Bundled string solutions of the Bethe ansatz equations in the non-Hermitian spin chain
BBundled string solutions of the Bethe ansatzequations in the non-hermitian spin chain
Yuki Ishigruo , Jun Sato , Katsuhiro Nishinari Department of Aeronautics and Astronautics,Faculty of Engineering,University ofTokyo,Hongo,Bunkyo-ku,Tokyo 113-8656,Japan Physics Department and Soft Matter Center, Ochanomizu University, Japan Research Center for Advanced Science and Technology, University of Tokyo,4-6-1Komaba,Meguro-ku,Tokyo 153-8904,JapanE-mail: [email protected]; [email protected];[email protected]
January 2021
Abstract.
The asymmetric simple exclusion process (ASEP) is a paradigmaticintegrable stochastic model describing nonequilibrium transport phenomena. Inthis paper, we investigate solutions of the Bethe equations for the ASEP in thethermodynamic limit. Many hermitian integrable systems, such as the Heisenbergspin chain, have regular solutions called string solutions. However, in the case of theASEP, the pattern of the string solution is changed due to its non-hermiticity. Wecall this new type of string solutions ”bundled string solutions”. We introduce andformulate the bundled string solutions, and then derive the extended Bethe-Takahashiequations.
1. Introduction
Understanding many-body systems out of equilibrium is one of the most importantchallenges in modern physics. Although a lot of researches have revealed some aspectsof nonequilibrium systems, its universal theory has not been constructed yet. In thecase of equilibrium systems, there are universal structures, on which the equilibriumthermodynamics is based. When systems are out of equilibrium, such thermodynamicstructures would be deformed or broken. If the thermodynamic structures are deformed,it may be possible to construct nonequilibrium thermodynamics by extending theequilibrium one. Therefore, it is important to understand how thermodynamicstructures are changed when systems transition from equilibrium to nonequilibrium.One of objectives of this research is to reveal such deformations of thermodynamicstructures by analysing a nonequilibrium integrable system called the asymmetric simpleexclusion process (ASEP) [1]. a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n undled string solutions in the non-hermitian spin chain
2. Asymmetric simple exclusion process
The asymmetric simple exclusion process (ASEP) is an asymmetric random walk modelwith the exclusion volume effect on the one-dimensional lattice. A schematic drawing undled string solutions in the non-hermitian spin chain p ( q ), ( p + q = 1). If the destination is not vacant, hoppingdoes not occur due to the exclusion volume effect. Despite of its simplicity, the ASEPdescribes interesting nonequilibrium phenomena, such as the boundary-induced phasetransition [8, 9]. Therefore, this model is applied to various transport phenomena, suchas biophysical transport [10, 11], traffic flow [12, 13] and surface growth phenomena[14]. Moreover, this model is exactly solvable. Its stationary state is constructed by thematrix product ansatz [15] and Hamiltonian is diagonalized by the Bethe ansatz [4, 5].Let us formulate the ASEP by the second quantization method. We consider theperiodic boundary condition. The number of lattice sites is L . We introduce the variable n j , which denotes the number of particles at site j . A local state of site j is denoted by2-dimensional vector | n j (cid:105) . The basis of a local states is ( | (cid:105) = (0 , T , | (cid:105) = (1 , T ). Ifa particle exists at site j , | n j (cid:105) = | (cid:105) . Otherwise, | n j (cid:105) = | (cid:105) . The basis of a configurationspace is | n (cid:105) = | n (cid:105) ⊗ | n (cid:105) ⊗ · · · ⊗ | n L (cid:105) . Then, a state of the system at time t is given by | ψ ( t ) (cid:105) = (cid:88) n ψ ( n, t ) | n (cid:105) , (1)where ψ ( n, t ) is the probability that the system is in a state | n (cid:105) at time t . The Markovmatrix of the ASEP ˆ H ASEP is given byˆ H ASEP = L (cid:88) j =1 (cid:110) p ˆ S + j ˆ S − j +1 + q ˆ S − j ˆ S + j +1 − p ˆ n j (1 − ˆ n j +1 ) − q (1 − ˆ n j )ˆ n j +1 (cid:111) . (2)where ˆ S ± j and ˆ n j are the ladder operators and the particle number operator, which aredefined by the Pauli matrices ˆ σ x,y,z as ˆ S ± j = (ˆ σ xj ± i ˆ σ yj ), ˆ n j = (1 − ˆ σ zj ). The timedevelopment of this state is described by the following master equation, ddt | ψ ( t ) (cid:105) = ˆ H ASEP | ψ ( t ) (cid:105) . (3)This master equation is the imaginary time Schr¨odinger equation, so we call the Markovmatrix ˆ H ASEP
Hamiltonian hereafter.
