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Edge modes and boundary impurities in the anisotropic Heisenberg spin chain

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A boundary spin-1/2 impurity rearranges the XXZ chain's fractional ±1/4 edge modes into effective quarter or three-quarter modes and, when integrability breaks, creates a new mid-gap phase.

desk verdict The integrable four-phase classification for impurities in the gapped XXZ chain is solid and worth refereeing, but the new non-integrable mid-gap phase and ferromagnetic screening claims rely on fixed-size numerics and need finite-size scaling before they can be taken as established. read the letter →

arxiv 2412.00194 v2 pith:Q262CJRQ submitted 2024-11-29 cond-mat.str-el cond-mat.stat-mechhep-thmath-phmath.MPquant-ph

Pradip Kattel , Parameshwar R. Pasnoori , J. H. Pixley , Natan Andrei This is my paper · ORCID
classification cond-mat.str-elcond-mat.stat-mechhep-thmath-phmath.MPquant-ph MSC 82B2382B2082B27
keywords XXZspinchainboundaryimpurityfractionaledgemodesKondoeffectBetheansatzmid-gapstatesdensitymatrixrenormalizationgroupscreening
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 establishes the full phase diagram for a single spin-1/2 impurity coupled to the edge of a gapped antiferromagnetic XXZ spin chain. In the integrable case, the impurity sits in one of four phases—Kondo screened, antiferromagnetic bound mode, ferromagnetic bound mode, or fully unscreened—each distinguished by how the impurity interacts with the fractionalized ±1/4 spin accumulations at the chain edges. The hallmark result is that these fractional edge modes survive the impurity and rearrange with it: antiferromagnetic coupling effectively flips the edge mode, while ferromagnetic coupling can align with it to form a ±3/4 mode or anti-align to leave an effective ±1/4 mode. When the boundary coupling is deliberately made non-integrable, an additional phase appears for antiferromagnetic coupling, characterized by a pair of degenerate mid-gap states below the bulk mass gap; even ferromagnetic coupling then drives the impurity toward screening rather than leaving it free. If correct, the work provides a controlled setting—partly exact, partly numerically corroborated—for understanding how boundary impurities reconfigure fractionalized edge degrees of freedom in gapped spin chains.

What carries the argument

The machine that carries the argument is the Bethe-ansatz solution of the open XXZ chain with integrable boundary impurities, together with the boundary string solution $\lambda_d = \pm i(\eta-2d)$, an imaginary root of the Bethe equations that describes an exponentially localized bound mode. Its energy $E_d(J,\eta,d)$, given by a convergent sum, determines the phase boundaries: $E_d$ is negative and below the single-spinon minimum in the ABM phase, positive in the FBM phase, and ceases to be finite in the unscreened phase. The fractional edge modes are defined through the operators $\hat{S}^z_L = \lim_{\alpha\to 0} \lim_{N\to\infty} \sum_j e^{-\alpha j} S_j^z$, whose eigenvalues are quantized sharp observables ($\pm1/4$) with vanishing variance; their stability under the impurity is checked numerically via a scaling ansatz for the variance. The impurity entropy difference (computed with finite-temperature tensor-network purification) distinguishes the phases by the flow from $\ln 2$ in the UV to $0$ (screened) or $\ln 2$ (free) in the IR. For the non-integrable Hamiltonian, the phase boundaries $J_{c1}^*(\eta)$ and $J_{c2}^*(\eta)$ are extracted from DMRG magnetization jumps and exact-diagonalization low-lying spectra, the latter showing the two-fold degenerate mid-gap states.

What would settle it

Track the two lowest excited states in the non-integrable antiferromagnetic chain at $\eta=2$, $J_{\rm imp}/J=0.3$ as the system size grows from 11 to 17 bulk sites; the claim of a mid-gap phase requires a pair of degenerate $S^z=\pm1$ states below the bulk mass gap, whereas a finite-size artifact would have their splitting grow or their energy rise above the gap.

