Universality and Quantum Criticality in Quasiperiodic Spin Chains

TL;DR

This paper reveals that generic quasiperiodic spin chains flow to discrete fixed points under renormalization, enabling exact analysis of their critical behavior and uncovering universal properties across different models.

Contribution

It demonstrates that generic quasiperiodic modulations in spin chains flow to discrete sequences, allowing for a functional renormalization group approach to analyze their criticality.

Findings

Quasiperiodic spin chains flow to discrete fixed points.

The approach applies to Heisenberg, Ising, and Potts models.

Provides insights into quasiperiodic many-body localization transition.

Abstract

Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems as well as disordered ones. Quasiperiodic criticality was previously understood only in the special limit where the couplings follow discrete quasiperiodic sequences. Here we consider generic quasiperiodic modulations; we find, remarkably, that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization group transformation. These discrete sequences are therefore fixed points of a \emph{functional} renormalization group. This observation allows for an asymptotically exact treatment of the critical points. We use this approach to analyze the quasiperiodic Heisenberg, Ising, and Potts spin chains, as well as a phenomenological model for the quasiperiodic many-body localization transition.

Equations

$$H=\sum _i{J}_i{\vec{S}}_i\cdot {\vec{S}}_{i+1},$$

$${\ell }_i^{\prime }={\ell }_{i-1}-{\ell }_i+{\ell }_{i+1}+c,$$

$${\ell }_j=-\ln J_j=a+\cos \left(2\pi \phi j+\theta \right),$$

$$A_m(i)$$

$$B_m(i)$$

$$H=-{\sum }_i{J}_i{\delta }_{n_i,n_{i+1}}-\sum _i\frac{h_i}{q}\sum _{n_i,n_i^{\prime }}|n_i⟩⟨n_i^{\prime }|,$$

$$⟨{\sigma }_0^{(a)}{\sigma }_r^{(a)}⟩\sim r^{-2{\Delta }_{\sigma }},\mathrm{with}{\Delta }_{\sigma }=\frac{\ln \left(1+2{\phi }^{-1}/3\right)}{3\ln \phi }.$$

$${\ell }_{\mathrm{new}}^{T/I}={\ell }_{i-1}^{T/I}+{\beta }^{I/T}{\ell }_i^{I/T}+{\ell }_{i+1}^{T/I},$$

Full text

Quenched randomness has dramatic effects on the thermodynamics and response of one-dimensional quantum systems. Infinitesimal randomness can completely change the critical properties of quantum phase transitions, if the correlation length exponent $\nu$ violates the Harris criterion $\nu>2$ Harris (1974). For a wide variety of models of quantum magnetism, the resulting random critical points are of the “infinite-randomness” type, for which asymptotically exact renormalization-group methods exist Fisher (1992, 1994, 1995); Iglói and Monthus (2005). Many properties of these critical points can be understood by central limiting arguments that rely on the uncorrelated nature of the disorder potential. Such infinite-randomness fixed points also seem to exist in two dimensions, although the theory is less controlled in this case Motrunich et al. (2000).

Many systems of current interest—such as quasicrystals Shechtman et al. (1984); Levine and Steinhardt (1984); Merlin et al. (1985), twisted bilayer graphene Bistritzer and MacDonald (2011); Cao et al. (2018a, b), and cold atoms in bichromatic laser potentials Roati et al. (2008); Deissler et al. (2010); Schreiber et al. (2015); Lüschen et al. (2017); Bordia et al. (2017)—are inhomogeneous, but with quasiperiodic rather than random modulation of the couplings. Quasiperiodic patterns are deterministic and have long-range spatial correlations: thus the central limiting arguments that describe random critical points fail in the quasiperiodic case. Clean critical points are more stable to quasiperiodic than to random potentials: the most general stability criterion, due to Luck, is $\nu\geq 1$ (in one dimension) Luck (1993). The Heisenberg and Potts models violate even this weaker criterion, so when their couplings are quasiperiodically modulated they flow to a critical point dominated by the modulation. The study of such quasiperiodic quantum critical points is in its infancy; while certain special cases (in which the couplings come from a binary substitution sequence) have been analyzed using renormalization-group (RG) methods Kohmoto et al. (1983); Satija and Doria (1988); Levitov (1989); Iglói et al. (1998); Vidal et al. (1999); Hermisson (1999); Hida (2004); Vieira (2005), the case of generic quasiperiodic modulation has only been addressed for free fermions Wilkinson (1984); Chandran and Laumann (2017); Crowley et al. (2018a, b).

