Multicritical structure and emergent symmetries of the interacting Majorana chain with competing interactions

The interacting Majorana chain with two competing short-range interactions — a four-Majorana coupling $g$ and the O'Brien–Fendley coupling $f$ — realizes at its self-dual point a phase diagram with nine phases, including three incommensurate floating (Luttinger-liquid) phases, two tri-critical Ising critical lines, an extended Lifshitz critical line with dynamical exponent $z=3$, and a commensurate line along which the floating phases collapse into a single multicritical point that DMRG numerics place in the eight-vertex universality class. Because this Hamiltonian lacks the microscopic particle-hole (spin-flip) symmetry that fixed the commensurate line in every previously known realization of eight-vertex criticality, the symmetry governing the multicritical point must be emergent — an assertion resting on finite-size evidence with no known mechanism. The 2023 source study left the entire middle of the $(g,f)$ phase diagram unresolved: the fate of the tri-critical Ising lines, the critical theory of the Lifshitz-line end-point where they presumably terminate, the mechanism and exact location of the emergent eight-vertex point, and the supersymmetry content of the floating phases (candidates $\mathcal{N}=(1,1)$ and $\mathcal{N}=(3,3)$, and lattice supersymmetry along the tri-critical Ising lines). This problem asks for a controlled, thermodynamic-limit determination of this multicritical structure and its emergent symmetries. No resolution of any component is known in the literature to date; the problem is open.

Background

Majorana fermions are operators that are their own antiparticles: Hermitian operators $\gamma_a=\gamma_a^\dagger$ obeying $\{\gamma_a,\gamma_{a'}\}=2\delta_{a,a'}$. Kitaev's exactly solvable chain of such fermions — equivalent to a one-dimensional spinless $p$-wave superconductor — hosts unpaired Majorana zero modes at its ends (A. Yu. Kitaev, Phys. Usp. 44, 131 (2001)). Through a Jordan–Wigner transformation, an exact lattice bijection between fermion operators and spin-1/2 Pauli matrices, the non-interacting Majorana chain maps onto the transverse-field Ising model at criticality. Interacting extensions are studied mainly with the density matrix renormalization group (DMRG), the adaptive tensor-network method of choice for one-dimensional quantum systems, since interactions generally destroy exact solvability.

The two shortest-range interaction terms are a four-Majorana coupling $g$ over four consecutive sites (in spin language $\sigma^z_j\sigma^z_{j+1}+\sigma^x_j\sigma^x_{j+2}$; in fermion language a nearest-neighbor density repulsion), and the O'Brien–Fendley term $f$ spanning four Majoranas over five sites, introduced as a lattice shortcut to supersymmetry at the tricritical Ising point, with an exactly supersymmetric frustration-free point at $f=1/2$ when $g=0$ (E. O'Brien and P. Fendley, Phys. Rev. Lett. 120, 206403 (2018)). Supersymmetry (SUSY) unifies bosonic and fermionic degrees of freedom into multiplets; a superconformal field theory is a conformal field theory (CFT) whose symmetry algebra includes fermionic generators, and the central charge $c$ of a CFT counts its effective degrees of freedom ($c=1/2$ for Ising, $c=1$ for a free boson).

The $g$-only model is well mapped (A. Rahmani, X. Zhu, M. Franz, I. Affleck, Phys. Rev. B 92, 235123 (2015); N. Chepiga and N. Laflorencie, SciPost Phys. 14, 152 (2023)): an Ising critical phase terminates at $g\approx0.29$ in a Lifshitz transition — a multicritical transition at which the gap closes while correlations simultaneously become incommensurate, their wave vector departing from a simple fraction of $\pi$, with dynamical critical exponent $z=3$ (the gap scales with system size as $\Delta\sim L^{-z}$) — into a floating phase, a gapless incommensurate Luttinger liquid (a critical phase described by a free compactified boson whose stiffness is encoded in the Luttinger parameter $K$), superposed with Ising criticality so that the total central charge is $c=3/2$. The floating phase ends near $g\approx1.3$; Ising criticality ends at a tricritical Ising (TCI) point near $g\approx3$. The TCI theory is the $c=7/10$ minimal model between Ising criticality and first-order behavior, and its continuum limit is supersymmetric; emergent supersymmetry was in fact found at several points of the $g$-only diagram (A. Rahmani et al., Phys. Rev. Lett. 115, 166401 (2015); Chepiga and Laflorencie 2023).