Watch

Extended reading notes

Core claim

The central discovery is a complete boundary phase diagram for the spin-1/2 XXZ chain with an impurity, described by a single impurity parameter $d$ in the integrable limit (coupling and boundary anisotropy linked by the integrability condition $J_q = J \sinh^2\eta \cosh d_q / (\sinh^2\eta - \sinh^2 d_q)$, $\Delta_q = \cosh \eta / \cosh d_q$). For antiferromagnetic coupling, the impurity is always screened in the ground state, either by multiparticle Kondo physics ($d$ purely imaginary or $0 < d < \eta/2$) or by an exponentially localized single-particle bound mode ($\eta/2 < d < \eta$, a purely imaginary Bethe root $\lambda_d = \pm i(\eta-2d)$). For ferromagnetic coupling the impurity remains free in the ground state, with a high-energy bound mode available for screening only when $\eta<d<3\eta/2$ (FBM) and no screening at all for $d>3\eta/2$ (US). In these phases, the fractionalized $\pm1/4$ edge modes—defined by sharply quantized local operators with vanishing variance—persist and rearrange: the antiferromagnetic impurity flips the adjacent quarter mode (turning $(1/4,-1/4)$ edge pairs into $(-1/4,-1/4)$ and so on), while a ferromagnetic impurity either flips it or aligns to create a three-quarter mode. When integrability is broken (same bulk and boundary anisotropy, arbitrary coupling ratio $J_{\rm imp}/J$), an antiferromagnetic impurity shows three phases: a mid-gap phase for $0 < J_{\rm imp}/J < J_{c1}^*$, a Kondo phase for $J_{c1}^* < J_{\rm imp}/J < J_{c2}^*$, and an ABM phase for $J_{\rm imp}/J > J_{c2}^*$; the mid-gap phase carries two degenerate states below the mass gap with edge spin accumulation $\pm 3/4$, $\pm 1/4$. Non-integrable ferromagnetic coupling, in contrast to the integrable case, removes the four-fold degeneracy and yields a screened ground state with impurity entropy flowing from $\ln 2$ to $0$.

Load-bearing premise

The load-bearing premise is that the non-integrable phase diagram seen in moderate-size numerical simulations persists in the thermodynamic limit.

Editorial extensions

If this is right

  • Each integrable impurity phase has a distinct impurity-entropy temperature flow: monotonic $\ln 2 \to 0$ in Kondo, non-monotonic to $0$ in ABM, constant $\ln 2$ in US, and a dip back to $\ln 2$ in FBM; these flows give experimentally testable thermodynamic signatures.
  • The fractional $\pm1/4$ edge modes are robust to a boundary impurity, so the predicted quarter/three-quarter spin accumulations should be observable as quantized local magnetization plateaus in cold-atom or solid-state realizations of the XXZ chain.
  • In the non-integrable antiferromagnetic case, the existence of two degenerate mid-gap states below the mass gap means the low-energy spectrum is not fixed by the bulk gap alone; impurities can create effective two-level systems that dominate low-temperature response.
  • Non-integrable ferromagnetic coupling fundamentally changes the impurity's fate: instead of a free local moment, the impurity is screened at low temperature, contradicting the naive expectation that ferromagnetic exchange leaves impurity spins unscreened.
  • With two impurities, each edge independently chooses one of the four phases, yielding 16 combined phases; the mass gap makes the two impurities effectively independent in the thermodynamic limit.

Reading between the lines

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

  • The mid-gap phase is plausibly a generic phenomenon in gapped spin chains with an impurity that breaks integrability: the same two-state sub-gap response is known in superconducting impurity problems, so the DMRG/ED machinery here could be exported to spin-$S$ chains that carry $\pm S/2$ edge modes.
  • The flips/aligned rearrangements suggest a route to effectively build spin-$3/2$ edge objects (a $1/4$ edge mode plus an aligned spin-$1/2$ impurity) that are sharp observables; this could be a way to locally engineer the ground-state manifold of the chain.
  • The thermodynamic fate of the non-integrable ferromagnetic impurity could be settled by measuring the impurity entropy at the lowest reachable temperature in a longer chain: the paper predicts it goes to $0$, whereas a free local moment would keep it at $\ln 2$.
  • The location of $J_{c1}^*(\eta)$ where the mid-gap states disappear might coincide with the formation of the Kondo screening cloud; checking whether the Kondo length diverges at that boundary would link the mid-gap phase to the onset of many-body screening.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies a spin-1/2 anisotropic Heisenberg (XXZ) chain in its gapped antiferromagnetic phase with boundary impurities. For integrable boundary couplings, the authors use Bethe ansatz to classify four impurity phases: Kondo, antiferromagnetic bound mode (ABM), ferromagnetic bound mode (FBM), and unscreened (US), and they describe how fractionalized ±1/4 edge modes rearrange with the impurity into effective quarter or three-quarter modes. For non-integrable couplings, the paper claims a richer phase diagram for antiferromagnetic couplings, including a new mid-gap phase, and that non-integrable ferromagnetic coupling also leads to impurity screening. The evidence combines Bethe ansatz string analyses, exact diagonalization at small system sizes, DMRG magnetization profiles, and finite-temperature purification entropy calculations.

