Complete Boundary Phase Diagram of the Spin-$\frac{1}{2}$ XXZ Chain with Boundary Fields in the Anti-Ferromagnetic Gapped Regime

J. H. Pixley; Junhyun Lee; Natan Andrei; Parameshwar R. Pasnoori; Patrick Azaria; Yicheng Tang

REVIEW 3 major objections 5 minor 76 references

Complete Boundary Phase Diagram of the Spin-$\frac{1}{2}$ XXZ Chain with Boundary Fields in the Anti-Ferromagnetic Gapped Regime

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The gapped spin-1/2 XXZ chain with boundary fields has a complete 36-region phase diagram, classified by ground-state spin and by boundary bound states that organize the Hilbert space into towers; the tower count jumps at hc1=Δ−1 and…

desk verdict Solid Bethe-ansatz phase diagram with real numerical support, but the printed central equation has a boundary factor equal to 1 and the 'complete' claim rests on an unproven string hypothesis. read the letter →

arxiv 2507.00386 v1 pith:7JFWSREE submitted 2025-07-01 cond-mat.str-el cond-mat.stat-mechquant-ph

Parameshwar R. Pasnoori , Yicheng Tang , Junhyun Lee , J. H. Pixley , Patrick Azaria , Natan Andrei This is my paper · ORCID

classification cond-mat.str-elcond-mat.stat-mechquant-ph MSC 82B2381R1282B20

keywords XXZchainBetheansatzboundaryboundstateseigenstatephasetransitionspinfractionalizationmagneticfieldsgappedantiferromagnetHilbert-spacetowers

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims a complete solution, via Bethe ansatz, of the gapped antiferromagnetic spin-1/2 XXZ chain with arbitrary diagonal boundary fields, yielding a boundary phase diagram with 36 regions. The phases are distinguished in two independent ways: by the ground state's total spin $S^z$ and by the number of boundary bound states exponentially localized at the edges. The number of bound states determines whether the Hilbert space decomposes into one, two, or four towers, and this tower count changes at the critical boundary fields $h_{c1}=\Delta-1$ and $h_{c2}=\Delta+1$, where a bound state leaks into the bulk as a spinon. Because the tower count can change while the ground state remains unchanged, the paper introduces an eigenstate phase transition, also called a Hilbert-space phase transition. DMRG and exact diagonalization are used to confirm the predicted $1/4$ boundary spin accumulation and its vanishing variance, so the fractional boundary spin is claimed to be a sharp quantum observable.

What carries the argument

The machinery is the coordinate and algebraic Bethe ansatz for the open XXZ chain, with the string hypothesis as the organizing assumption. Bethe roots are real quasi-momenta describing bulk spinons, while the boundary bound states appear as purely imaginary boundary-string roots $\lambda_{\mathrm{bs}\alpha}=\pi\pm i\gamma(1-\tilde{\epsilon}_\alpha)$ in the low-field phases and $\lambda'_{\mathrm{bs}\alpha}=\pm i\gamma(1-\tilde{\epsilon}_\alpha)$ in the high-field phases. Fourier-transformed root densities $\hat{\rho}(\omega)$ supply the total spin $S^z=N/2-M$ and the boundary-string energies $m_\beta$ and $m'_\beta$, whose equality with the spinon mass $m$ or band height $M$ fixes the critical fields. The towers are labeled by the bound-state parities $P_{L,R}$, and the physical mechanism of the transition is a bound state leaking into the bulk: at $h_{c1}$ the boundary-string energy equals the mass gap $m=E_{\theta\to\pi}$, while at $h_{c2}$ it equals the band height $M=E_{\theta\to 0}$. The paper uses DMRG and exact diagonalization to verify the $1/4$ boundary spin accumulation and to estimate the variance through the fitted ansatz of Eq. (16).

What would settle it

Numerically solve the Bethe equations for a finite chain such as $N=20$ and $\Delta=5$ at representative fields in each of the 36 regions and count all solution sectors against the predicted root types; if any eigenstate corresponds to an unclassified complex root configuration, the tower classification fails. More directly, compute the boundary-spin variance for the ground state across increasing system sizes and check whether it extrapolates to zero as $L\to\infty$ according to the ansatz in Eq. (16); if it does not vanish, the $1/4$ boundary spin is not a sharp observable.

