195.3.1 Associated Lindblad equation 19

6 A comment on scaling limits 19

7 Some unsuccessful maps 207.1 Perk-Schultz models 207.2 Higher conservation laws 217.3 Alcaraz-Bariev model 21

8 Discussion 23

A Structure of the Liouvillian for the most general jump operator acting on a bond 24

References 25

1 Introduction

Weak couplings to an environment can have very interesting effects on the dynamics of many-particle quantum systems. In particular they can result in desirable non-equilibrium steadystates [1–5]. In order to arrive at a tractable theoretical description it is customary to employa Markovian approximation that assumes that the characteristic times scales associated withthe environment are much shorter than those of the many-particle system of interest. Theabsence of a back action of the system onto its environment then facilitates a well definedmathematical description of open many-particle systems. In the quantum case this a prioriresults in a Markovian quantum stochastic many-particle system [6–9], which is however dif-ficult to analyze. The customary approach is therefore to focus on the dynamics averaged overthe environment, which leads to a description by the Lindblad master equation [10] for thetime-dependent reduced density matrix ρ(t)

dρd t= i[ρ, H] +

∑

a

γa

�

LaρL†a −

12{L†

a La,ρ}�

. (1)

Here H is the system Hamiltonian, La are jump operators that encode the coupling to the en-vironment and γa > 0. While much progress has been made in analyzing Lindblad equationsfor many-particle systems by employing e.g. perturbative [11, 12] and matrix product statesmethods [13–16] it clearly is highly desirable to have exact solutions in specific, and hope-fully representative, cases. In the context of master equations for classical stochastic many

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particle systems an example of such a solvable paradigm is the asymmetric simple exclusionprocess [17–22]. In the quantum case it has been known for some time that certain Lind-blad equations describing many-particle systems can be represented by Liouvillians that arequadratic in fermionic or bosonic creation and annihilation operators, which makes it possi-ble to solve them exactly by elementary means [23–26]. Very recently examples of Lindbladequations with Liouvillians related to interacting Yang-Baxter integrable models have beenfound [27–29]. This opens the door for bringing quantum integrability methods to bear onobtaining exact results for the dynamics of open many-particle quantum systems. An obviousquestion is whether the known cases are exceptional, or whether there are other examples ofYang-Baxter integrable Lindblad equations. In this work we report on the results of a searchfor integrable cases among a particular class of Lindblad equations for translationally invariantmany-particle quantum systems.

2 Lindblad equations for lattice models

We now turn to the precise definition of the class of quantum master equations we will beinterested in. We consider one dimensional lattice models with local Hilbert spaces that caninclude bosonic as well as fermionic degrees of freedom. A basis of the local Hilbert space isformed by N bosonic and M fermionic quantum states

|α⟩ j , α= 1, . . . , N +M . (2)

We denote the fermion parity of the state |α⟩ j by εα

εα =

¨

0 if α is bosonic

1 if α is fermionic. (3)

An orthonormal basis of the full Hilbert space HL on an L-site chain is then given by the states

|α⟩ ≡ ⊗Lj=1|α j⟩ j , α j ∈ {1, . . . , N +M} . (4)

We define the fermion parity of the states (4) by

εα =L∑

j=1

εα j. (5)

A basis of the space of linear operators acting on site j is then provided by

Eαβj = |α⟩ j j⟨β | , α,β ∈ {1, . . . , N +M} . (6)

These are often referred to as Hubbard operators. Their fermion parity is εα + εβ mod 2, i.e.

they are fermionic if either the state |α⟩ or the state |β⟩ is fermionic. The operators Eαβn acton the states |α⟩ as

Eαβn |α⟩= (−1)(εα+εβ )∑n−1

j=1 εα jδβ ,αn|α′⟩ , α′ = α1, . . . ,αn−1,α,αn+1, . . . ,αL . (7)

Minus signs are acquired when moving fermionic operators past fermionic states. The opera-tors defined in this way either commute or anticommute on different sites

Eαβj Eγδk = (−1)(εα+εβ )(εγ+εδ)Eγδk Eαβj , k 6= j . (8)

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For later convenience we define a graded permutation operator on sites j and j + 1

Π j, j+1 =∑

α,β

(−1)εβ Eαβj Eβαj+1 . (9)

It acts on states as

Π j, j+1|β⟩ j|α⟩ j+1 = (−1)εαεβ |α⟩ j|β⟩ j+1 , (10)

i.e. it permutes the states and generates a minus sign if both states are fermionic.

2.1 A useful decomposition for N = n2B + n2

F , M = 2nBnF for integer nB, nF

The general local Hilbert space VN+M introduced above has N bosonic and M fermionic basisstates. If N and M are such that they can be expressed as N = n2

B + n2F and M = 2nBnF for

integer nB, nF it is possible to express VN+M as a graded tensor product of two n = nB + nF -dimensional spaces VN+M = Vn⊗Vn. Here nB and nF are the numbers of bosonic and fermionicbasis states of Vn. Denoting the basis of Vn by {|1⟩, . . . , |n⟩} we can express the N + M basisstates of VN+M as

|α⟩= |eα⟩ ⊗ |α⟩ , α= 1, . . . , N +M , (11)

where 1≤ α, eα≤ n are related to α by

α= α mod n+ nδα mod n,0 , eα=j α

n+ 1

k

+ 1 . (12)

We note that α = n(eα − 1) + α and that the fermion parities are related by εα = εeα + εα.Defining operators

eeeαeβ

j = |eα⟩ j j⟨eβ | , eαβj = |α⟩ j j⟨β | , (13)

we may express Eαβj in the form

Eαβj = |α⟩⟨β |= |eα⟩|α⟩⟨β |⟨eβ |= (−1)εeβ (εα+εβ ) eeeαeβ

j eαβj . (14)

We will use this decomposition in several models considered below. In the purely bosonic caseM = 0 such decompositions are possible for N = n2 with integer n.

2.2 Super-operator formalism for Lindblad equations

We now consider a Lindblad equation (1) with a Hamiltonian H and jump operators La actingon HL defined above. We are ultimately interested in cases where the Hamiltonian density andLa have local expansions in terms of the Eαβj . To start with we will assume for simplicity thatall jump operators are bosonic. The cases where some of the jump operators are fermionic willbe discussed later. The reduced density matrix can be expressed in terms of the basis statesdefined above as

ρ =∑

α,β

ρα,β |α⟩⟨β | . (15)

The matrix elements are related to particular Green’s functions of the operators Eαβj

ρα,β = (−1)∑L−1

j=1

∑Lk= j+1 εβ j

�

εβk+εαk

�

Tr�

ρ EβLαLL . . . Eβ1α1

1

�

. (16)

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In terms of components the Lindblad equation reads

dd tρα,β = i

∑

γ

ρα,γHγ,β −Hα,γργ,β

+∑

a

γa

¦∑

γ,δ

�

La

�

α,γργ,δ�

L†a

�

δ,β −12

∑

γ

�

L†a La

�

α,γργ,β +ρα,γ

�

L†a La

�

γ,β

©

,

(17)

where we have introduced the following notations for the matrix elements of an operator O

⟨α|O|β⟩=Oα,β . (18)

We can view the density matrix as a state in a (N+M)2L dimensional Hilbert spaceHS =HL⊗HLwith basis states

|α⟩ |β⟩⟩= |α1⟩1 . . . |αL⟩L |β1⟩⟩1 . . . |βL⟩⟩L . (19)