Significance. If the non-integrable phase diagram survives thermodynamic-limit scrutiny, the paper provides a useful classification of boundary impurity phases in a gapped spin chain and a concrete setting in which fractionalized edge modes coexist with Kondo-like screening. The integrable part is a clear strength: the Bethe ansatz root analysis is detailed, the bound-mode energy expressions are explicit, and the DMRG/ED data are consistent with the calculated phase structure. The paper also gives a helpful physical distinction between many-body Kondo screening and single-particle bound-mode screening, and it carefully contrasts the XXZ edge modes with topological Haldane-chain edge modes. The main weakness is that the new non-integrable claims, which are central to the abstract and conclusion, are inferred from fixed-size numerics without finite-size scaling or error bars.

major comments (3)
  1. [§V A, Figs. 23-26] The non-integrable mid-gap phase and the phase boundaries J*_c1(eta) and J*_c2(eta) are extracted from DMRG at a single system size (N=500) and exact diagonalization at Nb=11/13, with no finite-size scaling and no error bars on the boundary locations. Because an impurity in a gapped chain generically produces in-gap bound states at finite size, the magnetization jump at h*_1 < Mg and the two low-lying states in Fig. 26 do not by themselves establish a thermodynamic phase. The authors should show that the two degenerate in-gap states remain separated from the two-spinon continuum as N increases, that the splitting between them decreases (or at least does not grow), and that the critical coupling J*_c1 does not flow to zero or merge with the Kondo crossover. Without such an extrapolation, the claimed third phase and the numerical phase boundaries in Fig. 25 are not established in the thermodynamic limit.
  2. [§V B, Figs. 28-29] The claim that non-integrable ferromagnetic coupling screens the impurity rests on the two-fold degeneracy of the ground state at Nb=11 and on the impurity-entropy curve in Fig. 29, for which the system size, bond dimension, and convergence criteria are not stated. A two-fold degenerate ground state can also arise from a boundary valence-bond-like polarization that is not Kondo screening, and the entropy curve alone does not distinguish these scenarios. The authors should provide a system-size and bond-dimension analysis of Simp(T), including the low-temperature limit, and confirm that the two-fold degeneracy survives in the thermodynamic limit before concluding that the impurity is screened.
  3. [§V A, Fig. 25] The phase boundary curves J*_c1(eta) and J*_c2(eta) are presented without error bars, without a description of the DMRG sweep resolution in the magnetic field, and without a statement of how the boundary is determined from the magnetization curves. Since these curves constitute the quantitative prediction for the non-integrable phase diagram, the extraction procedure and its uncertainties need to be documented, and ideally the boundaries should be checked against a second diagnostic (for example, an energy-gap or entanglement-based criterion).
minor comments (5)
  1. [Fig. 26 caption and §V A text] The text states that the exact diagonalization for the mid-gap phase is performed with Nb=11, while the caption of Fig. 26 reports Nb=13; this inconsistency should be corrected.
  2. [§VI] There is a typo in the conclusion: “magimum energy” should be “maximum energy.”
  3. [Fig. 25] The phrase “computed for various values of the crossing parameter eta” is vague; the range of eta and the number of data points used to draw the phase boundaries should be stated in the caption or text.
  4. [Fig. 29] The impurity-entropy plot lacks axis labels, units, and a statement of the bond dimension and total system size used in the purification calculation; these details are needed to judge convergence.
  5. [General] A data/code availability statement would be helpful, since several numerical results (DMRG profiles, finite-temperature purification curves, and the extraction of J*_c1/J*_c2) are not reported with enough detail to be independently reproduced without additional information.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the central phase diagrams are derived from Bethe ansatz and independent numerics, with only a minor self-cited ansatz used for the edge-mode variance extrapolation.