The central result of this work is that, for a wide class of models, generic quasiperiodic modulations flow under renormalization to discrete substitution sequences. These sequences are attractors in the space of quasiperiodic modulations. Since generic patterns flow to substitution sequences, for which asymptotically exact real-space RG schemes exist, we can construct asymptotically exact descriptions of generic quasiperiodic quantum critical points, and describe many of their properties (such as critical exponents) analytically. In what follows, we explain why (and under what conditions) substitution sequences are attractors, using the illustrative example of the Heisenberg spin chain; we then extend our analysis to the Ising and Potts models, and finally to a toy model of the MBL transition Zhang et al. (2016); Goremykina et al. (2019).

Quasiperiodic Heisenberg chain.— Although our results are very general and apply to a variety of one-dimensional systems, for concreteness we will illustrate the approach on a paradigmatic example of quantum magnetism: the antiferromagnetic spin-$\frac{1}{2}$ Heisenberg spin chain

[TABLE]

with $J_{i}>0$. In the clean case $J_{i}=J$, this spin chain is gapless and is described at low energy by a $SU(2)$ symmetric Luttinger liquid with Luttinger parameter $g=\frac{1}{2}$. Disorder in the $J_{i}$ couplings is a relevant perturbation Giamarchi and Schulz (1988) that leads to a quantum critical random-singlet state Fisher (1992). In the random case, the low-energy physics can be captured by a strong disorder real space RG with the following iterative procedure Ma et al. (1979); Dasgupta and Ma (1980): one identifies the strongest remaining coupling $J_{i}$, forms a singlet out of the spins $i,i+1$ (thus eliminating them from the problem), and generates a new effective coupling $J_{\rm eff}=\frac{J_{i-1}J_{i+1}}{2J_{i}}$—at second order in perturbation theory—between spins $i-1$ and $i+2$. This procedure is accurate so long as $J_{i}\gg J_{i-1},J_{i+1}$; in the random case, the ratio of neighboring couplings flows to infinity, so the procedure is asymptotically exact Fisher (1994).

Here, we are interested instead in quasiperiodic modulations of the couplings, with $J_{i}=f(i)$ with $f(x)=f(x+\varphi^{-1})$ for irrational $\varphi$ [in the bulk of this text we take $\varphi=\frac{1+\sqrt{5}}{2}$]. We take $f>0$ to be a general smooth function with a smooth logarithm. To understand whether this perturbation is relevant at the Heisenberg “critical point” we recall that the Heisenberg chain is a critical point separating two inequivalent dimerized phases, which occur when the even and odd bonds have different strengths. The correlation length exponent for the dimerization transition is $\nu=2/3<1$; thus, weak quasiperiodicity is relevant Vidal et al. (1999); Hida (1999); Vidal et al. (2001) by the Harris-Luck criterion Luck (1993), and the system flows to a quasiperiodicity-dominated fixed point.

Flow to discrete sequences.—We now apply the standard real-space decimation procedure (described above for the random case) to this quasiperiodic model. It is convenient to write the decimation rule as

[TABLE]

where we have defined $\ell_{j}=-\ln{J_{j}}$, $c=\ln 2$ (for the Heisenberg model), and $\ell_{i}=\min\{\ell_{j}\}$. We will be interested in other values of $c$ below, in the context of the Potts model, so we will treat it as a parameter. For simplicity, we consider the following quasiperiodic potential

[TABLE]

with $\varphi$ the golden ratio, $\theta$ is a random phase and $a$ is an arbitrary constant. However, we emphasize that our conclusions also hold for any sufficiently regular quasiperiodic potential with frequency $\varphi$ sup .