Watch

Extended reading notes

Core claim

The central discovery is a complete classification of the gapped XXZ chain with diagonal boundary fields. In every region of the $(h_L,h_R)$ plane, the exact Bethe-ansatz root structure yields the ground state and the full set of boundary bound states: two bound states in the A, E, and F phases, one in the B and D phases, and none in the C phases. The Hilbert space splits into four, two, or one towers, with towers labeled by the parities $P_{L,R}=(-1)^{N_{L,R}}$ of the number of bound states at each edge. Crossing $h_{c1}=\Delta-1$ or $h_{c2}=\Delta+1$ destroys or creates a bound state whose energy coincides with the spinon mass $m$ or the band height $M$, so the bound state becomes a bulk spinon with rapidity $\theta\to\pi$ or $\theta\to 0$; this is the eigenstate phase transition. The ground-state spin is determined separately by which sub-phase is selected by the field directions, giving first-order level-crossing transitions between sub-phases within each alphabet phase. The paper also argues that each boundary carries average spin $\pm 1/4$ and that the variance of this boundary spin vanishes in the thermodynamic limit, making it a genuine observable.

Load-bearing premise

The tower counting and the completeness of the 36-region diagram rest on the string hypothesis, the assumption that every eigenstate of the open chain is captured by real roots plus boundary strings, bulk strings, quartets, and spinons, and the paper gives no proof of this completeness.

Editorial extensions

If this is right

If the classification is complete, the 36-region diagram is the full zero-temperature phase diagram of the gapped XXZ chain with diagonal boundary fields, with no missing phases in that parameter plane.
The eigenstate phase transition implies that global properties of the entire Hilbert space, such as the number of towers, can change discontinuously across a boundary-field line even when the ground-state spin and energy do not change.
In the A and B phases the boundary bound states lie below the bulk mass gap, so the lowest excitations of the open chain are boundary bound states rather than bulk spinons.
The $1/4$ boundary spin is a sharp observable only if its variance vanishes in the thermodynamic limit; the presented DMRG data and the fitted ansatz support this conclusion, making the fractional boundary spin measurable.
The parity labels $P_{L,R}=(-1)^{N_{L,R}}$ give a simple bookkeeping for the entire spectrum: all eigenstates fall into towers labeled by which edges host a bound state, so each phase carries a definite tower structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the eigenstate phase transition is robust, it should be visible in dynamics: boundary operators at zero or infinite temperature should show a nonanalytic signature when the tower count changes, since the towers have different boundary parity content.
The tower-counting logic may extend to other integrable open chains with boundary fields, suggesting a general principle that the number of Hilbert-space towers equals the number of boundary bound states.
Because this transition is purely spectral, it offers a case where ground-state probes such as energy or correlation functions can miss a phase transition; observables built from boundary occupation numbers may be better diagnostics.
A direct numerical count of all Bethe-root sectors for a finite chain at representative fields in each of the 36 regions could turn the completeness claim from a working assumption into a checkable statement.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take on arXiv:2507.00386. The paper delivers a genuine large piece of work: a Bethe-ansatz solution of the open XXZ chain with boundary fields in the gapped regime, classifying the boundary-field plane into 36 regions, with each phase characterized by ground-state spin and by the number of boundary bound states, plus a Hilbert-space tower decomposition. The bound-state energy formula matches DMRG in Fig. 4, and the spin-profile calculations support the 1/4 edge-spin picture. That is real, reproducible evidence, and the tower classification is internally coherent. The 'eigenstate phase transition' is mostly a new name for a bound state merging with the continuum; the tower-count perspective is useful, but it is not a new mechanism.

Now the soft spots. The biggest is the printed Bethe equation, Eq. (17): the boundary factor is sin(1/2(...))/sin(1/2(...)) with identical arguments, equal to 1. The boundary fields drop out of the central displayed equation. The appendix equations (A1 etc.) have the correct-looking structure, so the authors clearly solved something else, but the main text does not state the equation actually solved. That must be fixed. Second, the 'complete' claim rests on the string-hypothesis classification in Appendix A with no completeness proof. For this model, that is a common working assumption, and the numerics corroborate the low-lying spectrum, but 'complete' is too strong. The paper should either prove completeness or explicitly flag it as an assumption. Third, the variance vanishing is supported by a fitted ansatz, Eq. (16), and there is another typo: Eq. (15) says lim α→∞, which should be α→0. Minor, but sloppy.