full rationale

The central integrable phase diagram is derived from the Bethe ansatz string analysis (Eqs. (3)-(5), (20)-(21), Appendix D): the Kondo, ABM, FBM, and US phases are distinguished by the sign and magnitude of the boundary string energy relative to the spinon gap, not by any fitted parameter. The non-integrable phase diagram is obtained from DMRG magnetization curves and exact-diagonalization spectra (Figs. 23-26); the mid-gap phase is identified by a magnetization jump below the mass gap and by two degenerate in-gap states. These are numerical observations with finite-size extrapolation concerns, but the data are not constructed from the phase boundaries they are used to infer. The only self-citation that enters the derivation is the variance ansatz Eq. (16) from [59,100], used to extrapolate the edge-mode variance to the thermodynamic limit and thereby assert that the fractional ±1/4 edge modes survive in the presence of impurities. That ansatz was verified in the clean chain in the cited prior work; in the impurity setting it is an adopted extrapolation form rather than a result re-derived here, so the edge-mode survival sub-claim is somewhat dependent on same-author prior work. This is a minor self-citation and ansatz-dependence, not a circular reduction: the ansatz does not assume the value of the extrapolated variance, and the central phase diagram does not rest on it. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is used to forbid alternatives. The limitations noted in the paper, such as the lack of an analytic bound-mode energy in a magnetic field and the deferral of detailed non-integrable ferromagnetic phases to future work, are correctness/completeness issues rather than circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper imports the fractional edge-mode formalism, the variance ansatz, and the isotropic phase structure from self-cited prior work. The main new load-bearing assumptions are the completeness of the Bethe string configurations and the thermodynamic reliability of fixed-size DMRG for the non-integrable phase boundaries.

free parameters (2)
  • A_alpha, B_alpha in variance ansatz Eq. (16) = not reported; fitted to DMRG variance data
    Used to extrapolate the vanishing variance of fractional edge spin operators and to claim sharp edge modes in the presence of impurities.
  • Non-integrable phase boundaries J*_c1(eta) and J*_c2(eta) = numerical DMRG curves in Fig. 25
    These curves define the non-integrable phase diagram and are reported without error bars or a closed-form expression.
assumptions (5)
  • domain assumption The Bethe ansatz string hypothesis: all low-energy states are captured by real roots plus the specific boundary strings lambda_bs = pi +/- i*eta and lambda_d = +/- i*(eta - 2d), with higher-order strings when needed.
    Used throughout Appendix D to construct ground states and compute bound-mode energies; no completeness proof is given.
  • domain assumption Spontaneous Z2 symmetry breaking gives two degenerate vacua in the thermodynamic limit, with a finite mass gap and exponentially decaying correlations.
    Used in Section III to set the bulk framework and in the impurity entropy arguments.
  • ad hoc to paper The variance extrapolation ansatz deltaS^2(N,alpha) = deltaS^2(infty,alpha) - A_alpha exp(-B_alpha N) remains valid with an impurity attached.
    Eq. (16), inherited from ref [59], is fitted to DMRG data to claim sharp fractional edge modes in the presence of impurities.
  • domain assumption DMRG with truncation cutoff 10^-10 and 100 sweeps converges to ground-state properties at N=500, and phase boundaries do not require finite-size extrapolation.
    Used for all non-integrable phase diagram claims in Section V.
  • standard math The double-row transfer matrix construction and Yang-Baxter relation guarantee integrability for the special boundary couplings of Eq. (2).
    Appendix B uses this to derive the Bethe equations; this is standard in algebraic Bethe ansatz.

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Cite this review

Pith. "Pith review of Edge modes and boundary impurities in the anisotropic Heisenberg spin chain." pith.science (2026). https://pith.science/paper/Q262CJRQ

@misc{pith2026241200194,
  author       = {Pith},
  title        = {Pith review of: Edge modes and boundary impurities in the anisotropic Heisenberg spin chain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q262CJRQ}},
  note         = {Machine review of arXiv:2412.00194}
}
abstract

We present a comprehensive analysis of boundary phenomena in a spin-$\frac{1}{2}$ anisotropic Heisenberg chain (XXZ-$\frac{1}{2}$) in the gapped antiferromagnetic phase, with a particular focus on the interplay between fractionalized spin-$\frac{1}{4} $ edge modes and a coupled spin-$\frac{1}{2}$ impurity at the edge. Employing a combination of Bethe Ansatz, exact diagonalization, and density matrix renormalization group (DMRG) methods, we explore the intricate phase diagram that emerges when the impurity is coupled either integrably or non-integrably to the chain. For integrable antiferromagnetic impurity couplings, we identify two distinct phases: the Kondo phase, where the impurity is screened by a multiparticle Kondo effect, and the antiferromagnetic bound mode phase, where an exponentially localized bound state screens the impurity in the ground state. When coupled ferromagnetically while maintaining integrability, the impurity behaves as a free spin-$\frac{1}{2}$, leading to either a ferromagnetic bound mode phase, where the impurity remains free in the ground state but may be screened at higher energy excitations or an unscreened (or local moment) phase where impurity remains unscreened in every eigenstate whereas for non-integrable ferromagnetic coupling, the impurity is not free. In the case of non-integrable antiferromagnetic coupling, a third phase emerges, characterized by mid-gap excitations with two degenerate states below the mass gap on top of the Kondo and antiferromagnetic bound mode phases, further enriching the phase diagram. Our findings highlight the nuanced behavior of boundary impurities in gapped antiferromagnetic systems, offering new insights into Kondo effects and impurity screening in the presence of fractionalized edge modes and bulk antiferromagnetic order.