Many properties of the decimation procedure can be understood analytically. It is helpful to introduce the notion of a local minimum: i.e., a site $j$ such that $\ell_{j}<\min(\ell_{j-1},\ell_{j+1})$. Any such coupling will get decimated before its neighbors, so we can decimate them all at once. It is crucial to note that, if we label minima as $B$ and all other sites as $A$, we will arrive at a two-letter Fibonacci sequence defined by the deflation rule $A\rightarrow AB,B\rightarrow A$ sup . With this observation in mind, we denote the set of minima as $B_{0}$ and the set of all other couplings as $A_{0}$. We now decimate all the $B_{0}$ couplings — we call this a Fibonacci RG step. This gives rise to a new sequence, in which, once again, we can identify the local minima, denote them $B_{1}$, decimate them, and so on. Remarkably, after a few steps, we find that all the $A(i)$ and $B(i)$ at a given step become increasingly similar in magnitude. Specifically, after $m$ Fibonacci RG steps, we find the effective couplings sup

[TABLE]

where $F_{n}$ is the nth Fibonacci number, and $\epsilon^{A,B}_{i,m}$ are site-dependent fluctuations that go to 0 exponentially as $m\to\infty$. With each Fibonacci step the fluctuations get smaller, and we obtain a sharper sequence which asymptotically becomes a perfect binary Fibonacci sequence. Even if the fluctuations are non-negligible in the initial steps of the RG, they are small enough for us to decimate all $B$ couplings at once (Fig. 1). This means that the complicated initial potential (3) flows under RG to a perfect binary Fibonacci sequence. Moreover, we have $A_{m}-B_{m}=mc+{o}(1)$ as $m\to\infty$, which indicates that the decimation rule (2) given by second order perturbation theory becomes asymptotically exact as $m\to\infty$, as in the random case. In contrast with the random case, where the fixed point is specified by a probability distribution of couplings, in the quasiperiodic case the fixed point Hamiltonian is specified by a specific, self-similar sequence of couplings: any sufficiently regular function $f(x)=f(x+\varphi^{-1})$ flows to a binary piecewise function, which is invariant (up to rescaling) under the RG.

We remark that although this flow to nearly discrete sequences is a general property of the RG rules, and occurs for any irrational number, the self-similarity of these discrete sequences is a special property of the Golden Ratio and other irrational numbers with recurring continued fraction expansions, such as the “metallic means” $1/(n+1/(n+\ldots))$ (the Golden Ratio is the case $n=1$). For more general irrational numbers the sequences do not repeat under the RG: instead, each level of the RG is governed by the coefficient of the continued fraction expansion at the corresponding level Wilkinson (1984); Levitov (1989); Szabó and Schneider (2018).

Quantum critical behavior.— The critical properties of the quasiperiodic chain (1) then follow straightforwardly from eq. (4), in agreement with previous works on discrete aperiodic sequences. The new fixed point has dynamical exponent $z=\infty$: a chain of length $L\sim\varphi^{3m}$ is fully decimated in $m$ Fibonacci steps, so the energy gap $\Delta E$ of the chain is set by the last coupling to be decimated, $\log\Delta E\sim-m^{2}c$, so that $\Delta E\sim e^{-\frac{c}{(3\ln\varphi)^{2}}\ln^{2}L}$ Hida (2004), where we have used (4). The correlation length exponent $\nu$ can be calculated by introducing an asymmetry between even and odd couplings at the new critical point. Let us assume a perfect sequence of $A_{0}$ and $B_{0}$ which is dimerized such that $A^{E}_{0}=A_{0}+\delta/2,\ B^{E}_{0}=B_{0}+\delta/2,\ A^{O}_{0}=A_{0}-\delta/2,\ B^{O}_{0}=B_{0}-\delta/2$. It is simple to check that if one constructs an effective bond made of $n$ microscopic bonds, its coupling will have an additive piece $\pm n\delta/2$ where the sign depends on whether the effective bond is even or odd. Thus, after the RG is iterated out to a scale $L$, the effective couplings at that scale consist of $O(L)$ microscopic couplings, so the asymmetry between the even and odd couplings is $\sim\delta L$. The asymmetry thus becomes of order unity when $L\sim 1/\delta$, and it follows that $\nu=1$ sup .