The ASEP could be regarded as the nonequilibrium Heisenberg spin chain [4, 16]. Here,we review the correspondence between the ASEP and the Heisenberg chain. When thehopping rate is symmetric p = q = 1 /
2, the Hamiltonian of the ASEP ˆ H ASEP isˆ H ASEP = L (cid:88) j =1 (cid:110) ˆ S xj ˆ S xj +1 + ˆ S yj ˆ S yj +1 + ˆ S zj ˆ S zj +1 (cid:111) − L H XXX , (4)where the spin operators are given by ˆ S x,y,zj = ˆ σ x,y,zj . This Hamiltonian is the same asthe one of the XXX model. Therefore, when hopping rate is symmetric, the ASEP isequilibrium system and equivalent to the XXX model. undled string solutions in the non-hermitian spin chain Figure 1.
Schematic drawing of the ASEP
When the hopping rate is asymmetric ( p (cid:54) = q ), the Hamiltonian of the ASEP ˆ H ASEP is a non-hermitian matrix. In this case, we can map the Hamiltonian ˆ H ASEP to theHeisenberg XXZ model with the twisted boundary condition. We define the operator U j = I ⊗ · · · ⊗ I i − ⊗ (cid:32) α i − (cid:33) ⊗ I i +1 ⊗ · · · ⊗ I L , (5)where I j is the 2 × α = (cid:113) qp . We consider the operator given by H = ( α + α − ) (cid:32) L (cid:89) i =1 U i (cid:33) (cid:18) ˆ H ASEP + L (cid:19) (cid:32) L (cid:89) i =1 U i (cid:33) − , (6)where I j is the 2 L × L identity matrix. Then, we rewrite the H as the Heisenberg XXZspin chain with the twisted boundary conditions as follows. H = 2 L (cid:88) j =1 (cid:26) ˆ S xj ˆ S xj +1 + ˆ S yj ˆ S yj +1 + α + α − S zj ˆ S zj +1 (cid:27) + α L ˆ S − L ˆ S +1 + α − L ˆ S + L ˆ S − +( α + α − ) ˆ S zL ˆ S z , (7)ˆ S + L +1 = α L ˆ S +1 , ˆ S − L +1 = α − L ˆ S − , ˆ S zL +1 = α L ˆ S z . In this sense, the ASEP is regarded as the nonequilibrium version of the Heisenbergchain. When the driven force of the ASEP is symmetric, it is identical with theHeisenberg XXX spin chain. As the asymmetricity of the driven force become larger,the Heisenberg XXX spin chain go away from equilibrium. In the spin chain description,the asymmetricity of the driven force of the ASEP is interpreted as the magnetic fieldat the boundary.
3. String solution
In this section, we investigate the solution of the Bethe equations in the thermodynamiclimit. First, we review the traditional string solution of the Heisenberg XXX spin chain.Then, we present the bundled string solutions of the ASEP. undled string solutions in the non-hermitian spin chain As is shown in the previous section, when the hopping rate is symmetric ( p = q ), theASEP is equivalent to the Heisenberg XXX spin chain. In the case of the HeisenbergXXX spin chain, there are regular solutions of the Bethe equations in the thermodynamiclimit. The Bethe equations of the Heisenberg XXX spin chain with the set of N rapidities { λ j } are given by (cid:18) λ j + i λ j − i (cid:19) L = N (cid:89) l (cid:54) = j λ j − λ l + iλ j − λ l − i , j = 1 , · · · , N. (8)We assume the complex solution λ j . In the thermodynamic limit L → ∞ , if Im λ j > λ j − λ l − i ∼ , if Im λ j > . (9)Similarly, if Im λ j <
0, the LHS of the Eq. (8) becomes zero in the thermodynamic limit L → ∞ . So, the numerator of the RHS of the Bethe equation becomes zero: λ j − λ l + i ∼ , if Im λ j < . (10)The Eqs. (9),(10) means that if λ j is a Bethe root, then λ j ± i is also the Bethe root. Thismechanism yields a series of Bethe roots on the line which is parallel to the imaginaryaxis. These solutions are called the string solutions. An n -string is given by λ n,jα = λ nα + i n + 1 − j ) , j = 1 , · · · , n. (11)Here, n is the string length, α is the index of the strings and λ nα is called the stringcenter. In reality, the string solutions contain deviations. These deviations from thestring hypothesis in the Heisenberg XXX spin chain are discussed in the ref. [17]. Hereafter we discuss the string solution of the Bethe equations in the ASEP, which aremain results of this paper. The Bethe equations of the ASEP with the set of N rapidities { z j } are given by z Lj = ( − N − N (cid:89) l (cid:54) = j p − z j + qz j z l p − z l + qz j z l , j = 1 , · · · , N. (12)Here, p, q are hopping rates ( p + q = 1). We introduce a parameter (cid:15) ∈ [0 ,