Who is this for? People working on boundary effects in integrable spin chains, and numerical benchmarkers. The phase diagram will likely be useful. It deserves a serious referee. I would accept it with major revision: fix Eq. (17), state the completeness assumption, and soften 'complete' unless proven.

Referee Report

3 major / 5 minor

Summary. The paper studies the spin-1/2 XXZ chain with diagonal boundary fields in the gapped antiferromagnetic regime and claims to obtain the complete boundary phase diagram via Bethe ansatz. The phase diagram is organized into 36 regions (A through F, with subscripts), classified by the ground-state spin and by the number of boundary bound states (zero, one, or two), which is argued to equal the number of Hilbert-space towers (one, two, or four). The authors identify two boundary critical fields, hc1 = Δ - 1 and hc2 = Δ + 1, and introduce an 'eigenstate phase transition' in which the tower structure changes without a change of the ground state. Numerical DMRG and exact diagonalization are used to confirm bound-state energies, spin-1/4 edge accumulation, and the vanishing of the edge-spin variance. The Bethe ansatz solution is presented in Section IV and Appendix A, with explicit root classifications and tower constructions for odd and even chains.

Significance. If the central claims are correct, the paper provides a fairly complete analytical description of an integrable model with boundary fields, including a concrete mechanism for a Hilbert-space reorganization ('eigenstate phase transition') that is distinct from a ground-state transition. The analytic bound-state energy in Eq. (10) is checked against DMRG in Fig. 4, and the predicted boundary spin accumulation of 1/4 is verified numerically. This goes beyond the usual low-energy effective description and could be useful for understanding boundary effects in integrable spin chains. However, the 'complete' nature of the phase diagram and the tower decomposition rest on unproven assumptions about completeness of the Bethe-root classification, and the printed Bethe equation contains a typo that currently removes the boundary fields. These issues need to be resolved before the paper's strongest claims can be accepted.

major comments (3)

[§IV, Eq. (17)] The boundary term in the displayed Bethe equation is identically equal to 1 as printed, because the numerator and denominator of the product over α are both sin(1/2(λ_j + iγ(1+ε_α))). Consequently, the boundary fields hL and hR drop out of the central equation, contradicting the rest of the paper and the boundary-dependent equations used in Appendix A (e.g., Eq. (A1), Eq. (A16)). The exact equations actually solved are therefore not stated in the main text. Please correct Eq. (17) and ensure the main-text equation matches the equations used in the appendix.
[§V and Appendix A] The claim that the Hilbert space splits into one, two, or four towers, and hence the claim that the phase diagram is 'complete', relies on the assumption that every Bethe eigenstate is captured by the classification in Appendix A: real roots, boundary strings, bulk strings, quartets, and spinons. No completeness proof or counting test is provided for the open chain with boundary fields. If additional complex root configurations exist, the tower counts and the location of the eigenstate phase transitions could change. Please either provide a proof or a systematic finite-size counting check that the classification is exhaustive, or explicitly soften the 'complete' claim to the class of solutions considered.
[§III.C, Eqs. (15)-(16)] The sharpness of the boundary spin-1/4 observable is supported by the claim that the variance vanishes in the thermodynamic limit. Eq. (15) writes δS² as lim_{α→∞} lim_{L→∞} δS²(L,α), which is inconsistent with the definition in Eq. (11) where the scaling limit is α→0. Also, Eq. (16) is an assumed Ornstein-Zernicke form with fitted parameters A and B; the extrapolation to α→0 is not derived. Since the vanishing variance is load-bearing for interpreting 1/4 as a sharp quantum observable, this requires a corrected limit definition and a more careful analysis of the fitting/error budget.

minor comments (5)