Quantum Potts model.—We now turn to the $q$-state quantum Potts model, governed by the Hamiltonian

[TABLE]

where $n_{i}$ is a variable on site $i$ that takes one of $q$ possible values. To treat this model in the RG scheme, one formally rewrites it as a chain with twice the number of links, and assigns the variables $J_{i}$ to even links and $h_{i}$ to odd links. The decimation step Senthil and Majumdar (1996) then takes the same form as Eq. (2) with $c=\log(q/2)$. When $q>2$, $c>0$, so once again the RG flows to discrete sequences. The main distinction between the Potts and Heisenberg models lies in the choice of initial couplings: in the Heisenberg model it was natural to draw all bonds from the same quasiperiodic sequence, whereas here it is natural to take the $h_{i}$ and $J_{i}$ from distinct quasiperiodic sequences with frequency $\varphi$ but different phases: $J_{i},h_{i}=a+\cos(2\pi\varphi i+\theta_{J,h})$, with $a>1$. This introduces a separate variable to the problem, viz. the relative phase $\theta\equiv\theta_{J}-\theta_{h}$ between the sequences for $h_{i}$ and $J_{i}$.

Numerically running the RG in this case leads to the following picture. When $\theta$ is close to the special values, we once again observe a flow to self-similar Fibonacci-like sequences. In that case, the results for the XXX spin chain carry over to Potts immediately – in particular, $\nu=1$. Other critical exponents can readily be computed analytically; for example, we find that correlation function of the order parameter $\sigma^{(a)}_{i}=\delta_{n_{i},a}-q^{-1}$ with $a$ a given Potts color, scales as sup

[TABLE]

However, for large $\theta$ we see abrupt transitions to different sequences. Thus, there appear to be multiple fixed-point sequences, with “transitions” between them occurring at special values of $\theta$. These fixed points have different length-energy scaling: in all cases, $\Delta E\sim e^{-a\ln^{2}L}$, but $a$ depends on $\theta$. However, all of the fixed points agree on the exponent $\nu=1$, as well as on the spin-spin correlation function (6) (Fig. 2).

Ising model.—We briefly remark on the $q=2$ Potts model, i.e., the Ising model. In this case, $c=0$ in the decimation rule (2). Thus the RG does not take arbitrary functions to sequences. Instead, nonsingular sequences of $\ell$ are generically squashed under the RG and become effectively constant after a few steps. This corresponds to the observation that weak quasiperiodic modulation is (marginally) irrelevant at the Ising critical point. To see a nontrivial transition in this case, one must take singular distributions of $\ell$; one can do this, e.g., by taking $J_{i},h_{i}=a+\cos(2\pi\varphi i+\theta_{J,h})$, with $0<a<1$ so $\ell_{2i}=-\ln|J_{i}|$, $\ell_{2i+1}=-\ln|h_{i}|$ is singular. In this model, the critical dynamics is strongly modified because the chain has a finite density of nearly broken links. Although our RG procedure is not controlled for this model, running the RG numerically yields a fixed point that is qualitatively similar to the numerically seen one Wilkinson (1984); Crowley et al. (2018a, b) (Fig. 2): in particular, it has a finite dynamical critical exponent $z\approx 1.6$, which is close to the numerical value $z\approx 2$. The remaining discrepancies are to be expected given that the procedure is not controlled; however, the RG correctly identifies the change in critical behavior that occurs when the distribution of couplings goes through zero.