In these notations we have|ρ⟩=

∑

α,β

ρα,β |α⟩|β⟩⟩ , (20)

and the “wave-functions” ρα,β correspond to Green’s functions in the original problem. TheLindblad equation (17) can be cast in the form

d|ρ⟩d t= L|ρ⟩ , (21)

where the Liouvillian L for bosonic jump operators La is given by

L= −iH + iH +∑

a

γa

�

La L†a −

12

�

L†a La + L†

a La

�

�

. (22)

Here we employ notations such that O =O⊗1 and have defined related operators O = 1⊗Oby

⟨⟨γ|O|β⟩⟩= ⟨β |O|γ⟩ . (23)

One can easily check that taking the scalar product of (21) with the state ⟨⟨β |⟨α| preciselyreproduces (17). A convenient basis for expanding operators O is constructed in terms ofoperators eEαβn defined as

eEαβn = 1⊗�

|α⟩⟩n n⟨⟨β |�

. (24)

These act on basis states according to

eEαβn |α⟩|β⟩⟩ = (−1)(εα+εβ )εα |α⟩ eEαβn |β⟩⟩

= (−1)(εα+εβ )εα(−1)(εα+εβ )∑n−1

j=1 εβ jδβ ,βn|α⟩‖β ′⟩⟩ , (25)

where |β ′⟩⟩ = |β1⟩1 , . . . , |α⟩n , . . . , |βL⟩L and εα has been defined in (5). We note that theoperators Eαβn act on HS as Eαβn ⊗1.

2.3 Fermionic jump operators

If some of the jump operators are fermionic the super-operator formalism needs to be modified.Let us denote the fermion parity of the jump operator La by εLa

∈ {0,1}. When written in

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components the Lindblad equation still takes the form (17). However, the Liouvillian (22) isnow replaced by

L = −iH + iH +∑

a

γa

�

(−i)εLa La L†a −

12

�

L†a La + L†

a La

�

�

. (26)

The state representing the density matrix is also modified and now takes the form

|ρ⟩=∑

α,β

ρα,β [(−i)εαP+ + iεβ P−] |α⟩|β⟩⟩ , (27)

where P± are projection operators onto states with even and odd fermion parity respectively

P± =1± (−1)F

2, (−1)F =

L∏

`=1

N+M∏

α=1εα=1

(1− 2Eαα` )(1− 2eEαα` ) . (28)

We have(−1)F |α⟩|β⟩⟩= (−1)εα+εβ |α⟩|β⟩⟩ . (29)

It is straightforward to check that inserting (26) and (27) into the equation

dd t|ρ⟩= L|ρ⟩ , (30)

and expanding it in a basis of states precisely recovers (17). We stress that in our constructionboth bosonic and fermionic jump operators can be accommodated as long as any given jumpoperator has a definite fermion parity.

3 Lindblad equations as non-Hermitian two-leg ladders

As we are interested in Liouvillians with local densities we focus on jump operators where theindex a runs either over the sites or the nearest-neighbour bonds of a one dimensional ring.

In this setting −iH −∑

aγa2 L†

a La and iH −∑

aγa2 L†

a La describe interactions along the two legsof the ladder, while

∑

a γa La L†a play the role of interactions between the two legs.

3.1 Single-site jump operators

In translationally invariant situations the most general bosonic single-site jump operator canbe written in the form

` j =∑

α,β

λαβ Eαβj , (31)

where λαβ = 0 unless (εα + εβ) mod 2 = 0. This generates “interaction terms” between thetwo legs of the form

` j`†j =

∑

αβ

∑

γδ

λαβ λ∗γδ Eαβj

eEγδj . (32)

The other jump operator terms in the Liouvillian generate “generalized magnetic field terms”acting on the two legs

`†j` j + `

†j` j =

∑

β ,γ

ΛβγEβγj +Λγβ eEβγj , (33)

where Λβγ =∑

αλ∗αβλαγ.

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3.2 Single-bond jump operators

The most general bosonic jump operator acting on a bond takes the form

L j =∑

α,β

λαβ Eαβj +λ′αβ Eαβj+1 +

∑

α,β ,γ,δ

µαβγδ Eαβj Eγδj+1 . (34)

This gives rise to quartic, cubic and quadratic “interaction terms” in the Liouvillian. The result-ing explicit expression is presented in Appendix A. The extension to fermionic jump operatorsis straightforward.

3.3 General form of the Liouvillian

In the following we will consider Liouvillians of the form

L= −iH + iH +L∑

j=1

∑

a

γa

�

L(a)j (L(a)j )

† −12

�

(L(a)j )† L(a)j + (L

(a)j )

† L(a)j

��

, (35)

where L(a)j are jump operators that act either on site j or the bond ( j, j + 1) and γa > 0. Ouraim is to identify cases which are Yang-Baxter integrable. In practice this means that we needto check whether any of the large number of integrable Hamiltonians that can be interpretedas two-leg ladder models can be cast in the particular form (35). An added complication isthat we should allow for general similarity transformations, i.e. consider

L′ = SLS−1 . (36)

The spatial locality of the Hamiltonian density of the various integrable models imposes strongrestrictions on the possible form of S. Transformations of the form

S =L∏

j=1

S j , (37)

where S j acts non-trivially only on site j are always compatible with the aforementioned localstructure.

4 Generalized Hubbard models

The first example of a Lindblad equation that is related to an interacting Yang-Baxter integrablemodel was presented in Ref. [27], where it was shown that the Lindblad equation for a tight-binding chain with dephasing noise can be mapped onto a fermionic Hubbard model withpurely imaginary interactions. We now briefly review some results obtained in that work.We then show that the mathematical structure that underlies the integrability of the Hubbardmodel quite naturally leads to a connection with a Lindblad equation.

4.1 SU(2) Hubbard model

The Hubbard Hamiltonian is given by

H = −tL∑

j=1

∑

σ=↑,↓

c†j,σc j+1,σ + c†

j+1,σc j,σ + UL∑

j=1

�

n j,↑ −12

��

n j,↓ −12

�

, (38)

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where n j,σ = c†j,σc j,σ. The model is integrable for any complex value of U/t [41]. In terms of

the notations of section 2.1 we can choose a basis such that

c†j,↑ = e21

j , c†j,↓ = ee

21j , n j,↑ = e22

j , n j,↓ = ee22j , (39)

and concomitantly

H(U) = −t∑

j

�

e21j e12

j+1 +ee21j ee

12j+1 + h.c.