Adding both interactions together produces the richest phase diagram of this family (N. Chepiga, Phys. Rev. B 108, 054509 (2023)). The model, given below, is self-dual — invariant under a Kramers–Wannier-type duality exchanging order and disorder variables. At the self-dual point its ground state contains nine phases: two Ising critical phases, three floating phases (one being the floating phase superposed with Ising criticality, $c=3/2$), and four gapped phases with three- or six-fold degenerate ground states. Transitions out of the floating phases are of Kosterlitz–Thouless (KT) type — infinite-order transitions governed by unbinding of domain-wall pairs, occurring when the Luttinger parameter reaches $K=1/4$ (translation-symmetry breaking by two sites) or $K=1/2$ (pairing) — or first order. Two TCI critical lines cross the diagram, and the Lifshitz line with $z=3$ extends over a finite interval.

The centerpiece is the commensurate line: the locus inside the incommensurate region where the wave vector of short-range correlations locks back to the commensurate value $q=\pi$. Along it the floating phases cannot exist and collapse into a single direct transition between gapped phases, ending at a multicritical point. The numerics of the source study show that this point obeys the eight-vertex universality class: Baxter's exactly solved eight-vertex lattice model (R. J. Baxter, Ann. Phys. 70, 193 (1972)) has a critical line along which exponents are not universal numbers but vary continuously, all controlled by a single parameter $\mu$ — the order-parameter exponent is $\beta=(\pi-\mu)/(4\mu)$ and the scaling dimension $d=(\pi-\mu)/(2\pi)$ — so eliminating $\mu$ yields the parameter-free prediction $d=2\beta/(1+4\beta)$, known as weak universality. The interacting Kitaev chain realizes this class at the integrable XYZ point $J_x=-J_z$ and, as established by DMRG, along the commensurate lines of non-integrable versions (N. Chepiga and F. Mila, Phys. Rev. B 107, L081106 (2023)); the eight-vertex/XYZ chain also hides a non-local lattice supersymmetry on the special line $J_xJ_y+J_xJ_z+J_yJ_z=0$ (C. Hagendorf and P. Fendley, J. Stat. Phys. 2012, DOI 10.1007/s10955-012-0430-0). Crucially, in every previous realization an explicit particle-hole symmetry of the Hamiltonian (in spin language the spin-flip $\sigma^z_i\to-\sigma^z_i$) pinned the commensurate line's exact location. The competing-interaction Hamiltonian has no such symmetry, so the eight-vertex identification there implies that the governing symmetry is emergent — a property of the critical point absent from the microscopic Hamiltonian. Emergent symmetries at one-dimensional criticality are documented (for example the emergent $U(1)$ symmetry in constrained chains, Z. Jouini, N. Chepiga, L. Herviou, F. Mila, Phys. Rev. B 108, 205145 (2023), and the emergent supersymmetry at Ising–KT multicritical points, L. Huijse, B. Bauer, E. Berg, Phys. Rev. Lett. 114, 090404 (2015)), but no mechanism is known for an emergent particle-hole symmetry at a commensurate-line eight-vertex point.

What the source study could not settle — and what remains open — is the middle of the $(g,f)$ diagram, where these objects meet. The TCI line could not be located for $g\lesssim1.2$ because incommensurability inflates finite-size effects; the central charge measured at the tip of the floating phase, $c\approx1.4$, is intermediate between $c=3/2$ (floating plus Ising) and $c=1$ (floating alone), leaving open whether Ising criticality survives up to the commensurate line; the end-point of the Lifshitz line — where at least one and probably both TCI lines were conjectured to terminate — has no identified critical theory; the eight-vertex multicritical point's exact coordinates (bracketed $0.35\lesssim f_c\lesssim0.39$ near $g\approx0.37$), its Baxter parameter $\mu$, and the mechanism of its emergent symmetry are unknown; and the supersymmetry conjectures — an $\mathcal{N}=(1,1)$ extended supersymmetry across the whole floating-plus-Ising phase, lattice supersymmetry along the TCI lines, and the higher $\mathcal{N}=(3,3)$ superconformal symmetry at the KT transition where Ising criticality enters the floating phase (realizable when the fermionic mode velocity does not exceed the bosonic one, per the analysis of Huijse, Bauer and Berg) — await velocity measurements that the source study did not perform.