1) and rewritethe hopping rate as p = 1 + (cid:15) , q = 1 − (cid:15) . (13) undled string solutions in the non-hermitian spin chain (cid:15) corresponds to the asymmetricity of the driven force. When the (cid:15) becomes large, the system goes away from equilibrium. Notice that we ignore the caseof the TASEP (totally asymmetric simple exclusion process) (cid:15) = 1 ( p = 1 , q = 0). Inthis case, the Bethe equations are singular and string solutions do not exist. The Betheequations of the TASEP are investigated in the refs. [18, 19]. Then the Bethe equationsare rewritten as z Lj = ( − N − N (cid:89) l (cid:54) = j (1 + (cid:15) ) − z j + (1 − (cid:15) ) z j z l (1 + (cid:15) ) − z l + (1 − (cid:15) ) z j z l , j = 1 , · · · , N. (14)We change the rapidities { z j } as z j = λ j + i λ j − i , (15)then the Bethe equations are (cid:18) λ j + i λ j − i (cid:19) L = N (cid:89) l (cid:54) = j (1 − (cid:15) ) λ j − (1 + (cid:15) ) λ l + i (1 + (cid:15) ) λ j − (1 − (cid:15) ) λ l − i , j = 1 , · · · , N. (16)When (cid:15) = 0, this equations are identical with the Eqs. (8). The Bethe equations of theHeisenberg XXX spin chain are deformed by the asymmetricity.Let’s investigate the Bethe roots of the Eq. (16) as we did in the case of theHeisenberg XXX spin chain. We assume the complex solutions: λ j = a j + ib j , a j , b j ∈ R . (17)In the thermodynamic limit L → ∞ , if the imaginary part of the solution b j >
0, theLHS of the Eq. (16) diverges to infinity. Therefore, the denominator of the RHS of theEq. (16) becomes zero:(1 + (cid:15) ) λ j − (1 − (cid:15) ) λ l − i ∼ , if b j > . (18)Similarly, when the imaginary part of the solution b j <
0, the LHS of the Eq. (16)becomes zero in the thrmodyniamic limit L → ∞ . So the numerator of the RHS of theEq. (16) becomes zero:(1 − (cid:15) ) λ j − (1 + (cid:15) ) λ l + i ∼ , if b j > . (19)These Eqs. (18),(19) means that when a λ j is a Bethe root, the λ l defined as follows isalso a Bethe root: λ l ∼ (cid:15) − (cid:15) λ j − i − (cid:15) ∼ (cid:15) − (cid:15) a j − i − (cid:15) { − (1 + (cid:15) ) b j } , if b j > λ l ∼ − (cid:15) (cid:15) λ j − i
11 + (cid:15) ∼ − (cid:15) (cid:15) a j − i (cid:15) { − (cid:15) ) b j } , if b j < undled string solutions in the non-hermitian spin chain λ l < b j if b j > λ l > b j if b j < b j < (cid:15) . (24)Therefore, the bundled string solutions are restricted in this region on the complexplane.We rewrite the Bethe roots in the bundled string to λ n,jα = a n,jα + ib n,jα , j = 1 , · · · n (25)where α is an index of bundled string, n is a string length, and j is an index of a Betheroot in a string. The indices { j } are ordered by descending order of the imaginary partsof the solutions. The example of the 5-bundled string solution is shown in the Fig. 2. Figure 2.
From the Eqs. (20),(21), we obtain recursion relations of the bundled stringsolutions λ n,jα : a n,j +1 α = 1 + (cid:15) − (cid:15) a n,jα (26) b n,j +1 α = 1 − (cid:15) (cid:15) b n,jα − − (cid:15) ) . (27)We define a parameter γ as γ = 1 + (cid:15) − (cid:15) (28) undled string solutions in the non-hermitian spin chain a n,jα = a n, α γ j − (29) b n,jα = (cid:18) b n, α − (cid:15) (cid:19) γ j − + 12 (cid:15) . (30)Therefore, bundled string solutions are discribed as λ n,jα = a n,jα + b n,jα = a n, α γ j − + i (cid:26)(cid:18) b n, α − (cid:15) (cid:19) γ j − + 12 (cid:15) (cid:27) , j = 1 , · · · , n. (31)From the Eqs. (29),(30), we get b n,jα = 2 (cid:15)b n, α − (cid:15)a n, α a n,jα + 12 (cid:15) . (32)This indicates that the bundled string solutions are located on the strait line, whoseslope is (cid:15)b n, α − (cid:15)a n, α and intercept is (cid:15) . Therefore, all strings are crossed at the point (0 , / (cid:15) ).That’s why we named this type of strings ”Bundled strings”. In the symmetric limit (cid:15) →
0, the intersection diverges infinity (0 , / (cid:15) ) → (0 , ∞ ) and the string becomesparallel to the imaginary axis. This is consistent with the traditional strings of theHeisenberg XXX spin chain. A schematic drawing of the bundled string solution andits symmetric limit is shown in Fig. 3. Figure 3.