[§III.C, after Eq. (16)] The sentence 'Then, δS² = lim α→0(∞, α) = S2(L, 0)' is garbled and appears to contain missing arguments and an inconsistent limit; it should be rewritten to state clearly how δS² is obtained from the fitted form.
[Table V caption] The caption says 'Values of the boundary fields corresponding to eight B phases' but the table lists D phases; the label should be corrected.
[§V.A.1] There is a typo 'excitated states' that should read 'excited states'.
[§IV, Eq. (18)] The definition of δ_α contains two cases that both give the same value; the formula as written appears to be missing a different sign or condition for the second case, and should be checked.
[§V.B.1, F2 sub-phase] The text refers to 'higher order boundary strings' but these are not explicitly defined in Appendix A; a definition or reference would help the reader verify the tower construction.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central phase diagram is derived from Bethe ansatz with DMRG as confirmation; self-citations are supporting and not load-bearing.

full rationale

The central phase diagram is not circular: the 36-region structure and the 4/2/1 tower decomposition are obtained from the Bethe-ansatz solution (Sec. IV and Appendix A) by classifying Bethe roots and constructing the low-lying sectors explicitly; the DMRG/ED calculations in Sec. III are confirmatory and are benchmarked against the Bethe-ansatz boundary bound-state energy (Fig. 4). The critical fields hc1=Delta-1 and hc2=Delta+1 are introduced as notation in Eq. (3), but their role as thresholds follows from the boundary-string solutions and bound-state energy formulas in the appendix (e.g., Eq. (A15), Eq. (22), Eq. (37)), not from a fitted parameter. The tower count is a consequence of the explicit construction; labeling towers by bound-state parities in Eq. (92) organizes already-constructed towers rather than defining the count. The paper does rely on its own prior work for the variance extrapolation ansatz (Eq. (16), citing [21]) and for the fractional-spin observability criterion [44], but these are supporting claims: the boundary bound-state spectrum is independently re-derived and numerically reproduced, and the fitted Ornstein-Zernike form does not by construction force the zero variance intercept, which is an extrapolated DMRG result. The main caveats—the unproven completeness of the string-hypothesis root classification and the apparent typo in Eq. (17) where the boundary factor equals 1—are correctness/verifiability issues, not circular reductions. No step of the derivation is equivalent to its input by definition.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central phase diagram rests on the Bethe ansatz classification of states. The paper introduces no fitted constants for the phase boundaries; the only fitted parameters appear in numerical extrapolations and profile fits.

free parameters (3)

B (variance ansatz exponent) = ≈2
Fitted to DMRG data in Eq. (16); used to extrapolate the boundary spin variance to the thermodynamic limit.
A (variance ansatz prefactor) = non-universal
Prefactor in Eq. (16) fitted to DMRG data; part of the extrapolation showing the variance vanishes.
Spin profile fit parameters (A1, A2, C1, C2, m1, m2, xi) = field-dependent
Parameters of the ansatz for the boundary spin deviation, Eq. (13), fitted to DMRG profiles; illustrative, not used in the phase diagram derivation.

assumptions (5)

domain assumption The open XXZ chain with diagonal boundary fields is Bethe ansatz integrable, and the Bethe equations (17) correctly describe the spectrum.
The paper adopts the Bethe equations from prior literature (Sklyanin, Kapustin-Skorik, Grijalva et al.) without re-deriving them; standard for this model.
domain assumption String hypothesis and completeness: all eigenstates are captured by real roots, boundary strings, bulk strings and quartets, and spinons.
The classification of phases and towers in Section V and Appendix A assumes no other types of Bethe roots exist. No completeness proof is provided; this is the main load-bearing assumption.
domain assumption Boundary bound states are exponentially localized and independent, so the left and right edge contributions add.
Used throughout, e.g., energies E0 + mL + mR, to build towers. This is expected in the gapped regime but is an assumption about the separation of scales.
domain assumption The relation between boundary field and Bethe parameter, h_alpha = -sinh(gamma) coth(epsilon_alpha gamma / 2) with the branch choices of Eq. (18), is correct.
The bound state energies and the critical values hc1, hc2 depend on this parameterization; it is stated without detailed derivation in the main text.
ad hoc to paper Ornstein-Zernicke ansatz for the variance finite-size corrections, Eq. (16).
Used to extrapolate the boundary spin variance to the thermodynamic limit and conclude it vanishes; the form is assumed, not derived.