�

+ U∑

j

�

e22j −

12

��

ee22j −

12

�

. (40)

4.1.1 Associated Lindblad equation

Let us consider a tight-binding model

H0 = −t∑

j

e12j e21

j+1 + h.c. , (41)

coupled to an environment by jump operators

L j = e22j . (42)

In the super-operator formalism the corresponding Liouvillian (22) is

L(γ) = i t∑

j

[e12j e21

j+1 −ee12j ee

21j+1 + h.c.] +

∑

j

γ

�

e22j ee

22j −

12(e22

j +ee22j )�

. (43)

This is related to the Hubbard Hamiltonian by [27]

L(γ) = −iU†H(iγ)U −γL4

, U =L/2∏

j=1

(e112 j − e22

2 j ) . (44)

4.2 Integrable structure of generalized Hubbard models and associated Lind-blad equations

The Hubbard model was embedded into the general framework of the Quantum Inverse Scat-tering Method [40] in seminal work by Shastry [31, 32]. This construction was subsequentlygeneralized to other classes of integrable models [35–39]. The construction is based on anR-matrix r12(λ) acting on the tensor product of two graded linear vector spaces V ⊗ V and aconjugation matrix C acting on V that fulfil the Yang-Baxter relation

r12(λ1 −λ2)r13(λ1 −λ3)r23(λ2 −λ3) = r23(λ2 −λ3)r13(λ1 −λ3)r12(λ1 −λ2) , (45)

as well as the “decorated” Yang-Baxter relation

r12(λ1 +λ2)C1r13(λ1 −λ3)r23(λ2 +λ3) = r23(λ2 +λ3)r13(λ1 −λ3)C1r12(λ1 +λ2) . (46)

In the cases considered below the r12(λ) is given by

r12(λ) =�

cos2�λ

2

�

− sin2�λ

2

�

C1C2

�

Π12 +sin(λ)

2[I⊗ I− C1C2] , (47)

where Π12 is a graded permutation operator (9) acting on V ⊗ V and

C = 2π−1 , (48)

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where π is a projection operator onto a subspace of V . The R-matrix of an integrable general-ized Hubbard model is then obtained by gluing together two copies [37,39,41]

R⟨12⟩⟨34⟩(λ1,λ2) = r13(λ1 −λ2)r24(λ1 −λ2)

+ α(λ1,λ2)r13(λ1 +λ2)C1r24(λ1 +λ2)C2 . (49)

Here the function α(λ,µ) is given by

α(λ,µ) =cos(λ−µ) sinh

�

h(λ)− h(µ)�

cos(λ+µ) cosh�

h(λ)− h(µ)� , (50)

where h(µ) is a solution of the equation

sinh�

2h(λ)�

= U sin(2λ) . (51)

The local Hamiltonian density of the integrable “fundamental spin model” [40] correspondingto this R-matrix is

H⟨12⟩⟨34⟩ =d

dλ

�

�

�

�

λ=u0

Π13Π24R⟨12⟩⟨34⟩(λ, u0)

= Π13r ′13(0) +Π24r ′24(0) +α′(u0, u0)Π13r13(2u0)C1Π24r24(2u0)C2 . (52)

Here we have generalized the construction of [37] by taking the logarithmic derivative of thetransfer matrix at a shifted point u0 following Ref. [43, 44]. Importantly the structure of theHamiltonians (52) is such that they all can be related to Liouvillians of Lindblad equations. Inthe following we discuss a number of examples.

4.3 USW model

As a first application we consider eqn (52) for the case of the Hubbard model R-matrix [44].The Hamiltonian of these models was first derived by Umeno, Shiroishi and Wadati in [43]and is of the form

HUSW(U) = −∑

j

�

e21j e12

j+1 +ee21j ee

12j+1 + h.c.

�

+U

cosh (2h(u0))

∑

j

B j, j+1B j, j+1 , (53)

where

B j, j+1 =�

cos2(u0)�

e11j − e22

j

�

− sin2(u0)�

e11j+1 − e22

j+1

�

+ sin(2u0)�

e21j e12

j+1 − e21j+1e12

j

��

, (54)

and B j, j+1 is obtained from B j, j+1 by replacing eαβn → eαβn . Here u0 is a free (complex) param-eter and the function h(u) is fixed by the requirement

sinh (2h(u0)) = U sin(2u0) . (55)

We note that the operators eαβj are related to spinful fermion creation and annihilation opera-tors by (39). The Hamiltonian (53) is SO(4) symmetric [43] and in particular commutes withthe total particle number

N =L∑

j=1

e22j + e22

j . (56)

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4.3.1 Associated Lindblad equation

The USW model is related to a Lindblad equation with a tight-binding Hamiltonian

H0 = −∑

j

e12j e21

j+1 + h.c. , (57)

and jump operators

L j = B j, j+1 , (58)

where the parameter u0 is taken to be purely imaginary. In the super-operator formalism thecorresponding Liouvillian (22) is

L(γ) = i∑

j

�

e12j e21

j+1 −ee12j ee

21j+1 + h.c.

�

+ γ∑

j

�

B j, j+1B∗j, j+1 − cos2(2u0)�

. (59)

This is related to the USW Hamiltonian by

L(γ) = −iU†HUSW(u)U − γ cos2(2u0)L , (60)

where the unitary transformation U is given by (44) and the parameter u is purely imaginaryand related to γ by

γ= −iu

cosh�

2h(u0)� . (61)

4.3.2 Differential equations for correlation functions

As the jump operators are Hermitian the Lindblad equation implies the following time evolu-tion for expectation values of (time independent) operators

dd t

Tr [ρ(t)O] = −iTr (ρ(t)[O, H0]) +γ

2

∑

j

Tr�

ρ(t) [[L j ,O], L j]�

. (62)

It is straightforward to verify that the jump operators (58) fulfil

[Ln, c j] = 2δn, j−1 sin(u0)�

cos(u0)c j−1 − sin(u0)c j

�

+ 2δn, j cos(u0)�

cos(u0)c j − sin(u0)c j+1

�

. (63)

This shows that n-particle Green’s functions fulfil simple, closed evolution equations. This isanalogous to the case of the imaginary-U Hubbard model [27]. For example, the single-particleGreen’s function

G j,k(t) = Tr�

ρ(t)c†j ck

�

(64)

has the following equation of motion

dd t

G j,k =∑

`,m

K`,mj,k G`,m ,

K`,mj,k = δ j,`δk−1,m

�

i −γ sin(4u0)

2

�

+δ j,`δk+1,m

�

i +γ sin(4u0)

2

�

− δ j−1,`δk,m

�

i −γ sin(4u0)

2

�

−δ j+1,`δk,m

�

i +γ sin(4u0)

2

�

− 4γδ j,`δk,m cos2(2u0)− 4γδ j,k

�

sin2(u0)M`,mj−1 − cos2(u0)M

`,mj

�

− 4γ�

δ j−1,k sin(u0) cos(u0)M`,mj−1 −δ j,k−1 sin(u0) cos(u0)M

`,mj

�

. (65)

Here we have defined

M`,mj = cos2(u0)δ`, jδm, j − sin(u0) cos(u0)

�

δ`, jδm, j+1 −δ`, j+1δm, j

�

− sin2(u0)δ`, j+1δm, j+1 .(66)

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4.4 Maassarani models

In [33, 34] Maassarani introduced a class of integrable 2n-state models that generalize theHubbard model along the lines set out in section 4.2 above. We now discuss these models inmore detail. A basis of the local Hilbert space is given by the tensor product

|a⟩ ⊗ |a⟩ , a, a = 1, . . . , n , (67)

where all states are bosonic, i.e. εa = 0= εa. While these models a priori are generalized spinmodels they can be related to interacting fermion models by Jordan-Wigner transformationsas is done for a simple case below. A basis of operators acting on these states is then given byeab

j ea bj . In terms of these (bosonic) operators Maassarani’s Hamiltonian reads

HMa,n(U) =L∑

j=1

P(n)j, j+1 + eP(n)j, j+1 + U

�

C j eC j − 1�

, (68)

where

P(n)j, j+1 =∑

a∈A

∑

b∈B

xabebaj eab

j+1 + x−1ab eab

j ebaj+1 ,

C j =∑

a∈A

eaaj −

∑

b∈B

ebbj . (69)