Quasiperiodic MBL transition.—The observation that generic quasiperiodic potentials flow to analytically-tractable substitution sequences under renormalization has broad implications beyond zero-temperature quantum criticality. In particular, isolated quantum systems can undergo eigenstate phase transitions Parameswaran et al. (2017), in which the properties of eigenstates (and correspondingly the nature of the dynamics) changes in a nonanalytic way as some parameter is tuned. A key example of such a transition is the highly studied many-body localization (MBL) Anderson (1958); Fleishman and Anderson (1980); Altshuler et al. (1997); Gornyi et al. (2005); Basko et al. (2006); Oganesyan and Huse (2007); Pal and Huse (2010); Nandkishore and Huse (2015); Vasseur and Moore (2016); Abanin et al. (2018) transition in one dimension, which separates localized and chaotic dynamical phases. In the random case, the nature of the transition remains debated Pal and Huse (2010); Serbyn et al. (2015); Kjäll et al. (2014); Luitz et al. (2015); Agarwal et al. (2015); Khemani et al. (2017a, b); Zhang and Yao (2018), but many RG schemes have been proposed Vosk et al. (2015); Potter et al. (2015); Dumitrescu et al. (2017); Thiery et al. (2018, 2017); Zhang et al. (2016); Goremykina et al. (2019), with the most recent works supporting a Kosterlitz-Thouless-like transition Goremykina et al. (2019); Dumitrescu et al. (2019); Morningstar and Huse (2019).

In comparison with the random case, the physics of MBL in quasiperiodic systems remains largely unexplored. Within the latest RG schemes, the mechanism driving the transition is the nucleation of chaos in rare, anomalously thermal regions Agarwal et al. (2015); Gopalakrishnan et al. (2015); Agarwal et al. (2017). Quasiperiodic systems seem to lack such regions, and might therefore have a qualitatively different MBL transition, but the nature of this transition is not fully understood (but see Ref. Zhang and Yao (2018)). It would be natural to adapt previous RG approaches to the present case; however, there are two inequivalent ways that one could do this. The first approach is to try to construct typical individual eigenstates; in the localized phase, these eigenstates look like random product states. Some RG schemes proceed by locating clusters of spins that are resonant; however, the resonance condition itself depends on the initial product state (e.g., via Hartree effects) and is effectively random. Including these random Hartree shifts would presumably lead to a transition that looks similar to the random one. However, an alternative approach is to disentangle the Hamiltonian by a series of unitary rotations, and thus construct localized integrals of motion (LIOMs) in the MBL phase Serbyn et al. (2013a); Huse et al. (2014); Serbyn et al. (2013b); Imbrie (2016a, b). Deep in the MBL phase, a full set of LIOMs exists; and at a critical disorder value, all (or most) of these LIOMs cease to exist. The properties of the LIOMs in a quasiperiodic system are clearly quasiperiodic. Phenomenological RG schemes that model a system in terms of thermal and localized “blocks” can capture this procedure, since they never explicitly invoke eigenstates.

Motivated by this observation, we now explore the simplest such scheme Zhang et al. (2016); Goremykina et al. (2019). This model also describes certain classical coarsening dynamics Rutenberg and Bray (1994), and as we shall see, its RG rules are similar to those discussed above. This model assumes that on some coarse-grained scale, the system can be described alternating thermal (T) and insulating blocks (I), characterized by a single parameter which we call “length” $\ell_{T,I}$. During the RG, the smallest I or T block is decimated and merged with its neighbors into a new T/I block, with in the simplest version of this RG Zhang et al. (2016), a renormalized length that is the sum of the decimated blocks $\ell^{T/I}_{\rm new}=\ell^{T/I}_{i-1}+\ell^{I/T}_{i}+\ell^{T/I}_{i+1}$. This RG rule is almost the same as (2), with $c=0$ and a different sign in front of $\ell_{i}$. For random initial distributions of the block sizes, this simple model has a second order phase transition with correlation length exponent $\nu\approx 2.5$. Starting from quasiperiodic initial lengths $\ell^{T/I}_{i}=W^{T/I}(1+\cos\left(2\pi\varphi i+\theta\right))$, we find that the lengths once again renormalize to a Fibonacci sequence for $W^{T}=W^{I}$.