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Cite this review

Pith. "Pith review of Complete Boundary Phase Diagram of the Spin-$\frac{1}{2}$ XXZ Chain with Boundary Fields in the Anti-Ferromagnetic Gapped Regime." pith.science (2026). https://pith.science/paper/7JFWSREE

@misc{pith2026250700386,
  author       = {Pith},
  title        = {Pith review of: Complete Boundary Phase Diagram of the Spin-$\frac12$ XXZ Chain with Boundary Fields in the Anti-Ferromagnetic Gapped Regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7JFWSREE}},
  note         = {Machine review of arXiv:2507.00386}
}

abstract

We consider the spin $\frac{1}{2}$ XXZ chain with diagonal boundary fields and solve it exactly using Bethe ansatz in the gapped anti-ferromagnetic regime and obtain the complete phase boundary diagram. Depending on the values of the boundary fields, the system exhibits several phases which can be categorized based on the ground state exhibited by the system and also based on the number of bound states localized at the boundaries. We show that the Hilbert space is comprised of a certain number of towers whose number depends on the number of boundary bound states exhibited by the system. The system undergoes boundary phase transitions when boundary fields are varied across certain critical values. There exist two types of phase transitions. In the first type the ground state of the system undergoes a change. In the second type, named the `Eigenstate phase transition', the number of towers of the Hilbert space changes, which is again associated with the change in the number of boundary bound states exhibited by the system. We use the DMRG and exact diagonalization techniques to probe the signature of the Eigenstate phase transition and the ground state phase transition by analyzing the spin profiles in each eigenstate.

Figures

Figures reproduced from arXiv: 2507.00386 by the authors.

**Figure 3.** FIG. 3: Qualitative phase diagram of the gapped XXZ spin [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** FIG. 4: (a) DMRG calculation comparing with Bethe Ansatz calculation for the energy gap ∆ [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5: Fitting the spin profile for ∆ = 3 with various boundary fields [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6: Exact diagonalization calculation for the manybody spectrum with the expectation value for edge spin operator [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7: Spin accumulation [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

76 extracted references · 65 canonical work pages

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In these cases both boundary magnetic fields point towards the same direction: along the positive z axis for the E1, E8 sub- phases and negative z axis for the E5, E4 sub-phases

Odd number of sites The (E1,E5) and (E8,E4) sub-phases. In these cases both boundary magnetic fields point towards the same direction: along the positive z axis for the E1, E8 sub- phases and negative z axis for the E5, E4 sub-phases. The pairs of sub-phases E1, E5 and E8, E4 are related by the isometry (20). Qualitatively speaking, in the sub-phases E1,8...

work page
[2]

In the sub- phase E1, the ground-state contains a bound state at the left edge and has total spin Sz = 0 and is represented by |0⟩L

Even number of sites The (E1,E5) and (E8,E4) sub-phases. In the sub- phase E1, the ground-state contains a bound state at the left edge and has total spin Sz = 0 and is represented by |0⟩L. (61) The energy of this state is E0 + mL. The state which contains a bound state at the right edge has total spin Sz = 0 and is is represented by |0⟩R. (62) The energy...

work page
[3]

(71) The ground state (71) generates a tower of excited states obtained by adding an arbitrary even number of spinons, bulk strings and quartets

Odd number of sites In the B1 phase, the ground state has total spin Sz = − 1 2 which corresponds to a static spin distribution and is represented by | −1 2 ⟩. (71) The ground state (71) generates a tower of excited states obtained by adding an arbitrary even number of spinons, bulk strings and quartets. Unlike in the A phases, there exists only a single ...

work page
[4]

This state is represented by |0⟩L

Even number of sites In the phase B1, the lowest energy state contains the bound state at the left edge and has total spin Sz = 0 with energy E0 + EL. This state is represented by |0⟩L. (75) 17 TABLE III: Energies and local fermionic parities of the ground state and the lowest energy states corresponding to each tower in all the B phases for odd number of...

work page
[5]