Here the two sets A and B form an arbitrary partition of {1, . . . , n} and xab are arbitrary complexparameters. In the following we will simply set them equal to 1. The operators eP(n)j, j+1 and eC j

are of the same forms as P(n)j, j+1 and C j respectively but with the replacement eabj → eeab

j .Maassarani’s models are related to Lindblad equations with Hamiltonians

H(n)0 = −∑

j

�

∑

a∈A

∑

b∈B

ebaj eab

j+1 + eabj eba

j+1

�

, (70)

and jump operatorsL j = c − C j , (71)

where c ∈ R is a free parameter. In the superoperator formalism the corresponding Liouvillianis

LMa,n(γ) = −i(H(n)0 − eH(n)0 ) + γ

∑

j

�

C j eC j − 1�

, (72)

where eH(n)0 is of the same form as H(n)0 but with eabj replaced by eeab

j . This is related to Maas-sarani’s Hamiltonian by

LMa,n(γ) = iUHMa,n(−iγ)U† , U =L/2∏

j=1

eC2 j . (73)

4.4.1 3-state Maassarani model

The simplest Maassarani model is obtained by considering a local Hilbert space of three bosonicstates. Choosing a decomposition A= {1}, B = {2, 3} gives

H(3)0 = −∑

j

e21j e12

j+1 + e31j e13

j+1 + h.c. . (74)

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In order to fermionize this model we embed it into an enlarged Hilbert space with four statesper site, and then employ the results of section 2.1. This gives

e12j = e12

j ee11j , e13

j = e11j ee

12j . (75)

Finally we carry out a Jordan-Wigner transformation

e21j =

j−1∏

`=1

(1− 2n`,↑)c†j,↑ , e21

j =L∏

`=1

(1− 2n`,↑)j−1∏

`=1

(1− 2n`,↓)c†j,↓ . (76)

After these transformations the Hamiltonian H(3)0 can be written in the form

H(3)0 = −P∑

j,σ

�

c†j+1,σc j,σ + h.c.

�

P , (77)

where

P =L∏

j=1

(1− n j,↑n j,↓) (78)

is a projection operator that ensures that all sites are at most singly occupied. The Hamiltonian(77) can be viewed as the U →∞ limit of the Hubbard model and is sometimes referred toas the t − 0 model. In terms of the fermionic operators the jump operator takes the form

L j = 1− 2(1− n j,↑)(1− n j,↓) + c . (79)

Choosing c = 1 we haveL j|0⟩= 0 , L jc

†j,σ|0⟩= 2c†

j,σ|0⟩ , (80)

which shows that the bath acts on the charge degrees of freedom. The Hamiltonian partH(3)0 has a free fermionic spectrum [45,46], but the creation operators of the non-interactingfermion degrees of freedom are related to the c†

j,σ in a non-local way [47,48]. As a result thesingle-particle Green’s function does not obey a simple evolution equation. The time evolutionis again given by the general expression (62), where the relevant commutators are

[[Ln, c j,σ], Ln]P = −4c j,σδ j,n P ,

P[c j,σ, H(3)0 ]P = P�

− (c j+1,σ + c j−1,σ)− c†j,σc j,σ(c j+1,σ + c j−1,σ)

�

P . (81)

4.4.2 4-state Maassarani model

In the 4-state case we can express the eabj in terms of two species of Pauli operators, cf. 2.1.

Choosing A= {1,2, 3} and B = {4} we then can interpret H(4)0 as the Hamiltonian of a two-legspin ladder model

H(4)0 =L∑

j=1

�

σ+j σ−j+1τ

+j τ−j+1 +σ

−j σ+j+1τ

−j τ+j+1 +

14(σ+j σ

−j+1 +σ

−j σ+j+1)(1−τ

zj)(1−τ

zj+1)

+14(τ+j τ

−j+1 +τ

−j τ+j+1)(1−σ

zj )(1−σ

zj+1)

�

. (82)

The jump operators become (setting again c = 1 in (71))

L j =12(1−σz

j )(1−τzj) . (83)

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4.4.3 Bethe Ansatz solution

The Maassarani models have been solved by Bethe Ansatz in Ref. [49]. Without loss of gener-ality we restrict our discussion to the case where the sets A and B in (69) are given by

A= {1,2, . . . , p} , B = {p+ 1, p+ 2, . . . , n} . (84)

The exact eigenstates of HMa,n(U) are then labelled by good quantum numbers as follows. Theoperators

Qa =L∑

j=1

eaaj , Qa =

L∑

j=1

eaaj , a = 1, . . . , n , (85)

commute with HMa,n(U) and with one another. Hence their eigenvalues Na, Na can be used asgood quantum numbers. Following Ref. [49] we introduce integers

NA =p∑

a=2

Na , NB =n∑

a=p+1

Na , NA =p∑

a=2

Na , NB =n−1∑

a=p+1

Na , (86)

and N ≥ NA+ NB + NA+ NB. We then define sets

MA = {1, . . . , NA} , MB = {NA+ 1, . . . , NA+ NB} ,

MA = {NA+ NB + 1, . . . , NA+ NB + NA} ,

MB = {NA+ NB + NA+ 1, . . . , NA+ NB + NA+ NB} , (87)

and finally introduce two non-intersecting ordered sets of integers 1≤ a j ≤ N ≤ L

AA = {a j| j ∈MA} , AA = {a j| j ∈ MA} , AA∩ AA =∅ , AA∪ ħAA ≡ A . (88)

By ordered we mean that a j < a j+1 if a j , a j+1 ∈ AA and similarly for AA. The eigenstatesof the Liouvillian LMa,n(γ) are then given in terms of rapidities {k1, . . . , kN}, {Λ j| j ∈ MB},{bm|m ∈ MB} and integers {n1, . . . , nNB−Nn−1

}, {n1, . . . , nNB−Nn} subject to the following set of

Bethe Ansatz equations [49]

eik j L = e2πiΦ∏

l∈MB

Λl − sin k j + γ

Λl − sin k j − γ, j ∈ [1, N]\A ,

N∏

j=1j /∈A

Λm − sin k j + γ

Λm − sin k j − γ= e2πiΨ

∏

l∈MBl 6=m

Λm −Λl + 2γΛm −Λl − 2γ

, m ∈MB , (89)

bNB+Nn`

=NB−Nn−1∏

j=1

e2πi

n jNB , 1≤ n1 < · · ·< nNB−Nn−1

≤ NB , ` ∈ MB ,

eik j(L−NB) = (−1)NA−1e2πi mαNA , mα ∈ [1, NA] , j ∈ AA ,

eik j(L−NB−Nn) = (−1)NA−1e2πi mα

NA , mα ∈ [1, NA] , j ∈ AA , (90)

where we require arg(b`)< arg(b`+1) and the phases Φ and Ψ are given by

e2πiΦ = (−1)NB+Nn−1∏

m∈MB

bm

∏

j∈MA

e−ika j ,

e2πiΨ = (−1)N−NA−NA

∏

j∈MA

e−ika j

∏

m∈MA

eikam

∏

`∈MB

b−1`

NB−Nn∏

s=1

e2πi nsNB ,

1≤ n1 < · · ·< nNB−Nn< NB . (91)

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The corresponding eigenvalues of LMa,n(γ) are

E = 2i∑

j∈MB∪MB

cos k j − 2γ(NB + NB + Nn) . (92)

