Emergent symmetries and interactions in high dimensions: An isolated fixed point versus a manifold of strongly interacting fixed points

Fei Zhou

Phys. Rev. B 109, 184503 – Published 2 May 2024

Abstract

In this article, we explore conditions of continuous emergent symmetries in gapless states, either as topological quantum critical points (TQCPs) or a stable phase with protecting symmetries and connections to smooth deformations of the gapped states around. For a wide class of gapless states that can be associated with fully isolated scale-invariant fixed points, we illustrate that there shall be emergent continuous symmetries that are directly related to smooth deformations of gapped states with symmetries lower than the protecting ones $G_{p}$ . A short-distance invariance of gapped states under deformations can descend to be an emergent continuous symmetry when approaching the gapless limit. Around one TQCP with $G_{p} = Z_{2}^{T}$ symmetry, we construct these deformations explicitly and show emergence of symmetries via fully gapped quaternion superconducting states that break the protecting symmetry $G_{p} = Z_{2}^{T}$ . For a 3D TQCP in DIII classes with $G_{p} = Z_{2}^{T}, U_{E M} = U (1)$ , and $N_{f} = \frac{1}{2}$ fermions but without charge $U (1)$ symmetry, we further explicitly construct a corresponding boundary representation based on a $4 D$ topological state with lattice symmetry $H = Z_{2}^{T} ⋉ U (1)$ and $N_{f} = 1$ fermions. The lattice model is shown to be dual to a conventional $4 D$ topological insulator. Although emergent continuous symmetries appear to be robust at weakly interacting TQCPs, we further show the breakdown of such one-to-one correspondence between deformations of gapped states and emergent continuous symmetries when gapless states become strongly interacting. In a strongly interacting limit, gapless states can be represented by a smooth manifold of conformal-field-theory fixed points rather than a fully isolated one. A smooth manifold of strong coupling fixed points hinders emergence of a continuous emergent symmetry in the strongly interacting gapless limit, as deformations no longer leave a gapless state or a TQCP invariant, unlike in the more conventional weakly interacting case. This typically reduces continuous emergent symmetries to a discrete symmetry originating from duality transformations under the protection symmetry $G_{p}$ .

Received 3 January 2024
Revised 11 April 2024
Accepted 12 April 2024

DOI:https://doi.org/10.1103/PhysRevB.109.184503

Physics Subject Headings (PhySH)

Symmetry protected topological states Topological phase transition Topological phases of matter Topological superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Fei Zhou

Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia V6T 1Z1, Canada

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Vol. 109, Iss. 18 — 1 May 2024