(79) The ground state (79) generates a tower of excited states obtained by adding an arbitrary even number of spinons, bulk strings and quartets

Odd number of sites In the D1 phase, the ground state has total spin Sz = − 1 2 which corresponds to a static spin distribution and is represented by | −1 2 ⟩. (79) The ground state (79) generates a tower of excited states obtained by adding an arbitrary even number of spinons, bulk strings and quartets. Unlike in the A phases, there exists only a single ...

work page
[6]

This state and has total 19 spin Sz = 0, −1 depending on the spin orientation of the spinon

Even number of sites In the phase D1, the lowest energy state contains a spinon with rapidity θ → π. This state and has total 19 spin Sz = 0, −1 depending on the spin orientation of the spinon. This state has energy E0 + m and is represented by |0⟩, | −1⟩. (83) The lowest excited states above (76) consist of a spinon branch with θ ̸= π. On top of this, th...

work page
[7]

The ground states in C1 , C3 are represented by | ∓1 2 ⟩ (87) respectively

Odd number of sites In the phases C1 , C3 , the ground state has total spin Sz = ∓ 1 2 respectively, which corresponds to a static spin distribution. The ground states in C1 , C3 are represented by | ∓1 2 ⟩ (87) respectively. The energy of these states is E0. On top of this, the state (87) generates a tower of excited states obtained by adding an arbitrar...

work page
[8]

Even number of sites In the phase C1 , the ground state contains a spinon with rapidity θ → π on top of the static spin distribution of the ground state in the phase C1 corresponding to odd number of sites case. It is two fold degenerate with energy E0 + m and and have total spin Sz = 0, Sz = −1 corresponding to the spin orientation of the spinon which is...

work page

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We use the notation of [19]) Appendix A: Bethe ansatz Solution In this section we construct the ground state and boundary excitations in each region of the phase diagram for both odd and even number of sites
[60]

This corresponds to ϵα = −˜ϵα + iπ, with ˜ϵα < 1, α = L, R

Region A1: odd number of sites The region A1 corresponds to the following values of the boundary magnetic fields: 0 < hL, hR < hc1. This corresponds to ϵα = −˜ϵα + iπ, with ˜ϵα < 1, α = L, R. First consider the state with all real λ, which take values between ( −π, π]. Applyin...
[61]

Region A1: Even number of sites The Bethe equations corresponding to all spin up reference state have two boundary string solutions λbsα, where λbsα = π + ±iγ(1 − ˜ϵα), α = L, R. (A15) Adding either of these two boundary strings to the Bethe equations (17) and taking logarithm...
[62]

C1 Odd and even number of sites In this region both hL, hR take the following values: hc1 < hL, hR < hc2. By starting with Bethe reference state with all spin down, and considering the state with all real λj, we obtain the following logarithmic form of Bethe equations (2N + 1)...
[63]

This corresponds to ϵα = −˜ϵα, with ˜ϵα < 1, α = L, R

F1 Odd number of sites The region F1 corresponds to the following values of the boundary magnetic fields: hc2 < hL, hR. This corresponds to ϵα = −˜ϵα, with ˜ϵα < 1, α = L, R. Starting with the Bethe equations corresponding to all spin up reference state and considering the sta...
[64]

F1: Even number of sites For even number of sites the lowest energy state is obtained by starting with Bethe equations corresponding to all spin down reference state and considering a state with all real roots and a spinon. We have (2N + 1)a(λ, 1) − X α=L,R a(λ, 1 + ˜ϵα) + a(λ...
[65]

This corresponds to ϵR = −˜ϵR, ϵL = iπ − ˜ϵLwith ˜ϵα < 1, α = L, R

E1 Odd number of sites The region E1 corresponds to the following values of the boundary magnetic fields: hc2 < hR, 0 < hL < hc1. This corresponds to ϵR = −˜ϵR, ϵL = iπ − ˜ϵLwith ˜ϵα < 1, α = L, R. Starting with the Bethe equations corresponding to all spin up reference state ...
[66]