4.4.4 String solutions and vanishing of the Liouvillian gap in the thermodynamic limit

The first two sets (89) of the Bethe Ansatz equations are the same as for the Hubbard modelwith imaginary interactions strength and twisted boundary conditions. This ensures that the“k-Λ string solutions” constructed in [27] are valid solutions for the n-state Maassarani modelsas well. A k-Λ string of length m corresponds to the following pattern of rapidities

k(m)α, j = arcsin(iΛ(m)α − (m− 2 j + 2)γ′) ,

k(m)α, j+m = π− arcsin(iΛ(m)α + (m− 2 j + 2)γ′) ,

Λ(m)α, j = iΛ(m)α + γ(m+ 1− 2 j) , 1≤ j ≤ m . (93)

Here the string centres Λ(m)α are real and γ′ = −γ sgn(Λ(m)α ).We now take NA = NA = 0 and consider a Bethe Ansatz state with a single k-Λ string of

length m� L. The corresponding eigenvalue of the Liouvillian is

ε= 4ImÇ

1− (i|Λ(m)α | −mγ)2 − 4γm . (94)

In the framework of the string hypothesis the equation that fixes the allowed positions of thestring centres Λ(m)α is obtained my “multiplying out the string” [50], which gives

exp

i L2m∑

j=1

k(m)j

!

= e2πim(2Φ+Ψ) . (95)

Taking logarithms this can be cast in the form

sgn(Λ(m))�

π− arcsin(iΛ(m) +mγ) + arcsin(iΛ(m) −mγ)�

=2πL

�

J (m)α +ϕ�

, (96)

where we have definedϕ = m(2Φ+Ψ) mod 1 . (97)

For even lattice lengths L the J (m)α are integers with range

−L + 1− 2m

2−ϕ < J (m) <

L + 1− 2m2

−ϕ . (98)

We now focus on the particular sequence of string states characterized by integers

J (m)α =L2−m−α , α= 1,2, · · · � L . (99)

In the limit of large system sizes L� 1 the corresponding string centres follow from (96)

Λ(n)α =mγL

π(m+α−ϕ)+O(1) . (100)

Substituting this into our expression (94) for the eigenvalue of the Liouvillian gives

ε(m)α = −2π2

mγL2(m+α−ϕ)2 +O(L−4) . (101)

This shows that in the large-L limit we have a band of Liouvillian eigenstates with eigenvaluesthat scale as L−2. This establishes that the Liouvillian gap vanishes in the thermodynamiclimit. Moreover, the scaling with system size suggests that the corresponding eigenmodesare diffusive. Our calculation does not rule out the existence of eigenstates with gaps thatapproach zero faster than L−2.

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4.5 GL(N , M) Maassarani models

As we already mentioned above in section 4.2 the Shastry-Maassarani construction can begeneralized to graded magnets based on GL(N , M). Following Ref. [37] we consider the classof Hamiltonians

HgMa(U) =∑

j

Π(n)j, j+1 + eΠ

(n)j, j+1 + U

�

C j eC j − 1�

, (102)

where

Π(n)j, j+1 =

∑

k 6=N ,K

�

EkNj ENk

j+1 − EkKj EKk

j+1 + (−1)εk(ENkj EkN

j+1 + EKkj EkK

j+1)�

,

C j = 1− 2EKKj − 2ENN

j , K = N +M . (103)

We can relate this to a Lindblad equation with Hamiltonian

H0 = −∑

j

Π(n)j, j+1 , (104)

and jump operatorsL j = 1− C j . (105)

4.5.1 3-state GL(1,2) model

The simplest example is the 3-state model based in GL(1, 2). Like in the case of the 3-stateMaassarani model considered above we may represent the Hamiltonian in terms of canonicalspinful fermion creation and annihilation operators by identifying the three states per site as

|1⟩ j = |0⟩ j , |2⟩ j = c†j,↑|0⟩ j , |3⟩ j = c†

j,↓|0⟩ j . (106)

Then H0 can be represented as

H0 = −PL∑

j=1

�

c†j,↑c j+1,↑ − S+j S−j+1 + h.c.

�

P , (107)

where P is the projection operator on singly occupied sites (78) and S+j = c†i,↑c j,↓. This de-

scribes correlated hopping of the up fermions, whereas the down fermions can only movethrough spin-flip processes. The jump operator is

L j = 2n j,↑ − 1 . (108)

5 Other integrable two-leg ladder models

The generalized Hubbard models considered above are all related to Lindblad equations witha single jump operator on each bond by virtue of their integrability structure. There are manyother integrable models that can be represented as two-leg ladders and a question we haveinvestigated at some length is whether some of them can be associated with Lindblad equationsas well.

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5.1 GL(N 2) magnets

We now consider generalized spin models on a local Hilbert space with N2 bosonic states. Awell-known class of integrable models is obtained by taking [51,52]

HGL(N2) =L∑

j=1

N2∑

α,β=1

Eαβj Eβαj+1 , (109)

where Pj, j+1 =∑N2

α,β=1 Eαβj Eβαj+1 is a permutation operator acting on nearest-neighbour latticesites

Pj, j+1|γ⟩ j|δ⟩ j+1 = |δ⟩ j|γ⟩ j+1 . (110)

The Hamiltonian H is GL(N2) symmetric and hence

[H,Qα,β] = 0 , Qα,β =L∑

j=1

Eαβj . (111)

5.1.1 Representation as a 2-leg ladder

The permutation models can be viewed as 2-leg ladders by employing the decomposition ofsection 2.1 for M = N . This provides a representation of the permutation operator as a tensorproduct

Pj, j+1 =

N∑

α,β=1

eeαβj eeβαj+1

N∑

γ,δ=1

eγδj eδγj+1

. (112)

It is clear from the representation (112) that

[H, Jαβ] = 0= [H, eJαβ] = 0 , (113)

where

eJαβ =L∑

j=1

eeαβj , Jαβ =L∑

j=1

eαβj , α,β = 1, . . . N . (114)

These operators are related to the GL(N2) symmetry generators by

Jαβ =N∑

γ=1

QN(γ−1)+α,N(γ−1)+β , eJαβ =N∑

γ=1

QN(α−1)+γ,N(β−1)+γ . (115)

5.1.2 Associated Lindblad equation

Consider now a Lindblad equation with Hamiltonian H0 and two sets of jump operators {L j}and {`αβj }

H0 =N∑

α,β=1

λαβ Jαβ , L j =

N∑

α,β=1

eαβj eβ αj+1

, `αβj = eαβj . (116)

Noting that L†j L j = 1 we conclude that the corresponding Liouvillian is

L=N2∑

α,β=1

fαβQα,β + γL∑

j=1

�

Pj, j+1 − 1�

, (117)

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where

N2∑

α,β=1

fαβQα,β =N∑

α,β=1

−iλαβ[Jαβ − eJ αβ] + γαβ

�

QN(α−1)+α,N(β−1)+β −J β β + eJ β β

2

�

. (118)

By construction the first term in (117) commutes with the second, which is γHGL(N2). AsHGL(N2) is invariant under all global GL(N2) rotations U we conclude that (117) is integrablefor choices of γαβ and λαβ such that

UN2∑

α,β=1

fαβQα,βU† =N2∑

α=1

gαQα,α , gα ∈ C . (119)