Figures

Figures reproduced from arXiv: 2412.00194 by the authors.

Figure 1
Figure 1. FIG. 1: Impurity entropy ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Energy of spinon for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Schematic representation of the spin accumula [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (29 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Schematic representation of the spin accumula [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Phase diagram for Hamiltonian ( [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7: When a spin- [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 6
Figure 6. Figure 6: FIG. 6: When a spin- [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The plot offers a visualization of the spectral [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Local impurity magnetization for various values [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Energy of the impurity bound mode exponen [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Schematic of low-lying excitation in the Kondo [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Local impurity magnetization for various values [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: The exponentially localized + [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Schematic of low-lying excitation in the anti [PITH_FULL_IMAGE:figures/full_fig_p013_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: The four-fold degenerate ground state a) and b) [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17: The left fractional spin accumulation on the four-fold degenerate ground state when [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]
Figure 21
Figure 21. Figure 21: FIG. 21: Schematic of low lying excitation in the un [PITH_FULL_IMAGE:figures/full_fig_p015_21.png]
Figure 19
Figure 19. Figure 19: FIG. 19: Energy of the impurity bound mode exponen [PITH_FULL_IMAGE:figures/full_fig_p016_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20: Schematic of low-lying excitation in the ferro [PITH_FULL_IMAGE:figures/full_fig_p016_20.png]
Figure 23
Figure 23. Figure 23: FIG. 23: Local impurity magnetization for various val [PITH_FULL_IMAGE:figures/full_fig_p017_23.png]
Figure 25
Figure 25. Figure 25: FIG. 25: The phase boundary differentiating the three [PITH_FULL_IMAGE:figures/full_fig_p018_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26: 50 lowest lying excitations of Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p018_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27: Left localized [PITH_FULL_IMAGE:figures/full_fig_p019_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28: 50 lowest lying excitations of Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p019_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29: The impurity entropy is ln(2) in the UV and 0 [PITH_FULL_IMAGE:figures/full_fig_p020_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30: The ground state of the Haldane chain with open boundary conditions is four-fold degenerate, with each [PITH_FULL_IMAGE:figures/full_fig_p040_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31: A single spin- [PITH_FULL_IMAGE:figures/full_fig_p040_31.png]
Figure 32
Figure 32. Figure 32: FIG. 32: Unique ground state in Haldane chain with two impurities are the edges that are coupled antiferromagneti [PITH_FULL_IMAGE:figures/full_fig_p041_32.png]
Figure 33
Figure 33. Figure 33: FIG. 33: The 50 low-lying entanglement spectra [PITH_FULL_IMAGE:figures/full_fig_p041_33.png]
Figure 34
Figure 34. Figure 34: FIG. 34: The spin profile in the two-fold degenerate ground state of spin-1 XXZ chain, both of which have bulk [PITH_FULL_IMAGE:figures/full_fig_p042_34.png]
Figure 35
Figure 35. Figure 35: FIG. 35: The spin profile in two-fold degenerate spin-1 XXZ chain with spin- [PITH_FULL_IMAGE:figures/full_fig_p043_35.png]
Figure 36
Figure 36. Figure 36: FIG. 36: The spin profile in two-fold degenerate spin-1 XXZ chain with spin- [PITH_FULL_IMAGE:figures/full_fig_p043_36.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. What is the topological dual of the XXZ spin Chain?

    cond-mat.str-el 2025-07 conditional novelty 5.0 of 10

    The U(1)-symmetric XXZ spin chain is dual, via a modified Jordan-Wigner transformation, to a local fermionic model whose two gapped phases are topological and which are separated by a critical Luttinger liquid.

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    Other phases a. ABM-ABM phase When both impurity parameters take values between η/2 and η, the model is in the ABM-ABM phase. The ground state is constructed by adding the boundary strings solution λd and λb on top of all the real roots of Bethe equations and the complex bound...

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