The ground state is obtained by adding λbsR′ to the state 1 2 E1

E1: Even number of sites The Bethe equations corresponding to all spin up reference state contain two boundary string solutions λbsR′ = ±iγ(1 − ˜ϵR), λbsL = π ± iγ(1 − ˜ϵL) . The ground state is obtained by adding λbsR′ to the state 1 2 E1 . Adding λbsR′ to the Bethe equations...
[67]

This region can be further divided into two regions depending on whether hc1 < hR < sinh γ and sinh γ < hR < hc2

B1: Odd number of sites Region B1 corresponds to the following values of the boundary fields: hc1 < hR < hc2, 0 < hL < hc1. This region can be further divided into two regions depending on whether hc1 < hR < sinh γ and sinh γ < hR < hc2. a. hc1 < hR < sinh γ In the case of hc1...
[68]

hc1 < hR < sinh γ In this case we have ϵR = −˜ϵR + iπ, ˜ϵR > 1

B1: Even number of sites a. hc1 < hR < sinh γ In this case we have ϵR = −˜ϵR + iπ, ˜ϵR > 1. The logarithmic form of Bethe equations corresponding to all spin up reference state take the following form (2N + 1)a(λ, 1) + a(λ − π, ˜ϵR − 1) − a(λ − π, 1 − ˜ϵL) + a(λ − π, 1) −2πδ(λ...
[69]

This region can be further divided into two regions depending on whether hc1 < hL < sinh γ and sinh γ < hL < hc2

D1: Odd number of sites Region D1 corresponds to the following values of the boundary fields: hc1 < hL < hc2, hR > hc2. This region can be further divided into two regions depending on whether hc1 < hL < sinh γ and sinh γ < hL < hc2. a. hc1 < hL < sinh γ In the case of hc1 < h...
[70]

hc1 < hL < sinh γ In this case we have ϵL = −˜ϵL + iπ, ˜ϵL > 1

D1: Even number of sites a. hc1 < hL < sinh γ In this case we have ϵL = −˜ϵL + iπ, ˜ϵL > 1. The logarithmic form of Bethe equations corresponding to all spin up reference state take the following form (2N + 1)a(λ, 1) + a(λ − π, ˜ϵL − 1) − a(λ, 1 − ˜ϵR) + a(λ − π, 1) −2πδ(λ) − ...
[71]

In this region the logarithmic form of the Bethe equations can be obtained from (A1) by the transformation ˜ϵL → −˜ϵL

A2: Odd and even number of sites The region A2 corresponds to the following values of the boundary magnetic fields: 0 < hR < hc1, −hc1 < hL < 0. In this region the logarithmic form of the Bethe equations can be obtained from (A1) by the transformation ˜ϵL → −˜ϵL. We have (2N +...
[72]

This region can be further split into four sub regions depending on whether the absolute values of the boundary fields are greater than or less than sinh γ

C2: Even and odd number of sites In this region both hL, hR take the following values: hc1 < hR < hc2, −hc2 < hL < −hc1. This region can be further split into four sub regions depending on whether the absolute values of the boundary fields are greater than or less than sinh γ....
[73]

This corresponds to ϵR = −˜ϵR, ϵL = ˜ϵL with |˜ϵα| < 1, α = L, R

F2: Even and odd number of sites The region F2 corresponds to the following values of the boundary magnetic fields: hc2 < hR, hL < −hc2. This corresponds to ϵR = −˜ϵR, ϵL = ˜ϵL with |˜ϵα| < 1, α = L, R. Making the transformation ˜ϵL → −˜ϵL and starting with the Bethe equations...
[74]

This corresponds to ϵR = −˜ϵR, ϵL = −iπ + ˜ϵLwith |˜ϵα| < 1, α = L, R

E2 Even and odd number of sites The region E2 corresponds to the following values of the boundary magnetic fields: hc2 < hR, 0 > hL > −hc1. This corresponds to ϵR = −˜ϵR, ϵL = −iπ + ˜ϵLwith |˜ϵα| < 1, α = L, R. We use the transformation ˜ϵL → −˜ϵL. Starting with the Bethe equa...
[75]

This region can be further divided into two regions depending on whether hc1 < hR < sinh γ and sinh γ < hR < hc2

B2: Even and odd number of sites Region B2 corresponds to the following values of the boundary fields: hc1 < hR < hc2, −hc1 < hL < 0. This region can be further divided into two regions depending on whether hc1 < hR < sinh γ and sinh γ < hR < hc2. 39 a. hc1 < hR < sinh γ In th...
[76]

This region can be further divided into two regions depending on whether −hc1 > hL > − sinh γ and − sinh γ > hL > −hc2

D2: Even and odd number of sites Region D2 corresponds to the following values of the boundary fields: −hc1 > hL > −hc2, hR > hc2. This region can be further divided into two regions depending on whether −hc1 > hL > − sinh γ and − sinh γ > hL > −hc2. We make the transformation...