5.1.3 Twisting the boundary conditions

As we have mentioned above, in general we need to consider similarity transformations whentrying to ascertain whether a Lindblad equation is related to an integrable Hamiltonian. Asimple example is provided by considering a Lindblad equation with vanishing Hamiltonianand jump operators

L j =

N∑

α,β=1

ei(ϕβ−ϕα)eαβj eβ αj+1

, ϕα ∈ R . (120)

The corresponding Liouvillian is

L = γ

L∑

j=1

N∑

eα,eβ ,α,β=1

ei(ϕβ−ϕα−ϕeβ+ϕeα)eαβj eβ αj+1 eeeαeβj ee

eβ eαj+1 − 1

= γ

L∑

j=1

N2∑

α,β=1

�

Eαβj Eβαj+1ei(φβ−φα) − 1�

, (121)

where we have used the decomposition 2.1 and fixed the phases φα by

φβ −φα = ϕβ −ϕα +ϕeα −ϕeβ , (122)

whereα,β , α, β , eα, eβ are related by (12). To relate this to the GL(N2)Hamiltonian we considerthe canonical transformation

U Eαβj U† = Eαβj e−i(φα−φβ ) j , (123)

under which the Liouvillian transforms as

ULU† =L∑

j=1

N2∑

α,β=1

�

Eαβj Eβαj+1 − 1�

, (124)

where we have imposed twisted boundary conditions

EβαL+1 = Eβα1 e−i(φα−φβ )L . (125)

We conclude that the Liouvillian is related to the integrable GL(N2) Hamiltonian with twistedboundary conditions

ULU† = γHGL(N2)

�

�

�

�

twisted bc. (126)

The integrability of twisted boundary conditions in the GL(N2)models is well known [53–55].

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5.1.4 Example: GL(4) spin ladder

As a specific example let us consider the GL(4) case

H = JL∑

j=1

Pj, j+1 +h2

�

Q1,1 +Q2,3 +Q3,2 +Q4,4�

, (127)

where we have added a particular generalized magnetic field term. Using 2.1 we can expressthis in terms of two species of Pauli operators, cf. [42]

H =14

L∑

j=1

J�

σ j .σ j+1 + 1� �

τ j .τ j+1 + 1�

+ h�

σ j .τ j + 1�

. (128)

The related Lindblad equation has no Hamiltonian and two sets of jump operators

L j =12

∑

a=x ,y,z

σajσ

aj+1 + 1 , {`(a)j = σ

aj |a = x , y, z} . (129)

The corresponding Liouvillian is

L= γL∑

j=1

�

Pj, j+1 − 1�

+ γ′L∑

j=1

(σxj eσ

xj −σ

yj eσ

yj +σ

zj eσ

zj − 3) . (130)

After a local basis rotation around the y-axis

τxj = −eσ

xj , τ

yj = eσ

yj , τz

j = −eσzj , (131)

this maps onto (128) (up to a constant contribution) if we identify γ= J/4 and h= −γ′.

5.2 GL(n2B + n2

F |2nBnF) magnets

We now turn to particular graded magnets, where we have n2B+n2

F bosonic and 2nBnF fermionicstates at a given site of the lattice, where nB,F ∈ N0. A much studied family of integrable mod-els is given by [52,56–61]

H =∑

j

Π j, j+1 +∑

j

∑

α

λαEααj , (132)

whereΠ j, j+1 is a graded permutation operator (9) and λα are generalized chemical potentials.The case nB = nF = 1 gives the EKS model (a.k.a. supersymmetric extended Hubbard model).We now employ the decomposition 2.1 and choose a tensor product basis for the local Hilbertspace as

|α⟩= |eα⟩ ⊗ |α⟩ , εα = εeα + εα , α, α= 1, . . . , nB + nF , (133)

where α= (nB + nF )(eα− 1) + α. The Eαβj ’s can then be expressed as

Eαβj = (−1)εeβ (εα+εβ ) eeeαeβ

j eαβj , (134)

which in turn leads to the following decomposition of the graded permutation operator

Π j, j+1 =

nB+nF∑

eα,eβ=1

(−1)εeβeeeαeβ

j eeβ eαj+1

nB+nF∑

α,β=1

(−1)εβ eαβj eβ αj+1

. (135)

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5.2.1 Associated Lindblad equation

Consider now a Lindblad equation with no Hamiltonian and Hermitian jump operators

L j =N∑

α,β=1

(−1)εβ eαβj eβ αj+1 . (136)

Noting thatL†

j L j = 1 , (137)

we conclude that the corresponding Liouvillian is

L= γL∑

j=1

(Π j, j+1 − 1) . (138)

We can slightly generalize this by following the construction for the GL(N2) case, e.g. we canadd a Hamiltonian

H =nB+nF∑

α=1

λα

L∑

j=1

eααj . (139)

5.3 Integrable spin ladder model of Refs [62,63]

The Hamiltonian of this model can be cast in the form of a two-leg spin ladder [42]

H(J) =14

L∑

j=1

�

�

σ j .σ j+1 + 1� �

τ j .τ j+1 + 1�

+ J�

σ j .τ j + 1�

+�

σ j .τ j + 1� �

σ j+1.τ j+1 + 1�

−�

σ j .τ j+1 + 1� �

τ j .σ j+1 + 1�

�

. (140)

5.3.1 Associated Lindblad equation

The Hamiltonian (140) is related to a Lindblad equation with no Hamiltonian part and a setof Hermitian jump operators

L j = σ j ·σ j+1 + 1 , A(a)j =∑

b,c

εabcσbjσ

cj+1 , B(a)j = σ

aj +σ

aj+1 , a = x , y, z . (141)

After a local basis rotationτa

j → τyj τ

ajτ

yj , a = x , y, z , (142)

and setting the γ parameters to be equal for all jump operator terms we arrive at a Liouvillian

L= 4γH(−4)− 12Lγ . (143)

6 A comment on scaling limits

A standard way of generating integrable QFTs is by taking appropriate scaling limits of in-tegrable lattice models. A paradigmatic example is the scaling limit of the Hubbard model,which gives rise to the integrable Yang-Gaudin model. An interesting question is whetherwe can carry out an analogous construction for our integrable Liouvillians and arrive at non-unitary integrable QFTs. The answer seems to be negative. Let us consider a lattice modelwith Hamiltonian

H0 = −t∑

j

c†j c j+1 + c†

j+1c j −µ∑

j

c†j c j , (144)

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where c j and c†j are annihilation and creation operators of spinless fermions, and jump oper-

atorsL j = n j = c†

j c j . (145)

These give rise to a Liouvillian

L = −iH0 + i eH0 + γ∑

j

n j n j −12(n j + n j) , (146)

where H0 is of the same form as H0 but written in terms of fermion annihilation and creationoperators c j and c†

j . The sign difference between H0 and H0 can be removed by a canonicaltransformation

c j → c j(−1) j . (147)

In analogy of what we do in order to obtain the Yang-Gaudin model from the Hubbard modelwe now consider the scaling limit

t →∞ , a0→ 0 , ta20 fixed. (148)

In this limit lattice fermion operators are replaced by continuum fields

c j 'p

a0Ψ↑(x) , c j 'p

a0Ψ↓(x) , x = ja0 . (149)

The Liouvillian becomes

L =h

i(2t +µ)−γ

2

i

∫

d x∑

σ

Ψ†σ(x)Ψσ(x)

+ i ta20

∫

d x∑

σ

Ψ†σ(x)∂

2x Ψσ(x) + γa0

∫

d xΨ†↑(x)Ψ↑(x)Ψ

†↓(x)Ψ↓(x) . (150)

A problem now occurs in the first term. If γ were purely imaginary, as is the case for theHubbard model, we could tune the chemical potential in such a way to ensure that the prefactorremains finite in the scaling limit. This would leave us with a Yang-Gaudin model at a finitedensity, but with imaginary interaction strength. However, given that γ is real and positive wecannot take γ →∞, but must keep it finite in order to describe states with finite real partsof their “energies”. This means that the only scaling limit is trivial as the interaction termdisappears. This would appear to be a more general feature, independent of integrability.