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Figure 1
(a) Deformations of gapped states around a gapless TQCP with protection symmetry $G_{p}$ . Only the horizontal axis $M_{1}$ is protected by $G_{p}$ but the rest of the plane where $M_{2}$ breaks the symmetry $G_{p}$ has a lower symmetry. Smooth deformations $U_{D} (λ)$ along the upper half of the dashed oval connects two topologically distinct gapped states $T 0, T 1$ (light blue dots) located along the horizontal axis $M_{1}$ . The lower half of the dashed oval describes a deformation $U_{D *} (λ)$ induced when applying $G_{p}$ to $U_{D} (λ)$ , or $U_{D *} = U_{G_{p}}^{- 1} (λ) U_{D} U_{G_{p}} (λ)$ . $U_{D}$ together with $U_{D *}$ trace out a closed loop (dashed oval) around the TQCP leading to an emergent symmetry at the TQCP. (b) Loop (dashed oval) traced out by smooth deformations of gapped states around a stable gapless state (red dot in the center) protected by a symmetry $G_{p}$ . Symmetry $G_{p}$ is only present along the third axis perpendicular to $M_{1} − M_{2}$ plane (the arrowed axis from the red dot in the center) where gapless states form a stable phase. The 3D volume away from the arrowed axis breaks the symmetry of $G_{p}$ and states are fully gapped except along the center arrowed axis.
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Figure 2
(a) Symmetries in two gapped topological phases with different topological invariants $N_{w}, N_{w}^{'}$ and at a typical TQCP where a topological phase transition occurs. The gapped phases are protected by symmetry $G_{p}$ , which is also the symmetry of the ground states. $H = G_{p} ⋉ U_{E M}$ and the symmetry at the gapless TQCP is higher than $G_{p}$ because of emergent symmetries, i.e., $G_{p} \subset H$ . (b) Near an ordered-disordered phase transition. symmetries in disordered and ordered phases are $G$ and $G_{S} \subset G$ respectively. At a generic gapless QCP, the symmetry group $H$ is equal to $G$ , but not higher ( $G$ shall also be the symmetry of interactions).
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Figure 3
A conventional order-disorder QCP (red dots) with Hamiltonian symmetry $G$ defined along the horizontal axis [either $m_{0}$ as in (a), (c), and (d) or $m_{1}$ in (b)]. In the rest of the plane, the symmetry $G$ can be either partially explicitly broken or simply lowered to $G_{S} \subset G$ . (a) A generic situation where a QCP along the axis of $m_{0}$ with symmetry $G$ is extended into an open line (red) of critical points into the $m_{0} − m_{2}$ plane when a subgroup symmetry in symmetry group $G$ is further explicitly broken. (b) Another situation where QCPs with symmetry $G$ become a closed loop (red) of critical points in the plane when a subgroup symmetry of the Hamiltonian symmetry $G$ is broken beyond the horizontal symmetry axis of $m_{1}$ . In this case, there can be a second QCP along the symmetric axis. As illustrated in Appendix pp1, an example of (b) is when the $Z_{2}^{T}$ subgroup in $G = Z_{2} \otimes Z_{2}^{T}$ along the $m_{1}$ axis is further broken in the $m_{1} - m_{2}$ plane when moving away from the horizontal axis of $m_{1}$ . In (c) and (d) when the symmetry $G$ is simply lowered into a smaller group $G_{S}$ in either the $m_{0} − m_{1}$ or $m_{0} − m_{2}$ plane away from the horizontal axis of $m_{0}$ , generically the QCP along the axis of $m_{0}$ with Hamiltonian symmetry $G$ becomes an intersecting point of two critical lines [see Appendix pp1 for discussions on $G = U (1) ⋊ Z_{2}^{T}$ lowered into $G_{S} = Z_{2} \otimes Z_{2}^{T}]$ . In (a)–(d), smooth deformations around a QCP (dashed oval enclosing the red dots) are strictly forbidden because of a line or lines of critical points emitted from the QCP located along the horizontal symmetry axis.
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Figure 4
TQCP as an isolated fixed point vs as a fixed point located in a smooth manifold of other fixed points without protecting symmetries. (a) A closed path (dashed oval) of deformations of gapped state around a TQCP (red dot) with protecting symmetry $G_{p}$ in the plane of $M_{1} − M_{2}$ ; the states along the axis of $M_{1}$ are symmetric under $G_{p}$ but the rest of the plane breaks the protecting symmetry $G_{p}$ . $T 0, T 1$ (blue dots) are two topologically distinct states with symmetry $G_{p}$ . (b) Two scenarios for a gapless TQCP (at $M_{1, 2} = 0$ ); the horizontal (1) and the vertical (2) axis are, respectively, for interactions with and without protecting symmetry $G_{p}$ . The top one is when the TQCP is an isolated infrared stable noninteracting fixed point that has the protecting symmetry $G_{p}$ and the gapless state is unique. Deformations leave it invariant. The bottom one is for the strongly interacting limit where there can be a smooth manifold of conformal field theory fixed points. Here a TQCP is identified with one of two fixed points (red dots along the horizontal axis) with protection symmetry $G_{p}$ . The deformations applied to a TQCP trace out all conformal fields along the ring-like manifold that break the protecting symmetry $G_{p}$ and hence do not leave the TQCP invariant. Two fixed points on the horizontal axis are dual to each other. Dashed lines here imply possible separatrices in renormalization flows towards the ring manifold. See Sec. 7 and Appendix pp4 for details.
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Figure 5
Examples of diagrams contribute to Eq. (49) and the ring of fixed points in Eq. (50). Dashed (blue) lines are for real fermion fields and the solid (black) ones for scalar fields. 1,2, and 4 numerate the type interaction vertices appearing in our calculations. 1,2 are for $g_{Y 1, Y 2}$ respectively while 4 is for $g_{4}$ .
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