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Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

In these cases both boundary magnetic fields point towards the same direction: along the positive z axis for the E1, E8 sub- phases and negative z axis for the E5, E4 sub-phases

Odd number of sites The (E1,E5) and (E8,E4) sub-phases. In these cases both boundary magnetic fields point towards the same direction: along the positive z axis for the E1, E8 sub- phases and negative z axis for the E5, E4 sub-phases. The pairs of sub-phases E1, E5 and E8, E4 are related by the isometry (20). Qualitatively speaking, in the sub-phases E1,8...

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[2] [2]

In the sub- phase E1, the ground-state contains a bound state at the left edge and has total spin Sz = 0 and is represented by |0⟩L

Even number of sites The (E1,E5) and (E8,E4) sub-phases. In the sub- phase E1, the ground-state contains a bound state at the left edge and has total spin Sz = 0 and is represented by |0⟩L. (61) The energy of this state is E0 + mL. The state which contains a bound state at the right edge has total spin Sz = 0 and is is represented by |0⟩R. (62) The energy...

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[3] [3]

(71) The ground state (71) generates a tower of excited states obtained by adding an arbitrary even number of spinons, bulk strings and quartets

Odd number of sites In the B1 phase, the ground state has total spin Sz = − 1 2 which corresponds to a static spin distribution and is represented by | −1 2 ⟩. (71) The ground state (71) generates a tower of excited states obtained by adding an arbitrary even number of spinons, bulk strings and quartets. Unlike in the A phases, there exists only a single ...

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[4] [4]

This state is represented by |0⟩L

Even number of sites In the phase B1, the lowest energy state contains the bound state at the left edge and has total spin Sz = 0 with energy E0 + EL. This state is represented by |0⟩L. (75) 17 TABLE III: Energies and local fermionic parities of the ground state and the lowest energy states corresponding to each tower in all the B phases for odd number of...

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[5] [5]

(79) The ground state (79) generates a tower of excited states obtained by adding an arbitrary even number of spinons, bulk strings and quartets

Odd number of sites In the D1 phase, the ground state has total spin Sz = − 1 2 which corresponds to a static spin distribution and is represented by | −1 2 ⟩. (79) The ground state (79) generates a tower of excited states obtained by adding an arbitrary even number of spinons, bulk strings and quartets. Unlike in the A phases, there exists only a single ...

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[6] [6]

This state and has total 19 spin Sz = 0, −1 depending on the spin orientation of the spinon

Even number of sites In the phase D1, the lowest energy state contains a spinon with rapidity θ → π. This state and has total 19 spin Sz = 0, −1 depending on the spin orientation of the spinon. This state has energy E0 + m and is represented by |0⟩, | −1⟩. (83) The lowest excited states above (76) consist of a spinon branch with θ ̸= π. On top of this, th...

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[7] [7]

The ground states in C1 , C3 are represented by | ∓1 2 ⟩ (87) respectively

Odd number of sites In the phases C1 , C3 , the ground state has total spin Sz = ∓ 1 2 respectively, which corresponds to a static spin distribution. The ground states in C1 , C3 are represented by | ∓1 2 ⟩ (87) respectively. The energy of these states is E0. On top of this, the state (87) generates a tower of excited states obtained by adding an arbitrar...

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[8] [8]

Even number of sites In the phase C1 , the ground state contains a spinon with rapidity θ → π on top of the static spin distribution of the ground state in the phase C1 corresponding to odd number of sites case. It is two fold degenerate with energy E0 + m and and have total spin Sz = 0, Sz = −1 corresponding to the spin orientation of the spinon which is...

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