7 Some unsuccessful maps

Most of the integrable ladder models we have considered cannot be associated in a straightfor-ward way with Lindblad equations. In the following we present some representative examples.

7.1 Perk-Schultz models

As an example we consider the N = 4 Perk-Schultz model [64,65]

HPS = J∑

j

�

cosh(η)∑

α

Eααj Eααj+1 +∑

α6=β

Eβαj Eαβj+1 + sgn(α− β) sinhη Eααj Eββj+1

�

. (151)

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This can be viewed as a q-deformation of the GL(4) Hamiltonian considered above. Using thedecomposition 2.1 we can rewrite HPS as

HPS = J∑

j

Pj, j+1 +cosh(η)− 1

4

�

1+σzjσ

zj+1

��

1+τzjτ

zj+1

�

+J sinh(η)

4

∑

j

�

σzj+1 −σ

zj

��

1+τzjτ

zj+1

�

. (152)

As the spectra of σzj+1−σ

zj and 1+τz

jτzj+1 are different the term in the second line cannot be

related to a jump operator structure in this representation.

7.2 Higher conservation laws

A well-known way of obtaining integrable spin-ladder models is by considering higher conser-vation laws [40, 66]. In case of the spin-1/2 Heisenberg XXX chain higher conservation lawsH(k+1) can be obtained from the transfer matrix by taking logarithmic derivatives at the “shiftpoint”. By construction we have [H(k), H(l)] = 0. The Hamiltonian we want to consider hereis H(b) = H(2) + bH(4) + const [66–68], which takes the form

H(b) = 4L∑

j=1

�

(1− b)S j · S j+1 +b2

S j · S j+2 + 2b�

S j−1 · S j+1

� �

S j · S j+2

�

−2b�

S j−1 · S j+2

� �

S j · S j+1

�

�

. (153)

This can be viewed as a zig-zag ladder model by associating all even (odd) sites with the first(second) leg, which gives

H(b) =L/2∑

j=1

(1− b)σ j ·�

τ j +τ j+1

�

+b2

¦

σ j ·σ j+1[τ j ·τ j+1 +τ j+1 ·τ j+2]

+ σ j ·σ j+1 +τ j ·τ j+1 −τ j ·σ j+1 σ j ·τ j+1 −σ j ·τ j+2 σ j+1 ·τ j+1

©

. (154)

This is asymmetric under leg exchange in a way that precludes a direct relation with a Lindbladequation.

7.3 Alcaraz-Bariev model

The Alcaraz-Bariev two-parameter families of integrable models [69] come in two classes de-noted by A± and B± respectively. The B± family contains the Hubbard model as a special limitand this is the only case in which we succeeded in obtaining an interpretation in terms of aLindblad equation. We now discuss why such a relation does not seem to exist in general forthe A± family of models. The Hamiltonian of the A± family can be cast in the form

H(ε)A =∑

j

T j, j+1 + T (1)j, j+1 + T (2)j, j+1 + gT (3)j, j+1 + cosθ[S j, j+1 − εT (p)j, j+1 + Vj, j+1 − εU j, j+1] , (155)

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SciPost Phys. 8, 044 (2020)

where g = (1+ ε)(1− sinθ ) and

T j, j+1 = −e21j e12

j+1 +ee21j ee

12j+1 + h.c. ,

T (1)j, j+1 = −(e21j e12

j+1 − e12j e21

j+1)ee22j (ε sinθ − 1) + (ee21

j ee12j+1 −ee

12j ee

21j+1)e

22j (sinθ − 1) ,

T (2)j, j+1 = −(e21j e12

j+1 − e12j e21

j+1)ee22j+1(sinθ − 1) + (ee21

j ee12j+1 −ee

12j ee

21j+1)e

22j+1(ε sinθ − 1) ,

T (3)j, j+1 = −e21j e12

j+1ee22j ee

22j+1 +ee

21j ee

12j+1e22

j e22j+1 + h.c. ,

S j, j+1 = e21j e12

j+1ee12j ee

21j+1 + h.c. ,

T (p)j, j+1 = −e21j e12

j+1ee21j ee

12j+1 + h.c. ,

Vj, j+1 = e−2ηe22j ee

22j+1 + e2η

ee22j e22

j+1 ,

U j, j+1 = e22j ee

22j + e22

j+1ee22j+1 . (156)

Here we have carried out a unitary transformation

Ueabj U† = eab

j (−1) j(a−b) (157)

on the Hamiltonian given in [69] in anticipation of relating it to a Liouvillian on a Lindbladequation. We start by noting that we require g = 0 for such an interpretation to be possible.

The reason is that the only way to generate T (3)j, j+1 is as a “cross-term” in ` j`†j with

` j = ae21j e12

j+1 + be12j e21

j+1 + ce22j e22

j+1 . (158)

However, such jump operators would also generate an unwanted contribution

|c|2e22j e22

j+1ee22j ee

22j+1. (159)

As this cannot be cancelled by introducing additional jump operators and does not feature inH(ε)A we conclude that we must have g = 0. Next we turn to the cubic terms T (1)j, j+1. Thesemust arise from jump operators of the form

L j = ae21j e12

j+1 + be12j e21

j+1 + ce22j . (160)

These jump operators give rise to inter-species interactions

L j L†j = |a|2e21

j e12j+1ee

21j ee

12j+1 + ab∗e21

j e12j+1ee

12j ee

21j+1 + a∗be12

j e21j+1ee

21j ee

12j+1

+ |b|2e12j e21

j+1ee12j ee

21j+1 + |c|

2e22j ee

22j + c∗(ae21

j e12j+1 + be12

j e21j+1)ee

22j

+ ce22j (a

∗ee21

j ee12j+1 + b∗ee12

j ee21j+1) , (161)

and intra-species interactions

L†j L j = |a|2(1− e22

j )e22j+1 + |b|

2e22j (1− e22

j+1) + |c|2e22

j − a∗ce12j e21

j+1 + c∗ae21j e12

j+1 ,

L†j L j = |a|2(1−ee22

j )ee22j+1 + |b|

2ee22

j (1−ee22j+1) + |c|

2ee22

j − ac∗ee12j ee

21j+1 + ca∗ee21

j ee12j+1 .(162)

In order to produce the cubic terms in H(ε)A we require

a = −b , ac∗ = 1− ε sinθ , ca∗ = sinθ − 1 . (163)

Combining these with the requirement that g = 0 leads to

ε= sinθ = 1 . (164)

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SciPost Phys. 8, 044 (2020)

In this case the A± model reduces to free fermions. We have also investigated whether carryingout a similarity transformation SH(ε)A S−1 with

S =L∏

j=1

exp�

ϕe22j ee

22j + j

�

ϕ1e11j +ϕ2e22

j + ϕ1ee11j + ϕ2ee

22j

��

(165)

may facilitate a Lindblad interpretation. The answer appears to be negative.

