Nonperturbative fate of interactions and the tricritical point on the infinite-randomness critical line of the random XYZ spin chain

The antiferromagnetic XYZ spin chain with quenched bond randomness has a critical line separating localized x- and y-polarized Ising antiferromagnetic phases. Roberts and Motrunich (2021) showed with unbiased tensor-network ("rigorous renormalization group") numerics that this line is governed by infinite-randomness fixed points with continuously varying exponents for disorder-averaged correlations, proved analogous varying exponents in a solvable "locally correlated" random XY model, and conjectured that the microscopic $J^z$ couplings — four-fermion interactions in the Majorana representation — are renormalization-group irrelevant for all coupling bandwidths $\tilde J^z<1$ beyond inducing local correlations between the renormalized transverse couplings. The conjecture and its corollaries remain open: no subsequent work has treated the interacting random XYZ chain nonperturbatively. The closest result (Chepiga and Laflorencie, 2024) established stability of the noninteracting infinite-randomness fixed point against interactions in the related single-chain Ising–Majorana model, which does not settle the two-chain, $(\mathbb Z_2)^2$-symmetric XYZ problem. The surviving open core is to determine nonperturbatively the fate of interactions along the entire critical line, the dynamical exponent $\psi$ and the quantitative map to the effective local-correlation parameter $\delta_{\rm eff}$, and the nature of the $S_3$-symmetric tricritical (XYZC) fixed point governing the transition into the z-antiferromagnetic phase.

Background

Strong-disorder renormalization group. The strong-disorder renormalization group (SDRG), introduced by Ma, Dasgupta, and Hu for random Heisenberg chains and systematized by Fisher, treats one-dimensional quantum spin chains with quenched randomness by repeatedly eliminating the largest local energy scale: the spins connected by the strongest coupling lock into a cluster, and the weaker couplings to their neighbors are renormalized perturbatively. For random antiferromagnetic XX and Heisenberg chains the flow reaches an infinite-randomness fixed point (IRFP), at which the distribution of coupling log-weights broadens without limit: the ground state is a random-singlet phase (spins paired into nearly independent singlets at all length scales), disorder-averaged spin correlations decay as power laws ($\eta=2$ for the XX chain) while typical correlations decay much faster, and the energy $E$ of a singlet of length $L$ obeys activated scaling $\ln(1/E)\sim L^{\psi}$ with tunneling exponent $\psi=1/2$ (formally an infinite dynamical exponent).

Random XXZ chains and multicriticality. Fisher extended this analysis to the random XXZ chain. The critical point of the random XY chain (between x- and y-polarized Ising-ordered phases) maps to two decoupled copies of the random transverse-field Ising chain, giving $\eta_x=\eta_y=4-2\phi$ with $\phi=(1+\sqrt 5)/2$ the golden ratio; weak in-plane anisotropy is a marginal perturbation about the XX point, and Fisher conjectured a line of fixed points connecting the XX and XY IRFPs. In the XXZ chain with dominant $J^z$, a transition into a z-polarized antiferromagnet occurs at a fixed point Fisher called XXZC, which is random-singlet-like. Damle and Huse analyzed a family of permutation-symmetric multicritical points at which $n$ types of locally ordered domains meet; their IRFPs have $\psi=1/n$ and $\nu=(\sqrt{4n+1}+1)/2$ (equivalently $2n/(\sqrt{4n+1}-1)$; for $n=2$ this reproduces the random-singlet value $\nu=2$).

The random XYZ chain. Consider the antiferromagnetic XYZ chain with independent random positive bond couplings

$$H=\sum_{j=1}^{N-1}\left(J^x_j\,\sigma^x_j\sigma^x_{j+1}+J^y_j\,\sigma^y_j\sigma^y_{j+1}+J^z_j\,\sigma^z_j\sigma^z_{j+1}\right),$$

with all couplings drawn from distributions of the same shape and bandwidths $\tilde J^{x,y,z}$. The model has a $\mathbb Z_2\times\mathbb Z_2$ symmetry (global spin flips about two axes) and, when all three bandwidths are equal, a statistical $S_3$ permutation symmetry; this statistically isotropic point (denoted XYZC) is a tricritical point where the x–y critical line meets the transitions into the z-antiferromagnetic (z-AFM) phase, which sets in for $\tilde J^z$ larger than the other two bandwidths. A Jordan–Wigner transformation (the standard exact mapping of spin-$1/2$ operators to fermions) maps the model onto two coupled hopping chains of Majorana fermions — real fermion operators that are their own Hermitian conjugates — (the X and Y chains): the $J^x$ and $J^y$ couplings become the alternating hopping amplitudes on the two chains, while the $J^z$ couplings become four-fermion interactions coupling them. At $J^z=0$ the problem is free-fermion; for $J^z\neq0$ it is interacting, and the SDRG generates an uncontrolled hierarchy of four-, six-, and higher-fermion terms.