Problem Statement

Consider the interacting Majorana chain with competing interactions studied by Chepiga (Phys. Rev. B 108, 054509 (2023)), defined (with $t=1$) by

$$\mathcal{H}=i\sum_{a}\gamma_{a}\gamma_{a+1}-g\sum_{a}\gamma_{a}\gamma_{a+1}\gamma_{a+2}\gamma_{a+3}-f\sum_{a}\gamma_{a}\gamma_{a+1}\gamma_{a+3}\gamma_{a+4},$$

with Hermitian Majorana operators $\{\gamma_a,\gamma_{a'}\}=2\delta_{a,a'}$, at its self-dual point; equivalently, after Jordan–Wigner transformation, $\mathcal{H}=\sum_j[-\sigma^x_j\sigma^x_{j+1}-\sigma^z_j+g(\sigma^z_j\sigma^z_{j+1}+\sigma^x_j\sigma^x_{j+2})+f(\sigma^z_j\sigma^x_{j+1}\sigma^x_{j+2}+\sigma^x_j\sigma^x_{j+1}\sigma^z_{j+2})]$.

The common objective is to determine, with controlled thermodynamic-limit statements, the multicritical structure and the emergent-symmetry content of the middle region of the $(g,f)$ phase diagram — the region where the two tri-critical Ising (TCI) lines, the extended Lifshitz line, the commensurate line, and the floating phases meet. The enumerated sub-questions are the specific unknowns that jointly constitute this characterization; they are not independent deliverables but intertwined aspects of the same region (for instance, settling the TCI trajectories requires identifying the Lifshitz end-point where they are conjectured to terminate, and the emergent-symmetry question concerns the eight-vertex point that sits at the center of this region).

(1) TCI-line topology: establish the thermodynamic-limit trajectories and terminations of the two TCI lines ($c=7/10$) through the middle of the diagram, where the source study could not locate them ($g\lesssim1.2$). Does the TCI line descending from small $g$ turn to meet the second TCI line at the end-point of the Lifshitz line (the scenario the source favors), or do the lines connect otherwise? In particular, does Ising criticality persist throughout the Floating-1 phase (the incommensurate Luttinger-liquid phase extending from small $f$ almost to the commensurate line) up to the commensurate line (central charge $c=3/2$), or does it terminate inside the floating phase (thermodynamic central charge dropping to $c=1$)?

(2) Lifshitz end-point: identify the critical theory governing the end-point of the $z=3$ Lifshitz line — in particular whether it is distinct from the Lifshitz, TCI, and eight-vertex critical theories already present in the diagram.

(3) Emergent eight-vertex symmetry: for the multicritical point on the commensurate line, where the floating phases collapse into a direct transition between gapped phases and the source's numerics indicate the eight-vertex universality class, establish — analytically or by numerics with controlled thermodynamic-limit extrapolation — whether and how the particle-hole (spin-flip $\sigma^z_i\to-\sigma^z_i$) symmetry, absent from the Hamiltonian, emerges at criticality; identify the microscopic mechanism of this emergence; and determine the exact location of the multicritical point in the $(g,f)$ plane, together with the corresponding Baxter parameter $\mu$ and critical exponents (the source gives only loose independent finite-size brackets for the exponents — $0.65\lesssim\beta\lesssim0.9$ from the order-parameter scaling and $0.392\lesssim d\lesssim0.457$ from the Friedel-oscillation fit, two separate estimates that need not jointly satisfy the eight-vertex relation $d=2\beta/(1+4\beta)$ — and places the critical point at $0.35\lesssim f_c\lesssim0.39$ near $g\approx0.37$).

(4) Supersymmetry: measure or derive the diagnostics deciding whether the Floating-1+Ising phase hosts an emergent $\mathcal{N}=(1,1)$ supersymmetry (conditions: preserved $\mathbb{Z}_2$ symmetry plus the emergent $U(1)$ symmetry that stabilizes the floating phase); whether the TCI lines realize lattice supersymmetry; and whether the Kosterlitz–Thouless transition between the Floating-1+Ising phase (the floating Luttinger liquid superposed with Ising criticality) and the Ising-2 phase (the second Ising-critical phase, at larger $g$) realizes the extended $\mathcal{N}=(3,3)$ supersymmetry (criterion: the fermionic mode velocity not exceeding the bosonic one). Identify the critical theory along the line where the TCI line is superposed with the floating phase, and at the point where it crosses the Kosterlitz–Thouless transition into the critical phase.

Answers must reach the thermodynamic limit with quantified control of finite-size and incommensurability effects, since these are precisely the limitations that prevented the source study from settling the questions. An answer may combine numerical determination with analytic field-theory reasoning; exploring the extended $(h,g,f)$ parameter space proposed in the source is admissible as a route but is not itself required.

References

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