Symmetric limit of the bundled string equation
4. Extended Bethe-Takahashi equation
In the thermodynamic Bethe ansatz, we usually treat all Bethe roots on the same stringas a quasi particle. The Bethe equations are reduced to the Bethe-Takahashi equation undled string solutions in the non-hermitian spin chain a nα of the bundled string as a intersection of thebundled string and the real axis as shown in the Fig. 3. Substitute b n,jα = 0 for the Eq.(32), then the string center a nα is given by a nα = − a n, α (cid:15) ( b n, α − (cid:15) ) (33)Here, we impose a nontrivial assumption. In the case of the Heisenberg XXX spinchain, the Bethe equations have self-conjugacy [20]. In other words, { λ j } = { ¯ λ j } . As aextension of self-conjugacy for the ASEP, we assume that the number of the solutionswhose imaginary part is positive is equal to the one whose imaginary part is negative.This assumption leads us to calculate below: b n, α = 12 (cid:15) (1 − γ − n − ) . (34)From the Eqs. (29),(30),(33),(34), we obtain a nα = a n, α γ n − (35) a n,jα = a nα γ j − n +12 (36) b n,jα = − (cid:15) γ j − n +12 + 12 (cid:15) . (37)Substituting the bundled string solutions { λ n,jα } into the Bethe equations (16), weobtain (cid:32) λ n,jα + i λ n,jα − i (cid:33) L = (cid:89) ( m,β ) (cid:54) =( n,α ) m (cid:89) k =1 (1 − (cid:15) ) λ n,jα − (1 + (cid:15) ) λ m,kβ + i (1 + (cid:15) ) λ n,jα − (1 − (cid:15) ) λ m,kβ − i × (cid:89) j (cid:48) (cid:54) = j (1 − (cid:15) ) λ n,jα − (1 + (cid:15) ) λ n,j (cid:48) α + i (1 + (cid:15) ) λ n,jα − (1 − (cid:15) ) λ n,j (cid:48) α − i , j = 1 , · · · , n. (38)By multiplying n Bethe equations on the bundled string, the last product of the Eq.(38) become n (cid:89) j =1 (cid:89) j (cid:48) (cid:54) = j (1 − (cid:15) ) λ n,jα − (1 + (cid:15) ) λ n,j (cid:48) α + i (1 + (cid:15) ) λ n,jα − (1 − (cid:15) ) λ n,j (cid:48) α − i = 1 . (39)Therefore, the product of the Bethe equations is given by n (cid:89) j =1 (cid:32) λ n,jα + i λ n,jα − i (cid:33) L = n (cid:89) j =1 (cid:89) ( m,β ) (cid:54) =( n,α ) m (cid:89) k =1 (1 − (cid:15) ) λ n,jα − (1 + (cid:15) ) λ m,kβ + i (1 + (cid:15) ) λ n,jα − (1 − (cid:15) ) λ m,kβ − i . (40) undled string solutions in the non-hermitian spin chain L = n (cid:89) j =1 λ n,jα + i λ n,jα − i = n (cid:89) j =1 γ a nα γ j − n +12 − − i (cid:15) { γ j − n +12 − − (1 − (cid:15) ) } a nα γ j − n +12 − i (cid:15) { γ j − n +12 − (1 − (cid:15) ) } = γ n a nα γ − n +12 − i (cid:15) { γ − n +12 − (1 − (cid:15) ) } a nα γ n − − i (cid:15) { γ n − − (1 − (cid:15) ) } = a nα − i (cid:15) { − (1 − (cid:15) ) n } a nα − i (cid:15) { − (1 − (cid:15) ) − n } . (41)Next, calculate the RHS of the Eq. (40). First we do the product over k : m (cid:89) k =1 (1 − (cid:15) ) λ n,jα − (1 + (cid:15) ) λ m,kβ + i (1 + (cid:15) ) λ n,jα − (1 − (cid:15) ) λ m,kβ − i = m (cid:89) k =1 (1 − (cid:15) ) λ n,jα − (1 + (cid:15) )( a m,kβ + ib m,kβ ) + i (1 + (cid:15) ) λ n,jα − (1 − (cid:15) )( a m,kβ + ib m,kβ ) − i = m (cid:89) k =1 γ − λ n,jα − a m,kβ γ k − m +12 +1 − i (cid:15) (1 − γ k − m +12 +1 ) λ n,jα − a m,kβ γ k − m +12 +1 − i (cid:15) (1 − γ k − m +12 +1 )= γ − m (cid:40) λ n,jα − a mβ γ m − − i (cid:15) (1 − γ m − ) λ n,jα − a mβ γ − m − − i (cid:15) (1 − γ − m − ) (cid:41)(cid:40) λ n,jα − a mβ γ m +12 − i (cid:15) (1 − γ m +12 ) λ n,jα − a mβ γ − m +12 − i (cid:15) (1 − γ − m +12 ) (cid:41) . (42)Here, we introduce the notation: ϕ l = a nα − a mβ γ l − i (cid:15) (1 − γ l ) , (43) undled string solutions in the non-hermitian spin chain j : n (cid:89) j =1 γ − m (cid:40) λ n,jα − a mβ γ m − − i (cid:15) (1 − γ m − ) λ n,jα − a mβ γ − m − − i (cid:15) (1 − γ − m − ) (cid:41)(cid:40) λ n,jα − a mβ γ m +12 − i (cid:15) (1 − γ m +12 ) λ n,jα − a mβ γ − m +12 − i (cid:15) (1 − γ − m +12 ) (cid:41) = n (cid:89) j =1 γ − m (cid:40) a nα − a mβ γ − j + n + m − i (cid:15) (1 − γ − j + n + m ) λ n,jα − a mβ γ − j + n − m +22 − i (cid:15) (1 − γ − j + n − m +22 ) (cid:41) n (cid:89) j =1 γ − m (cid:40) a nα − a mβ γ − j + n + m +22 − i (cid:15) (1 − γ − j + n + m +22 ) λ n,jα − a mβ γ − j + n − m − i (cid:15) (1 − γ − j + n − m ) (cid:41) = n (cid:89) j =1 γ m (cid:18) ϕ n + m − j ϕ n − m +2 − j (cid:19) (cid:18) ϕ n + m +2 − j ϕ n − m − j (cid:19) = (cid:18) γ − m ϕ m − n ϕ − ( m − n ) (cid:19) (cid:40)(cid:18) γ − m ϕ m − n +2 ϕ − ( m − n +2) (cid:19) (cid:18) γ − m ϕ m − n +4 ϕ − ( m − n +4) (cid:19) · · ·· · · (cid:18) γ − m ϕ m + n − ϕ − ( m + n − (cid:19) (cid:41) (cid:18) γ − m ϕ m + n ϕ − ( m + n ) (cid:19) = Φ nm ( a nα , a mβ ) , (44)where we defineΦ nm ( a nα , a mβ ) = (cid:18) γ − m ϕ m − n ϕ − ( m − n ) (cid:19) (cid:40)(cid:18) γ − m ϕ m − n +2 ϕ − ( m − n +2) (cid:19) (cid:18) γ − m ϕ m − n +4 ϕ − ( m − n +4) (cid:19) · · ·· · · (cid:18) γ − m ϕ m + n − ϕ − ( m + n − (cid:19) (cid:41) (cid:18) γ − m ϕ m + n ϕ − ( m + n ) (cid:19) . (45)From the Eqs. (41),(44), we obtain the Bethe-Takahashi equation for the ASEP: (cid:32) a nα − i (cid:15) { − (1 − (cid:15) ) n } a nα − i (cid:15) { − (1 − (cid:15) ) − n } (cid:33) L = (cid:89) ( m,β ) (cid:54) =( n,α ) Φ nm ( a nα , a mβ ) . (46)In the symmetric limit (cid:15) →
0, this equation becomes the Bethe-Takahashi equation forthe Heisenberg XXX spin chain.
5. Conclusion
In this paper, we investigated the Bethe ansatz equation for the ASEP in order tounderstand the deformation of the thermodynamic structures of out-of-equilibriumsystem. The ASEP is an essentially nonequilibrium model, and its Hamiltoniancontains the information of nonequilibrium. Therefore, we expect the deformation of undled string solutions in the non-hermitian spin chain (cid:15) = 0, the strings are parallel to the imaginary axis. However,when (cid:15) >
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