We now consider perturbing away from these fixed points, e.g., driving the system into the insulating phase by making the $I$ blocks slightly larger than the $T$ blocks. We can do this with the same dimerization perturbation as we previously had. A key thing to note is that this perturbation is actually irrelevant at the Fibonacci-sequence fixed point: a small perturbation does not change the sequence of decimations, and the perturbation strength decreases under decimation, as a consequence of the crucial sign-change in the RG rule. This situation changes, however, for continuous initial distributions: here, there are pairs of adjacent couplings that are nearby in energy; thus, there are minima (at the critical point) that cease to be minima when the system is perturbed. Thus, even small perturbations change the order in which sites are decimated, and thus disrupt the fixed-point sequence. In this sense, dimerization is a dangerous irrelevant perturbation at the quasiperiodic MBL fixed point.

Interestingly, regardless of this subtlety, we once again find a correlation length exponent $\nu=1$ for smoothly varying initial conditions. The argument is as follows: define a defect as a bond that was a local minimum for the critical pattern but ceased to be one after being perturbed (or vice versa). If a defect is present at position $i=0$, then another defect will be created at $i=F_{n}$ with $n$ large enough, since $\varphi F_{n}$ is almost an integer. Going away from criticality by an amount $\delta W$ induces defects if we choose $n$ so that the deviation of $F_{n}$ from being the exact period (given by the fractional part of $\varphi F_{n}$) satifies $\{\varphi F_{n}\}=\varphi^{-n}\sim\delta W$ sup . The distance between the defects then sets the correlation length $\xi\sim F_{n}\approx\varphi^{n}\sim\delta W^{-1}$, implying that $\nu=1$. An interesting consequence is that one can tunably vary $\nu$ by introducing singularities at appropriate points in the original sequence; we have checked that this is in fact true sup .

This toy RG is perfectly symmetric between the $T$ and $I$ phases; in realistic models, this symmetry is absent, as thermal blocks are much more “infectious” than insulating blocks. To introduce some asymmetry between the two phases, we follow Ref. Goremykina et al. (2019), and introduce a new parameter $\beta$ in the RG rules

[TABLE]

with $\beta^{I}=1/\beta^{T}=\beta$. This modified RG is analytically solvable for random distributions, and captures much of the phenomenology of the MBL transition in the maximally asymmetric limit $\beta\to\infty$, with a KT-type transition ($\nu=\infty$). In the quasiperiodic case, weak asymmetry is irrelevant at the fixed point; however, as the asymmetry is cranked up, eventually the order of decimations changes, and one flows to a new fixed-point sequence. Thus, as $\beta$ is increased, the system discontinuously jumps between a number of increasingly complicated self-similar fixed-point sequences which we detail in the Supplemental Material sup . Perturbations away from criticality lead to defects in these self-similar sequences, with the same exponent $\nu=1$ from the argument above. We therefore conjecture that the value $\nu=1$ remains exact as $\beta\to\infty$. This exponent $\nu=1$ is in agreement with exact diagonalization results Khemani et al. (2017b); Lee et al. (2017), and supports the scenario of two distinct universality classes for the quasiperiodic MBL transition Khemani et al. (2017b).

Conclusion.—To summarize, we have shown that under quite general conditions, conventional real-space RG rules attract generic initial conditions to fixed-point sequences that are self-similar under RG. The simplest case is in the Heisenberg spin chain, where the fixed-point sequence is given by the simple Fibonacci deflation rule; many results were previously known for this sequence, but our work shows that these results apply far more generally than had been appreciated. For the Potts model, we again found attracting sequences, but unlike the Heisenberg model we found multiple distinct attractors. Finally, for the toy MBL transition models, we found a much more unexpected type of RG flow, in which the control parameter driving the transition is in fact dangerously irrelevant at the critical point, but nevertheless the exponent $\nu=1$. It would be interesting to investigate whether our results can be extended to two-dimensional quasiperiodic systems Jagannathan (2004).

Acknowledgments.— We thank A. Chandran, P. Crowley, D. Huse, V. Khemani and C. Laumann for stimulating discussions. We also thank S. Gazit, B.M. Kang and J. Pixley for useful discussions and for sharing with us their unpublished Quantum Monte Carlo results. This work was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, under Early Career Award No. DE-SC0019168 (U.A. and R.V.), the Sloan Foundation through a Sloan Research Fellowship (R.V.), and by NSF Grant No. DMR-1653271 (S.G.).