8 Discussion

In this work we have reported our findings for a search for Yang-Baxter integrable Lindbladequations. We have focused on translationally invariant situations where jump operators acton bonds or sites of a one dimensional chain. We have derived a superoperator representationfor lattice models with both fermionic and bosonic degrees of freedom, and jump operatorswhich can be bosonic or fermionic. In this representation the Lindblad equation takes theform of a imaginary time Schrödinger equation with a non-Hermitian “Hamiltonian” with lo-cal density, which can be thought of in terms of a two-leg ladder model of interacting spins orfermions. We have then investigated which Yang-Baxter integrable two-leg ladder models canbe related to such Lindblad equations in a “direct” way. Our main result is that a wide classof generalized Hubbard models can be interpreted as Liouvillians of Lindblad equations. Wetraced this back to their integrability structure, which is based on gluing together certain so-lutions of the Yang-Baxter equation in a particular way. Some of the corresponding dissipativemodels are physically meaningful, an example being the infinite-U Hubbard model subject toon-site dephasing noise. As the jump operators in this class of models are Hermitian, the com-pletely mixed state is a steady state in all cases. Using the Bethe Ansatz solution we have shownfor a subclass of generalized Hubbard models that the Liouvillian gap vanishes like L−2 as thethermodynamic limit is approached. The corresponding eigenstates correspond to particle-like“excitations” with quadratic dispersions, which suggests that the late-time behaviour in thesemodels is likely to be diffusive.

We have identified a few Yang-Baxter integrable Lindblad equations that are not general-ized Hubbard models by showing that certain known integrable Hamiltonians can be cast inthe form of Liouvillians associated with a Lindblad equation. However, in most cases we haveconsidered such mappings are not possible. As this is often difficult to see we have presenteda non-trivial case of such a failure in the Alcaraz-Bariev two-parameter family of integrablemodels.

We stress that in this work we have focused on a particular “direct” relation between Liou-villians of Lindblad equations and Hamiltonians of Yang-Baxter integrable models. There areknown cases where it is possible to establish such relationships by means of more complicated(non-local) maps [29]. Moreover, as we pointed out in section 3.3, one ought to allow for sim-ilarity transformations that maintain locality of the Hamiltonian density in integrable modelswhen trying to establish relations with Lindblad equations. A systematic way of doing this isby considering invariances of the Yang-Baxter equation, cf. Chapter 12.2.5 of Ref. [41]. Forexample, given a solution R(λ,µ) ∈ End(C⊗C) of the Yang-Baxter equation other solutionscan be obtained as

�

V (µ)⊗ V (λ)�

R(λ,µ)�

V−1(λ)⊗ V−1(µ)�

, (166)

where V (λ) is an invertible n × n matrix. This allows one to introduce additional free pa-rameters in the resulting Hamiltonian. The latter will generally be non-Hermitian, but this isnot a problem in the present context of Lindblad equations. It would be interesting to pursue

23

SciPost Phys. 8, 044 (2020)

this line of enquiry further and a good starting point will be the models successfully related toLindblad equations in this work.

In this we work we focused on identifying integrable Lindblad equations and only brieflyexplored using methods of quantum integrability to obtain physical properties. A good start-ing point for this is to determine the spectrum of the Liouvillian, which is given in terms ofthe solutions of the relevant Bethe Ansatz equations. It is well understood that the natureof solutions to Bethe Ansatz equations changes quite substantially when a parameter is madecomplex, as this results in the “scattering phases” acquiring magnitudes different from unity.In practice this means that the structure of solutions to the Bethe Ansatz equations, whichis usually encoded in appropriate string hypotheses, must be revisited and typically becomesmore involved. Even in the simplest case of the Hubbard model the structure of Bethe Ansatzroots for Liouvillian eigenstates with eigenvalues that have large real parts and non-zero imag-inary parts appears to be non-trivial. We plan to report on this issue in a future publication.Ultimately one would like to determine the dynamics of general Green’s functions

Tr�

ρ(t)Eα1β1j1

. . . Eαnβnjn

�

(167)

for evolution from a given initial density matrix ρ(0). In some of the cases discussed abovethis is relatively simple because the equations of motion for these Green’s functions decoupleand for two-point functions can thus either be integrated numerically or determined from theexact Liouvillian eigenstates in the two-particle sector [70]. In cases like the 3-state Maassaranimodel a more involved analysis is required and it would be interesting to investigate this casein more detail.

Acknowledgements

We are grateful to F. Göhmann, H. Katsura and T. Prosen for very helpful discussions. Thiswork was supported by the EPSRC under grant EP/S020527/1.

A Structure of the Liouvillian for the most general jump operatoracting on a bond

The most general two site bosonic jump operator with nearest-neighbour interactions is

L j =∑

αβ

�

λαβ Eαβj +λ′

αβ Eαβj+1

�

+∑

αβγδ

µαβγδEαβj Eγδj+1 . (168)

This gives rise to interaction terms between the two legs of the ladder

L j L†j = I(2)j + I(3)j + I(4)j , (169)

where I(n)j involves n Hubbard operators Eαβj , eEαβj . The interaction along a single rung of theladder is

I(2)j =∑

α1β1α2β2

�

λα1β1λ∗α2β2

Eα1β1j

eEα2β2j +λα1β1

λ′∗α2β2

Eα1β1j

eEα2β2j+1

+λ′

α1β1λ∗α2β2

Eα1β1j+1

eEα2β2j +λ

′

α1β1λ′∗α2β2

Eα1β1j+1

eEα2β2j+1

�

,

(170)

24

SciPost Phys. 8, 044 (2020)

while the three and four point interactions on a given plaquette are given by

I(3)j =∑

α1β1γ1δ1α2β2

µα1β1γ1δ1Eα1β1

j Eγ1δ1j+1

�

λ∗α2β2eEα2β2

j +λ′∗α2β2

eEα2β2j+1

�

+∑

α1β1α2β2γ2δ2

µ∗α2β2γ2δ2

�

λα1β1Eα1β1

j +λ′

α1β1Eα1β1

j+1

�

eEα2β2j

eEγ2δ2j+1 ,

I(4)j =∑

α1β1γ1δ1α2β2γ2δ2

µα1β1γ1δ1µ∗α2β2γ2δ2

Eα1β1j Eγ1δ1

j+1eEα2β2

jeEγ2δ2

j+1 . (171)

There are also interaction terms along the two legs of the ladder

L†j L j =

∑

βγ

��

∑

α

λαβλ∗αγ

�

Eγβj +�

∑

α

λ′

αβλ′∗αγ

�

Eγβj+1

�

+∑

αβγδ

�

fαβγδEαβj Eγδj+1 + h.c.�

,

L†j L j =

∑

βγ

��

∑

α

λ∗αβλαγ�

eEγβj +�

∑

α

λ′∗αβλ

′

αγ

�

eEγβj+1

�

+∑

αβγδ

�

fβαδγeEγδj+1eEαβj + h.c.

�

, (172)

where

fαβγδ = λ∗βαλ

′

γδ +∑

η

�

λ∗ηαµηβγδ +λ′∗ηγµαβηδ

�

+12

∑

ην

(−1)(εα+εβ )(εη+εγ)µνβηδµ∗ναηγ . (173)

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