Universal freezing transitions of dipole-conserving chains

We demonstrate the existence of a universal phase diagram of quantum chains with range- <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>k</a:mi> </a:math> interactions subject to the conservation of a total charge and its dipole moment. These systems exhibit “freezing” transitions between strongly and weakly Hilbert-space-fragmented phases as the charge filling <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>ν</b:mi> </b:math> is varied. We show that these continuous phase transitions occur at a critical charge filling of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:msub> <c:mi>ν</c:mi> <c:mi>c</c:mi> </c:msub> <c:mo>=</c:mo> <c:msup> <c:mrow> <c:mo>(</c:mo> <c:mi>k</c:mi> <c:mo>−</c:mo> <c:mn>2</c:mn> <c:mo>)</c:mo> </c:mrow> <c:mrow> <c:mo>−</c:mo> <c:mn>1</c:mn> </c:mrow> </c:msup> </c:mrow> </c:math> of the on-site Hilbert-space dimension <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>d</d:mi> </d:math> . To this end, we analytically prove that, for any <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>d</e:mi> </e:math> , any state with <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>ν</f:mi> <f:mo>&lt;</f:mo> <f:msub> <f:mi>ν</f:mi> <f:mi>c</f:mi> </f:msub> </f:mrow> </f:math> hosts a finite density of sites belonging to “blockages,” which we define as subregions of the chain across which transport of charge and dipole moment cannot occur. Some blockages arise from sequences of frozen sites, i.e., sites with an unchanging on-site charge, while others do not involve frozen sites at all. We prove that the presence of blockages implies strong fragmentation of typical symmetry sectors into Krylov subspaces, each of which forms an exponentially vanishing fraction of the total sector. By studying the distribution of blockages we analytically characterize how typical states are subdivided into dynamically disconnected local “active bubbles” and prove that typical eigenstates at these charge fillings exhibit area-law entanglement entropy, while there exist rare eigenstates featuring non-area-law scaling. We also numerically show that for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>ν</g:mi> <g:mo>&gt;</g:mo> <g:msub> <g:mi>ν</g:mi> <g:mi>c</g:mi> </g:msub> </g:mrow> </g:math> and arbitrary <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>d</h:mi> </h:math> , typical symmetry sectors are weakly fragmented, with their dominant Krylov sectors constituted of states that are free of blockages. We analytically derive some critical exponents characterizing the transition and numerically determine the density of blockages at <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mi>ν</i:mi> <i:mo>=</i:mo> <i:msub> <i:mi>ν</i:mi> <i:mi>c</i:mi> </i:msub> </i:mrow> </i:math> , with clear implications for transport at the critical point. Finally, we investigate the properties of special-case models for which no phase transitions occur.

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