Status from the source paper. Slagle, You, and Xu studied this chain with the spectrum bifurcation renormalization group (SBRG), an excited-state SDRG applicable to Hamiltonians written as sums of products of Pauli operators, and reported continuously varying exponents along the critical line, interpreting the transition as "marginal many-body localization" — eigenstates that fail to thermalize while showing a logarithmic violation of the entanglement area law. This interpretation relies on many-body localization (MBL) stability at finite energy density, which resonance-based arguments have since called into question. Roberts and Motrunich re-examined the ground state with the "rigorous renormalization group" (RRG), an unbiased tensor-network method targeting the low-energy subspace, up to $N=80$ spins. They confirmed infinite-randomness phenomenology with continuously varying exponents for averaged correlations and an unchanged effective central charge ($c_{\rm eff}=c\ln 2$, entering $S(n)\sim(c_{\rm eff}/6)\ln n$). A self-consistent Hartree–Fock (mean-field) treatment of the $J^z$ interactions captures much of this physics, and an exactly solved "locally correlated" random XY model — in which the $J^x$ and $J^y$ bond strengths on each link have correlation $\delta\in[0,1]$ — proves continuously varying exponents via a mapping to two-dimensional random walks: $\eta_z^e=\pi/\arccos(-\delta)$ and $\eta_z=2\pi/\arccos(-\delta)$ for end-to-end and bulk $z$-correlations, interpolating from the XY values ($\delta=0$) to the XX values ($\delta=1$), with $\psi=1/2$ along the whole free-fermion line. Their central conjecture: along the interacting critical line ($\tilde J^x=\tilde J^y\ge\tilde J^z$), the generated interactions flow to zero while inducing a nonzero local correlation $\delta_{\rm eff}$ between the renormalized transverse couplings, so that the long-distance physics is controlled by the free line of IRFPs with exponents fixed by $\delta_{\rm eff}$, for all $\tilde J^z<1$ up to the transition into the z-AFM phase. RRG gap scaling gives $\psi$ close to but possibly drifting from $1/2$, and at $\tilde J^z\to1$ appears inconsistent with $n=3$ Damle–Huse universality, though the authors regard this evidence as finite-size limited and state that their methods cannot access the XYZC fixed point.

Problem Statement

Determine, nonperturbatively and at strong disorder, the critical behavior of the random antiferromagnetic XYZ chain

$$H=\sum_{j=1}^{N-1}\left(J^x_j\,\sigma^x_j\sigma^x_{j+1}+J^y_j\,\sigma^y_j\sigma^y_{j+1}+J^z_j\,\sigma^z_j\sigma^z_{j+1}\right),$$

with independent positive random couplings of identical distribution shape — following the source, $p(J^\alpha_i)=\frac{1}{\Gamma\tilde J^\alpha}(J^\alpha_i)^{1/\Gamma-1}$ for $J^\alpha_i\in[0,(\tilde J^\alpha)^\Gamma]$, where $\Gamma>0$ is the shape parameter of the common distribution family (the source's numerics use $\Gamma=2$; the specific value should have little effect on the universal low-energy physics for large enough systems) — and bandwidths $\tilde J^{x,y,z}$, along the critical line $\tilde J^x=\tilde J^y\ge\tilde J^z$ separating the localized x-AFM and y-AFM phases, in the normalization $\tilde J^x=\tilde J^y=1$ used by the source, so that statistical isotropy sits at $\tilde J^z=1$, and at its endpoint. The components of the common objective are:

(i) Fate of interactions. Establish or refute the conjecture that the four-fermion and higher interaction terms generated under renormalization are irrelevant for all $\tilde J^z<1$, so that the line is controlled at long distances by the free-Majorana line of infinite-randomness fixed points with locally correlated hoppings. In particular, determine — e.g. by a systematic numerical SDRG that tracks all generated multi-fermion terms and decimates interaction terms when they are the strongest, by ground-state-projected SBRG, or by controlled unbiased numerics at larger sizes — whether the interaction strengths progressively decrease relative to the Majorana hoppings under the flow, or whether a genuine crossover to a new interacting infinite-randomness fixed point occurs at some finite $\tilde J^z<1$.

(ii) Exponents along the line. Determine whether the tunneling exponent $\psi$ (the exponent in the activated dynamical scaling $\ln(1/E)\sim L^{\psi}$ relating energy and length scales) equals $1/2$ along the entire interacting critical line or varies with $\tilde J^z$, separating true drift from slow crossover; and, should the irrelevance scenario of component (i) be confirmed, compute the quantitative map $\delta_{\rm eff}(\tilde J^z,\Gamma)$ from the microscopic bandwidths and disorder distribution (of shape parameter $\Gamma$) to the renormalized local-correlation parameter, so that the disorder-averaged correlation exponents follow from the locally correlated model (bulk and end-to-end $z$-correlation exponents $\eta_z=2\pi/\arccos(-\delta_{\rm eff})$ and $\eta_z^e=\pi/\arccos(-\delta_{\rm eff})$, with the transverse exponents obtained by the same characterization).

(iii) Tricritical point. Characterize the $S_3$-symmetric tricritical (XYZC) fixed point at statistical isotropy that controls the transition into the z-AFM phase, and the universality class of that transition — free-fermion-like IRFP, Damle–Huse-type multicritical IRFP (for $n=3$: $\psi=1/3$, $\nu=(\sqrt{13}+1)/2$), or a distinct non-free-fermion IRFP.

An answer must resolve components (i)–(iii) for the stated Hamiltonian; partial resolutions of individual components do not complete the problem.

References

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